A Fermion Primer

1. Introduction

Fermions present surprising and unfamiliar aspects, with respect to standard macroscopic objects. When we take a turn of 360^o we see the same world as before turning. For a fermion this is not the case: in our anthropic language we say that it has to turn twice, i.e. 720^o, in order to return to the initial situation. A fermion sees a world twice as large as our world. We say that fermions have half-integer spin, at variance with the other set of elementary particles, the bosons, which have integer spin. The consequence of this peculiarity is that the way we represent fermions is drastically different from the way we represent bosons. In order to represent fermions we use mathematical objects called spinors. The latter are called that way because they belong to representations of a spin group (see below), which has precisely to do with the revolving property alluded to above. Spin one-half spinors in 4d can be represented by two or four complex numbers, usually ordered in a column vector. As a consequence the action on them is represented by matrices, the Pauli or gamma matrices. The spinor wave functions or spinor fields are thus represented by columns of two or four functions. There are three distinct types of spinor wave-functions or fields: Weyl, Majorana and Dirac. The equations of motion are matrices of linear differential operator acting on them. The non-standard mathematics of spinors brings into the game some problems which do not seem to have been uniformly elaborated in the literature, or, at least, are resolved in ambiguous ways. This is a review where we discuss such issues and the way we have resolved them. We utilize in part material taken or ri-elaborated from our past research [1,2,3].

A first problem arises concerning the mass term for spinors, which can be vanishing or non-vanishing. Massive spinors are either Dirac or Majorana. They are bispinors with four components. In the Majorana case they can be cast in real form. Massless spinors can be of three types: Weyl spinors with two complex components or chiral bispinors; self-conjugate Majorana bispinors; Dirac bispinors. The Majorana mass term leads inevitably to a self-conjugate Majorana bispinor, for a Weyl spinor has perforce zero mass the two chiral components being decoupled (the mass terms does in fact couple them, as is well known). If a spinor has mass it cannot be Weyl, and viceversa. This is what we try to clarify in the simplest possible way, also to remove the debris generated by the illegal identification of massless Majorana and Weyl fermions.

The second problem arises when one quantizes a theory containing Weyl fermions. Quantization requires the existence of the corresponding propagator. A propagator is the inverse of the kinetic operator, and for Weyl fermions formally it does not exist because the kinetic operator contains a chiral projector. In perturbative field theory one resolves the puzzle by assigning to Weyl fermions the same propagator as for massless Dirac fermions (which does exist). As will be commented upon below this is an arbitrary way of proceeding and one does not have any warranty that the result so obtained makes any sense. In fact experience shows that this trick may work as long as one does not meet critical issues such as anomalies (for Weyl fermions anomalies are the symptoms that a proper propagator does not exist). We will argue below that in such cases this procedure is based on a logical paradox. Nevertheless this may work in perturbative field theory because the fermionic chiral nature is preserved by vertices which contain the chiral projector. An explanation will be provided for the cases in which this trick leads to correct results. In general it has to be assumed that the perturbative approach (Feynman diagrams) applied to Weyl fermion, in which the propagator employed is the Dirac one, is not reliable. For nonperturbative approaches (such as the heat kernel one) the same trick may lead astray because it changes the nature of the problem as will be explained below.

Additional problems and ambiguities arise when one comes to the effective Action of a Weyl fermion theory. One can compute it perturbatively or non-perturbatively. In any case one faces the issue of regularization. In the perturbative case one needs propagators and vertices, which, as we have just seen, becomes a formidable obstacle for Weyl fermions. The tentative remedy is to use the Dirac propagator instead of the missing Weyl one. Once this choice is made one faces the issue of regularization. We have several choices: dimensional [4,5,6] Pauli-Villars [7,8] or cutoff regularizaton - below we consider these three in a simple enough example. Here we make clear a fundamental point: a PV regularized propagator is not the inverse of any kinetic term of whatsoever classical local Action, independently of its being Dirac or Majorana (whose propagators coincide). A somewhat similar problem arises with the technique of Wick rotation: one may be tented to do the Wick rotation in the original classical Minkowski theory and formulate an equivalent problem in a Euclidean spacetime. However we will show that the so obtained Euclidean theory, provided it exists, has very little to do with the original Minkowski one. In simple words, one cannot regularize the original local form of the Action. The regularization process is something separate, which has to do with the mathematical (distribution theoretical) properties of Feynman diagrams. In the case of Weyl fermions this process is particularly delicate, one must make sure that throughout it chirality is preserved. Careful not to be victim of a semantic trap: it is standard lore that in the regularization process a symmetry may be broken, but chirality is not a symmetry, it is the defining status of the fields in the theory. If `chirality is broken’ it means that one, without realizing it, has landed in a foreign theory. For this reason it is of upmost importance how we define the relevant Feynman diagrams or the relevant heat kernel operator. From this point of view PV regularization must be particularly under scrutiny. It needs masses and, as anticipated above, massive fermions can only be Dirac or Majorana, which carry both chiralities. PV can be therefore applied only as a mathematical algorithm to regularize chiral fermion integrals (see an example below), where masses are simple regularizing parameters, without any reference to actual fields and particles. In general a good amount of skepticism is in order when using a perturbative method based on Feynman diagrams for a theory involving Weyl fermions. Better methods are available.

The paper is organized as follows. Section 2 is devoted to a general introduction of fermions as representations of a Clifford algebra and a spin group. This Section can be skipped by those who are not specifically interested in it without prejudice for the comprehension of the subsequent Sections. Section 3 contains a very basic introduction to Weyl, Dirac and Majorana spinors in 4d1. Weyl fermions are the basis for the construction also of Dirac and Majorana ones. Classical solutions, symmetries and canonical quantization, with observables and causal Green’s functions, are explicitly derived and exhibited. In Section 4 a summary is given of Dirac, Majorana and Weyl fermions in 4D with their properties in the language more appropriate for quantum field theory. Section 5 is devoted to explicit examples of regularizations (dimensional, Pauli-Villars and cutoff) for Weyl fermions. The subject of Section 6 are Dirac and Weyl propagators, together with the severe problems carried along by the latter, of which both false and true solutions are discussed. Two more critical issues are also discussed in this Section: the difference between Weyl and massless Majorana fermions, and the Dirac and Majorana mass terms. Section 7 is devoted to Euclidean fermion field theories and, in particular, to the impossibility to write down a real Action for Weyl fermions. Finally in Section 8 the Atiyah-Singer family’s index theorem is presented and its relevance for the invertibility of the Dirac-Weyl operator (=existence of propagator) is highlighted.

Notation. We use a metric $g_{\mu \nu }$ with mostly - signature. The gamma matrices satisfy $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2g^{\mu \nu }$ and

$$\begin{array}{c}\hfill {\gamma }_{\mu }^{†}={\gamma }_0{\gamma }_{\mu }{\gamma }_0\end{array}$$

The generators of the Lorentz group are ${\Sigma }_{\mu \nu }={\displaystyle \frac{1}{4}}[{\gamma }_{\mu },{\gamma }_{\nu }]$. The charge conjugation operator C is defined to satisfy

$$\begin{array}{c}\hfill {\gamma }_{\mu }^T=-C^{-1}{\gamma }_{\mu }C,C{C}^{*}=-1,C{C}^{†}=1\end{array}$$

(1)

The chiral matrix ${\gamma }_5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ has the properties

$$\begin{array}{c}\hfill {\gamma }_5^{†}={\gamma }_5,{\left({\gamma }_5\right)}^2=1,C^{-1}{\gamma }_{5}C={\gamma }_5^T\end{array}$$

In the sequel, when more convenient, we shall use a more explicit representation of the gamma matrices, see beginning of section 3.

2. Clifford Algebras and Spinors

At the basis of all the elaborations on spinor fields in quantum field theories is the mathematical concept of Clifford algebra and its representations. Spinors are in fact representations both of a Clifford algebra and of a spin group, which is in turn a subgroup of the Clifford algebra. This Section is a short introduction to this subject and to its connection with spin and spinor fields (for more complete accounts see, for instance,[11,12]). The uninterested reader can skip it without prejudice. The only essential thing to be kept in mind for the sequel is the table at the end.

Given a vector space V with a quadratic form $q\left(v\right)$ consider the tensor algebra $\mathcal{T}\left(V\right)={\sum }_{r=0}^{\infty }{\otimes }^{r}V$. The quotient by the ideal ${\mathcal{I}}_q\left(V\right)$ formed by all the elements of the type $v\otimes v-q\left(v\right)1$ defines the Clifford algebra

$$\begin{array}{c}\hfill C\ell (V,q)=\mathcal{T}\left(V\right)/{\mathcal{I}}_q\left(V\right)\end{array}$$

It can be defined alternatively as the algebra generated by V (and 1), subject to the relation

$$\begin{array}{c}\hfill v·w+w·v=2g(v,w)\end{array}$$

(2)

for $u,w\in V$, where $2g(v,w)=q(v+w)-q\left(v\right)-q\left(w\right)$. The dot in (2) is the product

$$\begin{array}{c}\hfill v·w=v\wedge w+g(v,w)\end{array}$$

(3)

If the quadratic form is the one defined on the basis elements $\left\{e_i\right\},i=1,\dots n$ of V by $g(e_i,e_j)={\eta }_{ij}$, where ${\eta }_{ij}$ is the diagonal matrix with r entries equal to 1 and s entries equal to -1, with $r+s=n$, the relation (2) becomes

$$\begin{array}{c}\hfill e_i·e_j+e_j·e_i=2{\eta }_{ij}\end{array}$$

(4)

One easily recognizes the form of the $\gamma$-matrix algebra (although the definition (2) is more general): in fact the field theory $\gamma$-matrices are a representation of the Clifford algebra. When we refer specifically to the canonical form (4), the Clifford algebra will be denoted $C{\ell }_{r,s}$. These algebras have been classified in terms of fields and real or complex $2\times 2$ matrix rings, $\mathcal{M}(2,\mathbb{R})$ and $\mathcal{M}(2,\mathbb{C})$, respectively. For instance

$$\begin{array}{c}\hfill C{\ell }_{3,0}=\mathcal{M}(2,\mathbb{C}),C{\ell }_{3,1}=\mathcal{M}(2,\mathbb{R})\otimes \mathcal{M}(2,\mathbb{R}),\mathrm{etc}.\end{array}$$

So far we have considered real Clifford algebras. Complex Clifford algebra (which are the complexification of real ones) are denoted $C{\ell }_{\mathbb{C}}\left(n\right)$. Their classification is rather simple

$$\begin{array}{c}\hfill C{\ell }_{\mathbb{C}}\left(2k\right)=\mathcal{M}(2^k,\mathbb{C}),C{\ell }_{\mathbb{C}}(2k+1)=\mathcal{M}(2^k,\mathbb{C})\oplus \mathcal{M}(2^k,\mathbb{C})\end{array}$$

The automorphism of $C\ell (V,q)$ induced by $v\to \tilde{v}=-v$ for $v\in V$, defines a decomposition into even and odd subspaces

$$\begin{array}{c}\hfill C\ell (V,q)=C{\ell }^0(V,q)\oplus C{\ell }^1(V,q)\end{array}$$

(5)

and gives $C\ell (V,q)$ a ${\mathbb{Z}}_2$-graded algebra structure: $C{\ell }^i(V,q)·C{\ell }^j(V,q)\subseteq C{\ell }^{i+j}(V,q)$, $i,j=0,1$.

Inside a Clifford algebra one can define various groups. The largest one is the group of invertible elements

$$\begin{array}{c}\hfill C{\ell }_{r,s}^{\times }=\{a\in C{\ell }_{r,s}|\exists a^{-1}\}\end{array}$$

(6)

An important subgroup is the Clifford-Lipschitz group

$$\begin{array}{c}\hfill {\mathcal{C}}_{r,s}=\{a\in C{\ell }_{r,s}^{\times }|av{a}^{-1}\in V,\forall v\in V={\mathbb{R}}^{r,s}\}\end{array}$$

(7)

This definition is based on the adjoint action in V. We can extend it to the full Clifford algebra

$$\begin{array}{c}\hfill {\sigma }_a\left(x\right)=ax{a}^{-1},\forall a\in {\mathcal{C}}_{r,s},\forall x\in C{\ell }_{r,s}\end{array}$$

(8)

${\sigma }_a$ is an automorphism of ${\mathcal{C}}_{r,s}$. A more instrumental operation is the twisted adjoint

$$\begin{array}{c}\hfill {\tilde{\sigma }}_a\left(x\right)=\tilde{a}x{a}^{-1}\end{array}$$

(9)

where $\tilde{a}$ has been defined before.

Let us define the group

$$\begin{array}{c}\hfill O(V,q)=\{t\in GL\left(V\right):t^{*}q=q\}\end{array}$$

where $t^{*}q$ is the pullback of q by $t:V\to V$. I.e. $O(V,q)$ is the group of linear invertible transformations of V that leave the quadratic form q unchanged. And let us define the subgroup

$$\begin{array}{c}\hfill SO(V,q)=\{t\in O(V,q):\det (t)=1\}\end{array}$$

As it turns out, we have the correspondences

$$\begin{array}{c}\hfill \tilde{\sigma }\left({\mathcal{C}}_{r,s}\right)=O(r,s),\tilde{\sigma }\left({\mathcal{C}}_{r,s}^0\right)=SO(r,s)\end{array}$$

(10)

where ${\mathcal{C}}_{r,s}^0={\mathcal{C}}_{r,s}\cap C{\ell }_{r,s}^0$ is the even Clifford subalgebra.

Now let us come to the spin groups. We can define a norm $N\left(a\right)$ for any element $a\in C\ell (V,q)$, such that $N\left(v\right)=q\left(v\right)$ for any $v\in V$. Then the spin group of V endowed with a quadratic form q is

$$\begin{array}{c}\hfill Spin(V,q)=\{a\in {\mathcal{C}}_{r,s}^0|N\left(a\right)=\pm 1\}\end{array}$$

(11)

The restricted spin group $Spi{n}^{+}(V,q)$ is identified by the condition $N\left(a\right)=1$.

One can prove that the following sequence is exact

$$\begin{array}{c}\hfill 0⟶{\mathbb{Z}}_2⟶Spin(V,q)\stackrel{\tilde{\sigma }}{⟶}SO(V,q)⟶0\end{array}$$

where the twisted adjoint map $\tilde{\sigma }$, for any $v,w\in V$, can also be represented as

$$\begin{array}{c}\hfill {\tilde{\sigma }}_v\left(w\right)=w-2{\displaystyle \frac{q(v,w)}{q\left(v\right)}}v\end{array}$$

(12)

By the above exact sequence $Spin(V,q)$ is a double covering of $SO(V,q)$.

A $\mathbb{K}$ ($\mathbb{K}$=$\mathbb{C}$ or $\mathbb{R}$) representation of $C\ell (V,q)$ is a $\mathbb{K}$-algebra homomorphism

$$\begin{array}{c}\hfill C\ell (V,q)⟶Ho{m}_{\mathbb{K}}(W,W)\end{array}$$

of $C\ell (V,q)$ into the linear transformations of a finite dimensional vector space W. W is called a $\mathbb{K}$-module of $C\ell (V,q)$. Clearly this defines also representations of $Spin(V,q)$ and $SO(V,q)$ as groups contained in $C\ell (V,q)$. In particular $C\ell (V,q)$ can be itself a representation space for the adjoint or twisted adjoint action defined above. Of course this will be a representation only of $C{\ell }^{\times }(V,q)$ or of its subgroups $Spin(V,q)$ and $SO(V,q)$.

We are now ready to define spinors. Among various definitions that can be met in the literature we choose the following one:

• Given a vector space $V={\mathbb{R}}^{r,s}$ with canonical quadratic form q and the associated Clifford algebra $C\ell (V,q)$, a spinor is any element of an irreducible representation space of the associated spin group $Spin(V,q)$.

A classification of spin representations can be found in the literature. Since, in the problems considered in this paper, we are interested in complex representations, we limit ourselves to them. They are as follows

• $n=2k,{\mathbb{C}}^{2^{k-1}}\oplus {\mathbb{C}}^{2^{k-1}}$
• $n=2k+1,{\mathbb{C}}^{2^k}$

($\mathbb{C}$ is the field of complex numbers).

The meaning of the decomposition in two irreducible representations when $n=2k$ is related to the volume element. In the complex case the latter is

$$\begin{array}{c}\hfill {\omega }_{\mathbb{C}}=i^k{e}_1\dots e_{2k},\end{array}$$

(13)

where $e_i$ is an orthonormal basis of $V={\mathbb{R}}^{r,s}$. It has the property ${\omega }_{\mathbb{C}}^2=1$, $\forall k$. The two irreducible representations correspond to the eigenvalues $\pm 1$ of ${\omega }_{\mathbb{C}}$ and are identified by the projectors ${\pi }_{\pm }={\displaystyle \frac{1}{2}}(1\pm {\omega }_{\mathbb{C}})$.

The element ${\omega }_{\mathbb{C}}$ is evidently an alias of the chirality operator in quantum field theory.

2.1. Spinor Representations in Even Dimension

The properties of spinors in even dimensional spaces are summarized in the following Table 1:

where $L,R$ denotes left-handed or right-handed spinors; the numbers in the columns with $\mathbb{C}$ or $\mathbb{R}$ tag denote the number of complex or real components according to whether they are in column $\mathbb{C}$ or $\mathbb{R}$, respectively. `Self’ means self-conjugate and `Mutual’ means that the L and R representations are conjugate to each other.

After this general and abstract introduction we focus in the sequel on 4d spinors and fields.

3. Spinors in 4 D

In this Section we shall analyze in depth the two basic forms of spinors: namely, the Weyl 2-component spinors, their chiral bispinor representations and the corresponding dynamical properties, as well as the charge self-conjugated Majorana bispinors. As we shall see, the longstanding well-known Dirac bispinors will be obtained as complex combinations of Majorana bispinors and in terms of a local Bogoliubov transformation. For the sake of clarity and completeness, we would like to specify that in the sequel we shall employ the following (more detailed) conventions and notations.

The Pauli spin matrices are the three Hermitean $2\times 2$ matrices

$${\sigma }_1=〔\begin{array}{cc}0& 1\\ 1& 0\end{array}〕{\sigma }_2=〔\begin{array}{cc}0& -i\\ i& 0\end{array}〕{\sigma }_3=〔\begin{array}{cc}1& 0\\ 0& -1\end{array}〕$$

which satisfy

$${\sigma }_j{\sigma }_k={\delta }_{jk}+i{\epsilon }_{jk\ell }{\sigma }_{\ell }(j,k,\ell =1,2,3)$$

where ${\epsilon }_{123}=1=-{\epsilon }^{123}$ so that

$$[{\sigma }_j,{\sigma }_k]=2i{\epsilon }_{jk\ell }{\sigma }_{\ell }\{{\sigma }_i,{\sigma }_j\}=2{\delta }_{ij}$$

The Dirac matrices in the Weyl, or spinorial, or even chiral representation are given by

$${\gamma }^0=〔\begin{array}{cc}0& \mathbf{1}\\ \mathbf{1}& 0\end{array}〕{\gamma }^k=〔\begin{array}{cc}0& {\sigma }_k\\ -{\sigma }_k& 0\end{array}〕{\gamma }_5=〔\begin{array}{cc}-\mathbf{1}& 0\\ 0& \mathbf{1}\end{array}〕$$

with the Hermitean conjugation property

$${\gamma }^{\mu †}={\gamma }_0{\gamma }^{\mu }{\gamma }_0\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2g^{\mu \nu }\mathbb{I}$$

so that

$$\beta ={\gamma }^0={\gamma }_0=〔\begin{array}{cc}0& \mathbf{1}\\ \mathbf{1}& 0\end{array}〕{\alpha }^k={\gamma }_0{\gamma }^k=〔\begin{array}{cc}-{\sigma }_k& 0\\ 0& {\sigma }_k\end{array}〕(k=1,2,3)$$

We denote by ${\psi }_L$ the 2-component spinor belonging to the irreducible bidimensional left representation $D({\displaystyle \frac{1}{2}},0)$ of the Lorentz group, while ${\psi }_R$ belongs to its nonequivalent right counterpart $D(0,{\displaystyle \frac{1}{2}}).$ The same holds true for the spin-states $u_L\left(\mathbf{p}\right)$ and $u_R\left(\mathbf{p}\right)$et cetera. The four-component spinors, or bispinors, $\psi \left(x\right)$ do belong to the reducible tetradimensional representation $D({\displaystyle \frac{1}{2}},0)⨁D(0,{\displaystyle \frac{1}{2}})$

$$\psi \left(x\right)=〔\begin{array}{c}{\psi }_L\left(x\right)\\ {\psi }_R\left(x\right)\end{array}〕$$

We denote the rank-two chiral projectors with ${\mathbb{P}}_{\pm }\equiv {\displaystyle \frac{1}{2}}(1\pm {\gamma }_5),$ in such a manner that for any four-component bispinor $\psi \left(x\right)$

$${\psi }_{-}\left(x\right)={\mathbb{P}}_{-}\psi \left(x\right)=〔\begin{array}{c}{\psi }_L\left(x\right)\\ 0\end{array}〕$$

$${\psi }_{+}\left(x\right)={\mathbb{P}}_{+}\psi \left(x\right)=〔\begin{array}{c}0\\ {\psi }_R\left(x\right)\end{array}〕$$

and the chiral spin-states $u_{\pm }\left(\mathbf{p}\right)$ et cetera are defined in the very same way.

3.1. Weyl Spinor Fields

Let us first analyze the massless and chiral Weyl spinor fields and their specific properties at the classical and quantum level.

Consider a massless Dirac bispinor, i.e., any complex and Grassmann valued bispinor in the Minkowski space, which is a solution of the differential equation

$$\partial /\psi \left(x\right)=0\psi \left(x\right)=〔\begin{array}{c}{\psi }_L\left(x\right)\\ {\psi }_R\left(x\right)\end{array}〕$$

In order to obtain the most general solution of the above equation, one has to sensibly move from the conventional approach and formalism that has been developed for the celebrated Dirac equation

$$i\partial /\psi \left(x\right)=M\psi \left(x\right)$$

In the chiral representation for the gamma matrices we can build up a very convenient set of spin-states as follows. Consider the matrix ${\alpha }^{\nu }\equiv {\gamma }^0{\gamma }^{\nu }=(\mathbb{I},\overrightarrow{\gamma })$ with matrix elements

$${\alpha }^{\nu }=〔\begin{array}{cc}{\sigma }^{\nu }& 0\\ 0& {\tilde{\sigma }}^{\nu }\end{array}〕$$

where we have set ${\sigma }^{\nu }=(\mathbb{I},-{\sigma }_k)$ with $k=1,2,3,$ while ${\tilde{\sigma }}^{\nu }=(\mathbb{I},{\sigma }_k).$ Then the massless Dirac equation can also be cast in the form

$$\alpha ·\partial \psi \left(x\right)=({\partial }_0+{\alpha }^k{\partial }_k)\psi \left(x\right)=0⟺\left\{\begin{array}{c}\sigma ·\partial {\psi }_L\left(x\right)=0\\ \overline{\sigma }·\partial {\psi }_R\left(x\right)=0\end{array}\right.$$

(14)

in such a manner that it can be rewritten à la Schrödinger

$$i\hslash {\displaystyle \frac{\partial \psi }{\partial t}}=H_0\psi$$

where the massless Dirac Hamiltonian operator reads, in physical units,

$$H_0=c{\alpha }^k(-i\hslash {\nabla }_k)=c{\alpha }^k{\widehat{p}}^k$$

Solving the massless Dirac equation is nothing but finding the general solutions of the pair of decoupled Weyl equations (14).

3.1.1. Spin-States

We can search the plane waves solutions of eq. (14) in the form

$$\psi \left(x\right)=\Gamma \left(p\right)exp\{-i{p}_{\mu }{x}^{\mu }\}=\Gamma \left(p\right)e^{-ip·x}{p}_{\mu }={\tilde{p}}^{\mu }=(p_0,-\mathbf{p})$$

(15)

where $\Gamma \left(p\right)$ are the so called spin-states associated to the plane wave solutions of the massless Dirac-Weyl equation (14). Now, if we set $H_0\left(\mathbf{p}\right)\equiv {\alpha }^k{p}^k=\overrightarrow{\alpha }·\overrightarrow{p},$ from the commutation relation $[{\gamma }_5,H_0]=0$ we see that we can always select a basis in the 4d complex linear space of the bispinor spin-states according to their chirality $\chi =\mp 1$ and energy (or frequency) $p_0=\pm \left|\mathbf{p}\right|$ pair of eigenvalues.

Thus, in order to realize a basis, we have to search among states of opposite chirality and frequency. Then, in so doing, we will be able to have at hand a complete and orthogonal quartet of spin-states for the 4d massless bispinor space. The construction goes as follows. Explicit evaluation of the relevant spin-matrices yields for $p_0=\left|\overrightarrow{p}\right|=\wp$

$$\sigma ·p={\sigma }_{\mu }{p}^{\mu }=\wp \mathbb{I}+{\sigma }_k{p}^k=\wp +\overrightarrow{\sigma }·\overrightarrow{p}=〔\begin{array}{cc}\wp +p_z& p_x-i{p}_y\\ p_x+i{p}_y& \wp -p_z\end{array}〕$$

$$\tilde{\sigma }·p={\sigma }_{\mu }{\tilde{p}}^{\mu }=\wp \mathbb{I}-{\sigma }_k{p}^k=\wp -\overrightarrow{\sigma }·\overrightarrow{p}=〔\begin{array}{cc}\wp -p_z& -p_x+i{p}_y\\ -p_x-i{p}_y& \wp +p_z\end{array}〕$$

$$\begin{array}{ccc}\hfill {\gamma }^{\nu }{p}_{\nu }& \equiv & p/=〔\begin{array}{cc}0& \tilde{\sigma }·p\\ \sigma ·p& 0\end{array}〕\hfill \\ & =& 〔\begin{array}{cccc}0& 0& \wp -p_z& -p_x+i{p}_y\\ 0& 0& -p_x-i{p}_y& \wp +p_z\\ \wp +p_z& p_x-i{p}_y& 0& 0\\ p_x+i{p}_y& \wp -p_z& 0& 0\end{array}〕\hfill \\ \\ \hfill {\gamma }^{\nu }{\tilde{p}}_{\nu }& \equiv & \tilde{p}/=〔\begin{array}{cc}0& \sigma ·p\\ \tilde{\sigma }·p& 0\end{array}〕\hfill \\ & =& 〔\begin{array}{cccc}0& 0& \wp +p_z& p_x-i{p}_y\\ 0& 0& p_x+i{p}_y& \wp -p_z\\ \wp -p_z& -p_x+i{p}_y& 0& 0\\ -p_x-i{p}_y& \wp +p_z& 0& 0\end{array}〕\hfill \end{array}$$

so that we find

$$\begin{array}{c}\hfill \sigma ·p\tilde{\sigma }·p=\tilde{\sigma }·p\sigma ·p=p^2={\wp }^2-{\overrightarrow{p}}^2=p{/}^2=\tilde{p}{/}^2=0\end{array}$$

(16)

$${(\sigma ·p)}^2=2\wp \sigma ·p{(\tilde{\sigma }·p)}^2=2\wp \tilde{\sigma }·p$$

Notice that ${\tilde{p}}^{\nu }=(\wp ,-\overrightarrow{p})=p_{\nu }$ is the dual light-cone tetra-vector which satisfies

$${\tilde{p}}^2=p^2=0p·\tilde{p}=2{\overrightarrow{p}}^2=2{\wp }^2$$

Moreover it is useful to define the on-shell $(p_0=\wp )$ Weyl-Dirac 2d projectors

$$\begin{array}{c}\hfill \Pi \left(\mathbf{p}\right)\equiv {\displaystyle \frac{\tilde{p}/p/}{4{\wp }^2}}={\displaystyle \frac{1}{2\wp }}〔\begin{array}{cc}\sigma ·p& 0\\ 0& \tilde{\sigma }·p\end{array}〕={\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{\overrightarrow{\alpha }·\overrightarrow{p}}{\wp }}\right)={\displaystyle \frac{\alpha ·p}{2\wp }}\end{array}$$

(17)

$$\begin{array}{c}\hfill \tilde{\Pi }\left(\mathbf{p}\right)\equiv {\displaystyle \frac{p/\tilde{p}/}{4{\wp }^2}}={\displaystyle \frac{1}{2\wp }}〔\begin{array}{cc}\tilde{\sigma }·p& 0\\ 0& \sigma ·p\end{array}〕={\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{\overrightarrow{\alpha }·\overrightarrow{p}}{\wp }}\right)={\displaystyle \frac{\alpha ·\tilde{p}}{2\wp }}\end{array}$$

(18)

which fulfill by construction

$$\begin{array}{c}\hfill \Pi ={\Pi }^{†}{\Pi }^2=\Pi \mathrm{Tr}\Pi =2\end{array}$$

(19)

$$\begin{array}{c}\hfill \tilde{\Pi }={\tilde{\Pi }}^{†}{\tilde{\Pi }}^2=\tilde{\Pi }\mathrm{Tr}\tilde{\Pi }=2\end{array}$$

(20)

$$\Pi \tilde{\Pi }=0=\tilde{\Pi }\Pi [{\gamma }_5,\Pi ]=[\tilde{\Pi },{\gamma }_5]=0$$

(21)

$${\gamma }_0\Pi {\gamma }_0=\tilde{\Pi }{\gamma }_0\tilde{\Pi }{\gamma }_0=\Pi$$

(22)

It means that $\Pi$ and $\tilde{\Pi }$ are projectors onto 2d orthogonal chirally invariant though parity exchanging spaces. The momentum space Weyl-Dirac equation (14) for the positive energy spin-states evidently reads $\Pi \left(\mathbf{p}\right)\Gamma (\wp ,\mathbf{p})=0,$ the general solution of which being $\Gamma (\wp ,\mathbf{p})=\tilde{\Pi }\left(\mathbf{p}\right)\xi ,$ where $\xi$ is any arbitrary constant bispinor. Hence, without loss of generality, we can set, for example, with $p^{\nu }=(\wp ,\mathbf{p})=(\wp ,\overrightarrow{p})$

$$\begin{array}{ccc}\hfill u_{-}\left(\mathbf{p}\right)& =& {\displaystyle \frac{1}{\sqrt{\wp -p_z}}}{\alpha }^{\nu }{\tilde{p}}_{\nu }\xi =〔\begin{array}{c}{u}_L\left(\mathbf{p}\right)\\ 0\\ 0\end{array}〕\hfill \\ & =& {\displaystyle \frac{1}{\sqrt{\wp -p_z}}}〔\begin{array}{c}\wp -p_z\\ -p_x-i{p}_y\\ 0\\ 0\end{array}〕\hfill \\ \hfill \xi & =& 〔\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}〕u_L\left(\mathbf{p}\right)={\displaystyle \frac{\sigma ·\tilde{p}}{\sqrt{\wp -p_z}}}\left|\begin{array}{c}1\\ 0\end{array}\right|\hfill \end{array}$$

(23)

The above spin-state is a positive frequency solution of equation (14) and exhibits negative chirality

$$\begin{array}{ccc}& & H_0\left(\mathbf{p}\right)u_{-}\left(\mathbf{p}\right)=\wp u_{-}\left(\mathbf{p}\right){\gamma }_5{u}_{-}\left(\mathbf{p}\right)=-u_{-}\left(\mathbf{p}\right)\hfill \end{array}$$

(24)

$$\begin{array}{ccc}& & \Pi u_{-}\left(\mathbf{p}\right)=0\tilde{\Pi }{u}_{-}\left(\mathbf{p}\right)=u_{-}\left(\mathbf{p}\right)\hfill \end{array}$$

(25)

Moreover, the 2-component left-handed Weyl spinor $u_L\left(\mathbf{p}\right)$ does fulfill

$$u_L^{†}\left(\mathbf{p}\right)u_L\left(\mathbf{p}\right)=2\wp u_L\left(\mathbf{p}\right)\otimes u_L^{†}\left(\mathbf{p}\right)=2\wp {\tilde{P}}_L(p_0=\wp )$$

(26)

where the rank-two square matrix

$${\tilde{P}}_L\equiv {\displaystyle \frac{\tilde{\sigma }·p}{2\wp }}={\displaystyle \frac{\sigma ·\tilde{p}}{2\wp }}$$

does satisfy for $p_0=\wp$

$$\sigma ·p{\tilde{P}}_L=0{\tilde{P}}_L^{†}={\tilde{P}}_L{\tilde{P}}_L^2={\tilde{P}}_L\mathrm{Tr}{\tilde{P}}_L=1$$

so that it corresponds to the one dimensional projector on positive energy and left-handed spin-states.

In order to set up a right-handed positive energy spin-state, we can involve the charge conjugated of the above constructed spin-state: namely,

$$u_{+}\left(\mathbf{p}\right)\equiv i{\gamma }^2{u}_{-}^{*}\left(\mathbf{p}\right)=u_{-}^c\left(\mathbf{p}\right)$$

(27)

As a matter of fact, it is easy to check that the chiral bispinor $u_{+}\left(\mathbf{p}\right)$ satisfies the distinguished features

$${\gamma }_5{u}_{+}\left(\mathbf{p}\right)=u_{+}\left(\mathbf{p}\right)H_0\left(\mathbf{p}\right)u_{+}\left(\mathbf{p}\right)=\wp u_{+}\left(\mathbf{p}\right)$$

(28)

Proof. We immediately find

$${\gamma }_5{u}_{+}\left(\mathbf{p}\right)=i{\gamma }_5{\gamma }^2{u}_{-}^{*}\left(\mathbf{p}\right)=-i{\gamma }^2{\left[{\gamma }_5{u}_{-}\left(\mathbf{p}\right)\right]}^{*}=i{\gamma }^2{u}_{-}^{*}\left(\mathbf{p}\right)=u_{+}\left(\mathbf{p}\right)$$

Furthermore we can write

$$\begin{array}{c}\hfill H_0\left(\mathbf{p}\right)u_{+}\left(\mathbf{p}\right)=H_0\left(\mathbf{p}\right)i{\gamma }^2{u}_{-}^{*}\left(\mathbf{p}\right)={\left[H_0^{*}\left(\mathbf{p}\right)i{\gamma }^2{u}_{-}\left(\mathbf{p}\right)\right]}^{*}\\ \hfill H_0^{*}\left(\mathbf{p}\right){\gamma }^2=({\alpha }^1{p}_x+{\alpha }^3{p}_z-{\alpha }^2{p}_y){\gamma }^2={\gamma }^2{H}_0\left(\mathbf{p}\right)\\ \hfill H_0\left(\mathbf{p}\right)u_{+}\left(\mathbf{p}\right)={\left[i{\gamma }^2\wp u_{-}\left(\mathbf{p}\right)\right]}^{*}=i{\gamma }^2\wp u_{-}^{*}\left(\mathbf{p}\right)=\wp u_{+}\left(\mathbf{p}\right)\end{array}$$

that proves the above statements (28).

On the other side, it is immediate to realize that a right-handed spin-state with negative energy is provided by

$$v_{+}\left(\mathbf{p}\right)\equiv {\gamma }_0{u}_{-}\left(\mathbf{p}\right)$$

(29)

which obviously fulfills

$${\gamma }_5{v}_{+}\left(\mathbf{p}\right)=v_{+}\left(\mathbf{p}\right)H_0\left(\mathbf{p}\right)v_{+}\left(\mathbf{p}\right)=-\wp v_{+}\left(\mathbf{p}\right)$$

(30)

its charge conjugated being left-handed, i.e.,

$$v_{-}\left(\mathbf{p}\right)\equiv i{\gamma }^2{v}_{+}^{*}\left(\mathbf{p}\right)=v_{+}^c\left(\mathbf{p}\right)$$

(31)

and satisfying

$${\gamma }_5{v}_{-}\left(\mathbf{p}\right)=-v_{-}\left(\mathbf{p}\right)H_0\left(\mathbf{p}\right)v_{-}\left(\mathbf{p}\right)=-\wp v_{-}\left(\mathbf{p}\right)$$

(32)

The ultimate meaning of the above construction is that, once we have fixed the initial left-handed spin-state of positive energy, the rest of the spin-state orthogonal basis can be fully determined by the charge conjugation and parity discrete symmetry operations.

It turns out that the above defined quartet of massless and chiral bispinor spin-states does fulfill orthogonality and closure relations. As a matter of fact, spin-states of opposite chirality and/or opposite frequency are orthogonal

$$\begin{array}{c}\hfill \left.\begin{array}{c}{u}_{\pm }^{†}\left(\mathbf{p}\right)u_{\mp }\left(\mathbf{q}\right)=0=v_{\pm }^{†}\left(\mathbf{p}\right)v_{\mp }\left(\mathbf{q}\right)\\ \\ u_{\pm }^{†}\left(\mathbf{p}\right)v_{\mp }\left(\mathbf{q}\right)=0\\ \\ u_{\pm }^{†}\left(\mathbf{p}\right)v_{\pm }(-\mathbf{p})=0=v_{\mp }^{†}\left(\mathbf{p}\right)u_{\mp }(-\mathbf{p})\end{array}\right\}\end{array}$$

(33)

Moreover we have

$$\begin{array}{c}\hfill u_{\pm }^{†}\left(\mathbf{p}\right)u_{\pm }\left(\mathbf{p}\right)=v_{\pm }^{†}\left(\mathbf{p}\right)v_{\pm }\left(\mathbf{p}\right)=2\wp \end{array}$$

(34)

that exhausts the orthogonality relations and furthermore we get

$$\begin{array}{c}\hfill {(2\wp )}^{-1}[u_{\pm }\left(\mathbf{p}\right)\otimes u_{\pm }^{†}\left(\mathbf{p}\right)+v_{\pm }(-\mathbf{p})\otimes v_{\pm }^{†}(-\mathbf{p})]={\mathbb{P}}_{\pm }\end{array}$$

(35)

with of course ${\mathbb{P}}_{\pm }={\displaystyle \frac{1}{2}}(\mathbb{I}\pm {\gamma }_5)$ which represent the closure or completeness relations for the massless and chiral bispinor spin-states basis quartet.

The corresponding plane-wave functions, which are normal-mode solutions of the massless Dirac equation, read

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_{\pm ,\mathbf{p}}\left(x\right)={[{\left(2\pi \right)}^{3}2\wp ]}^{-{\displaystyle \frac{1}{2}}}{u}_{\pm }\left(\mathbf{p}\right)e^{-i\wp t+i\mathbf{p}·\mathbf{x}}\\ \\ v_{\pm ,\mathbf{p}}\left(x\right)={[{\left(2\pi \right)}^{3}2\wp ]}^{-{\displaystyle \frac{1}{2}}}{v}_{\pm }\left(\mathbf{p}\right)e^{i\wp t-i\mathbf{p}·\mathbf{x}}\end{array}\right.\end{array}$$

(36)

and fulfill in turn orthonormality and closure relation with respect to the usual Poincaré invariant inner product: for instance

$$\begin{array}{c}\hfill \int \mathrm{d}\mathbf{x}{u}_{ı,\mathbf{p}}^{†}\left(x\right)v_{j,{\mathbf{p}}^{\prime }}\left(x\right)=0\\ \hfill \int \mathrm{d}\mathbf{x}{u}_{ı,\mathbf{p}}^{†}\left(x\right)u_{j,{\mathbf{p}}^{\prime }}\left(x\right)={\delta }_{ıj}\delta (\mathbf{p}-{\mathbf{p}}^{\prime })=\int \mathrm{d}\mathbf{x}{v}_{ı,\mathbf{p}}^{†}\left(x\right)v_{j,{\mathbf{p}}^{\prime }}\left(x\right)\\ \hfill (\forall ı,j=+,-\vee \mathbf{p},{\mathbf{p}}^{\prime }\in {\mathbb{R}}^3)\end{array}$$

et cetera.

We recall that, according to the general principles of quantum field theory, the plane wave functions $v_{\pm ,\mathbf{p}}\left(x\right)$ of negative energy or frequency are associated to the anti-particles of the corresponding spin ${\displaystyle \frac{1}{2}}$ charged quantum field.

In the Standard Model of the fundamental interactions the neutrino and anti-neutrino particles are supposed to be massless2 and charge conjugated one to each other, the charge being identified with the so called lepton number. Moreover, the statu quo of the present phenomenological frame [10] does require that neutrino particles are always left-handed and with a negative helicity, while anti-neutrinos always appear to right-handed and with a positive helicity. Sic stantibus rebus it turns out to be natural to associate $u_{-}\left(\mathbf{p}\right),u_{-,\mathbf{p}}\left(x\right)$ to the left-handed spin-state and normal mode, respectively, of the Standard Model neutrino Weyl field, while $v_{+}\left(\mathbf{p}\right),v_{+,\mathbf{p}}\left(x\right)$ will refer to the corresponding negative frequency spin-state and normal modes, which will be related to the anti-neutrino particles of opposite chirality and helicity, as far as their masses can eventually be neglected. It is worthwhile to stress once more that this mandatory strong correlation between chirality and helicity does actually occur only for the Weyl spinor field and never for any other kind of relativistic spin ${\displaystyle \frac{1}{2}}$ wave fields.

The remaining pair of chiral spin-states and normal-modes $u_{+},v_{-}$ would correspond to hypothetical right-handed neutrinos and related left-handed anti-neutrino spin-states and normal modes, that have never been observed so far, although their existence does represent, if any, one of the most intriguing and exciting actual possibilities to go beyond the Standard Model.

3.1.2. Symmetries

The classical Lagrange density for the massless Dirac field can be written in different forms: namely,

$$\begin{array}{ccc}\hfill {\mathcal{L}}_0& =& {\psi }^{†}\left(x\right)i\alpha ·\partial \psi \left(x\right)\doteq \frac{1}{2}{\psi }^{†}\left(x\right)i\alpha ·\overleftrightarrow{\partial }\psi \left(x\right)\hfill \end{array}$$

(37)

$$\begin{array}{ccc}& \doteq & \frac{1}{2}{\psi }_L^{†}\left(x\right)i\sigma ·\overleftrightarrow{\partial }{\psi }_L\left(x\right)+\frac{1}{2}{\psi }_R^{†}\left(x\right)i\overline{\sigma }·\overleftrightarrow{\partial }{\psi }_R\left(x\right)\hfill \end{array}$$

(38)

up to irrelevant tetra-divergences, where the spinor fields are understood as Grassmann valued functions on the Minkowski space, whereas

$$\psi \left(x\right)=〔\begin{array}{c}{\psi }_L\left(x\right)\\ {\psi }_R\left(x\right)\end{array}〕{\alpha }^{\nu }=〔\begin{array}{cc}{\sigma }^{\nu }& 0\\ 0& {\overline{\sigma }}^{\nu }\end{array}〕$$

The Euler-Lagrange field equations are

$$\alpha ·\partial \psi \left(x\right)=0\iff \sigma ·\partial {\psi }_L\left(x\right)=\overline{\sigma }·\partial {\psi }_R\left(x\right)=0$$

the general plane-wave solutions of which have been obtained and discussed in the previous Section. The canonical energy-momentum tensor reads

$$\begin{array}{c}\hfill T_{\mu \nu }={\psi }^{†}\left(x\right){\alpha }_{\mu }i{\partial }_{\nu }\psi \left(x\right)={\psi }_L^{†}\left(x\right){\sigma }_{\mu }i{\partial }_{\nu }{\psi }_L\left(x\right)+{\psi }_R^{†}\left(x\right){\overline{\sigma }}_{\mu }i{\partial }_{\nu }{\psi }_R\left(x\right)\end{array}$$

(39)

which is not symmetric and drives to the conserved energy-momentum vector

$$\begin{array}{ccc}\hfill P_0& =& \int \mathrm{d}\mathbf{x}{\psi }^{†}\left(x\right)i{\partial }_0\psi \left(x\right)=\int \mathrm{d}\mathbf{x}{\psi }^{†}\left(x\right)H_0\psi \left(x\right)\hfill \end{array}$$

(40)

$$\begin{array}{ccc}& =& \int \mathrm{d}\mathbf{x}\left[{\psi }_L^{†}\left(x\right)H_L{\psi }_L\left(x\right)+{\psi }_R^{†}\left(x\right)H_R{\psi }_R\left(x\right)\right]\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill \mathbf{P}& =& \int \mathrm{d}\mathbf{x}{\psi }^{†}\left(x\right)(-i\hslash \nabla )\psi \left(x\right)\hfill \end{array}$$

(42)

Notice that the energy-momentum tensor is trace-less $g^{\mu \nu }{T}_{\mu \nu }=0,$ owing to the validity of the field equations. The canonical total angular momentum density tensor can always be written in the form

$$\begin{array}{c}\hfill M^{\lambda \mu \nu }=x^{\mu }{T}^{\lambda \nu }-x^{\nu }{T}^{\lambda \mu }+S^{\lambda \mu \nu }\end{array}$$

(43)

$$\begin{array}{c}\hfill S^{\lambda \mu \nu }=\frac{1}{2}{\psi }^{†}\left(x\right){\gamma }^0\{{\gamma }^{\lambda },{\sigma }^{\mu \nu }\}\psi \left(x\right)\end{array}$$

(44)

so that from Noether theorem related to the Lorentz symmetry we get

$$T_{\nu \mu }-T_{\mu \nu }={\partial }^{\lambda }{S}_{\lambda \mu \nu }$$

which notoriously entails the helicity conservation in any direction along which the energy-momentum tensor appears to be symmetric. In order to determine the helicity assignements we can proceed as follows. Since the generators of the rotation group for both the irreducible Weyl spinor representations are $S_{jk}={\displaystyle \frac{1}{2}}{\epsilon }_{jk\ell }{\sigma }_{\ell }$ it follows that the corresponding Pauli-Lubanski operator reads

$$W_0=\mathbf{P}·\mathbf{S}=\overrightarrow{p}·\frac{1}{2}\hslash \overrightarrow{\sigma }$$

$$W_j=\frac{1}{2}\wp {\sigma }_j-\frac{1}{2}i{\epsilon }_{jk\ell }{p}_k{\sigma }_{\ell }$$

which entails

$$\begin{array}{c}\hfill g^{\mu \nu }{W}_{\mu }{W}_{\nu }=\frac{1}{4}{p}_j{p}_k{\sigma }_j{\sigma }_k-\frac{1}{4}\left(\wp {\sigma }_j-i{\epsilon }_{jk\ell }{p}_k{\sigma }_{\ell }\right)\left(\wp {\sigma }_j-i{\epsilon }_{jrs}{p}_r{\sigma }_s\right)\\ \hfill =-\frac{1}{2}{\wp }^2+\frac{1}{4}\left({\delta }_{kr}{\delta }_{\ell s}-{\delta }_{ks}{\delta }_{\ell r}\right)p_k{p}_r{\sigma }_{\ell }{\sigma }_s=0\end{array}$$

where the $2\times 2$ identity matrix has been understood. Then, thanks to the light-like nature of the Pauli-Lubanski operator, we can identify the helicity operator to be

$$\mathrm{h}\equiv {\displaystyle \frac{W_0}{\wp }}={\displaystyle \frac{\left|\mathbf{W}\right|}{\wp }}=\frac{1}{2}\hslash \widehat{n}·\overrightarrow{\sigma }\widehat{n}=\overrightarrow{p}/\wp$$

together with the Weyl Hamiltonian operators

$$H_0=〔\begin{array}{cc}{H}_L& 0\\ 0& H_R\end{array}〕H_R=c\hslash \overrightarrow{p}·\overrightarrow{\sigma }=-H_L$$

in such a manner that we find $H_L=-\hslash c\wp \mathrm{h},H_R=\hslash c\wp \mathrm{h}$ and consequently

$$\begin{array}{ccc}& & H_L{u}_L\left(\mathbf{p}\right)=\hslash c\wp u_L\left(\mathbf{p}\right)\mathrm{h}{u}_L\left(\mathbf{p}\right)=-\frac{1}{2}{u}_L\left(\mathbf{p}\right)\hfill \\ & & H_L{v}_L\left(\mathbf{p}\right)=-\hslash c\wp v_L\left(\mathbf{p}\right)\mathrm{h}{v}_L\left(\mathbf{p}\right)=\frac{1}{2}{v}_L\left(\mathbf{p}\right)\hfill \\ \\ & & H_R{u}_R\left(\mathbf{p}\right)=\hslash c\wp u_R\left(\mathbf{p}\right)\mathrm{h}{u}_R\left(\mathbf{p}\right)=\frac{1}{2}{u}_R\left(\mathbf{p}\right)\hfill \\ & & H_R{v}_R\left(\mathbf{p}\right)=-\hslash c\wp v_R\left(\mathbf{p}\right)\mathrm{h}{v}_R\left(\mathbf{p}\right)=-\frac{1}{2}{v}_R\left(\mathbf{p}\right)\hfill \end{array}$$

which means that :

• neutrino spin-states carry negative helicity, while the anti-neutrino spin-states carry positive helicity, i.e. ${\gamma }_5{u}_{-}\left(\mathbf{p}\right)=-{\gamma }_5{v}_{+}\left(\mathbf{p}\right)=-1.$
• The situation is reversed for the remaining pair of the spin-states. Thus, positive helicity is exhibited by positive frequency particles - the would-be right neutrinos - while negative helicity by negative frequency anti-particles, i.e. ${\gamma }_5{u}_{+}\left(\mathbf{p}\right)=-{\gamma }_5{v}_{-}\left(\mathbf{p}\right)=+1.$

The Weyl Action integrals

$$S_L=\int {\mathrm{d}}^{4}x\frac{1}{2}{\psi }_L^{†}\left(x\right)i\sigma ·\overleftrightarrow{\partial }{\psi }_L\left(x\right)S_R=\int {\mathrm{d}}^{4}x\frac{1}{2}{\psi }_R^{†}\left(x\right)i\overline{\sigma }·\overleftrightarrow{\partial }{\psi }_R\left(x\right)$$

and the massless Dirac Action integral, which is the sum of the left- and right-handed Weyl Action integrals, are invariant under scale or dilatation symmetry. The latter acts upon any inertial coordinate system and spinor field variables on the Minkowski space according to

$$\begin{array}{c}\hfill x^{\lambda }⟼x^{\prime \lambda }=e^{-\varrho }{x}^{\lambda }(\varrho \in \mathbb{R})\end{array}$$

(45)

$$\begin{array}{c}\hfill \Psi \left(x\right)⟼{\Psi }^{\prime }\left(x\right)=e^{{\displaystyle \frac{3}{2}}\varrho }\Psi \left(e^{\varrho }x\right)(\Psi =\psi ,{\psi }_L,{\psi }_R)\end{array}$$

(46)

In order to obtain the infinitesimal operators of the scale transformations we consider an infinitesimal dilatation/contraction, viz.,

$$x^{\prime \mu }=(1-\delta \varrho +\dots )x^{\mu }\left(\right|\delta \varrho |\ll 1)$$

$$\begin{array}{c}\hfill {\Psi }^{\prime }\left(x\right)=\left(1+\frac{3}{2}\delta \varrho +\dots \right)\Psi (x+x\delta \varrho +\dots )\\ \hfill =\left(1+\frac{3}{2}\delta \varrho \right)\left[\Psi \left(x\right)+x·\partial \Psi \left(x\right)\delta \varrho \right]+\dots \\ \hfill =\Psi \left(x\right)+\frac{3}{2}\delta \varrho \Psi \left(x\right)+x·\partial \Psi \left(x\right)\delta \varrho +\dots \end{array}$$

and thereby

$$\delta \Psi \left(x\right)=\delta \varrho \left[\frac{3}{2}+x·\partial \right]\Psi \left(x\right)=\frac{3}{2}\delta \varrho \Psi \left(x\right)-\mathrm{d}\Psi \left(x\right)$$

because $\delta x=-x\delta \varrho .$ Then we can write the infinitesimal total variation of the spinor fields in the form

$$\Delta \Psi \left(x\right)=(\delta +\mathrm{d})\Psi \left(x\right)=\frac{3}{2}\delta \varrho \Psi \left(x\right)$$

(47)

in accordance with the well-known general rule that connects local ($\delta$) and total ($\Delta$) variations of any classical relativistic wave field. Hence, we are allowed to identify

$${\mathrm{D}}_{\Psi }\equiv \frac{3}{2}+x·\partial$$

(48)

with the generator of the local scale transformations on the classical spinor fields. Now, since ${\Psi }^{\prime }\left(x^{\prime }\right)=e^{{\displaystyle \frac{3}{2}}\varrho }\Psi \left(x\right),$ it follows that the Lagrangian (37) does satisfy

$${\mathcal{L}}_0^{\prime }\left(x^{\prime }\right)=e^{4\varrho }{\mathcal{L}}_0\left(x\right)$$

so that the Action integral for the massless Dirac spinor field appears to be invariant, viz.,

$$S=\int {\mathrm{d}}^4{x}^{\prime }{\mathcal{L}}_0^{\prime }\left(x^{\prime }\right)=\int {\mathrm{d}}^{4}x{\mathcal{L}}_0\left(x\right)$$

(49)

The dilatation tetra-current can be readily obtained from Noether theorem and can be written in different though equivalent forms, viz.,

$$\begin{array}{ccc}\hfill {\mathcal{D}}^{\mu }\left(x\right)& =& {\displaystyle \frac{\delta {\mathcal{L}}_0}{\delta {\partial }_{\mu }\psi \left(x\right)}}{\mathrm{D}}_{\psi }\psi \left(x\right)-{\mathcal{L}}_0\left(x\right)x^{\mu }\hfill \\ & =& \frac{\delta {\mathcal{L}}_0}{\delta {\partial }_{\mu }\psi \left(x\right)}\frac{3}{2}\psi \left(x\right)+x_{\nu }{T}^{\mu \nu }\left(x\right)\hfill \\ & =& \frac{3}{2}{\psi }^{†}\left(x\right)i{\alpha }^{\mu }\psi \left(x\right)+x_{\nu }{T}^{\mu \nu }\left(x\right)\hfill \end{array}$$

(50)

so that

$$\partial ·\mathcal{D}=g^{\mu \nu }{T}_{\mu \nu }=0$$

(51)

Finally, it turns out that the massless Dirac Action integral turns out to be invariant under the U(1) groups of the ordinary and chiral phase changes,viz.,

$$\begin{array}{c}\hfill \psi \left(x\right)⟼\left\{\begin{array}{c}{\psi }^{\prime }\left(x\right)=e^{-i\theta }\psi \left(x\right)\\ \\ {\psi }^{\prime }\left(x\right)=e^{-i{\gamma }_5\theta }\psi \left(x\right)\end{array}\right.(0\le \theta <2\pi )\end{array}$$

(52)

or equivalently

$$\begin{array}{c}\hfill {\psi }_L\left(x\right)⟼\left\{\begin{array}{c}{\psi }_L^{\prime }\left(x\right)=e^{-i\theta }{\psi }_L\left(x\right)\\ \\ {\psi }_L^{\prime }\left(x\right)=e^{i\theta }{\psi }_L\left(x\right)\end{array}\right.(0\le \theta <2\pi )\end{array}$$

(53)

$$\begin{array}{c}\hfill {\psi }_R\left(x\right)⟼{\psi }_R^{\prime }\left(x\right)=e^{-i\theta }{\psi }_R\left(x\right)(0\le \theta <2\pi )\end{array}$$

(54)

in such a manner that Noether theorem drives to the internal currents

$$J^{\mu }\left(x\right)={\psi }^{†}\left(x\right){\alpha }^{\mu }\psi \left(x\right)J_5^{\mu }\left(x\right)={\psi }^{†}\left(x\right){\alpha }^{\mu }{\gamma }_5\psi \left(x\right)$$

(55)

which satisfy the continuity equation so that the internal charges

$$\begin{array}{c}\hfill \mathcal{Q}=\int \mathrm{d}\mathbf{x}\left[{\psi }_L^{†}\left(x\right){\psi }_L\left(x\right)+{\psi }_R^{†}\left(x\right){\psi }_R\left(x\right)\right]\end{array}$$

(56)

$$\begin{array}{c}\hfill {\mathcal{Q}}_5=\int \mathrm{d}\mathbf{x}\left[-{\psi }_L^{†}\left(x\right){\psi }_L\left(x\right)+{\psi }_R^{†}\left(x\right){\psi }_R\left(x\right)\right]\end{array}$$

(57)

are conserved in time $\dot{\mathcal{Q}}={\dot{\mathcal{Q}}}_5=0.$ To sum up, for a massless Dirac field we have seen until now the occurrence of a 13-parameter Lie group of space-time and internal symmetries leading to thirteen conserved charges. Actually it can be shown, as we shall see later on, that scale invariance implies in fact the invariance under a larger 15-parameter symmetry group called the conformal group, which includes the ten dimensional Poincaré group. Scale invariance implies conformal invariance if and only if there exists in the considered theory a symmetric traceless conserved energy-momentum tensor ${\Theta }_{\mu \nu },$ called the improved energy-momentum tensor, such that $\partial ·\mathcal{D}=g^{\mu \nu }{\Theta }_{\mu \nu }=0.$ It turns out that this is true for all power-counting renormalizable massless models involving fields of spin $\le 1.$ Hence, the massless Dirac Action integral is actually invariant under a 17-parameter Lie group involving space-time as well as internal symmetries.

Concerning discrete symmetries, it turns out that the Action integral $S=\int {\mathrm{d}}^{4}x{\mathcal{L}}_0\left(x\right)$ does respect $\mathcal{C},\mathcal{P}$ and $\mathcal{T}$ transformations separately. The charge conjugation $\mathcal{C}$ is an internal discrete symmetry which acts on the classical Dirac spinors according to

$$\psi \left(x\right)⟼{\psi }^c\left(x\right)={\gamma }^2{\psi }^{*}\left(x\right)$$

while parity $\mathcal{P}$ transforms yields ${\psi }^{\prime }(t,-\mathbf{x})={\gamma }^0\psi \left(x\right),$ whereas time reversal $\mathcal{T}$ is provided by ${\psi }^{\prime }(-t,\mathbf{x})=\Theta \psi \left(x\right),\Theta ={\gamma }^3{\gamma }^1.$ It is straightforward to verify by direct inspection that the Action integral is invariant. Notice that the above mentioned discrete transforms are not even well defined for chiral spinors, for they mix the chiral components and necessarily drive out of the specific chiral spinor spaces of the two irreducible nonequivalent Weyl representations of the Lorentz group $\mathrm{D}({\displaystyle \frac{1}{2}},0)$ and $\mathrm{D}(0,{\displaystyle \frac{1}{2}}).$

3.1.3. Quantum Theory

The setting up of a canonical quantum theory for the massless Dirac-Weyl spinor field is far from being straightforward and trivial. First of all it is not at all unique: there are several options available in the market, the best buy to be selected and picked up through the attainment to the general principles of quantum field theory as well as by the correspondence with the actual phenomenology. In order to be appropriate and applicable to the concrete high energy and particle Physics context, the operation of parity and charge conjugation must be actually well defined and truly implementable: this basic requirement clearly rules out a priori the possibility to formulate the quantum theory in terms of the 2-component Weyl spinor field ${\psi }_L$ and ${\psi }_R,$ as the afore mentioned operations are not supported by the 2d Weyl spinors. Hence, the only possible consistent and physically meaningful formulation is in terms of the chiral bispinors ${\psi }_{\mp }.$

The massless Dirac quantum field is the operator valued tempered distribution

$$\begin{array}{c}\hfill \psi \left(x\right)={\psi }_{-}\left(x\right)+{\psi }_{+}\left(x\right)\end{array}$$

(58)

$$\begin{array}{c}\hfill {\psi }^{(-)}\left(x\right)=\int \mathrm{d}\mathbf{p}{[{\left(2\pi \right)}^{3}2\wp ]}^{-{\displaystyle \frac{1}{2}}}\left[c_{-,\mathbf{p}}{u}_{-}\left(\mathbf{p}\right)+c_{+,\mathbf{p}}{u}_{+}\left(\mathbf{p}\right)\right]e^{i\mathbf{p}·\mathbf{x}-i\wp t}\end{array}$$

(59)

$$\begin{array}{c}\hfill {\psi }^{(+)}\left(x\right)=\int \mathrm{d}\mathbf{p}{[{\left(2\pi \right)}^{3}2\wp ]}^{-{\displaystyle \frac{1}{2}}}\left[d_{-,\mathbf{p}}^{†}{v}_{-}\left(\mathbf{p}\right)+d_{+,\mathbf{p}}^{†}{v}_{+}\left(\mathbf{p}\right)\right]e^{i\wp t-i\mathbf{p}·\mathbf{x}}\end{array}$$

(60)

$$\begin{array}{c}\hfill {\psi }_{\mp }\left(x\right)={\mathbb{P}}_{\mp }\psi \left(x\right)=\int \mathrm{d}\mathbf{p}\left[c_{\mp ,\mathbf{p}}{u}_{\mp ,\mathbf{p}}\left(x\right)+d_{\mp ,\mathbf{p}}^{†}{v}_{\mp ,\mathbf{p}}\left(x\right)\right]\end{array}$$

(61)

$$\begin{array}{c}\hfill {\psi }_{\mp }^{†}\left(y\right)={\psi }^{†}\left(y\right){\mathbb{P}}_{\mp }=\int \mathrm{d}{\mathbf{p}}^{\prime }\left[c_{\mp ,{\mathbf{p}}^{\prime }}^{†}{u}_{\mp ,{\mathbf{p}}^{\prime }}^{\top *}\left(y\right)+d_{\mp ,{\mathbf{p}}^{\prime }}{v}_{\mp ,{\mathbf{p}}^{\prime }}^{\top *}\left(y\right)\right]\end{array}$$

(62)

which corresponds to the most general solution of the Weyl-Dirac equations (14), in which the creation and destruction operators satisfy the canonical anti-commutation relations

$$\{c_{\mp ,\mathbf{p}},c_{\mp ,{\mathbf{p}}^{\prime }}^{†}\}=\{d_{\mp ,\mathbf{p}},d_{\mp ,{\mathbf{p}}^{\prime }}^{†}\}=\delta (\mathbf{p}-{\mathbf{p}}^{\prime })$$

(63)

all the other anti-commutators being null. It follows that, for example,

$$\begin{array}{ccc}& & {\{{\psi }_{-}\left(x\right),{\psi }_{-}^{†}\left(y\right)\}}_{x_0=y_0}=\int \mathrm{d}\mathbf{p}\int \mathrm{d}{\mathbf{p}}^{\prime }\hfill \\ & \times & {\left[\{c_{-,\mathbf{p}},c_{-,{\mathbf{p}}^{\prime }}^{†}\}{u}_{-,\mathbf{p}}\left(x\right)u_{-,{\mathbf{p}}^{\prime }}^{†}\left(y\right)+\{d_{-,\mathbf{p}}^{†},d_{-,{\mathbf{p}}^{\prime }}\}{v}_{-,\mathbf{p}}\left(x\right)v_{-,{\mathbf{p}}^{\prime }}^{†}\left(y\right)\right]}_{x_0=y_0}\hfill \\ & =& \int \mathrm{d}\mathbf{p}{\left[u_{-,\mathbf{p}}\left(x\right)u_{-,\mathbf{p}}^{†}\left(y\right)+v_{-,\mathbf{p}}\left(x\right)v_{-,\mathbf{p}}^{†}\left(y\right)\right]}_{x_0=y_0}\hfill \\ & =& \int {\displaystyle \frac{\mathrm{d}\mathbf{p}}{{\left(2\pi \right)}^{3}2\wp }}\left[u_{-}\left(\mathbf{p}\right)u_{-}^{†}\left(\mathbf{p}\right)e^{i\mathbf{p}·(\mathbf{x}-\mathbf{y})}+v_{-}(-\mathbf{p})v_{-}^{†}(-\mathbf{p})e^{-i\mathbf{p}·(\mathbf{x}-\mathbf{y})}\right]\hfill \\ & =& \int {\displaystyle \frac{\mathrm{d}\mathbf{p}}{{\left(2\pi \right)}^{3}2\wp }}\left[u_{-}\left(\mathbf{p}\right)u_{-}^{†}\left(\mathbf{p}\right)+v_{-}\left(\mathbf{p}\right)v_{-}^{†}\left(\mathbf{p}\right)\right]e^{i\mathbf{p}·(\mathbf{x}-\mathbf{y})}\hfill \\ & =& \int {\displaystyle \frac{\mathrm{d}\mathbf{p}}{{\left(2\pi \right)}^{3}2\wp }}2\wp {\mathbb{P}}_{-}{e}^{i\mathbf{p}·(\mathbf{x}-\mathbf{y})}={\mathbb{P}}_{-}\delta (\mathbf{x}-\mathbf{y})\hfill \end{array}$$

and quite analogously

$${\{{\psi }_{+}\left(x\right),{\psi }_{+}^{†}\left(y\right)\}}_{x_0=y_0}={\mathbb{P}}_{+}\delta (\mathbf{x}-\mathbf{y})$$

while all the remaining anti-commutators do vanish at arbitrary times, e.g. $\{{\psi }_{-}\left(x\right),{\psi }_{+}^{†}\left(y\right)\}=0$ et cetera. It follows that the canonical equal time anticommutation relations for the massless Dirac quantum field (58) will be provided by

$${\{\psi \left(x\right),{\psi }^{†}\left(y\right)\}}_{x_0=y_0}=\delta (\mathbf{x}-\mathbf{y}){\{\psi \left(x\right),\overline{\psi }\left(y\right)\}}_{x_0=y_0}={\gamma }_0\delta (\mathbf{x}-\mathbf{y})$$

while the general canonical anticommutation relations take the standard usual form

$$\begin{array}{c}\hfill \{\psi \left(x\right),\psi \left(y\right)\}=0=\{\overline{\psi }\left(x\right),\overline{\psi }\left(y\right)\}\end{array}$$

(64)

$$\{\psi \left(x\right),\overline{\psi }\left(y\right)\}\equiv S_0(x-y)={\gamma }^{\rho }\partial D_0(x-y)/\partial x^{\rho }$$

(65)

where $D_0(x-y)$ is the well known massless Pauli-Jordan distribution, which fulfill the D’Alembert wave equation and endorses the microcausality property

$${\gamma }_0{S}_0(x-y)={\partial }_0{D}_0(x-y){(x-y)}^2<0$$

(66)

Now the main point. A consistent and reliable canonical quantum theory for a free non interacting field must give rise to the causal Green’s function, or Feynman propagator, in the absence of which the perturbative expansion and the renormalization procedure cannot be implemented, so that any meaningful physical interpretation and predictions are prevented. The causal Green’s functions is the time-ordered product of field operators and provides the specific inversion of the kinetic operator which guarantees causality: namely, first a particle of the quantum field is created out of the vacuum and only later it is destroyed. Moreover, the causal Green’s function always admits the smooth transition to the Euclidean formulation, which is essential in order to evaluate the Feynman integrals in perturbation theory, in particular the divergent and regularized ones. As we shall see below, it turns out that only the canonical quantum theory of the massless Dirac spinor field can provide this featuring bonus: the Weyl chiral bispinor quantum fields () or any other combination of those ones but (58) can do the job. Let’s analyze this crucial issue in more detail.

3.1.4. Causal Green’s Functions

Consider the rank-two square matrix of the vacuum expectation value of the time ordered product of a pair of e.g. right-handed quantum Weyl fields: namely,

$$S_c(x-y)\equiv \langle 0|T{\psi }_R\left(x\right){\psi }_R^{†}\left(y\right)|0\rangle$$

(67)

After insertion of the normal-mode expansion we get

$$\begin{array}{ccc}\hfill S_c(x-y)& =& \theta (x_0-y_0){\langle {\psi }_R\left(x\right){\psi }_R^{†}\left(y\right)\rangle }_0-\theta (y_0-x_0){\langle {\psi }_R^{†}\left(y\right){\psi }_R\left(x\right)\rangle }_0\hfill \\ & =& \theta (x_0-y_0)\int \mathrm{d}\mathbf{p}{[{\left(2\pi \right)}^{3}2\wp ]}^{-1}{u}_R\left(\mathbf{p}\right)\otimes u_R^{†}\left(\mathbf{p}\right)\hfill \\ & \times & exp\left\{-i\wp (x_0-y_0)+i\mathbf{p}·(\mathbf{x}-\mathbf{y})\right\}\hfill \\ & -& \theta (y_0-x_0)\int \mathrm{d}\mathbf{p}{[{\left(2\pi \right)}^{3}2\wp ]}^{-1}{v}_R(-\mathbf{p})\otimes v_R^{†}(-\mathbf{p})\hfill \\ & \times & exp\left\{i\wp (x_0-y_0)-i\mathbf{p}·(\mathbf{x}-\mathbf{y})\right\}\hfill \\ & =& \theta (x_0-y_0)i\partial ·\sigma \int \mathrm{d}\mathbf{p}{[{\left(2\pi \right)}^{3}2\wp ]}^{-1}exp\left\{-i\wp (x_0-y_0)+i\mathbf{p}·(\mathbf{x}-\mathbf{y})\right\}\hfill \\ & +& \theta (y_0-x_0)i\partial ·\sigma \int \mathrm{d}\mathbf{p}{[{\left(2\pi \right)}^{3}2\wp ]}^{-1}exp\left\{i\wp (x_0-y_0)-i\mathbf{p}·(\mathbf{x}-\mathbf{y})\right\}\hfill \end{array}$$

Now, if we recall the definitions of the massless Wightman distributions [15]

$$\begin{array}{ccc}\hfill i{D}_0^{(\pm )}(x-y)& \equiv & {\pm \int Dpexp\left\{\pm ip·x\right\}⌋}_{p_0=\wp }\hfill \\ & =& \pm \int {\displaystyle \frac{\mathrm{d}\mathbf{p}}{{\left(2\pi \right)}^{3}2\wp }}exp\left\{\pm i\wp (x_0-y_0)\mp i\mathbf{p}·(\mathbf{x}-\mathbf{y})\right\}\hfill \end{array}$$

then we obtain

$$\begin{array}{ccc}\hfill S_c(x-y)& =& \theta (x_0-y_0)i\partial ·\sigma (-i)D_0^{(-)}(x-y)+\theta (y_0-x_0)i\partial ·\sigma i{D}_0^{(+)}(x-y)\hfill \\ & =& i\partial ·\sigma [\theta (y_0-x_0)i{D}_0^{(+)}(x-y)-\theta (x_0-y_0)i{D}_0^{(-)}(x-y)]\hfill \\ & -& \delta (x_0-y_0)[D_0^{(-)}(x-y)+D_0^{(+)}(x-y)]\hfill \\ & =& i{\partial }_x·\sigma D_0^c(x-y)\hfill \end{array}$$

where $D_0\left(\xi \right)=D_0^{(-)}\left(\xi \right)+D_0^{(+)}\left(\xi \right)$ is the massless Pauli-Jordan distribution [15], which fulfills $D_0(0,\overrightarrow{\xi })=0$, while

$$D_0^c(x-y)\equiv \theta (y_0-x_0)i{D}_0^{(+)}(x-y)-\theta (x_0-y_0)i{D}_0^{(-)}(x-y)$$

is the Feynman propagator, or causal Green’s function, of a massless scalar field, its Fourier representation being

$$D_0^c(x-y)={\displaystyle \frac{i}{{\left(2\pi \right)}^4}}\int {\mathrm{d}}^{4}p{\displaystyle \frac{e^{-ip·(x-y)}}{p^2+i\epsilon }}$$

so that $i{D}_0^c\left(\xi \right)$ acts as the unique fully causal inverse of the d’Alembert differential operator.

As a consequence we eventually find [ in this paragraph we use the notation ${\tilde{\sigma }}_{\mu }={\overline{\sigma }}_{\mu }$ ]

$$\overline{\sigma }·{\partial }_x{S}_c(x-y)={\delta }^{\left(4\right)}(x-y)$$

(68)

which means that the rank-two matrix-valued causal Green’s function $S_c$ is the fully causal inverse of the right-handed differential operator $\overline{\sigma }·\partial .$ By following step-by-step the above derivation mutatis mutandis, it is not difficult to prove that after setting

$${\overline{S}}_c(x-y)\equiv \langle 0|T{\psi }_L\left(x\right){\psi }_L^{†}\left(y\right)|0\rangle$$

(69)

we get

$$\begin{array}{c}\hfill {\overline{S}}_c(x-y)=i{\partial }_x·\overline{\sigma }{D}_0^c(x-y)\sigma ·{\partial }_x{\overline{S}}_c(x-y)={\delta }^{\left(4\right)}(x-y)\end{array}$$

(70)

Consider now a change of the inertial reference frame, i.e. $x^{\prime \mu }={\Lambda }_{\phantom{\mu }\nu }^{\mu }{(x+\mathrm{a})}^{\nu }$. We have

$$D_0^{c\prime }(x^{\prime }-y^{\prime })=D_0^c(x-y)$$

$${\Lambda }_L^{-1}{\overline{\sigma }}^{\mu }{\Lambda }_R{\partial }_{\mu }^{\prime }={\Lambda }_R^{†}{\overline{\sigma }}^{\mu }{\Lambda }_R{\partial }_{\mu }^{\prime }={\Lambda }_{\phantom{\mu }\kappa }^{\mu }{\overline{\sigma }}^{\kappa }{\Lambda }_{\mu }^{\phantom{\mu }\rho }{\partial }_{\rho }={\overline{\sigma }}^{\mu }{\partial }_{\mu }$$

and thereby

$${\overline{\sigma }}^{\mu }{\partial }_{\mu }^{\prime }={\Lambda }_L{\overline{\sigma }}^{\mu }{\partial }_{\mu }{\Lambda }_R^{-1}$$

It follows that the transformation law of the Weyl causal Green’s functions

$$\begin{array}{c}\hfill {\overline{S}}_c^{\prime }(x^{\prime }-y^{\prime })={\Lambda }_L{\overline{S}}_c(x-y){\Lambda }_R^{-1}\end{array}$$

(71)

$$\begin{array}{c}\hfill S_c^{\prime }(x^{\prime }-y^{\prime })={\Lambda }_R{S}_c(x-y){\Lambda }_L^{-1}\end{array}$$

(72)

does necessarily involve the $SL(2,\mathbb{C})$ matrices of both IRREPs $D(0,{\displaystyle \frac{1}{2}})$ and $D({\displaystyle \frac{1}{2}},0).$ To better grasp this subtle point, let us rewrite e.g. the causal Green’s function in its Fourier integral representation in its most suggestive form, viz.,

$$\begin{array}{ccc}& & {\overline{S}}_c(x-y)={\displaystyle \frac{i}{{\left(2\pi \right)}^4}}\int {\mathrm{d}}^{4}p{\displaystyle \frac{p_0-\overrightarrow{\sigma }·\overrightarrow{p}}{p_0^2-{\overrightarrow{p}}^2+i\epsilon }}{e}^{-ip·(x-y)}\hfill \\ & =& {\displaystyle \frac{i}{{\left(2\pi \right)}^4}}\int {\mathrm{d}}^{4}p{e}^{-ip·(x-y)}{\displaystyle \frac{p_0-\overrightarrow{\sigma }·\overrightarrow{p}}{(p_0-\wp +i\epsilon )(p_0+\wp -i\epsilon )}}\hfill \\ & =& \int {\displaystyle \frac{{\mathrm{d}}^{4}p}{{\left(2\pi \right)}^4}}{e}^{-ip·(x-y)}\left({\displaystyle \frac{i}{p_0-\wp +i\epsilon }}-{\displaystyle \frac{i}{p_0+\wp -i\epsilon }}\right){\displaystyle \frac{p_0-\overrightarrow{\sigma }·\overrightarrow{p}}{2\wp }}\hfill \end{array}$$

It turns out that in the above off-shell expression the rank-two square matrix

$$\mathbb{W}={\displaystyle \frac{p_0-\overrightarrow{\sigma }·\overrightarrow{p}}{2\wp }}$$

is no longer a projector on left-handed two-component spin states because

$${\mathbb{W}}^2={\displaystyle \frac{p_0^2-2p_0\overrightarrow{\sigma }·\overrightarrow{p}+{\overrightarrow{p}}^2}{4{\overrightarrow{p}}^2}}\ne \mathbb{W}\mathrm{Tr}\mathbb{W}={\displaystyle \frac{p_0}{\wp }}\in \mathbb{R}$$

as it was true for the corresponding on-shell matrix with $p_0=\wp .$ Hence, in the process of inversion for the Weyl differential operators, leading to the Weyl causal Green’s functions, the very notion of chirality is definitely and unavoidably lost, just owing to the nature of the time ordered products of the Weyl quantum fields.

In order to get a bona fide fully causal inversion for the kinetic differential operator of a massless spinor we have to turn back to Dirac bispinors. Consider in fact the quantity

$$\begin{array}{ccc}\hfill S_0^c(x-y)& =& \langle 0|T\psi \left(x\right)\overline{\psi }\left(y\right)|0\rangle =〔\begin{array}{cc}0& {\overline{S}}_c(x-y)\\ S_c(x-y)& 0\end{array}〕\hfill \\ & =& 〔\begin{array}{cc}0& i\partial ·\overline{\sigma }\\ i\partial ·\sigma & 0\end{array}〕D_0^c(x-y)\hfill \end{array}$$

so that

$$\begin{array}{c}\hfill i\partial /S_0^c(x-y)=(-1)〔\begin{array}{cc}\square & 0\\ 0& \square \end{array}〕D_0^c(x-y)=i{\delta }^{\left(4\right)}(x-y)\end{array}$$

(73)

which shows, as expected, that the massless Schwinger propagator $S_0(x-y)$ is the unique inverse of the massless Dirac differential operator $\partial /$, that fulfills causality forward and backward in time, its Fourier representation being

$$S_0^c(x-y)={\displaystyle \frac{i}{{\left(2\pi \right)}^4}}\int {\mathrm{d}}^{4}p{\displaystyle \frac{p/}{p^2+i\epsilon }}{e}^{-ip·(x-y)}$$

(74)

3.1.5. Observables

As we have seen above, the consistent quantum theory of the massless Weyl-Dirac free field necessarily involves four types of quanta, although the phenomenological constraints seem to indicate that only two of them are physical. The 1-particle state associated to a real left-handed neutrino of momentum $\mathbf{p}$ and a negative helicity is given by $c_{-,\mathbf{p}}^{†}|0\rangle ,$ while a real right-handed anti-neutrino of momentum $\mathbf{q}$ and positive helicity is provided by $d_{+,\mathbf{q}}^{†}|0\rangle .$ The plane wave function that describes an incoming neutrino in a scattering experiment or decay process is therefore

$$\langle 0|\psi \left(x\right)c_{-,\mathbf{p}}^{†}|0\rangle =u_{-,\mathbf{p}}\left(x\right)$$

(75)

Quite analogously, an incoming right-handed anti-neutrino of momentum $\mathbf{q}$ and positive helicity will be described by the plane wave function

$$\langle 0|\overline{\psi }\left(x\right)d_{+,\mathbf{q}}^{†}|0\rangle ={\overline{v}}_{+,\mathbf{q}}\left(x\right)$$

(76)

The outgoing states for the very same physical particles will be clearly described by the plane wave functions

$$\langle 0|c_{-,\mathbf{p}}\overline{\psi }\left(x\right)|0\rangle ={\overline{u}}_{-,\mathbf{p}}\left(x\right)\langle 0|d_{+,\mathbf{q}}^{†}\psi \left(x\right)|0\rangle =v_{+,\mathbf{q}}\left(x\right)$$

(77)

All the remaining 1-particle state are not physical, as far as we know from experiments, in such a manner that the Standard Model gauge interactions will forbid the transitions from physical massless left-handed neutrinos and right-handed anti-neutrinos to unphysical Weyl particles, as it does.

Let us come now to the evaluation of the Observables - or better to say invariant and conserved charges - in the massless spinor quantum theory. For example, since we have

$$\begin{array}{c}\hfill {\psi }_{\pm }^{†}\left(y\right)i{\partial }_{\mu }{\psi }_{\pm }\left(x\right)=:{\psi }_{\pm }^{†}\left(y\right)i{\partial }_{\mu }{\psi }_{\pm }\left(x\right):-\int \mathrm{d}\mathbf{p}{p}_{\mu }{v}_{\pm ,\mathbf{p}}^{\top *}\left(y\right)v_{\pm ,\mathbf{p}}\left(x\right)\end{array}$$

with $p_0=\wp ,$ one can easily obtain

$$\begin{array}{c}\hfill P^{\mu }=\sum _{\iota =-,+}\int \mathrm{d}\mathbf{p}{p}^{\mu }{\left[c_{\iota ,\mathbf{p}}^{†}{c}_{\iota ,\mathbf{p}}+d_{\iota ,\mathbf{p}}^{†}{d}_{\iota ,\mathbf{p}}\right]}_{p_0=\wp }=P_L^{\mu }+P_R^{\mu }\end{array}$$

(78)

In order to set up the spin operator, consider for instance a left Weyl field ${\psi }_{-}(t,z)$ travelling along the $Oz-$axis so that

$$T_{-}^{12}(t,z)-T_{-}^{21}(t,z)=0={\partial }_{\nu }{S}_{-}^{\nu 21}(t,z)$$

$${\Sigma }_{-}={\int }_{-\infty }^{\infty }\mathrm{d}z:S_{-}^{021}(t,z):{\dot{\Sigma }}_{-}=0$$

$${\Sigma }_{-}=-{\displaystyle \frac{1}{2}}{\int }_{-\infty }^{\infty }\mathrm{d}z:{\psi }_{-}^{†}(0,z){\Sigma }_3{\psi }_{-}(0,z):=-{\displaystyle \frac{1}{2}}{\int }_{-\infty }^{\infty }\mathrm{d}z:{\psi }_L^{†}(0,z){\sigma }_3{\psi }_L(0,z):$$

Now, since for a 1d motion we have for $\wp =p_z\mathrm{sgn}\left(p_z\right)$

$$\begin{array}{ccc}& & {\psi }_L(t,z)={\int }_{-\infty }^{\infty }\mathrm{d}{p}_z{[4\pi \wp ]}^{-{\displaystyle \frac{1}{2}}}\hfill \\ & \times & \left[c_{-,p_z}{u}_L\left(p_z\right)e^{-i\wp t+iz{p}_z}+d_{-,p_z}^{†}{v}_L\left(p_z\right)e^{i\wp t-iz{p}_z}\right]\hfill \\ \\ & & u_L\left(p_z\right)={\displaystyle \frac{\wp -{\sigma }_3{p}_z}{\sqrt{\wp -p_z}}}\left|\begin{array}{c}1\\ 0\end{array}\right|v_L\left(p_z\right)={\displaystyle \frac{\wp +{\sigma }_3{p}_z}{\sqrt{\wp -p_z}}}\left|\begin{array}{c}0\\ 1\end{array}\right|\hfill \end{array}$$

and thereby

$${\sigma }_3{u}_L\left(p_z\right)=u_L\left(p_z\right),{\sigma }_3{v}_L\left(p_z\right)=-v_L\left(p_z\right),$$

$$\begin{array}{ccc}& & {\sigma }_3{\psi }_L(t,z)={\int }_{-\infty }^{\infty }\mathrm{d}{p}_z{[4\pi \wp ]}^{-{\displaystyle \frac{1}{2}}}\hfill \\ & \times & \left[c_{-,p_z}{u}_L\left(p_z\right)e^{-i\wp t+iz{p}_z}-d_{-,p_z}^{†}{v}_L\left(p_z\right)e^{i\wp t-iz{p}_z}\right]\hfill \end{array}$$

This entails that the general form of the spin operator for a one dimensional motion along e.g. the $Oz-$axis becomes

$$\begin{array}{ccc}\hfill {\Sigma }_z& =& -{\displaystyle \frac{1}{2}}{\int }_{-\infty }^{\infty }\mathrm{d}{p}_z\left[c_{-,p_z}^{†}{c}_{-,p_z}-d_{+,p_z}^{†}{d}_{+,p_z}\right.\hfill \\ & -& \left.c_{+,p_z}^{†}{c}_{+,p_z}+d_{-,p_z}^{†}{d}_{-,p_z}\right]\hfill \end{array}$$

(79)

which endorses the fact that neutrino particles are of negative helicity, while the antineutrinos exhibit positive helicity, owing to the opposite verse of the wave vector, the situation being reversed for the hypothetical right-handed neutrinos of positive chirality.

Turning to the internal symmetries and charges we find

$$\begin{array}{ccc}\hfill \mathcal{Q}& =& \int \mathrm{d}\mathbf{x}:{\psi }_{-}^{†}\left(x\right){\psi }_{-}\left(x\right)+{\psi }_{+}^{†}\left(x\right){\psi }_{+}\left(x\right):\equiv {\mathcal{Q}}_{-}+{\mathcal{Q}}_{+}\hfill \\ \hfill {\mathcal{Q}}_{\mp }& =& \int \mathrm{d}\mathbf{x}\int \mathrm{d}{\mathbf{p}}^{\prime }:\left[c_{\mp ,{\mathbf{p}}^{\prime }}^{†}{u}_{\mp ,{\mathbf{p}}^{\prime }}^{\top *}\left(x\right)+d_{\mp ,{\mathbf{p}}^{\prime }}{v}_{\mp ,{\mathbf{p}}^{\prime }}^{\top *}\left(x\right)\right]\hfill \\ & \times & \int \mathrm{d}\mathbf{p}\left[c_{\mp ,\mathbf{p}}{u}_{\mp ,\mathbf{p}}\left(x\right)+d_{\mp ,\mathbf{p}}^{†}{v}_{\mp ,\mathbf{p}}\left(x\right)\right]:\hfill \end{array}$$

(80)

$$\begin{array}{ccc}& =& \int \mathrm{d}\mathbf{p}\left[c_{\mp ,\mathbf{p}}^{†}{c}_{\mp ,\mathbf{p}}-d_{\mp ,\mathbf{p}}^{†}{d}_{\mp ,\mathbf{p}}\right]\hfill \end{array}$$

(81)

$$\begin{array}{ccc}\hfill {\mathcal{Q}}_5& =& {\mathcal{Q}}_{+}-{\mathcal{Q}}_{-}\hfill \end{array}$$

(82)

Finally, the dilatation charge actually reads

$$\begin{array}{ccc}\hfill \mathcal{D}& =& \int \mathrm{d}\mathbf{x}:\frac{3}{2}i{\psi }^{†}\left(x\right)\psi \left(x\right)+x_{\nu }{T}^{0\nu }\left(x\right):\hfill \\ & =& x_0{P}_0+i\left\{\frac{3}{2}\mathcal{Q}-\int \mathrm{d}\mathbf{x}{x}^k:{\psi }^{†}\left(x\right){\nabla }_k\psi \left(x\right):\right\}\hfill \end{array}$$

(83)

We find

$$\begin{array}{ccc}\hfill {\partial }_0\mathcal{D}& =& P_0+\int \mathrm{d}\mathbf{x}{x}^k:{\left[i{\partial }_0\psi \left(x\right)\right]}^{†}{\nabla }_k\psi \left(x\right):-\int \mathrm{d}\mathbf{x}{x}^k:{\psi }^{†}\left(x\right){\nabla }_{k}i{\partial }_0\psi \left(x\right):\hfill \\ & =& P_0+\int \mathrm{d}\mathbf{x}{x}^k:{\psi }^{†}\left(x\right)H_0{\nabla }_k\psi \left(x\right):-\int \mathrm{d}\mathbf{x}{x}^k:{\psi }^{†}\left(x\right){\nabla }_k{H}_0\psi \left(x\right):\hfill \\ & =& P_0\hfill \end{array}$$

Since we have the well-known relation

$${\partial }_0\mathcal{D}=i[\mathcal{D},P_0]=P_0$$

it follows that we can write, as expected,

$$\dot{\mathcal{D}}={\displaystyle \frac{\mathrm{d}\mathcal{D}}{\mathrm{d}t}}={\partial }_0\mathcal{D}+i[P_0,\mathcal{D}]=0$$

so that the dilatation charge is globally time independent.

From the above analysis and correspondences, it follows that the 1-particle states $c_{-}^{†}\left(\mathbf{p}\right)|0\rangle$ and $d_{+}^{†}\left(\mathbf{p}\right)|0\rangle$ of the Fock space of the massless Dirac spinor field do respectively represent the physical particles with the following quantum number assignements: namely,

i) opposite chirality $\chi =\mp 1$;

ii) the same energy-momentum $p^{\nu }=(\wp ,\mathbf{p});$

iii) opposite internal charge, called lepton number, $\ell =\pm 1;$

iv) opposite helicity $\mathrm{h}=\mp 1.$

3.1.6. Discrete Symmetries

Turning to the discrete symmetries beside the Nöther theorem, we start with the charge conjugation unitary transformation of the quantum theory for a massless Dirac spinor field, which amounts to the simultaneous exchange between chirality and particle-antiparticle operators, up to a phase factor, viz.,

$$\begin{array}{c}\hfill \mathcal{C}{c}_{\mp ,\mathbf{p}}{\mathcal{C}}^{-1}=e^{i\theta }{d}_{\pm ,\mathbf{p}}\mathcal{C}{d}_{\mp ,\mathbf{p}}{\mathcal{C}}^{-1}=e^{-i\theta }{c}_{\pm ,\mathbf{p}}\end{array}$$

(84)

$$\begin{array}{c}\hfill {\mathcal{C}}^2=\mathbb{I}\mathcal{C}={\mathcal{C}}^{†}={\mathcal{C}}^{-1}0\le \theta <2\pi \end{array}$$

(85)

that yields

$$\mathcal{C}{\psi }_{\mp }\left(x\right)\mathcal{C}=e^{i\theta }\int \mathrm{d}\mathbf{p}\left[d_{\pm ,\mathbf{p}}{u}_{\mp ,\mathbf{p}}\left(x\right)+c_{\pm ,\mathbf{p}}^{†}{v}_{\mp ,\mathbf{p}}\left(x\right)\right]$$

Now, from the charge conjugation relations for the plane wave functions, viz.,

$$u_{\mp ,\mathbf{p}}\left(x\right)=i{\gamma }^2{v}_{\pm ,\mathbf{p}}^{*}\left(x\right)v_{\mp ,\mathbf{p}}\left(x\right)=i{\gamma }^2{u}_{\pm ,\mathbf{p}}^{*}\left(x\right)$$

(86)

we can write

$$\begin{array}{ccc}\hfill \mathcal{C}{\psi }_{\mp }\left(x\right)\mathcal{C}& \equiv & {\psi }_{\mp }^c\left(x\right)=e^{i\theta }\int \mathrm{d}\mathbf{p}\left[d_{\pm ,\mathbf{p}}{u}_{\mp ,\mathbf{p}}\left(x\right)+c_{\pm ,\mathbf{p}}^{†}{v}_{\mp ,\mathbf{p}}\left(x\right)\right]\hfill \\ & =& i{\gamma }^2{e}^{i\theta }\int \mathrm{d}\mathbf{p}\left[c_{\pm ,\mathbf{p}}^{†}{u}_{\pm ,\mathbf{p}}^{*}\left(x\right)+d_{\pm ,\mathbf{p}}{v}_{\pm ,\mathbf{p}}^{*}\left(x\right)\right]\hfill \\ & =& {\gamma }^2{\left({\psi }_{\pm }^{†}\left(x\right)\right)}^{\top }\mathrm{for}\theta =-\pi /2\hfill \end{array}$$

Now, since we have $\psi \left(x\right)={\psi }_{-}\left(x\right)+{\psi }_{+}\left(x\right)$, we come to the result

$${\psi }^c\left(x\right)=\mathcal{C}\psi \left(x\right)\mathcal{C}={\gamma }^2{\left({\psi }^{†}\left(x\right)\right)}^{\top }$$

(87)

which is nothing but the quantum generalization of the charge conjugation rule for classical spinor fields, i.e. ${\psi }^c\left(x\right)={\gamma }^2{\psi }^{*}\left(x\right),$ as expected.

Thus the charge conjugated of a particle of e.g. negative chirality, negative helicity and positive lepton number is an anti-particle of opposite chirality, positive helicity and negative lepton number. It turns out that the charge conjugation transformation necessarily involves both chiralities, in order to be implemented. Hence the $\mathcal{C}$ transform is well defined only for 4-component bispinors and not for 2-component Weyl spinors belonging to the two irreducible 2d representations.

The parity transformation is a discrete space-time symmetry of the Action integral $S=\int {\mathrm{d}}^{4}x{\mathcal{L}}_0\left(x\right)$ that can be implemented at the quantum field theory level as

$$\begin{array}{c}\hfill \mathcal{P}{c}_{\mp ,\mathbf{p}}{\mathcal{P}}^{†}=e^{i{\eta }_{\mp }}{c}_{\pm ,-\mathbf{p}}\mathcal{P}{d}_{\mp ,\mathbf{p}}{\mathcal{P}}^{†}=e^{i{\theta }_{\mp }}{d}_{\pm ,-\mathbf{p}}\\ \hfill 0\le {\eta }_{\mp }<2\pi 0\le {\theta }_{\mp }<2\pi \end{array}$$

(88)

After insertion of the normal mode expansion of the massless Dirac quantum field, then we find with $x^{\prime \mu }={\tilde{x}}^{\mu }=(x_0,-\overrightarrow{x})=x_{\mu }$

$$\begin{array}{ccc}\hfill {\psi }^{\prime }\left(x^{\prime }\right)& =& \mathcal{P}\psi \left(\tilde{x}\right){\mathcal{P}}^{†}\hfill \\ & =& \int \mathrm{d}\mathbf{p}\left[\mathcal{P}{c}_{\mp ,\mathbf{p}}{\mathcal{P}}^{†}{u}_{\mp ,\mathbf{p}}\left(\tilde{x}\right)+\mathcal{P}{d}_{\mp ,\mathbf{p}}^{†}{\mathcal{P}}^{†}{v}_{\mp ,\mathbf{p}}\left(\tilde{x}\right)\right]\hfill \\ & =& \int \mathrm{d}\mathbf{p}\left[e^{i{\eta }_{\mp }}{c}_{\pm ,-\mathbf{p}}{u}_{\mp ,\mathbf{p}}\left(\tilde{x}\right)+e^{-i{\theta }_{\mp }}{d}_{\pm ,-\mathbf{p}}^{†}{v}_{\mp ,\mathbf{p}}\left(\tilde{x}\right)\right]\hfill \\ & =& \int \mathrm{d}\mathbf{p}\left[e^{i{\eta }_{\mp }}{c}_{\pm ,\mathbf{p}}{u}_{\mp ,-\mathbf{p}}\left(\tilde{x}\right)+e^{-i{\theta }_{\mp }}{d}_{\pm ,\mathbf{p}}^{†}{v}_{\mp ,-\mathbf{p}}\left(\tilde{x}\right)\right]\hfill \end{array}$$

Now we have

$$\begin{array}{c}\hfill u_{\mp ,-\mathbf{p}}\left(\tilde{x}\right)={[{\left(2\pi \right)}^{3}2\wp ]}^{-{\displaystyle \frac{1}{2}}}{u}_{\mp }(-\mathbf{p})e^{-i\wp t+i\mathbf{p}·\mathbf{x}}=-{\gamma }^0{u}_{\pm ,\mathbf{p}}\left(x\right)\\ \\ \hfill v_{\mp ,-\mathbf{p}}\left(\tilde{x}\right)={[{\left(2\pi \right)}^{3}2\wp ]}^{-{\displaystyle \frac{1}{2}}}{v}_{\mp }(-\mathbf{p})e^{i\wp t-i\mathbf{p}·\mathbf{x}}={\gamma }^0{v}_{\pm ,\mathbf{p}}\left(x\right)\end{array}$$

so that, after choosing

$${\eta }_{\mp }=\mp k\pi {\theta }_{\mp }=\mp 2k\pi (k\in \mathbb{Z})$$

we definitely obtain the usual transformation law

$$\begin{array}{ccc}\hfill {\psi }_{\mp }^{\prime }\left(x^{\prime }\right)& =& \mathcal{P}{\psi }_{\mp }\left(\tilde{x}\right)\mathcal{P}={\gamma }^0{\psi }_{\pm }\left(x\right)\hfill \end{array}$$

(89)

$$\begin{array}{ccc}\hfill {\psi }^{\prime }\left(x^{\prime }\right)& =& \mathcal{P}\psi \left(\tilde{x}\right)\mathcal{P}={\gamma }^0\psi \left(x\right)\hfill \end{array}$$

(90)

$$\begin{array}{ccc}& & {\mathcal{P}}^2=\mathbb{I}\mathcal{P}={\mathcal{P}}^{†}={\mathcal{P}}^{-1}\hfill \end{array}$$

(91)

Once again, it is worthwhile to remark that even the parity transform is not a well defined operation for a two-component Weyl fermion, because it necessarily involves both chiralities, i.e. both irreducible 2d representations of the proper Lorentz group.

In quantum field theory the time reversal operator $\mathcal{T}$ corresponds to an antilinear and antiunitary transformation [13,14] which acts as follows: namely,

$$\begin{array}{c}\hfill [P_0,\mathcal{T}]=0\{\overrightarrow{P},\mathcal{T}\}=0\{M^{\rho \sigma },\mathcal{T}\}=0\end{array}$$

(92)

where, as usual, $(P^{\mu },M^{\rho \sigma })$ are the ten generators of the Poincaré group. It follows therefrom that the antilinear and antiunitary time reversal operation does reverse the signs of all particle momenta and spin angular momenta, the same being true for antiparticles. In this regard we need to explain what is meant by spin-flip [14]. Consider for example the spin state

$$\begin{array}{ccc}\hfill u_{-}\left(\mathbf{p}\right)& =& 〔\begin{array}{c}{u}_L\left(\mathbf{p}\right)\\ 0\\ 0\end{array}〕={\displaystyle \frac{1}{\sqrt{\wp -p_z}}}〔\begin{array}{c}\wp -p_z\\ -p_x-i{p}_y\\ 0\\ 0\end{array}〕\hfill \\ \hfill u_L\left(\mathbf{p}\right)& =& {\displaystyle \frac{\sigma ·\tilde{p}}{\sqrt{\wp -p_z}}}\left|\begin{array}{c}1\\ 0\end{array}\right|\hfill \end{array}$$

(93)

This positive energy, left-handed spin-state of negative chirality does involve the spin-up constant spinor $\left|\begin{array}{c}1\\ 0\end{array}\right|$ and to our purpose it will be better labelled as $u_{-}(\mathbf{p},\uparrow ).$ Its corresponding spin reversed state $u_{-}(\mathbf{p},\downarrow )$ has the very same structure, but for the constant spinor that will be turned to the spin-down $\left|\begin{array}{c}0\\ 1\end{array}\right|.$ Thus we can denote the pair of normalized spin flip states according to

$$\begin{array}{c}\hfill u_{-}(\mathbf{p},\uparrow )={\displaystyle \frac{1}{\sqrt{\wp -p_z}}}〔\begin{array}{c}\wp -p_z\\ -p_x-i{p}_y\\ 0\\ 0\end{array}〕\end{array}$$

(94)

$$\begin{array}{c}\hfill u_{-}^{*}(-\mathbf{p},\downarrow )={\displaystyle \frac{1}{\sqrt{\wp -p_z}}}〔\begin{array}{c}{p}_x+i{p}_y\\ \wp -p_z\\ 0\\ 0\end{array}〕\end{array}$$

(95)

and quite analogous relationships for the remaining spin states. Then we have

$$\begin{array}{ccc}& & \mathcal{T}{c}_{\mp ,\mathbf{p}}{\mathcal{T}}^{-1}=e^{i{\eta }_{\mp }}{c}_{\mp ,-\mathbf{p}}\mathcal{T}{d}_{\mp ,\mathbf{p}}{\mathcal{T}}^{-1}=e^{i{\theta }_{\mp }}{d}_{\mp ,-\mathbf{p}}\hfill \\ & & {\psi }_{\mp }^{\prime }\left(x^{\prime }\right)=\mathcal{T}{\psi }_{\mp }(-x_0,\overrightarrow{x}){\mathcal{T}}^{-1}\hfill \end{array}$$

(96)

$$\begin{array}{ccc}& =& \int \mathrm{d}\mathbf{p}\left[e^{i{\eta }_{\mp }}{c}_{\mp ,-\mathbf{p}}{u}_{\mp ,\mathbf{p}}^{*}(-\tilde{x})+e^{-i{\theta }_{\mp }}{d}_{\mp ,-\mathbf{p}}^{†}{v}_{\mp ,\mathbf{p}}^{*}(-\tilde{x})\right]\hfill \end{array}$$

(97)

$$\begin{array}{ccc}& =& \int \mathrm{d}\mathbf{p}\left[e^{i{\eta }_{\mp }}{c}_{\mp ,\mathbf{p}}{u}_{\mp ,-\mathbf{p}}^{*}(-\tilde{x})+e^{-i{\theta }_{\mp }}{d}_{\mp ,\mathbf{p}}^{†}{v}_{\mp ,-\mathbf{p}}^{*}(-\tilde{x})\right]\hfill \end{array}$$

(98)

the complex conjugation being due to the antilinear and antiunitary nature of the $\mathcal{T}-$operation, while the spin-flip operation in the spinor plane wave functions is understood. Consider now, for instance,

$$\begin{array}{ccc}& & \mathcal{T}{\psi }^{(-)}(-t,\mathbf{x}){\mathcal{T}}^{-1}=\int \mathrm{d}\mathbf{p}{[{\left(2\pi \right)}^{3}2\wp ]}^{-{\displaystyle \frac{1}{2}}}\hfill \\ & \times & \left[e^{i{\eta }_{-}}{c}_{-,\mathbf{p}}{u}_{-}^{*}(-\mathbf{p}\downarrow )+e^{i{\eta }_{+}}{c}_{+,\mathbf{p}}{u}_{+}^{*}(-\mathbf{p}\downarrow )\right]e^{-i\wp t+i\mathbf{p}·\mathbf{x}}\hfill \end{array}$$

It can be readily seen that the right-hand side of the above equation becomes a local expression for the quantum field at the instant t, notably $\Theta {\psi }^{(-)}(t,\mathbf{x}),$ if a $4\times 4$ matrix can be found such that

$$e^{i{\eta }_{\mp }}{u}_{\mp }^{*}(-\mathbf{p}\downarrow )=\Theta u_{\mp }(\mathbf{p}\uparrow )$$

(99)

From the explicit form (94-) it turns out that the above relations hold true for ${\eta }_{\mp }=0.$ The othogonality and closure relations for the spin-states require the unitarity property $\Theta {\Theta }^{†}=\mathbb{I}={\Theta }^{†}\Theta .$ Moreover, from the spin-state equations we obtain

$$\begin{array}{c}\hfill \overrightarrow{\alpha }·\overrightarrow{p}{u}_{\mp }\left(\overrightarrow{p}\right)=\wp u_{\mp }\left(\overrightarrow{p}\right)\\ \hfill -{\overrightarrow{\alpha }}^{*}·\overrightarrow{p}{u}_{\mp }^{*}(-\overrightarrow{p})=\wp u_{\mp }^{*}(-\overrightarrow{p})\\ \hfill -{\overrightarrow{\alpha }}^{*}·\overrightarrow{p}\Theta u_{\mp }\left(\overrightarrow{p}\right)=\wp \Theta u_{\mp }\left(\overrightarrow{p}\right)\\ \hfill -{\Theta }^{†}{\overrightarrow{\alpha }}^{*}\Theta =\overrightarrow{\alpha }\end{array}$$

The unitary solution of the very last equation is unique except for an irrelevant phase factor. In the Weyl representation of the Clifford algebra, the real matrix

$$\Theta =-{\gamma }^1{\gamma }^3=-i{\Sigma }_2$$

(100)

is a solution with the important property ${\Theta }^{*}\Theta +\mathbb{I}=0,$ which does not depend on the specific representation of the Dirac matrices. Thus we definitely find

$${\psi }_{\pm }^{\prime }\left(x^{\prime }\right)=\mathcal{T}{\psi }_{\pm }(-t,\mathbf{x}){\mathcal{T}}^{-1}=\Theta {\psi }_{\pm }(t,\mathbf{x})(x^{\prime }=-\tilde{x})$$

(101)

which shows that the time reversal is a chirality preserving transformation. Moreover, time reversal is well defined even for two component Weyl spinors, since the spin-flip operation does not mix left and right components of a massless Dirac bispinor. This means that also the $\mathcal{CP}$ and $\mathcal{CPT}$ transformations are well defined for two-component Weyl spinors. In summary, the quantum theory which arises from the classical Action integral

$$S_0\equiv \int {\mathrm{d}}^{4}x{\mathcal{L}}_0\left(x\right)=\int {\mathrm{d}}^{4}x{\psi }^{†}\left(x\right)i\alpha ·\partial \psi \left(x\right)$$

is invariant under the discrete $\mathcal{C},\mathcal{P},\mathcal{T}$ transformation separately, while the Weyl Action integrals

$$S_L=\int {\mathrm{d}}^{4}x\frac{1}{2}{\psi }_L^{†}\left(x\right)i\sigma ·\overleftrightarrow{\partial }{\psi }_L\left(x\right)S_R=\int {\mathrm{d}}^{4}x\frac{1}{2}{\psi }_R^{†}\left(x\right)i\overline{\sigma }·\overleftrightarrow{\partial }{\psi }_R\left(x\right)$$

and related quantum theories are invariant under $\mathcal{T}$ and $\mathcal{CP}$ transforms, which are well defined and good symmetries for left and right-handed Weyl spinors.

3.2. Majorana Spinor Field

One can write a relativistic invariant field equation for a massive 2-component spinor field: to this aim, let us start from a left Weyl spinor ${\psi }_L\in D({\displaystyle \frac{1}{2}},0)$ that transforms according to the $SL(2,\mathbb{C})$ matrix ${\Lambda }_L.$ Call such a 2-component spinor field ${\chi }_a\left(x\right)(a=1,2)$.

3.2.1. Definitions and Lagrangians

Let us consider the Weyl spinor wave field as a classical anti-commuting field, i.e. a Graßmann valued left Weyl spinor field function over the Minkowski space which satisfies

$$\{{\chi }_a\left(x\right),{\chi }_b\left(y\right)\}=0\left(x,y\in \mathcal{M}|a,b=1,2\right)$$

together with the complex conjugation rule

$${\left({\chi }_1{\chi }_2\right)}^{*}={\chi }_2^{*}{\chi }_1^{*}=-{\chi }_1^{*}{\chi }_2^{*}$$

(102)

so as to imitate the Hermitean conjugation of quantum fields. A Majorana classical spinor field is a self-conjugated bispinor, that can be constructed, for example, out of the left-handed spinor ${\chi }_a\left(x\right)(a=1,2)$ as follows: namely,

$${\chi }_M\left(x\right)=〔\begin{array}{c}\chi \left(x\right)\\ -{\sigma }_2{\chi }^{*}\left(x\right)\end{array}〕={\chi }_M^c\left(x\right)$$

the charge conjugation rule for any classical bispinors $\psi$ being defined by the general relationship

$${\psi }^c\left(x\right)=e^{i\theta }{\gamma }^2{\psi }^{*}\left(x\right)(0\le \theta <2\pi )$$

which is a discrete internal - i.e. space-time point independent - symmetry transformation. Here below we shall suitably choose $\theta =0.$ The Majorana bispinor has a right-handed lower Weyl spinor component $-{\sigma }_2{\chi }^{*}\in D(0,{\displaystyle \frac{1}{2}}),$ albeit functional dependent, due to the charge self-conjugation constraint, in such a manner that ${\chi }_M$ possesses both chiralities and polarizations, at variance with its left-handed Weyl building spinor $\chi \left(x\right).$ There is another kind of self-conjugated Majorana bispinor, which can be set-up out of a right-handed Weyl building spinor $\phi \in D(0,{\displaystyle \frac{1}{2}})$

$${\phi }_M\left(x\right)=〔\begin{array}{c}{\sigma }_2{\phi }^{*}\left(x\right)\\ \phi \left(x\right)\end{array}〕={\phi }_M^c\left(x\right)$$

From the Majorana self-conjugated bispinors, one can readily construct the most general Poincaré invariant and power counting renormalizable Lagrangian. For instance, by starting from the bispinor ${\chi }_M\left(x\right)$ we have

$${\mathcal{L}}_M=\frac{1}{4}{\chi }_M^{†}\left(x\right){\alpha }^{\mu }i{\overleftrightarrow{\partial }}_{\mu }{\chi }_M^c\left(x\right)-\frac{1}{2}m{\overline{\chi }}_M\left(x\right){\chi }_M^c\left(x\right)$$

where ${\alpha }^{\nu }={\gamma }_0{\gamma }^{\nu },$ while the employed notation reminds us that the upper and lower components of a Majorana bispinor can never be treated as functionally independent, even formally, due to the presence of the self-conjugation constraint. It follows that the Majorana mass term can be written in the two equivalent forms

$$\begin{array}{ccc}\hfill {\mathcal{L}}_M^m& =& \frac{1}{2}m\left[{\chi }^{\top }\left(x\right){\sigma }_2\chi \left(x\right)+{\chi }^{†}\left(x\right){\sigma }_2{\chi }^{*}\left(x\right)\right]\hfill \\ & =& -im\left[{\chi }_1\left(x\right){\chi }_2\left(x\right)+{\chi }_1^{*}\left(x\right){\chi }_2^{*}\left(x\right)\right]\hfill \\ & =& im\left[{\chi }_2^{*}\left(x\right){\chi }_1^{*}\left(x\right)+{\chi }_2\left(x\right){\chi }_1\left(x\right)\right]={\left({\mathcal{L}}_M^m\right)}^{*}\hfill \end{array}$$

whence it is clear that the corresponding integral $\int {\mathrm{d}}^{4}x{\mathcal{L}}_M^m\left(x\right)$ is not invariant under the internal $U\left(1\right)$ phase transformation

$$\chi \left(x\right)\mapsto {\chi }^{\prime }\left(x\right)=\chi \left(x\right)e^{i\theta }(0\le \theta <2\pi )$$

Concerning the kinetic term, from the relations

$${\chi }_M^{†}=〔{\chi }^{†}-{\chi }^{\top }{\sigma }_2〕{\chi }^{†}={\left({\chi }^{*}\right)}^{\top }$$

together with

$${\alpha }^{\nu }={\gamma }_0{\gamma }^{\nu }=〔\begin{array}{cc}{\sigma }^{\nu }& 0\\ 0& {\overline{\sigma }}^{\nu }\end{array}〕{\sigma }_{\mu }={\overline{\sigma }}^{\mu }=(1,\overrightarrow{\sigma })$$

$${\alpha }^{\nu }{\gamma }^2=〔\begin{array}{cc}0& {\sigma }^{\nu }{\sigma }_2\\ -{\overline{\sigma }}^{\nu }{\sigma }_2& 0\end{array}〕$$

one can obtain the Majorana kinetic term in the 2-component formalism

$$\begin{array}{ccc}& & \frac{1}{4}{\chi }_M^{†}\left(x\right){\alpha }^{\mu }{\gamma }^{2}i{\overleftrightarrow{\partial }}_{\mu }{\chi }_M^{*}\left(x\right)\hfill \\ & =& \frac{1}{4}{\chi }^{†}\left(x\right){\sigma }^{\mu }i{\overleftrightarrow{\partial }}_{\mu }\chi \left(x\right)+\frac{1}{4}{\chi }^{\top }\left(x\right){\sigma }_2{\overline{\sigma }}^{\mu }{\sigma }_{2}i{\overleftrightarrow{\partial }}_{\mu }{\chi }^{*}\left(x\right)\hfill \\ & =& \frac{1}{4}{\chi }^{†}\left(x\right){\sigma }^{\mu }i{\overleftrightarrow{\partial }}_{\mu }\chi \left(x\right)+\frac{1}{4}{\chi }^{\top }\left(x\right){\left({\overline{\sigma }}^{\mu }\right)}^{*}i{\overleftrightarrow{\partial }}_{\mu }{\chi }^{*}\left(x\right)\hfill \\ & =& \frac{1}{2}{\chi }^{†}\left(x\right){\sigma }^{\mu }i{\partial }_{\mu }\chi \left(x\right)+\frac{1}{2}{\chi }^{\top }\left(x\right){\left({\overline{\sigma }}^{\mu }\right)}^{*}i{\partial }_{\mu }{\chi }^{*}\left(x\right)\hfill \\ & =& {\chi }^{†}\left(x\right){\sigma }^{\mu }i{\partial }_{\mu }\chi \left(x\right)-\frac{1}{2}i{\partial }_{\mu }\left[{\chi }^{†}\left(x\right){\sigma }^{\mu }\chi \left(x\right)\right]\hfill \end{array}$$

(103)

which coincides with the Lagrangian for a left-handed Weyl spinor, as expected, where the above complex conjugation rule (102) for Grassmann valued field has been used.

It turns out that even the Majorana Action integral

$$\begin{array}{ccc}\hfill S_M& =& \frac{1}{4}\int {\mathrm{d}}^{4}x\left[{\chi }_M^{†}\left(x\right){\alpha }^{\mu }{\gamma }^{2}i{\overleftrightarrow{\partial }}_{\mu }{\chi }_M^{*}\left(x\right)-2m{\chi }_M^{†}\left(x\right){\alpha }^2{\chi }_M^{*}\left(x\right)\right]\hfill \end{array}$$

(104)

$$\begin{array}{ccc}& =& \int {\mathrm{d}}^{4}x\left\{{\chi }^{†}\left(x\right){\sigma }^{\mu }i{\partial }_{\mu }\chi \left(x\right)+\frac{1}{2}m\left[{\chi }^{†}\left(x\right){\sigma }_2{\chi }^{*}\left(x\right)+{\chi }^{\top }\left(x\right){\sigma }_2\chi \left(x\right)\right]\right\}\hfill \end{array}$$

(105)

is not invariant under the overall phase transformation ${\chi }_M^{\prime }\left(x\right)=e^{i\theta }{\chi }_M\left(x\right)$ of the Majorana bispinor, just like the previously discussed mass term. Hence, as it will be further endorsed after the transition to the Majorana representation of the Dirac matrices, there is no invariant scalar charge or lepton number for a Majorana massive spinor. According to the general Noether theorem for a Lie group of global internal symmetry transformations

$${\partial }_{\mu }{J}_a^{\mu }=\delta \mathcal{L}/\delta {\omega }_a(a=1,2,\dots ,s)$$

where ${\omega }_a$ are the parameters of the internal symmetry Lie group of dimension s, we find

$$J_M^{\mu }=(\delta {\mathcal{L}}_M/\delta {\partial }_{\mu }\chi )i\chi =-{\chi }^{†}{\sigma }^{\mu }\chi \delta {\mathcal{L}}_M/\delta \theta =im\left({\chi }^{\top }{\sigma }_2\chi -{\chi }^{†}{\sigma }_2{\chi }^{*}\right)$$

(106)

so that

$${\partial }_{\mu }{J}_M^{\mu }={\partial }_{\mu }\left({\chi }^{†}{\sigma }^{\mu }\chi \right)=im({\chi }^{†}{\sigma }_2{\chi }^{*}-{\chi }^{\top }{\sigma }_2\chi )$$

(107)

We recall that the canonical dimensions of any kind of spinor $\psi$ in physical units are $\left[\psi \right]=\sqrt{\mathrm{eV}}{\mathrm{cm}}^{-1}.$ It follows that the Noether charge is the lepton number, a pure number $\ell ,$ which can be identified as follows: namely,

$$\ell \left(t\right)=-{\displaystyle \frac{1}{\hslash c}}\int \mathrm{d}\mathbf{x}{\chi }^{†}(t,\mathbf{x})\chi (t,\mathbf{x})$$

(108)

which satisfies the evolution equation

$${\displaystyle \frac{\mathrm{d}\ell }{\mathrm{d}t}}={\displaystyle \frac{im}{{\hslash }^2}}\int \mathrm{d}\mathbf{x}\left[{\chi }^{\top }\left(x\right){\sigma }_2\chi \left(x\right)-{\chi }^{†}\left(x\right){\sigma }_2{\chi }^{*}\left(x\right)\right]$$

(109)

that leads to the lepton number conservation only in the massless case.

Actually, as it will be discussed and clarified in the sequel, there is a relic continuous U(1) symmetry only for Majorana massless spinors, which drives to the existence of a conserved pseudo-scalar charge, the meaning of which will be better focused further on.

Another key observation is now in order. Starting with a left-handed Weyl 2-component spinor $\chi \left(x\right)\in D({\displaystyle \frac{1}{2}},0)$ one can also build up a chiral left-handed bispinor

$${\nu }_L\left(x\right)=P_L{\chi }_M\left(x\right)=〔\begin{array}{c}\chi \left(x\right)\\ 0\end{array}〕P_R{\nu }_L=\frac{1}{2}(1+{\gamma }_5){\nu }_L=0$$

together with its right-handed counterpart

$${\nu }_L^c\left(x\right)={\gamma }^2{\nu }_L^{*}\left(x\right)=〔\begin{array}{c}0\\ -{\sigma }_2{\chi }^{*}\left(x\right)\end{array}〕\equiv {\nu }_R\left(x\right)P_L{\nu }_R=\frac{1}{2}(1-{\gamma }_5){\nu }_R=0$$

which still involves only the very same left-handed Weyl 2-component spinor $\chi \left(x\right)\in D({\displaystyle \frac{1}{2}},0).$ With the help of those two chiral left- and right-handed bispinors, one can suitably describe a massive neutrino, once we employ the related Lagrangian density

$$\begin{array}{ccc}\hfill {\mathcal{L}}_L& =& \frac{1}{2}{\overline{\nu }}_L\left(x\right){\gamma }^{\mu }i{\overleftrightarrow{\partial }}_{\mu }{\nu }_L\left(x\right)-\frac{1}{2}m{\overline{\nu }}_L\left(x\right){\nu }_R\left(x\right)+\mathrm{c}.\mathrm{c}.\hfill \end{array}$$

(110)

$$\begin{array}{ccc}& =& \frac{1}{2}{\chi }^{†}\left(x\right){\sigma }^{\mu }i{\partial }_{\mu }\chi \left(x\right)+\frac{1}{2}m{\chi }^{†}\left(x\right){\sigma }_2{\chi }^{*}\left(x\right)+\mathrm{c}.\mathrm{c}.\hfill \end{array}$$

(111)

However, it turns out that the above Lagrangian does actually coincides with the Majorana Lagrangian in (104) in such a manner that the field equations for a chiral left-handed massive bispinor ${\nu }_L\left(x\right)=P_L{\chi }_M\left(x\right)$ are nothing but the very same as the massive Majorana spinor $\chi \left(x\right),$ the general solutions of which will be fully discussed in the sequel, after the transition to the Majorana representation.

The full Majorana Lagrangian in the 2-component formalism reads

$$\begin{array}{c}\hfill {\widehat{\mathcal{L}}}_M=\frac{1}{2}{\chi }^{†}\left(x\right){\sigma }^{\mu }i{\partial }_{\mu }\chi \left(x\right)+\frac{1}{2}m{\chi }^{\top }\left(x\right){\sigma }_2\chi \left(x\right)+\mathrm{c}.\mathrm{c}.\end{array}$$

(112)

so that the Euler-Lagrange field equation becomes

$$i{\sigma }^{\mu }{\partial }_{\mu }\chi \left(x\right)+m{\sigma }_2{\chi }^{*}\left(x\right)=0$$

(113)

which is known as the Majorana field equation. Multiplying to the left by ${\sigma }_2$ and taking complex conjugation yields

$$i{\sigma }_2{\partial }_0{\chi }^{*}\left(x\right)-i{\sigma }_2{\sigma }_k^{*}{\partial }_k{\chi }^{*}\left(x\right)+m\chi \left(x\right)=0$$

Remembering that we have ${\sigma }_2\overrightarrow{\sigma }{\sigma }_2=-\overrightarrow{\sigma }$ and that ${\overline{\sigma }}^{\mu }=(1,\overrightarrow{\sigma })$ we come to the equivalent form of the Majorana wave equation, viz.,

$$i{\overline{\sigma }}^{\mu }{\sigma }_2{\partial }_{\mu }{\chi }^{*}\left(x\right)+m\chi \left(x\right)=0$$

(114)

Now, if we act from the left with the operator $i{\overline{\sigma }}^{\nu }{\partial }_{\nu }$ to equation (113) and use equation (114) we find

$${\overline{\sigma }}^{\nu }{\sigma }^{\mu }{\partial }_{\nu }{\partial }_{\mu }\chi \left(x\right)+m^2\chi \left(x\right)=(\square +m^2)\chi \left(x\right)=0$$

which means that the left-handed spinor $\chi \in D({\displaystyle \frac{1}{2}},0),$ i.e. the building block of the self-conjugated Majorana bispinor, is actually solution of the Klein-Gordon wave equation. Moreover, by understanding the Majorana bispinor to be defined by the self-conjugation constraint ${\chi }_M\left(x\right)={\chi }_M^c\left(x\right),$ it is easy to check that the pair of coupled wave equations (113) and (114) is equivalent to the single bispinor wave equation

$$({\alpha }^{\nu }i{\partial }_{\nu }-\beta m){\chi }_M\left(x\right)=0$$

(115)

where use has been made of the Dirac notation $\beta ={\gamma }_0,$ while the Majorana Lagrangian can be recast in a further 4-component form

$${\mathcal{L}}_M=\frac{1}{4}{\chi }_M^{†}\left(x\right){\alpha }^{\mu }i{\overleftrightarrow{\partial }}_{\mu }{\chi }_M^c\left(x\right)-\frac{1}{2}m{\chi }_M^{†}\left(x\right)\beta {\chi }_M^c\left(x\right)$$

(116)

It is immediate to verify that the bispinor form (115) of the field equations does coincide with the pair of functionally dependent and equivalent forms (113) and (114) of the Majorana spinor wave equation. Notice that the Majorana self-conjugated bispinor transforms under the Poincaré group according to

$${\chi }_M^{\prime }\left(x^{\prime }\right)={\Lambda }_{{\displaystyle \frac{1}{2}}}{\chi }_M^c\left(x\right){\Lambda }_{{\displaystyle \frac{1}{2}}}=〔\begin{array}{cc}{\Lambda }_L& 0\\ 0& {\Lambda }_R\end{array}〕$$

(117)

with $x^{\prime }=\Lambda (x+\mathrm{a}).$ Hence it follows that the generators of the Lorentz group acting upon any kind of bispinor are provided by the usual expressions

$$S^{0k}={\displaystyle \frac{i\hslash }{2}}{\gamma }_0{\gamma }^k={\displaystyle \frac{\hslash }{2}}〔\begin{array}{cc}-i{\sigma }_k& 0\\ 0& i{\sigma }_k\end{array}〕$$

$$S^{jk}=\frac{i\hslash }{4}[{\gamma }^j,{\gamma }^k]=\frac{1}{2}\hslash {\epsilon }^{jk\ell }〔\begin{array}{cc}{\sigma }_{\ell }& 0\\ 0& {\sigma }_{\ell }\end{array}〕=\frac{1}{2}\hslash {\epsilon }^{jk\ell }{\Sigma }^{\ell }$$

in such a manner that we obtain the infinitesimal variation

$$\Delta {\chi }_M\left(x\right)=-\frac{1}{2}i{\sigma }_{\mu \nu }{\chi }_M\left(x\right)\delta {\omega }^{\mu \nu }{\sigma }_{\mu \nu }={\displaystyle \frac{i}{4}}[{\gamma }_{\mu },{\gamma }_{\nu }]$$

3.2.2. The Majorana Real Representation

Let us now search the general complex solutions of the Majorana linear wave equation (115) for a self-conjugated bispinor ${\chi }_M\left(x\right).$ Once obtained, we can readily get the general complex solution for the massive chiral left-handed neutrino ${\nu }_L\left(x\right)$ after a projection on the upper component of ${\chi }_M\left(x\right).$ To implement this task, it appears utmost convenient to employ the so called Majorana representation [16]. It turns out that, by definition, the Majorana bispinor ${\chi }_M\left(x\right)={\chi }_M^c\left(x\right)$ must fulfill the self-conjugation constraint, which linearly relates the lower spinor component to the complex conjugate of the upper spinor component. Then, a representation must exist which makes the Majorana bispinor real, with the previously introduced two independent complex functions ${\chi }_a\in \mathbb{C}(a=1,2)$ replaced by the four real functions ${\psi }_{M,\alpha }\in \mathbb{R}(\alpha =1,2,3,4).$ To obtain this real representation, we notice that for any complex bispinor $\psi$ we find

$${\psi }^c={\gamma }^2{\psi }^{*}{\psi }^{*}=-{\gamma }^2{\psi }^c$$

in such a manner that for the self-conjugated complex bispinor ${\chi }_M$ we can write

$${\chi }_M^{*}\left(x\right)=-{\gamma }^2{\chi }_M\left(x\right)$$

Then a linear transformation to a real bispinor field ${\psi }_M={\psi }_M^{*}$ can be found by noting that

$${\chi }_M=S{\psi }_M{\chi }_M^{*}=S^{*}{\psi }_M=S^{*}{S}^{-1}{\chi }_M{\gamma }^2+S^{*}{S}^{-1}=0$$

The explicit form of the unitary transformation matrix can be readily found to be

$$S={\displaystyle \frac{\sqrt{2}}{2}}〔\begin{array}{cc}1& {\sigma }_2\\ -{\sigma }_2& 1\end{array}〕={\displaystyle \frac{\sqrt{2}}{2}}(1+{\gamma }^2)S^{†}=S^{-1}={\displaystyle \frac{\sqrt{2}}{2}}(1-{\gamma }^2)$$

(118)

or even more explicitly

$$S={\displaystyle \frac{\sqrt{2}}{2}}\left(\begin{array}{cccc}1& 0& 0& -i\\ 0& 1& i& 0\\ 0& i& 1& 0\\ -i& 0& 0& 1\end{array}\right)$$

From the above relation ${\psi }_M=S^{-1}{\chi }_M={\psi }_M^{*}$ one can immediately obtain the correspondence rule between the complex and real forms of the self-conjugated Majorana bispinor: namely,

$$\begin{array}{c}\hfill \begin{array}{cc}{\psi }_{M1}=\sqrt{2}\Re \mathrm{e}{\chi }_1& {\psi }_{M2}=\sqrt{2}\Re \mathrm{e}{\chi }_2\\ {\psi }_{M3}=\sqrt{2}\Im \mathrm{m}{\chi }_2& {\psi }_{M4}=-\sqrt{2}\Im \mathrm{m}{\chi }_1\end{array}\end{array}$$

(119)

$$\begin{array}{c}{\chi }_1=({\psi }_{M1}-i{\psi }_{M4})/\sqrt{2}\\ {\chi }_2=({\psi }_{M2}+i{\psi }_{M3})/\sqrt{2}\end{array}$$

(120)

Thus we can suitably make use of the so called Majorana representation for the gamma matrices, which is purely imaginary, and provided by a similarity transformation acting on the $\gamma -$matrices in the Weyl representation, viz.,

$${\gamma }_M^{\mu }\equiv S^{†}{\gamma }^{\mu }S$$

that yields

$${\gamma }_M^0=〔\begin{array}{cc}-{\sigma }_2& 0\\ 0& {\sigma }_2\end{array}〕{\gamma }_M^5=〔\begin{array}{cc}0& -{\sigma }_2\\ -{\sigma }_2& 0\end{array}〕$$

$${\gamma }_M^1=〔\begin{array}{cc}-i{\sigma }_3& 0\\ 0& -i{\sigma }_3\end{array}〕{\gamma }_M^2=〔\begin{array}{cc}0& {\sigma }_2\\ -{\sigma }_2& 0\end{array}〕{\gamma }_M^3=〔\begin{array}{cc}i{\sigma }_1& 0\\ 0& i{\sigma }_1\end{array}〕$$

which satisfy by direct inspection the Clifford algebra

$$\{{\gamma }_M^{\mu },{\gamma }_M^{\nu }\}=2g^{\mu \nu }\{{\gamma }_M^{\nu },{\gamma }_M^5\}=0$$

$${\gamma }_M^0={\beta }_M^{†}{\gamma }_M^k=-{\gamma }_M^{k†}{\gamma }_M^5={\gamma }_M^{5†}$$

together with

$${\gamma }_M^{\nu }=-{\gamma }_M^{\nu *}{\gamma }_M^5=-{\gamma }_M^{5*}$$

The result is that, at the place of a complex self-conjugated bispinor, which has been constructed out of a left-handed Weyl spinor ${\chi }_M\left(x\right),$ one can safely and more suitably employ a real Majorana bispinor: namely,

$${\chi }_M\left(x\right)={\chi }_M^c\left(x\right)\leftrightarrow {\psi }_M\left(x\right)=S^{†}{\chi }_M\left(x\right)={\psi }_M^{*}\left(x\right)$$

a quite analogous construction being obviously there, had we started from a right-handed Weyl spinor $\phi \in D(0,{\displaystyle \frac{1}{2}}).$ Then, the Majorana Lagrangian and the ensuing Majorana wave equation take the manifestly real forms

$${\mathcal{L}}_M=\frac{1}{4}{\psi }_M^{\top }\left(x\right){\alpha }_M^{\nu }{\overleftrightarrow{\partial }}_{\nu }{\psi }_M\left(x\right)+\frac{1}{2}im{\psi }_M^{\top }\left(x\right){\beta }_M{\psi }_M\left(x\right)$$

(121)

$$(i\partial {/}_M-m){\psi }_M\left(x\right)=0{\psi }_M\left(x\right)={\psi }_M^{*}\left(x\right)$$

(122)

$${\alpha }_M^{\nu }={\gamma }_M^0{\gamma }_M^{\nu }{\alpha }_M^0=\mathbb{I}{\beta }_M\equiv {\gamma }_M^0$$

with the real symmetric matrices

$${\alpha }_M^1=〔\begin{array}{cc}-{\sigma }_1& 0\\ 0& {\sigma }_1\end{array}〕{\alpha }_M^2=〔\begin{array}{cc}0& -1\\ -1& 0\end{array}〕{\alpha }_M^3=〔\begin{array}{cc}-{\sigma }_3& 0\\ 0& {\sigma }_3\end{array}〕$$

which manifestly fulfill ${\overrightarrow{\alpha }}_M={\overrightarrow{\alpha }}_M^{*}={\overrightarrow{\alpha }}_M^{\top }.$ It turns out that, from the manifestly real form of the Majorana Lagrangian, the only relic internal symmetry of the Majorana’s Action integral - but in the massless case - is the discrete ${\mathbb{Z}}_2$ symmetry, i.e. ${\psi }_M\left(x\right)⟼-{\psi }_M\left(x\right).$

Singular massless limit. Actually, it turns out that only in the massless casethere exists a further accidental invariance of the Action integral under the U(1) internal symmetry group

$${\psi }_M\left(x\right)\mapsto {\psi }_M^{\prime }\left(x\right)=exp\left\{\pm i\theta {\gamma }_M^5\right\}{\psi }_M\left(x\right)(0\le \theta <2\pi )$$

the imaginary unit being convenient to keep the reality of the trasformed Majorana bispinor. From the Noether theorem we get the corresponding real tetra-current, which satisfies the continuity equation

$$j_5^{\mu }\left(x\right)=\frac{1}{2}{\psi }_M^{\top }\left(x\right){\alpha }_M^{\mu }i{\gamma }_M^5{\psi }_M\left(x\right)\partial ·j_5\left(x\right)=0$$

as well as the ensuing conserved pseudo-scalar charge

$$\pm Q_5=\pm {\displaystyle \frac{1}{2}}\int \mathrm{d}\mathbf{x}{\psi }_M^{\top }(t,\mathbf{x})i{\gamma }_M^5{\psi }_M(t,\mathbf{x}){\dot{Q}}_5=0$$

the overall ± sign being conventional and irrelevant. In order to better understand the meaning of this quantity we notice that, if we turn to the complex form ${\chi }_M\left(x\right)$ of the self-conjugated bispinor, we can immediately realize that the above U(1) internal symmetry of the massless case is nothing but the ordinary phase transformation for the independent building spinor $\chi \left(x\right).$ As a matter of fact, the phase transformation ${\chi }^{\prime }\left(x\right)=e^{-i\theta }\chi \left(x\right)$ just induces the chiral transformation ${\chi }_M^{\prime }\left(x\right)=e^{i\theta {\gamma }_5}{\chi }_M\left(x\right)$ on the complex form of the Majorana bispinor. Notice that the quantity

$${\psi }_M^{\top }i{\gamma }_M^5{\psi }_M=〔{\psi }_{M\uparrow }{\psi }_{M\downarrow }〕〔\begin{array}{cc}0& -i{\sigma }_2\\ -i{\sigma }_2& 0\end{array}〕〔\begin{array}{c}{\psi }_{M\uparrow }\\ {\psi }_{M\downarrow }\end{array}〕$$

does vanish for ordinary real functions ${\psi }_{M,\alpha }\in \mathbb{R}(\alpha =1,2,3,4)$ while in the case of anti-commuting Grassmann-valued functions we get

$$\pm Q_5=\pm \int \mathrm{d}\mathbf{x}\left[{\psi }_{M,1}(t,\mathbf{x}){\psi }_{M,4}(t,\mathbf{x})-{\psi }_{M,2}(t,\mathbf{x}){\psi }_{M,3}(t,\mathbf{x})\right]$$

the integrated quantity being nothing but than $-i{\chi }^{†}\left(x\right)\chi \left(x\right)$ as expected.

A very important remark is now in order. The Majorana Lagrangian (121) and wave equation (122) have been derived starting from an original two-component left-handed spinor $\chi \left(x\right)$ or, equivalently, a self-conjugated bispinor ${\chi }_M\left(x\right)={\chi }_M^c\left(x\right).$ However, a closely related and still real Lagrange density, just leading to the very same field equation, can be immediately written even for any arbitraty complex bispinor ${\Psi }_M\left(x\right):$ namely,

$${\mathcal{L}}_M=\frac{1}{2}{\Psi }_M^{†}\left(x\right){\alpha }_M^{\nu }i{\overleftrightarrow{\partial }}_{\nu }{\Psi }_M\left(x\right)-m{\Psi }_M^{†}\left(x\right){\beta }_M{\Psi }_M\left(x\right)$$

(123)

$$(i\partial {/}_M-m){\Psi }_M\left(x\right)=0$$

(124)

At variance with (121) the Lagrange density (123) does enjoy the invariance under the global U(1) symmetry transformations ${\Psi }_M^{\prime }\left(x\right)=e^{-i\theta }{\Psi }_M\left(x\right)(0\le \theta <2\pi )$ driving eventually to the charge or number conservation. It follows therefrom that the real Majorana wave equation will admit in general both real as well as complex solutions, with quite different physical properties and behaviors.

3.2.3. Solution of the Majorana Wave Equation

One can rewrite the Majorana wave equation (122) in the Schrödinger form and in physical units

$$i\hslash {\displaystyle \frac{\partial {\psi }_M}{\partial t}}=m{c}^2{\beta }_M{\psi }_M-\hslash c{\alpha }_M^{k}i{\nabla }_k{\psi }_M=H_M{\psi }_M\left(x\right)$$

whence we can read the Majorana one-particle Hamiltonian operator

$$H_M={\alpha }_M^k{\widehat{p}}^{k}c+m{c}^2{\beta }_M({\widehat{p}}^k=-i\hslash {\nabla }_k)$$

(125)

To solve the Majorana linear and real wave equation (122) we turn to the Fourier representation

$${\psi }_M\left(x\right)=\int {\displaystyle \frac{{\mathrm{d}}^{4}p}{{\left(2\pi \right)}^{3/2}}}{\tilde{\psi }}_M\left(p\right)exp\{-ip·x\}=\int {\displaystyle \frac{{\mathrm{d}}^{4}p}{{\left(2\pi \right)}^{3/2}}}{\tilde{\psi }}_M(-p)exp\{ip·x\}$$

so that from (122) we get

$$\left\{\begin{array}{c}(p{/}_M-m){\tilde{\psi }}_M\left(p\right)=0\\ (p{/}_M+m){\tilde{\psi }}_M(-p)=0\end{array}\right.p{/}_M\equiv p_{\nu }{\gamma }_M^{\nu }$$

(126)

which implies

$$\begin{array}{ccc}& & {\tilde{\psi }}_{M,\alpha }\left(p\right)={(p{/}_M+m)}_{\alpha \beta }{\tilde{\varphi }}_{\beta }\left(p\right)\hfill \\ & & \left(p^2-m^2\right){\tilde{\varphi }}_{\alpha }\left(p\right)=0(\alpha =1,2,3,4)\hfill \\ & & {\tilde{\varphi }}_{\alpha }\left(p\right)=\delta \left(p^2-m^2\right)f_{\alpha }\left(p\right)\hfill \end{array}$$

and in turn

$$\begin{array}{ccc}\hfill {\tilde{\psi }}_{M,\alpha }(-p)& =& {(m-p{/}_M)}_{\alpha \beta }{\tilde{\varphi }}_{\beta }(-p)={(m-p{/}_M)}_{\alpha \beta }{f}_{\beta }(-p)\delta \left(p^2-m^2\right)\hfill \end{array}$$

The momentum space Majorana equations (126) are a pair of algebraic equation

$$\left\{\begin{array}{c}(p_0{\beta }_M-p^k{\gamma }_M^k-mc){\tilde{\psi }}_M\left(p\right)=0\\ (p_0{\beta }_M-p^k{\gamma }_M^k+mc){\tilde{\psi }}_M(-p)=0\end{array}\right.c{p}_0=\pm \hslash {\omega }_{\mathbf{p}}=\pm \sqrt{{\mathbf{p}}^2{c}^2+m^2{c}^4}$$

which can be turned into a pair of energy eigenvalue problems

$${\tilde{H}}_M\left(\mathbf{p}\right){\tilde{\psi }}_M\left(p\right)=\pm \hslash {\omega }_{\mathbf{p}}{\tilde{\psi }}_M\left(p\right)c{p}^k{\alpha }_M^k+m{c}^2{\beta }_M\equiv {\tilde{H}}_M\left(\mathbf{p}\right)$$

(127)

where ${\tilde{H}}_M\left(\mathbf{p}\right)$ is the 1-particle Majorana Hamiltonian in the momentum space representation, which satisfies the property ${\tilde{H}}_M^2\left(\mathbf{p}\right)={\omega }_{\mathbf{p}}^2.$

In order to find explicit solutions in momentum space, let us first consider the following matrix-valued functions: namely,

$${\mathcal{E}}_M^{\pm }\left(\mathbf{p}\right)=\left(mc\pm p{/}_M\right)/2mc\mathrm{with}c{p}_0=\pm \hslash {\omega }_{\mathbf{p}}=\pm \sqrt{{\mathbf{p}}^2{c}^2+m^2{c}^4}$$

which do satisfy the following relationships

$${\left[{\mathcal{E}}_M^{\pm }\left(\mathbf{p}\right)\right]}^{*}={\mathcal{E}}_M^{\mp }\left(\mathbf{p}\right){\left[{\mathcal{E}}_M^{\pm }\left(\mathbf{p}\right)\right]}^{†}={\mathcal{E}}_M^{\pm }(-\mathbf{p})$$

$${\left[{\mathcal{E}}_M^{\pm }\left(\mathbf{p}\right)\right]}^2={\mathcal{E}}_M^{\pm }\left(\mathbf{p}\right){\mathcal{E}}_M^{\pm }\left(\mathbf{p}\right){\mathcal{E}}_M^{\mp }\left(\mathbf{p}\right)=0$$

$$\mathrm{tr}{\mathcal{E}}_M^{\pm }\left(\mathbf{p}\right)=2{\mathcal{E}}_M^{+}\left(\mathbf{p}\right)+{\mathcal{E}}_M^{-}\left(\mathbf{p}\right)=\mathbb{I}$$

The above matrix-valued functions can be fully understood as projectors, if we simultaneously perform the matrix Hermitean conjugation and the full inversion, or parity transform, on the function variable $\mathbf{p}\to -\mathbf{p}.$ The physical meaning of those projectors can be better featured and focused on the Majorana particle rest frame $\mathbf{p}=0,$ in which $p{/}_M=mc{\beta }_M.$ To this concern, we recall the spin operators $S_k={\displaystyle \frac{1}{2}}\hslash {\Sigma }_k(k=1,2,3)$ in the Majorana representation where

$${\Sigma }_{M,1}=i{\gamma }_M^2{\gamma }_M^3=〔\begin{array}{cc}0& i{\sigma }_3\\ -i{\sigma }_3& 0\end{array}〕{\Sigma }_{M,3}=i{\gamma }_M^1{\gamma }_M^2=〔\begin{array}{cc}0& -i{\sigma }_1\\ i{\sigma }_1& 0\end{array}〕$$

$${\Sigma }_{M,2}=i{\gamma }_M^3{\gamma }_M^1=〔\begin{array}{cc}{\sigma }_2& 0\\ 0& {\sigma }_2\end{array}〕$$

From the relationships

$${\beta }_M^2=1={\Sigma }_{M,2}^2[{\beta }_M,{\Sigma }_{M,2}]=0$$

it appears that one can actually set up a complete orthogonal tetrad of bispinors, which are common eigenvectors of the Hermitean commuting matrices ${\beta }_M$ and ${\Sigma }_{M,2}.$ It turns out that the spin states ${\xi }_r(r=\pm )$ are the pair of degenerate eigenstates of the Majorana 1-particle Hamiltonian ${\tilde{H}}_M\left(\mathbf{p}\right)$ in the particle rest frame $\mathbf{p}=0,$ with positive eigenvalue $p_0=m$ and with opposite spin projections on the $OY$ axis, viz.,

$${\beta }_M{\xi }_{\pm }={\xi }_{\pm }\left({\Sigma }_{M,2}\mp 1\right){\xi }_{\pm }=0{\xi }_{\mp }^{†}{\xi }_{\pm }=0$$

In the very same way we obtain

$${\beta }_M{\eta }_{\pm }=-{\eta }_{\pm }\left({\Sigma }_{M,2}\mp 1\right){\eta }_{\pm }=0{\eta }_{\mp }^{†}{\eta }_{\pm }=0$$

which means that the spin states ${\eta }_r(r=\pm )$ are in fact the two degenerate eigenstates of the Majorana 1-particle Hamiltonian in the particle rest frame $\mathbf{p}=0,$ with negative eigenvalue $p_0=-m$ and with opposite spin projections on the $OY$ axis. Altogether we can write

$${\xi }_r^{†}{\xi }_s=2{\delta }_{rs}={\eta }_r^{†}{\eta }_s{\xi }_r^{†}{\eta }_s=0\forall r,s=\pm$$

Notice that, from the relations $\{{\gamma }_5,{\gamma }_0\}=0=[{\gamma }_5,{\Sigma }_2]$, it follows that the chiral matrix ${\gamma }_5$ maps the positive energy space into the negative one, and viceversa, in the Majorana particle rest frame $\mathbf{p}=0.$ Hence, without loss of generality, we can always assume

$${\gamma }_M^5{\xi }_{\pm }={\eta }_{\pm }{\gamma }_M^5{\eta }_{\pm }={\xi }_{\pm }$$

In general, the freedom in the choice of an orthogonal pair, in the positive and negative energy planes of the rest frame $\mathbf{p}=0,$ is provided by the $SO(2,\mathbb{R})$ Abelian symmetry little group

$$F_{\pm }\left(\theta \right)=exp\{\pm i{\Sigma }_{M,2}\theta \}=\mathbf{I}cos\theta \pm {\gamma }_M^1{\gamma }_M^{3}sin\theta {\left[F_{\pm }\left(\theta \right)\right]}^{-1}={\left[F_{\pm }\left(\theta \right)\right]}^{\top }=F_{\pm }(-\theta )$$

because ${\beta }_M^{\top }+{\beta }_M=0,{\gamma }_M^{k\top }={\gamma }_M^k,$ in such a manner that

$${\xi }_r^{\prime }=\sum _{s=+,-}{\left[F_{+}\left(\theta \right)\right]}_{rs}{\xi }_s{\eta }_r^{\prime }=\sum _{s=+,-}{\left[F_{-}\left(\vartheta \right)\right]}_{rs}{\eta }_s$$

(128)

To sum up, one can set up the positive and negative energy spaces projectors ${\epsilon }_{\pm }$, in the rest frame of the massive Majorana particle, which satisfy

$$\begin{array}{c}\hfill {\epsilon }_{+}=\frac{1}{2}({\xi }_{+}\otimes {\xi }_{+}^{†}+{\xi }_{-}\otimes {\xi }_{-}^{†}){\epsilon }_{-}=\frac{1}{2}({\eta }_{+}\otimes {\eta }_{+}^{†}+{\eta }_{-}\otimes {\eta }_{-}^{†})\end{array}$$

(129)

$$\begin{array}{c}\hfill {\epsilon }_{\pm }^2={\epsilon }_{\pm }{\epsilon }_{\pm }{\epsilon }_{\mp }=0{\epsilon }_{+}+{\epsilon }_{-}=1\mathrm{tr}{\epsilon }_{\pm }=2\end{array}$$

(130)

Now, in order to move from the rest frame to an arbitrary inertial frame we can proceed as follows: consider the so called spin-states of positive and negative energy, viz.,

$$\left\{\begin{array}{c}{u}_r\left(\mathbf{p}\right)\equiv 2m{(2{\omega }_{\mathbf{p}}+2m)}^{-{\displaystyle \frac{1}{2}}}{\mathcal{E}}_M^{+}\left(\mathbf{p}\right){\xi }_r\\ \\ v_r\left(\mathbf{p}\right)\equiv 2m{(2{\omega }_{\mathbf{p}}+2m)}^{-{\displaystyle \frac{1}{2}}}{\mathcal{E}}_M^{-}\left(\mathbf{p}\right){\eta }_r\end{array}\right.(r=+,-)$$

(131)

which again satisfy

$${\gamma }_M^5{u}_r\left(\mathbf{p}\right)=v_r\left(\mathbf{p}\right){\gamma }_M^5{v}_r\left(\mathbf{p}\right)=u_r\left(\mathbf{p}\right)(r=+,-)$$

The above spin-states do fulfill by construction

$$\left\{\begin{array}{c}2m{\mathcal{E}}_M^{-}\left(\mathbf{p}\right)u_r\left(\mathbf{p}\right)=(m-p{/}_M)u_r\left(\mathbf{p}\right)=0\\ \\ 2m{\mathcal{E}}_M^{+}\left(\mathbf{p}\right)v_r\left(\mathbf{p}\right)=(m+p{/}_M)v_r\left(\mathbf{p}\right)=0\end{array}\right.(p_0={\omega }_{\mathbf{p}},r=+,-)$$

(132)

and they are the two pair of degenerate eigenstates of the positive and negative energy projectors

$${\mathcal{E}}_M^{+}\left(\mathbf{p}\right)u_r\left(\mathbf{p}\right)=u_r\left(\mathbf{p}\right){\mathcal{E}}_M^{-}\left(\mathbf{p}\right)v_r\left(\mathbf{p}\right)=v_r\left(\mathbf{p}\right)(r=+,-)$$

(133)

as well as of the Majorana 1-particle Hamiltonian in momentum space, viz.,

$${\tilde{H}}_M\left(\mathbf{p}\right)u_r\left(\mathbf{p}\right)={\omega }_{\mathbf{p}}{u}_r\left(\mathbf{p}\right){\tilde{H}}_M(-\mathbf{p})v_r\left(\mathbf{p}\right)=-{\omega }_{\mathbf{p}}{v}_r\left(\mathbf{p}\right)(r=+,-)$$

(134)

By direct inspection one can readily verify the orthogonality relations

$$\left\{\begin{array}{cc}{u}_r^{†}\left(\mathbf{p}\right)u_s\left(\mathbf{p}\right)=2{\omega }_{\mathbf{p}}{\delta }_{rs}& {\overline{u}}_r\left(\mathbf{p}\right)u_s\left(\mathbf{p}\right)=2m{\delta }_{rs}\\ \\ v_r^{†}\left(\mathbf{p}\right)v_s\left(\mathbf{p}\right)=2{\omega }_{\mathbf{p}}{\delta }_{rs}& {\overline{v}}_r\left(\mathbf{p}\right)v_s\left(\mathbf{p}\right)=-2m{\delta }_{rs}\end{array}\right.(r,s=+,-)$$

(135)

together with

$$u_r^{†}\left(\mathbf{p}\right)v_s(-\mathbf{p})={\overline{u}}_r\left(\mathbf{p}\right)v_s\left(\mathbf{p}\right)=0(r,s=+,-)$$

(136)

For example we have

$$\begin{array}{ccc}& & v_r^{†}\left(\mathbf{p}\right)v_s\left(\mathbf{p}\right)={(2{\omega }_{\mathbf{p}}+2m)}^{-1}{\eta }_r^{†}(m-\widetilde{p/})(m-p/){\eta }_s\hfill \\ & =& {\omega }_{\mathbf{p}}{({\omega }_{\mathbf{p}}+m)}^{-1}{\eta }_r^{†}({\omega }_{\mathbf{p}}+\overrightarrow{p}·\overrightarrow{\gamma }{\beta }_M-m{\beta }_M){\eta }_s=2{\omega }_{\mathbf{p}}{\eta }_r^{†}{\eta }_s=2{\omega }_{\mathbf{p}}{\delta }_{rs}\hfill \end{array}$$

where use has been made of the reations ${\beta }_M{\eta }_s=-{\eta }_s$ and ${\eta }_r^{†}{\overrightarrow{\gamma }}_M{\eta }_s\equiv 0.$ Furthermore we get

$$\begin{array}{ccc}& & {\overline{u}}_r\left(\mathbf{p}\right)u_s\left(\mathbf{p}\right)={(2{\omega }_{\mathbf{p}}+2m)}^{-1}{\xi }_r^{†}(m+\widetilde{p/}){\beta }_M(m+p/){\xi }_s\hfill \\ & =& {(2{\omega }_{\mathbf{p}}+2m)}^{-1}{\xi }_r^{†}{(m+p/)}^2{\xi }_s=2m{(2{\omega }_{\mathbf{p}}+2m)}^{-1}{\xi }_r^{†}(m+p/){\xi }_s=2m{\delta }_{rs}\hfill \\ \\ & & {\overline{u}}_r\left(\mathbf{p}\right)v_s\left(\mathbf{p}\right)={(2{\omega }_{\mathbf{p}}+2m)}^{-1}{\xi }_r^{†}(m+\widetilde{p/}){\beta }_M(m-p/){\eta }_s\hfill \\ & =& {(2{\omega }_{\mathbf{p}}+2m)}^{-1}{\xi }_r^{†}(m+p/)(m-p/){\eta }_s\equiv 0\hfill \\ \\ & & u_r^{†}\left(\mathbf{p}\right)v_s(-\mathbf{p})={(2{\omega }_{\mathbf{p}}+2m)}^{-1}{\xi }_r^{†}(m+\widetilde{p/})(m-\widetilde{p/}){\eta }_s\equiv 0\hfill \end{array}$$

Finally, from the relation ${\beta }_M{\eta }_{\pm }=-{\eta }_{\pm }$ we get for $N={(2{\omega }_{\mathbf{p}}+2m)}^{-1/2}$

$$\begin{array}{ccc}\hfill v_{\pm }\left(\mathbf{p}\right)& =& -N(m-p{/}_M){\beta }_M{\eta }_{\pm }=N({\omega }_{\mathbf{p}}-m{\beta }_M+p^k{\alpha }_M^k){\eta }_{\pm }=N[{\omega }_{\mathbf{p}}-{\tilde{H}}_M(-\mathbf{p})]{\eta }_{\pm }\hfill \end{array}$$

in such a manner that from the relationship ${\tilde{H}}_M^2(-\mathbf{p})={\omega }_{\mathbf{p}}^2$ we can immediately obtain

$$\begin{array}{c}\hfill {\tilde{H}}_M(-\mathbf{p})v_r\left(\mathbf{p}\right)=N{\omega }_{\mathbf{p}}[{\tilde{H}}_M(-\mathbf{p})-{\omega }_{\mathbf{p}}]{\eta }_r=-{\omega }_{\mathbf{p}}{v}_r\left(\mathbf{p}\right)(r=+,-)\end{array}$$

as expected.

Moreover, it is possible to prove the closure relations

$$\left\{\begin{array}{c}\sum _{r=+,-}{u}_r\left(\mathbf{p}\right)\otimes {\overline{u}}_r\left(\mathbf{p}\right)=p/+m\\ \\ \sum _{r=+,-}{v}_r\left(\mathbf{p}\right)\otimes {\overline{v}}_r\left(\mathbf{p}\right)=p/-m\end{array}\right.(p_0={\omega }_{\mathbf{p}})$$

(137)

Proof. Consider for instance the closure relation for the negative energy spin-states. It turns out that

$$\begin{array}{c}\hfill \sum _{r=+,-}{v}_r\left(\mathbf{p}\right)\otimes {\overline{v}}_r\left(\mathbf{p}\right)v_s\left(\mathbf{p}\right)=-2m{v}_s\left(\mathbf{p}\right)(p/-m)v_s\left(\mathbf{p}\right)=-2m{\mathcal{E}}_M^{-}\left(\mathbf{p}\right)v_s\left(\mathbf{p}\right)=-2m{v}_s\left(\mathbf{p}\right)\end{array}$$

$$\sum _{r=+,-}{v}_r\left(\mathbf{p}\right)\otimes {\overline{v}}_r\left(\mathbf{p}\right)u_s\left(\mathbf{p}\right)=0=(p/-m)u_s\left(\mathbf{p}\right)$$

which means that the two matrix-valued functions on both sides of the above equality do share the same rank two, the same degenerate eigenvalues $(-2m,0)$ and the corresponding very same eigenvectors. A quite analogous conclusion holds obviously true even concerning closure relation for the positive energy spin-states. It follows that the matrix-valued functions in the equality (137) are proportional. However, if we multiply by ${\beta }_M$ from the right both closure relations and take the trace, then we get the same result $4{\omega }_{\mathbf{p}},$ which means that the proportionality constant is one, as required.

As a corollary of the closure relations, after direct substitution, we come to the nontrivial general identity

$$(m\pm p/){\epsilon }_{\pm }(m\pm p/)=(p/\pm m)({\omega }_{\mathbf{p}}+m)$$

(138)

Concerning the Majorana spinor wave functions, it turns out that a complete orthonormal set of complex plane wave bispinor solutions of the Majorana wave equation (122) is provided by

$$\left\{\begin{array}{c}{u}_{\mathbf{p},r}\left(x\right)\equiv {\left[{\left(2\pi \right)}^{3}2{\omega }_{\mathbf{p}}\right]}^{-{\displaystyle \frac{1}{2}}}{u}_r\left(\mathbf{p}\right)exp\{-i{\omega }_{\mathbf{p}}t+i\mathbf{p}·\mathbf{x}\}\\ v_{\mathbf{p},r}\left(x\right)\equiv {\left[{\left(2\pi \right)}^{3}2{\omega }_{\mathbf{p}}\right]}^{-{\displaystyle \frac{1}{2}}}{v}_r\left(\mathbf{p}\right)exp\{i{\omega }_{\mathbf{p}}t-i\mathbf{p}·\mathbf{x}\}\end{array}\right.(r=+,-)$$

(139)

which fulfill the energy eigenvalue problem

$$H_M{u}_{\mathbf{p},r}\left(x\right)={\omega }_{\mathbf{p}}{u}_{\mathbf{p},r}\left(x\right)H_M{v}_{\mathbf{p},r}\left(x\right)=-{\omega }_{\mathbf{p}}{v}_{\mathbf{p},r}\left(x\right)$$

(140)

as well as the orthogonality and closure relations: namely,

$$\int \mathrm{d}\mathbf{x}{u}_{\mathbf{p},r}^{†}\left(x\right)u_{\mathbf{q},s}\left(x\right)={\delta }_{rs}\delta (\mathbf{p}-\mathbf{q})=\int \mathrm{d}\mathbf{x}{v}_{\mathbf{p},r}^{†}\left(x\right)v_{\mathbf{q},s}\left(x\right)$$

(141)

$$\begin{array}{c}i\sum _{r=+,-}\int \mathrm{d}\mathbf{p}{u}_{\mathbf{p},r}\left(x\right)\otimes {\overline{u}}_{\mathbf{p},r}\left(y\right)=(i\partial {/}_M+m)D^{(-)}(x-y)\\ \\ i\sum _{r=+,-}\int \mathrm{d}\mathbf{p}{v}_{\mathbf{p},r}\left(x\right)\otimes {\overline{v}}_{\mathbf{p},r}\left(y\right)=(i\partial {/}_M+m)D^{(+)}(x-y)\end{array}$$

(142)

where $D^{(\pm )}(x-y)$ are the well known tempered distribution

$$D^{(\pm )}(x-y)={\displaystyle \frac{\pm 1}{i{\left(2\pi \right)}^3}}\int {\mathrm{d}}^{4}pexp\{\pm ip·(x-y)\}\delta (p^2-m^2)\theta \left(p_0\right)$$

Proof. By direct replacement we find

$$\begin{array}{ccc}& & i\sum _{r=+,-}\int \mathrm{d}\mathbf{p}{u}_{\mathbf{p},r}\left(x\right)\otimes {\overline{u}}_{\mathbf{p},r}\left(y\right)\hfill \\ & =& i\sum _{r=+,-}\int \mathrm{d}\mathbf{p}{\left[{\left(2\pi \right)}^{3}2{\omega }_{\mathbf{p}}\right]}^{-1}{u}_r\left(\mathbf{p}\right)\otimes {\overline{u}}_r\left(\mathbf{p}\right)exp\{-i{\omega }_{\mathbf{p}}(x_0-y_0)+i\mathbf{p}·(\mathbf{x}-\mathbf{y})\}\hfill \\ & =& i(i{\gamma }_M^{\mu }{\partial }_{\mu ,x}+m)\int \mathrm{d}\mathbf{p}{\left[{\left(2\pi \right)}^{3}2{\omega }_{\mathbf{p}}\right]}^{-1}exp\{-i{\omega }_{\mathbf{p}}(x_0-y_0)+i\mathbf{p}·(\mathbf{x}-\mathbf{y})\}\hfill \\ & =& {\displaystyle \frac{i}{{\left(2\pi \right)}^3}}(m+i\partial {/}_M)\int {\mathrm{d}}^{4}pexp\{-ip·(x-y)\}\delta (p^2-m^2)\theta \left(p_0\right)=(i\partial {/}_M+m)D^{(-)}(x-y)\hfill \\ \\ & & i\sum _{r=+,-}\int \mathrm{d}\mathbf{p}{v}_{\mathbf{p},r}\left(x\right)\otimes {\overline{v}}_{\mathbf{p},r}\left(y\right)\hfill \\ & =& i\sum _{r=+,-}\int \mathrm{d}\mathbf{p}{\left[{\left(2\pi \right)}^{3}2{\omega }_{\mathbf{p}}\right]}^{-1}{v}_r\left(\mathbf{p}\right)\otimes {\overline{v}}_r\left(\mathbf{p}\right)exp\{i{\omega }_{\mathbf{p}}(x_0-y_0)-i\mathbf{p}·(\mathbf{x}-\mathbf{y})\}\hfill \\ & =& -i(i{\gamma }_M^{\mu }{\partial }_{\mu ,x}+m)\int \mathrm{d}\mathbf{p}{\left[{\left(2\pi \right)}^{3}2{\omega }_{\mathbf{p}}\right]}^{-1}exp\{i{\omega }_{\mathbf{p}}(x_0-y_0)-i\mathbf{p}·(\mathbf{x}-\mathbf{y})\}\hfill \\ & =& {\displaystyle \frac{-i}{{\left(2\pi \right)}^3}}(m+i\partial {/}_M)\int {\mathrm{d}}^{4}pexp\{ip·(x-y)\}\delta (p^2-m^2)\theta \left(p_0\right)=(i\partial {/}_M+m)D^{(+)}(x-y)\hfill \end{array}$$

Now a key observation is in order, for the genius of Ettore Majorana to be fully recognized and appreciated. It turns out that only in the Majorana representation of the purely imaginary Dirac matrices, two further complete orthonormal sets of positive and negative frequency solutions of the field equation (122) can be immediately obtained by complex conjugation of the previously obtained ones. As a matter of fact we find

$$\begin{array}{c}\hfill v_r^{*}\left(\mathbf{p}\right)\equiv 2m{(2{\omega }_{\mathbf{p}}+2m)}^{-{\displaystyle \frac{1}{2}}}{\mathcal{E}}_M^{+}\left(p\right){\eta }_r^{*}(r=\pm \vee p_0={\omega }_{\mathbf{p}})\end{array}$$

which manifestly satisfies

$${\tilde{H}}_M\left(\mathbf{p}\right)v_{\pm }^{*}\left(\mathbf{p}\right)={\omega }_{\mathbf{p}}{v}_{\pm }^{*}\left(\mathbf{p}\right)$$

owing to ${\beta }_M{\eta }_{\pm }^{*}={\eta }_{\pm }^{*}$ and ${\beta }_M{\xi }_{\pm }^{*}=-{\xi }_{\pm }^{*}.$ In a quite analogous way, the spin states

$$u_r^{*}\left(\mathbf{p}\right)\equiv 2m{(2{\omega }_{\mathbf{p}}+2m)}^{-{\displaystyle \frac{1}{2}}}{\mathcal{E}}_M^{-}\left(p\right){\xi }_r^{*}(r=\pm \vee p_0={\omega }_{\mathbf{p}})$$

do satisfy

$${\tilde{H}}_M(-\mathbf{p})u_{\pm }^{*}\left(\mathbf{p}\right)=-{\omega }_{\mathbf{p}}{u}_{\pm }^{*}\left(\mathbf{p}\right)$$

Notice en passant that, e.g., ${\gamma }_M^5{u}_{\pm }^{*}\left(\mathbf{p}\right)=-v_{\pm }^{*}\left(\mathbf{p}\right),$ whilst for the adjoint spin-states we find ${\overline{u}}_{\pm }^{*}\left(\mathbf{p}\right)={\left[u_{\pm }^{†}\left(\mathbf{p}\right){\beta }_M\right]}^{*}=-u_{\pm }^{\top }\left(\mathbf{p}\right){\beta }_M.$ It turns out that any spin-state is truly orthogonal to its complex conjugate because, e.g.,

$${\overline{u}}_r\left(\mathbf{p}\right)u_s^{*}\left(\mathbf{p}\right)={\xi }_r^{†}(m+\widetilde{p/}){\beta }_M(m-p/){\xi }_s^{*}{(2m+2{\omega }_{\mathbf{p}})}^{-1}\equiv 0(r,s=+,-)$$

while the closure relation can be directly obtained from the complex conjugation of eq.s (137)

$$\sum _{r=+,-}{u}_r^{*}\left(\mathbf{p}\right)\otimes {\overline{u}}_r^{*}\left(\mathbf{p}\right)=-\sum _{r=+,-}{u}_r^{*}\left(\mathbf{p}\right)\otimes u_r^{\top }\left(\mathbf{p}\right){\beta }_M=-p/+m$$

(143)

It follows therefrom that the most general classical solutions of the real Majorana wave field equation (122) might appear in both real and complex forms: namely,

$${\psi }_M\left(x\right)=\sum _{r=+,-}\int \mathrm{d}\mathbf{p}{a}_{\mathbf{p},r}{u}_{\mathbf{p},r}\left(x\right)+\mathrm{c}.\mathrm{c}.={\psi }_M^{*}\left(x\right)$$

(144)

$${\widehat{\psi }}_M\left(x\right)=\sum _{r=+,-}\int \mathrm{d}\mathbf{p}{b}_{\mathbf{p},r}{v}_{\mathbf{p},r}^{*}\left(x\right)+\mathrm{c}.\mathrm{c}.={\widehat{\psi }}_M^{*}\left(x\right)$$

(145)

$${\Psi }_D\left(x\right)=\sum _{r=+,-}\int \mathrm{d}\mathbf{p}[c_{\mathbf{p},r}{u}_{\mathbf{p},r}\left(x\right)+d_{\mathbf{p},r}^{*}{v}_{\mathbf{p},r}\left(x\right)]$$

(146)

where all the coefficients $a_{\mathbf{p},\pm },b_{\mathbf{p},\pm },c_{\mathbf{p},\pm },d_{\mathbf{p},\pm }$ are Grassmann valued complex numbers. As already noticed, the dynamics and the symmetry properties related to the above real and complex solutions of the Majorana real wave field equations (122) and (124) will be quite different and will be analyzed in the sequel, after the transition to the quantum theory. The complex solutions of the Majorana real wave equation are usually called Dirac bispinors, or spinors, whilst the real solutions are the self-conjugated Majorana bispinors, or spinors.

3.2.4. Canonical Quantum Theory

The transition to the quantum theory is achieved by passing from the complex Grassmann valued c-number coefficients $a_{\mathbf{p},\pm },b_{\mathbf{p},\pm },c_{\mathbf{p},\pm },d_{\mathbf{p},\pm }$ to the corresponding creation-annihilation operators, which satisfy the canonical anti-commutation relations, viz.,

$$\begin{array}{c}\hfill \{a_{\mathbf{p},r},a_{\mathbf{q},s}^{†}\}=\delta (\mathbf{p}-\mathbf{q}){\delta }_{rs}\{b_{\mathbf{p},r},b_{\mathbf{q},s}^{†}\}=\delta (\mathbf{p}-\mathbf{q}){\delta }_{rs}\end{array}$$

(147)

$$\begin{array}{c}\hfill \{c_{\mathbf{p},r},c_{\mathbf{q},s}^{†}\}=\delta (\mathbf{p}-\mathbf{q}){\delta }_{rs}\{d_{\mathbf{p},r},d_{\mathbf{q},s}^{†}\}=\delta (\mathbf{p}-\mathbf{q}){\delta }_{rs}\end{array}$$

(148)

all the other anti-commutators being null. In so doing, it appears that the Majorana quantum fields, which arise out of the real classical solutions ${\psi }_M\left(x\right)$ and ${\widehat{\psi }}_M\left(x\right),$ will become Hermitean and neutral quantum fields, whilst that one arising from ${\Psi }_M\left(x\right)$ will represent a charged quantum field. Consider in fact the neutral, Hermitean, self-conjugated quantum spinor field

$${\psi }_M\left(x\right)=\sum _{\mathbf{p},r}\left[a_{\mathbf{p},r}{u}_{\mathbf{p},r}\left(x\right)+a_{\mathbf{p},r}^{†}{u}_{\mathbf{p},r}^{*}\left(x\right)\right]={\left[{\psi }_M^{†}\left(x\right)\right]}^{\top }\equiv {\psi }_M^c\left(x\right)$$

(149)

$${\overline{\psi }}_M\left(x\right)={\psi }_M^{†}\left(x\right){\beta }_M=\sum _{\mathbf{p},r}\left[a_{\mathbf{p},r}^{†}{\overline{u}}_{\mathbf{p},r}\left(x\right)+a_{\mathbf{p},r}{u}_{\mathbf{p},r}^{\top }\left(x\right){\beta }_M\right]={\psi }_M^{\top }\left(x\right){\beta }_M$$

(150)

in which we have introduced the shorthand notation

$$\sum _{\mathbf{p},r}\equiv \int \mathrm{d}\mathbf{p}\sum _{r=+,-}$$

Here the dagger symbol means simultaneously full Hermitean conjugation in the Fock space, as well as in the complex bispinor part. Let us first evaluate its charge or lepton number

$$\begin{array}{ccc}& & \int \mathrm{d}\mathbf{x}\rho \left(x\right)=\int \mathrm{d}\mathbf{x}:{\psi }_M^{†}\left(x\right){\psi }_M\left(x\right):\hfill \\ & =& \int \mathrm{d}\mathbf{x}\sum _{\mathbf{p},r}\sum _{\mathbf{q},s}:\left[a_{\mathbf{p},r}^{†}{u}_{\mathbf{p},r}^{†}\left(x\right)+a_{\mathbf{p},r}{u}_{\mathbf{p},r}^{\top }\left(x\right)\right]\left[a_{\mathbf{q},s}{u}_{\mathbf{q},s}\left(x\right)+a_{\mathbf{q},s}^{†}{u}_{\mathbf{q},s}^{*}\left(x\right)\right]:\hfill \\ & =& \sum _{\mathbf{p},r}\sum _{\mathbf{q},s}\delta (\mathbf{p}-\mathbf{q}){\delta }_{rs}:a_{\mathbf{p},r}^{†}{a}_{\mathbf{q},s}+a_{\mathbf{p},r}{a}_{\mathbf{q},s}^{†}:=\sum _{\mathbf{p},r}\left(a_{\mathbf{p},r}^{†}{a}_{\mathbf{p},r}-a_{\mathbf{p},r}^{†}{a}_{\mathbf{p},r}\right)\equiv 0\hfill \end{array}$$

which endorses the neutral nature of the Hermitean and self-conjugate Majorana spinor quantum field. Needless to say, the above equality is a formal one and deserves a regularization procedure - such as the point-splitting e.g. - but it is eventually and essentially true, thanks to the normal ordering prescription and the canonical anticommutation relations.

Let us determine the canonical anticommutators for the Hermitean Majorana quantum field, which is an operator valued tempered distribution. In order to be as clear as possible, it is better to recall that the Majorana spinor quantum field is a Hermitean operator valued tempered distribution, which also lies in the four dimensional bispinor space ${\psi }_{M,\alpha }\left(x\right)(\alpha =1,2,3,4).$ More precisely, it turns out that the Majorana bispinor quantum field do actually live in a two dimensional subspace of the four dimensional linear bispinor space, as it satisfies the constraints

$$\begin{array}{c}\hfill {\mathfrak{E}}_M^{-}{\psi }_M\left(x\right)=0{\mathfrak{E}}_M^{+}{\psi }_M\left(x\right)={\psi }_M^c\left(x\right)\end{array}$$

(151)

$${\mathfrak{E}}_M^{\pm }\equiv (m\pm i\partial {/}_M)/2m$$

(152)

thanks to the basic relations (132) and (133). It is worthwhile to remark that the rank-two differential operators ${\mathfrak{E}}_M^{\pm }$ are real in the Majorana representation, owing to the purely imaginary nature of the gamma matrices, so that

$${\left[{\mathfrak{E}}_M^{\pm }\right]}^{*}=(m\mp i{\gamma }_M^{\nu *}{\partial }_{\nu })/2m={\mathfrak{E}}_M^{\pm }$$

Then we can write

$$\begin{array}{ccc}& & \{{\psi }_M\left(x\right),{\overline{\psi }}_M\left(y\right)\}=\{{\psi }_M\left(x\right),{\psi }_M^{†}\left(y\right)\}{\beta }_M\hfill \\ & =& \sum _{\mathbf{p},r}\sum _{\mathbf{q},s}\left\{\left[a_{\mathbf{p},r}{u}_{\mathbf{p},r}\left(x\right)+a_{\mathbf{p},r}^{†}{u}_{\mathbf{p},r}^{*}\left(x\right)\right],\left[a_{\mathbf{q},s}^{†}{u}_{\mathbf{q},s}^{†}\left(y\right)+a_{\mathbf{q},s}{u}_{\mathbf{q},s}^{\top }\left(y\right)\right]\right\}{\beta }_M\hfill \\ & =& \sum _{\mathbf{p},r}\sum _{\mathbf{q},s}\left[\{a_{\mathbf{p},r},a_{\mathbf{q},s}^{†}\}{u}_{\mathbf{p},r}\left(x\right)\otimes {\overline{u}}_{\mathbf{q},s}\left(y\right)+\{a_{\mathbf{p},r}^{†},a_{\mathbf{q},s}\}{u}_{\mathbf{p},r}^{*}\left(x\right)\otimes u_{\mathbf{q},s}^{\top }\left(y\right){\beta }_M\right]\hfill \\ & =& \sum _{\mathbf{p},r}\left[u_{\mathbf{p},r}\left(x\right)\otimes u_{\mathbf{p},r}^{†}\left(y\right)+u_{\mathbf{p},r}^{*}\left(x\right)\otimes u_{\mathbf{p},r}^{\top }\left(y\right)\right]{\beta }_M\hfill \\ & =& \int \mathrm{d}\mathbf{p}{\left[{\left(2\pi \right)}^{3}2{\omega }_{\mathbf{p}}\right]}^{-1}{\left[(p/+m)e^{-ip·(x-y)}+(p/-m)e^{-ip·(x-y)}\right]}_{p_0={\omega }_{\mathbf{p}}}\hfill \\ & =& (m+i\partial {/}_M)\int \mathrm{d}\mathbf{p}{\left[{\left(2\pi \right)}^{3}2{\omega }_{\mathbf{p}}\right]}^{-1}{\left[e^{-ip·(x-y)}-e^{ip·(x-y)}\right]}_{p_0={\omega }_{\mathbf{p}}}\hfill \\ & =& (-i)(m+i\partial {/}_M)D(x-y)\equiv S_M(x-y)\hfill \end{array}$$

(153)

where we have set $\partial {/}_M\equiv {\gamma }_M^{\mu }{\partial }_{\mu },$ while use has been made of the closure relation (143) whereas $D(x-y)$ is the well known Pauli-Jordan tempered distribution

$$D(x-y)={\displaystyle \frac{i}{{\left(2\pi \right)}^3}}\int {\mathrm{d}}^{4}kexp\{-ik·(x-y)\}\delta (k^2-m^2)\mathrm{sgn}\left(k_0\right)$$

(154)

As a matter of fact, the canonical spinor anticommutator $\{\psi \left(x\right),\overline{\psi }\left(y\right)\}=S(x-y)$ is generally and uniquely determined by 1. Lorentz covariance 2. the field equation $(m-i\partial /)S(x-y)=0$ and 3. the general spinor microcausality property

$${\gamma }_{0}S(x-y)={\partial }_{0}D(x-y){(x-y)}^2<0⟹S(0,\mathbf{x}-\mathbf{y})={\gamma }_0\delta (\mathbf{x}-\mathbf{y})$$

It is important to emphasize that the above mentioned noteworthy featured properties do not depend at all upon the specific Dirac matrices representations, so that they keep holding true for both Dirac and Majorana spinor quantum fields, in spite of their deep difference in the internal symmetries, the same conclusion being also valid, mutatis mutandis, for the off-shell causal Green’s functions or Feynman propagators. However, it turns out that only for the Majorana spinor we get $S_M(x-y)+S_M^{*}(x-y)=0,$ thanks to the Majorana spinor self-conjugation and the purely imaginary nature of the Dirac matrices in the Majorana representation.

A further neutral, self-conjugate, Hermitean quantum Majorana spinor can be obtained out of the classical solution (145), viz.,

$${\widehat{\psi }}_M\left(x\right)=\sum _{\mathbf{p},r}\left[b_{\mathbf{p},r}{v}_{\mathbf{p},r}^{*}\left(x\right)+b_{\mathbf{p},r}^{†}{v}_{\mathbf{p},r}\left(x\right)\right]={\left[{\widehat{\psi }}_M^{†}\left(x\right)\right]}^{\top }\equiv {\widehat{\psi }}_M^c\left(x\right)$$

(155)

$${\widehat{\psi }}_M^{†}\left(x\right){\beta }_M=\sum _{\mathbf{q},s}\left[b_{\mathbf{q},s}^{†}{v}_{\mathbf{q},s}^{\top }\left(x\right){\beta }_M+b_{\mathbf{q},s}{\overline{v}}_{\mathbf{q},s}\left(x\right)\right]$$

(156)

which again satisfies the canonical anticommutation relations

$$\{{\widehat{\psi }}_M\left(x\right),{\widehat{\psi }}_M^{†}\left(y\right)\}{\beta }_M=S_M(x-y)\mathrm{with}\{{\widehat{\psi }}_M\left(x\right),{\overline{\psi }}_M\left(y\right)\}=0$$

Consider now the non-Hermitean combination

$${\Psi }_D\left(x\right)=[{\psi }_M\left(x\right)+i{\widehat{\psi }}_M\left(x\right)]/\sqrt{2}$$

(157)

which still satisfies the Majorana wave equation (122). Its normal modes expansion reads

$$\begin{array}{ccc}\hfill {\Psi }_D\left(x\right)& =& 2^{-1/2}\sum _{\mathbf{p},r}\left[a_{\mathbf{p},r}{u}_{\mathbf{p},r}\left(x\right)+i{b}_{\mathbf{p},r}{v}_{\mathbf{p},r}^{*}\left(x\right)+a_{\mathbf{p},r}^{†}{u}_{\mathbf{p},r}^{*}\left(x\right)+i{b}_{\mathbf{p},r}^{†}{v}_{\mathbf{p},r}\left(x\right)\right]\hfill \\ & =& \sum _{\mathbf{p},r}\left[c_{\mathbf{p},r}{w}_{\mathbf{p},r}\left(x\right)+d_{\mathbf{p},r}^{†}{z}_{\mathbf{p},r}\left(x\right)\right]\hfill \end{array}$$

whence the identity follows

$$\begin{array}{c}\hfill a_{\mathbf{p},r}{u}_{\mathbf{p},r}\left(x\right)+i{b}_{\mathbf{p},r}{v}_{\mathbf{p},r}^{*}\left(x\right)=\sqrt{2}{c}_{\mathbf{p},r}{w}_{\mathbf{p},r}\left(x\right)\end{array}$$

(158)

$$\begin{array}{c}\hfill a_{\mathbf{p},r}^{†}{u}_{\mathbf{p},r}^{*}\left(x\right)+i{b}_{\mathbf{p},r}^{†}{v}_{\mathbf{p},r}\left(x\right)=\sqrt{2}{d}_{\mathbf{p},r}^{†}{z}_{\mathbf{p},r}\left(x\right)\end{array}$$

(159)

which is nothing but a local Bogoliubov transformation, as we shall check here below. As a matter of fact, after taking inner products with some suitable spin-state wave functions, it is possible to prove that the creation/annihilation operators $c_{\mathbf{p},r},c_{\mathbf{p},r}^{†},d_{\mathbf{q},s},d_{\mathbf{q},s}^{†}$ do still satisfy the canonical anticommutation relations, which endorses the local nature of the above Bogoliubov transformation.

Proof. Consider the inner product

$$\int \mathrm{d}\mathbf{x}{u}_{\mathbf{q},s}^{†}\left(x\right)\left[a_{\mathbf{p},r}{u}_{\mathbf{p},r}\left(x\right)+i{b}_{\mathbf{p},r}{v}_{\mathbf{p},r}^{*}\left(x\right)\right]=\sqrt{2}{c}_{\mathbf{p},r}\int \mathrm{d}\mathbf{x}{u}_{\mathbf{q},s}^{†}\left(x\right)w_{\mathbf{p},r}\left(x\right)$$

We find

$$\begin{array}{c}\hfill \int \mathrm{d}\mathbf{x}{u}_{\mathbf{q},s}^{†}\left(x\right)v_{\mathbf{p},r}^{*}\left(x\right)={\left(2{\omega }_{\mathbf{p}}\right)}^{-1}{u}_s^{†}\left(\mathbf{p}\right)v_r^{*}\left(\mathbf{p}\right)\delta (\mathbf{p}-\mathbf{q})=\frac{1}{2}{\xi }_s^{†}{\eta }_r^{*}\delta (\mathbf{p}-\mathbf{q})\\ \hfill {\left(2{\omega }_{\mathbf{p}}\right)}^{-1}{u}_s^{†}\left(\mathbf{p}\right)v_r^{*}\left(\mathbf{p}\right)={\left(2{\omega }_{\mathbf{p}}\right)}^{-1}{(2m+2{\omega }_{\mathbf{p}})}^{-1}{\xi }_s^{†}(m+{\omega }_{\mathbf{p}}{\beta }_M-p_k{\gamma }_M^k)(m+{\omega }_{\mathbf{p}}{\beta }_M+p_{\ell }{\gamma }_M^{\ell }){\eta }_r^{*}\\ \hfill ={\left(2{\omega }_{\mathbf{p}}\right)}^{-1}{(2m+2{\omega }_{\mathbf{p}})}^{-1}{\xi }_s^{†}[2{\omega }_{\mathbf{p}}^2+2m{\omega }_{\mathbf{p}}{\beta }_M+{\omega }_{\mathbf{p}}{\beta }_M{p}_{\ell }{\gamma }_M^{\ell }-p_k{\gamma }_M^k{\omega }_{\mathbf{p}}{\beta }_M]{\eta }_r^{*}\\ \hfill ={\left(2{\omega }_{\mathbf{p}}\right)}^{-1}{(2m+2{\omega }_{\mathbf{p}})}^{-1}{\xi }_s^{†}[2{\omega }_{\mathbf{p}}^2+2m{\omega }_{\mathbf{p}}+{\omega }_{\mathbf{p}}{p}_{\ell }{\gamma }_M^{\ell }-p_k{\gamma }_M^k{\omega }_{\mathbf{p}}]{\eta }_r^{*}=\frac{1}{2}{\xi }_s^{†}{\eta }_r^{*}\end{array}$$

because ${\xi }_s^{†}{\beta }_M={\xi }_s^{†},{\beta }_M{\eta }_r^{*}={\eta }_r^{*}$ and ${\xi }_s^{†}{\gamma }_M^k{\eta }_r^{*}={\xi }_s^{†}\{{\beta }_M,{\gamma }_M^k\}{\eta }_r^{*}\equiv 0.$ The two orthogonal pair ${\xi }_{\pm },{\eta }_{\pm }^{*}$ of the positive energy bispinors in the Majorana particle rest frame $\mathbf{p}=0$ are evidently connected by the $SO(2,\mathbb{R})$ transform (128) so that $\frac{1}{2}{\xi }_s^{†}{\eta }_r^{*}=\frac{1}{2}{\xi }_s^{†}{\left[F_{+}\left(\theta \right)\right]}_{rt}{\xi }_t\equiv f_{rs}\left(\theta \right)$ in which $f_{rs}^{-1}\left(\theta \right)=f_{sr}\left(\theta \right)=f_{rs}(-\theta )(r,s=+,-)$ and the summation over repeated indexes is understood. Then we obtain

$$a_{\mathbf{p},r}+i{b}_{\mathbf{p},t}{f}_{tr}\left(\theta \right)=\sqrt{2}{c}_{\mathbf{p},t}{f}_{tr}\left(\overline{\theta }\right)a_{\mathbf{q},s}^{†}-i{b}_{\mathbf{q},t}^{†}{f}_{ts}\left(\theta \right)=\sqrt{2}{c}_{\mathbf{q},t}^{†}{f}_{ts}\left(\overline{\theta }\right)$$

(160)

$$\sqrt{2}{c}_{\mathbf{p},r}=a_{\mathbf{p},t}{f}_{tr}(-\overline{\theta })+i{b}_{\mathbf{p},t}{f}_{tr}(\theta -\overline{\theta })\sqrt{2}{c}_{\mathbf{q},s}^{†}=a_{\mathbf{q},u}^{†}{f}_{us}(-\overline{\theta })-i{b}_{\mathbf{q},u}^{†}{f}_{us}(\theta -\overline{\theta })$$

(161)

$$\begin{array}{ccc}\hfill \{c_{\mathbf{p},r},c_{\mathbf{q},s}^{†}\}& =& \frac{1}{2}\{a_{\mathbf{p},t}{f}_{tr}(-\overline{\theta })+i{b}_{\mathbf{p},t}{f}_{tr}(\theta -\overline{\theta }),a_{\mathbf{q},u}^{†}{f}_{us}(-\overline{\theta })-i{b}_{\mathbf{q},u}^{†}{f}_{us}(\theta -\overline{\theta })\}\hfill \\ & =& \delta (\mathbf{p}-\mathbf{q}){\delta }_{rs}\hfill \end{array}$$

Analogous procedures drive to show the full validity of the canonical commutation relation $\{d_{\mathbf{p},r},d_{\mathbf{q},s}^{†}\}=\delta (\mathbf{p}-\mathbf{q}){\delta }_{rs},$ as well as the vanishing of all the remaining ones.

It follows that a Dirac spinor, which represents a charged and non-Hermitean quantum spinor field, can always be expressed as a complex combination of two independent neutral, self-conjugated and Hermitean Majorana quantum spinor fields, i.e.,

$${\psi }_D\left(x\right)=[{\psi }_M\left(x\right)+i{\widehat{\psi }}_M\left(x\right)]/\sqrt{2}$$

(162)

From the similarity transformation rule ${\gamma }^{\mu }=S{\gamma }_M^{\mu }{S}^{-1}$ we can go back to the Weyl representation of the gamma matrices and thereby

$${\psi }_D\left(x\right)=\sum _{\mathbf{p},r}\left[c_{\mathbf{p},r}S{w}_{\mathbf{p},r}\left(x\right)S^{-1}+d_{\mathbf{p},r}^{†}S{z}_{\mathbf{p},r}\left(x\right)S^{-1}\right]$$

(163)

The normal modes - in the Weyl representation of the gamma matrices - are provided by

$$\left\{\begin{array}{c}{w}_{\mathbf{p},r}\left(x\right)\equiv {\left[{\left(2\pi \right)}^{3}2{\omega }_{\mathbf{p}}\right]}^{-{\displaystyle \frac{1}{2}}}{w}_r\left(\mathbf{p}\right)exp\{-i{\omega }_{\mathbf{p}}t+i\mathbf{p}·\mathbf{x}\}\\ \\ z_{\mathbf{p},r}\left(x\right)\equiv {\left[{\left(2\pi \right)}^{3}2{\omega }_{\mathbf{p}}\right]}^{-{\displaystyle \frac{1}{2}}}{z}_r\left(\mathbf{p}\right)exp\{i{\omega }_{\mathbf{p}}t-i\mathbf{p}·\mathbf{x}\}\end{array}\right.(r=+,-)$$

(164)

where

$$w_r\left(\mathbf{p}\right)={(2m+2{\omega }_{\mathbf{p}})}^{-1/2}(m+p/){\varsigma }_r{z}_r\left(\mathbf{p}\right)={(2m+2{\omega }_{\mathbf{p}})}^{-1/2}(m-p/){\zeta }_r$$

(165)

where

$${\gamma }_0{\varsigma }_{\pm }={\varsigma }_{\pm }{\gamma }_0{\zeta }_{\pm }=-{\zeta }_{\pm }({\Sigma }_2\mp 1){\varsigma }_{\pm }=0({\Sigma }_2\pm 1){\zeta }_{\pm }=0$$

with ${\Sigma }_2=i{\gamma }_3{\gamma }_1.$ The above spin states do satisfy $z_{\pm }\left(\mathbf{p}\right)=i{\gamma }_2{w}_{\pm }^{*}\left(\mathbf{p}\right)$ and $w_{\pm }\left(\mathbf{p}\right)=i{\gamma }_2{z}_{\pm }^{*}\left(\mathbf{p}\right).$

Proof. After setting $N={(2m+2{\omega }_{\mathbf{p}})}^{-1/2}$ we find

$$\begin{array}{c}\hfill i{\gamma }_2{w}_{\pm }^{*}\left(\mathbf{p}\right)=iN{\gamma }_2(m+{\gamma }_0{\omega }_{\mathbf{p}}+{\gamma }_1{p}_x+{\gamma }_3{p}_z-{\gamma }_2{p}_y){\varsigma }_{\pm }^{*}=N(m-p/)i{\gamma }_2{\varsigma }_{\pm }^{*}\\ \hfill {\gamma }_{0}i{\gamma }_2{\varsigma }_{\pm }^{*}=-i{\gamma }_2{\varsigma }_{\pm }^{*}{\Sigma }_{2}i{\gamma }_2{\varsigma }_{\pm }^{*}=\mp i{\gamma }_2{\varsigma }_{\pm }^{*}⟹i{\gamma }_2{\varsigma }_{\pm }^{*}\equiv {\zeta }_{\pm }⟹z_{\pm }\left(\mathbf{p}\right)=i{\gamma }_2{w}_{\pm }^{*}\left(\mathbf{p}\right)\end{array}$$

Charge conjugation is conventionally defined as the operation in which particles and antiparticles are interchanged, up to an arbitrary overall phase factor. It follows thereby that if we set

$$\mathcal{C}{c}_{\mathbf{p},r}{\mathcal{C}}^{-1}={\mathrm{e}}^{i\eta }{d}_{\mathbf{p},r}\mathcal{C}{d}_{\mathbf{p},s}{\mathcal{C}}^{-1}={\mathrm{e}}^{-i\eta }{c}_{\mathbf{p},s}$$

(166)

$$\forall r,s=+,-\mathbf{p}\in {\mathbb{R}}^3$$

then the repeated application of the charge conjugation operation yields

$$\begin{array}{c}\hfill \mathcal{C}\left(\mathcal{C}{c}_{\mathbf{p},r}{\mathcal{C}}^{-1}\right){\mathcal{C}}^{-1}={\mathrm{e}}^{i\eta }\mathcal{C}{d}_{\mathbf{p},r}{\mathcal{C}}^{-1}=c_{\mathbf{p},r}\end{array}$$

so that we can always assume the following properties: namely,

$${\mathcal{C}}^2=\mathbb{I}⟹\mathcal{C}={\mathcal{C}}^{†}={\mathcal{C}}^{-1}$$

It turns out that the above defined spin-states do fulfill the remarkable relationship

$$z_{\pm }\left(\mathbf{p}\right)=i{\gamma }_2{w}_{\pm }^{*}\left(\mathbf{p}\right)w_{\pm }\left(\mathbf{p}\right)=i{\gamma }_2{z}_{\pm }^{*}\left(\mathbf{p}\right)$$

Hence, the charge conjugation transformation law can be rewritten as

$$\begin{array}{ccc}\hfill {\psi }_D^c\left(x\right)& =& \sum _{\mathbf{p},r}{\mathrm{e}}^{i\eta }\left[d_{\mathbf{p},r}{w}_{\mathbf{p},r}\left(x\right)+c_{\mathbf{p},r}^{†}{z}_{\mathbf{p},r}\left(x\right)\right]\hfill \\ & =& e^{i(\eta -\pi /2)}\sum _{\mathbf{p},r}\left[d_{\mathbf{p},r}{\gamma }^2{z}_{\mathbf{p},r}^{*}\left(x\right)+c_{\mathbf{p},r}^{†}{\gamma }^2{w}_{\mathbf{p},r}^{*}\left(x\right)\right]\hfill \\ & =& e^{i(\eta -\pi /2)}{\gamma }^2{\left({\psi }_D^{†}\left(x\right)\right)}^{\top }\hfill \end{array}$$

(167)

Thus, if we choose $\eta =\pi /2$ then we obtain the transformation rule

$${\psi }_D^c\left(x\right)={\gamma }^2{\left({\psi }_D^{†}\left(x\right)\right)}^{\top }$$

(168)

which precisely corresponds to the classical transformation ${\psi }^c\left(x\right)={\gamma }^2{\psi }^{*}\left(x\right).$

3.2.5. Observables

Let us now analyze and discuss the observables of a Majorana quantum field and the related properties of the Majorana particles, if eventually detected to any concern in some forthcoming experiment.

First of all, it is worthwhile to notice that from the orthogonality relations

$$\begin{array}{c}\hfill \left(u_{\mathbf{p},r},u_{\mathbf{q},s}\right)=\int \mathrm{d}\mathbf{x}{\overline{u}}_{\mathbf{p},r}\left(x\right){\beta }_M{u}_{\mathbf{q},s}\left(x\right)={\delta }_{rs}\delta (\mathbf{p}-\mathbf{q})\\ \hfill \left(u_{\mathbf{p},r},u_{\mathbf{q},s}^{*}\right)=\int \mathrm{d}\mathbf{x}{\overline{u}}_{\mathbf{p},r}\left(x\right){\beta }_M{u}_{\mathbf{q},s}^{*}\left(x\right)=0\\ \hfill \left(u_{\mathbf{p},r}^{*},u_{\mathbf{q},s}^{*}\right)=\int \mathrm{d}\mathbf{x}{u}_{\mathbf{p},r}^{\top }\left(x\right){\beta }_M{u}_{\mathbf{q},s}^{*}\left(x\right)=-{\delta }_{rs}\delta (\mathbf{p}-\mathbf{q})\end{array}$$

one can easily obtain all the observable quantities involving the Majorana massive quantum spinor field. For example the energy-momentum tetra-vector takes the form

$$\begin{array}{ccc}\hfill P_{\mu }& =& {\displaystyle \frac{i}{4}}\int \mathrm{d}\mathbf{x}:{\overline{\psi }}_M\left(x\right){\beta }_M{\overleftrightarrow{\partial }}_{\mu }{\psi }_M\left(x\right):\hfill \\ & =& {\displaystyle \frac{i}{4}}\int \mathrm{d}\mathbf{x}:\sum _{\mathbf{q},s}\left[a_{\mathbf{q},s}^{†}{\overline{u}}_{\mathbf{q},s}\left(x\right)+a_{\mathbf{q},s}{\overline{u}}_{\mathbf{q},s}^{*}\left(x\right)\right]\hfill \\ & \times & {\beta }_M{\overleftrightarrow{\partial }}_{\mu }\sum _{\mathbf{p},r}\left[a_{\mathbf{p},r}{u}_{\mathbf{p},r}\left(x\right)+a_{\mathbf{p},r}^{†}{u}_{\mathbf{p},r}^{*}\left(x\right)\right]:\hfill \\ & =& {\displaystyle \frac{1}{2}}\sum _{\mathbf{p},r}{p}_{\mu }:a_{\mathbf{p},r}^{†}{a}_{\mathbf{p},r}-a_{\mathbf{p},r}{a}_{\mathbf{p},r}^{†}:\hfill \\ & =& \sum _{\mathbf{p},r}{p}_{\mu }{a}_{\mathbf{p},r}^{†}{a}_{\mathbf{p},r}(p_0={\omega }_{\mathbf{p}})\hfill \end{array}$$

where the normal ordering prescription has been customarily adopted.

Concerning the total angular momentum operator, owing to the rotation invariance of the theory and without loss of generality, one can always arrange the Observer reference frame in such a manner that the direction of the motion of a Majorana particle lies along the $Oy-$axis. In such a circumstances, for example, if ${\partial }_x{\psi }_M={\partial }_z{\psi }_M=0$ that implies ${\psi }_M(t,y)={\psi }_M(t,0,y,0),$ then we obtain for the energy-momentum tensor

$$2i{T}^{13}(t,y)={\overline{\psi }}_M(t,y){\gamma }_M^1{\overleftrightarrow{\partial }}_z{\psi }_M(t,y)=0$$

$$2i{T}^{31}(t,z)={\overline{\psi }}_M(t,y){\gamma }_M^3{\overleftrightarrow{\partial }}_x{\psi }_M(t,y)=0$$

and thereby

$${\partial }_{\mu }{M}_{13}^{\mu }={\partial }_{\mu }{S}_{13}^{\mu }=0$$

whence it follows that the helicity - the projection of the spin angular momentum on the direction of the motion - is conserved in time. After insertion of the normal modes expansion one gets

$$\begin{array}{ccc}\hfill \mathrm{h}& =& {\int }_{-\infty }^{\infty }\mathrm{d}y:\frac{1}{2}{\psi }_M(t,y){\Sigma }_{M,2}{\psi }_M(t,y):\hfill \\ & =& {\int }_{-\infty }^{\infty }\mathrm{d}y:\sum _{p,r}\left[a_{p,r}{u}_{p,r}(t,y)+a_{p,r}^{†}{u}_{p,r}^{*}(t,y)\right]\hfill \\ & \times & \frac{1}{2}{\Sigma }_{M,2}\sum _{q,s}\left[a_{q,s}^{†}{u}_{q,s}^{*}(t,y)+a_{q,s}{u}_{q,s}(t,y)\right]:\hfill \end{array}$$

in which we have set

$$\mathbf{p}=(0,p,0)\mathbf{q}=(0,q,0){\omega }_p=\sqrt{p^2+m^2}$$

$$u_{p,r}(t,y)={\left[4\pi {\omega }_p\right]}^{-1/2}{u}_r\left(p\right)exp\{ipy-it{\omega }_p\}(r=+,-)$$

the normalization being now consistent with the occurrence that the spinor plane waves are independent of the transverse spatial coordinates $x_{\perp }=(x^1,x^3).$ From the commutation relation

$$[{\omega }_p{\gamma }_M^0-p{\gamma }_M^2,{\Sigma }_{M,2}]=0$$

together with the definition

$$u_{\pm }\left(p\right)\equiv {(2{\omega }_p+2m)}^{-1/2}\left(m+{\omega }_p{\gamma }_M^0-p{\gamma }_M^2\right){\xi }_{\pm }$$

it can be readily derived that

$$({\Sigma }_{M,2}\mp 1){\xi }_{\pm }=0\Rightarrow ({\Sigma }_{M,2}\mp 1)u_{\pm }\left(p\right)=0$$

which yields in turn

$$\mathrm{h}=\frac{1}{2}\hslash {\int }_{-\infty }^{\infty }\mathrm{d}p\left[a_{p,+}^{†}{a}_{p,+}-a_{p,-}^{†}{a}_{p,-}\right]$$

It follows therefrom that the 1-particle states $a_{\mathbf{p},\pm }^{†}|0\rangle$ do represent charge self-conjugated, or neutral, Majorana massive particles with energy-momentum $p^{\mu }=({\omega }_{\mathbf{p}},\mathbf{p})$ and two positive/negative helicity states, that has to be eventually detected with equal probability.3

Atiah-Singer index. We recall that, only for massless self-conjugated spinors, there exists an accidental U(1) internal symmetry for the Majorana quantum spinor field, the corresponding conserved pseudo-scalar charge being expressed by

$$Q_5=\pm \frac{1}{2}\int \mathrm{d}\mathbf{x}:{\overline{\psi }}_M(t,\mathbf{x}){\beta }_M{\gamma }_M^5\psi (t,\mathbf{x}):{\dot{Q}}_5=0$$

the overall sign being conventional and irrelevant. After insertion of the normal mode expansion we get

$$\begin{array}{ccc}\hfill Q_5& =& \pm \int \mathrm{d}\mathbf{x}:\sum _{\mathbf{q},s}\left[a_{\mathbf{q},s}^{†}{\overline{\upsilon }}_{\mathbf{q},s}\left(x\right)+a_{\mathbf{q},s}{\overline{\upsilon }}_{\mathbf{q},s}^{*}\left(x\right)\right]\hfill \\ & \times & \frac{1}{2}{\beta }_M{\gamma }_M^5\sum _{\mathbf{p},r}\left[a_{\mathbf{p},r}{\upsilon }_{\mathbf{p},r}\left(x\right)+a_{\mathbf{p},r}^{†}{\upsilon }_{\mathbf{p},r}^{*}\left(x\right)\right]\hfill \end{array}$$

where we have denoted by

$$\begin{array}{c}\hfill {\upsilon }_r\left(\mathbf{p}\right)=p{/}_M{\upsilon }_r/\sqrt{2\wp }(\wp \equiv |\mathbf{p}|,r=+,-)\\ \hfill {\upsilon }_{+}=〔\begin{array}{c}1\\ -i\\ 1\\ -i\end{array}〕{\upsilon }_{-}=〔\begin{array}{c}i\\ -1\\ i\\ -1\end{array}〕\\ \hfill {\upsilon }_{r,\mathbf{p}}\left(x\right)={[{\left(2\pi \right)}^{3}2\wp ]}^{-{\displaystyle \frac{1}{2}}}{\upsilon }_r\left(\mathbf{p}\right)exp\{-it\wp +i\mathbf{p}·\mathbf{x}\}\end{array}$$

the spin-states and plane wave-functions respectively of the massless self-conjugated spinor field. The spin-states satisfy the conventional orthogonality and closure relations as they correspond to the regular limit ${lim}_{m\to 0}{w}_r\left(\mathbf{p}\right)={\upsilon }_r\left(\mathbf{p}\right).$ Now we have

$$\begin{array}{c}\hfill {\gamma }_M^5=〔\begin{array}{cccc}0& 0& 0& i\\ 0& 0& -i& 0\\ 0& i& 0& 0\\ -i& 0& 0& 0\end{array}〕\end{array}$$

so that

$$\begin{array}{c}{\gamma }_M^5{\upsilon }_{\pm }=\pm {\upsilon }_{\pm }{\gamma }_M^5{\upsilon }_{\pm }\left(\mathbf{p}\right)=\mp {\upsilon }_{\pm }\left(\mathbf{p}\right)\end{array}$$

which means that the spin-states in the massless case are eigenstates of the chiral matrix in the Majorana representation.

It turns out that in the 1d case, i.e. the dynamics along e.g. the $Oy-$axis, which is the relevant situation for the construction of the helicity operator, we actually have

$${\upsilon }_{\pm }\left(p_y\right)=({\beta }_M\wp -{\gamma }_M^2{p}_y){\upsilon }_{\pm }/\sqrt{2\wp }[\wp =p_y\mathrm{sgn}\left(p_y\right)]$$

in such a manner that the above spin-states are common eigenstates of the chiral and spin matrices because

$$[{\beta }_M\wp -{\gamma }_M^2{p}_y,{\Sigma }_M^2]=0[{\Sigma }_M^2,{\gamma }_M^5]=0$$

Now we obtain

$$\begin{array}{c}\hfill \int \mathrm{d}\mathbf{x}{\upsilon }_{r,\mathbf{p}}^{†}(,\mathbf{x}){\gamma }_M^5{\upsilon }_{s,\mathbf{q}}^{*}(t,\mathbf{x})={\displaystyle \frac{1}{2\wp }}{\upsilon }_r^{†}\left(\mathbf{p}\right){\gamma }_M^5{\upsilon }_s^{*}(-\mathbf{p})exp\{2i\wp t\}\\ \hfill {\upsilon }_r^{†}\left(\mathbf{p}\right){\gamma }_M^5{\upsilon }_s^{*}(-\mathbf{p})=-{\displaystyle \frac{{\tilde{p}}^2}{2\wp }}{\upsilon }_r^{†}{\gamma }_M^5{\upsilon }_s^{*}\equiv 0(\forall r,s=+,-\vee {\tilde{p}}^{\nu }=p_{\nu })\end{array}$$

Moreover we find

$$\int \mathrm{d}\mathbf{x}{\upsilon }_{r,\mathbf{p}}^{†}(,\mathbf{x}){\gamma }_M^5{\upsilon }_{s,\mathbf{q}}(t,\mathbf{x})=\frac{1}{2}{\upsilon }_r^{†}{\gamma }_M^5{\upsilon }_s=\frac{1}{2}{\delta }_{rs}({\delta }_{r-}-{\delta }_{s+})(\forall r,s=+,-)$$

whence it follows, after a suitable choice of the overall sign, that the above introduced pseudo-scalar charge $Q_5$ is nothing but the quantum counterpart of the Atiah-Singer index for a Majorana spinor field

$$Q_5={\nu }_M\equiv \int \mathrm{d}\mathbf{p}\left(a_{\mathbf{p},+}^{†}{a}_{\mathbf{p},+}-a_{\mathbf{p},-}^{†}{a}_{\mathbf{p},-}\right)$$

i.e. the difference between the number of quanta with positive and negative helicity.

We can therefore conclude that the 1-particle states $a_{\mathbf{p},\pm }^{†}|0\rangle$ do represent neutral, i.e. charge self-conjugated, spin ${\displaystyle \frac{1}{2}}$ Majorana particles, massive or massless, with energy-momentum $p_{\mu }=({\omega }_{\mathbf{p}},\mathbf{p})$ and two equally probable positive/negative helicity states.

From the Majorana Lagrangian and related wave equation, one can immediately realize that the special distributions for the Majorana neutral spinor field are the very same as for a Dirac charged quantum field: namely,

$$\begin{array}{c}\hfill \left\{{\psi }_M\left(x\right),{\overline{\psi }}_M\left(y\right)\right\}=S(x-y;m)\\ \hfill \langle 0|T{\psi }_M\left(x\right){\overline{\psi }}_M\left(y\right)|0\rangle =S^F(x-y;m)\end{array}$$

the massless limit being smooth.

4. Dirac, Majorana and Weyl Fermions in 4D

In this Section we briefly summarize and highlight the main properties of the Dirac, Weyl and Majorana fermion fields that we have thoroughly discussed at length in the previous Sections.

Let us start from a few basic definitions and properties of spinors on a 4d Minkowski space. A 4-component Dirac fermion $\psi$ under a Lorentz transformation transforms as

$$\begin{array}{c}\hfill \psi \left(x\right)\to {\psi }^{\prime }\left(x^{\prime }\right)=exp\left[-{\displaystyle \frac{1}{2}}{\lambda }^{\mu \nu }{\Sigma }_{\mu \nu }\right]\psi \left(x\right),\end{array}$$

(169)

for $x^{\prime \mu }={\Lambda }^{\mu }{}_{\nu }{x}^{\nu }$. Here ${\lambda }^{\mu \nu }+{\lambda }^{\nu \mu }=0$ are six real canonical coordinates for the Lorentz group, ${\Sigma }_{\mu \nu }$ are the generators in the 4d reducible representation of Dirac bispinors, while ${{\Lambda }^{\mu }}_{\nu }$ are the Lorentz matrices in the irreducible vector representation $D({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}})$. The invariant kinetic Lagrangian for a free Dirac field is

$$\begin{array}{c}\hfill i\overline{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi =i\overline{\psi }\gamma ·\partial \psi \end{array}$$

(170)

where $\overline{\psi }={\psi }^{†}{\gamma }_0.$ A Dirac fermion admits a Lorentz invariant mass term $m\overline{\psi }\psi .$

A Dirac bispinor can be seen as the direct sum of two Weyl spinors4

$$\begin{array}{c}\hfill {\psi }_L=P_L\psi ,{\psi }_R=P_R\psi ,\mathrm{where}{P}_L={\displaystyle \frac{1-{\gamma }_5}{2}},P_R={\displaystyle \frac{1+{\gamma }_5}{2}}\end{array}$$

with opposite chiralities

$$\begin{array}{c}\hfill {\gamma }_5{\psi }_L=-{\psi }_L,{\gamma }_5{\psi }_R={\psi }_R.\end{array}$$

A left-handed Weyl fermion admits a Lagrangian kinetic term

$$\begin{array}{c}\hfill i({\psi }_L,\gamma ·\partial {\psi }_L)\equiv {\overline{\psi }}_L{\gamma }^{\mu }{\partial }_{\mu }{\psi }_L\end{array}$$

(171)

but not a mass term, because $({\psi }_L,{\psi }_L)=0$, since ${\gamma }_5{\gamma }^0+{\gamma }^0{\gamma }_5=0$. Hence, a Weyl fermion is massless and this property is protected by chirality conservation.

For classical Majorana spinors we need the notion of Lorentz covariant charge conjugate spinor ${\psi }^c$

$$\begin{array}{c}\hfill {\psi }^c=\mathcal{C}\psi {\mathcal{C}}^{-1}={\eta }_C{\gamma }^2{\psi }^{*}\end{array}$$

(172)

where ${\eta }_C$ is a phase which, for simplicity, in the sequel we set equal to 1. It is not hard to show that if $\psi$ transforms like (169), then

$$\begin{array}{c}\hfill {\psi }^c\left(x\right)\to exp\left[-{\displaystyle \frac{1}{2}}{\lambda }^{\mu \nu }{\Sigma }_{\mu \nu }\right]{\psi }^c\left(x\right).\end{array}$$

(173)

Therefore one can impose on $\psi$ the condition

$$\begin{array}{c}\hfill \psi ={\psi }^c\end{array}$$

(174)

because both sides transform in the same way. By definition, a spinor satisfying (174) is a Majorana spinor. As we have seen before it admits both kinetic and mass term.

The Lorentz group spinor representation properties are as follows: the chiral matrix ${\gamma }_5$ commutes with Lorentz transformations matrix $exp\left[-{\displaystyle \frac{1}{2}}{\lambda }^{\mu \nu }{\Sigma }_{\mu \nu }\right]$ and so do $P_L$ and $P_R.$ This means that the Dirac representation is reducible. Multiplying the spinors by $P_L$ and $P_R$ selects the Weyl irreducible representations. To state it more precisely, Weyl representations are irreducible representations of the group $SL(2,C)$, which is the covering group of the proper ortochronous Lorentz group. They are usually denoted $({\displaystyle \frac{1}{2}},0)$ and $(0,{\displaystyle \frac{1}{2}})$ in the $SU\left(2\right)\times SU\left(2\right)$ notation of the $SL(2,C)$ irreducible representations. As we have seen in (173), Lorentz transformations commute also with the charge conjugation operation (172). This also implies that Dirac spinors are reducible and suggests another possible reduction: by imposing (174) we single out another irreducible representation, the Majorana one. The Majorana representation is the minimal irreducible representation of a (one out of eight) covering of the complete Lorentz group [17,18]. It is evident, and well-known, that the Majorana and Weyl representations in 4d are incompatible.

Let us consider next the charge conjugation and parity discrete symmetries and recall the relevant properties of a Weyl spinor. We have

$$\begin{array}{c}\hfill \mathcal{C}{\psi }_L{\mathcal{C}}^{-1}=P_L\mathcal{C}\psi {\mathcal{C}}^{-1}=P_L{\psi }^c={\left({\psi }^c\right)}_L\end{array}$$

(175)

The charge conjugate of a Majorana field is, by definition, itself. While the Action of a Majorana field is invariant under charge conjugation, for a Weyl spinor we have [ ${\gamma }^{\mu }{\partial }_{\mu }\equiv \gamma ·\partial$ ]

$$\begin{array}{c}\hfill \mathcal{C}\left(\int \overline{{\psi }_L}i\gamma ·\partial {\psi }_L\right){\mathcal{C}}^{-1}=\int \overline{{\left({\psi }^c\right)}_L}i\gamma ·\partial {\psi }_L^c=\int \overline{{\psi }_R}i\gamma ·\partial {\psi }_R\end{array}$$

(176)

i.e. a Weyl fermion is, so to say, maximally non-invariant.

The parity operation is defined by

$$\begin{array}{c}\hfill \mathcal{P}{\psi }_L(t,\overrightarrow{x}){\mathcal{P}}^{-1}={\eta }_P{\gamma }_0{\psi }_R(t,-\overrightarrow{x})\end{array}$$

(177)

where ${\eta }_P$ is a phase, which we set to 1. In terms of the Action we have

$$\begin{array}{c}\hfill \mathcal{P}\left(\int \overline{{\psi }_L}i\gamma ·\partial {\psi }_L\right){\mathcal{P}}^{-1}=\int \overline{{\psi }_R}i\gamma ·\partial {\psi }_R\end{array}$$

(178)

For a Majorana spinor the Action is parity invariant.

This also suggests a useful representation for a Majorana fermion. Let ${\psi }_R=P_R\psi$ be a generic Weyl spinor: as seen above, we have $P_R{\psi }_R={\psi }_R$ and it is easy to prove that $P_L{\left({\psi }_R\right)}^c={\left({\psi }_R\right)}^c$, i.e. ${\left({\psi }_R\right)}^c$ is left-handed. Therefore the sum ${\psi }_M={\psi }_R+{\left({\psi }_R\right)}^c$ is a Majorana spinor because it satisfies (174) and any Majorana fermion can be represented in this way.

Considering next the CP discrete symmetry. From the above it follows that the Action of a Majorana spinor is obviously invariant under it. On the other hand for a Weyl spinor we have

$$\begin{array}{c}\hfill \mathcal{CP}{\psi }_L(t,\overrightarrow{x}){\left(\mathcal{CP}\right)}^{-1}={\gamma }_0{P}_R{\psi }^c(t,-\overrightarrow{x})={\gamma }_0{\left({\psi }^c\right)}_R(t,-\overrightarrow{x})\end{array}$$

(179)

Applying now the CP transformation to the Action for a Weyl spinor, one gets

$$\begin{array}{ccc}& & \mathcal{CP}\int \overline{{\psi }_L}i\gamma ·\partial {\psi }_L{\left(\mathcal{CP}\right)}^{-1}=\int \overline{{\left({\psi }^c\right)}_R}(t,-\overrightarrow{x})i{\gamma }^{†}·\partial {\left({\psi }^c\right)}_R(t,-\overrightarrow{x})\hfill \\ & =& \int \overline{{\left({\psi }^c\right)}_R}(t,\overrightarrow{x})i\gamma ·\partial {\left({\psi }^c\right)}_R(t,\overrightarrow{x})\hfill \end{array}$$

(180)

But one can prove as well that

$$\begin{array}{c}\hfill \int \overline{{\left({\psi }^c\right)}_R}(t,\overrightarrow{x})i\gamma ·\partial {\left({\psi }^c\right)}_R(t,\overrightarrow{x})=\int \overline{{\psi }_L}\left(x\right)i\gamma ·\partial {\psi }_L\left(x\right)\end{array}$$

(181)

It follows therefore that the Action for a Weyl spinor is CP invariant. Moreover, as we have seen in the previous Section, it is also, separately, T invariant and so CPT invariant.

The transformation properties of the Weyl and Majorana spinor fields are summarized in the following table:

$$\begin{array}{c}\hfill \begin{array}{cccc}& \mathrm{Majorana}& & \mathrm{Weyl}\\ \mathrm{P}:& \mathcal{P}\psi (t,\overrightarrow{x}){\mathcal{P}}^{-1}={\gamma }_0\psi (t,-\overrightarrow{x})& & \mathcal{P}{\psi }_L(t,\overrightarrow{x}){\mathcal{P}}^{-1}={\gamma }_0{\psi }_R(t,-\overrightarrow{x})\\ \mathrm{C}:& \mathcal{C}\psi {\mathcal{C}}^{-1}={\gamma }^2{\psi }^{*}=\psi & & \mathcal{C}{\psi }_L{\mathcal{C}}^{-1}=P_L{\psi }^c={\left({\psi }^c\right)}_L\\ \mathrm{CP}:& \mathcal{CP}\psi (t,\overrightarrow{x}){\left(\mathcal{CP}\right)}^{-1}=\psi (t,-\overrightarrow{x})& & \mathcal{CP}{\psi }_L(t,\overrightarrow{x}){\left(\mathcal{CP}\right)}^{-1}={\gamma }_0{\left({\psi }^c\right)}_R(t,-\overrightarrow{x})\end{array}\end{array}$$

(182)

In the conventional Standard Model of the fundamental interactions the neutrino and antineutrino fields are treated as massless chiral Weyl bispinors: according to our conventions and notations we can set

$$\psi \left(x\right)={\nu }_L\left(x\right)+{\nu }_R\left(x\right)=\int \mathrm{d}\mathbf{p}\left[c_{-,\mathbf{p}}{u}_{-,\mathbf{p}}\left(x\right)+d_{+,\mathbf{p}}^{†}{v}_{+,\mathbf{p}}\left(x\right)\right]+{\nu }_R\left(x\right)$$

(183)

where $u_{-},v_{+}$ are fixed and independent left- and right-handed spinors that, without loss of generality, we can identify with those ones of eq. (36). It is worthwhile to stress that, in spite of being a spin ${\displaystyle \frac{1}{2}}$ field, the left-handed neutrino field necessarily exhibits positive frequency plane waves with only negative helicity and negative frequency plane waves with solely positive helicity. Contrary to some popular belief, this characteristic striking constraint for the Weyl spinors ${\psi }_{\mp }$ is not merely due to their masslessness, because Majorana and Dirac massless spinors do behave quite differently.

We can interpret the Fourier expansion (183) of the Standard Model neutrino field as follows: $c_{-,\mathbf{p}}$ annihilates a left-handed particle with negative helicity, while $d_{+,\mathbf{p}}^{†}$ creates a right-handed anti-particle with positive helicity. Needless to say, it appears to be evident that in this game right-handed neutrinos ${\nu }_R$ and left-handed anti-neutrinos play only the role of sterile and non-interacting spectators, this role being confined to provide a consistent causal Green’s function: it turns out that the latter have never been detected neither in laboratory experiments, nor in astrophysical or cosmological observations.

5. Regularizations for Weyl Spinors

The classical Lagrange density for a Weyl (left) spinor in the four component formalism

$$\psi \left(x\right)={\chi }_L\left(x\right)=〔\begin{array}{c}\chi \left(x\right)\\ 0\end{array}〕$$

reads

$$\mathcal{K}\left(x\right)=\overline{\psi }\left(x\right)i\partial /\psi \left(x\right)={\chi }_L^{†}\left(x\right){\alpha }^{\nu }i{\partial }_{\nu }{\chi }_L\left(x\right)$$

It follows that the corresponding matrix valued Weyl differential operator

$$w_L\equiv {\alpha }^{\nu }i{\partial }_{\nu }{P}_L[P_L=\frac{1}{2}(\mathbb{I}-{\gamma }_5)]$$

is singular and does not possess any rank-four inverse. After minimal coupling with a real massless vector field $A^{\mu }\left(x\right)$ we come to the classical Lagrangian

$$\mathcal{L}={\chi }_L^{†}{\alpha }^{\nu }i{\partial }_{\nu }{\chi }_L+g{A}^{\nu }{\chi }_L^{†}{\alpha }_{\nu }{\chi }_L-\frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }$$

where $0\le g<1$ is a non-negative small parameter, while $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }.$ It turns out that the classical Action

$$S=\int {\mathrm{d}}^{4}x\mathcal{L}$$

is invariant under the Poincaré group, as well as under the internal U(1) phase transformations ${\chi }_L\left(x\right)\mapsto e^{ig\theta }{\chi }_L\left(x\right).$ The Action is invariant under the so called scale or dilatation transformations, viz.,

$$\begin{array}{c}\hfill x^{\prime \mu }=e^{-\varrho }{x}^{\mu }{\chi }_L^{\prime }\left(x\right)=e^{{\displaystyle \frac{3}{2}}\varrho }{\chi }_L\left(e^{\varrho }x\right)A^{\prime \mu }\left(x\right)=e^{\varrho }{A}^{\mu }\left(e^{\varrho }x\right)\end{array}$$

with $\varrho \in \mathbb{R}$, as well as with respect to the local phase or gauge transformations

$${\chi }_L^{\prime }\left(x\right)=e^{ig\theta \left(x\right)}{\chi }_L\left(x\right)A_{\nu }^{\prime }\left(x\right)=A_{\nu }\left(x\right)+{\partial }_{\nu }\theta \left(x\right)$$

which amounts to the ordinary U(1) phase transform in the limit of constant phase. It follows therefrom that there are twelve conserved charges in this model at the classical level and, in particular, owing to scale and gauge invariance, no mass term is allowed for both spinor and vector fields. The question naturally arises if those symmetries hold true after the transition to the quantum theory and, in particular, if they are protected against loop radiative corrections within the perturbative approach. Now, in order to develop perturbation theory, one has to face the problem of the lack of an inverse for both the Weyl and gauge fields, owing to chirality and gauge invariance. In order to solve this problem, it is expedient to add to the Lagrangian non-interacting terms, which are fully decoupled from any physical quantity. These terms break chirality and gauge invariance, albeit in a harmless way, just to allow the setting up of a Feynman propagator, or causal Green’s functions, for both the Weyl and gauge quantum fields. The simplest choice, which keeps Poincaré and internal U(1) phase change symmetries, is provided by

$${\mathcal{L}}^{\prime }={\phi }_R^{†}{\alpha }^{\nu }i{\partial }_{\nu }{\phi }_R-\frac{1}{2}{(\partial ·A)}^2$$

where

$$\psi \left(x\right)={\phi }_R\left(x\right)=〔\begin{array}{c}0\\ \phi \left(x\right)\end{array}〕$$

is a left-chirality breaking right-handed Weyl spinor field. Notice en passant that the modified Lagrangian $\mathcal{L}+{\mathcal{L}}^{\prime }$ does exhibit the further U(1) internal symmetry under the so called chiral phase transformations

$${\psi }^{\prime }\left(x\right)=(cos\theta +igsin\theta {\gamma }_5)\psi \left(x\right)\psi \left(x\right)=〔\begin{array}{c}\chi \left(x\right)\\ \phi \left(x\right)\end{array}〕$$

so that the modified theory involves another conserved charge at the classical level. From the modified Lagrange density we get the Feynman propagators for the massless Dirac field $\psi \left(x\right)$, as well as for the massless vector field in the so called Feynman gauge: namely,

$$S\left(p\right)={\displaystyle \frac{ip/}{p^2+i\epsilon }}{D}_{\mu \nu }\left(k\right)={\displaystyle \frac{-i{g}_{\mu \nu }}{k^2+i\epsilon }}$$

and the vertex $ig{\gamma }^{\nu }{P}_L,$ with $k+p-q=0$, which involves a vector particle of momentum k and a Weyl pair of particle and anti-particle of momenta p and q respectively and of opposite helicity.5

The lowest order 1-loop correction to the Weyl kinetic term $p/P_L$ is formally provided by the Feynman rules in the Minkowski space, viz.,

$${\Sigma }_2(p/)=-i{g}^2\int {\displaystyle \frac{{\mathrm{d}}^4\ell }{{\left(2\pi \right)}^4}}{\gamma }^{\mu }{D}_{\mu \nu }(p-\ell )S(\ell ){\gamma }^{\nu }{P}_L$$

By naïve power counting the above 1-loop integral turns out to be UV divergent. Hence a regularization procedure is mandatory to give a meaning and evaluate the radiative correction ${\Sigma }_2(p/)$ to the Weyl kinetic operator. Here in the sequel we shall examine in detail the dimensional, Pauli-Villars and UV cut-off regularizations.

5.0.1. Dimensional Regularization

In a $2\omega -$dimensional space-time the radiative correction to the Weyl kinetic term takes the form

$$\mathtt{reg}{\Sigma }_2(p/)=-i{g}^2{\mu }^{2ϵ}\int {\displaystyle \frac{{\mathrm{d}}^{2\omega }\ell }{{\left(2\pi \right)}^{2\omega }}}{D}_{\mu \nu }(\ell ){\gamma }^{\mu }S(\ell +p){\gamma }^{\nu }{P}_L$$

where $ϵ=2-\omega >0$ is the shift with respect to the Minkowski space. Since the above expression is traceless and has the canonical engineering dimension of a mass in natural units, it is quite apparent that the latter cannot generate any mass term, which would be proportional to the unit matrix. Hence, mass is forbidden and it is convenient to set and evaluate

$$\begin{array}{ccc}\hfill \mathtt{reg}{\Sigma }_2(p/)& \equiv & f\left(p^2\right)p/P_L\mathrm{tr}[p/\mathtt{reg}{\Sigma }_2(p/)]=\frac{1}{2}{2}^{\omega }{p}^{2}f\left(p^2\right)\hfill \\ \hfill \mathrm{tr}[p/\mathtt{reg}{\Sigma }_2(p/)]& =& g^2{\mu }^{2ϵ}{\left(2\pi \right)}^{-2\omega }\int {\mathrm{d}}^{2\omega }\ell {\displaystyle \frac{(-i)\mathrm{tr}\left(p/{\gamma }^{\lambda }\ell /{\gamma }_{\lambda }{P}_L\right)}{\left[{(\ell -p)}^2+i\epsilon \right]\left({\ell }^2+i\epsilon \right)}}\hfill \end{array}$$

Concerning dimensional regularization, we collect here below the definitions and key properties for the $2^{\omega }\times 2^{\omega }$$\gamma -$matrices in a $2\omega -$dimensional space-time with a Minkowski signature

$$\begin{array}{ccc}& & {\gamma }^{\mu }=\left\{\begin{array}{cc}{\overline{\gamma }}^{\mu }& \mu =0,1,2,3\\ {\widehat{\gamma }}^{\mu }& \mu =4,\dots ,2\omega -4\end{array}\right.\hfill \\ & & \left\{{\overline{\gamma }}^{\mu },{\overline{\gamma }}^{\nu }\right\}=2{\overline{g}}^{\mu \nu }\mathbb{I}\left\{{\widehat{\gamma }}^{\mu },{\widehat{\gamma }}^{\nu }\right\}=2{\widehat{g}}^{\mu \nu }\mathbb{I}\left\{{\overline{\gamma }}^{\mu },{\widehat{\gamma }}^{\nu }\right\}=0\hfill \\ & & ∥\overline{g}∥=\mathrm{diag}(+,-,-,-)∥\widehat{g}∥=-\widehat{\mathbb{I}}\hfill \\ & & {\gamma }_5\equiv i{\overline{\gamma }}^0{\overline{\gamma }}^1{\overline{\gamma }}^2{\overline{\gamma }}^3{\gamma }_5^2=\mathbb{I}\left\{{\overline{\gamma }}^{\mu },{\gamma }_5\right\}=0=\left[{\widehat{\gamma }}^{\mu },{\gamma }_5\right]\hfill \end{array}$$

Taking all the above listed equations into account, it is not difficult to check the following trace formulas: namely,

$$\begin{array}{ccc}\hfill \mathrm{tr}\left({\gamma }^{\mu }{\gamma }^{\nu }\right)& =& g^{\mu \nu }\mathrm{tr}\mathbb{I}=2^{\omega }{g}^{\mu \nu }\hfill \\ \hfill 2^{-\omega }\mathrm{tr}\left({\gamma }^{\kappa }{\gamma }^{\lambda }{\gamma }^{\mu }{\gamma }^{\nu }\right)& =& g^{\kappa \lambda }{g}^{\mu \nu }-g^{\kappa \mu }{g}^{\lambda \nu }+g^{\kappa \nu }{g}^{\lambda \mu }\hfill \\ \hfill \mathrm{tr}\left({\overline{\gamma }}^{\kappa }{\overline{\gamma }}^{\lambda }{\widehat{\gamma }}^{\mu }{\widehat{\gamma }}^{\nu }\right)& =& 2^{\omega }{\overline{g}}^{\kappa \lambda }{\widehat{g}}^{\mu \nu }\hfill \\ \hfill \mathrm{tr}\left({\gamma }_5{\overline{\gamma }}^{\mu }{\overline{\gamma }}^{\lambda }{\overline{\gamma }}^{\rho }{\overline{\gamma }}^{\nu }\right)& =& -i{2}^{\omega }{ϵ}^{\mu \lambda \rho \nu }\hfill \end{array}$$

Traces involving an odd number of Dirac matrices do vanish. Then we get

$$\mathrm{tr}\left(p/{\gamma }^{\lambda }\ell /{\gamma }_{\lambda }{P}_L\right)=2^{\omega }(ϵ-1)p·\ell$$

and thereby

$$p^{2}f\left(p^2\right)=i{g}^2{\mu }^{2ϵ}{\displaystyle \frac{ϵ-1}{{\left(2\pi \right)}^{2\omega }}}\int {\displaystyle \frac{2p·\ell {\mathrm{d}}^{2\omega }\ell }{\left[{(\ell -p)}^2+i\epsilon \right]\left({\ell }^2+i\epsilon \right)}}$$

Turning to the Feynman parametric representation we obtain

$$p^{2}f\left(p^2\right)=i{g}^2{\mu }^{2ϵ}{\displaystyle \frac{ϵ-1}{{\left(2\pi \right)}^{2\omega }}}{\int }_0^1\mathrm{d}x\int {\displaystyle \frac{2p·\ell {\mathrm{d}}^{2\omega }\ell }{{\left[{\ell }^2-2xp·\ell +x{p}^2+i\epsilon \right]}^2}}$$

Completing the square in the denominator and after shifting the momentum ${\ell }^{\prime }\equiv \ell -xp,$ dropping the linear term in ${\ell }^{\prime }$ in the numerator owing to symmetric integration, we have

$$f\left(p^2\right)=2i{g}^2{\mu }^{2ϵ}{\displaystyle \frac{ϵ-1}{{\left(2\pi \right)}^{2\omega }}}{\int }_0^1\mathrm{d}xx\int {\displaystyle \frac{{\mathrm{d}}^{2\omega }\ell }{{\left[{\ell }^2+x(1-x)p^2+i\epsilon \right]}^2}}$$

When $p^2<0$ one can perform the Wick rotation and readily get the result

$$\begin{array}{ccc}\hfill f\left(p^2\right)& =& -2g^2{\mu }^{2ϵ}{\displaystyle \frac{ϵ-1}{{\left(4\pi \right)}^{\omega }}}{\int }_0^1\mathrm{d}xx{\int }_0^{\infty }\mathrm{d}\tau {\tau }^{ϵ-1}{e}^{-\tau x(1-x)p_E^2}\hfill \\ & =& 2{\left({\displaystyle \frac{g}{4\pi }}\right)}^2[\Gamma \left(ϵ\right)-\Gamma (1+ϵ)]{\left(-{\displaystyle \frac{4\pi {\mu }^2}{p^2}}\right)}^{ϵ}{\int }_0^1\mathrm{d}x{x}^{1-ϵ}{(1-x)}^{-ϵ}\hfill \\ & =& 2{\left({\displaystyle \frac{g}{4\pi }}\right)}^2[\Gamma \left(ϵ\right)-\Gamma (1+ϵ)]{\left(-{\displaystyle \frac{4\pi {\mu }^2}{p^2}}\right)}^{ϵ}B(2-ϵ,1-ϵ)\hfill \end{array}$$

Expansion around $ϵ=0$ yields

$$\begin{array}{ccc}\hfill f\left(p^2\right)& =& {\left({\displaystyle \frac{g}{4\pi }}\right)}^2\left[{\displaystyle \frac{1}{ϵ}}-\mathbf{C}-1+\dots \right]\left[1+ϵln\left(-{\displaystyle \frac{4\pi {\mu }^2}{p^2}}\right)+\dots \right]\hfill \\ & \times & \left\{1-ϵ\left[\psi \right(2)+\psi (1)-2\psi (3\left)\right]+\dots \right\}\hfill \\ & =& {\left({\displaystyle \frac{g}{4\pi }}\right)}^2\left[{\displaystyle \frac{1}{ϵ}}+1+3\mathbf{C}+ln\left(-{\displaystyle \frac{4\pi {\mu }^2}{p^2}}\right)\right]+\mathrm{evanescent}\hfill \end{array}$$

5.0.2. Pauli-Villars Regularization

Let us now repeat the very same calculation for the left Weyl spinor self-energy in the Pauli-Villars regularization. The latter is simply implemented by the following replacement of the massless Dirac propagator

$$\mathtt{reg}{\Sigma }_2(p/)=-i{g}^2\int {\displaystyle \frac{{\mathrm{d}}^4\ell }{{\left(2\pi \right)}^4}}{\gamma }^{\mu }{D}_{\mu \nu }(p-\ell ){\displaystyle \sum _{s=0}^S}{C}_{s}S(\ell ,M_s){\gamma }^{\nu }{P}_L$$

where $M_0=0,C_0=1$ while $\{M_s\equiv {\lambda }_{s}M|{\lambda }_s\gg 1(s=1,2,\dots ,S)\}$ is a collection of very large auxiliary masses. The set of constants $C_s$ will be suitably selected, as we shall see in the sequel, in such a manner to obtain a specific and mathematically meaningful form for the ultraviolet divergences that will manifest themselves in the limit ${\lambda }_s\to \infty .$ Since we have

$$S(\ell ,M_s)={\displaystyle \frac{i(\ell /+M_s)}{{\ell }^2-M_s^2+i\epsilon }}(s=1,2,\dots ,S)$$

it is now convenient to set

$$\mathtt{reg}{\Sigma }_2(p/)\equiv f\left(p^2\right)p/P_L-M\upsilon \left(p^2\right)$$

in such a manner that we can write

$$\begin{array}{ccc}\hfill \mathrm{tr}\left[\mathtt{reg}{\Sigma }_2(p/)\right]& =& -4M\upsilon \left(p^2\right)\hfill \\ & =& -i{g}^2\int {\displaystyle \frac{{\mathrm{d}}^4\ell }{{\left(2\pi \right)}^4}}{\displaystyle \sum _{s=1}^S}{C}_s{\displaystyle \frac{8M_s}{[{(\ell -p)}^2+i\epsilon ]\left({\ell }^2-M_s^2+i\epsilon \right)}}\hfill \\ \hfill \mathrm{tr}\left[p/\mathtt{reg}{\Sigma }_2(p/)\right]& =& 2p^{2}f\left(p^2\right)\hfill \\ & =& i{g}^2\int {\displaystyle \frac{{\mathrm{d}}^4\ell }{{\left(2\pi \right)}^4}}{\displaystyle \sum _{s=0}^S}{C}_s{\displaystyle \frac{4p·\ell }{[{(\ell -p)}^2+i\epsilon ]\left({\ell }^2-M_s^2+i\epsilon \right)}}\hfill \end{array}$$

Once again, if we take profit of the Feynman parametric formula we obtain

$$\begin{array}{ccc}\hfill \upsilon \left(p^2\right)& =& \int {\displaystyle \frac{{\mathrm{d}}^4\ell }{{\left(2\pi \right)}^4}}{\displaystyle \sum _{s=1}^S}{C}_s{\int }_0^1\mathrm{d}x{\displaystyle \frac{2i{g}^2{\lambda }_s}{{\left[{\ell }^2-2xp·\ell +x{p}^2-(1-x)M_s^2+i\epsilon \right]}^2}}\hfill \\ & =& {\int }_0^1\mathrm{d}x\int {\displaystyle \frac{{\mathrm{d}}^4{\ell }^{\prime }}{{\left(2\pi \right)}^4}}{\displaystyle \sum _{s=1}^S}{\displaystyle \frac{2i{g}^2{C}_s{\lambda }_s}{{\left[{\ell }^{\prime 2}+x(1-x)p^2-(1-x)M_s^2+i\epsilon \right]}^2}}\hfill \\ & =& -2g^2{\int }_0^1\mathrm{d}x\int {\displaystyle \frac{{\mathrm{d}}^4{\ell }_E}{{\left(2\pi \right)}^4}}{\displaystyle \sum _{s=1}^S}{\displaystyle \frac{C_s{\lambda }_s}{{\left[{\ell }_E^2+x(1-x)p_E^2+(1-x)M_s^2\right]}^2}}\hfill \end{array}$$

In order to proceed further on it is convenient to consider the generating integral

$$I_n\left(z_E\right)\equiv {(-1)}^n\int {\displaystyle \frac{{\mathrm{d}}^4{\ell }_E}{{\left(2\pi \right)}^4}}{\displaystyle \sum _{s=1}^S}{C}_s{\lambda }_s{\displaystyle \frac{exp\{i{\ell }_E·z_E\}}{{({\ell }_E^2+{\Delta }_s^2)}^n}}$$

where ${\Delta }_s^2=(1-x)[x{p}_E^2+M^2{\lambda }_s^2]>0,\forall s=1,2,\dots ,S.$ Explicit calculation yields

$$I_n\left(z_E\right)={\displaystyle \frac{{(-1)}^n}{8{\pi }^2\Gamma \left(n\right)}}{\displaystyle \sum _{s=1}^S}{C}_s{\lambda }_s{\left({\displaystyle \frac{2{\Delta }_s}{|z_E|}}\right)}^{2-n}{K}_{2-n}({\Delta }_s|z_E\left|\right)$$

(184)

where $z_E=(\mathbf{z},z_4)$ while $|z_E|=\sqrt{{\mathbf{z}}^2+z_4^2},$ whence we immediately obtain

$$I_2\left(z_E\right)={\displaystyle \frac{1}{8{\pi }^2}}{\displaystyle \sum _{s=1}^S}{C}_s{\lambda }_s{K}_0\left(z_s\right)(z_s={\Delta }_s|z_E\left|\right)$$

where $K_0$ is the modified Bessel function of the third kind and of order zero, the series representation of which is provided by

$$K_0\left(z\right)={\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{1}{{(k!)}^2}}{\left({\displaystyle \frac{z}{2}}\right)}^{2k}\left[\psi (k+1)-ln{\displaystyle \frac{z}{2}}\right]=ln{\displaystyle \frac{2}{z}}-\mathbf{C}+O(z^{2}lnz)$$

Then we can write

$${\displaystyle \frac{1}{4M}}\mathrm{tr}\left[\mathtt{reg}{\Sigma }_2(p/)\right]=-\upsilon \left(p^2\right)=2g^2{\int }_0^1\mathrm{d}x\underset{z_E\to 0}{lim}{I}_2\left(z_E\right)$$

whence it is clear that the limit exists, so that the regularization works, iff the Pauli-Villars condition is fulfilled, viz.,

$${\displaystyle \sum _{s=1}^S}{C}_s{\lambda }_s=0$$

which provides in turn $\upsilon =0$, i.e., no mass term is allowed in the 1-loop Weyl kinetic term, both in dimensional and Pauli-Villars regularizations, as expected.

Let us turn now to the other form factor

$$\begin{array}{ccc}& & \mathrm{tr}\left[p/\mathtt{reg}{\Sigma }_2(p/)\right]=2p^{2}f\left(p^2\right)\hfill \\ & =& i{g}^2\int {\displaystyle \frac{{\mathrm{d}}^4\ell }{{\left(2\pi \right)}^4}}{\displaystyle \sum _{s=0}^S}{C}_s{\displaystyle \frac{4p·\ell }{[{(\ell -p)}^2+i\epsilon ]\left({\ell }^2-M_s^2+i\epsilon \right)}}\hfill \\ & =& \int {\displaystyle \frac{{\mathrm{d}}^4\ell }{{\left(2\pi \right)}^4}}{\displaystyle \sum _{s=0}^S}{C}_s{\int }_0^1\mathrm{d}x{\displaystyle \frac{4i{g}^{2}p·\ell }{{\left[{\ell }^2-2xp·\ell +x{p}^2-(1-x)M_s^2+i\epsilon \right]}^2}}\hfill \\ & =& 4i{g}^2{\int }_0^1\mathrm{d}x\int {\displaystyle \frac{{\mathrm{d}}^4{\ell }^{\prime }}{{\left(2\pi \right)}^4}}{\displaystyle \sum _{s=0}^S}{\displaystyle \frac{C_{s}x{p}^2}{{\left[{\ell }^{\prime 2}+x(1-x)p^2-(1-x)M_s^2+i\epsilon \right]}^2}}\hfill \\ & =& -2g^2{p}^2{\int }_0^1\mathrm{d}xx\int {\displaystyle \frac{{\mathrm{d}}^4{\ell }_E}{{\left(2\pi \right)}^4}}{\displaystyle \sum _{s=0}^S}{\displaystyle \frac{C_s}{{\left[{\ell }_E^2+x(1-x)p_E^2+(1-x)M_s^2\right]}^2}}\hfill \end{array}$$

whence we obtain

$$\begin{array}{ccc}\hfill f\left(p^2\right)& =& -g^2{\int }_0^1\mathrm{d}xx\int {\displaystyle \frac{{\mathrm{d}}^4{\ell }_E}{{\left(2\pi \right)}^4}}{\displaystyle \sum _{s=0}^S}{\displaystyle \frac{C_s}{{\left[{\ell }_E^2+{\Delta }_s^2\right]}^2}}\hfill \\ & =& -g^2{\int }_0^1\mathrm{d}xx\int {\displaystyle \frac{{\mathrm{d}}^4{\ell }_E}{{\left(2\pi \right)}^4}}{\displaystyle \sum _{s=0}^S}{C}_s{\int }_0^{\infty }\mathrm{d}\tau \tau exp\left\{-\tau {\ell }_E^2-\tau {\Delta }_s^2\right\}\hfill \\ & =& -{\displaystyle \frac{g^2}{16{\pi }^2}}{\int }_0^1\mathrm{d}xx{\int }_0^{\infty }{\displaystyle \frac{\mathrm{d}\tau }{\tau }}{\displaystyle \sum _{s=0}^S}{C}_{s}exp\left\{-\tau (1-x)[x{p}_E^2+M^2{\lambda }_s^2]\right\}\hfill \\ & =& -{\displaystyle \frac{g^2}{16{\pi }^2}}{\int }_0^1\mathrm{d}xx{\int }_0^{\infty }{\displaystyle \frac{\mathrm{d}\tau }{\tau }}exp\left\{\tau x(1-x)p^2\right\}\hfill \\ & \times & {\displaystyle \sum _{s=0}^S}{C}_s\left[1-{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{{\tau }^n}{n!}}{(1-x)}^n{M}^{2n}{\lambda }_s^{2n}\right]\hfill \end{array}$$

Integrability requires the further Pauli-Villars condition

$${\displaystyle \sum _{s=1}^S}{C}_s=-1$$

and consequently

$$\begin{array}{ccc}\hfill f\left(p^2\right)& =& -{\displaystyle \frac{g^2}{16{\pi }^2}}{\int }_0^1\mathrm{d}xx{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{n!}}{(1-x)}^n{M}^{2n}{\displaystyle \sum _{s=1}^S}{C}_s{\lambda }_s^{2n}\hfill \\ & \times & {\int }_0^{\infty }{\displaystyle \frac{\mathrm{d}\tau }{\tau }}{\tau }^{n}exp\left\{\tau x(1-x)p^2\right\}\hfill \\ & =& -{\displaystyle \frac{g^2}{16{\pi }^2}}{\displaystyle \sum _{s=1}^S}{C}_s{\int }_0^1\mathrm{d}xx{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{1}{n}}{\left({\displaystyle \frac{{\lambda }_s^2{M}^2}{x{p}^2}}\right)}^n\hfill \\ & =& {\displaystyle \frac{g^2}{16{\pi }^2}}{\displaystyle \sum _{s=1}^S}{C}_s{\int }_0^1\mathrm{d}xxln\left(1+{\displaystyle \frac{{\lambda }_s^2{M}^2}{-x{p}^2}}\right)\hfill \\ & =& {\displaystyle \frac{g^2}{16{\pi }^2}}{\int }_0^1\mathrm{d}xx\left\{{\displaystyle \sum _{s=1}^S}{C}_{s}ln\left({\lambda }_s^2-{\displaystyle \frac{x{p}^2}{M^2}}\right)-ln\left(-{\displaystyle \frac{x{p}^2}{M^2}}\right)\right\}\hfill \\ & =& {\displaystyle \frac{g^2}{16{\pi }^2}}\left\{-{\displaystyle \frac{1}{2}}ln\left(-{\displaystyle \frac{p^2}{M^2}}\right)-{\int }_0^1\mathrm{d}xxlnx+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{s=1}^S}{C}_{s}ln{\lambda }_s^2\right.\hfill \\ & +& \left.{\int }_0^1\mathrm{d}xx{\displaystyle \sum _{s=1}^S}{C}_{s}ln\left(1-{\displaystyle \frac{x{p}^2}{M_s^2}}\right)\right\}\hfill \\ & =& {\left({\displaystyle \frac{g}{4\pi }}\right)}^2\left[{\displaystyle \sum _{s=1}^S}{C}_{s}ln{\lambda }_s+{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{2}}ln\left(-{\displaystyle \frac{M^2}{p^2}}\right)\right]+\mathrm{evanescent}\hfill \\ & \sim & {\left({\displaystyle \frac{g}{4\pi }}\right)}^2\left[{\displaystyle \frac{1}{ϵ}}+1+3\mathbf{C}+ln\left(-{\displaystyle \frac{4\pi {\mu }^2}{p^2}}\right)\right]+\mathrm{evanescent}\hfill \end{array}$$

where, in the last line, a comparison with the result in dimensional regularization has been exhibited. It follows that by identifying

$${\displaystyle \frac{1}{ϵ}}={\displaystyle \sum _{s=1}^S}{C}_{s}ln{\lambda }_s$$

in which the two Pauli-Villars conditions6 hold true, viz.,

$${\displaystyle \sum _{s=1}^S}{C}_s=-1{\displaystyle \sum _{s=1}^S}{C}_s{\lambda }_s=0$$

the coefficient of the divergent part turns out to be the same for both regulators, as expected.

5.0.3. Cut-Off Regularization

Let us repeat once more the one loop calculation of the Weyl spinor self-energy with a physical very large cut-off regulator, e.g. $K=l_P^{-1}$ with $l_P=\sqrt{\hslash G_N/c^3}$ the Planck length. Of course, once again, no mass term is allowed for the trace of an odd number of gamma matrices is null. Moreover, for the previously introduced coefficient of the Weyl kinetic term we find with ${\ell }^{\mu }=({\ell }_0,\overrightarrow{\ell })$

$$\begin{array}{ccc}\hfill p^{2}f\left(p^2\right)& =& {\displaystyle \frac{-i{g}^2}{{\left(2\pi \right)}^4}}\int {\displaystyle \frac{2p·\ell \theta \left(K^2-{\overrightarrow{\ell }}^2\right){\mathrm{d}}^4\ell }{\left[{(\ell -p)}^2+i\epsilon \right]\left({\ell }^2+i\epsilon \right)}}\hfill \\ & =& {\displaystyle \frac{-i{g}^2}{{\left(2\pi \right)}^4}}{\int }_0^1\mathrm{d}x\int {\displaystyle \frac{2p·\ell \theta \left(K^2-{\overrightarrow{\ell }}^2\right){\mathrm{d}}^4\ell }{{\left[{\ell }^2-2xp·\ell +x{p}^2+i\epsilon \right]}^2}}\hfill \\ & =& {\displaystyle \frac{-i{g}^2}{{\left(2\pi \right)}^4}}{\int }_0^1\mathrm{d}x\int \mathrm{d}\overrightarrow{\ell }\theta \left(K^2-{\overrightarrow{\ell }}^2\right)\hfill \\ & \times & \underset{\eta \to 0}{lim}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\eta }}{\int }_{-\infty }^{\infty }\mathrm{d}{\ell }_0{\displaystyle \frac{2p_0{\ell }_0-2\overrightarrow{p}·\overrightarrow{\ell }}{{\ell }^2-2xp·\ell +x{p}^2-\eta +i\epsilon }}\hfill \end{array}$$

After translation of the integration variable $\ell \mapsto \ell -xp$, for a suitably large cut-off K and space-like momentum $p^2<0$ we get

$$\begin{array}{ccc}\hfill p^{2}f\left(p^2\right)& =& {\displaystyle \frac{-i{g}^2}{{\left(2\pi \right)}^4}}{\int }_0^1\mathrm{d}x\int \mathrm{d}\overrightarrow{\ell }\theta \left(K^2-{(\overrightarrow{\ell }+x\overrightarrow{p})}^2\right)\hfill \\ & \times & \underset{\eta \to 0}{lim}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\eta }}{\int }_{-\infty }^{\infty }\mathrm{d}{\ell }_0{\displaystyle \frac{2p_0{\ell }_0-2\overrightarrow{p}·\overrightarrow{\ell }+2x{p}^2}{{\ell }^2+x(1-x)p^2-\eta +i\epsilon }}\hfill \\ & =& {\displaystyle \frac{-i{g}^2}{{\left(2\pi \right)}^4}}{\int }_0^1\mathrm{d}x\int \mathrm{d}\overrightarrow{\ell }\theta \left(4K^2-{\overrightarrow{\ell }}^2\right)\hfill \\ & \times & \underset{\eta \to 0}{lim}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\eta }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{\mathrm{d}{\ell }_{0}2x{p}^2}{{\ell }_0^2-{\overrightarrow{\ell }}^2+x(1-x)p^2-\eta +i\epsilon }}\hfill \\ & =& {\displaystyle \frac{-g^2{p}^2}{{\left(2\pi \right)}^3}}{\int }_0^1\mathrm{d}xx\int \mathrm{d}\overrightarrow{\ell }\theta \left(4K^2-{\overrightarrow{\ell }}^2\right)\hfill \\ & \times & \underset{\eta \to 0}{lim}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\eta }}{\left[{\overrightarrow{\ell }}^2-x(1-x)p^2-\eta \right]}^{-1/2}\hfill \\ \hfill f\left(p^2\right)& =& {\displaystyle \frac{g^2}{4{\pi }^2}}{\int }_0^1\mathrm{d}xx{\int }_0^{2K}\mathrm{d}\ell {\ell }^2{\left[{\ell }^2-x(1-x)p^2\right]}^{-3/2}\hfill \\ & =& {\displaystyle \frac{g^2}{4{\pi }^2}}{\int }_0^1\mathrm{d}xx{\int }_0^1\mathrm{d}y{y}^2{\left[y^2-x(1-x){\displaystyle \frac{p^2}{4K^2}}\right]}^{-3/2}\hfill \end{array}$$

Consider $u=\sqrt{a+y^2},a=x(1-x)(-p^2/4K^2)>0$, then from GR2.272 4. p. 86 we get the elementary result

$$\begin{array}{c}\hfill {\int }_0^1\mathrm{d}y{\displaystyle \frac{y^2}{u^3}}={\left[-{\displaystyle \frac{y}{u}}+ln(y+u)\right]}_0^1={\displaystyle \frac{-1}{\sqrt{a+1}}}+ln\left(1+\sqrt{a+1}\right)-ln\sqrt{a}\end{array}$$

For a very large cut-off $K\to \infty$ we obtain

$$\begin{array}{ccc}\hfill f\left(p^2\right)& =& {\displaystyle \frac{g^2}{4{\pi }^2}}{\int }_0^1\mathrm{d}xx\left\{-1+ln2+{\displaystyle \frac{1}{2}}ln\left(-{\displaystyle \frac{4K^2}{p^2}}\right)-{\displaystyle \frac{1}{2}}lnx(1-x)\right\}+\dots \hfill \\ & =& {\left({\displaystyle \frac{g}{4\pi }}\right)}^2\left[ln\left(-{\displaystyle \frac{4K^2}{p^2}}\right)+ln4\right]+\mathrm{evanescent}\hfill \end{array}$$

To sum up, we have verified that the 1-loop correction to the (left) Weyl spinor self-energy has the general form, which is universal, i.e. regularization independent: namely,

$$\begin{array}{ccc}\hfill \mathtt{reg}{\Sigma }_2(p/)& \equiv & f\left(p^2\right)p/P_L\hfill \\ \hfill f\left(p^2\right)& :=& {\left({\displaystyle \frac{g}{4\pi }}\right)}^2\left[{\displaystyle \frac{1}{ϵ}}+1+3\mathbf{C}+ln\left(-{\displaystyle \frac{4\pi {\mu }^2}{p^2}}\right)\right]\left(\mathrm{DR}\right)\hfill \\ & :=& {\left({\displaystyle \frac{g}{4\pi }}\right)}^2\left[{\displaystyle \sum _{s=1}^S}{C}_{s}ln{\lambda }_s+{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{2}}ln\left(-{\displaystyle \frac{M^2}{p^2}}\right)\right]\left(\mathrm{PV}\right)\hfill \\ & :=& {\left({\displaystyle \frac{g}{4\pi }}\right)}^2\left[ln\left(-{\displaystyle \frac{4K^2}{p^2}}\right)+ln4\right](\mathrm{CUT}-\mathrm{OFF})\hfill \end{array}$$

Some final remarks and comments are in order.

1.

In the present model of a left Weyl spinor minimally coupled with a gauge vector potential, no mass term can be generated by the radiative corrections in any regularization scheme. The left-handed part of the classical kinetic term does renormalize, while its right-handed part does not undergo any radiative correction and keeps on being free. The latter has to be necessarily introduced in order to define a Feynman propagator for the massless spinor field, like the gauge fixing term to invert the kinetic term of the gauge potential. The (one loop) renormalized Lagrangian for a Weyl fermion minimally coupled with a gauge vector potential has the universal - i.e. regularization independent - form

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{1-\mathrm{loop}}& =& \mathcal{L}={\chi }_L^{†}{\alpha }^{\nu }i{\partial }_{\nu }{\chi }_L+g{A}^{\nu }{\chi }_L^{†}{\alpha }_{\nu }{\chi }_L-\frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }\hfill \\ & +& {\phi }_R^{†}{\alpha }^{\nu }i{\partial }_{\nu }{\phi }_R-\frac{1}{2}{(\partial ·A)}^2-(Z_3-1)\frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }\hfill \\ & +& (Z_2-1){\chi }_L^{†}{\alpha }^{\nu }i{\partial }_{\nu }{\chi }_L+(Z_1-1)g{A}^{\nu }{\chi }_L^{†}{\alpha }_{\nu }{\chi }_L\hfill \\ \hfill (Z_2-1)& =& -{\left({\displaystyle \frac{g}{4\pi }}\right)}^2\left[{\displaystyle \frac{1}{ϵ}}+F_2(ϵ,p^2/{\mu }^2)\right]+\dots \hfill \\ & =& -{\left({\displaystyle \frac{g}{4\pi }}\right)}^2\left[{\displaystyle \sum _{s=1}^S}{C}_{s}ln{\lambda }_s+{\tilde{F}}_2({\lambda }_s,p^2/M^2)\right]+\dots \hfill \\ & =& -{\left({\displaystyle \frac{g}{4\pi }}\right)}^2\left[ln\left(-{\displaystyle \frac{4K^2}{p^2}}\right)+ln4+{\widehat{F}}_2(K^2/p^2)\right]+\dots \hfill \end{array}$$

where conventional notations have been employed. Notice that, as usual, the arbitrary finite parts $F_2,{\tilde{F}}_2,{\widehat{F}}_2$ of the countertems are analytic for $ϵ\to 0$ and ${\lambda }_s,K\to \infty$, respectively, and have to be uniquely fixed by the renormalization prescription.

2.

The interaction definitely preserves left chirality and scale invariance of the counterterms in the transition from the classical to the (perturbative) quantum theory: no mass coupling between the left-handed (interacting) Weyl spinor ${\chi }_L$ and right-handed (free) Weyl spinor ${\phi }_R$ can be generated by radiative loop corrections.

3.

While the cut-off and dimensional regularized theory does admit a local formulation in $D=4$ or $D=2\omega$ space-time dimensions, there is no such local formulation for the Pauli-Villars regularization. The reason is that the PV spinor propagator

$${\displaystyle \sum _{s=0}^S}{C}_{s}S(\ell ,M_s)$$

where $M_0=0,C_0=1$ while $\{M_s\equiv {\lambda }_{s}M|{\lambda }_s\gg 1(s=1,2,\dots ,S)\}$, cannot be the inverse of any local differential operator of the Calderon-Zygmund type. Hence, there is no local Action involving a bilinear spinor term that can produce, after a suitable causal inversion, the Pauli-Villars regularized spinor propagator.

One such pitfall (that is confusing the inverse of a sum of differential operators with the sum of their inverses) is not rare in the literature and can be found in several approaches nominally utilizing a PV regularization. They have nothing to do with the Pauli-Villars regularization in the present calculation, notwithstanding their misleading denomination.

Now the main point concerning the chiral anomaly and the unitarity of the collision operator. As we have seen in details, in the present model of a left-handed Weyl spinor interacting with a gauge vector potential, all the symmetries of the classical Action do survive the divergent and finite parts of the counter-terms, for any choice of the regulators and renormalization prescription. As it is well-known since a long time, this is not so for the famous chiral left-handed triangle diagram, which turns out to be UV finite and yields

$$\begin{array}{ccc}& & {\left.{\tilde{\Gamma }}_{{\mu }_1{\mu }_2{\mu }_3}^{\left(L\right)}(k_1,k_2,k_3)\right|}_{1-\mathrm{loop}}\hfill \\ & =& {\displaystyle \frac{g^3}{12{\pi }^2}}{\epsilon }_{\sigma {\mu }_1{\mu }_2{\mu }_3}\left(k_1^{\sigma }-k_2^{\sigma }\right)k_3^2{I}_{12}(k_1,k_2,k_3)+\mathrm{cyclic}\mathrm{permutations}\hfill \\ & +& {\displaystyle \frac{g^3}{4{\pi }^2}}{\epsilon }_{\sigma \tau {\mu }_1{\mu }_2}{k}_1^{\sigma }{k}_2^{\tau }{k}_3^{{\mu }_3}{I}_{12}(k_1,k_2,k_3)+\mathrm{cyclic}\mathrm{permutations}\hfill \\ & & (k_1+k_2+k_3=0)\hfill \\ \\ & & I_{rs}(k_1,k_2,k_3)\hfill \\ & =& -{\int }_0^1\mathrm{d}{x}_1{\int }_0^1\mathrm{d}{x}_2{\int }_0^1\mathrm{d}{x}_3{\displaystyle \frac{2x_r{x}_s\delta (1-x_1-x_2-x_3)}{k_1^2{x}_2{x}_3+k_2^2{x}_3{x}_1+k_3^2{x}_1{x}_2}}\hfill \\ & & (r,s=1,2,3)\hfill \end{array}$$

The one loop chiral left-handed triangular amplitude does satisfy the anomalous Ward identity

$$k_1^{\sigma }{\tilde{\Gamma }}_{\sigma {\mu }_2{\mu }_3}^{\left(L\right)}(k_1,k_2,k_3)=-{\displaystyle \frac{g^3}{12{\pi }^2}}{\epsilon }_{\sigma \tau {\mu }_2{\mu }_3}{k}_2^{\sigma }{k}_3^{\tau }$$

and related cyclic permutations.

The above expression for the 1-loop triangular amplitude is finite, it does not depend at all upon regulators, or counterterms or even renormalization prescriptions and it does not undergo any higher loop corrections (Adler-Bardeen theorem).However, it turns out that, owing to the anomalous Ward identity, the left-handed Weyl current does no longer satisfy the continuity equation, at variance with the classical case, viz.,

$${\partial }_{\nu }{\chi }_L^{†}{\alpha }^{\nu }{\chi }_L=\frac{1}{3}{\left(\frac{g}{4\pi }\right)}^2{F}_{*}^{\mu \nu }{F}_{\mu \nu }{F}_{*}^{\mu \nu }=\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{F}_{\rho \sigma }$$

This finite and universal quantum breaking of the U(1) symmetry under phase transformations on the left Weyl spinor field does actually put in jeopardy the unitarity issue. As a matter of fact the Lorentz invariant quantum theory of a gauge vector field unavoidably involves a Fock space of states with indefinite norm. Now, in order to select a physical Hilbert subspace of the Fock space a subsidiary condition is necessary. In the Abelian case, when the fermion current satisfies the continuity equation then the equations of motion lead to $\square (\partial ·A)=0$, so that a subspace of states of non-negative norm can be selected through the auxiliary condition

$$\partial ·A^{(-)}\left(x\right)|\mathrm{phys}\rangle =0$$

$A^{(-)}\left(x\right)$ being the destruction, positive frequency part of a d’Alembert quantum field. Conversely, in the present chiral model we find

$$-g^{-1}\square (\partial ·A)={\partial }_{\nu }{\chi }_L^{†}{\alpha }^{\nu }{\chi }_L={\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{g}{4\pi }}\right)}^2{F}_{*}^{\mu \nu }{F}_{\mu \nu }\ne 0$$

in such a manner that nobody knows how to select a physical subspace of states with non negative norm, if any, where a unitary restriction of the collision operator S could be defined.

6. Weyl and Dirac Propagators

Equipped with the previous definitions, clarifications and results, in the rest of the paper we cope with some critical issues one comes across in quantum field theories involving Weyl fermions. We start with the fermion functional integral.

We denote by $\cancel{D}$ the Dirac operator: namely, the massless matrix-valued differential operator applied in general to Dirac spinors on the 4d curved space with Minkowski signature $(+,-,-,-)$

$$\begin{array}{c}\hfill \cancel{D}=i\left(\cancel{\partial }+\cancel{V}\right)\end{array}$$

(185)

where $V_{\mu }$ is any real vector potential, valued in a some Lie algebra. We understand a spin connection when in presence of a non-trivial background metric. The Dirac fermions are represented in the four component formalism. The functional integral, i.e. the effective Action for a quantum Dirac spinor in the presence of a classical background vector potential

$$\begin{array}{c}\hfill \mathcal{Z}\left[V\right]=\int \mathcal{D}\psi \mathcal{D}\overline{\psi }{e}^{i\int d^{4}x\sqrt{g}\overline{\psi }\cancel{D}\psi }\end{array}$$

(186)

is formally understood as the determinant of $\cancel{D}:$ $\det \left(\cancel{D}\right)={\prod }_i^{\infty }{\lambda }_i.$ Concretely, the latter can be operationally defined in two alternative ways: either in perturbation theory, i.e. as the sum of an infinite number of 1-loop Feynman diagrams, some of which containing UV divergences by naive power counting, or in a non-perturbative approach, i.e. as the suitably regularized infinite product of the eigenvalues of $\cancel{D}$ by means of the analytic continuation tool. In the perturbative approach, in order to assign a meaning to a finite number of UV divergent 1-loop diagrams by naive power counting, one needs a regulator (dimensional, cutoff, PV or others). In the non-perturbative framework the complex power construction and the analytic continuation tool, if available, may provide by themselves the necessary to set up for the infinite product of eigenvalues of a normal operator, without need of any further regulator, in such a manner to produce the Heat-Kernel expansion and related topics [22,23].

Any variation of (186) with respect to a parameter contained in the Dirac operator requires the existence of an inverse of the kinetic operator. Formally

$$\begin{array}{c}\hfill \delta \det \left(\cancel{D}\right)=\det \left(\cancel{D}\right)\mathrm{tr}\left({\cancel{D}}^{-1}\delta \cancel{D}\right)\end{array}$$

(187)

In other words the existence of the inverse of $\cancel{D}$ is essential for the definition itself of the functional integral. It turns out that, in the case of Dirac fermions, an inverse of $\cancel{D}$ does exist and, if full causality is required in forwards and backwards time evolution on e.g. Minkowski space, it is the Feynman propagator or Schwinger distribution $\cancel{S}$, which is unique and characterized by the well-known Feynman prescription, which satisfies

$$\begin{array}{c}\hfill {\cancel{D}}_x\cancel{S}(x-y)=\delta (x-y),\cancel{D}\cancel{S}=1,\end{array}$$

(188)

The second equation is a synthetic notation7.

An example is the way we extract the trace anomaly from the functional integral. It the response of the latter under a Weyl (or even a scale) transform ${\delta }_{\omega }{g}_{\mu \nu }=2\omega g_{\mu \nu }:$

$$\begin{array}{c}\hfill {\delta }_{\omega }log\mathcal{Z}=\int d^{4}x\omega \left(x\right)g_{\mu \nu }\left(x\right)\langle T^{\mu \nu }\left(x\right)\rangle \end{array}$$

(189)

which is nothing but a particular case of (187). Here $g_{\mu \nu }\left(x\right)\langle T^{\mu \nu }\left(x\right)\rangle$ is the quantum trace of the energy-momentum tensor. Again, the latter can be calculated in various ways with perturbative or non-perturbative methods, that is with the Feynman diagram technique or with the so-called analytic functional method, respectively. The latter include the Schwinger’s proper-time method [21], the heat kernel method [22,23], the Seeley-DeWitt [19,24,25] and the zeta-function regularization [26].

The results concerning the trace anomaly for Dirac (and Majorana) fermions are well established. It is worth recalling that they rely on the possibility to turn to the Euclidean formulation, which is allowed thanks to the above mentioned Feynman prescription to get the Schwinger causal Green’s function or Feynman propagator for the quantum Dirac spinor field. In the absence of such a transition to an Euclidean formulation, the whole construction becomes mathematically meaningless and unreliable.

For Weyl fermions things drastically change. Let us denote a left-handed Weyl fermion by ${\psi }_L=P_L\psi$, where $P_L={\displaystyle \frac{1}{2}}\left(1-{\gamma }_5\right)$. Then, for example, the classical Action on the 4d Minkowski space reads

$$\begin{array}{c}\hfill S_L=\int {\mathrm{d}}^{4}x{\overline{\psi }}_L\cancel{D}{\psi }_L\end{array}$$

(190)

The Dirac operator, acting on left-handed spinors maps them to right-handed ones. Hence, the eigenvalue problem itself is not even well posed, so that the Weyl determinant cannot be defined at all. This is reflected in the fact that the inverse of

$$\begin{array}{c}\hfill {\cancel{D}}_L=\cancel{D}{P}_L=P_R\cancel{D}\end{array}$$

(191)

does not exist, since it is the product of an invertible operator times a projector. The full propagator of a Weyl fermion does not exist in this naive form (this problem can be circumvented in a more sophisticated approach, see below). It is incorrect to pretend that the propagator is ${\cancel{S}}_L=\cancel{S}{P}_R=P_L\cancel{S}$. First because such an inverse does not exist, second because, even formally,

$$\begin{array}{c}\hfill {\cancel{D}}_L{\cancel{S}}_L=P_R,\mathrm{and}{\cancel{S}}_L\overleftarrow{{\cancel{D}}_L}=P_L\end{array}$$

(192)

The inverse of the Weyl kinetic operator is not the inverse of the Dirac operator multiplied by a chiral projector. Therefore the propagator for a Weyl fermion is not the Feynman propagator for a Dirac fermion multiplied by the same projector. The impossibility to define eigenvalues of the kinetic operator or the lack of an inverse for the chiral Weyl kinetic term is a very severe problem. It is clear that the corresponding usual formulas for the chiral Weyl quantum theory do not exist at all.

How can we cope with these problem? In perturbation theory, in order to construct Feynman diagrams, one usually employs the ordinary free Feynman propagator for Dirac fermions. The use of a free Dirac propagator is formally justified, because one can add a free right-handed fermion to allow the inversion of the kinetic operator, as was done above. This is an easy way out, but it immediately raises several questions: in this way one explicitly breaks gauge and Lorentz invariance, and, if a background metric is minimally coupled to the Weyl fermion also diffeomorphisms and local Lorentz invariances are explicitly broken; moreover how can one preserve in this way the notion of chirality, which is a distinctive feature of Weyl spinors? There is no intrinsic way to guarantee that it is preserved all the way through the nontrivial steps of the calculations. In any case one must check a posteriori that diffeomorphism invariance is satisfied8.

The reason why one does such an expedient is because one hopes that the information about chirality is preserved by the fermion-boson-fermion vertex in (190), which contains the $P_L$ projector. As it has been shown in the example of Section 4, this method is believed to work correctly as long as it does not come across anomalies. Therefore absence of anomalies is a warranty that we are not introducing unwanted (opposite-handed) degrees of freedom. But this precisely cast doubts on the validity of results when anomalies are present. How can we trust an anomaly calculation when we know that the calculation is reliable only when anomalies are absent? We face a logical ouroboros9. We cannot trust our calculations precisely when we wish to compute anomalies. It sounds very much like the liar’s paradox: we are asking true information to somebody we know may be a liar. This logical short-circuit is not surprising because the procedure we have outlined has a bag at the very beginning: when we add a free right-handed fermion to the left-handed one, the partition function of which we aim to compute, the resulting Dirac operator has well defined eigenvalues, as peculiar as they may be. Hence, the use of the path integral definition as a product of eigenvalues is justified, but here lies the contradiction: this is not what we are looking for, because what we want to compute is the determinant of an operator whose eigenvalues are not even defined.

When we pass to nonperturbative approaches it is not possible to simply replace the missing Weyl propagator with the Dirac one. The reason is that the latter does not contain the correct information about chirality. To follow closely the perturbative approach we should consider the inverse of

$$\begin{array}{c}\hfill i\left(\cancel{\partial }+\cancel{V}{P}_L\right)\end{array}$$

(193)

The operator (193), as it stands, just treats the two chiralities asymmetrically. We will return to this point later on, but what matters here is to remark that the differential operator (193) is not the inverse of (185) multiplied by $P_L$. If one pretends to replace the full Weyl propagator with the full Dirac propagator, one loses any information concerning the chirality, and trying to recover it by multiplying the Dirac propagator by $P_L$, rather than a method, is a desperate deception.

Even though (193) can be used to compute consistent gauge anomalies yielding satisfactory results (for reasons that will be explained further on), it shares the same drawbacks as the just above outlined perturbative method. One should always and firmly keep in mind that all these approaches are based on the above manipulations, where we assume that the inverse of the relevant kinetic operator exists and if, by any chance, the kinetic operator is not invertible we pretend that it is and go on undeterred with the formalism. This attitude, based on the time-honored faith on practical calculations, may however hide unpleasant implications such as the logical loophole pointed out above. The inevitable conclusion is that all the results (even ours own) obtained with such procedure - that understands the invertibility of the kinetic operator when this is not the case - are intrinsically contradictory and should be considered with a good amount of skepticism.

There is however a way to avoid the logical trap pointed out before. A little thought shows what is critical in the previous procedures: there is a discontinuous jump between say (193) and the operator (191) which we should be computing. It is somewhat like studying a function only at the point where it is singular. There is no continuous parameter that connects one to the other. An elegant way out to bypass the difficulties with Weyl fermions is precisely following this suggestion; it consists in enlarging the space of potentials, by adding a spectator axial vector field, as was first done by W.A.Bardeen [27]. After all the most practical way to overcome the liar’s paradox is to ask many people.

Limiting ourselves for simplicity to the Abelian gauge case the Action integral for a Dirac fermion coupled to a vector $V_{\mu }$ and an axial potential $A_{\mu }$ is

$$\begin{array}{c}\hfill S[V,A]=\int d^{4}xi\overline{\psi }\left(\cancel{\partial }-i\cancel{V}-i\cancel{A}{\gamma }_5\right)\psi \end{array}$$

(194)

It is invariant under the following vector and axial-vector transformations

$$\begin{array}{ccc}& & {\delta }_{\alpha }{V}_{\mu }={\partial }_{\mu }\alpha ,{\delta }_{\alpha }{A}_{\mu }=0\hfill \\ & & {\delta }_{\beta }{V}_{\mu }=0,{\delta }_{\beta }{A}_{\mu }={\partial }_{\mu }\beta \hfill \end{array}$$

(195)

One can now do perturbative quantization of this theory without any difficulty since the fermion is Dirac and its propagator is the standard one. Simultaneously the definition of the functional integral as the product of eigenvalues makes sense because the eigenvalue problem is well defined for it, as complicated as these eigenvalues may be. After carrying out all the necessary calculations one can extract the results relevant for the Weyl fermion case by taking the limit $V\to V/2,A\to -V/2$. For in this limit in (194) one recovers the kinetic operator (193). This yields the bonafide correct results for a theory of Weyl fermions.

This method can be easily extended to non-Abelian potentials. The calculation of the effective Action generated out of (194) can be carried out along the traditional lines. For the perturbative approach everything needed for a correct computation is well defined: the propagator is the one for Dirac fermions, the vertices are of two types: for fermion-fermion-vector potential and for fermion-fermion-axial potential. The approach is the appropriate and traditional one for an operator that has a well defined spectrum of eigenvalues, like the Dirac operator discussed at the beginning of this Section. No poetic license is needed and any logical loophole is avoided. Now it is also evident why the very common (and basically contradictory) approach based on the simple replacement of the (missing) Weyl propagator with the Dirac one, may yield correct results. It is because the calculation steps are the same as in the (194) case as long as one imitates the substitution $A_{\mu }=0$ for the covariant case and $V\to V/2,A\to -V/2$ for the consistent case at the right points: in most cases a risky but often successful approach.

The extension to calculations with a non-trivial background metric requires a further step. This is easily understandable by looking at the kinetic operator (193) which is supposed to be applied to Dirac fermions. In this case we must add a spin connection ${\omega }_{\mu }$, i.e. $\cancel{V}$ is replaced by $\cancel{V}+\cancel{\omega }$, and ${\gamma }^{\mu }={\gamma }^a{e}_a^{\mu }$ where $e_a^{\mu }$ is the inverse tetrad. It is immediately evident that the corresponding theory is not invariant under diffeomorphisms. If, on the other hand, we replace $\cancel{V}{P}_L$ with $\cancel{\omega }+\cancel{V}{P}_L$ diffeomorphisms are preserved in the classical Action, but we loose the effect the chirality can have on the metric dependence. The surplace can be overcome by introducing a second spectator metric. The corresponding construction is conceptually simple but requires a laborious extension of the formalism. We give up the idea of presenting it here, and refer to the literature. It was first formulated in [28]. The interested reader can found an extensive review with applications in [29]. One thing is however worth being reminded: in this formalism a central role is played by diffeomorphism invariance. There are unfortunately in the literature examples of calculations where such invariance has simply been explicitly broken, thus compromising the validity of the results.

6.1. Weyl and Massless Majorana Fermions

The second criticality is related to the (wrong) identification between Weyl and Majorana massless spinors. If this identification were possible the problem discussed in the previous Section would not exist, because we could simply replace the Weyl kinetic operator with the massless Majorana one, and the missing Weyl propagator with the existing Majorana propagator (identical to the Dirac one). We have already explained in detail above that Weyl and Majorana spinors belong to different representations of the Lorentz group. A Weyl spinor is a complex spinor with definite chirality (an eigenstate of ${\gamma }_5$) which corresponds to its helicity, constituting a minimal representation of the Lorentz group. A Majorana spinor is a self-conjugate bispinor, which can be cast in real form with a suitable choice of the gamma matrices. A Majorana spinor can be represented as a superposition of two Weyl spinors, conjugate to each other and thus with two opposite chirality. Therefore a Majorana spinor has undefined chirality, so that the relation with its helicity is also undefined. Moreover a parity operation maps the Majorana Action into itself, while it maps the Weyl Action (171) into the same Action for the opposite chirality. The same holds for the charge conjugation operator.

There is no possibility of confusing a Weyl spinor with a Majorana massless spinor. Therefore one may wonder how such misunderstanding may have arisen and spread. It certainly comes from a superficial knowledge of the mathematical properties of spinors in 4d, in which a false analogy with 2d, where Weyl and Majorana spinors coexist, may have played a role. But, above all, we suspect, the common properties of the two types of spinors may be at the origin of the misunderstanding. Beside being both massless they have other properties in common. For instance we can decompose any (massless) Dirac fermion into the sum of two Weyl spinors of opposite chirality or into the sum of two independent Majorana spinors. In this sense in model building it is equivalent to use Weyl or Majorana, provided the end result consists of Dirac spinors. Another reason why they are sometimes considered the same object may be due to the fact that we can establish a one-to-one correspondence between the components of a Weyl spinor and those of a Majorana spinor in such a way that the Lagrangian, in two-component notation, looks the same. If, in the chiral representation, we write ${\psi }_L$ as $\left(\begin{array}{c}\omega \\ 0\end{array}\right)$, where $\omega$ is a two component spinor, then (171) above becomes

$$\begin{array}{c}\hfill i{\omega }^{†}{\overline{\sigma }}^{\mu }{\partial }_{\mu }\omega \end{array}$$

(196)

which has the same form (up to an overall factor) as a massless Majorana Action (see (103)). But the Action is not everything in a theory, it must be accompanied by a set of specifications. Even though numerically the Actions coincide, the way they respond to a variation of the Weyl and Majorana fields is different. One leads to the Weyl, the other to the Majorana equation of motion. The delicate issue is precisely this: when we take the variation of an Action with respect to a field in order to extract the equations of motion, we must make sure that the variation respects the symmetries and the properties that are expected in the equations of motion10. If we wish the eom to preserve chirality we must use variations that preserve chirality, i.e. must be eigenfunctions of ${\gamma }_5$. If instead we wish the eom to transform in the Majorana representation we have to use variations that transform suitably, i.e. must be eigenfunctions of the charge conjugation operation. If we do so we obtain two different results, which are irreducible to each other, no matter what Action we use.

At this point it is perhaps not useless to clarify an issue concerning the already mentioned U(1) continuous symmetry of Weyl fermions. The latter is often confused with an axial $\mathbb{R}$ symmetry of Majorana fermions and used to justify the identification of Weyl and massless Majorana fermions. To start with, let us consider a free massless Dirac fermion $\psi$. Its free Action is clearly invariant under the transformation $\delta \psi =i(\alpha +{\gamma }_5\beta )\psi$, where $\alpha$ and $\beta$ are real numbers. This symmetry can be gauged by minimally coupling $\psi$ to a vector potential $V_{\mu }$ and an axial potential $A_{\mu }$, in the combination $V_{\mu }+{\gamma }_5{A}_{\mu }$, so that $\alpha$ and $\beta$ become arbitrary real functions. For convenience let us choose the Majorana representation for gamma matrices, so that all of them, including ${\gamma }_5$, are imaginary. If we now impose $\psi$ to be a Majorana fermion, its four components will be real and only the symmetry parametrized by $\beta$ makes sense in the Action (let us call it $\beta$ symmetry). If instead we impose $\psi$ to be Weyl, say $\psi ={\psi }_L$, then, since ${\gamma }_5{\psi }_L={\psi }_L$, the symmetry transformation will be $\delta {\psi }_L=i(\alpha -\beta ){\psi }_L$.

This may be the origin of the confusion, because it looks like we can merge the two parameters $\alpha$ and $\beta$ into a unique one and identify it with the $\beta$ of the Majorana axial $\beta$ symmetry. However this is not possible because for a right handed Weyl fermion the symmetry transformation is $\delta {\psi }_R=i(\alpha +\beta ){\psi }_R$. Forgetting $\beta$, the Majorana fermion does not transform. Forgetting $\alpha$, both Weyl and Majorana fermions transform, but the Weyl fermions transform with opposite signs for opposite chiralities. In terms of anomalies, it is well-known that the axial $\mathbb{R}$ Majorana symmetry is anomalous: this is the well-known covariant or ABJ anomaly $\sim \int d^{4}x\beta F_A\wedge F_A$ ($F_A$ is the curvature of A). For a left (right) Weyl fermion the symmetry with parameter $\alpha -\beta$ ($\alpha +\beta$) is anomalous. This is a consistent anomaly, which, in the Abelian case we are considering, coincides in form with the covariant anomaly (but not in the non-Abelian case), although with a different coefficient and with opposite signs for opposite chiralities. Since a Dirac fermion can be regarded as a sum of two Weyl fermions with opposite chiralities, we see that the anomalies triggered by the $\alpha$ transformation cancel out, while the anomalies triggered by $\beta$ add up. This is consistent with the well-known fact that for a Dirac fermion the $U\left(1\right)$$\alpha$ symmetry is not anomalous, while the axial $\beta$ symmetry is anomalous and corresponds to twice the anomaly of a Majorana fermion. As we see the symmetries and anomalies of Majorana fermions are different from the symmetries and anomalies of Weyl fermions.

No, there is no room for confusing massless Majorana spinors with chiral Weyl spinors. Actually, a Majorana spinor describes neutral spin 1/2 objects - not yet detected in Nature - and consequently there is no phase transformation (U(1) continuous symmetry) involving self-conjugated Majorana spinors, independently of the presence or not of a mass term. Hence, e.g., its particle states do not admit antiparticles of opposite charge, simply because charge does not exist at all for charge self-conjugated spinors (actually, this was the surprising discovery of Ettore Majorana, after the appearance of the Dirac equation and the positron detection). The general solution of the wave field equations for a free Majorana spinor always entails the presence of two polarization states with opposite helicity. On the contrary, it is well known that a chiral Weyl spinor, describing massless quarks and leptons (before the electroweak symmetry breaking11) in the Standard Model, admits only one polarization or helicity state. It always involves antiparticles of opposite helicity and it always carries a conserved internal quantum number such as the lepton number, which is opposite for particles and antiparticles.

Notwithstanding their common properties, the difference between Weyl and massless Majorana spinors becomes crucial in certain instances. One of these is the functional measure in the path integral. The functional measure is well defined for Dirac or Majorana fermions, while it is undefined for Weyl fermions. Here the problem of course arises in nonperturbative calculations. K. Fujikawa invented a method [30] to compute ABJ anomalies (i.e. anomalies that appear in theories with Dirac or Majorana fermions). It is a heuristic method based upon the calculation of the symmetry variation of the functional measure. It works very well for Dirac and Majorana fermions. But the functional measure is computed relying on the decomposition of the fermion fields into a complete set of eigenvectors of the relevant Dirac operator. We know that the eigenvalue problem is not even defined in the case of Weyl fermions, thus it is mandatory to resort to other methods.

There are other more flexible methods, beside Fujikawa’s one, which in addition are mathematically well founded. These are the functional method alluded to above, which we denote with the term functional or `heat kernel’ methods. They translate into a mathematically reliable language the idea of determinant of the kinetic operator. Since a formulation of these methods with a linear Dirac operator is not known (yet), one resorts to a quadratic elliptic operator (for instance the square of the Dirac operator) and bases all calculations on the square root of the resulting path integral. For the method to be valid, however, such a quadratic operator must preserve all the properties of the linear one. Now, it is not difficult to concoct an elliptic operator out of the Weyl-Dirac one, but it is impossible to do it while preserving all the critical information, i.e. while preserving the chirality of the model and its classical symmetries12.

There are in fact insurmountable difficulties. As a matter of fact, first of all, in order to turn the square of the Dirac operator into an elliptic normal differential and matrix-valued operators, the transition to the Euclidean formulation is mandatory. The latter is absolutely legitimate and viable for Dirac fermions, whilst it does not exist at all for Weyl fermions. The very reason is deep and sharp: the Euclidean 4d symmetry group is $O(4,\mathbb{R})$ which is locally isomorphic to the direct product $O(3,\mathbb{R})\times O(3,\mathbb{R})$, in such a manner that any item in this theory must be invariant with respect to the exchange of any spin representation of the two identical and equivalent orthogonal trivial factors, no room being left to the very concept of chirality, which requires two non-equivalent irreducible 2d representations of the Lorentz group.

The solution to these series of problems exists without straining any fundamental principle or lore. But before coming to that we have to clarify another concept, that of Dirac mass and Majorana mass.

6.2. Dirac and Majorana Mass Terms

Another particularly confusing concept seems to be connected with mass terms for fermions. Let us start from a basic element, the mass term $\overline{\psi }\psi$ for a Dirac spinor, which is uncontroversial. It can be also rewritten by projecting the latter into its chiral components

$$\begin{array}{c}\hfill \overline{\psi }\psi =\overline{{\psi }_L}{\psi }_R+\overline{{\psi }_R}{\psi }_L\end{array}$$

(197)

If $\psi$ is a Majorana spinor this can be written

$$\begin{array}{c}\hfill \overline{\psi }{\psi }^c=\overline{{\left({\psi }^c\right)}_L}{\psi }_R+\overline{{\psi }_R}{\left({\psi }^c\right)}_L,\end{array}$$

(198)

which is therefore well defined and Lorentz invariant by construction. Now, using the Lorentz covariant conjugate we can rewrite (198) as

$$\begin{array}{c}\hfill {\left({\psi }_L\right)}^T{C}^{-1}{\psi }_L+{\psi }_L^{†}C{\left({\psi }_L\right)}^{*},\end{array}$$

(199)

which is expressed solely in terms of ${\psi }_L$. The above expression (199) for the mass term may create the illusion that there exists a mass term also for Weyl fermions. But this is not the case. If we add this term to the kinetic term (171) we obtain an Action which gives rise to a field equation that is not Lorentz covariant: the kinetic and mass terms in the field equation belong to two different representations. To be some more explicit, a massive Dirac equation of motion for a Weyl fermion would be

$$\begin{array}{c}\hfill i{\gamma }^{\mu }{\partial }_{\mu }{\psi }_L-m{\psi }_L=0,\end{array}$$

(200)

but this equation breaks Lorentz covariance because the first piece transforms according to a right-handed representation while the second piece according to a left-handed one, in such a manner that it cannot be derived from a Lorentz invariant Lagrange density.

Instead of the second term in the LHS of (192) one could use $mC{\overline{{\psi }_L}}^T$, which has the right Lorentz properties, but the corresponding Lagrangian term would not be self-adjoint and one would be forced to introduce the adjoint term and end up again with (199). The reason of all this is, of course, that (199) is not expressible in the same canonical form as (171), while it is an allowed mass term for a Majorana fermion. This implies, in particular, that there does not exist such a thing as a “massive Weyl propagator”, that is a massive propagator involving only one chirality. This is something that, in particular, renders the use of the Pauli-Villars regularization problematic. In fact it would be natural to regularize the theory by means of massive Weyl ghost fields. But this is impossible. Sometimes in place of the missing massive Weyl propagator a Dirac or Majorana propagator is used. But this is a leap in the dark: since in these propagators both left and right chiralities appear, one has no warranty that chirality is preserved throughout the calculations and thus that the result one obtains is the correct one. A minimal precaution, in such a case, would be to crosscheck the results obtained with this regularization by comparing them with those obtained with others. In Sectiom 5, for instance, we have shown that the masslessness result of the Weyl kinetic term is preserved at one loop with three different regularization, including a PV one. These results, which are clearly visible in the four component formalism used so far, are much less recognizable in the two-component formalism.

It turns out that, on the one hand, the renowned and famed Goldhaber et al. series of experiments [10] unequivocally proves that neutrinos are always left-handed and of negative helicity, while anti-neutrinos are distinguished and charge conjugated, since they always appear to be right-handed and of positive helicity: it means, beyond any reasonable doubt, that neutrinos and anti-neutrinos are truly Weyl fermions. On the other hand, soon after the discovery of the neutrino flavor oscillations it has been promptly suggested to understand the latter in terms of a Majorana mass term which provides, as we have seen (109), a breakdown of the lepton number conservation and an open door towards flavor changes. Unfortunately, as we have thoroughly discussed and elucidated in the present review, the hypothesis of a Majorana massive neutrino badly and strongly contradicts the Goldhaber et al. long ago well established and solid results: it is impossible to simultaneously keep a full barrel and a drunk wife. The deep reasons to explain and understand neutrino oscillations are elsewhere but in the Majorana mass, at least if we believe in the principles of quantum field theory and the Standard Model construction and if we trust in [10].

Finally, going to nonperturbative approaches, we have already noted that the fermion functional integral measures are different in the two cases. This is a rather heuristic way of formulating the issue. One can express the same idea and calculations in terms of more rigorous functional or heat-kernel like methods. In fact in the literature one meets various different formulations of this problem, which oscillate between rigorous and heuristic approaches. Anyhow, the main central object is the quadratic operator that should represent the square of the kinetic operator, the Dirac operator in our case. Since, as we have already remarked, a version of the heat-kernel involving the linear Dirac operator is not available, one considers a quadratic version, i.e. a functional integral in which the linear Dirac operator is replaced by its square, and then takes the square root of it. This is the logical extension to continuous operators of the determinant of a matrix represented by the product of its eigenvalues. The square of the matrix is represented by the product of the squared eigenvalues.

Once this procedure has been accepted, the most important move is the choice of the quadratic operator. Let us explain this point in some detail. In the case of a Dirac fermion, by applying twice the Dirac operator makes sense because, for instance, for a gauge transformation $\delta \psi =i\alpha \psi$ with $\alpha ={\alpha }^a{T}^a$, where $T^a$ are the Hermitean generators of a Lie algebra, we have

$$\begin{array}{c}\hfill \delta \cancel{D}\psi =i\alpha \cancel{D}\psi \end{array}$$

Thus it makes sense to apply twice $\cancel{D}$ to $\psi$, because $**D*{*}^2\psi$ under a gauge transformation has the same properties as $\cancel{D}\psi$. This was the meaning of our statement in the previous Section whereby the quadratic operator must respect the symmetries and features of the original kinetic operator. In many cases one meets in the literature this is not the case.

Let us consider a simple example. The functional integral for a free Dirac fermion (170) $\cancel{D}=i\cancel{\partial }+\cancel{V}$ (where V denotes any potential), is the (suitably regularized) product of its eigenvalues. This is true for a massless Majorana fermion, while for a Weyl fermion it is not so straightforward. Since the Dirac operator anticommutes with ${\gamma }_5$, it maps a left-handed spinor to a right-handed one. Therefore the eigenvalue problem is not even defined for ${\cancel{D}}_L=\cancel{D}{P}_L$. We may replace the looked for $\det {\cancel{D}}_L$ with ${\left(\det \left({\cancel{D}}_L^{†}{\cancel{D}}_L\right)\right)}^{{\displaystyle \frac{1}{2}}}$. If $\cancel{D}$ is self-adjoint (after a Wick rotaton) this reduces to ${\left(\det \left({\cancel{D}}^2{P}_L\right)\right)}^{{\displaystyle \frac{1}{2}}}$. But this is deceivingly simple. It is true that the symmetries are preserved but the most important structure, i.e. the interaction vertex, is lost. Even if the quadratic operator contains a chiral projector the latter is totally ineffective. It is analogous to stating that the inverse of the Dirac-Weyl operator is the inverse of the Dirac operator multiplied by chiral projector. To protect the interaction vertex from disappearing one should use (193) as basic kinetic operator. But, as already noticed, (193) breaks gauge and diffeomorphism covariance. The puzzle can be solved with the already mentioned Bardeen’s approach. The real important point to be retained from this discussion is that the squared Dirac operator one is obliged to use in the functional methods must preserve the same symmetries as the linear Dirac operator which appear in the defining Action.

In most of the nonperturbative functional methods, after turning to a Euclidean setting, one uses the dimensional regularization or a regularization based on analytic continuation such as the zeta function one. At variance with these standard treatments, among the heat-kernel- like approaches there have been also attempts to use masses as regulators, much as in the PV regularization method. We have already remarked that the presence of massive terms introduce in general two chiralities into the game, so that the PV regularization does not seem the most suitable method for calculations in which the chirality must be preserved. In the perturbative case, however, an educated use of PV can lead to the correct result. But, due to its original sin on which it is based, the result must always be confirmed with other regularizations.

The nonperturbative use of masses as regulators consists in choosing instead of a massless quadratic operator, the square of a massive operator, which can only be a Dirac-Majorana massive operator. This method is sometime sold as PV. Now, first of all, as already noticed several times, the quadratic operator inevitably breaks the conservation of chirality, since it involves both chiralities in a balanced form and excludes any possibility to capture, for instance, consistent chiral anomalies. Moreover, the perturbative 1-loop corrections to the classical Weyl Action, as obtained from the original true Pauli-Villars regularization method, by no means can generate any dynamical mass term, neither Dirac nor Majorana, since a PV regularization preserves gauge invariance and U(1) internal chiral phase transformation invariance, so that it protects classical scale invariance and forbids any mass term at the quantum level. Beside other less formal reserves, it is roughly inappropriate to call this type of approach a PV regularization.

To sum up, the use of a Majorana-Dirac operator inevitably drives to a chirally symmetric operator, which cannot produce any left-right asymmetry, i.e. in particular no chiral split anomaly. i.e. with opposite sign for opposite chiralities. Such methods, in particular, are unable to intercept consistent chiral anomalies.

7. Euclidean Fermion Field Theories

As explained in the introduction, the technique of the Wick rotation to regularize the divergent integrals appearing in Feynman diagrams cannot be applied to the original classical Action. Below we explain why, but to do this job we need a short introduction to the Euclidean field theory for fermions, with the warning that the problem of the transition from a Minkowski fermion theory to a Euclidean counterpart [31,32,33,34,35,36] does not admit a unique solution. Below we outline a simple approach.

Let us start from the theory of a massive Dirac fermion and its spinor causal Green’s function, or Schwinger function13 . We can safely perform the natural replacements

$$i{p}_0=p_{4}i{x}_0=x_4$$

(201)

so as to obtain a positive definite denominator in the Fourier transform, that is

$$\begin{array}{c}\hfill S_{\alpha \beta }^F(-i{x}_4,\mathbf{x})=i{\left(2\pi \right)}^{-4}\int \mathrm{d}\mathbf{p}{\int }_{-\infty }^{\infty }\mathrm{d}{p}_4{\displaystyle \frac{exp\{i{p}_4{x}_4+i{p}_j{x}_j\}}{p_4^2+{\mathbf{p}}^2+M^2}}{({\gamma }^0{p}_4-i{\gamma }^k{p}^k+iM)}_{\alpha \beta }\end{array}$$

Next, we define the Euclidean Dirac Matrices

$$\begin{array}{c}\hfill {\overline{\gamma }}_4\equiv {\gamma }^0,{\overline{\gamma }}_k\equiv i{\gamma }_k=-i{\gamma }^k(k=1,2,3)\end{array}$$

(202)

$$\begin{array}{c}\hfill {\overline{\gamma }}_{\mu }=\left({\overline{\gamma }}_k,{\overline{\gamma }}_4\right)={\overline{\gamma }}_{\mu }^{†},\left\{{\overline{\gamma }}_{\mu },{\overline{\gamma }}_{\nu }\right\}=2{\delta }_{\mu \nu }\end{array}$$

(203)

so that we can write

$$\begin{array}{ccc}\hfill S_{\alpha \beta }^F(-i{x}_4,\mathbf{x})& =& {\displaystyle \frac{i}{{\left(2\pi \right)}^4}}\int {\mathrm{d}}^4{p}_{E}exp\{i{p}_E·x_E\}{〔{\displaystyle \frac{{\overline{\gamma }}_{\mu }{p}_{E\mu }+iM}{p_E^2+M^2}}〕}_{\alpha \beta }\hfill \\ & =& {\displaystyle \frac{i}{{\left(2\pi \right)}^4}}\int {\mathrm{d}}^4{p}_{E}exp\{i{p}_E·x_E\}{〔{\displaystyle \frac{1}{{\cancel{p}}_E-iM}}〕}_{\alpha \beta }\hfill \\ & \equiv & -S_{\alpha \beta }^E\left(x_E\right)\hfill \end{array}$$

(204)

where we understand

$$\mathbf{p}=(p^1,p^2,p^3)=(p_x,p_y,p_z)={\overrightarrow{p}}_E=({\overline{p}}_1,{\overline{p}}_2,{\overline{p}}_3)$$

together with

$$\begin{array}{c}\hfill p_4{\overline{\gamma }}_4+{\overline{\gamma }}_k{\overline{p}}_k={\overline{\gamma }}_{\mu }{p}_{E\mu }\equiv p{/}_E\\ \hfill \partial {/}_E\equiv {\overline{\gamma }}_{\mu }{\partial }_{E\mu }={\overline{\gamma }}_k{\partial }_k+{\overline{\gamma }}_4{\partial }_4\end{array}$$

The above move, however, is not cost-free. The Schwinger function (204) conflicts with hermiticity. The way out suggested by Osterwalder and Schrader, which ensures reflection positivity, is to assume that the proper classical variables for a Euclidean formulation of the Dirac spinor field theory, should be two distinct Euclidean bispinors, which we may denote again ${\psi }_E$ and ${\overline{\psi }}_E$, but obey

$$\{{\psi }_E\left(x\right),{\psi }_E\left(y\right)\}=\{{\overline{\psi }}_E\left(x\right),{\overline{\psi }}_E\left(y\right)\}=\{{\psi }_E\left(x\right),{\overline{\psi }}_E\left(y\right)\}=0$$

(205)

for all points x and y of the four-dimensional Euclidean space ${\mathbb{R}}^4$, while there is a non-trivial equal time canonical bracket between ${\psi }_E$ and the Hermitean conjugate of ${\overline{\psi }}_E$. The last of the (205) relations is crucial, for it implies that ${\overline{\psi }}_E$ does not coincide with the Hermitean conjugate of ${\psi }_E$ times some matrix ${\gamma }_4.$ Thus, if we want to set up a meaningful Euclidean formulation for the Dirac spinor field theory, then we can treat ${\psi }_E$ and ${\overline{\psi }}_E$ as totally independent classical Grassmann valued fields. This independence is the main novelty of the Euclidean fermion field theory, the rest of the construction being straightforward.

For instance, we use the definition of the Euclidean matrices ${\overline{\gamma }}_{\mu }$ to derive the $O\left(4\right)$ transformation law for ${\psi }_E$ in the usual way. As a matter of fact we have the six Hermitean generators

$$\begin{array}{c}\hfill {\Sigma }_{\mu \nu }={\displaystyle \frac{i}{4}}\left[{\overline{\gamma }}_{\mu },{\overline{\gamma }}_{\nu }\right]{\Sigma }_{ı}\equiv {\epsilon }_{ıjk}{\Sigma }_{jk}(ı,j,k=1,2,3)\end{array}$$

(206)

$$\begin{array}{c}\hfill {\Sigma }_{23}\equiv {\Sigma }_1=\frac{1}{2}〔\begin{array}{cc}{\sigma }_1& 0\\ 0& {\sigma }_1\end{array}〕{\Sigma }_{31}\equiv {\Sigma }_2=\frac{1}{2}〔\begin{array}{cc}{\sigma }_2& 0\\ 0& {\sigma }_2\end{array}〕\end{array}$$

(207)

$$\begin{array}{c}\hfill {\Sigma }_{12}\equiv {\Sigma }_3=\frac{1}{2}〔\begin{array}{cc}{\sigma }_3& 0\\ 0& {\sigma }_3\end{array}〕{\Sigma }_{k4}\equiv {\overline{\Sigma }}_k=\frac{1}{2}〔\begin{array}{cc}{\sigma }_k& 0\\ 0& -{\sigma }_k\end{array}〕\end{array}$$

(208)

$so\left(4\right)-$Lie algebra. The six $O(4,\mathbb{R})$ generators satisfy the following Lie algebra commutation relations, viz.,

$$\begin{array}{c}\hfill [{\Sigma }_j,{\Sigma }_k]=i{\epsilon }_{jk\ell }{\Sigma }_{\ell }=[{\overline{\Sigma }}_j,{\overline{\Sigma }}_k][{\Sigma }_j,{\overline{\Sigma }}_k]=i{\epsilon }_{jk\ell }{\overline{\Sigma }}_{\ell }\end{array}$$

which can be suitably recast into the four-vector notations to yield

$$\begin{array}{c}\hfill [{\Sigma }_{\mu \nu },{\Sigma }_{\kappa \lambda }]=i{\delta }_{\mu \kappa }{\Sigma }_{\nu \lambda }-i{\delta }_{\lambda \mu }{\Sigma }_{\nu \kappa }+i{\delta }_{\nu \lambda }{\Sigma }_{\mu \kappa }-i{\delta }_{\kappa \nu }{\Sigma }_{\mu \lambda }\end{array}$$

(209)

It follows that we have the following orthogonal transformation rules for the two-component spinors and four-component bispinors of the Euclidean formulation,

$$\begin{array}{c}\hfill {\psi }_{\uparrow }^{\prime }\left(x_E^{\prime }\right)=exp\{(i/2)\overrightarrow{\sigma }·(\overrightarrow{\alpha }+\overrightarrow{\eta })\}{\psi }_{\uparrow }\left(x_E\right)\end{array}$$

(210)

$$\begin{array}{c}\hfill {\psi }_{\downarrow }^{\prime }\left(x_E^{\prime }\right)=exp\{(i/2)\overrightarrow{\sigma }·(\overrightarrow{\alpha }-\overrightarrow{\eta })\}{\psi }_{\downarrow }\left(x_E\right)\end{array}$$

(211)

with $x_E^{\prime }=R{x}_E,R\in O\left(4\right)$14 while $0\le |\overrightarrow{\alpha }|<2\pi$ and $0\le |\overrightarrow{\eta }|<2\pi ,$ that yields

$${\psi }_E\left(x_E\right)=〔\begin{array}{c}{\psi }_{\uparrow }\left(x_E\right)\\ {\psi }_{\downarrow }\left(x_E\right)\end{array}〕$$

$$\begin{array}{c}\hfill {\psi }^{\prime }\left(x_E^{\prime }\right)=exp\left\{(i/2){\Sigma }_{\mu \nu }{\theta }_{\mu \nu }\right\}{\psi }_E\left(x_E\right)\end{array}$$

(212)

$$\begin{array}{c}\hfill {\theta }_{k4}=-{\eta }_k{\theta }_{ıj}=-{\epsilon }_{ıjk}{\alpha }_k\end{array}$$

(213)

It is crucial to gather that the upper and lower spinor components of a bispinor transform according to equivalent fundamental representations $D_{{\displaystyle \frac{1}{2}}}$ of SU(2), at variance with left and right Weyl spinors on the Minkowski space, so that the Euclidean bispinor ${\psi }_E\left(x_E\right)$ belongs to the 4d reducible representation $D_{{\displaystyle \frac{1}{2}}}\oplus D_{{\displaystyle \frac{1}{2}}}.$ Thus, the very concept of non-equivalent Weyl spinors of different chirality is lost in this Euclidean formulation of the spinor field theory. Moreover, ${\overline{\psi }}_E$ transforms just like the Hermitean conjugated of ${\psi }_E,$ by definition. As a matter of fact, from the unitary property

$$\begin{array}{c}\hfill U\left(\theta \right)=exp\left\{(i/2){\overline{\Sigma }}_{\mu \nu }{\theta }_{\mu \nu }\right\}{U}^{†}\left(\theta \right)=U^{-1}\left(\theta \right)=U(-\theta )\end{array}$$

(214)

we have

$$\begin{array}{c}\hfill \left({\overline{\psi }}_E^{\prime },{\psi }_E^{\prime }\right)=\int {\mathrm{d}}^4{x}_E^{\prime }{\overline{\psi }}_E^{\prime }\left(x_E^{\prime }\right){\psi }_E^{\prime }\left(x_E^{\prime }\right)=\int {\mathrm{d}}^4{x}_E{\overline{\psi }}_E\left(x_E\right)U^{†}\left(\theta \right)U\left(\theta \right){\psi }_E\left(x_E\right)=({\overline{\psi }}_E,{\psi }_E)\end{array}$$

with $x_E^{\prime }=R{x}_E,R^{\top }=R^{-1}.$ In this regard, let us focus on the conversion of the Minkowski space chiral matrix

$${\gamma }_5=i{\gamma }_0{\gamma }^1{\gamma }^2{\gamma }^3={\gamma }_5^{†}\{{\gamma }_5,{\gamma }^{\mu }\}=0$$

The standard definition in the Euclidean $D-$dimensional Clifford algebra is

$${\overline{\gamma }}_5\equiv {\overline{\gamma }}_1{\overline{\gamma }}_2{\overline{\gamma }}_3{\overline{\gamma }}_4=-{\gamma }_5\{{\overline{\gamma }}_5,{\overline{\gamma }}_{\mu }\}=0$$

(215)

in such a manner that the projector on the upper and lower components of a Euclidean bispinor become

$${\mathbb{P}}_{\uparrow }=\frac{1}{2}(1+{\overline{\gamma }}_5){\mathbb{P}}_{\downarrow }=\frac{1}{2}(1-{\overline{\gamma }}_5)$$

the upper and lower spinor components belonging to equivalent fundamental representations of SU(2).

The Euclidean Action integral for the bispinor field is given by

$$S_E[{\psi }_E,{\overline{\psi }}_E]=\int {\mathrm{d}}^4{x}_E{\overline{\psi }}_E\left(x_E\right)\left(\partial {/}_E+M\right){\psi }_E\left(x_E\right)$$

(216)

where ${\psi }_E$ is a column bispinor, whereas ${\overline{\psi }}_E$ is an independent transposed or row bispinor. Here the overall sign as well as any overall factor are actually irrelevant and arbitrary, so that we could always absorb them into e.g. ${\overline{\psi }}_E$ - remember that we are allowed to change ${\overline{\psi }}_E$ and not ${\psi }_E$ or viceversa. Conversely, the lack of the factor i in front of the derivative term is not at all conventional: it is there just to ensure that the Euclidean fermion propagator, a.k.a. the 2-point Schwinger’s function, is proportional to $(ip{/}_E-M)/(p_E^2+M^2);$ if it were not for this $i,$ then we would have tachyon poles after transition back to the Minkowski space.

It is worth to notice that the above spinor Euclidean Action integral can be obtained from the corresponding one in the Minkowski space, after the customary standard replacements (201) and (202)

$$x_4=i{x}_0,{\overline{\gamma }}_k\equiv -i{\gamma }^k(k=1,2,3),{\overline{\gamma }}_4\equiv {\gamma }_0$$

$$\psi \left(x\right)\mapsto {\psi }_E\left(x_E\right)\overline{\psi }\left(x\right)\mapsto e^{i\theta }{\overline{\psi }}_E\left(x_E\right)(0\le \theta <2\pi )$$

in such a manner that we can always set

$$\begin{array}{ccc}\hfill iS[\psi ,\overline{\psi }]& \mapsto & S_E[{\psi }_E,{\overline{\psi }}_E]=\int {\mathrm{d}}^4{x}_E{\overline{\psi }}_E\left(x_E\right)\left(\partial {/}_E+M\right){\psi }_E\left(x_E\right)\hfill \end{array}$$

(217)

Furthermore, the Euclidean Dirac operator $\left(\partial {/}_E+M\right)$ is precisely the one that gives, according to the definition (204), the 2-point Schwinger function inversion formula

$${\left(\partial {/}_E+M\right)}_{\alpha \beta }{S}_{\beta \eta }^E\left(x_E\right)=\delta \left(x_E\right){\delta }_{\alpha \eta }$$

Finally we summarize the basic formulas

$$\begin{array}{ccc}& & S_E[\overline{\psi },{\psi }_E]=\int {\mathrm{d}}^4{x}_E\overline{\psi }\left(x_E\right)\left(\partial {/}_E+M\right){\psi }_E\left(x_E\right)\hfill \\ & & S_{\alpha \beta }^F(-i{x}_4,\mathbf{x})\to -S_{\alpha \beta }^E\left(x_E\right)\hfill \end{array}$$

(218)

$$\begin{array}{ccc}& & S_{\alpha \beta }^E\left(x_E\right)=\int {\displaystyle \frac{{\mathrm{d}}^4{p}_E}{{\left(2\pi \right)}^4}}exp\{i{p}_E·x_E\}{〔{\displaystyle \frac{i}{-p{/}_E+iM}}〕}_{\alpha \beta }\hfill \\ & & {\left(\partial {/}_E+M\right)}_{\alpha \beta }{S}_{\beta \eta }^E\left(x_E\right)=\delta \left(x_E\right){\delta }_{\alpha \eta }\hfill \end{array}$$

(219)

It is important to highlight that the Euclidean Action integral for the bispinor field is by no means a real quantity.

What we have just presented is a possible solution for a Euclidean field theory obtained via a Wick rotation from a Minkowski field theory. The conclusion is that, as long as we consider the prescription here introduced, we end up with the doubling of the fermionic fields: the $\overline{\psi }$ in the Action has to be replaced with a bispinor independent of $\psi$, moreover we have to abandon the expectation of a real Action integral. Moreover, in this presentation there is no room for a Euclidean field theory of Weyl fermions. There are however more refined prescriptions, [37,38,39,40,41,42,43], which, for instance, supplement the Wick rotation with a further rotation of the spinor fields. In these new approaches one can impose hermiticity on the Action for a Euclidean Dirac spinor. It is also possible to define a Euclidean Action for Weyl spinors, but it remains impossible to comply with hermiticity: in the latter case the Action is not Hermitean and, as a consequence, the integrand in the path integral is complex.

The gist of this Section can be summarized with the following warning: when a field theory contains Weyl fermions one may be tempted to turn it into a classical Euclidean field theory and quantize this theory, instead of the original Minkowski one, with the aim of obtaining automatically Wick-rotated Feynman diagrams, already prepared for dimensional regularization. Nonetheless, this procedure does not make sense, because we have seen above that the Euclidean field theory one obtains in this way does represent a completely different physics, if any. This fact shares some close resemblance with the problem, illustrated above, of PV regularization: one cannot formulate a local field theory that automatically produces PV regularized Feynman diagrams, for the trivial reason that the inverse of the sum of kinetic operators is not the sum of their inverses.

8. The Family’S Index Theorem and Weyl Fermion Propagators

The existence or non-existence of a fermion propagator in a theory of Weyl fermions is related to the appearance of a special kind of anomalies which we call type O (O stands for obstructive). The latter are also referred to, somewhat neutrally, as consistent chiral anomalies. They are considered as a hazard for a theory, but they are only a symptom of a disease: the true disease is the lack of the Weyl fermion propagator. A fermion primer which does not cover this important correspondence would be largely incomplete; on the other hand an acceptable treatment of this issue would require a long digression in advanced mathematics, which is outside the scope of this paper. Therefore we end it with a summary we hope will be enough to appreciate the importance of the point and the rule that comes out of it.

For simplicity we limit here to the stage of gauge field theories. In a gauge field theory we have to do with spinors that transform according to definite representations of the gauge group $\mathsf{G}$. So, we consider the tensor product of a spinor bundle $S_{\mathbb{C}}^{\pm }$ with a vector bundle E corresponding to a representation $\rho$ of the structure group $\mathsf{G}$ of $\mathsf{P}(X,\mathsf{G})$: $S_{\mathbb{C}}^{\pm }\otimes E\equiv S_{\mathbb{C}}\left(E\right)$. The relevant connection will be the spin connection plus a gauge connection $V=V_{\mu }d{x}^{\mu }$ valued in the representation $\rho$ of the Lie algebra of $\mathsf{G}$ with antihermiten generators. The corresponding Dirac operator

$$\begin{array}{c}\hfill \cancel{D}=\cancel{D}+i\cancel{V}\end{array}$$

(220)

acts on the space of sections $S_{\mathbb{C}}\left(E\right)$, i.e. the space of spinor fields, and maps it to itself. The latter splits as

$$\begin{array}{c}\hfill S_{\mathbb{C}}\left(E\right)=S_{\mathbb{C}}^{+}\left(E\right)\oplus S_{\mathbb{C}}^{-}\left(E\right)\end{array}$$

(221)

where ⁺ and ⁻ denote chiralities, and eq.(220) splits accordingly as

$$\begin{array}{c}\hfill \cancel{D}=\left(\begin{array}{cc}0& {\cancel{D}}^{-}\\ {\cancel{D}}^{+}& 0\end{array}\right)\end{array}$$

(222)

The Atiyah-Singer index theorem reads:

$$\begin{array}{c}\hfill Ind\left({\cancel{D}}^{+}\right)\equiv dim\left(ker{\cancel{D}}^{+}\right)-dim\left(ker{\cancel{D}}^{-}\right)={\int }_{X}ch\left(E\right)\widehat{A}\left(X\right)\end{array}$$

(223)

$ker{\cancel{D}}^{\pm }$ denotes the space of zero modes of ${\cancel{D}}^{\pm }$ in a specific spacetime manifold X. The symbol $ch\left(E\right)$ indicates the Chern character of the E bundle, i.e. the rational characteristic class given in terms of the curvature F of V by

$$\begin{array}{c}\hfill ch\left(E\right)=r+{\displaystyle \frac{i}{2\pi }}\mathrm{tr}F+{\displaystyle \frac{i^2}{2{\left(2\pi \right)}^2}}\mathrm{tr}{F}^2+\dots +\dots \end{array}$$

(224)

where r is the dimension of the representation $\rho$. The symbol $\widehat{A}\left(X\right)$ denotes the $\widehat{A}$-genus, which is the (rational) Pontryagin characteristic class of X. It can be expressed in terms of the Riemann curvature R as follows

$$\begin{array}{ccc}\hfill \widehat{A}\left(X\right)& =& 1+{\displaystyle \frac{1}{{\left(4\pi \right)}^2}}{\displaystyle \frac{1}{12}}\mathrm{tr}{R}^2+{\displaystyle \frac{1}{{\left(4\pi \right)}^4}}\left[{\displaystyle \frac{1}{288}}\mathrm{tr}{\left(R^2\right)}^2+{\displaystyle \frac{1}{360}}\mathrm{tr}{R}^4\right]+\dots \hfill \end{array}$$

(225)

This is however the index theorem for a fixed potential V. In gauge field theories we have to do with a family of potentials $\mathcal{A}$, forming orbits of gauge transformations, in which the relevant space is the quotient $\mathcal{Q}={\displaystyle \frac{\mathcal{A}}{\mathcal{G}}}$, the moduli space of the theory. In this case the $ker{\cancel{D}}^{\pm }$ , i.e. the spaces of zero modes of ${\cancel{D}}^{\pm }$ is not fixed, but varies in dimension from point to point of $\mathcal{Q}$, i.e. they are bundles over $\mathcal{Q}$. Therefore the index in this case is the difference of two bundles. Differences of bundles are described by K theory. Using the latter Atiyah and Singer could write down the formula for the index of a family of operators.

In the gauge theory case just outlined in which the Dirac operator acts between two families of sections $\Gamma (S_{\mathbb{C}}^{+}\otimes V)$ and $\Gamma (S_{\mathbb{C}}^{-}\otimes V)$ the index theorem can be expressed in the form

$$\begin{array}{c}\hfill ch\left(ind\left({\cancel{D}}^{+}\right)\right)={\int }_{\mathsf{M}}ch\left(V\right)·\widehat{A}\left(T\mathcal{Q}\right)\end{array}$$

(226)

where $ch$ is the Chern character, which for a vector bundle V is defined by

$$\begin{array}{c}\hfill ch\left(V\right)=rank\left(V\right)+c_1\left(V\right)+{\displaystyle \frac{1}{2}}(c_1{\left(V\right)}^2-2c_2\left(V\right))+\dots \end{array}$$

(227)

In (226), $\mathsf{M}$ is a generic spacetime manifold, $\mathcal{Q}\equiv {\displaystyle \frac{\mathcal{A}}{\mathcal{G}}}$ is the orbit space of connections, $T\mathcal{Q}$ is its tangent space and V is the gauge bundle. The Chern character of the index measures the extent to which $ker{\cancel{D}}^{+}$ differs from $coker{\cancel{D}}^{+}=ker{\cancel{D}}^{-}$. It is intuitive that, as long as, $ind\left({\cancel{D}}^{+}\right)$ differs from 0, the inverse of ${\cancel{D}}^{+}$ cannot exists

The cohomology classes relevant to a gauge theory in 4d are of order 4 and 6. The order 6 ones, via the transgression formula, give rise to the consistent gauge anomalies. Therefore we see that consistent gauge anomalies are in one to one correspondence with cohomology classes that obstruct the invertibility of the Dirac-Weyl operator and for this reason are called obstructive (type O). But there are also the classes of order 4: they are proportional to $\mathrm{tr}\left(R^2\right)$ and $\mathrm{tr}\left(F^2\right)$, where R is the Riemann curvature and F, as usual, the gauge curvature. They are known as Pontryagin and Chern classes, respectively. In field theory they may appear as odd parity trace anomalies. It is clear that, independently of the field theory interpretation, the corresponding obstructions must be canceled in order to permit the inversion of the Dirac-Weyl operator, i.e. the existence of the corresponding propagator.

Some specifications are in order. When the potential A is a fixed one and $\mathsf{M}$ a precise space-time manifold, like in the case of the ordinary index theorem (223), one can, disregarding the zero modes, still define a product of nonzero eigenvalues and interpret it as a partition function. But in the case considered here the zero modes vary from point to point of $\mathcal{Q}$ and $\mathsf{M}$ represents all possible manifolds, see [3,29], such an operation is generically impossible. The integration over $\mathsf{M}$ should not mislead the reader. Contrary to the case of the ordinary index theorem where the topology of spacetime plays a major role, the family’s index theorem is formulated in a universal form, in terms of classifying space and classifying bundles, meaning that it holds independently of the topology of the spacetime. This matches exactly the sense of perturbative approach to field theory where locality requires to make the calculations in a local patch, so that the topology of spacetime is irrelevant.

Another due specification is about the proof of the family’s index theorem: to the best of our knowledge it is available only in the Euclidean version. Following a well established and sensible tradition we considers its statement valid also in Minkowski spacetimes. This does not contradict what has been said in the previous Section: in the present Section we consider the problem of inverting the Dirac-Weyl kinetic operator, in the previous section the object was the construction of a canonical Action involving the same operator as kinetic operator. Both issues are studied in a Euclidean setting, but are different.

Once these clarifications are set forth, the family’s index theorem establishes an uncontroversial rule for perturbative calculations in local field theory. Any method or procedure that concerns the computations of effective Actions, conservation laws and anomalies must be in agreement with this theorem. The family’s index theorem provides the backbone for the calculations of this type.

9. Conclusion

The purpose of this paper was trying to clarify a few controversial or unclear issues concerning fermions, their definition and their use. For this reason we have started in Section 2 and 3 with a review of the their basic properties. In particular, Section 3 is an elementary and detailed construction of Weyl fermion solutions of the 4d Dirac equations, by means of which both Dirac and Majorana massless spinors have been constructed. The corresponding quantum fields have been introduced, their specific properties under discrete symmetry transformations have been established and their causal Green’s functions properly constructed. The general properties of Dirac, Weyl and Majorana fields have been summarized in Section 4. All this preparatory material has been brought to bear in the subsequent Sections. Section 5 is about the perturbative calculation of the effective Action of a Weyl fermion coupled to an Abelian gauge potential. This example exhibits one of the difficult issues considered in this review: the non-existence of the Weyl fermion propagator. This problem is tackled in the most popular way met in the literature: one adds to the original theory a free Weyl fermion with opposite chirality.

The example carried out in Section 5 with three different regularization (dimensional, PV and cutoff) shows that an accurate treatment lead to the same result for all three, as it should, but especially that no mass term can arise from quantization and that the added Weyl fermion of opposite chirality remain free after quantization, so that chirality of the original theory is preserved. The case considered in Section 5 has been then analyzed in general in Section 6. The main result of this analysis has been that the practice of replacing the Weyl propagator, which does not exist, with a Dirac propagator, is however flawed by an underlying logical loophole, so that this trick may accidentally lead correct results but does not have any probatory value. It has been shown that there are ways to circumvent this obstacle by enlarging the space of potentials, such as for instance adding an axial potential to the original vector one.

Section 6 has been devoted to dispelling a not infrequent identification of massless Majorana with Weyl fermions. It is argued that while these two types of fermions have a few characteristics in common, they are crucially different in other fundamental aspects: Weyl fermions are eigenstates of the chirality matrix, while Majorana fermions do not have any definite chirality and no definite helicity. The chirality issue is fundamental in the framework of anomalies: all anomalies with opposite sign for opposite chiralities (the dangerous ones) are absent for Dirac and Majorana fermions. For the very same reason Dirac and Majorana fermions can have a mass term, while for the Weyl ones a mass is forbidden. The difference between the Dirac and Majorana mass terms has been subsequently clarified. The main outcome of our analysis is that, in spite of being build-up by the very same two-component Weyl spinor, the chiral Weyl bispinor and the massless Majorana spinor do exhibit quite different and distinguished mathematical and physical properties and behavior.

There are two cases in which locality of the classical Action is put in jeopardy. One is when the PV regularization is inappropriately transferred from the realm of Feynman diagrams to that of local Actions by confusing the sum of inverses of differential operators (the PV ghost propagators) with the inverse of the sum of the corresponding kinetic differential operators to construct a local Action. This point has been insisted on throughout the paper. There is a similar issue related to Wick rotation. One may be tempted to turn to the Euclidean metric already in the classical Action, but it was shown in Section 7 that the theory one ends up with in this way is a physical system different from the starting one. The Wick rotation is an algorithm concerning the regularization of Feynman diagrams and related amplitudes, which cannot be transferred back mechanically to te original Action integral.

The final Section is devoted to the family’s index theorem. Although anomalies are not a central issue in this review, they nevertheless loom all through it. The anomalies called obstructive (the dangerous ones) pop up in chiral theories as a signal of a disease: the lack of chiral fermion propagators. The connection between anomalies and (non-existence of) chiral propagators is provided by the Atiyah-Singer index theorem. For this reason we have deemed it useful to give a summary review of it, certainly insufficient but hopefully enough to stir the reader’s interest.