A State-Space Symmetry in the Electroweak Interaction Hamiltonian

1. Introduction

As one of the foundational principles in modern physics, interactions among elementary particles are assumed to possess certain basic symmetries, particularly, the Lorentz symmetry and gauge symmetry. Constraints from the symmetries are so strong that they fix major features of the interaction Lagrangians of particles in the so-called standard model (SM), the most successful quantum field theory (QFT) that has ever been developed (see, e.g., textbooks [1,2,3]). However, serious difficulties are met within the SM and, in fact, going beyond SM as a topic has attracted lots of attention in recent decades (see, e.g., Refs.[4,5,6,7,8,9,10,11,12,13,14,15,16]).

One question of persistent interest is about possible other symmetry that may be of fundamental importance. The aim of this paper is to discuss such a possibility. It is related to a property of the form of the ${\gamma }^{\mu }$ matrices that is used in practical computations; that is, their matrix elements remain invariant under Lorentz transformations applied simultaneously to both the spinor and vector indices. Within the present theoretical framework, this invariance is brought primarily for the sake of convenience in practical derivations and computations. In other words, it is usually regarded as an “accidental” symmetry, not of any fundamental relevance.

The ${\gamma }^{\mu }$ matrices essentially function as connecting certain Dirac-spinor indices and four-component vector indices. In other words, they provide a connection for the related spaces. As is known, a powerful mathematical theory for dealing with spinor spaces is the theory of spinors, which is based on the so-called $SL(2,\mathbb{C})$ group — a covering group of the proper orthochronous Lorentz group [17,18,19,20,21]. 1 In the spinor theory, the most fundamental spinors are two-component (Weyl) spinors, from which all other types of spinors can be constructed. And, Weyl spinors and four-component vectors are connected by the so-called Enfeld-van der Waerden symbols, in short, EvdW symbols. The EvdW symbols are invariant under $SL(2,\mathbb{C})$ transformations, that is, they also possess the aforementioned symmetric property of the ${\gamma }^{\mu }$ matrices. In fact, in their chiral representation, the ${\gamma }^{\mu }$ matrices are constructible from the EvdW symbols.

The EvdW symbols possess a clear geometric meaning regarding spinor spaces. That is, they describe an isomorphic map between the direct product of two Weyl-spinor spaces (for the left-handed (LH) and right-handed (RH) parts of Dirac spinors, respectively) and a four-component-vector space. And, this map is faithfully described by an operator form of the EvdW symbols.

The main goal of this paper is to show that a major part of the Glashow-Weinberg-Salam (GWS) interaction Hamiltonian may be reformulated in a form, in which the EvdW-symbol operator plays a central role. 2 For the sake of simplicity in discussion, we are to discuss the first generation of leptons only. The GWS electroweak interaction Hamiltonian denoted by $H_{\mathrm{int}}^{\mathrm{gws}}$, which is obtained after the Higgs mechanism has been used to introduce particle masses, consists of two parts,

$$\begin{array}{c}\hfill H_{\mathrm{int}}^{\mathrm{gws}}=H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}+H_{\mathrm{int},\mathrm{II}}^{\mathrm{gws}},\end{array}$$

(1)

where $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ includes all those interactions that are between leptons and vector bosons, while, $H_{\mathrm{int},\mathrm{II}}^{\mathrm{gws}}$ includes interactions merely among vector bosons. The aforementioned part of the GWS Hamiltonian to be reformulated refers to $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$. For the purpose here, it proves convenient to discuss in the Schrödinger picture, which is to be adopted.

To achieve the above mentioned goal, a main observation is that quantized fields may be written in forms sharing a state-space geometric feature, which is characterized by the related identity operators. More exactly, as to be explained in detail in later sections, formally and in the ket-bra notation, such a field may be written as a projection of the identity operator for the related single-particle state space on certain basis. We are to show that this feature enables a reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$, which exhibits a simple geometric structure whose core is the EvdW-symbol operator.

The paper is organized as follows. The operator form of the EvdW symbols, as well as basic properties of spinors, are discussed in Section 2. Basic results of this paper, particularly the above mentioned reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$, are presented in Section 3. Derivation of the reformulation is given in Section 4. Finally, a summary and discussions are given in Section 5.

2. Operator form of EvdW Symbols

In this section, we discuss the operator form of the EvdW symbols, which is to be denoted by $\sigma$, written in the abstract ket-bra form. 3 For this purpose, detailed properties of Weyl spinors and four-component vectors are needed, which are to be recalled in Section 2.1. Bras for spinors are discussed in Section 2.2 and the operator $\sigma$ is introduced in Section 2.3.

Concerning notation, the convention of repeated index implying summation is to be obeyed, unless otherwise stated. And, we are to use a tilde above a spinor to indicate its complex conjugate; and, similar for symbols related to spinors. 4

2.1. Spinors and Their Spaces

Spinors are classified according to their behaviors under transformations of the $SL(2,\mathbb{C})$ group. 5 Below are main properties of spinors to be used in later discussions.

(a)

Weyl spinors and Dirac spinors.

The two smallest nontrivial representation spaces of the $SL(2,\mathbb{C})$ group are denoted by $\mathcal{W}$ and $\tilde{\mathcal{W}}$, which are spanned by two-component Weyl spinors. 6 The two spaces are the complex conjugate of each other. 7 An arbitrary spinor in $\mathcal{W}$ possesses two components written as, say, ${\kappa }_A$, where A is the spinor index with $A=0,1$. The complex conjugate of ${\kappa }_A$ gives a Weyl spinor in the space $\tilde{\mathcal{W}}$, written as ${\tilde{\kappa }}_{A^{\prime }}$ with a primed index $A^{\prime }=0^{\prime },1^{\prime }$.

As is known, a Dirac spinor is written as a direct sum of two Weyl spinors, one in $\mathcal{W}$ and the other in $\tilde{\mathcal{W}}$. For example, a Dirac spinor $U^r\left(\mathbf{p}\right)$, as a solution of the free Dirac equation, is written as

$$\begin{array}{c}{U}^r\left(\mathbf{p}\right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}{u}^{r,A}\left(\mathbf{p}\right)\\ {\tilde{v}}_{B^{\prime }}^r\left(\mathbf{p}\right)\end{array}\right).\end{array}$$

(2)

where $u^{r,A}\left(\mathbf{p}\right)\in \mathcal{W}$ and ${\tilde{v}}_{B^{\prime }}^r\left(\mathbf{p}\right)\in \tilde{\mathcal{W}}$. Here, r ($r=0,1$) is the ordinary spin label, which is raised by the Kroneck symbol ${\delta }^{rs}$ and lowered by ${\delta }_{rs}$.

(b)

Space of four-component vectors.

We use $\mathcal{V}$ to indicate the spinor space of four-component vectors, which is in fact spanned by spin states of vector bosons. The space $\mathcal{V}$ is isomorphic to $\mathcal{W}\otimes \tilde{\mathcal{W}}$, as known in the theory of spinors. This implies that the complex-conjugate space of $\mathcal{V}$ is just itself, i.e., $\tilde{\mathcal{V}}=\mathcal{V}$.

(c)

Raising and lowering Weyl-spinor indices.

Two symbols of ${ϵ}^{AB}$ and ${ϵ}_{AB}$ are used for raising and lowering indices of spinors in $\mathcal{W}$, respectively, which have the matrix expression of

$$\begin{array}{c}\hfill \left[{ϵ}^{AB}\right]=\left[{ϵ}_{AB}\right]=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right).\end{array}$$

(3)

For example,

$$\begin{array}{c}\hfill {\kappa }^A={ϵ}^{AB}{\kappa }_B,{\kappa }_A={\kappa }^B{ϵ}_{BA}.\end{array}$$

(4)

The corresponding symbols for $\tilde{\mathcal{W}}$ are written as ${ϵ}^{A^{\prime }{B}^{\prime }}$ and ${ϵ}_{A^{\prime }{B}^{\prime }}$, respectively, described by the same matrix.

(d)

Kets of spinors in the spaces of $\mathcal{W},\tilde{\mathcal{W}}$, and $\mathcal{V}$.

We use a ket $|S^A\rangle$$(A=0,1)$ to indicate a basis in the space $\mathcal{W}$. Thus, a Weyl spinor ${\kappa }_A\in \mathcal{W}$ is written as

$$\begin{array}{c}\hfill |\kappa \rangle ={\kappa }_A|S^A\rangle .\end{array}$$

(5)

Correspondingly, a basis in $\tilde{\mathcal{W}}$, as the complex conjugate of $|S^A\rangle$, is written as $|{\tilde{S}}^{A^{\prime }}\rangle$. The complex conjugate of $|\kappa \rangle$, namely $|\tilde{\kappa }\rangle \in \tilde{\mathcal{W}}$, is written as $|\tilde{\kappa }\rangle ={\tilde{\kappa }}_{A^{\prime }}|{\tilde{S}}^{A^{\prime }}\rangle$.

A basis in a four-component vector space $\mathcal{V}$ is written as $|T^{\mu }\rangle$ ($\mu =0,1,2,3$). On this basis, an arbitrary ket $|K\rangle \in \mathcal{V}$ is expanded as $|K\rangle =K_{\mu }|T^{\mu }\rangle$. For example, the ordinary polarization vectors ${\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right)$ ($\lambda =0,1,2,3$) are written as $|{\epsilon }^{\lambda }\left(\mathbf{k}\right)\rangle ={\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right)|T^{\mu }\rangle$. The index $\mu$ is raised and lowered by the Minkovski metric $g^{\mu \nu }$ and $g_{\mu \nu }$, respectively [cf. Eq.(A137) in Appendix A.3], and similar for the label $\lambda$.

(e)

The direct-product space $\mathcal{W}\otimes \tilde{\mathcal{W}}$.

Basis spinors in this space are written as $|S_{A{B}^{\prime }}\rangle \equiv |S_A\rangle |{\tilde{S}}_{B^{\prime }}\rangle$. The following anticommutation relation is to be used,

$$\begin{array}{c}\hfill |S_A\rangle |{\tilde{S}}_{B^{\prime }}\rangle =-|{\tilde{S}}_{B^{\prime }}\rangle |S_A\rangle |,\end{array}$$

(6)

in computations that will be carried out later for the interaction amplitudes. 8

2.2. Bras for Spinors

One main purpose of introducing bras in quantum mechanics is to express inner product in a convenient way. However, in the theory of spinors, of primary importance is the concept of scalar product under $SL(2,\mathbb{C})$ transformations, but not inner product. This requires that bras should be first of all introduced for scalar products (see, e.g., discussions given in Ref.[21]). Below, we call such a bra a scalar-product-based bra and use the symbol of “$\langle \langle ·|$” to indicate it. For example, such a bra for $|S^A\rangle$ is written as $\langle \langle S^A|$. 9

In the theory of spinors, the most fundamental scalar product is that of Weyl spinors. In the space $\mathcal{W}$, it is written as ${\kappa }_A{\chi }^A$ for two arbitrary Weyl spinors ${\kappa }^A$ and ${\chi }^A$. (This scalar product is not an inner product.) In the ket-bra form, this product is written as

$$\langle \langle \kappa |\chi \rangle \equiv {\kappa }_A{\chi }^A.$$

(7)

For a ket $|\kappa \rangle$ expanded in Eq.(5), this requires that the scalar-product-based bra $\langle \langle \kappa |$ should be written as

$$\begin{array}{c}\hfill \langle \langle \kappa |=\langle \langle S^A|{\kappa }_A.\end{array}$$

(8)

Making use of Eq.(4), it is straightforward to find that

$$\langle \langle S^A|S^B\rangle ={ϵ}^{AB}.$$

(9)

The antisymmetry of ${ϵ}^{AB}$ implies that

$$\langle \langle \chi |\kappa \rangle \equiv {\chi }_A{\kappa }^A=-\langle \langle \kappa |\chi \rangle \equiv -{\kappa }_A{\chi }^A.$$

(10)

It is easy to check that [cf. Eq.(A101)]

$${\kappa }^A=\langle \langle S^A|\kappa \rangle ,{\kappa }_A=\langle \langle S_A|\kappa \rangle .$$

(11)

The above discussions are also applicable to spinors in the space $\tilde{\mathcal{W}}$.

For the space $\mathcal{V}$, we use $\langle \langle T^{\mu }|$ to indicate a scalar-product-based bra basis, corresponding to the basis of $|T^{\mu }\rangle$ in $\mathcal{V}$. The scalar product of two four-component vectors $K^{\mu }$ and $J^{\mu }$, which has the form of $K_{\mu }{J}^{\mu }$, is written as $\langle \langle K|J\rangle$ in the abstract notation. Similar to Eq.(8), the scalar-product-based bra of $|K\rangle =K_{\mu }|T^{\mu }\rangle$ is expanded as

$$\begin{array}{c}\hfill \langle \langle K|=\langle \langle T^{\mu }|K_{\mu },\end{array}$$

(12)

with the bases satisfying the following relation,

$$\begin{array}{c}\hfill \langle \langle T^{\mu }|T^{\nu }\rangle =g^{\mu \nu }.\end{array}$$

(13)

It is easy to check that $\langle \langle K|J\rangle =\langle \langle J|K\rangle$ and

$$\begin{array}{cc}\hfill & K^{\mu }=\langle \langle T^{\mu }|K\rangle ,K_{\mu }=\langle \langle T_{\mu }|K\rangle .\hfill \end{array}$$

(14)

Now, we discuss bras useful for single-particle spin states. As is well known, transferring kets for single-particle states to the corresponding bras should involve a procedure of complex conjugation. However, complex conjugation is not involved in Eqs.(8) and (12). 10 Hence, for the purpose of constructing spinor bras for single-particle states from scalar-product-based bras, further procedures are needed as to be discussed below.

For bosons, whose spin states are described by four-component vectors, the above problem is easily solved by the fact that the space $\mathcal{V}$ coincides with its complex-conjugate space. That is, for a ket $|K\rangle$, the bra useful for single-particle states, which is to be written in the ordinary notation of $\langle K|$, is simply given by

$$\begin{array}{c}\hfill \langle K|:=\langle \langle \tilde{K}|.\end{array}$$

(15)

Thus, e.g., the bra of a single-boson ket $|B\rangle =|\mathbf{k}\rangle |{\epsilon }_{\lambda }\left(\mathbf{k}\right)\rangle$ is written as $\langle B|=\langle {\tilde{\epsilon }}_{\lambda }\left(\mathbf{k}\right)|\langle \mathbf{k}|$, in consistency with the usual description.

For fermions, whose spin states are described by Dirac spinors as direct sums of Weyl spinors — one in $\mathcal{W}$ and the other in $\tilde{\mathcal{W}}$, the situation is a little more complicated than boson, because the complex conjugate of $\mathcal{W}$ is not itself. To solve this problem, one may follow the standard procedure in the construction of inner product of Dirac spinors, in which positions of the two Weyl spinors in a Dirac spinor is converted. In the abstract notation, the bra thus obtained is called a hat-bra of Dirac spinor, as discussed in Ref.[21] and briefly recalled in Appendix A.2. For example, the hat-bra of a Dirac spinor $\left|U\right(\mathbf{p})\rangle$ is indicated as $\langle \widehat{U}\left(\mathbf{p}\right)|$ [see Eq.(A124a)] and the bra of a single-fermion ket $|f\rangle =|\mathbf{p}\rangle \left|U\right(\mathbf{p})\rangle$ is written as $\langle f|=\langle \widehat{U}\left(\mathbf{p}\right)|\langle \mathbf{p}|$.

2.3. $\sigma$ as an Invariant Isomorphic-Map Operator

Now, we are ready to present the mathematical expression of $\sigma$, as the operator form of the EvdW symbols. It is written as 11

$$\sigma :=|T_{\mu }\rangle {\sigma }_{A{B}^{\prime }}^{\mu }\langle \langle {\tilde{S}}^{B^{\prime }}|\langle \langle S^A|,$$

(16)

where ${\sigma }_{A{B}^{\prime }}^{\mu }$ are the EvdW symbols. Clearly, $\sigma$ maps the space of $\mathcal{W}\otimes \tilde{\mathcal{W}}$ to the space of $\mathcal{V}$. Properties of the EvdW symbols ${\sigma }^{\mu A{B}^{\prime }}$ guarantees that $\sigma$ generates an isomorphic map between the two spaces.

As is known, in the chiral representation, the ${\gamma }^{\mu }$-matrices have the following form in terms of the EvdW symbols [17,18,19,20,21], i.e.,

$$\begin{array}{c}{\gamma }^{\mu }=\left(\begin{array}{cc}0& {\sigma }^{\mu A{B}^{\prime }}\\ {\tilde{\sigma }}_{A^{\prime }B}^{\mu }& 0\end{array}\right).\end{array}$$

(17)

And, in consistency with the Lorenz invariance of the ${\gamma }^{\mu }$ matrices, ${\sigma }^{\mu A{B}^{\prime }}$ are constant matrices in the sense of being invariant under $SL(2,\mathbb{C})$ transformations, namely, under simultaneous rotations in the three spaces of $(\mathcal{W},\tilde{\mathcal{W}},\mathcal{V})$ [see Eqs.(A165)-(A166)]. In fact, mathematically, it is more convenient to set ${\sigma }^{\mu A{B}^{\prime }}$ as constant matrices and, then, from this requirement derive properties of the transformations of the space $\mathcal{V}$. It turns out that the transformations constitute the (restricted) Lorentz group and, hence, the space $\mathcal{V}$ is composed of four-component vectors. (Appendix B).

Clearly, the above discussed invariance of the EvdW symbols is not due to the Lorentz symmetry, which merely requires that the symbols should change covariantly under Lorentz transformations, not imposing invariance. In other words, the invariance reflects a symmetry, which is stronger than the Lorentz symmetry.

3. Reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$

In this section, we present the main result of this paper, as a reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ which shows the key role played by $\sigma$ in an explicit way. Its derivation will be given in the next section. Specifically, the ordinary formulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ is recalled in Section 3.1, with analysis for a basic strategy to be adopted in later discussions. Descriptions of single-particle states, following the above mentioned basic strategy, are discussed in Section 3.2. The reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ is presented in Section 3.3 and further explanations to its contents are given in Section 3.4.

3.1. Ordinary Formulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ and Some Analysis

In this section, we first recall the ordinary formulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$, which is written in the mass representation obtained by the Higgs mechanism. Then, we give some analysis in a potential geometric interpretation to quantized fields, as well as the Hamiltonian, which leads to a basic strategy to be employed in later discussions.

For fermions, we are to discuss only those in the first generation of leptons — electron, positron, electron neutrino, and electron antineutrino, which are to be indicated as f. For bosons, we discuss the four mediating vector bosons — photon, $Z^0$ boson, and $W^{\pm }$ bosons, which are to be indicated as B.

In the ordinary formulation, $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ contains six terms which are to be labelled by $k=1,2,3,4,5L,5R$; that is,

$$\begin{array}{cc}\hfill H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}& =\sum _k{H}_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}.\hfill \end{array}$$

(18)

Densities of $H_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}$, denoted by ${\mathcal{H}}_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}$ with $H_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}=\int d^{3}x{\mathcal{H}}_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}$, are written as [22]

$$\begin{array}{cc}\hfill & {\mathcal{H}}_{\mathrm{int},\mathrm{I},1}^{\mathrm{gws}}=:{\xi }_1{A}_{\mu }{\psi }_e^{†}{\gamma }^0{\gamma }^{\mu }{\psi }_e:,\hfill \end{array}$$

(19a)

$$\begin{array}{cc}& {\mathcal{H}}_{\mathrm{int},\mathrm{I},2}^{\mathrm{gws}}=:{\xi }_2{W}_{\mu }^{-}{\psi }_{eL}^{†}{\gamma }^0{\gamma }^{\mu }{\psi }_{\nu L}:,\hfill \end{array}$$

(19b)

$$\begin{array}{cc}& {\mathcal{H}}_{\mathrm{int},\mathrm{I},3}^{\mathrm{gws}}=:{\xi }_3{W}_{\mu }^{+}{\psi }_{\nu L}^{†}{\gamma }^0{\gamma }^{\mu }{\psi }_{eL}:,\hfill \end{array}$$

(19c)

$$\begin{array}{cc}& {\mathcal{H}}_{\mathrm{int},\mathrm{I},4}^{\mathrm{gws}}=:{\xi }_4{Z}_{\mu }^0{\psi }_{\nu L}^{†}{\gamma }^0{\gamma }^{\mu }{\psi }_{\nu L}:,\hfill \end{array}$$

(19d)

$$\begin{array}{cc}& {\mathcal{H}}_{\mathrm{int},\mathrm{I},5\mathrm{L}}^{\mathrm{gws}}=:{\xi }_{5L}{Z}_{\mu }^0{\psi }_{eL}^{†}{\gamma }^0{\gamma }^{\mu }{\psi }_{eL}:,\hfill \end{array}$$

(19e)

$$\begin{array}{cc}& {\mathcal{H}}_{\mathrm{int},\mathrm{I},5\mathrm{R}}^{\mathrm{gws}}=:{\xi }_{5R}{Z}_{\mu }^0{\psi }_{eR}^{†}{\gamma }^0{\gamma }^{\mu }{\psi }_{eR}:,\hfill \end{array}$$

(19f)

where “$:\dots :$” indicate normal product. Here, ${\psi }_e$ and ${\psi }_{\nu }$ represent the electron and electron neutrino fields, respectively, which are Dirac fields; the subscripts “L” and “R” indicate the LH and RH parts of a field, respectively; $A_{\mu }$, $W_{\mu }^{\pm }$, and $Z_{\mu }^0$ represent fields for photon, $W^{\pm }$, and $Z^0$ bosons, respectively, all of which are four-component vector fields. The prefactors ${\xi }_k$ include two coupling constants of $e_0$ and g, which are connected by the Weinberg angle ${\theta }_W$;

$$\begin{array}{cc}\hfill & {\xi }_1=e_0=gsin{\theta }_W,\hfill \end{array}$$

(20a)

$$\begin{array}{cc}& {\xi }_2={\xi }_3=-{\displaystyle \frac{g}{\sqrt{2}}},\hfill \end{array}$$

(20b)

$$\begin{array}{cc}& {\xi }_4=-{\displaystyle \frac{g}{2cos{\theta }_W}},\hfill \end{array}$$

(20c)

$$\begin{array}{cc}& {\xi }_{5L}=-{\displaystyle \frac{g}{cos{\theta }_W}}(-{\displaystyle \frac{1}{2}}+{sin}^2{\theta }_W),\hfill \end{array}$$

(20d)

$$\begin{array}{cc}& {\xi }_{5R}=-{\displaystyle \frac{g}{cos{\theta }_W}}{sin}^2{\theta }_W.\hfill \end{array}$$

(20e)

All of the operators ${\mathcal{H}}_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}$ in Eq.(19) share a common formal feature, each being built from two fermionic fields and one bosonic field, connected by the ${\gamma }^{\mu }$-matrices. That is,

$$\begin{array}{c}\hfill {\mathcal{H}}_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}=:{\xi }_k{B}_{\mu }{\psi }_{f^{\prime }c}^{†}{\gamma }^0{\gamma }^{\mu }{\psi }_{fc}:,\end{array}$$

(21)

where

$$\begin{array}{c}\hfill B_{\mu }=\left\{\begin{array}{cc}{A}_{\mu },& \mathrm{for}k=1,\hfill \\ W_{\mu }^{-},& \mathrm{for}k=2,\hfill \\ W_{\mu }^{+},& \mathrm{for}k=3,\hfill \\ Z_{\mu }^0,& \mathrm{for}k=4,5L,5R,\hfill \end{array}\right.\end{array}$$

(22)

c is a chiral index,

$$\begin{array}{c}\hfill c:=\left\{\begin{array}{cc}T,\hfill & \mathrm{for}\mathrm{the}\mathrm{total}\mathrm{Dirac}\mathrm{spinor},\hfill \\ L,\hfill & \mathrm{for}\mathrm{the}\mathrm{LH}\mathrm{part}\mathrm{of}\mathrm{Dirac}\mathrm{spinor},\hfill \\ R,\hfill & \mathrm{for}\mathrm{the}\mathrm{RH}\mathrm{part}\mathrm{of}\mathrm{Dirac}\mathrm{spinor},\hfill \end{array}\right.\end{array}$$

(23)

and explicit dependence of $(f^{\prime },f)$ and c on k is as follows,

$$\begin{array}{c}\hfill (f^{\prime }c,fc)=\left\{\begin{array}{cc}(eT,eT)& \mathrm{for}k=1,\hfill \\ (eL,\nu L)& \mathrm{for}k=2,\hfill \\ (\nu L,eL)& \mathrm{for}k=3,\hfill \\ (\nu L,\nu L)& \mathrm{for}k=4,\hfill \\ \left(eL\right(R),eL(R\left)\right)& \mathrm{for}k=5L\left(R\right).\hfill \end{array}\right.\end{array}$$

(24)

The above fields can be expanded in terms of creation and annihilation operators. For example,the electron and photon fields are written as follows, 12

$$\begin{array}{c}{\psi }_e\left(\mathbf{x}\right)=\int d\check{p}{b}_r\left(\mathbf{p}\right)U^r\left(\mathbf{p}\right)e^{i\mathbf{p}·\mathbf{x}}+d_r^{†}\left(\mathbf{p}\right)V^r\left(\mathbf{p}\right)e^{-i\mathbf{p}·\mathbf{x}},\end{array}$$

(25a)

$$\begin{array}{c}{\psi }_e^{†}\left(\mathbf{x}\right)=\int d\check{p}{b}_r^{†}\left(\mathbf{p}\right)U^{†r}\left(\mathbf{p}\right)e^{-i\mathbf{p}·\mathbf{x}}+d_r\left(\mathbf{p}\right)V^{†r}\left(\mathbf{p}\right)e^{i\mathbf{p}·\mathbf{x}},\end{array}$$

(25b)

$$\begin{array}{c}{A}_{\mu }\left(\mathbf{x}\right)=\int d\check{k}{a}_{\lambda }\left(\mathbf{k}\right){\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right)e^{i\mathbf{k}·\mathbf{x}}+a_{\lambda }^{†}\left(\mathbf{k}\right){\epsilon }_{\mu }^{\lambda *}\left(\mathbf{k}\right)e^{-i\mathbf{k}·\mathbf{x}},\end{array}$$

(25c)

where $b_r\left(\mathbf{p}\right)$, $d_r\left(\mathbf{p}\right)$, and $a_{\lambda }\left(\mathbf{k}\right)$ indicate annihilation operators for electron, positron, and photon, respectively, with their Hermitian conjugates as creation operators. These operators satisfy well known (anti-)commutation relations, such as

$$\begin{array}{cc}\hfill & \{b_r^{†}\left(\mathbf{p}\right),b_s^{†}\left(\mathbf{q}\right)\}=0,\{d_r^{†}\left(\mathbf{p}\right),d_s^{†}\left(\mathbf{q}\right)\}=0,\hfill \end{array}$$

(26a)

$$\begin{array}{cc}& \{b_r^{†}\left(\mathbf{p}\right),d_s^{†}\left(\mathbf{q}\right)\}=0,\{b_r\left(\mathbf{p}\right),d_s^{†}\left(\mathbf{q}\right)\}=0,\hfill \end{array}$$

(26b)

$$\begin{array}{cc}& [a_{\lambda }^{†}\left(\mathbf{k}\right),a_{{\lambda }^{\prime }}^{†}\left({\mathbf{k}}^{\prime }\right)]=0,\hfill \end{array}$$

(26c)

and

$$\begin{array}{c}\{b_r\left(\mathbf{p}\right),b_s^{†}\left(\mathbf{q}\right)\}=p^0{\delta }^3(\mathbf{p}-\mathbf{q}){\delta }_{rs},\end{array}$$

(27a)

$$\begin{array}{c}\{d_r\left(\mathbf{p}\right),d_s^{†}\left(\mathbf{q}\right)\}=p^0{\delta }^3(\mathbf{p}-\mathbf{q}){\delta }_{rs},\end{array}$$

(27b)

where $p^0=\sqrt{{\mathbf{p}}^2+m^2}$ with m the electron mass.

Below, we give some analysis, showing a state-space geometric feature of the fields. Let us consider the first part of the electron field ${\psi }_e$ on the right-hand side (rhs) Eq.(25a). Equivalently, it may be written in the ket-bra notation, with $b_r\left(\mathbf{p}\right)\to \langle e_{\mathbf{p}r}|$ and $U^r\left(\mathbf{p}\right)e^{i\mathbf{p}·\mathbf{x}}=\langle \mathbf{x}|\langle S_D|e_{\mathbf{p}}^r\rangle$, where $|e_{\mathbf{p}}^r\rangle =|U^r\left(\mathbf{p}\right)\rangle |\mathbf{p}\rangle$ and $\langle S_D|$ represents a Dirac spinor basis. Then, this part is written as 13

$$\begin{array}{cc}\hfill & \int d\check{p}{b}_r\left(\mathbf{p}\right)U^r\left(\mathbf{p}\right)e^{i\mathbf{p}·\mathbf{x}}\hfill \\ & \to \langle \mathbf{x}{S}_D|\left(\int d\check{p}|e_{\mathbf{p}}^r\rangle \langle e_{\mathbf{p}r}|\right)\equiv \langle \mathbf{x}{S}_D|I_e,\hfill \end{array}$$

(28)

where $I_e$ represents the identity operator on the single-electron state space. It is seen that this part of the electron field has a quite simple geometric meaning in view of state space; i.e., it is essentially given by the identity operator for the single-electron state space, with a “projection” on a basis of $\langle \mathbf{x}{S}_D|$. One notes that the identity operator is a characteristic operator for the state space. Moreover, it is not difficult to see that the photon field $A_{\mu }\left(\mathbf{x}\right)$ may be treated in a similar way, getting a similar geometric interpretation in terms of the identity operator on the single-photon state space.

However, the above treatment does not work directly for the second part of ${\psi }_e$, because $V^r\left(\mathbf{p}\right)$ possesses a negative scalar product with itself, while positron’s anticommutation possesses a positive sign as seen in Eq.(). Later, we’ll show that this problem is solvable by a modification to the description for positron’s spin states, which is in a direct consistency with Eq.() and does not change amplitudes of the interaction Hamiltonian. 14

With the quantized fields written in forms that manifest the above discussed geometric feature, one may find that some part of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ can be written in a form with a simple geometric interpretation. But, this treatment does not work for the rest parts of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$. The obstacle lies in that the ${\gamma }^{\mu }$ matrices do not match the identity operators in an appropriate way for the rest parts. We are to show that this problem is solvable, if negative solutions of the free Dirac equation are employed, too. That is, with the help of these solutions, $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ can be reformulated in a form showing a quite simple geometric structure in the state space.

3.2. Description of Single-Particle States

In this section, we discuss descriptions of single-particle states, which are to be used for getting the aforementioned reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$. Particularly, compared with ordinary descriptions, two changes are to be made for reasons discussed above.

The first change is a reinterpretation to negative-$p^0$ states of fermions. We are to use a label $\varrho$ to indicate the sign of the zeroth component $p^0$ of a four-momentum $p^{\mu }$ ($\mu =0,1,2,3$) of solutions of the free Dirac equation, satisfying $p^{\mu }{p}_{\mu }=m^2$ with mass m. 15 16 Specifically, $\varrho$ is equal to either 1 or $-1$,

$$\begin{array}{c}\hfill p^0=\varrho \left|p^0\right|.\end{array}$$

(29)

The convention of repeated index implying summation is not obeyed for the label $\varrho$.

One merit of considering negative-$p^0$ fermionic states is that they allow us to introduce a concept, which is to be called fundamental vacuum fluctuations (FVF). The basic idea behind this concept is that there is no physical reason to deny the possibility for a fermion-antifermion pair, which possesses net zero four-momentum and net zero angular momentum, to emerge from the vacuum or vanish into the vacuum. We use FVF to refer to such a process. The fermionic pair involved is called an FVF pair. 17

To be consistent with the fact that no negative-$p^0$ fermionic state has ever been directly observed experimentally, we impose a rule for negative-$p^0$ fermionic state, which states that no free fermion may lie in a negative-$p^0$ state. 18 This rule implies that negative-$p^0$ fermions should behave differently from positive-$p^0$ fermions in a qualitative way.

The second change is a modification to the spin space for positron. Specifically, it is to be taken as the complex-conjugate space of the spin space of electron, the latter of which is the ordinarily used one. Similarly, the spin space of electron antineutrino is the complex conjugate of that of electron neutrino, the latter of which is similar to that of electron. As to be shown later, this modification changes neither the (anti)commutators of creation and annihilation operators, nor the amplitudes in the interaction Hamiltonian $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$; hence, it brings no change to predictions for experimental results.

Now, we discuss notations for single-particle states. A single-particle state of a fermion f, with a three-momentum $\mathbf{p}$, a sign $\varrho$, and spin label r, is written as $|f_{\mathbf{p}r\varrho }\rangle$. And, a single-particle state of a vector boson B, with a three-momentum $\mathbf{k}$ and a polarization index $\lambda$, is written as $|B_{\mathbf{k}\lambda }\rangle$. The state space of a fermion f with a fixed sign $\varrho$ is indicted as ${\mathcal{E}}_{f\varrho }$, spanned by $|f_{\mathbf{p}r\varrho }\rangle$; and that of a boson B is indicated as ${\mathcal{E}}_B$, spanned by $|B_{\mathbf{k}\lambda }\rangle$. The above states satisfy the usual anticommutation and commutation relations, i.e.,

$$\begin{array}{c}|f_{\mathbf{p}r\varrho }\rangle |f_{{\mathbf{p}}^{\prime }{r}^{\prime }{\varrho }^{\prime }}^{\prime }\rangle =-|f_{{\mathbf{p}}^{\prime }{r}^{\prime }{\varrho }^{\prime }}^{\prime }\rangle |f_{\mathbf{p}r\varrho }\rangle ,\hfill \end{array}$$

(30a)

$$\begin{array}{cc}& |B_{\mathbf{k}\lambda }\rangle |B_{{\mathbf{k}}^{\prime }{\lambda }^{\prime }}^{\prime }\rangle =|B_{{\mathbf{k}}^{\prime }{\lambda }^{\prime }}^{\prime }\rangle |B_{\mathbf{k}\lambda }\rangle .\hfill \end{array}$$

(30b)

It proves convenient to use a unified notation for antifermion. In particular, the antifermion of f is to be indicated as $\tilde{f}$, with $\tilde{\tilde{f}}\equiv f$ implied. Specifically, electron, positron, electron neutrino, and electron antineutrino are to be indicated as e, $\tilde{e}$, $\nu$ and $\tilde{\nu }$, respectively. 19

Each single-particle state is written as a direct product of a momentum part and a spin part. Their explicit expressions are as follows,

$$\begin{array}{c}|f_{\mathbf{p}r\varrho }\rangle =|\mathbf{p}\rangle |U_{r\varrho }\left(\mathbf{p}\right)\rangle (f=e,\nu ),\hfill \end{array}$$

(31a)

$$\begin{array}{c}|{\tilde{f}}_{\mathbf{p}r\varrho }\rangle =|\mathbf{p}\rangle |{\tilde{U}}_{r\varrho }\left(\mathbf{p}\right)\rangle (\tilde{f}=\tilde{e},\tilde{\nu }),\hfill \end{array}$$

(31b)

$$\begin{array}{c}|B_{\mathbf{k}\lambda }\rangle =|\mathbf{k}\rangle |{\epsilon }_{\lambda }\left(\mathbf{k}\right)\rangle .\hfill \end{array}$$

(31c)

Here, $|U_{r\varrho }\left(\mathbf{p}\right)\rangle$ of $\varrho =1$ is just the ket form of the ordinary Dirac spinor $U_r\left(\mathbf{p}\right)$, as a stationary solution of the free Dirac equation with a positive $p^0$ 20; $|U_{r\varrho }\left(\mathbf{p}\right)\rangle$ of $\varrho =-1$ corresponds to a negative-$p^0$ solution of the same Dirac equation 21; and $|{\epsilon }_{\lambda }\left(\mathbf{k}\right)\rangle$ is the ket form of the polarization vector ${\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right)$. From Eq.(), it is seen that spin states of positron are described by $|{\tilde{U}}_{r\varrho }\left(\mathbf{p}\right)\rangle$, as the ket form of ${\tilde{U}}_{r\varrho }\left(\mathbf{p}\right)$, which lie in the complex-conjugate space of the spin space for electron. The notation of $\tilde{f}$ for the antiparticle of f is in consistency with this property.

Bras of the kets in Eq.(31) may be introduced basically in the usual way, though a little more treatment is needed in the abstraction notation as discussed in Section 2.2. Scalar products of single-particle states have the ordinary forms except for the orthogonality imposed to the label $\varrho$, that is,

$$\begin{array}{cc}\hfill & \langle {f^{\prime }}_{{\mathbf{p}}^{\prime }{r}^{\prime }{\varrho }^{\prime }}|f_{\mathbf{p}r\varrho }\rangle =\varrho \left|p^0\right|{\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime }){\delta }_{r{r}^{\prime }}{\delta }_{f{f}^{\prime }}{\delta }_{\varrho {\varrho }^{\prime }},\hfill \end{array}$$

(32a)

$$\begin{array}{c}\hfill \langle B_{\mathbf{k}\lambda }|B_{{\mathbf{k}}^{\prime }{\lambda }^{\prime }}^{\prime }\rangle =\left|k^0\right|{\delta }^3(\mathbf{k}-{\mathbf{k}}^{\prime })g_{\lambda {\lambda }^{\prime }}{\delta }_{B{B}^{\prime }}.\end{array}$$

(32b)

Here, three-momentum states are normalized as follows,

$$\begin{array}{c}\langle \mathbf{q}|\mathbf{p}\rangle =|p^0|{\delta }^3(\mathbf{p}-\mathbf{q}),\end{array}$$

(33)

the rhs of which is Lorentz invariant and, hence, is a scalar under $SL(2,\mathbb{C})$-group transformations.

Making use of the above scalar products, it is easy to see that $\varrho I_{f\varrho }$ is the identity operator that acts on the space of ${\mathcal{E}}_{f\varrho }$, where

$$\begin{array}{cc}\hfill & I_{f\varrho }:=\int d\check{p}|f_{\mathbf{p}\varrho }^r\rangle \langle f_{\mathbf{p}r\varrho }|.\hfill \end{array}$$

(34)

Here and hereafter, $d\check{p}$ indicates the following abbreviation (similar for $d\check{k}$),

$$\begin{array}{c}d\check{p}:={\displaystyle \frac{1}{|p^0|}}{d}^{3}p.\end{array}$$

(35)

Similarly, $\varrho {\varrho }^{\prime }{I}_{f\varrho ,f^{\prime }{\varrho }^{\prime }}$ is the identity operator that acts on the space of ${\mathcal{E}}_{f\varrho }\otimes {\mathcal{E}}_{f^{\prime }{\varrho }^{\prime }}$, where

$$\begin{array}{cc}\hfill & I_{f\varrho ,f^{\prime }{\varrho }^{\prime }}:=\int d\check{p}d{\check{p}}^{\prime }|f_{\mathbf{p}\varrho }^r\rangle |{f^{\prime }}_{{\mathbf{p}}^{\prime }{\varrho }^{\prime }}^{r^{\prime }}\rangle \langle f_{{\mathbf{p}}^{\prime }{r}^{\prime }{\varrho }^{\prime }}^{\prime }|\langle f_{\mathbf{p}r\varrho }|.\hfill \end{array}$$

(36)

And, $I_B$ as the identity operator on the space ${\mathcal{E}}_B$ is written as

$$\begin{array}{c}{I}_B=\int d\check{k}|B_{\mathbf{k}\lambda }\rangle \langle B_{\mathbf{k}}^{\lambda }|.\end{array}$$

(37)

3.3. A Reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$

In this section, we present the major result of this paper as a reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$. It is written as

$$\begin{array}{c}\hfill H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}=\sum _k\sum _{\omega ,\varrho ,{\varrho }^{\prime }}{\xi }_k\overbrace{V_{\omega }\left({\mathcal{P}}_{k\varrho {\varrho }^{\prime }}^{\mathrm{FIO}1}\right)}+\mathrm{H}.\mathrm{c}.,\end{array}$$

(38)

where $(\varrho ,{\varrho }^{\prime })$ is for the fermionic pair $(f,f^{\prime })$, which is involved in ${\mathcal{H}}_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}$, and “H.c.” stands for Hermitian conjugate. Note that dependence of the fermionic species $(f,f^{\prime })$ on k is given in Eq.(24), while, the bosonic species B involved can be directly seen from Eq.(19). Below, we explain basic meanings of other symbols used in the above reformulation, with further explanations to be given in the next section.

On the rhs of Eq.(38), $V_{\omega }$ represents effects of FVF, the explicit mathematical expression of which will be given later [Eqs.(47) and (50)]. Here, $\omega =(n_f^V,n_{f^{\prime }}^V)$, where $n_f^V$ indicates the number of FVF pairs that are composed of ($f,\tilde{f})$, and $n_{f^{\prime }}^V$ indicates the number for $(f^{\prime },{\tilde{f}}^{\prime })$.

The symbol ${\mathcal{P}}_{k\varrho {\varrho }^{\prime }}^{\mathrm{FIO}1}$ represents a mapping operator, which maps a direct-product space ${\mathcal{E}}_{f\varrho }\otimes {\mathcal{E}}_{f^{\prime }{\varrho }^{\prime }}$ ($f\ne f^{\prime }$) to a space ${\mathcal{E}}_B$. For reasons that will become clear later (see Section 4.2), this operator is to be called a fundamental interaction operator (FIO) of FIO1. (FIO2 represents the reverse map, whose effects are included in the H.c. term in Eq.(38).) Explicitly, the FIO1 is written as

$${\mathcal{P}}_{k{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}=I_B{\mathcal{G}}_k^{\mathrm{FIO}1}{I}_{f\varrho ,f^{\prime }{\varrho }^{\prime }}\forall k,$$

(39)

where $I_B$ and $I_{f\varrho ,f^{\prime }{\varrho }^{\prime }}$ (multiplied by $\varrho {\varrho }^{\prime }$) are identity operators on the spaces of ${\mathcal{E}}_B$ and ${\mathcal{E}}_{f\varrho }\otimes {\mathcal{E}}_{f^{\prime }{\varrho }^{\prime }}$, respectively, and ${\mathcal{G}}_k^{\mathrm{FIO}1}$ connects kets in $I_{f\varrho ,f^{\prime }{\varrho }^{\prime }}$ to bras in $I_B$ and generates an amplitude. 22 Thus, a net effect of FIO1 is to annihilate two fermions and generate one boson, namely, $(f,f^{\prime })\to B$.

More exactly, the FIO1 ${\mathcal{P}}_{k{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$ is written as

$$\begin{array}{cc}\hfill & {\mathcal{P}}_{k{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}=\int d\check{p}d{\check{p}}^{\prime }d\check{k}|B_{\mathbf{k}\lambda }\rangle G^{\mathrm{mom}}{G}_k^{\mathrm{spin}}\langle f_{{\mathbf{p}}^{\prime }{r}^{\prime }{\varrho }^{\prime }}^{\prime }|\langle f_{\mathbf{p}r\varrho }|,\hfill \end{array}$$

(40)

where the amplitude $\langle B_{\mathbf{k}}^{\lambda }|{\mathcal{G}}_k^{\mathrm{FIO}1}|f_{\mathbf{p}\varrho }^r\rangle |{f^{\prime }}_{{\mathbf{p}}^{\prime }{\varrho }^{\prime }}^{r^{\prime }}\rangle$ has been written in the following form,

$$\begin{array}{cc}\hfill & \langle B_{\mathbf{k}}^{\lambda }|{\mathcal{G}}_k^{\mathrm{FIO}1}|f_{\mathbf{p}\varrho }^r\rangle |{f^{\prime }}_{{\mathbf{p}}^{\prime }{\varrho }^{\prime }}^{r^{\prime }}\rangle \equiv G^{\mathrm{mom}}{G}_k^{\mathrm{spin}},\hfill \end{array}$$

(41)

with momentum part and spin part separated. The momentum part $G^{\mathrm{mom}}$ represents the momentum conservation,

$$\begin{array}{cc}\hfill & G^{\mathrm{mom}}={\delta }^3(\mathbf{p}+{\mathbf{p}}^{\prime }-\mathbf{k}),\hfill \end{array}$$

(42)

and the spin part is written as

$$\begin{array}{cc}\hfill & G_k^{\mathrm{spin}}:=\langle {\epsilon }^{\lambda }\left(\mathbf{k}\right)|\sigma \overrightarrow{{\mathcal{S}}_c}|U_{\varrho }^r\left(\mathbf{p}\right){\tilde{U}}_{{\varrho }^{\prime }}^{r^{\prime }}\left({\mathbf{p}}^{\prime }\right)\rangle .\hfill \end{array}$$

(43)

Here, $\overrightarrow{{\mathcal{S}}_c}$ is a symbol defined as follows,

$$\begin{array}{c}\overrightarrow{{\mathcal{S}}_c}|X\tilde{W}\rangle \equiv \left\{\begin{array}{cc}|\chi \rangle |\tilde{w}\rangle +|\tilde{\kappa }\rangle |z\rangle ,\hfill & \mathrm{for}c=T,\hfill \\ |\chi \rangle |\tilde{w}\rangle ,\hfill & \mathrm{for}c=L,\hfill \\ |\tilde{\kappa }\rangle |z\rangle ,\hfill & \mathrm{for}c=R,\hfill \end{array}\right.\end{array}$$

(44)

where $|X\rangle$ and $|W\rangle$ are two arbitrary Dirac spinors written as direct sums of Weyl spinors, 23

$$\begin{array}{c}|X\rangle =\left(\begin{array}{c}|\chi \rangle \\ |\tilde{\kappa }\rangle \end{array}\right),|W\rangle =\left(\begin{array}{c}|w\rangle \\ |\tilde{z}\rangle \end{array}\right),\end{array}$$

(45)

and c is the chiral index [Eq.(23)]. Note that $|\tilde{W}\rangle =\left(\begin{array}{c}|\tilde{w}\rangle \\ |z\rangle \end{array}\right)$.

The term “$\overbrace{V_{\omega }\left({\mathcal{P}}_{k\varrho {\varrho }^{\prime }}^{\mathrm{FIO}1}\right)}$” represents a combination of $V_{\omega }$ and ${\mathcal{P}}_{k\varrho {\varrho }^{\prime }}^{\mathrm{FIO}1}$, where the overbrace indicates that the combination should be subject to the rule for negative-$p^0$ fermionic state, according to which a negative-$p^0$ fermionic state may exist only in FIOs and FVFs. Thus, all the “incoming” and “outgoing” fermions in $\overbrace{V_{\omega }\left({\mathcal{P}}_{k\varrho {\varrho }^{\prime }}^{\mathrm{FIO}1}\right)}$ should possess positive $p^0$. Further explanations to the combination will be given in the next section.

To summarize, the reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ in Eq.(38) shows that this interaction Hamiltonian contains six kernels as ${\mathcal{P}}_{k\varrho {\varrho }^{\prime }}^{\mathrm{FIO}1}$ of six k. Further, Eq.(43) shows that the spin parts of the six kernels share one common core: the operator $\sigma$, which maps the direct-product Weyl-spinor space $\mathcal{W}\otimes \tilde{\mathcal{W}}$ to the four-component vector space $\mathcal{V}$. As discussed previously, the central part of the operator $\sigma$, namely the EvdW symbols, are invariant under $SL(2,\mathbb{C})$ transformations, which implies that the form of $\sigma$ is independent of the bases employed in the spaces of $\mathcal{W}\otimes \tilde{\mathcal{W}}$ and $\mathcal{V}$.

3.4. FVF and Its Combination with FIO1

In this section we discuss mathematical description of FVF, as well as its combination with the FIO1 ${\mathcal{P}}_{k{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$.

Firstly, we discuss description of FVF. For the emergence of an FVF pair from the vacuum, which contains a fermion f and the antifermion $\tilde{f}$, it is not difficult to check that states of the two fermions, written as kets, may generically be written as $|f_{\mathbf{q}s-}\rangle$ and $|{\tilde{f}}_{\overline{\mathbf{q}}s+}\rangle$. Here and hereafter, an overline above a three-momentum $\mathbf{p}$ indicates the opposite momentum, i.e.,

$$\begin{array}{c}\hfill \overline{\mathbf{p}}:=-\mathbf{p}.\end{array}$$

(46)

We use $V_f\left(\mathcal{O}\right)$ to indicate the combination of an operator $\mathcal{O}$ and an emerging FVF pair. By definition, the combination is written as

$$\begin{array}{cc}\hfill V_f\left(\mathcal{O}\right)& \equiv \int d\check{q}|{\tilde{f}}_{\overline{\mathbf{q}}s+}\rangle \mathcal{O}|f_{\mathbf{q}s-}\rangle .\hfill \end{array}$$

(47)

One easily checks that

$$\begin{array}{c}\hfill V_f\left[V_{f^{\prime }}(\dots )\right]=V_{f^{\prime }}\left[V_f(\dots )\right].\end{array}$$

(48)

To describe the combination with a vanishing FVF pair, one needs to use bras for the FVF pair. Specifically, one may use the conjugate of $V_f$, denoted by $V_f^{†}$, to describe the combination of the FVF pair and an operator ${\mathcal{O}}^{†}$,

$$\begin{array}{cc}\hfill V_f^{†}\left({\mathcal{O}}^{†}\right)& \equiv \int d\check{q}\langle f_{\mathbf{q}s-}|{\mathcal{O}}^{†}\langle {\tilde{f}}_{\overline{\mathbf{q}}s+}|.\hfill \end{array}$$

(49)

Secondly, we define $V_{\omega }\left({\mathcal{P}}_{k\varrho {\varrho }^{\prime }}^{\mathrm{FIO}1}\right)$ on the rhs of Eq.(38), with $\omega =(n_f^V,n_{f^{\prime }}^V)$. As the combination of one FIO1, $n_f$ FVFs of the fermionic species f, and $n_{f^{\prime }}$ FVFs of $f^{\prime }$, it is written as 24

$$\begin{array}{cc}\hfill & V_{\omega }\left(\mathrm{FIO}1\right):=\underbrace{V_{f^{\prime }}\{\dots }\left[\underbrace{V_f(\dots }\left(\mathrm{FIO}1\right)\dots )\right]\dots \}.\hfill \\ n_{f^{\prime }}^V{n}_f^V\hfill \end{array}$$

(50)

Thirdly, we discuss the rule for negative-$p^0$ fermionic state, which is indicated by the overbrace on the rhs of Eq.(38). This rule imposes a mathematical restriction to possible combination of FIO and FVF. As an example, consider an FIO1 ${\mathcal{P}}_{k\varrho {\varrho }^{\prime }}^{\mathrm{FIO}1}$, in which one fermion lies in a negative-$p^0$ state. The rule requires that this FIO1 should be combined with some FVF or some other FIO, in such a way that the negative-$p^0$ fermionic state (effectively) appears only in a scalar product, which is formed by it and some other negative-$p^0$ fermionic state. For the same reason, an FVF pair must be combined with some FIO and/or FVF.

To illustrate the exact meaning of the above discussed requirement of the rule, let us consider an operator $\mathcal{O}=|\psi \rangle \langle f_1|\langle f_2|$, where $|\psi \rangle$ represents a bosonic state and $\langle f_1|$ and $\langle f_2|$ are fermionic states with $\langle f_2|$ possessing a negative $p^0$. The action of the operator on a negative-$p^0$ state $|f_{\mathbf{q}s-}\rangle$, subject to the above requirement, is written as

$$\begin{array}{c}\hfill \overbrace{\mathcal{O}|f_{\mathbf{q}s-}\rangle }=\left\{\begin{array}{cc}\langle f_2|f_{\mathbf{q}s-}\rangle |\psi \rangle \langle f_1|,\hfill & \mathrm{if}{p}^0>0\mathrm{for}\langle f_1|;\hfill \\ 0,\hfill & \mathrm{if}{p}^0<0\mathrm{for}\langle f_1|.\hfill \end{array}\right.\end{array}$$

(51)

Generically to say, in the computation of a sequence of kets and bras under an overbrace, those results in which each negative-$p^0$ fermionic state belongs to some scalar product are retained, while, others are abandoned.

Finally, making use of Eqs.(47) and (51), together with the fact that one FIO1 contains only two fermionic bras, it is not difficult to check that $\overbrace{V_{\omega }\left(\mathrm{FIO}1\right)}=0$ if $n_f^V>1$ or $n_{f^{\prime }}^V>1$. As a consequence, there are only four possibilities of $\omega$ for nonvanishing result, namely, $\omega =(0,0),(1,0),(0,1),(1,1)$.

4. Derivation of the Reformulation

In this section, we give derivation for the reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ in Eq.(38). Firstly, the ordinary formulation of the QED interaction Hamiltonian, $H_{\mathrm{int}}^{\mathrm{QED}}=H_{\mathrm{int},\mathrm{I},1}^{\mathrm{gws}}$, is recalled in Section 4.1. Then, a $V_{\omega }$-reformulation of $H_{\mathrm{int}}^{\mathrm{QED}}$ is given in Section 4.2, which is further written in a form with FIO1 in Section 4.3. Finally, in Section 4.4, Eq.(38) is proved by generalizing the results obtained for QED.

4.1. Ordinary Formulation of QED

The ordinary formulation of $H_{\mathrm{int}}^{\mathrm{QED}}$ contains only fermions with positive $\varrho$ and hence $\varrho$ is omitted in this section. It is written as

$$\begin{array}{c}\hfill H_{\mathrm{int}}^{\mathrm{QED}}=\int d^{3}x:{\psi }_e^{†}\left(\mathbf{x}\right){\gamma }^0{\gamma }^{\mu }{\psi }_e\left(\mathbf{x}\right)A_{\mu }\left(\mathbf{x}\right):.\end{array}$$

(52)

Here, for brevity, a prefactor of $H_{\mathrm{int}}^{\mathrm{QED}}$ is not written explicitly 25. Substituting Eq.(Section 3.1) into Eq.(25), one gets eight terms, denoted by $H_i$ of $i=1,\dots ,8$, with a superscript “QED” omitted for brevity. They are written as follows, in which creation and annihilation operators have been replaced by their equivalent forms of kets and bras,

$$\begin{array}{c}\hfill H_{\mathrm{int}}^{\mathrm{QED}}=\sum _{i=1}^8{H}_i,\end{array}$$

(53)

where 26

$$\begin{array}{cc}\hfill & H_1=\int d\check{p}d\check{q}d\check{k}|A_{\mathbf{k}\lambda }\rangle \langle {\tilde{e}}_{\mathbf{q}s}|\langle e_{\mathbf{p}r}|h_1,\hfill \end{array}$$

(54a)

$$\begin{array}{c}\hfill H_2=\int d\check{p}d\check{q}d\check{k}|e_{\mathbf{q}s}\rangle |{\tilde{e}}_{\mathbf{p}r}\rangle \langle A_{\mathbf{k}\lambda }|h_2,\end{array}$$

(54b)

$$\begin{array}{c}\hfill H_3=\int d\check{p}d\check{q}d\check{k}|A_{\mathbf{k}\lambda }\rangle |e_{\mathbf{q}s}\rangle \langle e_{\mathbf{p}r}|h_3,\end{array}$$

(54c)

$$\begin{array}{c}\hfill H_4=\int d\check{p}d\check{q}d\check{k}|e_{\mathbf{q}s}\rangle \langle e_{\mathbf{p}r}|\langle A_{\mathbf{k}\lambda }|h_4,\end{array}$$

(54d)

$$\begin{array}{c}\hfill H_5=-\int d\check{p}d\check{q}d\check{k}|A_{\mathbf{k}\lambda }\rangle |{\tilde{e}}_{\mathbf{p}r}\rangle \langle {\tilde{e}}_{\mathbf{q}s}|h_5,\end{array}$$

(54e)

$$\begin{array}{c}\hfill H_6=-\int d\check{p}d\check{q}d\check{k}|{\tilde{e}}_{\mathbf{p}r}\rangle \langle {\tilde{e}}_{\mathbf{q}s}|\langle A_{\mathbf{k}\lambda }|h_6,\end{array}$$

(54f)

$$\begin{array}{c}\hfill H_7=\int d\check{p}d\check{q}d\check{k}|A_{\mathbf{k}\lambda }\rangle |e_{\mathbf{q}s}\rangle |{\tilde{e}}_{\mathbf{p}r}\rangle h_7,\end{array}$$

(54g)

$$\begin{array}{c}\hfill H_8=\int d\check{p}d\check{q}d\check{k}\langle {\tilde{e}}_{\mathbf{q}s}|\langle e_{\mathbf{p}r}|\langle A_{\mathbf{k}\lambda }|h_8.\end{array}$$

(54h)

Here, $h_i$ of $i=1,\dots 8$ indicate interaction amplitudes. Explicitly, $h_i$ of odd i are written as

$$\begin{array}{c}{h}_1=V^{†s}\left(\mathbf{q}\right){\gamma }^0{\gamma }^{\mu }{U}^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda *}\left(\mathbf{k}\right){\delta }^3(\mathbf{p}+\mathbf{q}-\mathbf{k}),\end{array}$$

(55a)

$$\begin{array}{c}{h}_3=U^{†s}\left(\mathbf{q}\right){\gamma }^0{\gamma }^{\mu }{U}^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda *}\left(\mathbf{k}\right)\delta (\mathbf{p}-\mathbf{q}-\mathbf{k}),\end{array}$$

(55b)

$$\begin{array}{c}{h}_5=V^{†s}\left(\mathbf{q}\right){\gamma }^0{\gamma }^{\mu }{V}^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda *}\left(\mathbf{k}\right){\delta }^3(\mathbf{q}-\mathbf{p}-\mathbf{k}),\end{array}$$

(55c)

$$\begin{array}{c}{h}_7=U^{†s}\left(\mathbf{q}\right){\gamma }^0{\gamma }^{\mu }{V}^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda *}\left(\mathbf{k}\right){\delta }^3(\mathbf{p}+\mathbf{q}+\mathbf{k});\end{array}$$

(55d)

and those of even i are

$$\begin{array}{c}{h}_2=U^{†s}\left(\mathbf{q}\right){\gamma }^0{\gamma }^{\mu }{V}^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right)\delta (\mathbf{p}+\mathbf{q}-\mathbf{k}),\end{array}$$

(56a)

$$\begin{array}{c}{h}_4=U^{†s}\left(\mathbf{q}\right){\gamma }^0{\gamma }^{\mu }{U}^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right){\delta }^3(\mathbf{q}-\mathbf{p}-\mathbf{k}),\end{array}$$

(56b)

$$\begin{array}{c}{h}_6=V^{†s}\left(\mathbf{q}\right){\gamma }^0{\gamma }^{\mu }{V}^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right){\delta }^3(\mathbf{p}-\mathbf{q}-\mathbf{k}),\end{array}$$

(56c)

$$\begin{array}{c}{h}_8=V^{†s}\left(\mathbf{q}\right){\gamma }^0{\gamma }^{\mu }{U}^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right){\delta }^3(\mathbf{p}+\mathbf{q}+\mathbf{k}).\end{array}$$

(56d)

Clearly, each $H_i$ corresponds to one basic Feynman diagram and $H_{i+1}=H_i^{†}$ for $i=1,3,5,7$.

4.2. $V_{\omega }$-Reformulation of $H_{\mathrm{int}}^{\mathrm{QED}}$

To write $H_{\mathrm{int}}^{\mathrm{QED}}$ in a form similar to the rhs of Eq.(38), a crucial point is to employ negative-$p^0$ solutions of the free Dirac equation, whose spinor parts are written as $U_{-}^r\left(\mathbf{p}\right)$ and $V_{-}^r\left(\mathbf{p}\right)$. (See Appendix C for their explicit expressions.) In particular, we are to make use of the following relations between positive-$p^0$ and negative-$p^0$ solutions [see Eqs.(A180)-(A181) of Appendix C], i.e.,

$$\begin{array}{cc}\hfill & V^r\left(\mathbf{p}\right)=U_{-}^r\left(\overline{\mathbf{p}}\right),\hfill \end{array}$$

(57a)

$$\begin{array}{c}\hfill U^r\left(\mathbf{p}\right)=V_{-}^r\left(\overline{\mathbf{p}}\right).\end{array}$$

(57b)

As mentioned previously, indicating the sign $\varrho$ of $p^0$ explicitly, the Dirac spinors are written as $U_{\varrho }^r\left(\mathbf{p}\right)$ and $V_{\varrho }^r\left(\mathbf{p}\right)$, with $U_{+}^r\left(\mathbf{p}\right)\equiv U^r\left(\mathbf{p}\right)$ and $V_{+}^r\left(\mathbf{p}\right)\equiv V^r\left(\mathbf{p}\right)$.

Making use of Eq.(57), it is straightforward to write $h_{3,5,7}$ [Eq.(55)] in a form like that of $h_1$, i.e.,

$$\begin{array}{c}{h}_3=V_{-}^{†s}\left(\overline{\mathbf{q}}\right){\gamma }^0{\gamma }^{\mu }{U}^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda *}\left(\mathbf{k}\right)\delta (\mathbf{p}+\overline{\mathbf{q}}-\mathbf{k}),\end{array}$$

(58a)

$$\begin{array}{c}{h}_5=V^{†s}\left(\mathbf{q}\right){\gamma }^0{\gamma }^{\mu }{U}_{-}^r\left(\overline{\mathbf{p}}\right){\epsilon }_{\mu }^{\lambda *}\left(\mathbf{k}\right){\delta }^3(\mathbf{q}+\overline{\mathbf{p}}-\mathbf{k}),\end{array}$$

(58b)

$$\begin{array}{c}{h}_7=V_{-}^{†s}\left(\overline{\mathbf{q}}\right){\gamma }^0{\gamma }^{\mu }{U}_{-}^r\left(\overline{\mathbf{p}}\right){\epsilon }_{\mu }^{\lambda *}\left(\mathbf{k}\right){\delta }^3(\overline{\mathbf{p}}+\overline{\mathbf{q}}-\mathbf{k}).\end{array}$$

(58c)

Similarly, $h_{4,6,8}$ are written as

$$\begin{array}{c}{h}_4=U^{†s}\left(\mathbf{q}\right){\gamma }^0{\gamma }^{\mu }{V}_{-}^r\left(\overline{\mathbf{p}}\right){\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right){\delta }^3(\mathbf{q}+\overline{\mathbf{p}}-\mathbf{k}),\end{array}$$

(59a)

$$\begin{array}{c}{h}_6=U_{-}^{†s}\left(\overline{\mathbf{q}}\right){\gamma }^0{\gamma }^{\mu }{V}^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right){\delta }^3(\mathbf{p}+\overline{\mathbf{q}}-\mathbf{k}),\end{array}$$

(59b)

$$\begin{array}{c}{h}_8=U_{-}^{†s}\left(\overline{\mathbf{q}}\right){\gamma }^0{\gamma }^{\mu }{V}_{-}^r\left(\overline{\mathbf{p}}\right){\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right){\delta }^3(\overline{\mathbf{p}}+\overline{\mathbf{q}}-\mathbf{k}),\end{array}$$

(59c)

sharing the same formal form as $h_2$. Due to the above formal similarities, we introduce the following amplitudes,

$$\begin{array}{cc}\hfill & h_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}:=V_{{\varrho }^{\prime }}^{†r^{\prime }}\left({\mathbf{p}}^{\prime }\right){\gamma }^0{\gamma }^{\mu }{U}_{\varrho }^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda *}\left(\mathbf{k}\right){\delta }^3(\mathbf{p}+{\mathbf{p}}^{\prime }-\mathbf{k}),\hfill \end{array}$$

(60a)

$$\begin{array}{c}\hfill h_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}2}:=U_{{\varrho }^{\prime }}^{†r^{\prime }}\left({\mathbf{p}}^{\prime }\right){\gamma }^0{\gamma }^{\mu }{V}_{\varrho }^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right){\delta }^3(\mathbf{p}+{\mathbf{p}}^{\prime }-\mathbf{k}).\end{array}$$

(60b)

It is straightforward to check that

$$\begin{array}{cc}\hfill & h_1=h_{++}^{\mathrm{FIO}1}\&h_2=h_{++}^{\mathrm{FIO}2},\hfill \end{array}$$

(61a)

$$\begin{array}{cc}& h_3=h_{-+}^{\mathrm{FIO}1}\&h_4=h_{+-}^{\mathrm{FIO}2},\hfill \end{array}$$

(61b)

$$\begin{array}{cc}& h_5=h_{+-}^{\mathrm{FIO}1}\&h_6=h_{-+}^{\mathrm{FIO}2},\hfill \end{array}$$

(61c)

$$\begin{array}{cc}& h_7=h_{--}^{\mathrm{FIO}1}\&h_8=h_{--}^{\mathrm{FIO}2}.\hfill \end{array}$$

(61d)

Then, we introduce two operators, denoted by $H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$ and $H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}2}$, as Hermitian conjugate of each other,

$$\begin{array}{cc}\hfill & H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}:=\int d\check{p}d{\check{p}}^{\prime }d\check{k}|A_{\mathbf{k}\lambda }\rangle \langle {\tilde{e}}_{{\mathbf{p}}^{\prime }{r}^{\prime }{\varrho }^{\prime }}|\langle e_{\mathbf{p}r\varrho }|h_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1},\hfill \end{array}$$

(62a)

$$\begin{array}{c}\hfill H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}2}:=\int d\check{p}d{\check{p}}^{\prime }d\check{k}|e_{{\mathbf{p}}^{\prime }{r}^{\prime }{\varrho }^{\prime }}\rangle |{\tilde{e}}_{\mathbf{p}r\varrho }\rangle \langle A_{\mathbf{k}\lambda }|h_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}2}.\end{array}$$

(62b)

After some derivations (see Appendix E), one finds that $H_i$ are expressible in terms of the above two operators and the two superoperators of $V_f$ and $V_f^{†}$ for FVFs in Eqs.(47) and (49). The results are

$$\begin{array}{cc}\hfill & H_1=H_{++}^{\mathrm{FIO}1},\hfill \end{array}$$

(63a)

$$\begin{array}{c}\hfill H_2=H_{++}^{\mathrm{FIO}2},\end{array}$$

(63b)

$$\begin{array}{c}\hfill H_3=\overbrace{V_{\tilde{e}}\left(H_{-+}^{\mathrm{FIO}1}\right)},\end{array}$$

(63c)

$$\begin{array}{c}\hfill H_4=\overbrace{V_{\tilde{e}}^{†}\left(H_{+-}^{\mathrm{FIO}2}\right)},\end{array}$$

(63d)

$$\begin{array}{c}\hfill H_5=\overbrace{V_e\left(H_{+-}^{\mathrm{FIO}1}\right)},\end{array}$$

(63e)

$$\begin{array}{c}\hfill H_6=\overbrace{V_e^{†}\left(H_{-+}^{\mathrm{FIO}2}\right)},\end{array}$$

(63f)

$$\begin{array}{c}\hfill H_7=\overbrace{V_{\tilde{e}}\left(V_e\left(H_{--}^{\mathrm{FIO}1}\right)\right)},\end{array}$$

(63g)

$$\begin{array}{c}\hfill H_8=\overbrace{V_{\tilde{e}}^{†}\left(V_e^{†}\left(H_{--}^{\mathrm{FIO}2}\right)\right)}.\end{array}$$

(63h)

In view of the above expressions, the operators $H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$ and $H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}2}$ are more fundamental than the eight $H_i$. In fact, as to be shown later [see Eq.(79)], $H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$ is just the FIO1 defined in Eq.(40) for QED, namely ${\mathcal{P}}_{1{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$. It is for this reason that we use the name of FIO for the operator of ${\mathcal{P}}_{1{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$.

One remark: Based on the physical meaning of FVF discussed previously, further interpretations to the above expressions of the operators $H_{3,\dots ,8}$ are obtainable, which are discussed in Appendix F in detail. 27

Below, we derive the following expression of $H_{\mathrm{int}}^{\mathrm{QED}}$,

$$\begin{array}{c}\hfill H_{\mathrm{int}}^{\mathrm{QED}}=\sum _{\omega ,\varrho ,{\varrho }^{\prime }}\overbrace{V_{\omega }\left(H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}\right)}+\mathrm{H}.\mathrm{c}.,\end{array}$$

(64)

where $V_{\omega }$ is defined in Eq.(50) with $\omega =(n_e^V,n_{\tilde{e}}^V)$. Note that $H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$ in Eq.(62a) contains two fermionic bras; meanwhile, $V_f$ [Eq.(47)] contains two fermionic kets, one with a negative $p^0$ and the other with a positive $p^0$. As a consequence, under the rule for negative-$p^0$ states, as discussed previously, there are only four cases of $\omega$ for which $\overbrace{V_{\omega }\left(H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}\right)}$ does not definitely vanish. That is, (i) $n_e^V=n_{\tilde{e}}^V=0$; (ii) $n_e^V=1\&n_{\tilde{e}}^V=0$; (iii) $n_e^V=0\&n_{\tilde{e}}^V=1$; and, (iv) $n_e^V=n_{\tilde{e}}^V=1$.

In the first case of $n_e^V=n_{\tilde{e}}^V=0$, $\overbrace{V_{\omega }\left(H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}\right)}=\overbrace{H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}}$. According the rule for negative-$p^0$ states, there is only one nonvanishing $\overbrace{H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}}$, namely $H_{++}^{\mathrm{FIO}1}$, which is equal to $H_1$ [Eq.(63a)].

In the second case of $n_e^V=1\&n_{\tilde{e}}^V=0$, $\overbrace{V_{\omega }\left(H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}\right)}=\overbrace{V_e\left(H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}\right)}$. Substituting Eq.(62a) into Eq.(47), one gets that

$$\begin{array}{cc}\hfill \overbrace{V_e\left(H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}\right)}& =\int d\check{q}d\check{p}d{\check{p}}^{\prime }d\check{k}|{\tilde{e}}_{\overline{\mathbf{q}}s+}\rangle |A_{\mathbf{k}\lambda }\rangle h_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}\hfill \end{array}$$

$$\begin{array}{c}\hfill \times \overbrace{\left(\langle {\tilde{e}}_{{\mathbf{p}}^{\prime }{r}^{\prime }{\varrho }^{\prime }}|\langle e_{\mathbf{p}r\varrho }|\right)|e_{\mathbf{q}s-}\rangle }.\end{array}$$

(65)

Under the rule for negative-$p^0$ states, for the rhs of Eq.(65) to be nonvanishing, $\langle e_{\mathbf{p}r\varrho }|$ must form a nonvanishing scalar product with $|e_{\mathbf{q}s-}\rangle$, then, noting Eq.(32a), one finds that $\varrho =-$; meanwhile, ${\varrho }^{\prime }=+$. As a consequence, there is only one nonvanishing result, which corresponds to $({\varrho }^{\prime },\varrho )=(+,-)$, i.e., $\overbrace{V_e\left(H_{+-}^{\mathrm{FIO}1}\right)}$. One is ready to see that this gives $H_5$ [see Eq.(63e)].

The third case with $n_e^V=0\&n_{\tilde{e}}^V=1$ is similar to the second case. Here, the only nonvanishing result is $\overbrace{V_{\tilde{e}}\left(H_{-+}^{\mathrm{FIO}1}\right)}$, which is equal to $H_3$ [Eq.(63c)].

In the fourth case of $n_e^V=n_{\tilde{e}}^V=1$, one considers the action of $V_{\tilde{e}}\left[V_e(\dots )\right]$. Applying $V_{\tilde{e}}$ to the left-hand side of Eq.(65) within the overbrace, one finds that the only nonvanishing result corresponds to $({\varrho }^{\prime },\varrho )=(-,-)$, namely $\overbrace{V_{\tilde{e}}\left[V_e\left(H_{--}^{\mathrm{FIO}1}\right)\right]}$, which is just $H_7$ [Eq.(63g)].

Summarizing the above discussions and taking into account the Hermitian conjugate terms, one gets Eq.(64) from Eq.(53).

4.3. Concise Expression of $H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$

In this section, it is shown that $H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}={\mathcal{P}}_{1{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$. 28 To achieve this goal, we need to write $h_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$ in Eq.(60a) in a ket-bra form.

Making use of the explicit expressions of the Dirac spinors $U_{+}^r\left(\mathbf{p}\right)$ and $V_{+}^r\left(\mathbf{p}\right)$ [see Eqs.(A107) and (A113)] and noting the relations in Eq.(Section 4.2), it is ready to find the following unified expressions of $U_{\varrho }^r\left(\mathbf{p}\right)$ and $V_{\varrho }^r\left(\mathbf{p}\right)$,

$$\begin{array}{c}{U}_{\varrho }^r\left(\mathbf{p}\right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}{u}^{rA}\left(\varrho \mathbf{p}\right)\\ \varrho {\tilde{v}}_{B^{\prime }}^r\left(\varrho \mathbf{p}\right)\end{array}\right),\end{array}$$

(66a)

$$\begin{array}{c}{V}_{\varrho }^r\left(\mathbf{p}\right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}{u}^{rA}\left(\varrho \mathbf{p}\right)\\ -\varrho {\tilde{v}}_{B^{\prime }}^r\left(\varrho \mathbf{p}\right)\end{array}\right),\end{array}$$

(66b)

where $u^{r,A}\in \mathcal{W}$ and ${\tilde{v}}_{B^{\prime }}^r\in \tilde{\mathcal{W}}$. In the chiral representation of the ${\gamma }^{\mu }$-matrices, From Eq.(17), one gets the following expression of the product ${\gamma }^0{\gamma }^{\mu }$, 29

$$\begin{array}{c}{\gamma }^0{\gamma }^{\mu }=\left(\begin{array}{cc}{\tilde{\sigma }}_{B^{\prime }A}^{\mu }& 0\\ 0& {\sigma }^{\mu B{A}^{\prime }}\end{array}\right),\end{array}$$

(67)

with the EvdW symbols ${\sigma }^{\mu A{B}^{\prime }}$. Making use of Eqs.(66)-(67) and the relation of ${\tilde{\sigma }}_{B^{\prime }A}^{\mu }={\sigma }_{A{B}^{\prime }}^{\mu }$ [Eq.(A149)], it is straightforward to write the Dirac-spinor part of $h_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$ [Eq.(60a)] in terms of Weyl spinors, getting that

$$\begin{array}{c}{V}_{{\varrho }^{\prime }}^{†r^{\prime }}\left({\mathbf{p}}^{\prime }\right){\gamma }^0{\gamma }^{\mu }{U}_{\varrho }^r\left(\mathbf{p}\right)\hfill \\ ={\tilde{u}}^{r^{\prime }{B}^{\prime }}\left({\varrho }^{\prime }{\mathbf{p}}^{\prime }\right){\tilde{\sigma }}_{B^{\prime }A}^{\mu }{u}^{rA}\left(\varrho \mathbf{p}\right)-v_B^{r^{\prime }}\left({\varrho }^{\prime }{\mathbf{p}}^{\prime }\right){\sigma }^{\mu B{A}^{\prime }}{\tilde{v}}_{A^{\prime }}^r\left(\varrho \mathbf{p}\right)\hfill \end{array}$$

$$\begin{array}{c}\hfill ={\sigma }_{A{B}^{\prime }}^{\mu }\left[{\tilde{u}}^{r^{\prime }{B}^{\prime }}\left({\varrho }^{\prime }{\mathbf{p}}^{\prime }\right)u^{rA}\left(\varrho \mathbf{p}\right)-v^{r^{\prime }A}\left({\varrho }^{\prime }{\mathbf{p}}^{\prime }\right){\tilde{v}}^{r{B}^{\prime }}\left(\varrho \mathbf{p}\right)\right],\end{array}$$

(68)

where Eq.(A96) has been used and some spinor labels have been rearranged in the derivation of the second equality. Making use of Eq.(11), one further writes 30

$$\begin{array}{cc}\hfill V_{{\varrho }^{\prime }}^{†r^{\prime }}& \left({\mathbf{p}}^{\prime }\right){\gamma }^0{\gamma }^{\mu }{U}_{\varrho }^r\left(\mathbf{p}\right)={\sigma }_{A{B}^{\prime }}^{\mu }\langle \langle {\tilde{S}}^{B^{\prime }}|\langle \langle S^A|\hfill \end{array}$$

$$\begin{array}{c}\hfill \left[|u^r\left(\varrho \mathbf{p}\right)\rangle |{\tilde{u}}^{r^{\prime }}\left({\varrho }^{\prime }{\mathbf{p}}^{\prime }\right)\rangle -|v^{r^{\prime }}\left({\varrho }^{\prime }{\mathbf{p}}^{\prime }\right)\rangle |{\tilde{v}}^r\left(\varrho \mathbf{p}\right)\rangle \right].\end{array}$$

(69)

According to Eq.(15), bras of polarization vectors are written as

$$\begin{array}{c}\hfill \langle {\epsilon }_{\lambda }\left(\mathbf{k}\right)|:=\langle \langle {\tilde{\epsilon }}_{\lambda }\left(\mathbf{k}\right)|.\end{array}$$

(70)

$\langle \langle {\epsilon }_{\lambda }\left(\mathbf{k}\right)|$ on the rhs of Eq.(70) is the scalar-product-based bra of $|{\epsilon }_{\lambda }\left(\mathbf{k}\right)\rangle$ [cf. Eq.(12)], i.e.,

$$\begin{array}{c}\hfill \langle \langle {\epsilon }^{\lambda }\left(\mathbf{k}\right)|=\langle \langle T^{\mu }|{\epsilon }_{\mu }^{\lambda }\left(\mathbf{k}\right).\end{array}$$

(71)

From Eqs.(14) and (70), one gets that

$$\begin{array}{c}\hfill {\epsilon }_{\mu }^{\lambda *}\left(\mathbf{k}\right)=\langle {\epsilon }^{\lambda }\left(\mathbf{k}\right)|T_{\mu }\rangle .\end{array}$$

(72)

The vectors $|{\epsilon }^{\lambda }\left(\mathbf{k}\right)\rangle$ of $\lambda =0,1,2,3$ constitute a basis in the space $\mathcal{V}$, too, like the basis $|T_{\mu }\rangle$. Making use of Eqs.(13) and (70) and noting Eq.(A150), it is ready to find that such a basis should satisfy

$$\langle {\epsilon }_{\lambda }\left(\mathbf{k}\right)|{\epsilon }_{{\lambda }^{\prime }}\left(\mathbf{k}\right)\rangle =g_{\lambda {\lambda }^{\prime }}.$$

(73)

It proves convenient to introduce a symbol $\overrightarrow{\mathcal{S}}$, defined by the following action on Dirac spinors, that is,

$$\begin{array}{c}\overrightarrow{\mathcal{S}}|X\tilde{W}\rangle \equiv |\chi \rangle |\tilde{w}\rangle +|\tilde{\kappa }\rangle |z\rangle ,\end{array}$$

(74)

where $|X\rangle$ and $|W\rangle$ are two arbitrary Dirac spinors as given in Eq.(45). Then, making use of Eqs.(69), (72), and Eq.(6), it is straightforward to find that $h_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$ in Eq.(60a) has the following expression,

$$\begin{array}{c}\hfill h_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}=\langle {\epsilon }^{\lambda }\left(\mathbf{k}\right)|\sigma \overrightarrow{\mathcal{S}}|U_{\varrho }^r\left(\mathbf{p}\right){\tilde{U}}_{{\varrho }^{\prime }}^{r^{\prime }}\left({\mathbf{p}}^{\prime }\right)\rangle {\delta }^3(\mathbf{p}+{\mathbf{p}}^{\prime }-\mathbf{k}).\end{array}$$

(75)

Here, the Dirac spinors $U_{\varrho }^r\left(\mathbf{p}\right)$ in Eq.(66) have been written in their ket form,

$$\begin{array}{cc}\hfill & |U_{\varrho }^r\left(\mathbf{p}\right)\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}|u^r\left(\varrho \mathbf{p}\right)\rangle \\ \varrho |{\tilde{v}}^r\left(\varrho \mathbf{p}\right)\rangle \end{array}\right).\hfill \end{array}$$

(76)

It is of interest to note that, in the expression of the amplitude $h_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$ in Eq.(75), the positron-spin state is represented by the spinor $|{\tilde{U}}_{{\varrho }^{\prime }}^{r^{\prime }}\left({\mathbf{p}}^{\prime }\right)\rangle$. This implies that, when the interaction amplitude is expressed in Eq.(75), one may use $|{\tilde{U}}_{\varrho }^r\left(\mathbf{p}\right)\rangle$ to represent positron-spin states. And, this is the reason of writing single-particle states of positron in the form of Eq.(). Moreover, employing this spinor description for positron-spin states has an additional merit: It possesses the following inner product,

$$\begin{array}{c}\hfill {\tilde{U}}_{r\varrho }^{†}\left(\mathbf{p}\right){\gamma }^0{\tilde{U}}_{s\varrho }\left(\mathbf{p}\right)=\varrho {\delta }_{rs},\end{array}$$

(77)

which is in consistency with the scalar product of single-positron states in Eq.(32a), as well as the anticommutator in Eq.() for $\varrho =+1$. In contrast, the ordinary spinor description has the property of $V_{r\varrho }^{†}\left(\mathbf{p}\right){\gamma }^0{V}_{s\varrho }\left(\mathbf{p}\right)=-\varrho {\delta }_{rs}$, showing a difference in the sign.

Substituting Eq.(75) into Eq.(62a) one finds that

$$\begin{array}{cc}\hfill H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}& =\int d\check{p}d{\check{p}}^{\prime }d\check{k}{\delta }^3(\mathbf{p}+{\mathbf{p}}^{\prime }-\mathbf{k})|A_{\mathbf{k}\lambda }\rangle \langle {\epsilon }^{\lambda }\left(\mathbf{k}\right)|\hfill \end{array}$$

$$\begin{array}{c}\hfill \times \sigma \overrightarrow{\mathcal{S}}|U_{\varrho }^r\left(\mathbf{p}\right){\tilde{U}}_{{\varrho }^{\prime }}^{r^{\prime }}\left({\mathbf{p}}^{\prime }\right)\rangle \langle {\tilde{e}}_{{\mathbf{p}}^{\prime }{r}^{\prime }{\varrho }^{\prime }}|\langle e_{\mathbf{p}r\varrho }|.\end{array}$$

(78)

Then, making use of the identity operators in Eqs.(36) and (37), one gets a concise expression of $H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$,

$$H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}=I_A{\mathcal{G}}^{\mathrm{FIO}1}{I}_{e\varrho ,\tilde{e}{\varrho }^{\prime }}.$$

(79)

Here, ${\mathcal{G}}^{\mathrm{FIO}1}$ is defined by the following relation, which connects kets in $I_{e\varrho ,\tilde{e}{\varrho }^{\prime }}$ to bras in $I_A$, that is,

$$\begin{array}{c}\hfill \langle A_{\mathbf{k}}^{\lambda }|{\mathcal{G}}^{\mathrm{FIO}1}|e_{\mathbf{p}\varrho }^r\rangle |{\tilde{e}}_{{\mathbf{p}}^{\prime }{\varrho }^{\prime }}^{r^{\prime }}\rangle \equiv G^{\mathrm{mom}}{G}_1^{\mathrm{spin}},\end{array}$$

(80)

where $\langle A_{\mathbf{k}\lambda }|=\langle {\epsilon }_{\lambda }\left(\mathbf{k}\right)|\langle \mathbf{k}|$, $G^{\mathrm{mom}}$ is defined in Eq.(42), and

$$\begin{array}{cc}\hfill & G_1^{\mathrm{spin}}=\langle {\epsilon }^{\lambda }\left(\mathbf{k}\right)|\sigma \overrightarrow{\mathcal{S}}|U_{\varrho }^r\left(\mathbf{p}\right){\tilde{U}}_{{\varrho }^{\prime }}^{r^{\prime }}\left({\mathbf{p}}^{\prime }\right)\rangle .\hfill \end{array}$$

(81)

It is then ready to sees that $H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}={\mathcal{P}}_{1{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$.

4.4. Derivation of Eq.(38)

In this section, Eq.(38) is proved by generalizing results obtained above for QED. Note that, for each fixed k, $H_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}$ may be divided into eight parts, denoted by $H_{k,i}$, in a way similar to the rhs of Eq.(53) for $H_{\mathrm{int}}^{\mathrm{QED}}$ (as $k=1$), that is,

$$\begin{array}{c}\hfill H_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}=\sum _{i=1}^8{H}_{k,i},\end{array}$$

(82)

where $H_{k,i}$ have forms similar to those of $H_i$ in Eq.(54), respectively. Following a procedure, which is almost the same as that carried out previously and leading to Eq.(63) for QED, one finds that, with corresponding changes of particle species, other $H_{k,i}$ ($k\ne 1$) may also be written in forms similar to the rhs of Eq.(63).

In the above generalization, FIO1 needs to be introduced for each $H_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}$, which is to be called the k-th electroweak FIO1 and indicated as $H_{{\varrho }^{\prime }\varrho ,k}^{\mathrm{FIO}1}$. Note that $H_{{\varrho }^{\prime }\varrho ,1}^{\mathrm{FIO}1}=H_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$ for QED in Eq.(62a). More exactly, the FIO1s are written as 31

$$\begin{array}{cc}\hfill & H_{{\varrho }^{\prime }\varrho ,k}^{\mathrm{FIO}1}:=\int d\check{p}d{\check{p}}^{\prime }d\check{k}|B_{\mathbf{k}\lambda }\rangle \langle {\tilde{f}}_{{\mathbf{p}}^{\prime }{r}^{\prime }{\varrho }^{\prime }}^{\prime }|\langle f_{\mathbf{p}r\varrho }|h_{{\varrho }^{\prime }\varrho ,k}^{\mathrm{FIO}1},\hfill \end{array}$$

(83)

where

$$\begin{array}{cc}\hfill & h_{{\varrho }^{\prime }\varrho ,k}^{\mathrm{FIO}1}={\xi }_k{V}_{{\varrho }^{\prime }{c}^{\prime }}^{†r^{\prime }}\left({\mathbf{p}}^{\prime }\right){\gamma }^0{\gamma }^{\mu }{U}_{\varrho c}^r\left(\mathbf{p}\right){\epsilon }_{\mu }^{\lambda *}\left(\mathbf{k}\right){\delta }^3(\mathbf{p}+{\mathbf{p}}^{\prime }-\mathbf{k}).\hfill \end{array}$$

(84)

Here, dependence of $U_{\varrho c}^r$ on c [as a function of k as shown in Eq.(24)] is given by

$$\begin{array}{c}\hfill U_{\varrho c}^r=\left\{\begin{array}{cc}{U}_{\varrho }^r,\hfill & \mathrm{for}c=T,\hfill \\ {\displaystyle \frac{1-{\gamma }^5}{2}}{U}_{\varrho }^r,\hfill & \mathrm{for}c=L,\hfill \\ {\displaystyle \frac{1+{\gamma }^5}{2}}{U}_{\varrho }^r,\hfill & \mathrm{for}c=R,\hfill \end{array}\right.\end{array}$$

(85)

and similar for $V_{{\varrho }^{\prime }{c}^{\prime }}^{r^{\prime }}$.

Then, following a procedure similar to that from Eq.(63) to the expression of $H_{\mathrm{int}}^{\mathrm{QED}}$ in Eq.(64), it is not difficult to check that $H_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}$, which describes the particle change of $(f,f^{\prime })\to B$, has the following expression,

$$\begin{array}{c}\hfill H_{\mathrm{int},\mathrm{I},k}^{\mathrm{gws}}=\sum _{{\varrho }^{\prime },\varrho ,\omega }{\xi }_k\overbrace{V_{\omega }\left(H_{{\varrho }^{\prime }\varrho ,k}^{\mathrm{FIO}1}\right)}+\mathrm{H}.\mathrm{c}.,\end{array}$$

(86)

with $\omega =(n_f^V,n_{f^{\prime }}^V)$. Clearly, the rhs of Eq.(86) takes the same form as that of Eq.(38) for a fixed k, except that the former contains a kernel $H_{{\varrho }^{\prime }\varrho ,k}^{\mathrm{FIO}1}$, while, the latter contains a kernel ${\mathcal{P}}_{k{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$.

The two kernels discussed above are in fact identical. To see this point, we note that, like the expression of the QED FIO1 in Eq.(79), $H_{{\varrho }^{\prime }\varrho ,k}^{\mathrm{FIO}1}$ ($k\ne 1$) may be written in a concise form. In fact, following a procedure similar to that of deriving the expression of $h_{{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}$ in Eq.(75) from Eq.(60a), one gets the following expression for the k-th amplitude $h_{{\varrho }^{\prime }\varrho ,k}^{\mathrm{FIO}1}$, i.e.,

$$\begin{array}{cc}\hfill & h_{{\varrho }^{\prime }\varrho ,k}^{\mathrm{FIO}1}={\xi }_k\langle {\epsilon }^{\lambda }\left(\mathbf{k}\right)|\sigma \overrightarrow{{\mathcal{S}}_c}|U_{\varrho }^r\left(\mathbf{p}\right){\tilde{U}}_{{\varrho }^{\prime }}^{r^{\prime }}\left({\mathbf{p}}^{\prime }\right)\rangle {\delta }^3(\mathbf{p}+{\mathbf{p}}^{\prime }-\mathbf{k}),\hfill \end{array}$$

(87)

where $\overrightarrow{{\mathcal{S}}_c}$ was defined in Eq.(44). Substituting Eq.(87) into Eq.(83), similar to Eq.(79) for QED, it is straightforward to find that $H_{{\varrho }^{\prime }\varrho ,k}^{\mathrm{FIO}1}$ is equal to the operator in Eq.(39), i.e.,

$$\begin{array}{c}\hfill H_{{\varrho }^{\prime }\varrho ,k}^{\mathrm{FIO}1}={\mathcal{P}}_{k{\varrho }^{\prime }\varrho }^{\mathrm{FIO}1}\forall k.\end{array}$$

(88)

This finishes a proof of Eq.(38).

5. Summary and Discussions

In this paper, a state-space geometric structure has been revealed for the GWS electroweak interaction Hamiltonian $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ of the first generation of leptons, by a reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ in terms of ket-bras for single-particle states. The geometric structure possesses a core as $\sigma$, the operator form of the EvdW (Enfeld-van der Waerden) symbols ${\sigma }_{A{B}^{\prime }}^{\mu }$, which maps isomorphically the direct-product Weyl-spinor space of $\mathcal{W}\otimes \tilde{\mathcal{W}}$ to the four-component vector space $\mathcal{V}$.

In order to get the above result, compared with the ordinary descriptions of single-particle states, two changes have been made. The first one is usage of negative-$p^0$ states of fermions, subject to a rule requiring that no free fermion may possess a negative $p^0$. The second one is a modification to the spin space of positron, which is given by the complex conjugate space of that of electron, and similar for electron neutrino and antineutrino. These two changes bring no change to predictions for experimental results, because they change neither the (anti)commutators of free single particles, nor the amplitudes in the interaction Hamiltonian $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$.

Two concepts are crucial in the above discussed reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$. One is FIO (fundamental interaction operator), which describes a map from the state space of two fermions to that of one boson, or the reverse (with the relationship of Hermitian conjugation). The other is FVF (fundamental vacuum fluctuation), which refers to either emergence from the vacuum or vanishing into the vacuum, of a fermion-antifermion pair that possesses net zero four-momentum and net zero angular momentum.

The EvdW-symbol matrices ${\sigma }_{A{B}^{\prime }}^{\mu }$ possess an invariance property, which underlies a similar property of the practically used form of the ${\gamma }^{\mu }$ matrices; that is, both matrices are invariant under Lorentz transformations applied to both the vector and spinor indices. Usually, the invariance property of the ${\gamma }^{\mu }$ matrices is regarded as being introduced for the sake of convenience in practical derivations and computations. However, the central role played by $\sigma$ in the reformulation of $H_{\mathrm{int},\mathrm{I}}^{\mathrm{gws}}$ suggests that this property of the EvdW-symbols, as well as of the ${\gamma }^{\mu }$ matrices, may reflect a symmetry of fundamental significance.

One should notice the difference between the above discussed symmetry and Lorentz symmetry. The Lorentz symmetry requires covariance of quantities, while, the above symmetry requires invariance of the matrices. As for the operator $\sigma$ which possesses no free spinor or vector index, the Lorentz symmetry requires invariance, namely, $\sigma =\breve{\sigma }$, where $˘$ indicates the result of a Lorentz transformation. While, the above discussed symmetry requires that

$$|T_{\mu }\rangle {\sigma }_{A{B}^{\prime }}^{\mu }\langle \langle {\tilde{S}}^{B^{\prime }}|\langle \langle S^A|=|{\breve{T}}_{\mu }\rangle {\sigma }_{A{B}^{\prime }}^{\mu }\langle \langle {\breve{\tilde{S}}}^{B^{\prime }}|\langle \langle {\breve{S}}^A|,$$

(89)

that is, the same matrices ${\sigma }_{A{B}^{\prime }}^{\mu }$ are used for different sets of bases. This property could be of interest physically, because it implies an independence of $\sigma$ on the mathematical bases employed in the spaces of $\mathcal{W}\otimes \tilde{\mathcal{W}}$ and $\mathcal{V}$.

One question, which deserves future investigation, is whether the above discussed symmetry related the operator $\sigma$ may be useful for deeply understanding topics related to electroweak interactions. For example, whether some type of relationship may exist between this symmetry and the gauge-symmetry; anyway, both symmetries impose certain (though different) restrictions to the descriptions of fermions and vector bosons, in the sense of invariance of the interaction Hamiltonian. If this could be possible, then, the symmetry discussed above might be of some relevance to both the Lorentz symmetry and the gauge-symmetry and may shed new light on issues related to, say, anomaly.

Appendix A.1. Basic Properties of Weyl Spinors

In the spinor theory, there are two smallest nontrivial representation spaces of the $SL(2,\mathbb{C})$ group, which are spanned by two types of two-component Weyl spinors, respectively, with the relationship of complex conjugation. In this section, we give a brief discussion for Weyl spinors written in the ket-bra notation. 32

We use $\mathcal{W}$ to denote one of the two spaces mentioned above. In terms of components, a Weyl spinor in $\mathcal{W}$ is written as, say, ${\kappa }_A$ with an index $A=0,1$. In the abstract notation of ket, a basis in the space $\mathcal{W}$ is written as $|S^A\rangle$ and the ket form of the above spinor is written as $|\kappa \rangle$, with the expansion

$$|\kappa \rangle ={\kappa }_A|S^A\rangle ,$$

(A90)

with a summation over A implied. 33

One may introduce a space that is dual to $\mathcal{W}$, composed of bras with a basis written as $\langle \langle S^A|$. This type of bra is called a scalar-product-based bra in the main text, because it may form a scalar product with a ket. More exactly, in order to construct a product that is a scalar under $SL(2,\mathbb{C})$ transformations, the bra dual to the ket $|\kappa \rangle$ should be written as

$$\langle \langle \kappa |=\langle \langle S^A|{\kappa }_A,$$

(A91)

which has the same components as $|\kappa \rangle$ in Eq.(A90), but not their complex conjugates. (See Appendix B for basic properties of $SL(2,\mathbb{C})$ transformations, particularly Eq.(A161).)

Scalar products of the basis spinors satisfy

$$\langle \langle S^A|S^B\rangle ={ϵ}^{AB},$$

(A92)

where

$${ϵ}^{AB}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right).$$

(A93)

It proves convenient to introduce another matrix ${ϵ}_{AB}$, which has the same elements as ${ϵ}^{AB}$. These two matrices can be used to raise and lower indexes of components, say,

$${\kappa }^A={ϵ}^{AB}{\kappa }_B,{\kappa }_A={\kappa }^B{ϵ}_{BA},$$

(A94)

as well as for the basis spinors, namely,

$$\begin{array}{c}|S^A\rangle ={ϵ}^{AB}|S_B\rangle ,|S_A\rangle =|S^B\rangle {ϵ}_{BA}.\end{array}$$

(A95)

It is not difficult to verify that (i) $\langle \langle S_A|S_B\rangle ={ϵ}_{AB}$; (ii)

$${f_{\dots }^A{\left(g\right)}^{\dots }}_A=-f_{\dots A}{\left(g\right)}^{\dots A};$$

(A96)

and (iii) the symbols ${ϵ}_C^A={ϵ}^{BA}{ϵ}_{BC}$ and ${ϵ}_C^A={ϵ}^{AB}{ϵ}_{BC}$ satisfy the relation

$$\begin{array}{c}{ϵ}_C^A=-{ϵ}_C^A={\delta }_C^A,\end{array}$$

(A97)

where ${\delta }_B^A=1$ for $A=B$ and ${\delta }_B^A=0$ for $A\ne B$. (The $\delta$-symbols for other types of labels to be discussed below are defined in the same way.)

The scalar product of two generic spinors $|\chi \rangle$ and $|\kappa \rangle$, written as $\langle \langle \chi |\kappa \rangle$, has the expression of

$$\langle \langle \chi |\kappa \rangle ={\chi }_A{\kappa }^A.$$

(A98)

The anti-symmetry of ${ϵ}_{AB}$ implies that

$$\langle \langle \chi |\kappa \rangle =-\langle \langle \kappa |\chi \rangle$$

(A99)

and, as a consequence, $\langle \langle \kappa |\kappa \rangle =0$ for all $|\kappa \rangle$. Moreover, we note the following two properties: (a) The identity operator in the space $\mathcal{W}$, denoted by $I_{\mathcal{W}}$, is written as

$$\begin{array}{c}\hfill I_{\mathcal{W}}=|S^A\rangle \langle \langle S_A|,\end{array}$$

(A100)

satisfying $I_{\mathcal{W}}|\kappa \rangle =|\kappa \rangle$ for all $|\kappa \rangle \in \mathcal{W}$; and (b) the components of $|\kappa \rangle$ have the following expressions,

$${\kappa }^A=\langle \langle S^A|\kappa \rangle ,{\kappa }_A=\langle \langle S_A|\kappa \rangle .$$

(A101)

An operation of complex conjugation may be introduced, which converts $\mathcal{W}$ to a space denoted by $\tilde{\mathcal{W}}$. $\tilde{\mathcal{W}}$ is the second representation space of the $SL(2,\mathbb{C})$ group mentioned in the beginning of this section. This operation changes spinors $|\kappa \rangle$ in $\mathcal{W}$ to spinors in $\tilde{\mathcal{W}}$, denoted by $|\tilde{\kappa }\rangle$. Corresponding to the basis $|S_A\rangle \in \mathcal{W}$, the space $\tilde{\mathcal{W}}$ has a basis denoted by $|{\tilde{S}}_{A^{\prime }}\rangle$ with a primed index $A^{\prime }=0^{\prime },1^{\prime }$. On the basis of $|{\tilde{S}}_{A^{\prime }}\rangle$, $|\tilde{\kappa }\rangle$ is written as

$$|\tilde{\kappa }\rangle ={\tilde{\kappa }}_{A^{\prime }}|{\tilde{S}}^{A^{\prime }}\rangle ,$$

(A102)

where

$${\tilde{\kappa }}^{A^{\prime }}:={\left({\kappa }^A\right)}^{*}.$$

(A103)

Similar to the $ϵ$ metrices discussed above, one introduces metrices ${ϵ}^{A^{\prime }{B}^{\prime }}$ and ${ϵ}_{A^{\prime }{B}^{\prime }}$, which have the same elements as ${ϵ}^{AB}$ and are used to raise and lower primed labels. The identity operator in the space $\tilde{\mathcal{W}}$, denoted by $I_{\tilde{\mathcal{W}}}$, has the form of $I_{\tilde{\mathcal{W}}}=|{\tilde{S}}^{A^{\prime }}\rangle \langle \langle {\tilde{S}}_{A^{\prime }}|$. When a spinor ${\kappa }^A$ is transformed by an $SL(2,\mathbb{C})$ matrix, the spinor ${\tilde{\kappa }}^{A^{\prime }}$ is transformed by the complex-conjugate matrix (see Appendix B).

We also consider the direct-product space $\mathcal{W}\otimes \tilde{\mathcal{W}}$. Basis spinors in this space are written as $|S_{A{B}^{\prime }}\rangle \equiv |S_A\rangle |{\tilde{S}}_{B^{\prime }}\rangle$. As mentioned in the main text, sometimes we write $|\kappa \rangle |\tilde{\chi }\rangle$ as $|\kappa \tilde{\chi }\rangle$. In this space, in consistency with the well-known anticommutation relation of fermionic states, it is natural to assume that

$$\begin{array}{c}\hfill |S_A\rangle |{\tilde{S}}_{B^{\prime }}\rangle =-|{\tilde{S}}_{B^{\prime }}\rangle |S_A\rangle |.\end{array}$$

(A104)

For the space dual to $\mathcal{W}\otimes \tilde{\mathcal{W}}$, we write the basis spinors as $\langle \langle S_{B^{\prime }A}|\equiv \langle \langle {\tilde{S}}_{B^{\prime }}|\langle \langle S_A|$.

Appendix A.2. Dirac Spinors in the Abstract Notation

In this section, we briefly discuss the abstract ket-bra notation for Dirac spinors as solutions to the Dirac equation. We write them as combinations of Weyl spinors. 34

As is well known, the Dirac equation for a free electron with a mass m has two plane-wave solutions labelled by a Lorentz invariant index $r=0,1$, i.e.,

$$\begin{array}{c}{\phi }_{\mathrm{elec}}^r\left(x\right)=U^r\left(\mathbf{p}\right)e^{-ipx},\end{array}$$

(A105)

where $\mathbf{p}$ indicates a three-momentum and p a four-momentum, $p\equiv p^{\mu }=(p^0,\mathbf{p})$ with $\mu =0,1,2,3$, satisfying $p^{\mu }{p}_{\mu }=m^2$ with $p^0>0$. Here, $U^r\left(\mathbf{p}\right)$ are four-component spinors satisfying

$$({\gamma }^{\mu }{p}_{\mu }-m)U^r\left(\mathbf{p}\right)=0.$$

(A106)

In the chiral representation of the ${\gamma }^{\mu }$-matrices, a four-component Dirac spinor $U^r\left(\mathbf{p}\right)$ is decomposed into two Weyl spinors, as its left-handed (LH) part and right-handed (RH) part, respectively [2,3,17,18,19]. Specifically, the spinor $U^r\left(\mathbf{p}\right)$ is written as

$$\begin{array}{c}{U}^r\left(\mathbf{p}\right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}{u}^{r,A}\left(\mathbf{p}\right)\\ {\tilde{v}}_{B^{\prime }}^r\left(\mathbf{p}\right)\end{array}\right).\end{array}$$

(A107)

With labels for two-component spinors, the ${\gamma }^{\mu }$-matrices are written as

$$\begin{array}{c}{\gamma }^{\mu }=\left(\begin{array}{cc}0& {\sigma }^{\mu A{B}^{\prime }}\\ {\tilde{\sigma }}_{A^{\prime }B}^{\mu }& 0\end{array}\right),\end{array}$$

(A108)

where ${\sigma }^{\mu A{B}^{\prime }}$ are the so-called Enfeld-van der Waerden symbols, in short, EvdW symbols [17,18,19,20,21]. Note that ${\tilde{\sigma }}_{A^{\prime }B}^{\mu }$ indicates the complex conjugate of ${\sigma }_{A{B}^{\prime }}^{\mu }$, namely ${\tilde{\sigma }}_{A^{\prime }B}^{\mu }={\left({\sigma }_{A{B}^{\prime }}^{\mu }\right)}^{*}$. A set of explicit expressions often used for these symbols is written as

$$\begin{array}{c}\hfill {\sigma }^{0A{B}^{\prime }}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),{\sigma }^{1A{B}^{\prime }}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),\\ \hfill {\sigma }^{2A{B}^{\prime }}=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),{\sigma }^{3A{B}^{\prime }}=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).\end{array}$$

(A109)

The stationary Dirac equation (A106) is, then, split into two equivalent subequations, namely,

$$\begin{array}{c}\hfill u^{r,A}\left(\mathbf{p}\right)={\displaystyle \frac{1}{m}}{p}_{\mu }{\sigma }^{\mu A{B}^{\prime }}{\tilde{v}}_{B^{\prime }}^r\left(\mathbf{p}\right),\end{array}$$

(A110a)

$$\begin{array}{c}\hfill {\tilde{v}}_{B^{\prime }}^r\left(\mathbf{p}\right)={\displaystyle \frac{1}{m}}{p}_{\mu }{\tilde{\sigma }}_{B^{\prime }A}^{\mu }{u}^{r,A}\left(\mathbf{p}\right).\end{array}$$

(A110b)

Similarly, a solution for a free positron with a four-momentum p ($p^0>0$) is usually written as

$$\begin{array}{c}{\phi }_{\mathrm{posi}}^r\left(x\right)=V^r\left(\mathbf{p}\right)e^{ipx},\end{array}$$

(A111)

where $V^r\left(\mathbf{p}\right)$ satisfies

$$({\gamma }^{\mu }{p}_{\mu }+m)V^r\left(\mathbf{p}\right)=0$$

(A112)

and is written as

$$\begin{array}{c}{V}^r\left(\mathbf{p}\right)={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}{u}^{r,A}\left(\mathbf{p}\right)\\ -{\tilde{v}}_{B^{\prime }}^r\left(\mathbf{p}\right)\end{array}\right).\end{array}$$

(A113)

When explicitly writing two-component-spinor labels, special attention should be paid to the symbol ${\gamma }^0$. In fact, this symbol functions in two different ways: one as a component of ${\gamma }^{\mu }$, and the other as an ingredient that appears in some Lorentz-covariant quantities, such as $U^{†}{\gamma }^0{U}^{\prime }$ and ${\psi }^{†}{\gamma }^0{\gamma }^{\mu }\psi K_{\mu }$. When playing the second function, this symbol can not take the expression of ${\gamma }^{\mu }$ in Eq.(A108) with $\mu =0$. Indeed, e.g., doing this would lead to the following expression for ${\gamma }^0{\gamma }^{\mu }$,

$$\begin{array}{c}\left(\begin{array}{cc}{\sigma }^{0A{B}^{\prime }}{\tilde{\sigma }}_{B^{\prime }C}^{\mu }& 0\\ 0& {\tilde{\sigma }}_{A^{\prime }B}^0{\sigma }^{\mu B{C}^{\prime }}\end{array}\right),\end{array}$$

(A114)

which implies that the spinor part of the product ${\psi }^{†}{\gamma }^0{\gamma }^{\mu }\psi K_{\mu }$ would contain a term like ${\tilde{u}}^{r,A^{\prime }}\left(\mathbf{p}\right){\sigma }^{0A{B}^{\prime }}{\tilde{\sigma }}_{B^{\prime }C}^{\mu }{u}^{s,C}\left(\mathbf{q}\right)K_{\mu }$; the point lies in that this term is not Lorentz invariant due to the two labels $A^{\prime }$ and A.

In fact, in the second function discussed above, the sole role of ${\gamma }^0$ is to exchange positions of the LH and RH parts of Dirac spinors. For the sake of clearness in presentation, we write ${\gamma }_c^0$ for ${\gamma }^0$ in this case. It has the following matrix expression, without involving any two-component-spinor label,

$$\begin{array}{c}{\gamma }_c^0=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right).\end{array}$$

(A115)

Then, the matrix product ${\gamma }^0{\gamma }^{\mu }$ used in the interaction Hamiltonian has the following form,

$$\begin{array}{c}{\gamma }_c^0{\gamma }^{\mu }=\left(\begin{array}{cc}{\tilde{\sigma }}_{B^{\prime }A}^{\mu }& 0\\ 0& {\sigma }^{\mu B{A}^{\prime }}\end{array}\right).\end{array}$$

(A116)

Below, we discuss the abstract notation. In this notation, the Weyl spinors $u^{r,A}\left(\mathbf{p}\right)$ and ${\tilde{v}}_{B^{\prime }}^r\left(\mathbf{p}\right)$ are written as

$$\begin{array}{cc}\hfill & |u^r\left(\mathbf{p}\right)\rangle =u_A^r\left(\mathbf{p}\right)|S^A\rangle =-u^{r,A}\left(\mathbf{p}\right)|S_A\rangle ,\hfill \end{array}$$

(A117a)

$$\begin{array}{cc}& |{\tilde{v}}^r\left(\mathbf{p}\right)\rangle ={\tilde{v}}_{B^{\prime }}^r\left(\mathbf{p}\right)|S^{B^{\prime }}\rangle .\hfill \end{array}$$

(A117b)

They satisfy the following relations,

$$\begin{array}{cc}\hfill & \langle \langle u^r\left(\mathbf{p}\right)|u^s\left(\mathbf{p}\right)\rangle =\langle \langle {\tilde{v}}^r\left(\mathbf{p}\right)|{\tilde{v}}^s\left(\mathbf{p}\right)\rangle ={ϵ}^{rs}.\hfill \end{array}$$

(A118a)

$$\begin{array}{c}\hfill \langle \langle v^r\left(\mathbf{p}\right)|u^s\left(\mathbf{p}\right)\rangle ={\delta }^{rs}.\end{array}$$

(A118b)

The Dirac spinors $U^r\left(\mathbf{p}\right)$ and $V^r\left(\mathbf{p}\right)$ in ket are written as

$$\begin{array}{c}\hfill |U^r\left(\mathbf{p}\right)\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}|u^r\left(\mathbf{p}\right)\rangle \\ |{\tilde{v}}^r\left(\mathbf{p}\right)\rangle \end{array}\right),\end{array}$$

(A119a)

$$\begin{array}{c}\hfill |V^r\left(\mathbf{p}\right)\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}|u^r\left(\mathbf{p}\right)\rangle \\ -|{\tilde{v}}^r\left(\mathbf{p}\right)\rangle \end{array}\right).\end{array}$$

(A119b)

To be consistent with the scalar-product-based bra in Eq.(A91) for Weyl spinors, scalar-product-based bras corresponding to the above two Dirac kets should be written as

$$\begin{array}{c}\langle \langle U^r\left(\mathbf{p}\right)|={\displaystyle \frac{1}{\sqrt{2}}}\left(\langle \langle u^r\left(\mathbf{p}\right)|,\langle \langle {\tilde{v}}^r\left(\mathbf{p}\right)|\right),\end{array}$$

(A120a)

$$\begin{array}{c}\langle \langle V^r\left(\mathbf{p}\right)|={\displaystyle \frac{1}{\sqrt{2}}}\left(\langle \langle u^r\left(\mathbf{p}\right)|,-\langle \langle {\tilde{v}}^r\left(\mathbf{p}\right)|\right),\end{array}$$

(A120b)

without taking complex conjugation for the two-component spinors. Direct derivation shows that

$$\begin{array}{c}\langle \langle U^r\left(\mathbf{p}\right)|U^s\left(\mathbf{p}\right)\rangle =\langle \langle V^r\left(\mathbf{p}\right)|V^s\left(\mathbf{p}\right)\rangle ={ϵ}^{rs}.\end{array}$$

(A121)

The complex conjugates of $|U^r\left(\mathbf{p}\right)\rangle$ and $\langle \langle U^r\left(\mathbf{p}\right)|$ are written as

$$\begin{array}{c}|{\tilde{U}}^r\left(\mathbf{p}\right)\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{c}|{\tilde{u}}^r\left(\mathbf{p}\right)\rangle \\ |v^r\left(\mathbf{p}\right)\rangle \end{array}\right),\end{array}$$

(A122a)

$$\begin{array}{c}\langle \langle {\tilde{U}}^r\left(\mathbf{p}\right)|={\displaystyle \frac{1}{\sqrt{2}}}\left(\langle \langle {\tilde{u}}^r\left(\mathbf{p}\right)|,\langle \langle v^r\left(\mathbf{p}\right)|\right),\end{array}$$

(A122b)

and similar for $|V^r\left(\mathbf{p}\right)\rangle$ and $\langle \langle V^r\left(\mathbf{p}\right)|$.

Although a product $\langle \langle U\left(\mathbf{p}\right)|U^{\prime }\left(\mathbf{p}\right)\rangle$ is a Lorentz scalar, it is not an inner product, because the procedure of taking a scalar-product-based bra does not involve complex conjugation. In fact, Eq.(A121) implies that $\langle \langle U\left(\mathbf{p}\right)\left|U\right(\mathbf{p})\rangle =0$. In the ordinary notation, the inner product of two Dirac spinors is written as, say, $U^{†}{\gamma }^0{U}^{\prime }$. As shown in Ref.[21], to write the inner product in the abstract notation, one may make use of the following matrix ${\gamma }_c$,

$${\gamma }_c=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),$$

(A123)

and introduce hat-bras as defined below, 35

$$\begin{array}{c}\langle \widehat{U}\left(\mathbf{p}\right)|:=\langle \langle \tilde{U}\left(\mathbf{p}\right)|{\gamma }_c=(\langle \langle v\left(\mathbf{p}\right)|,-\langle \langle \tilde{u}\left(\mathbf{p}\right)\left|\right),\end{array}$$

(A124a)

$$\begin{array}{c}\langle \widehat{V}\left(\mathbf{p}\right)|:=\langle \langle \tilde{V}\left(\mathbf{p}\right)|{\gamma }_c=(-\langle \langle v\left(\mathbf{p}\right)|,-\langle \langle \tilde{u}\left(\mathbf{p}\right)\left|\right).\end{array}$$

(A124b)

It is straightforward to check the following scalar products,

$$\begin{array}{c}\langle {\widehat{U}}^r\left(\mathbf{p}\right)|U^s\left(\mathbf{p}\right)\rangle ={\delta }^{rs},\end{array}$$

(A125a)

$$\begin{array}{c}\langle {\widehat{V}}^r\left(\mathbf{p}\right)|V^s\left(\mathbf{p}\right)\rangle =-{\delta }^{rs},\end{array}$$

(A125b)

$$\begin{array}{c}\langle {\widehat{U}}^r\left(\mathbf{p}\right)|V^s\left(\mathbf{p}\right)\rangle =0,\end{array}$$

(A125c)

$$\begin{array}{c}\langle {\widehat{V}}^r\left(\mathbf{p}\right)|U^s\left(\mathbf{p}\right)\rangle =0.\end{array}$$

(A125d)

It is seen that $\langle {\widehat{U}}^r\left(\mathbf{p}\right)|U^s\left(\mathbf{p}\right)\rangle$ for electron is an inner product. While, $\langle {\widehat{V}}^r\left(\mathbf{p}\right)|V^s\left(\mathbf{p}\right)\rangle$ for positron is a negative inner product, due to the minus sign on the rhs of Eq.().

One remark: In fact, functions of ${\gamma }_c^0$ and ${\gamma }_c$ are essentially equivalent. Their difference lies in that ${\gamma }_c^0$ is used when Dirac spinors are written in the component notation, while, ${\gamma }_c$ is used in the abstract notation.

To introduce Dirac spinors with indices written in the lower position, due to the relation in Eq.(A125a), one may do in the ordinary way. That is, a label r in the upper position is lowered by ${\delta }_{rs}$; reversely, r in the lower position is raised by ${\delta }^{rs}$. Explicitly, one writes

$$\begin{array}{c}|U_s\left(\mathbf{p}\right)\rangle =|U^r\left(\mathbf{p}\right)\rangle {\delta }_{rs},|U^s\left(\mathbf{p}\right)\rangle ={\delta }^{sr}|U_r\left(\mathbf{p}\right)\rangle .\end{array}$$

(A126)

For the sake of consistency, lower labels of the spinors $|V\rangle$ should be defined in the same way as in Eq.(A126). Thus, one gets that

$$\begin{array}{c}\langle {\widehat{U}}^r\left(\mathbf{p}\right)|U_s\left(\mathbf{p}\right)\rangle ={\delta }_s^r,\langle {\widehat{V}}^r\left(\mathbf{p}\right)|V_s\left(\mathbf{p}\right)\rangle =-{\delta }_s^r.\end{array}$$

(A127)

Making use of Eq.(A127), it is easy to verify that the identity operator on the four-dimensional space of Dirac spinors, denoted by $I_{\mathcal{D}}$, has the following expression,

$$\begin{array}{c}{I}_{\mathcal{D}}=|U^r\left(\mathbf{p}\right)\rangle \langle {\widehat{U}}_r\left(\mathbf{p}\right)|-|V^r\left(\mathbf{p}\right)\rangle \langle {\widehat{V}}_r\left(\mathbf{p}\right)|.\end{array}$$

(A128)

Appendix A.3. Basic Properties of Four-Component Vectors

In this section, we recall basic properties of four-component vectors given in the theory of spinors [17,18,19]. We use the ordinary notation in this section and will discuss the abstract notation in the next section.

A basic point is a one-to-one mapping, given by the EvdW symbols discussed above, between the direct-product space $\mathcal{W}\otimes \tilde{\mathcal{W}}$ and a four-dimensional space denote by $\mathcal{V}$. For example, a spinor ${\varphi }_{A{B}^{\prime }}$ in the space $\mathcal{W}\otimes \tilde{\mathcal{W}}$ is mapped to a vector $K^{\mu }$ in the space $\mathcal{V}$ by

$$K^{\mu }={\sigma }^{\mu A{B}^{\prime }}{\varphi }_{A{B}^{\prime }}.$$

(A129)

In the space $\mathcal{V}$, of particular importance is a symbol denoted by $g^{\mu \nu }$, which is defined by the following relation to the $ϵ$-symbols discussed previously,

$$g^{\mu \nu }={\sigma }^{\mu A{B}^{\prime }}{\sigma }^{\nu C{D}^{\prime }}{ϵ}_{AC}{ϵ}_{B^{\prime }{D}^{\prime }}.$$

(A130)

One may introduce a lower-indexed symbol $g_{\mu \nu }$, which has the same matrix elements as $g^{\mu \nu }$, namely, $\left[g^{\mu \nu }\right]=\left[g_{\mu \nu }\right]$. These two symbols g, like the symbols $ϵ$ for the space $\mathcal{W}$, may be used to raise and lower indexes, e.g.,

$$K_{\mu }=K^{\nu }{g}_{\nu \mu },K^{\mu }=g^{\mu \nu }{K}_{\nu }.$$

(A131)

Making use of the antisymmetry of the symbol $ϵ$, it is easy to verify that $g^{\mu \nu }$ is symmetric, i.e.,

$$g^{\mu \nu }=g^{\nu \mu }.$$

(A132)

Due to this symmetry, the upper/lower positions of repeated indexes ($\mu$) are exchangeable, namely

$${F_{\dots }^{\mu }{\left(f\right)}^{\dots }}_{\mu }=F_{\dots \mu }{\left(f\right)}^{\dots \mu }.$$

(A133)

The EvdW symbols have the following properties,

$${\sigma }_{\mu }^{A{B}^{\prime }}{\sigma }_{C{D}^{\prime }}^{\mu }={\delta }_{C{D}^{\prime }}^{A{B}^{\prime }},{\sigma }_{A{B}^{\prime }}^{\mu }{\sigma }_{\nu }^{A{B}^{\prime }}={\delta }_{\nu }^{\mu },$$

(A134)

where ${\delta }_{C{D}^{\prime }}^{A{B}^{\prime }}:={\delta }_C^A{\delta }_{D^{\prime }}^{B^{\prime }}$. Making use of the relations in Eq.(A134), it is not difficult to check that the map from $\mathcal{W}\otimes \tilde{\mathcal{W}}$ to $\mathcal{V}$ given in Eq.(A129) is reversible. Moreover, using Eq.(A97), one finds that

$$\begin{array}{c}{\sigma }_{\mu A{B}^{\prime }}{\sigma }_{C{D}^{\prime }}^{\mu }={ϵ}_{AC}{ϵ}_{B^{\prime }{D}^{\prime }}.\end{array}$$

(A135)

Then, substituting the definition of $g^{\mu \nu }$ in Eq.(A130) into the product $g^{\mu \nu }{g}_{\nu \lambda }$, after simple algebra, one gets that

$$\begin{array}{c}{g}^{\mu \nu }{g}_{\nu \lambda }=g_{\lambda }^{\mu }=g_{\lambda }^{\mu }={\delta }_{\lambda }^{\mu }.\end{array}$$

(A136)

When an $SL(2,\mathbb{C})$ transformation is carried out on the space $\mathcal{W}$, a related transformation should be applied to the space $\mathcal{V}$. Requiring invariance of the EvdW symbols, transformations on the space $\mathcal{V}$ can be fixed, which turn out to constitute a (restricted) Lorentz group and the space $\mathcal{V}$ is a four-component vector space (see Appendix B). In fact, substituting the explicit expressions of the EvdW symbols in Eq.(A20) into Eq.(A41), one gets

$$\begin{array}{c}\hfill g^{\mu \nu }={\sigma }_{A{B}^{\prime }}^{\mu }{\sigma }^{\nu A{B}^{\prime }}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right),\end{array}$$

(A137)

which is just the Minkovski’s metric.

As shown in Appendix B [Eq.(A173)], the following product,

$$\begin{array}{c}{J}_{\nu }{K}^{\nu }=J^{\mu }{g}_{\mu \nu }{K}^{\nu },\end{array}$$

(A138)

is a scalar under Lorentz transformations. Physically, of more interest is a product, in which one of the two vectors takes a complex-conjugate form, say,

$$\begin{array}{c}{J}_{\nu }^{*}{K}^{\nu }=J^{\mu *}{g}_{\mu \nu }{K}^{\nu }.\end{array}$$

(A139)

Similarly, one finds that this product is also a scalar.

Appendix A.4. Abstract Notation for Four-Component Vectors

In the abstract notation of ket, a basis in the space $\mathcal{V}$ is written as $|T_{\mu }\rangle$. The index of the basis may be raised by $g^{\mu \nu }$, i.e., $|T^{\mu }\rangle =g^{\mu \nu }|T_{\nu }\rangle$, and similarly $|T_{\mu }\rangle =g_{\mu \nu }|T^{\nu }\rangle$. A generic four-component vector $|K\rangle$ in the space $\mathcal{V}$ is expanded as

$$|K\rangle =K_{\mu }|T^{\mu }\rangle =K^{\mu }|T_{\mu }\rangle .$$

(A140)

In consistency with the scalar-product-based bra in Eq.(A91), the scalar-product-based bra corresponding to $|K\rangle$ is written as

$$\begin{array}{c}\langle \langle K|=\langle \langle T_{\mu }|K^{\mu }.\end{array}$$

(A141)

Similar to the case of Weyl spinors in Eq.(A92), one requires that

$$\begin{array}{c}\hfill \langle \langle T_{\mu }|T_{\nu }\rangle =g_{\mu \nu }.\end{array}$$

(A142)

Then, it is easy to check that the scalar product $J_{\nu }{K}^{\nu }$ is written as $\langle \langle J|K\rangle$, namely,

$$\begin{array}{c}\langle \langle J|K\rangle =J_{\nu }{K}^{\nu }.\end{array}$$

(A143)

It is not difficult to verify the following properties. (i) Making use of Eq.(A136), one finds that the identity operator in the space $\mathcal{V}$, denoted by $I_{\mathcal{V}}$, is written as

$$I_{\mathcal{V}}=|T_{\mu }\rangle \langle \langle T^{\mu }|=|T^{\mu }\rangle \langle \langle T_{\mu }|.$$

(A144)

(ii) The components $K^{\mu }$ and $K_{\mu }$ have the following expressions,

$$K^{\mu }=\langle \langle T^{\mu }|K\rangle ,K_{\mu }=\langle \langle T_{\mu }|K\rangle .$$

(A145)

And (iii) the symmetry of $g^{\mu \nu }$ implies that $\langle \langle T_{\mu }|T_{\nu }\rangle =\langle \langle T_{\nu }|T_{\mu }\rangle$, as a result,

$$\begin{array}{c}\hfill \langle \langle K|J\rangle =\langle \langle J|K\rangle .\end{array}$$

(A146)

Since $\widetilde{|S_{A{B}^{\prime }}\rangle }=|S_{A^{\prime }B}\rangle =-|S_{B{A}^{\prime }}\rangle$, with $|S_{A{B}^{\prime }}\rangle \equiv |S_A\rangle |{\tilde{S}}_{B^{\prime }}\rangle$, the operation of complex conjugation maps the space $\mathcal{V}$ into itself. We use $\widetilde{|T_{\mu }\rangle }$ to denote the complex conjugate of $|T_{\mu }\rangle$. Since $\widetilde{|T_{\mu }\rangle }$ and $|T_{\mu }\rangle$ lie in the same space, it is unnecessary to introduce any change to the label $\mu$. Hence, $\widetilde{|T_{\mu }\rangle }$ can be written as $|{\tilde{T}}_{\mu }\rangle$ with the label $\mu$ unchanged.

It proves convenient to introduce an operator related to the EvdW symbols, denoted by $\sigma$, namely,

$$\sigma :=|T_{\mu }\rangle {\sigma }^{\mu A{B}^{\prime }}\langle \langle S_{B^{\prime }A}|.$$

(A147)

This operator $\sigma$ has a simple geometric meaning; that is, it maps a product space $\mathcal{W}\otimes \tilde{\mathcal{W}}$ to a vector space $\mathcal{V}$. Using $\tilde{\sigma }$ to indicate the complex conjugate of $\sigma$, one has

$$\tilde{\sigma }=|{\tilde{T}}_{\mu }\rangle {\tilde{\sigma }}^{\mu A^{\prime }B}\langle \langle S_{B{A}^{\prime }}|,$$

(A148)

where ${\tilde{\sigma }}^{\mu A^{\prime }B}\equiv {\left({\sigma }^{\mu A{B}^{\prime }}\right)}^{*}$. Making use of the explicit expressions of the EvdW symbols in Eq.(A109), it is easy to verify that

$${\tilde{\sigma }}_{\mu }^{B^{\prime }A}={\sigma }_{\mu }^{A{B}^{\prime }}.$$

(A149)

There is some freedom in the determination of the relationship between $\sigma$ and $\tilde{\sigma }$ and, relatedly, between $|T_{\mu }\rangle$ and $|{\tilde{T}}_{\mu }\rangle$. The simplest assumption is that $|T_{\mu }\rangle$ is “real”, i.e.,

$$|{\tilde{T}}_{\mu }\rangle =|T_{\mu }\rangle .$$

(A150)

Making use of Eqs.(16), (A104), and (A148)-(A149), it is not difficult to verify that

$$\tilde{\sigma }=-\sigma .$$

(A151)

Thus, the complex conjugates of $|K\rangle$ and $\langle \langle K|$ are written as

$$\begin{array}{c}|\tilde{K}\rangle =K^{\mu *}|T_{\mu }\rangle \&\langle \langle \tilde{K}|=\langle \langle T_{\mu }|K^{\mu *}.\end{array}$$

(A152)

Then, the scalar product in Eq.(A139) is written as

$$\begin{array}{c}\langle \langle \tilde{J}|K\rangle =J_{\mu }^{*}{K}^{\mu }.\end{array}$$

(A153)

It is easy to verify that

$$\begin{array}{c}\langle {\langle \tilde{K}|J\rangle }^{*}=\langle \langle \tilde{J}|K\rangle .\end{array}$$

(A154)

We use ${\sigma }^T$ to denote the transposition of $\sigma$,

$${\sigma }^T:=|S_{A{B}^{\prime }}\rangle {\sigma }^{\mu A{B}^{\prime }}\langle \langle T_{\mu }|.$$

(A155)

Computing the product of $\sigma$ and ${\sigma }^T$, with the help of Eqs.(A142), (A134), and (A100), it is easy to verify that

$${\sigma }^T\sigma =\sigma {\sigma }^T=I,$$

(A156)

which implies that ${\sigma }^T$ is the reverse of $\sigma$.