Submitted:
06 November 2025
Posted:
11 November 2025
Read the latest preprint version here
Abstract
Physical processes are usually described using four-dimensional vector quantities - coordinate vector, momentum vector, current vector. But at the fundamental level they are characterized by spinors - coordinate spinors, momentum spinors, spinor wave functions. The propagation of fields and their interaction takes place at the spinor level, and since each spinor uniquely corresponds to a certain vector, the results of physical processes appear before us in vector form. For example, the relativistic Schrödinger equation and the Dirac equation are formulated by means of coordinate vectors, momentum vectors and quantum operators corresponding to them. In the Dirac equation a step forward is taken and the wave function is a spinor with complex components, but still coordinates and momentum are vectors. For a closed description of nature using only spinor quantities, it is necessary to have an equation similar to the Dirac equation in which momentum, coordinates and operators are spinors. It is such an equation that is presented in this paper. Using the example of the interaction between an electron and an electromagnetic field, we can see that the spinor equation contains more detailed information about the interaction than the vector equations. This is not new for quantum mechanics, since it describes interactions using complex wave functions, which cannot be observed directly, and only when measured goes to probabilities in the form of squares of the moduli of the wave functions. In the same way spinor quantities are not observable, but they completely determine observable vectors. In Section 2 of the paper, we analyze the quadratic form for an arbitrary four-component complex vector based on Pauli matrices. The form is invariant with respect to Lorentz transformations including any rotations and boosts. The invariance of the form allows us to construct on its basis an equation for a free particle combining the properties of the relativistic wave equation and the Dirac equation. For an electron in the presence of an electromagnetic potential it is shown that taking into account the commutation relations between the momentum and coordinate components allows us to obtain from this equation the known results describing the interactions of the electron spin with the electric and magnetic field. In the presence of a potential the momentum components cease to commute with each other. To neutralize this effect, the Schrödinger equation is supplemented by several equations with mixed derivatives on coordinates. In section 3 of the paper this quadratic form is expressed through momentum spinors, which makes it possible to obtain an equation for the spinor wave function in spinor coordinate space by replacing the momentum spinor components by partial derivative operators on the corresponding coordinate spinor component. Section 4 presents a modification of the theory of the path integral, which consists in considering the path integral in the spinor coordinate space. The Lagrangian densities for the scalar field and for the electron field, along with their corresponding propagators, are presented. An equation of motion for the electron is proposed that is relativistically invariant, in contrast to the Dirac equation, which lacks this invariance. This novel equation permitted the construction of an actually invariant procedure for the second quantization of the fermion field in spinor coordinate space. Furthermore, it is demonstrated that the field operators are a combination of plane waves in spinor or vector space, with the coefficients of which being pseudospinors or pseudovectors. Each of these pseudovectors or pseudospinors corresponds to one of the particles presented in the theory of electrodynamics. Furthermore, each plane wave possesses an additional coefficient in the form of a creation or annihilation operator. In vector space, these operators commute, whereas in spinor space they anticommutate. The paper presents the spinor and vector representations of the field operators in explicit form, comprising sets of 16 pseudospinors or 4 pseudovectors corresponding to particles represented in electrodynamics. An explicit form of the symmetric traceless tensor with spin two, zero mass and two polarizations is presented, which can serve as a model of the graviton. The results obtained may prompt changes in some aspects of the construction of Feynman diagrams. Among other things, it presents a purely mathematical derivation of Maxwell's inhomogeneous equations without reference to empirical data on the action of electric current, which is usually referred to when deriving equations.
Keywords:
Dirac equation
; Pauli matrices
; Schrödinger equation
; second quantization
; path integral
; graviton
1. Introduction
Nowadays, the interest to study applications of the Dirac equation to different situations and to find out the conditions of its generalization is not weakening. In particular, in [1] new versions of an extended Dirac equation and the associated Clifford algebra are presented. In [2] a study of the Schrödinger-Dirac covariant equation in the presence of gravity, where the non-commuting gamma matrices become space-time-dependent, is carried out. In [3] an idea is discussed that the visible properties of the electron, including rest mass and magnetic moment, are determined by a massless charge spinning at light speed within a Compton domain. In [4] some aspects of conformal rescaling in detail are explored and the role of the "quantum" potential is discussed as a natural consequence of non-inertial motion and is not exclusive to the quantum domain. Author establishes the fundamental importance of conformal symmetry, in which rescaling of the rest mass plays a vital role. Thus, the basis for a radically new theory of quantum phenomena based on the process of mass-energy flow is proposed. In [5] author has derived the covariant fourth-order/one-function equivalent of the Dirac equation for the general case of an arbitrary set of γ-matrices.
Supporting these search aspirations, in our work we propose a deeper understanding of the Dirac equation with an emphasis on the direct use of the principles of symmetry and invariance to Lorentz transformations. For the first time we present a formulation of the Dirac and Schrödinger equations in spinor coordinate space.
2. Generalized Dirac Type Equation
Let us introduce notations, which will be used further on. The speed of light and the rationalized Planck’s constant will be considered as unity.
Pauli matrices
Matrices constructed from Pauli matrices
A set of arbitrary complex numbers and a vector of its three components
𝛬 = exp(𝛼1R1)exp(𝛽1K1)exp(𝛼2R2)exp(𝛽2K2)exp(𝛼3R3)exp(𝛽3K3)
Let's define a 4×4 matrix
In fact, we consider a quaternion with complex coefficients, which we multiply by its conjugate quaternion (due to the complexity of the coefficients, these are biquaternions, but we still use quaternionic conjugation, without complex conjugation).
Let us subject the set of complex numbers to the Lorentz transformation
Since the matrices anticommutate with each other, for a vector whose components commute with each other, we have just the simplest case with a diagonal matrix with on the diagonal. But if the components of vector do not commute, the matrix already has a more complex structure and carries additional physical information compared to . For example, the vector may include the electron momentum vector and the electromagnetic potential vector. The four-component potential vector is a function of the four-dimensional coordinates of Minkowski space. The components of the four-component momentum do not commute with the components of the coordinate vector, respectively, and the coordinate function does not commute with the momentum components, and their commutator is expressed through the partial derivative of this function by the corresponding coordinate. If the components of the vector do not commute, the matrix will no longer be invariant with respect to Lorentz transformations.
Assume that the complex numbers we are considering commute with all matrices, and consider that the matrices are pairwise anticommutative and their squares are equal to the unit 4×4 matrix I
Taking into account the commutation relations of the components of the momentum vector and the coordinate vector, the commutator of the momentum component and the coordinate function is expressed through the derivative of this function by the corresponding coordinate, e.g.
Along with the original form
The matrix in the general case has complex elements and is not diagonal, and in the Dirac equations instead of it is substituted the product of the unit matrix by the square of mass , the physical meaning of such a substitution is not obvious. Apparently it is implied that it is the square of the mass of a free electron. But the square of the length of the sum of the lengths of the electron momentum vectors and the electromagnetic potential vector is not equal to the sum of the squares of the lengths of these vectors, that is, it is not equal to the square of the mass of the electron, even if the square of the length of the potential vector were zero. But, for example, in the case of an electrostatic central field, even the square of the length of one potential vector is not equal to zero. Therefore, it is difficult to find a logical justification for using the mass of a free electron in the Dirac equation in the presence of an electromagnetic field. Due to the noted differences, the solutions of the generalized equation can differ from the solutions arising from the Dirac equation.
In the case when there is a constant magnetic field directed along the z-axis, we can write down
When the influence of the electromagnetic field was taken into account, no specific characteristics of the electron were used. When deriving a similar result using the Dirac equation, it is assumed that since the electron equation is used, the result is specific to the electron. In our case Pauli matrices and commutation relations are used, apparently these two assumptions or only one of them characterize the properties of the electron, distinguishing it from other particles with non-zero masses.
The proposed equation echoes the Dirac equation, at least from it one can obtain the same formulas for the interaction of spin and electromagnetic field as with the Dirac equation, and in the absence of a field the proposed equation is invariant to the Lorentz transformations. In contrast, to prove the invariance of the Dirac equation even in the absence of a field, the infinitesimal Lorentz transformations are used, but the invariance at finite angles of rotations and boosts is not demonstrated. The proof of invariance of the Dirac equation is based on the claim that a combination of rotations at finite angles can be represented as a combination of infinitesimal rotations. But this is true only for rotations or boosts around one axis, and if there are at least two axes, this statement is not true because of non-commutability of Pauli matrices, which are generators of rotations, so that the exponent of the sum is not equal to the product of exponents if the sum includes generators of rotations or boosts around different axes. By a direct check we can verify that the invariance of the Dirac equation takes place at any combination of rotations, but only under the condition of zero boosts, i.e., only in a rest frame of reference, any boost violates the invariance.
A test case for any theory is the model of the central electrostatic field used in the description of the hydrogen atom, in which the components of the vector potential are zero
All the above deductions are also valid when replacing 4×4 matrices by 2×2 matrices , since their commutative and anticommutative properties are the same. The corresponding generalized equation is of the form
One would seem to expect similar results from other representations of the momentum operator, e.g., [6], formula (24.15).
If one consistently adheres to the Heisenberg approach and does not involve the notion of wave function, it is not very clear how to search for solutions of the presented equations. The Schrödinger approach with finding the eigenvalues of the matrix and their corresponding eigenfunctions can help here.
Thus, we have arrived at an equation containing a matrix which is non-diagonal, complex and in general depends on the coordinates . After the standard procedure of separating the time and space variables, we can go to a stationary equation in which there will be no time dependence, but the dependence the matrix on the coordinates will remain. It is possible to ignore the dependence of on the coordinates and its non-diagonality and simply replace this matrix by a unit matrix with a coefficient in the form of the square of the free electron mass. Then the equation will give solutions coinciding with those of the Dirac equation. But this solution can be considered only approximate and the question remains how far we depart from strict adherence to the principle of invariance with respect to Lorentz transformations and how far we deviate from the hypothetical true solution, which is fully consistent with this principle. To find this solution, we need to approach this equation without simplifying assumptions and look for a set of solutions, each of which represents an eigenvalue matrix of arbitrary form and its corresponding four-component eigenfunction.
Let us return to the question of Lorentz invariance of the expression
It is the fulfilment of these equations that causes the mass and to acquire the properties we tend to expect of it, namely that the mass is not only invariant under Lorentz transformations, but does not change under accelerations either.
We can introduce tensor notations
To accurately calculate the fields at a moving point, it is necessary to use equations that take into account the Lorentz transformations for the given external electromagnetic potential and corresponding transformations for external fields.
In the simplest case, when the speed is constant, the expressions for converting fields at an immovable point into fields at a moving point are as follows.
Nevertheless, let us assume that external electric and magnetic fields are specified for a immovable point. From these fields we form electromagnetic tensor. We also need to know the instantaneous momentum vector and find the Lorentz transformation that results in this vector from the momentum vector of a fixed points. This Lorentz transformation acts not only on the momentum vector, but also on the coordinate vector. We must apply the same transformation to electromagnetic tensor and extract from it the values of the electric and magnetic fields for moving point that we need.
For quantum mechanics we can replace the momentum components in all equations by the derivative operators
The equations proposed here initially take into account the non-commutability of momentum components, their derivation relies only on the unconditional fulfilment (even in coupled systems) of the requirement of invariance to Lorentz transformations for the product of conjugate quaternions with arbitrary coefficients
The conditions expressed by additional strings of our equations may be too strong, since they require that each pair of brackets with derivatives is zero. But invariance can also be achieved with a weaker requirement that only their sum as a whole is zero. That is, each pair of brackets can deviate from zero; the main thing is that these deviations are compensated in the total sum. This can work both in classical and quantum mechanics. A hint on the validity of this approach is given by Maxwell's equations, in which conditions are imposed not on individual derivatives, but on their sums. In addition, it is intuitively clear that the components having similarity to velocity should be considered in the sum in order not to depend on the rotations of the coordinate system.
If not to substitute the coordinate derivative instead of the momentum component and to remain in the framework of classical physics, the system of equations
By means of the antisymmetric Levy-Civita symbol we transform antisymmetric tensors into dual tensors
There derivation is as follows. The homogeneous part of Maxwell’s equations
It is noteworthy that the four-dimensional divergence of the right-hand side of the proposed equation
Let us clarify our understanding of the interaction between the electromagnetic field and the momentum of a charged particle. The charged particle creates in the surrounding space lagging potentials and fields, which depend on the nature of its movement - it is motionless, moves evenly or accelerated. In any case, we are talking about the field at points not coinciding with the location of the particle itself. It’s another thing when we look at a field where the charged particle is. If there is an external electromagnetic field at this point, the charge interacts with it and moves with acceleration, while the required equality of the derivatives of the momentum and potential at the point where the charge is located is observed, no additional field is created.
If the particle is additionally accelerated or slowed under the influence of external mechanical force, then an additional field, either amplifying or attenuating it, arises to comply with the required equality of derivatives. This additional field under some special conditions can be a source of electromagnetic waves, which is the solution of an inhomogeneous wave equation in the right part of which as a source is just this additional field. But if an additional field at the point of a particle is created at any acceleration of its external force, for example, it takes place in an electrical generator, and then the radiation waves do not generally occur. To generate waves, the field at the point of the particle’s presence must be variable, that is, its first derivative must be different from zero, for example, in time, which corresponds to the second derivative of potential. So there must be different from zero also a second derivative of the momentum of the particle, that is, a derivative of acceleration, which is sometimes called a jerk. The fields in these waves have a different origin than that inherent in the moving particle of the changing field of lagging potentials. Thus, in the vicinity of a charged particle there is a superposition of three types of electromagnetic fields, if you take into account the external field. Let us emphasize that the creation of electromotive force in an electric generator is not a field in the vicinity of the particle, but at the point of its location.
Note that the expression
Thus, with the help of a sequence of electrical impulses we can precisely turn the electron at arbitrary angles around any axes. But we have three more components of the magnetic field standing in the same positions of the electromagnetic tensor as the derivatives of the pulse, if some of its components are substituted by derivatives.
Here we have Newton’s law already for rotations, i.e. the impulse of the magnetic field directly performs the rotation of the electron around the corresponding axis. While the pulse of the magnetic field rotates a particle at a fixed angle, the constant magnetic field rotates it at a constant speed. In a quantum computer, you can use both control of the rotation angle of a particle by an electric or magnetic field, and control of its constant rotation by means of a magnetic field. With the combination of four electrical pulses described above, it is also possible to change the direction of rotation of the particle, and even to stop the rotation. Because Newton’s law works in both directions, by means of electrical or magnetic pulses it is possible not only to initialize the state of the particle, but also to read the parameters of this state after performing manipulations in a quantum computer.
If we recall that in Schrödinger equation, we replace momentum with the derivative of the scalar wave function
Let us consider the matrix
For the system with mixed derivatives we have applied the most stringent requirements possible, equating to zero each of the expressions with matrices . But invariance can also be achieved with less stringent requirements
When transitioning from classical interpretation to quantum mechanics, the equations of motion are applied in a different way. In this context, let us clarify the relationship between the Lagrangian, the equations of motion, and the translation operator. This operator transforms the quantum state and its corresponding wave function in coordinate representation from one point to another infinitely close point, acting either in configuration space or in Minkowski real space
Thus, if we assume that nature at a fundamental level is described by the translation operator over an infinitesimally small distance and by the equations of motion, and these two laws are absolute truths, then the integral over trajectories method using the Lagrangian is only an approximate truth, which can be arrived at only at the cost of several controversial assumptions.
Let us formulate the phase and mass invariance requirement in a more general form using the metric tensor of the flat Minkowski space
If we replace the metric tensor of the Minkowski space by the metric tensor of the space curved by the action of gravitation of the general theory of relativity, will the phase and mass remain invariant to the Lorentz transformations? Since the curvature of space is caused only by the action of external masses, we can assume that the invariance principle is absolute. Then we have at our disposal the equations imposing restrictions on the metric of curved space.
The matrix may be rewritten in another form
Let’s go back to our equation
By the way, the basic equation given earlier
Extend the formulated principle to the gravitational field, for which we define a covariant derivative including affine connection and act on the vector field
Use familiar expressions to find the commutator of a covariant derivative
According to the proposed principle, components of a covariant derivative must commute, so
Similarly, as the equation of motion of a charged particle is an addition to the basic equation that includes mass, so the equation of motion of particles in a gravitational field is an addition to some equation that also includes mass. Suppose that in the case of gravity this equation has the analogous form, including a diagonal matrix with a square mass on the diagonal
For example, in Yang-Mills theory, the covariant derivative for a field Ψ, which can be viewed as a vector in a certain vector space and on which gauge fields act, is written as follows
3. Equation for the Spinor Coordinates Space
Let us consider the set of arbitrary complex numbers, for simplicity we will call it a vector
And since we do not doubt the truth of the theory of relativity and quantum mechanics, we cannot doubt the reality of spinor space, which by means of the simplest arithmetic operations generates our space and time.
The quantity is invariant under the Lorentz transformation simultaneously applied to the momentum and coordinate spinor, which automatically transforms both corresponding vectors as well
Let us apply the differential operator to the spinor analog of a plane wave
Since
When substantiating the Schrödinger equation for a plane wave in four-dimensional vector space, an assumption is made (further confirmed in the experiment) about its applicability to an arbitrary wave function. Let us make a similar assumption about the applicability of the reduced spinor equation to an arbitrary function of spinor coordinates, that is, we will consider this equation as universal and valid for all physical processes.
Let us clarify that by the derivative on a complex variable from a complex function we here understand the derivative from an arbitrary stepped complex function using the formula that is valid at least for any integer degrees
It is very important to emphasize that we consider the complex variable and the variable conjugate to it to be independent, so when finding the derivative of a complex variable from some function, we treat all the quantities which are conjugate to our variable and which are included in this function, as ordinary constants.
It is not by chance that we denote the eigenvalue by the symbol m, because if we form the momentum vector from the momentum spinor included in the expression for the plane wave
For the momentum spinor of a fermion-type particle we can consider another form in the rest system
Further we need an expression for the commutation relation between the components of the momentum spinor, to which is added the corresponding component of the electromagnetic potential spinor, which is a function of the spinor coordinates
An interesting fact is that time is always a positive quantity. As an assumption it can be noted that since we observe that time value goes forward, i.e. the value of t grows, and it is possible only due to scaling of all components of spinor space, such scaling leads to increase of distance between any two points of Minkowski space. As a result, with the passage of time the Minkowski space should expand, herewith at first relatively quickly, and then more and more slowly.
The advantages of considering physical processes in spinor coordinate space may not be limited to electrodynamics. It may turn out, for example, that the spinor space is not subject to curvature under the influence of matter, as it takes place in the general theory of relativity for the vector coordinate space. On the contrary, it can be assumed that it is when the components of vector coordinate space are computed from the coordinate spinor that the momentum spinor with a multiplier of the order of the gravitational constant is added to this spinor. This results in a warp that affects other massive bodies.
To account for the electron spin, we will further represent the electron wave function as a four-component spinor function of four-component spinor coordinates
We will search for the solution of the wave equation considered in the first part of this paper
Let us use the model of a plane wave in spinor space
In this case we can consider the matrix in the right part to be diagonal with the same elements on the diagonal , then the equation can be rewritten as an equation for the problem of finding eigenvalues and eigenfunctions
Since we have freedom of choice of the basis, it is reasonable to choose the spinor for the wave function as some set of momentum spinor components, for example
The mass of electron and the phase of the plane spinor wave
For a fermion, which can be an electron or a positron in the rest frame takes place , so the quantity
For the momentum spinor of a boson, such as a photon, it is true that , so its mass is zero
taking its values in the corresponding point of physical space with coordinates
Let us summarize the relations between quantum-mechanical quantities for the spinor space
The arbitrary choice of the basis of the linear space of the eigenvectors of the matrix takes place only for a free particle. In the general case the matrix K is not zero, the wave equation has no solution in the form of plane waves in spinor space and ceases to be invariant with respect to Lorentz transformations, and the eigenvalues become nondegenerate.
We propose to extend the scope of applicability of the presented equation consisting of differential operators in the form of partial derivatives on the components of coordinate spinors to case of a nonzero matrix K
This equation will be called the equation for the spinor wave function defined on the spinor coordinate space. Here the matrix is, generally speaking, neither diagonal nor real, but it does not depend on the coordinates and is determined solely by the parameters of the electromagnetic field. Only in the case of a plane wave it is diagonal and has on the diagonal the square of the mass of the free particle. We can try to simplify the problem and require that the matrix is diagonal with the same elements , then the equation can be rewritten in the form of the equation for the problem of search of eigenvalues and eigenfunctions for any quantum states
We are of the opinion that the spinor equation is more fundamental than the relativistic Schrödinger and Dirac equations, it is not a generalization of them, it is a refinement of them, because it describes nature at the spinor level, and hence is more precise and detailed than the equations for the wave function defined on the vector space.
Let us consider the proposed equation for the special case when the particle is in an external electromagnetic field, which we will also represent by a four-component spinor function at a point of the spinor coordinate space
After the addition of the electromagnetic field the components of the momentum spinor do not commute, the corresponding commutators are found above
Earlier we defined a matrix of commutators
We will not use the given considerations further in the paper, leaving them as an idea requiring a separate consideration.
Let's solve the equation
We can also consider the case of a constant magnetic field directed along the z-axis
While the Dirac equation is sometimes referred to as extracting the square root of the Klein-Gordon equation, here we see a different way of doing it.
Let us check the truth of the relation
Let us compare the phases of plane waves in vector and spinor spaces. Let us hypothesize that the plane wave in spinor space has a more complicated form than it was supposed earlier in the paper, namely, it contains an additional conjugate multiplier
Now we can define the function
4. Path Integral and Second Quantization in Spinor Coordinate Space
Based on the above, we can modify the theory of the path integral. We will consider it in the notations in which it is presented in [9]. For a free scalar field with sources J(X) the path integral has the form
Substituting the Lagrangian density into the Euler equation
The free field theory is developed for a special kind of polynomial
Now we have to find the path integral, which, along with the Lagrangian density, includes the sources
It is possible to recover Planck's constant, which provides a transition to the classical limit
We note at once that there is no simple correspondence between the so defined phase of a plane wave in spinor space and the phase of a plane wave in vector space, e.g.
One can see the difference between the propagators, since in one case is real and positive, while in spinor space m is complex in general. We can use the relation
We can find the derivatives of the scalar by the components of the coordinate spinor
What are the advantages of the transition from path integral in vector space to path integral in spinor space? A possible answer is that there are new conditions for working with divergent integrals. Now integration is performed over spinor space, so that in the numerator there is a four-dimensional differential element instead of element in the case of vector space. The spinor element has the order of magnitude instead of for the vector element, whish decreases the order of magnitude of the numerator, while the order of magnitude of the denominator does not change.
If the spinor coordinate space is indeed more fundamental, and the vector coordinate space is an offspring of it, then we ma y benefit from this transition in any case.
Now let us move from the scalar field to the field of an electron, that is, the field of a particle with half-integer spin. We will use gamma matrices in the Weyl basis
As a result, the propagator has the form
Let's return to the question about the use of completely relativistically invariant Lagrangian density
Let us consider in detail the derivation of the expression for the fermion propagator in [9], Sec. II.2. It is based on the assumption of relativistic invariance of the Dirac equation and therefore the calculations are carried out in the rest frame, and then the result is extended to an arbitrary frame of reference. Thus for the field spinor u the spinor u_ is defined and it is asserted that the value of
Nevertheless, this fact and the fact of non-invariance of the Dirac equation itself do not cancel the value of the second quantization procedure and the final form of the fermion propagator, which allows to make accurate predictions of the experimental results.
We hope that the proposed Lagrangian density for the spinor coordinate space can find application in the calculation of the path integral, but already in the spinor space. Whether such a calculation in spinor space has an advantage over the calculation of the path integral in vector space can be shown by their real comparison.
By analogy with the propagator of a photon, more precisely of a massive vector meson, given in [9], formula I.5.3
Let us formulate again the difference between the equations, the second of which is derived from the Dirac equation with gamma matrices in the Weyl basis
By analogy with [9], Chapter II.2, we will carry out the procedure of second quantization of the fermion field. Let us write the equation
It is important that all the above deductions are valid in any frame of reference, while the proof of anticommutativity of the fermion field in [9] is carried out for the rest frame.
Let us calculate the total energy of the fermion field
The following relations were taken into account in the derivation
In [10] the quantization procedure includes the use of one definite Lorentzian reference frame, i.e. it is not invariant. In our case all deductions are valid in any reference frame in the spinor space, and it means invariance to change of reference frames in the Minkowski space also.
The following relations are used in the transformations
In the spinor coordinate space, we can express the translational invariance of the field operator by the relations
Both operators act on the same state, but in one case the state is labeled by spinor coordinates and in the other by vector coordinates. The translation mechanism of the operators is essentially the same, but it is not possible to replace the action of one translation operator by some combination of actions of the other. Because of this, the question arises as to which of these operators better describes nature. Our point of view is that the translation operator in spinor space is primary, and the operator in vector space just successfully copies it, without being exact, but being some approximation. It attracted the attention of physicists first because vector space is more accessible for investigation. When integrating over a four-dimensional vector space in some cases there is a divergence, then use renormalization. When integrating over four-dimensional spinor space, the differential element has two orders of magnitude of the vector momentum component smaller, while the denominator in the integrand remains of the same order as when integrating over vector space. This difference possibly affects the convergence.
Let us calculate the total mass of the fermion field
creates and destroys a particle with mass , spin up and momentum in the interval , is the operator of the number of such particles
creates and destroys a particle with mass , spin up and momentum in the interval , is the operator of the number of such particles
creates and destroys a particle with mass , spin up and momentum in the interval , is the operator of the number of such particles
Note that and are translated into each other by a linear transformation, this is also true for other pairs of spinors
The properties of all particles and operators are summarized in a table
| creates | destroys | particle spinor | vector | number | mass | spin | wave sign |
|---|---|---|---|---|---|---|---|
| up | + | ||||||
| down | - | ||||||
| up | - | ||||||
| down | + | ||||||
| up | + | ||||||
| down | - | ||||||
| up | - | ||||||
| down | + | ||||||
| up | + | ||||||
| down | - | ||||||
| up | - | ||||||
| down | + | ||||||
| up | + | ||||||
| down | - | ||||||
| up | - | ||||||
| down | + |
Here the column “vector” shows the vector obtained from the corresponding spinor by the formula of the form
By the words destroys the particle it should be understood that this operator transforms this particle into the particle, and the operator performs the reverse transformation of into . The action of the operator on any other particle gives zero. Since both these particles have the same mass, the total mass of the fermionic field does not change from these transformations. The mass itself can have any sign or even be complex.
Although we call the spinors presented in the table particles, they actually describe the same particle whose characteristic property is a mass with a certain sign. A particle with mass of opposite sign is described by the other sixteen spinors. Let's compare the momenta of two particles with different mass signs
In the table below two last columns with a set of spinors and vectors corresponding to the particle with opposite sign of mass are added. It is supposed that at annihilation the spinors of the particle and antiparticle, which are in the same row of the table, are summed. The set of 16 spinors remains the same, but the order of their arrangement changes when the sign of mass changes.
| creates | destroys | particle spinor | vector | mass | spin | wave sign | antiparticle spinor | vector |
|---|---|---|---|---|---|---|---|---|
| up | + | |||||||
| down | - | |||||||
| up | - | |||||||
| down | + | |||||||
| up | + | |||||||
| down | - | |||||||
| up | - | |||||||
| down | + | |||||||
| up | + | |||||||
| down | - | |||||||
| up | - | |||||||
| down | + | |||||||
| up | + | |||||||
| down | - | |||||||
| up | - | |||||||
| down | + |
It is possible to assume that the reason and condition of distinction between particles with different charge is the presence of their non-zero mass. If the mass is zero, then in the given table there are no differences between spinors of the particle and antiparticle, i.e. there is no mechanism for formation of the internal degree of freedom, which we treat as charge.
Let us see what result we get if we apply another definition of anticommutativity of the fermionic field.
It is noteworthy that the antisymmetric treatment, whether or not complex conjugation is considered, yields a diagonal matrix that is invariant in one case but not in the other. It is encouraging to observe that the set of reference spinors remain consistent.
It is crucial to note that the proposed invariant approach cannot be realized within the Minkowski vector space. To achieve this, it is necessary to transition to the spinor space. This reiterates the secondary role of the Minkowski space in comparison to the spinor space.
Dirac's equation can be expressed in both spinor and vector spaces, a fact that led Dirac to discover it. In contrast, the invariant equation can be written in spinor space but not in vector space, which explains why it was unknown.
Let us write down the propagator of the fermionic field and the fermionic field invariant equation of motion using the proposed matrices
Let us look for a representation of the electromagnetic field operator in vector space without first referring to spinor space. Let us define four vectors expressed through the components of the momentum vector
An alternative view of the selected four vectors is possible. The initial 16 spinors can be interpreted in the spirit of quantum mechanics as vectors in a Hilbert space. They describe pure states and form a complete basis, since the sum of their density matrices (projectors) is equal to the diagonal matrix with the following value on its diagonal
It is interesting that if one does not use complex conjugation when forming the tensor product of spinors, then the sum of 16 such products will also be a diagonal matrix with the following value on the diagonal
When we consider the 16 states expressed through the components of the momentum spinor, the question arises as to how to interpret them if the electron is in an atom. It should be noted that in this case, the components of the electron's momentum are replaced by derivatives of the wave function in the spinor coordinate representation with respect to the corresponding spinor coordinate. In this case, each spatial mode is assigned 16 pseudo-spinors. This interpretation provides a physical basis for understanding the nature of the spin of an electron in an atom.
The polarization vectors for all other pairs do not have such a simple form and change under rotations and boosts. But among the spinors there are still pairs with interesting properties. For example, the state u4, which corresponds to the polarization vector U4, and v1, which corresponds to the vector V1. Their singlet and triplet states correspond to the following polarization vectors
Since electrons tend to form pairs (in atoms or in superconductors), it is logical to consider the tensor product of two Hilbert spaces. This product is itself a Hilbert space, and 256 tensor products of the original 16 spinors form a complete basis of states in this space. The sum of all projection operators is equal to the diagonal matrix with a diagonal element
The representation as vector composed of momentum components
For the selected vectors the following relations are valid
Let us find the commutation relations for the wave function and its time derivative, which in this case play the role of canonical momentum
Let us calculate the square of the field energy
As we have seen, neither electron current vectors nor electromagnetic field vectors are true vectors. When transforming the coordinate system, the same transformation acts on the components of the momentum vector, from these transformed components in each frame of reference the pseudovectors of the field are formed. But we know that the interaction between current and electromagnetic field is described by an additional term in the Lagrangian density of the electrodynamics theory. This term is the scalar product of the current and the electromagnetic potential and it is necessary for this product to be a scalar. But to form a scalar using a metric tensor, two true vectors are needed, and these are not available. There remains only one way to provide the scalar, it is necessary that signs of components in pseudovectors of current and field coincide, then they will compensate each other, and in fact we will get the scalar product of two vectors, and hence we will get a scalar.
Thus, there is a direct interrelation between the spinor description of the field and its vector description. 16 pseudospinors pass into 4 pseudovectors, moreover, the modulus of the complex mass in spinor space is equal to the mass in vector space. At all this by the value of the phase of a plane wave in spinor space by any direct way it is not possible to calculate the phase of a plane wave in vector space. Hence the assumption arises that operators in spinor space describe nature exactly, while operators in vector space provide only an approximate description. This may partly explain the problems with divergence when integrating in vector space.
To describe the evolution of the field state, we consider the vacuum averaged expression having the sense of the propagator. Before we do so, let us explain the meaning of operators included in the field decomposition
Let us define a vacuum state of the field with zero filling numbers of particles of each of four varieties by specifying its properties with respect to the action of annihilation operators
Note that the matrix entering the matrix element has no inverse, so we do not try to find the equation of motion or Lagrangian density, they are not necessary in this case, since we have an explicit expression for the field operator. We do not have to worry about following the principles of Lorentzian covariance, gauge invariance, or following ideas of symmetry. Instead, we rely only on the fulfilment of canonical commutation relations for the field operator. The field operator is written identically in any frame of reference, and to pass to another frame it is enough to know how the momentum vector is transformed, which is transformed by exactly the same law as the coordinate vector, which ensures the invariance of the phase of the plane wave. In other words, the field is not a vector but a set of pseudovectors (pseudospinors in spinor space), only momentum and coordinate are vectors (spinor).
We can make our reasoning more intuitively clear if we define the creation and annihilation operators of the field particle
Let us define the total particle number operator in the form
The fact that for the creation and annihilation operator’s commutation relations are fulfilled, allows to conclude that quanta of the field obey Bose statistics, therefore a single action of the creation operator increases the number of particles in the field by one, and the action of the annihilation operator decreases this number by one. Hence, by means of these operators it is possible to write the matrix element not only for the case when the initial and final states are vacuum, but also for the initial state with an arbitrary number of particles
Separate application of the creation and annihilation operators more corresponds to the ideology of second quantization than their use only as a sum, i.e. only as a field operator
Moreover, now the matrix element can be not bound to the vacuum state, but can be applied to the field state with arbitrary number of particles n>0. The application of the sum of operators to some state makes sense only in the case when all operators except one give as a result zero. Therefore, at the usual approach we have to work only with the vacuum state so that at calculation of the matrix element the annihilation operator gives zero. In our approach this restriction is removed, the operators are not summed, but only multiplied, and they can be applied to a state with any number of particles. For this purpose, let us take into account the following relations
We can use the creation and annihilation operators instead of the field operator, and we can apply them to an arbitrary state, not just the vacuum state. So, we don't need to calculate the vacuum mean and apply Wick's theorem.
We would like the matrix element to have properties of the Green's function, i.e., to satisfy the equation
Similarly, we use the commutator of the annihilation operator
If the mass is zero, then in any state the number of particles is zero, i.e., for example, the electromagnetic field in spinor space, where it should be fermionic, simply has no particles. The absence of particles does not contradict the presence of the field, which is represented by the same 16 spinors, this field obeys Fermi statistics, and it has no charge and can be treated as a Majorana fermion. This field interacts with electrons in spinor space, and the result of the interaction manifests itself in vector space.
With the help of the creation and annihilation operators we can write the matrix element for the situation when the initial and final states are states with arbitrary number of particles
We can repeat the above calculations, keeping the annihilation operator, but defining the creation operator differently
Note that in this revision the creation and annihilation operators are conjugate to each other
There is another way to give the properties of the Green's function to the propagator by introducing the necessary multiplier not in the definition of the creation and annihilation operators, but directly in the propagator. In this case we can proceed from two similar equations
It may seem artificial to add an additional multiplier to the denominator of the plane wave. But then we must remember where the plane wave itself came from, namely, that it is a solution of the homogeneous wave equation. But the wave cannot arise from nothing, it must have a source, that is, it must be a solution of the inhomogeneous wave equation. In the case of a source in the form of a delta function, this solution includes just such a multiplier in the denominator. Furthermore, the introduction into the denominator of a factor proportional to energy is often justified by the need to ensure the Lorentz invariance of the integral.
Using the found field operators and propagators, one can make corrections in the rules of construction of Feynman diagrams. In particular, one of 16 pseudospinors should be compared to the external lines of the fermion, and one of four pseudovectors should be compared to the external lines of the boson instead of the polarization vector. One should also correct the type of propagators of the fermion and the boson. All these corrections can be done even within the classical Feynman diagrams in vector space. Then the outer lines will be explicitly expressed through the momentum components, and the diagram will look the same in any reference frame. A complete transition from integration in vector space to integration in spinor space would be a more radical step, the consequences of which are yet to be studied.
In particular, when integrating over the momenta associated with the inner lines of Feynman diagrams in vector space, a divergence of the form , for example, arises. In spinor space the denominator remains the same, but the numerator in the integral is of lower order, since the component of the momentum vector is a bilinear form of the component of the momentum spinor.
In light of the existence of operators capable of creating and annihilating fermions and bosons that are constrained to a particular point in space, a re-evaluation of the concept of elementary particle interaction becomes imperative. Without recourse to prior reasoning or justification, it can be assumed that the interaction is described by the multiplication of any set of such operators bound to the same point in space. This product, which also incorporates constant multipliers to account for the degree of interaction, is then integrated over the entire coordinate space. The result of this integration is a multiple integral over the momentum space, which in vector space often diverges, but in spinor space perhaps not.
Previously defined operators
The simplest recipe for the description of the electron-positron field seems to be the use in its equation of motion of the matrix, which we obtained in the construction of the anticommuting field operator, namely
Replacing the momentum components by derivatives gives us an analogue of the Schrödinger equation
We can go further and assume that the picture of interaction of fields is more symmetric and is described by the mixed equation
However, if, following Heisenberg [12], Chapter 3, Paragraph 1, we index the field components differently
Let us suggest that the coordinate and momentum spinor spaces can also be indexed in a similar way
Let's rewrite the previously used quantities with new variables
Summarizing, we can formulate the following theses. The initial coordinate space is described by complex quantities, which can be represented as a square matrix
Each plane wave in superposition has a multiplier in the form of a matrix
For the field, we can obtain the equation of motion as an equation in partial derivatives on the complex variables by substituting the derivatives on these variables instead of the derivatives on in the previously discussed equations.
We can also consider the decomposition of the field by the previously considered plane waves of the form
Thus, a plane wave with invariant phase in spinor space is a more general concept than a plane wave in vector space, although the concept of invariant mass cannot be introduced for it in the general case.
Let us consider constructing an explicit expression for the tensor with spin two to propose a model for the gravitational field carrier. A tensor formed by the direct production of vectors does not give the desired result, because its spin remains equal to one. A different approach is needed.
By means of gamma matrices in the Weyl basis
With the 16 momentum spinors at our disposal we can define 256 tensors of the form, e.g.
We need to take a closer look at Maxwell's inhomogeneous equations. Let's repeat the relations that we will analyze
It can be argued that the expression
By analogy, we can recall that in Einstein's equations, the Riemann tensor also contracts to the Ricci tensor with a loss of information. It can be assumed that by simultaneously convolving all the terms in the original equation, it is possible to strictly derive Einstein's inhomogeneous equation, as is the case with Maxwell's inhomogeneous equations.
The expression we propose looks like the definition of a current vector by a given momentum tensor. But why do we convolve the tensor and reduce it to a vector? After all, when performing convolution, some of the information contained in the tensor is lost. For example, any information about the internal rotations of a particle described by the momentum rotor is definitely lost. One of the reasons we are forced to do this is that, in the early days of electrodynamics, current was described exclusively as a vector, and not as a tensor. Therefore, in order to write the equations, it was necessary to convert the left-hand side into a vector using a convolution, which is also combined with taking derivatives. With the development of quantum electrodynamics, this approach was reinforced by the fact that for an electron, the Dirac spinors describing it are combined in such a way that the current again turns out to be a vector. As a result, the system using the current vector turns out to be closed.
But if electric current is the movement of electrons, and electrons have spin, then spin does not disappear anywhere in electric current. Having spin, electrons are a source of magnetic fields, and when convolved into the current vector, this property of the source magnetic field source disappears, and there are no magnetic field sources on the right side of Maxwell's equations. This means that describing current as a vector is insufficient; we would like to define current as a tensor formed from the original spinor of the electron. The following definition is proposed
Using the 16 spinor at our disposal, we can define 16 tensors of the form, for example
We can also write the relation for momentum
Now, we can propose the following definition of the Lagrangian density for electrodynamics
Conclusions
An alternative approach to analyze relativistic and quantum effects inherent in charged particles in the presence of an electromagnetic field is proposed. Two ways of describing the electron behavior in the electromagnetic field are considered: by means of the vector equation, which is based on the plane wave model for a free electron, and the spinor equation, which is based on the representation of the electron as a plane wave in spinor space. For both equations, which are valid for a free particle, their applicability to an arbitrary physical situation is postulated, in particular to describe the behavior of a particle in the presence of an electromagnetic field. The presented equations are intended to fulfill the same role as the Schrödinger equation and the Dirac equation. At the same time, in our opinion, the spinor equations more accurately describe the details of the interaction between fields and particles.
According to the results of consideration the following picture emerges. Physical processes occur in the spinor coordinate space, which is perceived by us as a vector Minkowski coordinate space, the coordinates of which are equal to bilinear combinations of spinor coordinates. In the spinor space propagate plane waves whose phase is equal to the scalar product of the momentum and coordinate spinors. As amplitudes of plane waves in fermions are linear combinations of 16 pseudospinors, whose components are components of the momentum spinor with changing signs of some of them. The sign of the charge of a fermion is determined by the sign of its mass, which is calculated from its momentum and invariant to Lorentz transformations. Scalar, vector and tensor fields also propagate as plane waves in spinor space, but the coefficients of these waves are scalars, vectors and tensors formed by simple combinations on the basis of Pauli matrices of 16 fermionic pseudospinors. In this case only four pseudovectors (corresponding to four polarizations) of the photon and only two pseudotensors (corresponding to two polarizations) of the graviton are different. The spinor field has spin equal to 1/2, the spin of scalar fields is automatically zero, that of vector fields is unitary, and that of tensor fields is double. The second term responsible for interaction, in a fixed electromagnetic field and with 16 different current tensors, equals four different quantities with a plus sign and the same four quantities with a minus sign.
References
- Marsch, E.; Narita, Y. A New Route to Symmetries through the Extended Dirac Equation. Symmetry 2023, 15, 492. [Google Scholar] [CrossRef]
- Fleury, N.; Hammad, F.; Sadeghi, P. Revisiting the Schrödinger–Dirac Equation. Symmetry 2023, 15, 432. [Google Scholar] [CrossRef]
- Jean Maruani, The Dirac Equation as a Privileged Road to the Understanding of Quantum Matter. Quantum Matter 2015, 4, 3–11. [CrossRef]
- Hiley, B.; Dennis, G. de Broglie, General Covariance and a Geometric Background to Quantum Mechanics. Symmetry 2024, 16, 67. [Google Scholar] [CrossRef]
- Andrey Akhmeteli, The Dirac equation as one fourth-order equation for one function - a general, manifestly covariant form, arXiv: 1502.02351v9 [quant-ph] 23 Apr 2022. arXiv:1502.02351v9 [quant-ph] 23 Apr 2022.
- Leonard, I. Schiff, Quantum Mechanics, Third edition. McGraw-Hill Book Company, 1959 - 545 p.
- P. A. M. Dirac. The principles of quantum mechanics (International Series of Monographs on Physics), Fourth edition, Oxford Science Publications.
- Lewis H. Ryder, Quantum Field Theory, University of Kent, Canterbury – 1996 г.
- A Zee, Quantum Field Theory in a Nutshell, 2nd Edition. – Princeton University Press, 2003.
- Paul A., M. Dirac, Lectures on quantum field theory, New York 1967.
- Claude Itzycson and Jean-Bernard Zuber, Quantum field theory, McGraw-Hill Book Company, 1980.
- W. Heisenberg, Introduction to the unified field theory of elementary particles, Interscience publishers, London, New York, Sydney, 1966.
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.