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.
Matrices constructed from Pauli matrices
A set of arbitrary complex numbers and a vector of its three components
Let us define a 2×2 matrix of Lorentz transformations given by the set of real rotation angles and boosts
and a similar 4×4 transformation matrix
We also define a 4×4 matrix of Lorentz transformations, where
μ and
ν take values 0,1,2,3
which can also be written explicitly using the 4×4 matrices of rotation generators
and boosts
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
Let us write a relation whose validity for an arbitrary set of complex numbers can be checked directly
The matrix
in the simplest case is diagonal with equal complex elements on the diagonal equal to the square of the length of the vector
in the metric of Minkowski space, which we denote
. Both
and
do not change under any rotations and boosts, in physical applications the invariance of
is usually used, in particular, for the four-component momentum vector this quantity is called the square of mass.
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 expressions for pairwise products of matrices, we obtain
Consider the case when
is the sum of the momentum vector and the electromagnetic potential vector, which is a function of coordinates
For now, we'll stick with the Heisenberg approach, that is, we will consider the components of the momentum vector
as operators for which there are commutation relations with coordinates or coordinate functions such as
. In this approach, the operators do not have to act on any wave function.
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.
As a result, we have the expression
does not change under Lorentz transformations involving any rotations and boosts.
Taking into account the electron charge we have
Let us summarize our consideration. There is a correlation
Let's analyze the obtained equality
Note that the quantity
is invariant to the Lorentz transformations irrespective of whether the momentum and field components commute or not. To solve this equation, we have to make additional simplifications. For example, to arrive at an equation similar to the Dirac equation, we must equate
with the matrix
, where
is the square of the mass of a free electron. Then
With this substitution the generalized equation almost coincides with the equation [
6], formula (43.25), the difference is that there is a plus sign before
, and instead of
there is
, in which the matrices
have the following form
A similar equation is given by Dirac in [
7], Paragraph 76, Equation 24; he does not use the matrices
, only the matrices
, but the signs of the contributions of the magnetic and electric fields are the same.
Along with the original form
it is possible to consider the form with a different order of the factors. It can be shown that this leads to a change in the sign of the electric field contribution
Since
, unlike
, is invariant to Lorentz transformations, it would be logical to replace it by
. At least both these matrices are diagonal, and in the case of a weak field their diagonal elements are close. Nevertheless, the approach based on the Dirac equation leads to solutions consistent with experiment.
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
Here
. Only when the field is directed along the z-axis, the matrix
is diagonal and real because the third Pauli matrix is diagonal and real. And if the field is weak,
can be approximated by the
matrix. This is probably why it is customary to illustrate the interaction of electron spin with the magnetic field by choosing its direction along the z-axis. In any other direction
is not only non-diagonal, but also complex, so that it is difficult to justify the use of
.
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
If again we equate the left part with
, we obtain
Introducing the notations (
)
we obtain
If we substitute operators acting on the wave function instead of momentum components into the equation, we obtain a generalized analog of the relativistic Schrödinger equation, in which the wave function has four components. Using the substitutions
the equation for the four-component wave function
before all transformations has the form
and after transformations
Once again, note that the matrix
is not diagonal and real.
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
and the equation for the now two-component wave function looks like
In deriving his equation, Dirac [
7], Paragraph 74, noted that as long as we are dealing with matrices with two rows and columns, we cannot obtain a representation of more than three anticommuting quantities; to represent four anticommuting quantities, he turned to matrices with four rows and columns. In our case, however, three anticommuting matrices are sufficient, so the wave function can also be two-component. Dirac also explains that the presence of four components results in twice as many solutions, half of which have negative energy. In the case of a two-component wave function, however, no negative energy solutions are obtained. Particles with negative energy in this case also exist, but they are described by the same equation in which the signs of all four matrices
or
are reversed.
One would seem to expect similar results from other representations of the momentum operator, e.g., [
6], formula (24.15).
under the assumption that this representation can describe a particle with spin one. But this expectation is not justified, since the last three matrices do not anticommutate, and therefore the quadratic form constructed on their basis is not invariant under Lorentz transformations.
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.
In the left-hand side are the operators acting on the wave function, and in the right-hand side is a constant matrix on which the wave function is simply multiplied. This equality must be satisfied for all values of the four-dimensional coordinates
at once. Then
is not fixed but can take a set of possible values, finding all these values is the goal of solving the equation.
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
As it was noted, this expression does not change at rotations and boosts in Minkowski space only if the components of
commute with each other. If they do not commute, the matrix
changes under Lorentz transformations. Two parts can be distinguished in this matrix
The first row represents the unit matrix multiplied by a value that still does not change under Lorentz transformations. All changes occur in the last three rows. In the particular case of electrodynamics, we have (
)
Here the first line is invariant, but the last three are not. The only way to ensure complete invariance of
is to require these three lines to be zero. Let us again consider the commutation relations, but now we will not assume that the momentum components commute with each other, only the potential components still commute with each other. Now we can write the relations of the form
Such values as
always enter
as a sum with the component of the field, in this case the electric one
If we formally define a new value
and suppose that
m does not change at rotations and boosts, and we also take into account the presence of charge at the electron, we can require for this and all other similar sums the fulfilment of the condition
The value
can be regarded as a component of velocity, and velocity not in the usual sense, as a derivative of the spatial coordinate by time, but simply as a component of momentum divided by the inertial mass
m. Then the above equality can be interpreted in the spirit of Newton's law, namely, that the acceleration multiplied by the mass is equal to the force acting on the side of the electric field. If all such equalities are fulfilled, only the first line will remain in the quantity
, and it will be invariant under Lorentz transformations. It is possible to go further, and to assume equality of the masses appearing here, namely
As a result, we obtain a system consisting of the basic equation
which can be briefly written as
In the general case it is necessary to take into account in the equations an external force, e.g. of mechanical nature, defined at each point of the coordinate space. The equations then have the form
The external force is taken as given, and the acceleration and field are mutually adjusted to nullify the right-hand side. This is true for a charge in an electric field that is fixed stationary, for an electric generator, for an electric motor, and so on.
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
The resulting system of equations describes not only uniform but also accelerated motion. The presence of an external field leads to a change in momentum, and vice versa, any change in momentum under the influence of an external force perturbs the potential and generates an electromagnetic field. The equations include only the derivatives of the momentum components and lack the values of the momentum components themselves. However, if the particle is moving, the external field at its location changes depending on the nature of the motion, so the instantaneous values of the momentum components are indirectly included in the equations as well.
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.
Of course, the very assumption of constant velocity contradicts the essence of Newton's equations, in which momentum and velocity change, so Lorentz transformations for fields in general may be more complex. On the other hand, the vector product describing the Lorentz force is usually used in calculations without reservations about the constancy of velocity.
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
This also applies to equations from the second group, where mixed derivatives arise
As a result, we obtain for the wave function a system of equations with second order derivatives, the innovation compared to the commonly used Schrödinger equation is the presence in the equations of mixed derivatives on all components of the coordinate vector.
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
Putting all equations together, we write a truly relativistic system of equations for the wave function
This system is a generalization of the relativistic Schrödinger equation. The essence of the generalization consists not only in taking into account the spin of the electron, which takes place already in the Dirac equation, but also takes into account the non-commutability of the momentum components. It can be assumed that the solutions of this generalized system will give exact values for stationary electron energy levels in the atom, for which no radiative corrections will be needed.
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
describes the motion of a macroscopic charged particle in the presence of an electromagnetic field.
By means of the antisymmetric Levy-Civita symbol we transform antisymmetric tensors into dual tensors
and we use Maxwell's equations written in compact form
Let us apply the derivative operator to our proposed equations
then taking into account Maxwell's equations, we obtain
If in the presence of an arbitrary potential there is no particle in the moving point, then our equations are the homogeneous Maxwell’s equations for an arbitrarily moving point. If a charge is placed in the point, we obtain inhomogeneous equations for an arbitrarily moving charge. The proposed equations can be considered as a derivation of Maxwell's equations.
There derivation is as follows. The homogeneous part of Maxwell’s equations
clearly follows from the definition of electromagnetic tensor. The inhomogeneous part
is a direct consequence of the approach we propose. The four-dimensional vector on the right describes the field sources determined by the motion of a charged particle. This is very same current vector that was included in the equation derived by Maxwell from empirical data. We do not need a reference to empirical data, since the generation of electric current by a charged particle with nonzero mass directly follows from the principle of invariance of its mass.
It is noteworthy that the four-dimensional divergence of the right-hand side of the proposed equation
is zero in itself, and not because it is equal to the left-hand side, which has zero divergence. That is, the current determined in this way automatically satisfies the conservation law required of the current in the usual interpretation of Maxwell’s equations.
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
illustrates the possibility of controlling parameters, such as an electron for quantum computing. The first line of the matrix shows that the pulse of the electric field along the
x-axis changes the component
x of the particle momentum, that is, produces a boost along this axis. After this you can run a boost along the
y-axis, then a boost along the
x-axis in the opposite direction and also a boost along the
y-axis, also in the opposite direction
Since the boosts on different axes do not commute, this sequence of boosts leads to a rotation, so the particle will rotate at some angle around the
z-axis. Incidentally, we note that the commutator of any pair of boosts and/or rotations with different matrices
and
is equal to a matrix
different from them with some real or imaginary coefficient
, for example
This is a special case of a general relation for arbitrary complex square matrices
and
, arbitrary complex numbers
α, β and some complex number
С
It’s interesting whether this relations holds for the case when
and
are operators, not matrices?
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.
The magnetic field on the right-hand side of the equality is the curl of the electromagnetic potential, and the corresponding positions in the tensor on the left can be interpreted as the curl of the momentum, describing the internal rotations of the particle. The presence of a curl in the electromagnetic potential leads to the appearance of a curl in the charged particle momentum. And conversely, due to the presence of a momentum curl of charged particles there is a constant magnetic field.
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
and, on the other hand, the vector potential is also surely a derivative of some field
then the relationship between the momentum rotor and the magnetic field does not seem strange
It means that the covariant derivatives of fields are not required to commute (if they commuted, there would be no magnetic field), but their commutators are required to compensate each other. In the Schrödinger equation itself, we use the construction
It is tempting to generalize it to a more symmetrical form, that is, the derivative of the product of fields
At the same time, it is unclear why we can usually ignore the electromagnetic field in the first term, perhaps because this field has zero mass?
Let us consider the matrix
which includes sets of arbitrary complex numbers
Let us subject these sets of complex numbers to the Lorentz transformation
In this case, the matrix
has invariant trace proportional to the scalar product of vectors
Having made the same calculations that were made earlier, it is possible to check that
Let us consider a special case of the phase of a plane wave in Minkowski space in the presence of an electromagnetic potential
We again apply the principle of absolute invariance and proceed from the assumption that the trace
must be invariant to the Lorentz transformations even in the presence of the field. The first line satisfies this condition, and the trace of the other three must be equated to zero, which leads to the equations for the momentum components. Let us perform the substitutions
We have obtained the equations for the wave function, which, together with the equations
form a complete system that describes a charged quantum particle in the presence of an electromagnetic field.
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
Let us formulate the essence of the proposed approach. For each point of Minkowski space we have equations arising from the requirement of invariance of mass to Lorentz transformations and to changes of momentum (simply speaking, to accelerations). They reduce to the equality to zero of the set of differences between the derivatives of momentum and the derivatives of some potential field with possible addition of an external force. In the conventional sense, these are the equations of motion. If we want to obtain a complete picture of the behavior of the system in the form of a trajectory in coordinate space, we must choose such a trajectory at each point of which the equations of motion are satisfied. To do this, we have to integrate the mentioned differences over some measure and choose the trajectory with the minimum of the integral. This way we get the only classical trajectory. It is imperative to emphasize that the equations of motion were not obtained from the principle of least action; rather, they were derived from the requirement of invariance of mass to Lorentz transformations and its invariance to accelerations.
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
Here, momentum is an operator, and coordinate is a number – one of the eigenvalues of the coordinate operator, since we assume that the state is described by a wave function in coordinate representation. To move a finite distance between two points, you can choose a trajectory connecting these points, and apply the translator translation to each step. The resulting operator will be the product of the operators, i.e., the product of exponents of the specified type
To achieve our goal – the transition to Lagrangian – we will sacrifice rigor and replace the product of the exponent with the exponent of the sum, turning a blind eye to the possible non-commutativity of the exponents with each other
Next, we will replace the momentum operators with their eigenvalues, interpreting the resulting quantity as the probability amplitude of the of transition along the selected trajectory. This substitution may also be incorrect due to the probable non-commutativity of the momentum component operators with each other. After that, we find the sum of the probability amplitudes for all possible trajectories
The total phase along the trajectory is influenced by the fact that at each point of the trajectory the momentum components are not independent; they are subject to constraints imposed by the equation of motion. To account for this constraints, we use the ratio between the Hamiltonian and the Lagrangian
Using the two controversial assumptions mentioned above, we arrived at the well-known scheme using the Lagrangian and the corresponding action. The transition to the Lagrangian simplifies the calculation of the phase, reducing integration over multidimensional coordinate space to integration over time. Moreover, since, for a given Lagrangian, the equation of motion is determined by the Euler-Lagrange equation, it is assumed that the constraints on the momentum components contained in the equation of motion are automatically taken into account in the Lagrangian. However, if the equations of motion are considered to be a more fundamental description of nature than the Lagrangian, then doubts remain as to whether all the constraints defined by these equations are taken into account in it.
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
Let us note an important difference. In the absence of an external field the matrix
is diagonal and invariant to Lorentz transformations, while the matrix
is not diagonal even in the absence of an external field and it changes under Lorentz transformations. However, its trace is invariant and equal to four invariant phases. So we can formulate the general principle as a requirement that at addition of an external field the trace of matrices in both cases remains invariant to Lorentz transformations.
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
has the meaning of the energy-momentum tensor. However, in this form it is not very clear how to add the external field carefully. Earlier we added a vector field to the momentum vector, but perhaps it makes sense to add at once some tensor field to the momentum-energy tensor. The tensor of this field may have a more general form than the direct product of the sums of the electromagnetic potential vector and the momentum vector. In this context the question arises whether it is possible in this way to take into account the influence of the gravitational field, which is described by a tensor, while requiring the conservation of mass invariance.
Let’s go back to our equation
and look at it from a different point of view, taking into account the correspondence of the momentum component and the derivatives according to the coordinates
or, when using covariant derivatives instead of contravariant derivatives
Then the previously obtained condition of absolute mass invariance will have the form
Add the vector field
to the covariant derivative
and find the commutator component of a covariant derivative acting on the scalar field
The commutator coincides exactly with the left part of our equation, providing absolute mass invariance. Thus, in order for mass to be invariant, it is necessary that the covariant derivative containing the vector field and acting on the scalar field has a zero commutator of its components. This condition determines the equation of motion of a charged particle in an electromagnetic field, and the commutator of the derivative includes an electromagnetic tensor containing components of the electromagnetic field voltage. This principle of commutability of a covariant derivative works both in the classical interpretation for a charged particle in an electromagnetic field
and is the same in quantum interpretation when the scalar field
is a scalar wave function, and the operator equation of motion is an additional condition for the Schrödinger equation.
By the way, the basic equation given earlier
can also be written using the covariant derivative
The requirement of the commutativity of the covariant derivative is now added to the basic equation
It is sometimes assumed that the components of the ordinary derivative and the corresponding components of the momentum are switched between each other. If this were true, then the components of the momentum, for example, could not change over time, that is, the particle could not accelerate, because the derivative of the momentum in time is equal to the momentum commutator with the energy operator. In fact, the components of a generalized derivative that is obtained after adding a compensating field are commutated. It is the addition of this field that provides commutability of the derivative and mass invariance.
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
Here used designations: torsion tensor
which plays the role of a gravitational field strength.
According to the proposed principle, components of a covariant derivative must commute, so
In the interpretation of quantum mechanics we have obtained an operator equation for the vector wave function
in coordinate representation. In the classical interpretation, it is necessary to replace the components of the operator derivative with the components of momentum, which in this case must have two indices, that is to say, be in some sense energy - momentum tensor
Again, let us consider that the commutator of the coordinate function and the operator of the momentum is equal to a derivative of this function according to the corresponding coordinate
In the left part there is a derivative momentum, in particular, for the time coordinate it gives acceleration, and in the right part the force of gravity that causes this acceleration. This expression is an analogue of Newton’s law and represents the equation of motion when a particle interacts with the gravitational field, the tension of which is described by Riemann’s tensor. Unlike the case of the electromagnetic field, in the right part there is also a component of the momentum that is Newton’s law contains non-linearity. The formula works in the opposite direction. If the particle is exposed to external influences, such as electromagnetic nature, then it accelerates, so at this point an additional gravitational field is created that adds to the external one. If the second derivative of the momentum is not zero, the source appears in the right part of some wave equation. Then gravitational waves are generated.
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
The invariance of this mass to changes in the momentum is provided precisely by meeting the commutability requirement of the generalized covariant derivative
Strong and weak interaction theories also use covariant derivatives with corresponding compensating fields. It is logical to extend to them the principle of the vanishing commutator of covariant derivatives. Then we will get the equations of motion for these theories not on the basis of the principle of least action, but on the basis of the requirement of absolute mass invariance.
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
where
is the coupling constant,
is the component of the gauge field,
are the generators of the gauge group. Then, in accordance with the principle of mass invariance (in some cases, the invariant mass may be zero), the field obeys two equations
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 let us consider arbitrary four-component complex spinors
Among all possible vectors, let us select a set of such vectors for which there is a representation of components through arbitrary complex spinors
and there is another way to calculate them
Let's narrow this space to the Minkowski space, for what further we will assume both spinors to be identical, then the vector constructed from them
has real components, and we will assume that this is the electron momentum vector constructed from the complex momentum spinor
Consider the complex quantity
where we introduce one more complex spinor, which in the future we will give the meaning of the complex coordinate spinor
Coordinate vector of the four-dimensional Minkowski space
is obtained from the coordinate spinor by the same formulas
Thus, the vector in the Minkowski space is not a set of four arbitrary real numbers, but only such that are the specified bilinear combinations of components of completely arbitrary complex spinor
Accordingly, the components of the vector in Minkowski space are interdependent, from this dependence automatically follow the relations of the special theory of relativity between space and time. For the same reason, the coordinates of Minkowski space cannot serve as independent variables in the equations. From the commutative properties of
matrices, which are generators of rotations and boosts with respect to which the length of vectors is invariant, quantum mechanics automatically follows. Indeed, the commutation relations between the components of momenta are related to the noncommutativity of rotations in some way, and from them the commutation relations between the components of coordinates and momenta are directly deduced. And from these relations the differential equations are derived.
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
This quantity does not change for any combination of rotations and boosts
Accordingly, the exponent
characterizes the propagation process of a plane wave in spinor space with phase invariant to Lorentz transformations.
Let us apply the differential operator to the spinor analog of a plane wave
Applying this operator at another definition of the phase gives the same eigenvalue
that is, two different eigenfunctions correspond to this eigenvalue, but in the second case the phase in the exponent is not invariant with respect to the Lorentz transformation, so we will use the first definition.
Since
are complex spinors, which, under the transformation
is affected by the same matrix
, then the complex quantity
is invariant under the action on the momentum spinor
of the transformation
.
is an eigenvalue of the differential operator, and the plane wave is the corresponding
m eigenfunction, which is a solution of the equation
Here
denotes the complex function of complex spinor coordinates.
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
In particular, this is true for the exponential function, which is an infinite power series.
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
then for the square of its length the following equality will be satisfied
That is the square of the modulus
m has the sense of the square of the mass of a free particle, which is described by a plane wave in spinor space as well as by a plane wave in vector space. For the momentum spinor of a fermionic type particle having in the rest frame the following form
is real and not equal to zero, and for the bosonic-type momentum spinor having in the rest frame the following form
i.e., the boson satisfies the plane wave equation in spinor space with zero eigenvalue.
For the momentum spinor of a fermion-type particle we can consider another form in the rest system
then the mass will be real and negative
This particle with negative mass can be treated as an antiparticle, and in the rest frame its energy is equal to its mass modulo, but it is always positive
To describe the behavior of an electron in the presence of an external electromagnetic field, it is common practice to add the electromagnetic potential vector to its momentum vector. We use the same approach at the spinor level and to each component of the momentum spinor of the electron we add the corresponding component of the electromagnetic potential spinor. For simplicity, the electron charge is equal to unity.
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
Let us replace the momenta by differential operators
and find the commutation relation
Let us apply the proposed equation to analyze the wave function of the electron in a centrally symmetric electric field, this model is used to describe the hydrogen-like atom. For the components of the vector potential of a centrally symmetric electric field it is true that
As a result, it is possible to accept
=
We are looking for a solution of the spinor equation; we do not consider the electron's spin yet
This equation can be interpreted in another way. Let us take the invariant expression
And let's do the substitution
We will consider this equation as an equation for determining the eigenvalues of
and the corresponding eigenfunctions
Let's introduce the notations
this quantity does not change under rotations and boosts and is some analog of the interval defined for Minkowski space and
this quantity represents time in four-dimensional vector space.
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.
As a result, we have an equation for determining the eigenvalues of
m and their corresponding eigenfunctions
Instead of looking for solutions to this equation directly, we can first try substituting already known solutions to the Schrödinger equation for the hydrogen-like atom. If
is one of these solutions, we need to find its derivatives over all spinor components
Let’s pay attention to the shift in priorities. In the Schrödinger equation one looks for energy eigenvalues, while here it is proposed to look for mass eigenvalues, it seem more natural to us. The mass of a free particle is an invariant of the Lorentz transformations, and in the bound state the mass of the particle has a discrete series of allowed values, each of which corresponds to an energy eigenvalue, and the eigenfunction of these eigenvalues is the same. But these energy eigenvalues are not the same as the energy eigenvalues of the Schrödinger equation, because the equations are different. When an electron absorbs a photon, their spinors sum up and the mass of the electron changes. If the new mass coincides with some allowed value, the electron enters a new state. The kay idea here is the assumption that the interaction of spinors occurs simply by summing them.
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
where the coefficients
are complex quantities independent of coordinates. In fact, as shown at the end of the paper, the wave function is a linear combination of such right-hand sides with operator coefficients.
We will search for the solution of the wave equation considered in the first part of this paper
Let's express the left part through the components of the momentum spinor
Let's distinguish the direct products of vectors in these matrices
Let's introduce the notations
Let us substitute differential operators instead of spinor components
Then the quantities included in the wave equation
Let us consider the case of a free particle and represent the electron field as a four-component spinor function of four-component spinor coordinates
For a free particle, the components of the momentum spinor commute with each other, so all components of the matrix
are zero.
Let us use the model of a plane wave in spinor space
Substituting the plane wave solution into the differential equation, we obtain the algebraic equation
Let us take into account the commutativity of the momentum components, besides, let us introduce the notations
for the quantities which are invariant under any rotations and boosts, then we obtain
Additionally, introducing notation for Lorentz invariant quantities
We see that in the case of a plane wave in spinor space, the matrix in the left part of the equation is diagonal and remains so at any rotations and boosts, the diagonal element also does not change.
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
Let us compare our equation with the Dirac equation [6, formula (43.16)]
In the rest frame of reference, the three components of momentum are zero and the equation is simplified
That is, in the rest frame the Dirac equation and the spinor equation analyzed by us look identically and contain a diagonal matrix. The corresponding problem on eigenvalues and eigenvectors of these matrices has degenerate eigenvalues, which correspond to the linear space of eigenfunctions. In this space, one can choose an orthogonal basis of linearly independent functions, and this choice is quite arbitrary. For example, in [
8], formula (2.127), solutions in the form of plane waves in the vector space have been proposed for the Dirac equation in the rest frame
and the following spinors are chosen as basis vectors
For transformation to a moving coordinate system in [
8], formula (2.133), the following formula is used
The basis spinors form a complete system, that is, any four-component complex spinor can be represented as their linear combination and this arbitrary spinor will be a solution to the problem on eigenvalues and eigenfunctions in a resting coordinate system. The choice of the given particular basis has disadvantages, because if to find a four-dimensional current vector from any of these basis functions
then this current in the rest frame of reference
has non-zero components, and the square of the length of the current vector is zero. It turns out that a resting electron creates a current, which contradicts physical common sense.
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
An exhaustive list of 16 spinors of this kind, each corresponding to some particle of the fermionic field, is given in the last section of the paper. The proportionality factor is chosen so that in the rest frame the zero component of the current is equal to the charge of, for example, an electron or a positron.
The mass of electron and the phase of the plane spinor wave
do not change at rotations and boosts. The matrix on the left side of the equation does not change either, remaining diagonal with
on the diagonal.
For a fermion, which can be an electron or a positron in the rest frame takes place , so the quantity
which, unlike the mass
M in the Dirac equation, is complex in the general case, is also real for the fermion and can be positive or negative. For simplicity it is possible to consider the mass of the electron as negative and that of the positron as positive.
For the momentum spinor of a boson, such as a photon, it is true that , so its mass is zero
The given constructions are not abstract, but describe the physical reality, since the results of the processes occurring in the spinor space are displayed in the Minkowski vector space. In particular, the momentum vector corresponding to the momentum spinor has the following parameters
the square of the length is equal to the square of the mass of the electron or positron
A spinor wave function
at some point in spinor space can be given a probabilistic interpretation by establishing its correspondence with the vector wave function
(
)
taking its values in the corresponding point of physical space with coordinates
We act within the classical concepts of quantum mechanics, simply to describe the state of a physical system we use spinor coordinate and momentum representations along with vector coordinate and momentum representations. Both types of representations equally have the right to be more substantial and in principle there is no need to express the wave function in one representation through the wave function in the other, both wave functions equally describe the same physical state. Moreover, since vector coordinates and momenta are simply expressed through spinor analogues, we would prioritize the spinor representations as the more fundamental ones.
Let us summarize the relations between quantum-mechanical quantities for the spinor space
The complete orthonormalized system of eigenvectors of the momentum operator
The complete orthonormalized system of eigenvectors of the coordinate operator
The relation between wave function in momentum and coordinate representations and the relation between eigenvectors of the coordinate operator and the momentum operator
The wave function in coordinate or momentum spinor representation has the classical interpretation adopted in quantum mechanics; the square of its modulus represents the probability density of a measurable quantity taking a particular eigenvalue.
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
that is not only to the case of a plane wave, but to any situation in general. This transition is analogous to the transition from the application of the Schrödinger equation to a plane wave in vector space to its application in a general situation. The legitimacy of such transitions should be confirmed by the results of experiments.
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
This approach is pleasant in the Dirac equation, where the mass is fixed and equated to the mass of a free particle, and at the same time results giving good agreement with experiment are obtained.
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
We will apply to the wave function of the electron the operators corresponding to the components of the momentum spinor, putting for simplicity the electron charge equal to unity
Note that the electromagnetic potential vector can be calculated from the electromagnetic potential spinor by the standard formula
The advantage of the spinor description over the vector description is that instead of summing up the components of the momentum and electromagnetic potential vectors as is usually done
now we sum the spinor components and then the resulting vector is
in addition to the usual momentum and field vectors, contains an additional term
taking real values and describing the mutual influence of the fields of the electron and photon.
After the addition of the electromagnetic field the components of the momentum spinor do not commute, the corresponding commutators are found above
Let's find commutators for other operators
Further we will use these and analogous relations
Earlier, by giving absoluteness to the requirement of invariance of the mass squared to the Lorentz transformations, we obtained a system of equations for interacting fields in electrodynamics in the case when these fields exist in vector space. But we can apply this approach to interacting fields in spinor space as well. Let us analyze again the formula
If we deal with the field of a free particle, then
But since we want to make the invariance principle absolute, we require commutability also in the presence of the electromagnetic field
This can be achieved if we take into account the dependence of the momentum spinor components on the coordinates and impose the condition
As in the case of vector space, we can treat in the spirit of Newton's law equations of the form
If an external field is applied, the momentum of the electron field changes, if the momentum of the electron changes for some reason, the electromagnetic potential is perturbed and an electromagnetic field is generated.
Earlier we defined a matrix of commutators
and noted that for a free particle it is equal to the zero-point matrix. We can require that this matrix is zero also in the presence of an arbitrary field. Absolutization of this requirement gives us an additional set of equations, besides the main one
to describe the interaction between the field and the charged particle in the presence of spin. It is guaranteed that the basic equation remains true both for a free particle and for a particle in an external field.
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
Since the second factor
in the left-hand side of the equation has a simpler structure than the first factor, perhaps as a first step we should find the eigenvalues and eigenfunctions of the equation
and use them when solving the equation as a whole.
Let's calculate the expressions included in the equation
It would be interesting in this context to consider for the presented spinor model the case of a centrally symmetric electric field and to find solutions of the spinor wave equation for the hydrogen-like atom, taking into account the presence of spin at the electron. For such a model we can take
As mentioned above, we can substitute into the equation the already known exact solutions of the Dirac equation for the hydrogen-like atom by expressing the components of the coordinate vector and derivatives on them through the components of the coordinate spinor and derivatives on them. It is likely that the solution of the Dirac equation would not make the spinor equation an identity; it would be evidence that more arbitrary assumptions are made in the Dirac equation than in the spinor equation, and that the latter claims to be a better description of nature.
We can also consider the case of a constant magnetic field directed along the z-axis
We see that the scalar potential
grows with time, but does not depend on spatial coordinates, and the vector potential does not depend on time, so that there is no electric field. In this case
The equation considered up to now is rather cumbersome, therefore we would like to have a simpler and compact relativistic invariant equation for the fermion, taking into account the presence of a half-integer spin. Such equation really exists; its derivation is given in Section 4 of the paper. Here we will give its form for the electron in the presence of the electromagnetic field
Let's return to the relations
In this form they are equivalent, but if an external field is added, a difference arises, since in one case the field is added at the vector level and in the other at the spinor level
These relations correspond to differential equations including the relativistic Schrödinger equation
The corresponding inhomogeneous equation is
where the delta function can be represented as
For a free particle the eigenfunctions and eigenvalues solving these equations should coincide, but in the presence of an external field the eigenvalues and the corresponding eigenfunctions will differ because of the above mentioned difference in summation in one case of vector components and in the other case of spinor components.
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
To obtain this result, we did not have to make assumptions about commutability of the spinor components among themselves. Accordingly, a similar expression takes place for the phase of a plane wave in vector space
Further we assume that the components of the momentum spinor commute, which takes place for a free particle, then we obtain
On the other hand, we can write
Thus, the results of calculations coincide.
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
The phase of the wave in this form is closer to the generally accepted phase of a plane wave in vector space. But the phases calculated by two methods do not coincide with each other, although both of them are invariant under Lorentz transformations
Let us slightly modify the expression for the phase of the plane wave
Let’s change the order of derivatives
and write the difference of the two equations
Add an imaginary unit to the phase
Thus, we obtained a differential equation with an eigenvalue independent of coordinates
to which corresponds the eigenfunction
which is a plane wave with imaginary phase and bounded amplitude.
Now we can define the function
which satisfies to equation
thus,
has the properties of the Green's 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
It includes the action of
and the Lagrangian density for the free field
For convenience and clarity, the following notations are introduced
For the general case the Lagrangian density has the form
where V
-polynomial over the field
.
Substituting the Lagrangian density into the Euler equation
the field equation of motion is obtained.
The free field theory is developed for a special kind of polynomial
In summary, Euler's equation defines the equation of motion
The notations used here are
Thus, there is a correspondence of the Lagrangian density and the equation of motion for the free field
Our proposal is to replace the Lagrangian density in vector coordinate space by the Lagrangian density in spinor coordinate space. For this purpose, we use the equation of motion in spinor coordinate space and we want to find the Lagrangian density for which the Euler equation defines this equation of motion
For the sake of clarity, we use the same notation for the spinor coordinate derivative as for the vector coordinate derivative; the context allows us to distinguish between them
Let us write the Lagrangian density plus sources in the form
And let's substitute the Lagrangian density into the Euler equation
For the case of a free field the derivative operators commute, so we can write
It is pleasant that the Euler equation in invariant form works also in this situation, so that we obtain the desired form of the equation of motion in the spinor coordinate space. It is important that the proposed Lagrangian density has a relativistically invariant form, even in the general case, and not only at commuting derivatives. The polynomial has the form
In the case of a free field we restrict ourselves to the first term of the polynomial
Then the Lagrangian density and the equation of motion for the scalar field in spinor coordinate space have the form
For a free field when the derivative operators commute, we obtain
In the spinor equation of motion there is a plus sign before the mass, although in the rest of the paper there was a minus sign. To return to the minus sign it is enough to put a plus sign in front of the polynomial
in the Lagrangian density.
Now we have to find the path integral, which, along with the Lagrangian density, includes the sources
The components of spinors are complex, and we have already noted that the derivatives on complex variables are applied to the degree functions, which, most likely, can describe physical fields, respectively, the finding of an indefinite integral for the function of a complex variable can be treated similarly, i.e. as an indefinite integral from the degree function.
It is possible to recover Planck's constant, which provides a transition to the classical limit
One of the steps in computing the path integral in [
9] is to find the free propagator from equation
the solution of which has the form
In our case, we want to find
After integration by parts by analogy with [
9], Chapter 1.3, we obtain for the special case of a free field
In the process of calculation, it is necessary to find the solution of the equation
For this purpose, we pass to the momentum space by means of the integral transformation
The assumed propagator has the form
which is verified by substitution into Eq. Here it is assumed that the representation of the delta function
We added a conjugate phase to the exponent
which, on the one hand, provides convergence of the integral, and on the other hand, it does not affect the result of differentiation on variables
.
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.
but both parts of the inequality are invariant under Lorentz transformations.
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
in which it is taken into account that the fermion mass is real. Now the propagator has the form
Since we call the field under consideration a scalar field, we expect its value to be invariant to Lorentz transformations. But how to formalize this statement and to what exactly does this transformation apply? We propose to consider that the value of a scalar field is the scalar product of the representatives of a spinor field, which is the most fundamental field in nature, and vectors, tensors and, among others, scalars are formed from the spinors representing it. The scalar product is defined by means of the metric tensor of the spinor space. From any two spinors we can obtain a scalar, in general the complex case. But if we want to obtain a scalar with real values, we must impose some restrictions on the original spinors. For example, to any spinor u we can correspond a scalar
U taking real values, whose value does not change under the action of the Lorentz transformation on the spinor and the action of the same transformation on the conjugate spinor
When a spinor and its conjugate spinor are simultaneously rotated or boosted by some angle, the scalar undergoes a rotation or boost by zero angle.
We can find the derivatives of the scalar by the components of the coordinate spinor
The components of the coordinate spinor are complex quantities, the derivative on them is taken formally, since physical fields can be represented by power functions of the components of the coordinate spinor and its conjugate.
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
Let us consider the linear combination of these matrices with components of the momentum vector as coefficients, substituting the expressions of the vector components through the components of the momentum spinor
Let us represent the matrix
as a sum of direct products of spinors
For a free field the components of the momentum spinor commute, therefore
does not change at rotations and boosts for an arbitrary complex spinor. Moreover, by a direct check it is possible to check that for an arbitrary spinor
For a free field, when all components of the momentum spinor commute, we can write the relativistic equation of motion of the fermionic field
Where the matrix of derivatives
is obtained from the matrix
by substitutions
However, it is generally accepted to write for this field another equation, the Dirac equation, which does not possess the invariance property anymore
And for the more general case, when the momentum components do not commute, we need to write the equation
Further we will consider the equation of motion for a free field
We again want to find the path integral
For this, we need the Lagrangian density, from which, by means of the Euler equation, the equation of motion is derived
It is suggested to use the Lagrangian density
Let us substitute the Lagrangian density into the Euler equation and obtain the equation of motion
Since the Lagrangian density includes, along with the derivatives of
, the derivatives of
, it is logical to use a different definition of Euler's equation
Then for the free field case when the derivative operators commute with each other, we obtain the equation of motion
If the derivative operators do not commute, additional terms will appear in the equation of motion in the form of matrices similar to the
matrix, and these additional terms will not necessarily coincide with
. In this connection it is necessary to consider the Lagrangian density as more fundamental notion than the equation of motion and to derive the equation of motion from the Lagrangian density, i.e. to take as a basis not the derivation of the equation of motion in momentum space, with what we started, but to take as an axiom the form of the Lagrangian density in the form of field derivatives in the relativistically invariant form. Then, if to follow the invariance principle quite strictly, we should start from the product of two matrices, i.e. to use the Lagrangian density
Or, not limited to fermions,
Nevertheless, further we will search for the path integral in the simplest case with the originally proposed Lagrangian density and in addition assume commutativity of all derivative operators
After integration by parts, we presumably obtain
Then it is necessary to find the solution of the equation
For this purpose, we pass to the momentum space by means of the integral transformation
Here we use the equality, which is valid for an arbitrary complex spinor
It is based on the correlation verified earlier in our work
it is also taken into account that we consider fermions whose mass is real.
As a result, the propagator has the form
here we assume the validity of the relation
In the case of a fermion, the mass in integration is a fixed real quantity, and it can be considered negative for the electron and positive for the positron. Theoretically, the mass can be complex or purely imaginary. If we put mass equal to zero, it may be possible to apply this Lagrangian density to describe massless particles. I wonder if there are particles with complex or purely imaginary mass. In the latter case, the square of the mass will still be positive and the particle will satisfy the Klein-Gordon equation. Such particles can interact among themselves, but not with particles whose mass is real.
Let's return to the question about the use of completely relativistically invariant Lagrangian density
Let's find the product of matrices
The assumption that the following equalities hold is used
Further we find the product of matrices
Again we use the equality
and consider that the mass of the fermion is real, i.e.
therefore, the relations are valid
But the main advantage of the obtained matrix is the following
This matrix does not change at rotations and boosts, so it can be stated that the equation of motion, e.g., in the form of
is truly relativistically invariant, respectively we can use the invariant Lagrangian density
to which corresponds the relativistically invariant propagator of the boson having a real mass, which is negative for the electron and positive for the positron
Let us compare the propagator in spinor space with the propagator of the fermion given in [
9], , formula II.2.22 and formula II.5.18
In [
9] this formula is obtained by applying the second quantization procedure or using Grassmann integrals. The results are similar, but the integration here is performed in the vector momentum space. The Dirac equation and the corresponding Lagrangian density are not relativistically invariant. Besides, here the mass is considered always real and positive, but then it is not clear how electron and positron differ from the point of view of this formula.
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
is a Lorentz scalar. But it is not so, since in the spinor space the scalar is formed exclusively by the scalar product of two spinors, where the metric tensor of the spinor space is included
there are no other ways to construct a scalar in the spinor space.
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
we can assume the propagator form in the spinor space without revealing for compactness the expression of the momentum vector components through the momentum spinor components
Among other things, the equation
can be modified to take into account the electromagnetic potential, the electron charge is taken as a unit
and apply, in particular, to analyze the radiation spectrum of a hydrogen-like atom.
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
The difference is, the matrix
(p) remains unchanged under any rotations and boosts applied to the spinor
, while the matrix
(p) changes under any rotations and boosts.
Equally radically different are the corresponding Lagrangian densities and propagators.
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
in the momentum space, for which we apply the integral transformation
Let's substitute the wave function into the equation and obtain
Let us define two sets of four reference spinors
And let's express the matrix through them
Developing the idea of invariance, we pass to the set of reference spinors with wider filling, but continuing to form matrices possessing the invariance property
Let's express through the reference spinors the matrix
Let us decompose the fermion field into plane waves with operator coefficients
Let's impose the anticommutation conditions on the operator coefficients
We consider the rest anticommutators to be equal to zero. Then we can write the expression for the anticommutator of the field
We will consider this relation as a proof of the anti-symmetry of the fermion wave function under the stipulated anticommutation relations.
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
Each summand in brackets represents the operator of the number of particles with a certain reference spinor. The operator's action consists of consecutive application of the annihilation operator and the operator of the creation of a particle. On initial examination, it would appear that the energy associated with zero-point fluctuations in the vacuum has been overlooked. However, an examination of the final expression reveals that the field always possesses a constant energy, regardless of the particles that contribute to it. This constant energy of the field can be interpreted as the energy of zero-point fluctuations of the vacuum.
The following relations were taken into account in the derivation
Let us perform a series of transformations analogous to those presented by Dirac in [
10], Lecture 11
Following Dirac's argument in [
10], the value of
can be treated as either a Hamiltonian or a total energy operator, with
representing the creation operator and
representing the annihilation operator.
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
The chain of reasoning can be organized in a slightly different way as well
Here it is taken into account that
Let us draw an analogy between our approach and the relations given in [
11], Volume 1, Chapter 3, Section 3.3.1. There it is noted that the creation and annihilation operators of the fermionic field must satisfy such commutation relations that the equality expressing translational invariance is satisfied
which in differential form is written as
The coordinates here are the components of the Minkowski vector space. On the basis of these relations the anticommutation relations between the creation and annihilation operators are derived.
In the spinor coordinate space, we can express the translational invariance of the field operator by the relations
It is interesting to find out in what relation these translational operators are - one operator acts in vector space, the other in spinor space. In both cases the following interpretation can be given. Suppose we know the result of an operator acting on an arbitrary state at a point in space 1, and we want to know the result of its action on a state at point 2. Then we translate the state from point 2 to point 1, act on it by the operator, and transfer the obtained result back to point 2.
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
The ratios used in the derivation are
Let us give an interpretation of the operator coefficients for this approach
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
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
It is known [
9], formula II.1.30 that the charge conjugation operation transforms an electron into a positron with a change of the sign of the charge. Let us apply the charge conjugation to the reference spinor
As a result
not only transforms to
, but also changes a sign of mass due to the imaginary unit in the charge conjugation matrix. This confirms our thesis that the charge conjugation synchronously changes signs of charge and mass.
The properties of all particles and operators are summarized in a table
| creates |
destroys |
particle spinor |
vector |
number |
mass |
spin |
wave sign |
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Here the column “vector” shows the vector obtained from the corresponding spinor by the formula of the form
Although we have used the term vector for quantities like
U1, they are not really vectors in the sense that if a Lorentz transformation is applied to a coordinate spinor and hence a coordinate vector, the true vector must undergo the same transformation. For a momentum vector this is the case, but if the sign of one or more components in the momentum vector is changed, it will no longer be transformed according to this law. For example, charge conjugation changes the signs of some components
so the electron current and the positron current cannot be vectors at the same time, and in fact, as can be seen from the table, neither is a vector.
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
If we add sixteen spinors of the field of one particle with sixteen corresponding spinors of another particle, it will look the same as if the momenta of the particles were summed directly. It is clear that the momenta themselves cannot be directly summed, but summing the fields does not look impossible and leads to the same result as adding the momenta directly. The result can be represented as a sum of two other momenta, the mass of each of which is zero
Perhaps, such summation is an adequate model for describing the phenomenon of annihilation of particles with different masses. As an illustrative example, consider the case of an electron and a positron at rest
As a result of addition and separation, two photons with zero mass are obtained, having oppositely directed spatial components of the momentum vector, i.e. flying apart. At this interaction the total energy, the total momentum and the total mass are conserved. We can also say that the total charge is conserved, although in our interpretation the charge is not a numerical characteristic that can be calculated, the sign of the charge is determined by the structure of the spinor. In turn, this structure is determined precisely by the sign of the mass. Thus, a change in the sign of the mass leads ultimately to a change in the sign of the charge.
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 |
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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.
In deriving this result, the following relations are taken into account
The last operation of taking the value
out from under the sign of the integral seems doubtful because of its dependence on the momentum over which the integration is performed. If one closes one's eyes to this, as is generally accepted in the literature, in particular in [
9], this relation is taken to be interpreted as a proof of the anti-symmetry of the fermion wave function under the stipulated anticommutation relations. The only situation where this is unquestionably true is when considering in a rest system where boosts are excluded, energy is equal to mass, and invariant to rotations.
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
The equation of motion has the form
The equation is relativistically invariant, respectively we can use the invariant Lagrangian density
to which corresponds the relativistically invariant fermion propagator
The equation can be modified to take into account the electromagnetic potential, the electron charge is taken as a unit
and apply, in particular, to analyze the emission spectrum of the hydrogen-like atom.
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
Why we have chosen these 4 vectors out of 8 possible combinations of signs of three spatial components? Because they are represented in the previously given table of variants of spinor particles.
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
16 spinors are necessary precisely to ensure completeness. This basis is not orthogonal.
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
The three spatial components of the four Minkowski space vectors presented above are nothing more than a three-dimensional polarization vector, defined as the average of the spin operator over one of the 16 initial states. The polarization vectors form two pairs in which their directions are opposite. If we take two states with opposite polarization vectors, we can treat them as two electrons with opposite spins and use the tensor product to form a singlet state from these electrons. For example, we can use the state
u1, which corresponds to the polarization vector
V1, and
, which corresponds to the opposite vector
U1
This pure state is characterized by an antisymmetric wave function, each of the electrons here is in a mixed state, end they are entangled with each other. Let us define the extended spin operator as four 8
8 matrices
and let us find its average value in the singlet state
As a result, we obtain a four-dimensional polarization vector that has the form
This vector does not change with any rotations of the initial spinors, and during boosts its form remaining the same, but the energy value changes. The polarization vector of the three triplet states of this spin pair has exactly the same properties with the same energy value
Spinor
forms a set of four states with the same properties with only one spinor from the remaining 14, which is the spinor
corresponding to the polarization vector
In general, for any of the 16 spinors, there are two other spinors with which such singlet and triplet states are formed. These states are similar to Bell states, but Bell states only include zeros and ones, and the states themselves are an idealized representation of the properties of electrons, the result of a thought experiment. In our case, the states are associated with electron momentum and describe a pair of real relativistic particles under the action of arbitrary Lorentz transformations. The hypothesis that the model of combining two electrons using the tensor product of states adequately describes nature is confirmed at least by the fact that these four states are found in this form in the helium atom.
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
All components of these polarization vectors change during rotations and boosts, but the structure of the vectors and their coincidence in the singlet and one of the triplet states remain unchanged.
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
This interpretation leads us to believe that physical fields and their corresponding particles are described at a basic level by the 16 spinor states presented. The momentum vectors available to our perception are the result of a kind of measurement we make of the average values of various operators, in particular, the spin operator. The picture of the world that we see is the result of a quantum measurement procedure applied to the state space. Which is more productive: describing nature using initial quantum states, or settling for the average values of a certain operators? At the same time, one may ask whether the real electron is a pure state in the form of a superposition of 16 basis states, or whether it is a mixed state in the form of a weighted sum of 16 projectors onto these states.
The representation as vector composed of momentum components
is suitable for description both articles with zero mass as photons, and particles with non-zero mass, such electron. In this form, they participate in interactions in vector space. For electrons, it is interesting to recall Newton’s law for rotations, which was derived earlier
The presence of a curl of the momentum vector in a magnetic field implies the presence of a curl of each of the four vector representations of the particle, but the difference in the distribution of signs in them leads to these representations rotating differently in magnetic field, which may explain the behavior of electrons with different spins.
For the selected vectors the following relations are valid
Let us decompose the field into plane waves with operator coefficients and let's find the commutation relations for them. We will use the next notation for the scalar product of vectors
Let us apply the following commutation relations
Here it is taken into account that
We will consider this relation as a proof of the symmetry of the wave function under the stipulated commutation relations.
Let us find the commutation relations for the wave function and its time derivative, which in this case play the role of canonical momentum
The commutation relations remain the same
As one would expect, the field has only two degrees of freedom. This relation is valid for any reference frame, but the values of the momentum components in each of them are different.
Let us calculate the square of the field energy
If we consider the photon field, the mass is zero, so that only the energy of the field remains in the formula. Each summand in brackets under the integral represents the operator of number of particles with a certain reference vector, its action consists in the consecutive application of the annihilation operator and the particle creation operator. The last summand describes the energy of zero-point fluctuations of vacuum. When there is no particle, we have the equality
In this connection it is logical to use the normalization for the wave operator
If the mass is not zero, then we can relate
and
to the current of electrons with different spins and, respectively, relate
and
to the current of positrons with different spins.
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
For example,
is an operator of annihilation of a particle with pseudovector
, similarly, other operators without asterisks annihilate particles with pseudovector which stands in expansion with these operators. Accordingly, the operator
is the operator of the number of particles with pseudovector
.
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
It follows from these relations that
Let us construct the amplitude of the field component, which is born at the point with coordinates
and annihilated at the point with coordinates
For the reasons given above, let us apply the normalization of the field operator
If the mass is zero, this expression is the matrix element between states of the photon.
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 find the commutation relations between the components of these operators
As we see, the commutation relations are satisfied for the creation and annihilation operators.
Let us define the total particle number operator in the form
Let's find the commutator
Let's define the vacuum state using the relations
Let's act on vacuum by the creation operator and for the obtained state we find eigenvalues of the particle number operator
If we apply normalization
then the eigenvalues will have the form
Note that in the case of the photon field, the matrix, taking into account the normalization, contains elements whose modulus is less than or equal to ½, since at zero mass
.
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
For illustration let us consider a one-particle state
and act on it with the creation operator. Again, let's take into account
The eigenvalue of the particle number operator has increased, instead of a one-particle state we have a two-particle state
Further application of the creation operator increases the number of particles to any value. Now let us find a commutator for the annihilation operator, without taking into account the normalization for the moment
The ratios have been taken into account here
Let's act on the two-particle state by the annihilation operator and for the obtained state we find the eigenvalues of the particle number operator
Here, the fact that without taking into account the rationing
Thus, the annihilation operator reduces the number of particles and puts the field into a single-particle state.
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
then the matrix element really acquires the sense of the amplitude of the probability that the particle is born at the origin and annihilated at the point with coordinates
.
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
The assumption used here is that the scalar products
are the same for any values of momentum. Taking into account the normalization
At non-zero number of particles we can change the order of operators and first apply the annihilation operator
After normalization we obtain
Let’s return to the previously used definition of the vacuum state by means of relations
As we have seen, the action of the creation operator transforms the zero-particle state into a one-particle state
At that, none of the operators of the number of particles with a particular value of momentum
has no definite meaning, since the particle is only one. In this connection it makes sense not to define the vacuum in such a detailed way, it is enough to define that the vacuum state is characterized by only one condition
At this approach the field energy is not equal to the sum of energies of partial oscillations, accordingly the question about the energy of zero-point oscillations of each oscillator constituting the field is removed. We get rid of the problem of infinite energy of the sum of zero-point vibrations of an infinite number of oscillators.
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
The solution of this equation has the form
Therefore, we complement the denominator of the integrand, for what we relate the creation and annihilation operators
After such normalization, one doubts the expediency of the normalization introduced earlier, namely, the inclusion of the multiplier in the formula
By analogy with the introduced creation and annihilation operators for fields in vector space, let us describe the corresponding operators for fields in spinor space. As an initial one we use the previously described field operator for the fermionic field
Let us define the creation and annihilation operators
Let's find anticommutation relations between components of these operators
Besides these relations, the following `anti-commutation relations take place between the components of the annihilation and creation operators
Let's define operators of the total number of particles in the form
Let's find the commutators
Instead of defining the vacuum state through its properties under the action of annihilation operators
which would entail the ratios
we will not require from operators all these properties, but we will be limited by a weaker and simpler definition of vacuum, namely, absence of particles in vacuum
Let's use the found commutator
On the obtained one-particle state let's act on the obtained one-particle state by the creation operator again
We have obtained a state with two particles and we can thus increase the number of particles to infinity. All particles are identical and indistinguishable from each other, each of them is in all allowed states, of which the free field has infinitely many. Electrons in an atom have fewer allowed states, but still any electron occupies all of them equally with the others. This theory describes both electron and positron, the difference between them being only in the sign of the mass, it being convenient to consider that the electron has a negative mass and the positron a positive one.
Similarly, we use the commutator of the annihilation operator
Thus, the action of the annihilation operator has transformed the two-particle state into a one-particle state. Using the same calculations, we obtain the result of the annihilation operator action on the one-particle state
And in the same way we define the result of its action on the null state
We obtain a state with the number of particles minus one, but we see that in fact it is a state with one particle whose mass is negative. Thus, the positron annihilation operator is also the electron creation operator. It destroys positrons until they run out, after which it starts creating electrons. The creation operator, on the contrary, destroys electrons, and when they run out, starts creating positrons. Thus, since there are many electrons in our universe, this operator cannot be creating positrons because it cannot destroy all electrons due to their number. Moreover, the operator of annihilation of positrons because of the absence of the latter, only creates more and more electrons.
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 would like the spinor matrix element to have properties of the Green's function, i.e. to satisfy the equations which for this case are given below and which can be combined into one equation
where the delta function can be represented as
The solution of the combined equation has the form
Therefore, we complement the denominator of the integrand, for what we relate the creation and annihilation operators
The electron and positron have different mass sign, so their matrix elements will be different.
We can repeat the above calculations, keeping the annihilation operator, but defining the creation operator differently
As a result, we obtain the anticommutator
As before, using the creation and annihilation operators, we construct the matrix element for a state with an arbitrary number of particles
Now instead of mass the matrix element includes energy, therefore such theory is applicable also to the field with zero mass, i.e. it can serve as a model not only for the electron, but also for the electromagnetic field in spinor space. The only problem is that if earlier the action of the annihilation operator on the zero-point state gave a particle with negative mass, now this action gives a particle with negative energy, which makes the interpretation of such theory more difficult.
Note that in this revision the creation and annihilation operators are conjugate to each other
The considered free field matrix elements describe the situation when there is a point source with coordinate
and a point sink with coordinate
. In the general case in the spinor space the distribution of source-stocks
J(x) can be given and the value of
which is used for finding the integral over the trajectories and which can be written using the Fourier transform for the spinor space
In quantum field theory it is customary to calculate a similar quantity
in which the coordinates, momenta and the Fourier transform connecting them belong to the vector space. In our opinion, the transition to spinor space, more fundamental than vector space, which is a superstructure over spinor space, can eliminate divergences in calculating integrals in the framework of the formalism of the integral over trajectories. In momentum space the similarity is even more obvious, the kernels of the integrals are the same, the only difference is in the space where the integration takes place and the way of calculating the Fourier transform - either in vector or in spinor space
The spinor space has the additional advantage that the integrand is factorized
This factorization in momentum space looks like a consequence of a more fundamental property of factorization in coordinate space
We can assume that first it makes sense to perform integration separately on
x and
y, and only then to perform multiplication
Since earlier we have obtained an explicit representation of field operators in both vector and spinor space, we do not need to refer to the equation of motion and the Lagrangian density. Proceeding from these representations, we define the creation and annihilation operators, and from them we construct the matrix element as a function of relative coordinates.
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
Their solutions have the form
The first variant is remarkable because in it the denominator of the integrand coincides with the similar denominator of the photon propagator and the Feynman propagator for the fermion. The advantage of the second variant is that the denominator consists of two conjugate terms, which can be included from the beginning in the mutually conjugate creation and annihilation operators.
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
create or annihilate the fermion at a precisely defined point of space, while the momentum of the fermion is not defined. Taking into account the symmetry between the coordinate and momentum operators in quantum mechanics, we can postulate similar operators for a fermion with exactly definite momentum and indefinite coordinate.
where
is the invariant interval of the spinor coordinate space. Now the difference between an electron and a positron is determined by the sign of the interval.
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
which after the substitutions
transforms into a differential operator
Thus, the equation of motion has the form
After transition to the momentum space by means of integral transformation
we get the equation of motion
The Green's function is a solution of the inhomogeneous equation
Let us note that the diagonality of the left part of the free field equation
allows to take into account in a simple way the influence of the external electromagnetic potential expressed in the spinor form
Due to the diagonality of the equation, the addition of an external field does not break its invariance, unlike in the vector space. It turns out that at the spinor level the fields interact additively. There arises a strange assumption that since the matrix is diagonal, there is no need to zero out the non-diagonal elements to ensure mass invariance, hence Newton's laws are not relevant in spinor space, and the components of the momentum spinor always commute.
Replacing the momentum components by derivatives gives us an analogue of the Schrödinger equation
If we get rid of the unit matrix, we obtain the same equation for spinor space that we considered at the beginning of the paper. It turned out to be exact, not requiring an addition to take into account the electron's spin, as it was necessary for the Schrödinger equation in vector space. In the spinor equation the spin of the electron is taken into account automatically.
We can go further and assume that the picture of interaction of fields is more symmetric and is described by the mixed equation
In the extended momentum space
the equation has the form
From the equation we can isolate the part responsible for the interaction
The solution of this homogeneous equation is a plane wave
The theory outlined in the article allows us to answer the question how the fermion field changes under the action of Lorentz transformations on the coordinates. Exactly, if we move to another frame of reference by rotations and boosts, the coordinate spinor changes. As a consequence, the momentum spinor changes, the components of which are the coefficients of the expansion on the new coordinates, and the momentum spinor undergoes exactly the same transformation as the coordinates, so that the phases of all plane waves in spinor space do not change. The components of the new momentum spinor are substituted into the 16 spinors describing the fermion field. Thus, there is no any uniform law of transformation of a spinor of the fermionic field, each of 16 spinors corresponding to the particles forming it, is transformed in its own way.
However, if, following Heisenberg [
12], Chapter 3, Paragraph 1, we index the field components differently
Then it can appear that this field
on the first index will be transformed by three spatial rotations and three boosts, and on the second index it will be transformed by three rotations in isotopic space. In this case the additional quantum number related to the sign of mass may be an isotopic spin.
Let us suggest that the coordinate and momentum spinor spaces can also be indexed in a similar way
Thus, we are in a space
that is subject to three rotations, three boosts, and three isotopic rotations. All of these transformations are equally real, but there is an imbalance due to the luck of isotopic boosts. After all, isotopic rotations, like spatial rotations, are generated by Pauli matrices; these rotations also do not form a group. Therefore, the full isotopic group must also consist of three rotations and three boosts.
Let's rewrite the previously used quantities with new variables
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Summarizing, we can formulate the following theses. The initial coordinate space is described by complex quantities, which can be represented as a square matrix
The field is a superposition of plane waves with complex phase
where the momentum coefficients of the decomposition are represented as
The phase of a plane wave is constructed using two metric tensors of spinor space and therefore does not change if
and
are affected by the same transformation, which is a combination of three rotations and three boosts with arbitrary angles at the first index and a combination of three rotations and three boosts with arbitrary angles at the second index. Any transformation is given by 12 real values representing the angles of the rotations and boosts. When we considered a four-component spinor, we made do with 6 angles, since we took the same rotation and boost angles for both indexes. Note also that only under this condition the mass invariance takes place.
Each plane wave in superposition has a multiplier in the form of a matrix
which may be any matrix of 16 momentum combinations given in the table, e.g.
Each of these matrices can be compared to some elementary particle, and at transformation of coordinate and momentum space it is transformed according to some inherent law. The field operator has the form
In addition, a complex conjugate version of the phase should be added to both exponents, as was done above, then there would be an imaginary value in the exponent.
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
For the simpler case of a scalar field these plane waves correspond to the Green's function
Recall that the transition from spinor space to vector space is performed by transformations
Lorentz transformations are given by 2×2 matrices with a set of valid rotation angles and boosts
After acting on both spinors of the Lorentz transformation with 12 arbitrary angles
and corresponding transformations in the vector space
there is still equality of masses
and invariance of the plane wave phase in spinor space
However, at arbitrary 12 angles, the mass is not invariant
and the phase of a plane wave in vector space also changes at Lorentz transformations
And only under the condition of equality of 6 corresponding angles in the transformation matrices, i.e. under equality
both these invariance properties are restored.
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
we define the following six matrices in the form of commutators
Using these matrices, we define the coordinate and momentum tensors
By this scheme it is possible to generate a tensor of higher rank also
but we will restrict ourselves to rank 2. The tensor is symmetric and has zero trace
here we have indicated, for example
We can rotate the initial spinor, and the vector and tensor constructed from it are rotated together with it. By direct calculations we can check that when we rotate the spinor around any of the axes by 720 degrees, it turns into itself, the vector turns into itself when we rotate by 360 degrees, and 180 degrees is enough to translate our tensor into itself, i.e. the tensor defined by us has spin two. Let us note some properties of the tensor. At change of a sign of mass of spinor
to the opposite one, all components of the tensor change. In the rest system at zero boosts
For fixed non-zero boosts, these sums are not equal to the integer square of the mass, but they are invariant to rotations.
With the 16 momentum spinors at our disposal we can define 256 tensors of the form, e.g.
If the spinor mass is not zero, each of these tensors is equal to one of the eight pseudotensors
If the mass of the momentum spinor is zero, the degree of symmetry of the field increases and the tensor obtained from the spinor has the form
and 256 pseudotensors reduce to only two possible polarizations in the form of two pairs of tensors with opposite overall signs
For completeness of the theory we could define four types of tensors
from which it is possible to construct a combination having a diagonal form
However, by our scheme we can construct only the tensors
and
from 16 spinors. Therefore, we can propose the following expressions for the gravitational field operator and for its propagator
We shall give a comparative description of second quantization of vector and tensor fields. The theory is based on the coordinate and momentum spinors
From them vectors and tensors are formed
We can propose a unified approach to the description of several kinds of fields. Let us write out their operators, relating them to the spinor coordinate space
Here we have added a scalar field, where, for instance
Now we can apply Feynman diagrams to describe the interaction of fields and perform all integration operations in spinor space. For each field we can write down the creation and annihilation operators and the propagator for the inner line. Then the product of several creation or annihilation operators with the corresponding interaction constant is compared to the interaction event at some point of the coordinate space. It is assumed that this event can occur at any point of the space and therefore the integral over the spinor coordinate space is calculated, which gives as a result a delta function from the sum of momentum spinors. This delta function expresses the law of conservation of momentum. The inner lines of the diagram are matched to the propagators. The remaining integrals are calculated over the whole momentum spinor space.
We need to take a closer look at Maxwell's inhomogeneous equations. Let's repeat the relations that we will analyze
On the one hand, the requirement of invariance of mass leads us to the equation
On the other hand, we have the universally recognized Maxwell's inhomogeneous equations
In the results, we are forced to recognize the validity of the relationship
On the left side is the vector obtained as a result of the convolution of the tensor consisting of the derivatives of the momentum vector. This tensor contains information about the internal rotations of the particle, for example, the spin of an electron. After convolution, all this information is explicitly absent. In addition, when convolving electromagnetic tensor to obtain the generally accepted four independent equations, the assumption is made that the derivatives with respect to different coordinate components commute, and, in addition, Lorentz gauge is applied. As a result, generality is violated and a too special type of equations is obtained. On the right-hand side is the current vector, which can be interpreted as the momentum vector multiplied by the charge. This view of current arose historically, since current was considered as the movement of charges. Quantities with the dimension of charge appear twice here, since current density also has this dimension. This is logical, since the left side is proportional to force, and it depends both on the charge generating the potential and on the charge of the particle on which the force acts. There is a clear analogy with Coulomb's law, which includes the product of interacting charges.
It can be argued that the expression
is a theoretically sound formulation of inhomogeneous Maxwell's equations, which follows from the requirement of mass invariance, while the expression
is an empirically established relationship, where the right-hand side contains a surrogate vector that historically arose from practical considerations.
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
This current tensor is antisymmetric and has zero trace
here we have again denoted, for example,
Direct calculations show that when the original spinor is rotated 360 degrees around any of the axes, the current tensor created from it becomes identical to itself, i.e., its spin is equal to one.
Using the 16 spinor at our disposal, we can define 16 tensors of the form, for example
If the spinor mass is not zero, then each of these tensors is equal to one of four pairs of pseudotensors; in a pair, they differ in their overall sign.
If the mass of the spinor is zero, then the tensor obtained from the spinor has the form
and 16 pseudotensors are reduced to possible polarizations in the form of two pairs of tensors with opposite overall signs
The structure of the current tensor at nonzero mass coincides with the structure of the electromagnetic tensor (antisymmetry, zero trace), which suggests writing the equation
where by symbol
we denote some combination of 16 current tensors of the type
. This manifold allows us to take into account the properties of the current, in particular the fact that the electrons that form it can have different spins. Like Maxwell's equations, this equation establishes a connection between a massless field and its sources in the form of massive particles, exhaustively characterizing their mutual influence. This equation does not contain a convolution, which leads to a loss of information in the inhomogeneous Maxwell’s equations. However, there is nothing to prevent us from taking derivatives and convolving both sides of this equation.
We can also write the relation for momentum
This relation has a different meaning and is rather not an equation, but an identity that connects the external manifestation of current in Minkowski space with its internal structure at the spinor level.
Now, we can propose the following definition of the Lagrangian density for electrodynamics
The quantity
is invariant under rotations but not under boosts and it is equal to zero in the rest frame. These properties are natural as boosts affect energy. This is also true for all 16 tensors of type
, which should be actually be substituted into the Lagrangian density. 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.
