2.1. Scalar Electrodynamics
This work heavily uses the results of [
2,
6], so let us summarize some of them here. In [
6] (see also [
20] and references therein), Dirac considers the following conditions of stationary action for the free electromagnetic field Lagrangian subject to the constraint
:
where
is the potential of the electromagnetic field, and
is a Lagrange multiplier. The constraint represents a nonlinear gauge condition. One can assume that the conserved current in the right-hand side of Equation (
1) is created by particles of mass
m, charge
e, and momentum (not generalized momentum!)
, where
is a constant. If these particles move in accordance with the Lorentz equations
where
is the electromagnetic field, and
is the proper time of the particle (
), then
Due to the constraint,
, so
Therefore, Equations (
2)–(
4) are consistent if
, and then
implies
(so far the discussion is limited to the case
).
Thus, Equation (
1) with the gauge condition
describes both the independent dynamics of the electromagnetic field and the consistent motion of charged particles in accordance with the Lorentz equations. The words “independent dynamics” mean the following: if values of the spatial components
of the potential (
) and their first derivatives with respect to
,
, are known in the entire space at some moment in time (
), then
,
may be eliminated using Equation (
5),
may be eliminated using Equation (
1) for
(the equation does not contain second derivatives with respect to
for
), and the second derivatives with respect to
,
, may be determined from Equation (
1) for
.
In his comment on Dirac’s work, Schrödinger [
2] considered the interacting scalar charged field
and electromagnetic field
with the Klein–Gordon–Maxwell equations of motion (scalar electrodynamics):
where
is some conserved external current (added by the present author for generality). The metric signature is
.
The complex charged matter field
in scalar electrodynamics (Equations (
6)–(
8)) can be made real by a gauge transform (at least locally), and the equations of motion in the relevant gauge (unitary gauge) for the transformed four-potential of electromagnetic field
and real matter field
are as follows [
2]:
Schrödinger emphasized two circumstances. Firstly, except for the missing constraint, the equations for the electromagnetic potentials coincide with Equation (
1) (if we replace
with
and
with
). Secondly, the fact that the scalar field can be made real by a change in gauge, although easy to understand, contradicts the widespread belief about charged fields requiring complex representation. This consideration may be regarded as a part of the rationale for the present work, where real fields are used to describe charged particles.
The following unexpected result was proven in [
4]: the equations obtained from Equations (
9)–(
11) after natural elimination of the matter field form a closed system of partial differential equations and thus describe independent dynamics of the electromagnetic field. The detailed wording is as follows: if components of the four-potential of the electromagnetic field and their first derivatives with respect to time are known in the entire space at some time point, the values of their second derivatives with respect to time can be calculated for the same time point, so the Cauchy problem can be posed, and integration yields the four-potential in the entire space-time.
To eliminate the matter field
from Equations (
9)–(
11), let us use a substitution
first. For example, as
we obtain
Multiplying Equation (
9) by
, we obtain the following equations in terms of
instead of Equations (
9)–(
11):
To prove that these equations describe the independent evolution of the electromagnetic field
, it is sufficient to prove that if components
of the potential and their first derivatives with respect to
(
) are known in the entire space at some time point
(that means that all spatial derivatives of these values are also known in the entire space at that time point), Equations (
14) and (
15) yield the values of their second derivatives,
, for the same value of
. Indeed,
can be eliminated using Equation (
15) for
, as this equation does not contain
for this value of
:
(Greek indices in the Einstein sum convention run from 0 to 3, and Latin indices run from 1 to 3). Then
(
) can be determined by the substitution of Equation (
16) into Equation (
15) for
:
Thus, to complete the proof, we only need to find
. Conservation of current implies
This equation determines
, as spatial derivatives of
can be found from Equation (
16). Differentiation of this equation with respect to
yields
After substitution of
from Equation (
16),
from Equation (
18), and
from Equation (
14) into Equation (
19), the latter equation determines
as a function of
,
and their spatial derivatives (again, spatial derivatives of
and
can be found from the expressions for
and
as functions of
and
). Thus, if
and
are known in the entire space at a certain value of
, then
can be calculated for the same
, so integration yields
in the entire space-time. Therefore, we have the independent dynamics of the electromagnetic field.
2.2. Dirac Equation as an Equation for One Function
It turns out, however, that Schrödinger’s results also hold in the case of spinor electrodynamics. In general, four complex components of the Dirac spinor function cannot be made real by a single gauge transform, but three complex components out of four can be eliminated from the Dirac equation in a general case, yielding a fourth-order partial differential equation for the remaining component, and the latter can be made real by a gauge transform [
8]. The resulting two equations for one real component can be replaced by one equation plus the current conservation equation, and the latter can be derived from the Maxwell equations.
Spinor electrodynamics is a more realistic theory than scalar electrodynamics, so it seems important that the charged field of spinor electrodynamics can also be described by one real function.
Let us consider the Dirac equation:
where, e.g.,
(the Feynman slash notation). For the sake of simplicity, a system of units is used where
, and the electric charge
e is included in
(
). In the chiral representation of
-matrices [
21]
where index
i runs from 1 to 3, and
are the Pauli matrices. If
has components
the Dirac Equation (
20) can be written in components as follows:
Obviously, Equations (
25) and (
26) can be used to express components
via
and eliminate them from Equations (
23) and (
24) (cf. [
22,
23]). The resulting equations for
and
are as follows:
Equations (
27) and (
28) can be rewritten as follows:
where
electric field
and magnetic field
are defined by the standard formulae
and the modified d’Alembertian
is defined as follows:
As Equation (
29) contains
, but not its derivatives, it can be used to express
via
, eliminate it from Equation (
30) and obtain an equation of the fourth order for
:
It should be noted that the coefficient at
in Equation (
27) is gauge-invariant (it can be expressed via electromagnetic fields). While this elimination could not be performed for zero electromagnetic field, this does not look like a serious limitation, as in reality there always exist electromagnetic fields in the presence of charged fields, although they may be very small. It is not clear how free field being a special case is related to the divergences in quantum electrodynamics. It should also be noted that the above procedure could be applied to any component of the spinor function, not just to
. The above results are presented in a more symmetric form in the next subsection.
Using a gauge transform, it is possible to make
real (at least locally). Then the real and the imaginary parts of Equation (
28) after the substitution of the expression for
will present two equations for
. However, it is possible to construct just one equation for
in such a way that the system containing this equation and the current conservation equation will be equivalent to Equation (
28).
2.3. Algebraic Elimination of Components from the Dirac Equation in a General Form
To find a manifestly covariant form of Equation (
34) [
9,
10], let us start again with the Dirac Equation (
20). However, this time we do not choose a specific representation of the
-matrices, but only assume that the set of
-matrices satisfies the standard hermiticity conditions [
21]:
where
Then a charge conjugation matrix
C can be chosen in such a way [
24,
25] that
where the superscript
T denotes transposition, and
I is the unit matrix. Multiplying both sides of Equation (
20) by
from the left and using Equation (
32) and notation
we obtain (see details in [
9,
10]):
(A similar equation can be found in the original article by Dirac [
26]. Feynman and Gell-Mann [
22] used a similar equation to eliminate two out of four components of the Dirac spinor function. See also an earlier article [
23]). We obtain:
where the modified d’Alembertian
is defined by Equation (
33) and
Note that
and
F are manifestly relativistically covariant.
Let us choose a component of the Dirac spinor
in the form
, where
is a constant spinor (so it does not depend on the space-time coordinates
, and
), and multiply both sides of Equation (
41) by
from the left:
To derive an equation for only one component
, we need to express
via
. To simplify this task, we demand that
is an eigenvector of
(in other words,
is either right-handed or left-handed). This condition is Lorentz-invariant [
9,
10].
Eigenvalues of
equal either
or
, so
. The linear subspace of eigenvectors of
with the same eigenvalue as
is two-dimensional, so we can choose another constant spinor
that is an eigenvector of
with the same eigenvalue as
in such a way that
and
are linearly independent. This choice is Lorentz-covariant [
9,
10].
Eventually, as proven in [
9,
10], we obtain:
This equation looks more complex than Equation (
34), but it is much more general and manifestly relativistically covariant. It can be slightly simplified if we require that
and
are normalized in such a way that
(this condition also implies linear independence of
and
). Further simplification can be achieved using the following notation for components of the electromagnetic field:
where
and
are some Dirac spinors. Then we obtain the following instead of Equation (
44):
A different choice of
yields an equivalent equation [
9,
10].
This equation for one component
is generally equivalent to the Dirac equation (if
): on the one hand, it was derived from the Dirac equation, and on the other hand, the Dirac spinor
can be restored if its component
is known [
9,
10].
2.4. An Approach to Elimination of the Spinor Field from the Equations of Spinor Electrodynamics
The following suggests that differential algebraic elimination of the spinor field from the equations of spinor electrodynamics is possible [
7]. The equations of spinor electrodynamics are as follows:
(see notation at the beginning of
Section 2.2). The chiral representation of
-matrices (Equation (
21)) is used.
Now, to fix the gauge, let us require that
, the first component of the spinor field
is real:
. As
generally does not vanish identically, this does fix the gauge (in a general case).
Using the elimination process of
Section 2.2, we obtain Equation (
34), which is an equation of the fourth order for
.
Note that this complex equation contains two real equations for the real component
. In particular, these two equations imply current conservation. The current conservation can be written in the following form [
8]:
where
is the left-hand side of Equation (
34).
Obviously, Equation (
34) also implies the following equation:
One can check that this equation is a PDE of the third order with respect to
.
In a general case,
does not vanish identically, so the system containing Equations (
51) and (
52) is equivalent to Equation (
34). On the other hand, current conservation (
51) follows also from the Maxwell Equation (
49). Thus, if we eliminate all spinor components except
from Equation (
49) and add Equation (
52), we will obtain a system (A) of five real PDEs of the third order with respect to the real indeterminate
, and this system of equations is generally equivalent to the equations of spinor electrodynamics (
47), (
48), (
49) under the gauge condition.
It is well-known that, if there are
d independent variables, the number of all different derivatives of order
r, where
, equals the number of combinations
In our case,
(the number of dimensions of space-time). If we apply all different derivatives of order
r, where
, to the equations of system (A), we will obtain a system (B) containing
equations. As the equations of system (A) are of the third order with respect to
, system (B) will contain, in the worst case, all different derivatives of
of order
r, where
. Therefore, system (B) will contain no more than
such derivatives, which we can regard as indeterminates of a system of polynomial equations. Therefore, there are more equations than indeterminates in system (B), so, hopefully, it is possible to use the system to express
via components of the four-potential of electromagnetic field and their derivatives and thus eliminate the spinor field from spinor electrodynamics. The above estimates may be naive, and more rigorous estimates, such as those of [
27] may be required.
2.6. Spinor Electrodynamics
While there are reasons to believe that differential algebraic elimination of the spinor field is possible (see
Section 2.4), so far it has only been achieved at the expense of the introduction of the complex four-potential of an electromagnetic field [
5]. We are again using the equations of spinor electrodynamics
and the notation at the beginning of
Section 2.2.
Let us apply the following “generalized gauge transform”:
where the new four-potential
is complex (cf. [
29]) but produces the same electromagnetic fields as
),
,
,
, and
,
are real. The imaginary part of the complex four-potential is a gradient of a certain function, so alternatively we can use this function instead of the imaginary components of the four-potential.
After the transform, the equations of spinor electrodynamics can be rewritten as follows:
If
and
have components
let us fix the “gauge transform” of Equations (
68) and (
69) somewhat arbitrarily by the following condition:
We then follow the approach and notation of
Section 2.2. The Dirac Equation (
70) can be written in components as follows:
Equations (
77) and (
76) can be used to express components
via
and eliminate them from Equations (
74) and (
75). The resulting equations for
and
are as follows:
Equation (
78) can be used to express
via
:
where the modified d’Alembertian
is defined as follows:
Equation (
79) can be rewritten as follows:
The substitution of
from Equation (
80) into Equation (
79) yields an equation of the fourth order for
:
Application of the gauge condition of Equation (
73) to Equations (
80)–(
83) yields the following equations:
Obviously, Equations (
73), (
76), (
77) and (
85) can be used to eliminate spinor
from the equations of spinor electrodynamics (
70) and (
71). It is then possible to eliminate
from the resulting equations. Furthermore, it turns out that the equations describe independent dynamics of the (complex four-potential of) electromagnetic field
. More precisely, if components
and their temporal derivatives (derivatives with respect to
) up to the second order
and
are known at some point in time in the entire 3D space
= const, the equations determine the temporal derivatives of the third order
, so the Cauchy problem can be posed, and the equations can be integrated (at least locally). This statement is proven in [
5].
Thus, the matter field can be eliminated from equations of scalar electrodynamics and spinor electrodynamics, and the resulting equations describe the independent evolution of the electromagnetic field (see precise wording above). It should be noted that these mathematical results allow different interpretations. For example, in the de Broglie—Bohm interpretation, the electromagnetic field, rather than the wave function, can play the role of the guiding field [
4]. Alternatively, one can also use the above results to remove the matter field altogether, as if it were just a ghost field, and leave just the electromagnetic field in an interpretation. Another interpretation of real charged fields is described in
Section 2.8.
2.7. Algebraic Elimination of Spinor Components from the Dirac Equation in the Yang–Mills Field
Surprisingly, the procedure of
Section 2.3 can also be successfully applied in the case of the Dirac equation in the Yang–Mills field [
13]. Let us start with the equation ( [
30], p. 493) in the following form (the notation is similar to that of
Section 2.3):
where
,
are generators of the SU(
n) group. Components
of the field
have an SU(
n) fundamental representation index (“color” index)
i and a spinor index
.
Multiplying both sides of Equation (
88) by
from the left and using notation
we obtain (cf. [
30], p. 173):
We used the fact that the contraction of a symmetric and an antisymmetric tensor vanishes.
We obtain:
where the modified d’Alembertian
is defined as follows:
and
Let us note that
and
F are manifestly relativistically covariant.
We assume that the set of
-matrices satisfies the same hermiticity conditions and the charge-conjugation matrix
C is chosen in the same way as in
Section 2.3.
We will consider “spinor components” of the field
having the form
, where
is a Dirac spinor. Such “spinor components” have “color” components
. Let us further assume that
is a constant spinor (so it does not depend on the space-time coordinates
, and
) and multiply both sides of Equation (
91) by
from the left:
To derive an equation for only one “spinor component”
, we need to express
via
. To simplify this task, we demand that
is an eigenvector of
; in other words,
is either right-handed or left-handed (the derivation can also be performed for such constant spinors multiplied by a function of space-time coordinates). This condition is Lorentz-invariant. Indeed, Dirac spinors
transform under a Lorentz transformation as follows:
where matrix
is non-singular and commutes with
if the Lorentz transformation is proper and anticommutes otherwise [
28]. Therefore, if
is an eigenvector of
, then
is also an eigenvector of
, although not necessarily with the same eigenvalue.
Eigenvalues of
equal either
or
, so
. The linear subspace of eigenvectors of
with the same eigenvalue as
is two-dimensional, so we can choose another constant spinor
that is an eigenvector of
with the same eigenvalue as
in such a way that
and
are linearly independent. This choice is Lorentz-covariant, as matrix
in Equation (
95) is non-singular.
Obviously, we can derive an equation similar to (
94) for
:
We can express
as
, where
are the generators of the SU(
n) group, then
, where
. If
, then
is a left eigenvector of
with an eigenvalue
, as
The same is true for spinors
(the proof is identical to that in (
97)),
, and
, as
commutes with
[
21]. As the subspace of left eigenvectors of
with an eigenvalue
is two-dimensional and includes spinors
,
,
, and
, where the two latter spinors are linearly independent (otherwise spinors
and
would not be linearly independent), there exist such
,
,
,
that
For each spinor
, the charge conjugated spinor
can be defined, and it has the same transformation properties under Lorentz transformations as
[
25]. We have
as
and
are, respectively, symmetric and antisymmetric (see Equation (
38)) with respect to the transposition of
and
.
Let us multiply Equations (
98) and (
99) by
and
from the right:
so
Let us note that
and
as
(see Equations (
37) and (
38)). Therefore,
Equations (
94), (
96), (
98) and (
99) yield
where
so
and
or
Substituting the expressions for
,
,
from Equation (
102) and using Equations (
103) and (
110), we finally obtain:
This equation for one “spinor component”
is generally equivalent to the Dirac equation in the Yang–Mills field (
88) (if
is not identically singular, which is generally the case): on the one hand, it was derived from the Dirac equation; on the other hand, the field
can be restored if its “spinor component”
is known, and the Dirac Equation (
88) can be derived from Equation (
88). This can be demonstrated along the same lines as in [
9,
10].
The only “spinor component” in Equation (
111) has
n “color” components; however, using a gauge transformation, all of them but one can be made zero and the remaining component can be made real, as the action of the SU(
n) group on the unit sphere in
is transitive for
[
31].
2.8. Plasma-like Model of Quantum Particles
There is a well-known analogy between quantum particles and plasma: the dispersion relation for the Klein–Gordon equation (
)
is similar to the dispersion relation for waves in a simple plasma model
However, to expand this analogy, we need a description of both negatively and positively charged particles. The description is based on the possibility of using a real, rather than complex, wave function to describe charged particles. This possibility is described in
Section 2.2,
Section 2.3 and
Section 2.4.
Using one real wave function instead of complex functions introduces some “symmetry” between positive and negative frequencies and, thus, particles and antiparticles. Therefore, a tentative description of such (one-particle electron) wave function was given in [
3,
5,
17]: the wave function can describe
electrons and
N positrons, where
N is very large. This collection of particles and antiparticles has the same total charge (and mass) as an electron, and the value of the wave function at some point (or some combination of the wave function and its derivatives at the point) is a measure of both “vacuum polarization” and the density of probability of finding an electron at this point (finding a positron at that point is also possible, but probably requires much more energy). An electron found during a measurement can be any of the
electrons. The results of the measurement on the specific collection can depend, say, on the precise coordinates of the particles in the collection. One can consider such a collection as a “dressed” electron with a well-defined total charge. The description assumes trajectories of the “bare” electrons and positrons, but the uncertainty principle is valid for the “dressed” electron. The charge density distribution of the “dressed” electron is defined by all charges of the “bare” electrons and positrons and can be very close to the charge density distribution built from the traditional wave function (see below in this subsection). If an electron is removed from the collection (for example, a precise position measurement means high momentum uncertainty, and as a result, some particle acquires high speed and quickly leaves the collection; the area around the place vacated by the particle will tend to be filled with the surrounding particles) and the energy of the remaining collection is not high enough for the collection to manifest as pairs, it is difficult to tell the collection from the vacuum. This may be the source of discreteness emphasized in [
32]. It is also important to note that the spreading of wave packets (which complicates, e.g., the de Broglie’s double solution approach [
33]) is not problematic for this description.
The description of this article only covers the unitary evolution of quantum theory but not the wave function collapse. The rationale for that is provided in the introduction of [
17].
One can object that the mass of such a collection of a large number of particles and antiparticles would be too large, as each pair should have a mass of at least two electron masses, but it is not necessarily so, as the energy of an electron and a positron that are very close to each other can be significantly less than their energy when the distance between them is large.
The above description is illustrated in
Figure 1, where electrons and positrons are represented by minus and plus signs, respectively. Collections (a) and (b) there are identical except for an extra electron (represented by a blue minus sign) in collection (a). The collections are difficult to visually distinguish from each other, but the total charges of the collections are one and zero electron charge, respectively.
Figure 1 is similar to M. Strassler’s Figure 3 in [
34], but the figure here describes an electron, rather than a nucleon (which is a composite particle), and the size of the collection is defined by the size of the volume where the wave function does not vanish.
Similarities of the plasma-like description with the Bohmian interpretation are described in [
17].
While single-particle systems are very important (for example, they are sufficient to describe the double-slit experiment), it is necessary to discuss many-particle systems. While the author does not have a complete description of second quantization for such systems, the Pauli principle for fermions can emerge in the following way: for identical wave functions, the relevant collections of discrete charges have identical or very similar coordinates of the charges, and thus a combination of such collections can have very high energy, for example, due to Coulomb interaction. Matter and radiation (and, eventually, Fermi and Bose fields) are treated differently in the plasma-like description. The second quantization for bosons can be performed using the approach described in
Section 2.9.
How accurately can a continuous charge density distribution for a specific wave function with a total charge equal to one electron charge be approximated by a collection of discrete charges with values of electron charge? It is obvious that a Fourier expansion of a point-like charge density distribution contains arbitrarily high spatial frequencies, whereas high-spatial-frequency Fourier components of smooth charge density distributions tend to zero fast; therefore, it is probably impossible to approximate a continuous charge density distribution by a finite number of discrete quantized charges with good accuracy. However, quantum field theories are typically considered to be just effective theories. Therefore, we can look for collections of discrete charges with values of electron charge that have the same Fourier components with spatial frequencies below some limit value as the initial smooth charge density distribution.
Let us illustrate this approach in the one-dimensional case assuming that the smooth charge density distribution is periodic, e.g., with a period of
, so we consider it on a segment
. Let us choose some normalized function
:
(the total charge in the charge density distribution is +1 (electron charge)). The charge density distribution does not have to be non-negative everywhere, as we have in mind applications not just to the Schrödinger equation and the Dirac equation but also to the Klein–Gordon equation, where the charge density does not have to be of the same sign everywhere.
Let us consider the Fourier expansion:
where
We will try to find a discrete charge density distribution approximating
. Let us assume that this distribution describes
particles with coordinates
, including
electrons and
j positrons, so the discrete charge density distribution is
Let us demand that the
k-th cosine Fourier components of distribution
coincide with
for
(the zeroth cosine Fourier component of
automatically coincides with
due to Equation (
114)) and that the
k-th sine Fourier components of distribution
coincide with
for
, where
and
are some natural numbers.
To find such distribution, one needs to solve a system of polynomial equations [
17]. It is possible to do that numerically, using the powerful homotopy continuation method for polynomial systems (see [
35] and references there). That was achieved in [
17] for a system of 98 equations.
One can look for discrete distributions containing a number of points that is much larger than the number of Fourier coefficients that need to be equal for the smooth and the discrete distributions, so it was conjectured [
17] (but not proven) that there is always a real solution, no matter how many Fourier coefficients need to be equal. We prove this here for the one-dimensional case.
Let
, where
, be a real function with a Fourier series
and
so
.
Then let us prove the following statement (A):
, such that the distribution
has the same partial Fourier sum
as the function
.
Let us define a crystal of order
m as a distribution
where
. Intuitively, a “crystal” contains an equidistant “comb” of
m positively charged point particles and an equidistant “comb” of
m negatively charged particles. The “positive” comb is displaced with respect to the “negative” comb.
Let us prove the above statement (A) by building the required distribution
inductively. We choose
as
, where
is an arbitrary number such that
. Let us assume that we have a distribution
(
) that is a sum of
and a non-negative integer number of crystals whose orders are less than
l, and
has the same partial Fourier sum
as the function
(distribution
satisfies these conditions for
).
To build the distribution
, let us calculate the Fourier coefficients
of
, the crystal of order
m (Equation (
122)):
It is obvious that
. We also obtain:
As the sum
does not change when multiplied by
, it vanishes if
and equals
m if
or
(we do not need the explicit expressions for the sum for
). Thus,
vanishes if
, and
if
.
Therefore, if we add an integer number of crystals
to the distribution
, the Fourier components of the latter will not change for
and will coincide with those for
. Let us prove that ∀ complex
z∃
, and integer
j such that
To do this, it is sufficient to prove that ∀ complex
∃
, such that
After that, we will be able to choose
and the sign and the magnitude of
j in such a way that
and Equation (
128) is satisfied.
We can find the required values of
and
from the following equations:
where
is such that
,
, and
. We obtain:
,
, and
,
, so
, and Equation (
128) is proven.
If
has the following Fourier series:
(where
for
), let us choose an integer
j and a crystal
in such a way that
. Let us then consider the following distribution:
Its Fourier components
coincide with those of
for
and for
. As
,
, and
are real and, therefore,
,
and
, we also have
. Therefore, statement (A) is proven.
It is not clear if a similar approach can be used to find a discrete distribution with all Fourier components coinciding with those of the smooth distribution (to this end, the summation over
j in the right-hand side of Equation (
120) would be from
to
).
The approach of this section can be useful for the interpretation of quantum phase-space distribution functions [
36], such as the Wigner distribution function, which is not necessarily non-negative. It is also interesting to compare the approach with the initial Schrödinger’s interpretation of the wave function (see, e.g., in [
37,
38] and references therein). Schrödinger’s interpretation of
as charge density meets some objections. For example, A. Khrennikov noted ([
39], p. 23): “Unfortunately, I was not able to find in Schrödinger’s papers any explanation of the impossibility to divide this cloud into a few smaller clouds, i.e., no attempt to explain the fundamental discreteness of the electric charge”. The plasma-like description suggests that
(and its analogs for the Klein–Gordon and Dirac equations) is just a smoothed charge density, and the description is immune to the above objection. Note that the negativity of the Wigner function is considered a resource for quantum computing [
40].
An example mathematical model (equations of motion) of collections of charged particles and antiparticles interacting with an electromagnetic field was built in [
17]. The model is based on the results described in
Section 2.1 and emulates arbitrarily well (for a sufficiently large cutoff constant) a quantum theory—the Klein–Gordon–Maxwell electrodynamics (scalar electrodynamics); therefore, one can reasonably expect that the model should successfully describe a wide spectrum of quantum phenomena. Modeling of specific experiments, such as the double-slit experiment, is left for future work.
Let us emphasize that one needs to choose the cutoff constant to fully define the mathematical model. While this constant can be arbitrarily high, the model has problems at very high temporal/spatial frequencies once this choice is made. However, these problems seem similar to those of standard quantum field theories [
41].
The following problem of the plasma-like description was resolved in [
17]. Composite particles, such as nucleons or large molecules, also demonstrate quantum properties [
42,
43]. It is, however, difficult to imagine that molecule–antimolecule pairs play a significant role in diffraction of large molecules (creation of such pairs is possible, but much less probable than the creation of electron–positron pairs). However, composite particles take part in some interactions (for example, electromagnetic or strong interactions), so the description can be modified as follows in that case: composite particles are accompanied by a large collection of fermion–antifermion pairs (for example, electron–positron pairs for electromagnetic interactions and quark–antiquark pairs for strong interactions; in some situations, it can be difficult to tell such pairs from force carriers, such as photons or gluons). Such fermions/pairs/force carriers are present at all locations where the wave function traditionally describing the composite particle does not vanish, so the dimensions of the collection are not limited by the range of the interaction (for example, the short range of strong interaction). Thus, the composite particle can be detected at all locations where the wave function does not vanish, although at most locations, it is fermions/pairs/force carriers of the collection that interact directly with the instrument, not the composite particle itself.
There is an obvious analogy between this description and plasma. As the dispersion relation for the Klein–Gordon equation
(in a natural system of units) is similar to a dispersion relation of a simple plasma model, such analogy was used previously (see, e.g., in [
44,
45,
46]). This analogy illustrates the effective long-range interaction within a collection.
In [
17], a preliminary estimate was made of the density of particles in a collection modeling, which is perceived as one particle in traditional quantum experiments. If
is the electron density in the collection, the plasma frequency
in the electron–positron plasma is
times greater than the traditional plasma frequency [
47], i.e.,
(we do not consider any renormalization of mass and charge in this preliminary treatment). It is natural to suggest that this plasma frequency is equal (maybe on the order of magnitude) to the angular frequency of Zitterbewegung
[
48,
49], so we obtain
where
is the fine structure constant. Thus,
cm
or 21.8 per cube with an edge length equal to the reduced Compton wavelength
. The high electron density suggests that there is low energy per particle of a collection. Let us also note that in this context, the Zitterbewegung frequency plays a role of a “natural frequency”, rather than a frequency of some “internal clock” [
50].