Some Remarks on the Gupta–Bleuler and Dirac–Fock Quantization Schemes

Abstract

The classical Gupta–Bleuler quantization scheme in quantum electromagnetic field theory, based on the so-called “weakened” Lorenz subsidiary condition and indefinite metric on the reduced Fock space, is revisited within the context of the canonical Dirac–Fock quantization approach. The latter, as applied to the Fermi–Fock–Podolsky Lagrangian, allows for both satisfaction of the strong Lorenz subsidiary condition on the Fock space and its averaged invariance in time to be shown, as well as the positive energy density definiteness of quantum photon states. The time non-invariance of the averaged “weakened” Lorenz subsidiary condition within the Gupta and Bleuler approach is also demonstrated, resulting in energy density positive non-definiteness, which is later replaced by the so-called BRST cohomology condition, imposed on a suitable construction of the Fock space. We also sketch the corresponding variational approach of N.N. Bogolubov (Sr.) to constructing a deformed strong Lorenz-type operator condition, both within the perturbative Scwinger’s S-matrix approach and the Fock multi-temporal paradigm. A short supplement is devoted to analysis of the classical Lorenz-type subsidiary condition and the related temporal invariance of the electric potential energy, providing a derivation of the complete classical system of Maxwell’s equations.

1. Introduction

It is well-known that the commonly accepted Dirac-type quantization [1,2,3,4] in quantum electrodynamics raises certain mathematical problems [3,5,6,7,8,9] related to the need to simultaneously satisfy the requirement for local positive energy density, the Lorenz auxiliary condition imposed on quantum states, and the transversality condition. Furthermore, the entire formulation of the theory must be relativistic [10,11]. As the corresponding photon quanta are massless, attempts to transfer the Dirac-type quantization scheme to electrodynamics—a vector operator field theory—faced fundamental difficulties, resulting in nonsensical expressions of physical observables and singularities. In the early 1950s, various attempts to overcome these problems were made by using certain mathematical tricks that were poorly justified from both theoretical and mathematical physical points of view. This unsatisfactory state of quantum electromagnetic theory regarding the validity of imposing the strong Lorenz operator condition on the electromagnetic operator vector field was severely criticized by Gupta [12] and Bleuler [13], who proposed the use of appropriately defined “weakened” conditions based on both the physically unmotivated indefinite metric [3,6,7,8,14,15] and the corresponding weak Lorenz-dependent constraint imposed on the Fock space.The theoretical and mathematical physics community has accepted [15,16,17,18,19,20,21] this way of solving quantum electromagnetic field problems, and it characterizes the state-of the-art in modern quantum electromagnetic theory. Unfortunately, the Gupta–Bleuler quantization scheme, when applied directly to the Fermi Lagrangian, leads to both a non-symmetric and a non-positive-definite Hamiltonian operator, as well as to non-time-invariance of the averaged Lorenz auxiliary condition when weakly imposed on the Fock space with an indefinite Krein-type metric. Taking into account that these physical inconsistencies arise from the zero charge density constraint, the latter has been non-canonically replaced by means of the corresponding Klein–Gordon equation, satisfied in the strong sense on the Fock space. These important aspects were later strongly re-analyzed within the so-called BRST approach [3,15,22,23,24,25], which generalized and suitably replaced the Gupta–Bleuler formalism. In particular, the so-called “weakened” condition on the corresponding Fock space introduced by Gupta and Bleuler became the abelian precursor to the closure condition of the so-called BRST basic cohomology [3], imposed on the Fock space and extended by means of a super-symmetry field, which allowed for the construction of a physical Hilbert space rather than an infinite-dimensional Fock-type space with indefinite metric state truncation. The corresponding Hamiltonian of the electromagnetic field in these covariant Lorenz-gauge and BRST schemes, being non-positive-definite on the full Krein indefinite metric space, becomes positive definite on physical transverse states once the physical subspace is properly factored or constrained.

In the late 1950s of the past century, the great mathematician and mathematical physicist N.N. Bogolubov (Sr.) wrote—but never published—a preprint [26] devoted to a variational approach to the quantum theory of the electromagnetic field. In this work, he attempted to resolve the related problem of the invariance within the Gupta–Bleuler quantization scheme and succeeded in deriving a perturbed Lorenz-type operator within the quantum interaction representation of the corresponding S-matrix expansion. The S-matrix theory aspect, as used in our manuscript, demonstrated the important historical fact of N. Bogolubov (Sr.) coping with the proposed Gupta–Bleuler quantization of electromagnetic field; which, from his point of view, suffered from certain mathematical problems and did not satisfy main quantization principles. In an attempt to resolve this quantization problem, he based his analytical quantum-theoretical reasonings on the main observable quantum field object, the S-matrix, describing the consequence of the asymptotically free electron quantum states resulting from its interaction with the quantum electromagnetic field. This well-known technique had been proposed some years before by W. Heisenberg [27], and was essentially developed by N. Bogolubov (Sr.) [7] himself. The main result obtained by N. Bogolubov is that the basic quantized Lorenz subsidiary condition ceases to exist, making up for the additional component owing to the electron–photon interaction. The latter had been first confirmed by V. Fock, although using a different approach. Inspired by the work of N.N. Bogolubov (Sr.), we decided to re-analyze the canonical Dirac-type quantization scheme applied to a suitable construction of the Fock space, which we then applied to a modified description of the Fermi–Fock–Podolsky Lagrangian electrodynamics [28,29,30] extended with the Lorenz subsidiary component. Based on the Dirac-type quantization [1,2,3] with primary and secondary constraints, we succeeded in demonstrating that the Lorenz-type auxiliary operator constraint is strongly satisfied on a suitably reduced Fock subspace and is time-invariant, which simultaneously guarantees both the positive definite energy density of photon quanta and the associated relativistic covariance. Furthermore, we re-examined the variational approach of N.N. Bogolubov (Sr.) to quantum electromagnetic field theory, and succeeded in constructing a suitably perturbed Lorenz-type operator expression within the classical multi-time Fock paradigm, based on the proper time reference frames assigned to electromagnetic and electron fields. In a short supplement, we analyze the classical Lorenz-type subsidiary condition and the related temporal invariance of the electric potential energy, providing a derivation of the complete classical system of Maxwell’s equations.

2. The Fock–Podolski Lagrangian Formalism and Dirac Quantization Revisited

On the Minkowski spacetime $M_{\tilde{t}}^4,$ specified by the proper temporal parameter $\tilde{t}\in {\mathbb{R}}_{\tilde{t}},$ consider the following Fermi–Fock–Podolsky [5,6,7,29,30] Lagrangian:

$$\mathcal{L}[\phi ,A]:={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}\left(\right|\nabla \phi +\partial A/\partial \tilde{t}{|}^2{-|\nabla \times A|}^2-{\left(\partial \phi /\partial \tilde{t}+⟨\nabla |A⟩\right)}^2)d^{3}x$$

(1)

for the electromagnetic vector fields $(\phi ,A)\in \mathcal{M}\simeq C^2(M^4;\mathbb{R}\times {\mathbb{R}}^3)$ in vacuum, and calculate the related action functional

$$S[\phi ,A]:={\int }_0^T\mathcal{L}[\phi ,A]d\tilde{t}$$

(2)

on a temporal interval $[0,T)\subset {\mathbb{R}}_{\tilde{t}}.$ The variational condition $\delta S[\phi ,A]=0$ on the functional phase space $\mathcal{M}$ under the constraints ${(\delta \phi ,\delta A)|}_{\tilde{t}=0}{=0=(\delta \phi ,\delta A)|}_{\tilde{t}=T}$ gives rise to the following set of differential Klein–Gordon–Fock wave relationships:

$${\partial }^2\phi /\partial {\tilde{t}}^2-\Delta \phi =0,{\partial }^{2}A/\partial {\tilde{t}}^2-\Delta A=0$$

(3)

in $M_{\tilde{t}}^4.$

From the Lagrangian function (1), we obtain the dual Hamiltonian representation

$$\begin{array}{cc}\hfill \partial \phi /\partial \tilde{t}& =\delta H[\phi ,A]/\delta p_0=-p_0-⟨\nabla |A⟩,\partial A/\partial \tilde{t}=\delta H[\phi ,A]/\delta p=p-\nabla \phi ,\hfill \\ \hfill \partial p_0/\partial \tilde{t}& =-\delta H[\phi ,A]/\delta \phi =-⟨\nabla |p⟩,\partial p/\partial \tilde{t}=\delta H[\phi ,A]/\delta A=\nabla \times (\nabla \times A)-\nabla p_0\hfill \end{array}$$

(4)

with respect to the canonical Poisson brackets

$$\begin{array}{cc}\hfill \left\{\phi \right(x),\phi (y\left)\right\}& =0=\{A\left(x\right),A\left(y\right)\},\{p_0\left(x\right),\phi \left(y\right)\}=\delta (x-y),\hfill \\ \hfill \left\{\phi \right(x),A(y\left)\right\}& =0,\left\{p\right(x),\otimes A(y\left)\right\}=I\delta (x-y),\hfill \\ \hfill \left\{p\right(x),p(y\left)\right\}& =0=\{p_0\left(x\right),p_0\left(y\right)\},\{p_0\left(x\right),p\left(y\right)\}=0,\hfill \\ \hfill \{p_0\left(x\right),A\left(y\right)\}& =0=\left\{p\right(x),\phi (y\left)\right\}=0\hfill \end{array}$$

(5)

for all $x,y\in {\mathbb{R}}^3,$ where, by definition,

$$\begin{array}{cc}\hfill p_0& :=\delta \mathcal{L}[\phi ,A]/\delta \left(\partial \phi /\partial \tilde{t}\right)=-⟨\nabla |A⟩-\partial \phi /\partial \tilde{t},\hfill \\ \hfill p& :=\delta \mathcal{L}[\phi ,A]/\delta \left(\partial A/\partial \tilde{t}\right)=\nabla \phi +\partial A/\partial \tilde{t},\hfill \end{array}$$

(6)

and the Hamiltonian function

$$H_b={\int }_{{\mathbb{R}}_x^3}{H}_b[\phi ,A]d^{3}x,H_b[\phi ,A]:=p_0\partial \phi /\partial \tilde{t}+⟨p|\partial A/\partial \tilde{t}⟩-\mathcal{L}[\phi ,A],$$

(7)

that is

$$H_b={\int }_{{\mathbb{R}}_x^3}[{\displaystyle \frac{1}{2}}{\left|p\right|}^2+{\displaystyle \frac{1}{2}}{|\nabla \times A|}^2-{\displaystyle \frac{1}{2}}{p}_0^2-p_0⟨\nabla |A⟩-⟨p|\nabla \phi ⟩]d^{3}x$$

(8)

is a priori time-invariant,

$$d{H}_b/d\tilde{t}=0,$$

(9)

for all $\tilde{t}\in {\mathbb{R}}_{\tilde{t}}.$ It is easy to observe that the Hamiltonian Equation (4) reduces exactly to the Maxwell Equations (117) and (118) in vacuum, if the Lorenz subsidiary condition

$$p_0=-(\partial \phi /\partial \tilde{t}+⟨\nabla |A⟩),$$

(10)

is invariant in time

$$d{p}_0/\partial \tilde{t}=-⟨\nabla |p⟩\Rightarrow 0$$

(11)

under the additional constraint

$$d⟨\nabla |p⟩/\partial \tilde{t}=-⟨\nabla |\nabla p_0⟩\Rightarrow 0,$$

(12)

which must be satisfied for all $\tilde{t}\in {\mathbb{R}}_{\tilde{t}},$ considering the imposed external zero electric charge density condition $\rho =⟨\nabla |E⟩=0$ in vacuum, where the vector momentum $p:=-E$ denotes the corresponding electric field.

Thus, we claim here that the classical electromagnetic field theory is compatible iff the main determining constraints (11) and (12) are satisfied for all $(\tilde{t},x)\in M_{\tilde{t}}^4.$ As these constraints are of the first class within the Dirac approach to constructing reduced Poisson brackets [1,2,3], the latter imposes the corresponding deformation of the canonical Poisson structures (5) to the following reduced form:

$$\begin{array}{cc}\hfill \left\{\phi \right(x),\phi (y\left)\right\}& =0=\{A\left(x\right),A\left(y\right)\},\{p_0\left(x_0\right),\phi \left(y\right)\}=0,\hfill \\ \hfill \left\{\phi \right(x),A(y\left)\right\}& =0,\left\{p\right(x),\otimes A(y\left)\right\}=I\delta (x-y),\hfill \\ \hfill \left\{p\right(x),p(y\left)\right\}& =0,\{p_0\left(x_0\right),p_0\left(x_0\right)\}=0,\{p_0\left(x_0\right),p\left(y\right)\}=0,\hfill \\ \hfill \{p_0\left(x_0\right),A\left(y\right)\}& =0=\left\{p\right(x),\phi (y\left)\right\}=0\hfill \end{array}$$

(13)

for all $x,y\in {\mathbb{R}}^3$ on the cotangent phase space $T^{\ast }\left(\mathcal{M}\right)$ to the functional manifold $\mathcal{M}\simeq C^{\infty }({\mathbb{R}}^3;{\mathbb{E}}^1\times {\mathbb{E}}^3),$ where we took into account that the Lorenz expression $p_0:M_{\tilde{t}}^4\to \mathbb{R}$ is constant on $M_{\tilde{t}}^4;$ that is, $p_0\left(x\right)=p_0\left(x_0\right)\in \mathbb{R}$ for some fixed point $x_0\in {\mathbb{R}}^3,$ which entails the vanishing of the delta function ${\delta (x-y)|}_{x=x_0}=0$ on the whole space ${\mathbb{R}}^3,$ as its one-point support $x_0\in {\mathbb{R}}^3$ is of zero measure in ${\mathbb{R}}^3.$ Moreover, the following temporal evolution

$$\begin{array}{c}0=\frac{\partial }{\partial \tilde{t}}\{p_0\left(x\right),\phi \left(y\right)\}=\{\frac{\partial }{\partial \tilde{t}}{p}_0\left(x\right),\phi \left(y\right)\}+\{p_0\left(x\right),\frac{\partial }{\partial \tilde{t}}\phi \left(y\right)\}=\\ =-\underset{0}{\underbrace{\{⟨\nabla |p\left(x\right)⟩,\phi \left(y\right)\}}}+\underset{0}{\underbrace{\{p_0\left(x\right),p_0\left(y\right)-⟨\nabla |A\left(y\right)⟩\}}}=0\end{array}$$

(14)

confirms the invariance of the zero bracket $\{p_0\left(x\right),\phi \left(y\right)\}=0$ for all $x,y\in {\mathbb{R}}^3$ and $\tilde{t}\in \mathbb{R}.$ Considering the above, it possible to apply the standard Dirac quantization scheme [1,3,4,31,32,33,34] to the cotangent phase space $T^{\ast }\left(\mathcal{M}\right)\stackrel{ħ}{\to }{\widehat{T}}^{\ast }\left(\mathcal{M}\right)$ via the quantum operator commutator relationships

$$\begin{array}{cc}\hfill \left[\phi \right(x),\phi (y\left)\right]& =0=\left[A\right(x),A(y\left)\right],\left[\phi \right(x),A(y\left)\right]=0,\hfill \\ \hfill [p_0\left(x\right),\phi \left(y\right)]& =0,[p\left(x\right),p\left(y\right)]=0,[p_0\left(x_0\right),A\left(y\right)]=0\hfill \\ \hfill [p_0\left(x\right),p_0\left(y\right)]& =0,[p\left(x\right),\otimes A\left(y\right)]={\displaystyle \frac{ħ}{i}}I\delta (x-y)\hfill \end{array}$$

(15)

for all $x,y\in {\mathbb{R}}^3,$ where ħ is the classical Planck constant and the adjoint pairs of operators $(\phi ,p_0)$ and $(A,p)\in {\widehat{T}}^{\ast }\left(\mathcal{M}\right)\simeq$ $\mathrm{End}$ ${\mathsf{\Phi }}_0$ act on a suitable construction [7,30,35] of the Fock space, ${\mathsf{\Phi }}_0,$ which should effectively describe the electromagnetic quantum states of real light particles (i.e., photons).

To make this quantization procedure physically reasonable, we first take into account that the canonical operator pairs $(\phi ,p_0)$ and $(A,p)\in {\widehat{T}}^{\ast }\left(\mathcal{M}\right)\simeq \mathrm{End}$ ${\mathsf{\Phi }}_0$ possess the natural spatial Fourier integral splitting in positive and negative wave vector components, as follows:

$$\begin{array}{cc}\hfill p_0\left(x\right)& =p_0^{+}\left(x\right)+p_0^{-}\left(x\right),p_0^{\pm }\left(x\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^{3/2}}}{\int }_{{\mathbb{R}}^3}{p}_{0,k}^{\pm }{e}^{\mp i⟨k|x⟩}{d}^{3}k,\hfill \\ \hfill \phi \left(x\right)& ={\phi }^{+}\left(x\right)+{\phi }^{-}\left(x\right),{\phi }^{\pm }\left(x\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^{3/2}}}{\int }_{{\mathbb{R}}^3}{\phi }_k^{\pm }{e}^{\mp i⟨k|x⟩}{d}^3,\hfill \\ \hfill p\left(x\right)& =p^{+}\left(x\right)+p^{-}\left(x\right),p^{\pm }\left(x\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^{3/2}}}{\int }_{{\mathbb{R}}^3}{p}_k^{\pm }{e}^{\mp i⟨k|x⟩}{d}^3,\hfill \\ \hfill A\left(x\right)& =A^{+}\left(x\right)+A^{-}\left(x\right),A^{\pm }\left(x\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^{3/2}}}{\int }_{{\mathbb{R}}^3}{A}_k^{\pm }{e}^{\mp i⟨k|x⟩}{d}^3,\hfill \end{array}$$

(16)

where, by definition, ${\left({\phi }_k^{-}\right)}^{\ast }={\phi }_k^{+},{\left(A_k^{-}\right)}^{\ast }=A_k^{+},{\left(p_{0,k}^{-}\right)}^{\ast }=p_{0,k}^{+}$, and ${\left(p_k^{-}\right)}^{\ast }=p_k^{+}$ for all momenta $k\in {\mathbb{R}}^3.$ Then, the corresponding Fock space $\mathsf{\Phi }$ can be defined as a cyclic Hilbert space endowed with a scalar product $(·|·)$ and generated by its vacuum vector state $\left|0\right)\in \mathsf{\Phi }$ via the following determining conditions:

$$\begin{array}{cc}\hfill \mathsf{\Phi }& ={\mathrm{span}}_{\begin{array}{c}({\lambda }_0,\lambda ))\in L_2{({\mathbb{R}}^3;\mathbb{C})}^2\\ ({\mu }_0,\mu )\in L_2{\left({\mathbb{C}}^3\right)}^2\end{array}}exp[{\lambda }_{0,k}{p}_{0,k}^{+}+{⟨{\mu }_{0.k}|p_k^{+}⟩}_{{\mathbb{E}}^3}+\hfill \\ \hfill & +{\lambda }_k{\phi }_k^{+}+{⟨{\mu }_k|A_k^{+}⟩}_{{\mathbb{E}}^3}\left]\right|0)\hfill \end{array}$$

(17)

where, by definition, the vacuum state $\left|0\right)\in \mathsf{\Phi }$ satisfies the following constraint:

$$\begin{array}{c}{\mathrm{span}}_{\begin{array}{c}({\lambda }_0,\lambda ))\in L_2{({\mathbb{R}}^3;\mathbb{C})}^2\\ ({\mu }_0,\mu )\in L_2{({\mathbb{R}}^3;{\mathbb{C}}^3)}^2\end{array}}\{{\lambda }_{0,k}{p}_{0,k}^{-}+{⟨{\mu }_{0.k}|p_k^{-}⟩}_{{\mathbb{E}}^3}+\\ +{\lambda }_k{\phi }_k^{-}+{⟨{\mu }_k|A_k^{-}⟩}_{{\mathbb{E}}^3}\left\}\right|0)=0\}\end{array}$$

(18)

for all $k\in {\mathbb{R}}^3.$ The canonical simultaneous operator commutations

$$\begin{array}{cc}\hfill [{\phi }_k^{\pm },{\phi }_s^{\pm }]& =0=[p_{0,k}^{\pm },p_{0,s}^{\pm }],[{\phi }_k^{\pm },A_s^{\pm }]=0,\hfill \\ \hfill [p_{0,k}^{-},{\phi }_s^{+}]& ={\displaystyle \frac{ħ}{2i}}{\delta }_{ks}=[p_{0,k}^{+},{\phi }_s^{-}],[p_{0,k}^{-},{\phi }_s^{-}]=0=[p_{0,k}^{+},{\phi }_s^{+}]\hfill \\ \hfill [A_k^{\pm },\otimes p_s^{\pm }]& =0,[p_k^{\pm },\otimes A_s^{\mp }]={\displaystyle \frac{ħ}{2i}}I{\delta }_{ks},[A_k^{\pm },\otimes A_s^{\pm }]=0=[p_k^{\pm },\otimes p_s^{\pm }],\hfill \end{array}$$

(19)

are jointly satisfied for all momenta $k,s\in {\mathbb{R}}^3.$ If we now take into account the quantized form of the Hamiltonian operator (8),

$$\begin{array}{c}{H}_b={\int }_{{\mathbb{R}}_x^3}[\frac{1}{2}|p^{-}+p^{+}{|}^2+\frac{1}{2}{|\nabla \times A^{+}+\nabla \times A^{-}|}^2-\\ -\frac{1}{2}{|p_0^{+}+p_0^{-}|}^2-(p_0^{+}+p_0^{-})⟨\nabla |A^{+}+A^{-}⟩-\\ -⟨p^{+}+p^{-}|\nabla {\phi }^{+}+\nabla {\phi }^{-}⟩]d^{3}x=\\ =2{\int }_{{\mathbb{R}}_k^3}[⟨p_k^{+}|p_k^{-}⟩-p_{0,k+}^{+}{p}_{0,k}^{-}+⟨k\times A_k^{+}|k\times A_k^{-}⟩-⟨p_k^{+}|ik{\phi }_k^{-}⟩+\\ +⟨p_k^{-}|ik{\phi }_k^{+}⟩-i{p}_{0,k}^{+}⟨k|A_k^{-}⟩+i⟨k|A_k^{+}⟩p_{0,k}^{-}]d^{3}k,\end{array}$$

(20)

one can observe that it becomes positive semi-definite on the reduced Fock subspace

$${\mathsf{\Phi }}^{\left(red\right)}:=\left\{\right|f)\in \mathsf{\Phi }:p_{0,k}^{\pm }\left|f\right)=0,⟨k|p_k^{\pm }⟩\left|f\right)=0,k\in {\mathbb{R}}^3\},$$

(21)

which is defined by means of imposing, on the Fock space $\mathsf{\Phi }$, the additional time-invariance constraint $⟨\nabla |p⟩\sim ⟨k|p_k^{\pm }⟩=0,k\in$${\mathbb{R}}^3,$ following from the Heisenberg operator equation

$$d⟨k|p_k^{\pm }⟩/d\tilde{t}={\displaystyle \frac{i}{ħ}}[H_b,⟨k|p_k^{\pm }⟩]=i{\left|k\right|}^2{p}_{0,k}^{\pm }=0,$$

(22)

which is satisfied on $\mathsf{\Phi }$ for arbitrary $k\in {\mathbb{R}}^3.$ The latter yields the symmetric and positive semi-definite reduced quantum Hamiltonian operator

$$H_{b,0}:=H_b{|}_{{\mathsf{\Phi }}^{\left(red\right)}}=2{\int }_{{\mathbb{R}}_x^3}⟨k\times A_k^{+}|k\times A_k^{-}⟩+⟨p_k^{+}|p_k^{-}⟩d^{3}k,$$

(23)

as considered on the reduced Fock subspace ${\mathsf{\Phi }}^{\left(red\right)},$ jointly with the suitably reduced commutator relationships

$$\begin{array}{cc}\hfill [{\phi }_k^{\pm },{\phi }_s^{\pm }]& =0,[{\phi }_k^{\pm },A_s^{\pm }]=0,\hfill \\ \hfill [A_k^{\pm },\otimes A_s^{\pm }]& =0=[p_k^{\pm }\otimes ,p_s^{\pm }],[A_k^{\pm },\otimes p_s^{\pm }]=0,\hfill \\ \hfill [p_k^{-}\left(x\right),\otimes A_s^{+}]& ={\displaystyle \frac{ħ}{2i}}I{\delta }_{ks}=[p_k^{+}\left(x\right),\otimes A_s^{-}],\hfill \end{array}$$

(24)

where ${\left(p_{0,k}^{-}\right)}^{\ast }=0=p_{0,k}^{+}{|}_{{\mathsf{\Phi }}^{\left(red\right)}}$ and $⟨k|p_k^{\pm }⟩{|}_{{\mathsf{\Phi }}^{\left(red\right)}}=0$ for all momenta $k\in {\mathbb{R}}^3$ on the reduced Fock subspace ${\mathsf{\Phi }}^{\left(red\right)}\subset \mathsf{\Phi }.$ Considering the reasonings above, the following proposition is formulated.

Proposition 1. 

The Dirac–Fock quantized electromagnetic field in vacuum presents a physically correct quantum field theory, compatibly defined on the reduced Fock subspace ${\mathsf{\Phi }}^{\left(red\right)}\subset \mathsf{\Phi },$ defined by the strong time-invariant quantized Lorenz subsidiary operator condition in vacuum, equivalent to the strong operator constraints

$$\begin{array}{cc}\hfill p_0(\tilde{t},x_0)& =-(⟨\nabla |A(\tilde{t},x)⟩+\partial \phi (\tilde{t},x)/\partial \tilde{t}){|}_{{\mathsf{\Phi }}^{\left(red\right)}}=0,\hfill \\ \hfill \nabla p_0(\tilde{t},x){|}_{{\mathsf{\Phi }}^{\left(red\right)}}& =0,⟨\nabla |p(\tilde{t},x)⟩{|}_{{\mathsf{\Phi }}^{\left(red\right)}}=0\hfill \end{array}$$

(25)

for all spacetime variables $(\tilde{t},x)\in M_{\tilde{t}}$ and an arbitrarily chosen point $x_0\in {\mathbb{R}}^3.$

This result, in particular, means that we can also easily construct the reduced temporal quantization relationships

$$\begin{array}{cc}\hfill [{\phi }^{\pm }(\tilde{t},x),{\phi }^{\pm }(\tilde{s},y)]& =0=[p_0^{\pm }(\tilde{t},x_0),p_0^{\pm }(\tilde{s},x_0)],\hfill \\ \hfill [p_0^{-}(\tilde{t},x_0),{\phi }^{+}(\tilde{s},y)]& =0=[p_0^{+}(\tilde{s},x_0),{\phi }^{-}(\tilde{t},x)],\hfill \\ \hfill [A^{\pm }(\tilde{t},x),\otimes A^{\pm }(\tilde{s},y)]& =0=[p^{\pm }(\tilde{t},x)\otimes ,p^{\pm }(\tilde{s},y)],\hfill \\ \hfill [A^{\pm }(\tilde{t},x),\otimes p^{\pm }(\tilde{s},y)]& =0,\hfill \\ \hfill [p^{-}(\tilde{t},x),\otimes A^{+}(\tilde{s},y)]& ={\displaystyle \frac{ħI}{2i}}{D}_0(\tilde{t}-\tilde{s},x-y),\hfill \end{array}$$

(26)

where $D_0(\tilde{t}-\tilde{s},x-y)$ denotes the so-called relativistic invariant Pauli–Jordan function [6,7,8,36], determined for arbitrary spacetime parameters $\tilde{t},\tilde{s}\in {\mathbb{R}}_{\tilde{t}}$ and $x,y\in {\mathbb{R}}_x^3,$ as it easily follows from the temporal Heisenberg-type operator evolution relationships. The reduced Fock subspace ${\mathsf{\Phi }}^{\left(red\right)},$ constructed above, satisfies the strong temporal Schrödinger-type evolution invariance

$$\partial \left|f\right)/\partial \tilde{t}-i/ħH_{b,0}\left|f\right)\in {\mathsf{\Phi }}^{\left(red\right)}$$

(27)

for all $\tilde{t}\in {\mathbb{R}}_{\tilde{t}}$ and any given $\left|f\right)\in {\mathsf{\Phi }}^{\left(red\right)},$ where the Hamiltonian operator $H_{b,0}\in \mathrm{End}$ ${\mathsf{\Phi }}^{\left(red\right)}$ is both symmetric and positive definite on the reduced Fock subspace ${\mathsf{\Phi }}^{\left(red\right)}\subset \mathsf{\Phi }.$ Moreover, the averaged Lorenz subsidiary condition ${\overline{p}}_0:=\left(f\right|p_0\left|f\right)$ over ${\mathsf{\Phi }}^{\left(red\right)}\subset {\mathsf{\Phi }}^{\left(red\right)},$ imitating the classical one, remains time-invariant; that is

$$\begin{array}{cc}\hfill d{\overline{p}}_0/d\tilde{t}& =(\partial f/\partial \tilde{t}|p_0|f)+(f|\partial p_0/\partial \tilde{t}|f)+(f|p_0|\partial f/\partial \tilde{t})=\hfill \\ \hfill & =\underset{=0}{(\partial f/\partial \tilde{t}|p_0|f)}+{\displaystyle \frac{i}{ħ}}(f|[H_{b,0},p_0]f)+\underset{=0}{(p_{0}f|\partial f/\partial \tilde{t})}=\hfill \\ \hfill & ={\displaystyle \frac{i}{ħ}}\underset{=0}{\left(f\right|H_{b,0}{p}_{0}f)}-{\displaystyle \frac{i}{ħ}}\underset{=0}{\left(f\right|p_0{H}_{b,0}f)}=0,\hfill \end{array}$$

(28)

which easily follow from condition (25) for all $\tilde{t}\in {\mathbb{R}}_{\tilde{t}}.$

Recalling now the generally accepted quantization scheme of Gupta [12] and Bleuler [13], the result above is in strong contrast to their so-called “weak Lorenz condition”, imposed on the corresponding Fock space $\mathsf{\Phi }$, defined by the relationships (17) and (18). The respectively reduced Fock subspace

$${\mathsf{\Phi }}^{+}:=\overline{ker{p}_0^{+}}\subset \mathsf{\Phi }\simeq \left\{\right|f)\in \mathsf{\Phi }:p_{0,k}^{+}\left|f\right)=0,k\in {\mathbb{R}}^3\}$$

(29)

neglects both the condition $\nabla p_0{|}_{{\mathsf{\Phi }}^{+}}$ and the zero charge density value ${⟨\nabla |p⟩|}_{{\mathsf{\Phi }}^{+}}=0$ constraint, which should be imposed on ${\mathsf{\Phi }}^{+}$. Moreover, with respect to the reduced Hamiltonian operator $H_b^{+}:=H_b{|}_{{\mathsf{\Phi }}_0^{+}}\in \mathrm{End}$ ${\mathsf{\Phi }}^{+}$, the temporal evolution of any state $\left|f\right)\in {\mathsf{\Phi }}^{+}$ does not assure the corresponding vanishing of the averaged Lorenz subsidiary constraint:

$${\overline{p}}_0^{+}:=\left(f\right|p_{0}f)=\left(f\right|p_0^{+}f+p_0^{-}f)=\left(p_0^{+}f\right|f)+\left(f\right|p_0^{+}\left|f\right)=0$$

(30)

for all $\tilde{t}\in {\mathbb{R}}_{\tilde{t}}.$ In particular, the expression (30) is not time-invariant:

$$\begin{array}{cc}\hfill d{\overline{p}}_0^{+}/d\tilde{t}& =(\partial f/\partial \tilde{t}|p_0^{+}|f⟩+\left(f\right|\partial p_0^{+}/\partial \tilde{t}|f⟩+(f|p_0^{+}|\partial f/\partial \tilde{t}⟩=\hfill \\ \hfill & =(\partial f/\partial \tilde{t}|p_0^{+}|f⟩+{\displaystyle \frac{i}{ħ}}(f|H_b^{+}{p}_0^{+}-p_0^{+}{H}^{+}|f⟩+(f|p_0^{+}|\partial f/\partial \tilde{t}⟩=\hfill \\ \hfill & =\underset{=0}{(\partial f/\partial \tilde{t}|p_0^{+}|f⟩}+\underset{=0}{{\displaystyle \frac{i}{ħ}}(f|H_b^{+}{p}_0^{+}|f⟩}-{\displaystyle \frac{i}{ħ}}(f|p_0^{\_}{H}_b^{+}|f⟩+(f|p_0^{+}|\partial f/\partial \tilde{t}⟩=\hfill \\ \hfill & =-{\displaystyle \frac{i}{ħ}}(p_0^{-}f|H_b^{+}|f⟩+(p_0^{-}f|\partial f/\partial \tilde{t}⟩=(p_0^{-}f|(\partial f/\partial \tilde{t}⟩-{\displaystyle \frac{i}{ħ}}{H}_b^{+\ast }|f⟩)\ne 0,\hfill \end{array}$$

(31)

This is to take into account the fact that the reduced Hamiltonian operator $H_b^{+}\in \mathrm{End}{\mathsf{\Phi }}^{+}$ is not both positive semi-definite and symmetric; that is, $H_b^{+}\ne H_b^{+\ast }$ on the reduced Fock subspace ${\mathsf{\Phi }}^{+}\subset \mathsf{\Phi }.$

Faced with these inconsistencies, Gupta [12] and Bleuler [13] proposed considering the Fock space $\mathsf{\Phi }$ rigged with a suitably constructed indefinite metric jointly with the non-canonical Klein–Gordon evolution equation, which is second-order in time, in order to satisfy the zero charge density constraint discussed above. However, all of this directly highlights the essence of the mathematical and conceptual failure of the Gupta–Bleuler quantization approach, addressing exactly why physicists were forced to abandon it for more sophisticated frameworks such as the BRST [3,15,22,23,25] quantization method; in particular, the ad hoc introduction of an indefinite metric to the Fock space to fix a manifestation of negative energy feels mathematically artificial and unconvincing. Moreover, forcing a second-order constraint, based on the Lorenz condition and the Klein–Gordon equation, into a first-order canonical framework completely violates the strict mathematical rules laid out in Dirac’s canonical quantization of constrained systems. This is why the BRST approach is fundamentally superior and resolves these two fatal flaws; however, we will not dwell on this in further detail here.

As written in the book [7] (p. 120) by N.N. Bogolubov (Sr.) and D.V. Shirkov, “… these conditions (29) guarantee that the Lorenz condition (30) is satisfied on the average, which is quite sufficient to establish correspondence with the classical field…” Yet, as follows from the abovementioned calculative result (31), it is generally not the case that the above-stated non-symmetricity and positive non-semi-definiteness of the suitably reduced Hamiltonian operator $H_{b,0}^{+}:$${\mathsf{\Phi }}^{+}\to {\mathsf{\Phi }}^{+}$ jointly make it possible to suggest that the Gupta–Bleuler quantization scheme—as applied to an electromagnetic field on a suitably reduced Fock subspace ${\mathsf{\Phi }}^{+}\subset \mathsf{\Phi }$—gives rise to a physically ill-posed quantum field theory. In contrast, as demonstrated above, the respectively applied canonical Dirac–Fock quantization scheme gives rise to a well-posed quantum electromagnetic field theory with a both symmetric and positive semi-definite Hamiltonian operator $H_{b,0}:$${\mathsf{\Phi }}^{\left(red\right)}\to {\mathsf{\Phi }}^{\left(red\right)}$ on a suitably reduced Fock subspace ${\mathsf{\Phi }}^{\left(red\right)}\subset \mathsf{\Phi }.$

Moreover, from a physical point of view, this result means that the constrained Fock subspace ${\mathsf{\Phi }}^{+}\subset \mathsf{\Phi }$ is not suitable for applying the canonical Dirac-type quantization scheme and does not allow one to construct an adequate and correct scattering theory as well as the related S-matrix theory of Schwinger, whose perturbative mathematical structure has been clarified and developed in detail [7,37] within the so-called Bogolubov–Parasyuk R-operation. Many years ago, concerning these quantization aspects, the great mathematician and mathematical physicist N.N. Bogolubov (Sr.) had written—but never published—a preprint [26] devoted to the variational approach to quantum electromagnetic field theory, in which he tried to resolve this non-invariance problem and related inconsistencies of the Gupta–Bleuler reduced Fock subspace (29) within the interaction representation of the corresponding perturbative S-matrix expansion.

N.N. Bogolubov (Sr.) wrote: “… Consider now a question about subsidiary conditions, which one needs to get subordinated electromagnetic field under interaction.“

We need then to proceed from the corresponding condition in the noninteracting field theory. In this case, as a rule, one uses the Lorenz condition

$$\sum _{\left(\alpha \right)}{\displaystyle \frac{\partial A_{\alpha }\left(\mathrm{x}\right)}{\partial {\mathrm{x}}_{\alpha }}}f\left(\mathrm{x}\right)=0$$

(32)

at any space-time point $x\in M^4,$ where a wave vector $f\in \tilde{\mathsf{\Phi }}$ and a mapping $(A_{\alpha }$:$\alpha =\overline{0,3})$:$M^4\to$$\mathbb{R}\times {\mathbb{E}}^3$ denotes the electromagnetic four-vector operator field on the Minkowski space-time. Yet, during past years in his works Gupta and Bleuler criticized the reasonableness of imposing condition (32) on the electromagnetic operator vector field and proposed to use a “weakened” condition of the type

$$\sum _{\left(\alpha \right)}{\displaystyle \frac{\partial A_{\alpha }^{+}\left(\mathrm{x}\right)}{\partial {\mathrm{x}}_{\alpha }}}f\left(\mathrm{x}\right)=0,$$

(33)

where $(A_{\alpha }^{-}$:$\alpha =\overline{0,3})$:$M^4\to \mathbb{R}\times {\mathbb{E}}^3$—the positive-frequency part of the electromagnetic field operators. Now, we demonstrate how the conditions (32) and (33) will change under interaction of the electromagnetic field with ambient charge distribution. Since the S-matrix operator $S\left(g\right),$ where $g:M^4\to R$ denotes a seed interaction parameter, establishes a connection between the wave vector for free fields and the wave vector in the presence of interaction, we see that the influence of the operator constraint

$$B\left(\mathrm{x}\right)f\left(\mathrm{x}\right)=0,$$

(34)

where either  $B\left(\mathrm{x}\right):={\sum }_{\left(\alpha \right)}{\displaystyle \frac{\partial A_{\alpha }\left(\mathrm{x}\right)}{\partial {\mathrm{x}}_{\alpha }}}$  or  ${\sum }_{\left(\alpha \right)}{\displaystyle \frac{\partial A_{\alpha }^{-}\left(\mathrm{x}\right)}{\partial {\mathrm{x}}_{\alpha }}},\mathrm{x}\in M^4,$  on the free particle wave vector  $f\in \tilde{\mathsf{\Phi }}$ can be written as a condition, imposed on the interacting wave vector $f\left(g\right)\in \tilde{\mathsf{\Phi }}:$

$$S\left(g\right)B\left(\mathrm{x}\right)S^{+}\left(g\right)f\left(g\right)=0.$$

(35)

Let us construct the expansion of (35) in the interaction seed parameter  $g:M^4\to \mathbb{R}$  of the operator expression $S\left(g\right)B\left(\mathrm{x}\right)S^{+}\left(g\right):\tilde{\mathsf{\Phi }}\to \tilde{\mathsf{\Phi }},\mathrm{x}\in M^4.$ Taking into account that

$$SB{S}^{+}=B+(SB-BS)S^{+}$$

(36)

and

$$\begin{array}{c}SB\left(\mathrm{x}\right)-B\left(\mathrm{x}\right)S=\sum _{n\in \mathbb{N}}{\displaystyle \frac{1}{n!}}{\int }_{\underset{n\mathrm{times}}{\underbrace{M^4\times \dots \times M^4}}}\{S_n({\mathrm{x}}_1,{\mathrm{x}}_{2,}\dots ,{\mathrm{x}}_n)B\left(\mathrm{x}\right)-\\ -B\left(\mathrm{x}\right)S_n({\mathrm{x}}_1,{\mathrm{x}}_{2,}\dots ,{\mathrm{x}}_n)\}\times g\left({\mathrm{x}}_1\right)g\left({\mathrm{x}}_2\right)\dots g\left({\mathrm{x}}_n\right)d{\mathrm{x}}_{1}d{\mathrm{x}}_2\dots d{\mathrm{x}}_n,\end{array}$$

(37)

as well as the electromagnetic field commutation conditions

$$[A_{\mu }\left(\mathrm{x}\right),A_{\nu }\left(\mathrm{y}\right)]={\delta }_{\mu \nu }{D}_0(\mathrm{x}-\mathrm{y}),$$

(38)

for arbitrary space-time points $\mathrm{x},\mathrm{y}\in M^4$ and $\mu ,\nu =\overline{0,3},$ where $D_0$—the already mentioned above invariant Pauli-Jordan function. Since for arbitrary $\nu =\overline{0,3}$

$$[\sum _{\left(\mu \right)}{\displaystyle \frac{\partial A_{\mu }\left(\mathrm{x}\right)}{\partial {\mathrm{x}}_{\mu }}},A_{\nu }\left(\mathrm{y}\right)]=i{\displaystyle \frac{\partial D_0(\mathrm{x}-\mathrm{y})}{\partial {\mathrm{x}}_{\nu }}}$$

(39)

in case (32), one obtains that

$$[B\left(\mathrm{x}\right),A_{\nu }\left(\mathrm{y}\right)]=i{\displaystyle \frac{\partial D_0(\mathrm{x}-\mathrm{y})}{\partial {\mathrm{x}}_{\nu }}},$$

(40)

and in case (33), one obtains

$$[B\left(\mathrm{x}\right),A_{\nu }\left(\mathrm{y}\right)]=i{\displaystyle \frac{\partial D_0^{+}(\mathrm{x}-\mathrm{y})}{\partial {\mathrm{x}}_{\nu }}},$$

(41)

and these expressions give rise to the following commutator relationships:

$$\begin{array}{c}{S}_n({\mathrm{x}}_1,{\mathrm{x}}_{2,}\dots ,{\mathrm{x}}_n)B\left(\mathrm{x}\right)-B\left(\mathrm{x}\right)S_n({\mathrm{x}}_1,{\mathrm{x}}_{2,}\dots ,{\mathrm{x}}_n)=\\ =-i\sum _{\left(k=\overline{1,n},\nu \right)}\frac{\partial D_0(\mathrm{x}-\mathrm{y})}{\partial {\mathrm{x}}_{\nu }}{\displaystyle \frac{\partial S_n({\mathrm{x}}_1,{\mathrm{x}}_{2,}\dots ,{\mathrm{x}}_n)}{\partial A_{\nu }\left({\mathrm{x}}_k\right)}}\end{array}$$

(42)

in the case (32) and similarly to the commutator relationship

$$\begin{array}{c}{S}_n({\mathrm{x}}_1,{\mathrm{x}}_{2,}\dots ,{\mathrm{x}}_n)B\left(\mathrm{x}\right)-B\left(\mathrm{x}\right)S_n({\mathrm{x}}_1,{\mathrm{x}}_{2,}\dots ,{\mathrm{x}}_n)=\\ =-i\sum _{\left(k=\overline{1,n},\nu \right)}\frac{\partial D_0^{-}(\mathrm{x}-\mathrm{y})}{\partial {\mathrm{x}}_{\nu }}{\displaystyle \frac{\partial S_n({\mathrm{x}}_1,{\mathrm{x}}_{2,}\dots ,{\mathrm{x}}_n)}{\partial A_{\nu }\left({\mathrm{x}}_k\right)}}\end{array}$$

(43)

in the case (33). Thus, the commutator (37) becomes

$$\begin{array}{c}SB\left(\mathrm{x}\right)-B\left(\mathrm{x}\right)S=-\sum _{n\in \mathbb{N}}\frac{i}{(n-1)!}{\int }_{\underset{n\mathrm{times}}{\underbrace{M^4\times \dots \times M^4}}}\frac{\partial D_0(\mathrm{x}-{\mathrm{x}}_1)}{\partial x_{\nu }}\times \\ \times \frac{\partial S_n({\mathrm{x}}_1,{\mathrm{x}}_{2,}\dots ,{\mathrm{x}}_n)}{\partial A_{\nu }\left(x_1\right)}g\left({\mathrm{x}}_1\right)g\left({\mathrm{x}}_2\right)\dots g\left({\mathrm{x}}_n\right)d{\mathrm{x}}_{1}d{\mathrm{x}}_2\dots d{\mathrm{x}}_n=\\ =-\sum _{n\in \mathbb{N}}\frac{i}{n!}{\int }_{\underset{n+1\mathrm{times}}{\underbrace{M^4\times \dots \times M^4}}}\frac{\partial D_0(\mathrm{x}-{\mathrm{x}}_1)}{\partial x_{\nu }}\frac{\partial S_{n+1}({\mathrm{x}}_1,{\mathrm{x}}_{2,}\dots ,{\mathrm{x}}_n,{\mathrm{x}}_{n+1})}{\partial A_{\nu }\left(x_1\right)}\times \\ \times g\left({\mathrm{x}}_1\right)g\left({\mathrm{x}}_2\right)\dots g\left({\mathrm{x}}_n\right)d{\mathrm{x}}_{1}d{\mathrm{x}}_2\dots d{\mathrm{x}}_{n}d{\mathrm{x}}_{n+1}\end{array},$$

(44)

giving rise to the following result:

$$\begin{array}{c}\left(SB\left(x\right)-B\left(x\right)S\right)S^{+}=-\sum _{\left(n\in \mathbb{N},\nu \right)}\frac{i}{n!}{\int }_{{\displaystyle \underset{n\mathrm{times}}{\underbrace{M^4\times \dots \times M^4}}}}\frac{\partial D_0(\mathrm{x}-\mathrm{y})}{\partial {\mathrm{x}}_{\nu }}\left\{K\left({\displaystyle \frac{{\mathrm{x}}_1-{\mathrm{x}}_k}{{\mathrm{x}}_{k+1}-{\mathrm{x}}_n}}\right)\frac{\partial S_{k+1}(\mathrm{y},{\mathrm{x}}_{1,}\dots ,{\mathrm{x}}_k)}{\partial A_{\nu }\left(\mathrm{y}\right)}\times \right.\\ \times \left.S_{n-k}^{+}({\mathrm{x}}_{k+1},{\mathrm{x}}_{k+2,}\dots ,{\mathrm{x}}_n)\right\}g\left({\mathrm{x}}_1\right)g\left({\mathrm{x}}_2\right)\dots g\left({\mathrm{x}}_n\right)g\left(\mathrm{y}\right)dyd{\mathrm{x}}_{1}d{\mathrm{x}}_2\dots d{\mathrm{x}}_{n}d{\mathrm{x}}_n=\\ =-\sum _{\left(n\in \mathbb{N},\nu \right)}\frac{i}{ħn!}{\int }_{\underset{n\mathrm{times}}{\underbrace{M^4\times \dots \times M^4}}}\frac{\partial D_0(\mathrm{x}-\mathrm{y})}{\partial {\mathrm{x}}_{\nu }}\frac{\delta H(\mathrm{y},{\mathrm{x}}_{1,}\dots ,{\mathrm{x}}_n),}{\delta A_{\nu }\left(\mathrm{y}\right)}\times \\ \times g\left({\mathrm{x}}_1\right)g\left({\mathrm{x}}_2\right)\dots g\left({\mathrm{x}}_n\right)g\left(\mathrm{y}\right)d\mathrm{y}d{\mathrm{x}}_{1}d{\mathrm{x}}_2\dots d{\mathrm{x}}_{n}d{\mathrm{x}}_n=\\ =-\sum _{\left(\nu \right)}\frac{1}{ħ}{\int }_{M^4}\frac{\partial D_0(\mathrm{x}-\mathrm{y})}{\partial {\mathrm{x}}_{\nu }}\frac{\delta H(\mathrm{y},\mathrm{g})}{\delta A_{\nu }\left(\mathrm{y}\right)}g\left(y\right)dy=-\sum _{\left(\nu \right)}\frac{1}{ħ}{\int }_{M^4}\frac{\partial D_0(\mathrm{x}-\mathrm{y})}{\partial {\mathrm{x}}_{\nu }}{J}_{\nu }(\mathrm{y};g)g\left(\mathrm{y}\right)d\mathrm{y}=\\ =\sum _{\left(\nu \right)}\frac{1}{ħ}{\int }_{M^4}\frac{\partial D_0(\mathrm{x}-\mathrm{y})}{\partial {\mathrm{y}}_{\nu }}{J}_{\nu }(\mathrm{y};g)g\left(\mathrm{y}\right)d\mathrm{y},\end{array}$$

(45)

where $K\left(\frac{{\mathrm{x}}_1-{\mathrm{x}}_k}{{\mathrm{x}}_{k+1}-{\mathrm{x}}_n}\right)$ denotes combinations from  $n\in \mathbb{N}$ marked elements into subsets of k—and $(n-k)$—element groups, and  $H(\mathrm{y};g),$ $\mathrm{y}\in M^4,$ represents the Hamiltonian operator of the interacting electromagnetic field, as well as when deriving (45); the classical variational relationship $\frac{\delta H(\mathrm{y},\mathrm{g})}{\delta A_{\nu }\left(\mathrm{y}\right)}=J_{\nu }(\mathrm{y};g),$ $\nu =\overline{0,3},$ holding at all points  $\mathrm{y}\in M^4$ was used. If we now take into account the charge conservation law ${\sum }_{\left(\nu \right)}\frac{\partial J_{\nu }(\mathrm{y};g)}{\partial {\mathrm{x}}_{\nu }}=0$ at all $\mathrm{y}\in M^4,$ we obtain

$$\left(SB\left(x\right)-B\left(x\right)S\right)S^{+}=-\sum _{\left(\nu \right)}{\displaystyle \frac{1}{ħ}}{\int }_{M^4}{D}_0(\mathrm{x}-\mathrm{y})J_{\nu }(\mathrm{y};g){\displaystyle \frac{\partial g\left(\mathrm{y}\right)}{\partial {\mathrm{y}}_{\nu }}}d\mathrm{y},$$

(46)

which jointly with the operator expression (36) reduces to the following operator relationship:

$$\left\{\sum _{\left(\mu \right)}{\displaystyle \frac{\partial A_{\mu }\left(\mathrm{x}\right)}{\partial {\mathrm{x}}_{\mu }}}-\sum _{\left(\mu \right)}{\displaystyle \frac{1}{ħ}}{\int }_{M^4}{D}_0(\mathrm{x}-\mathrm{y})J_{\nu }(\mathrm{y};g){\displaystyle \frac{\partial g\left(\mathrm{y}\right)}{\partial {\mathrm{y}}_{\nu }}}d\mathrm{y}\right\}f\left(g\right)=0,$$

(47)

or similarly for the case (33)

$$\left\{\sum _{\left(\mu \right)}{\displaystyle \frac{\partial A_{\mu }^{+}\left(\mathrm{x}\right)}{\partial {\mathrm{x}}_{\mu }}}-\sum _{\left(\mu \right)}{\displaystyle \frac{1}{ħ}}{\int }_{M^4}{D}_0^{+}(\mathrm{x}-\mathrm{y})J_{\nu }(\mathrm{y};g){\displaystyle \frac{\partial g\left(\mathrm{y}\right)}{\partial {\mathrm{y}}_{\nu }}}d\mathrm{y}\right\}f\left(g\right)=0.$$

(48)

Unfortunately, these notes of N.N. Bogoluvov (Sr.) on the quantization scheme of Gupta and Bleuler for electromagnetic fields were never finished, lacking any further suggestions or inferences concerning the real mathematical physical meaning of the carried-out analysis of the problem. Nonetheless, as a natural consequence of these calculations of N.N. Bogolubov (Sr.) and the reasoning presented above, one can clearly state that the Lorenz subsidiary operator condition (32) for the interacting electromagnetic operator field should be deformed by some electric current operator components.

To refine the analytical form of this subsidiary Lorenz operator deformation, as predicted by N.N. Bogolubov (Sr.), we follow the approach devised in [30] and modified in [38,39] in the following section.

3. The Interacting Electromagnetic Field and Quantum Lorenz-Type Condition Deformation

Consider the fermionic electron creation and annihilation operator fields $\psi ,{\psi }^{+}:\mathsf{\Phi }\to {\mathsf{\Phi }}^4$, together with the related bosonic electromagnetic potential operator field $(\phi ,A):\mathsf{\Phi }\to {\mathsf{\Phi }}^4$ acting on the related Fock space $\mathsf{\Phi }$ and commuting with each other. We will treat these within the Fock many-temporal description approach [30,40], subject to the common reference frame $M_{\tilde{t}}^4$ specified by the temporal parameter $\tilde{t}\in {\mathbb{R}}_{\tilde{t}}.$ Based on the classical spin symmetry properties [41,42,43,44] of electrons, the “minimum interaction” principle [31,38,39,45] and the Lagrangian functional (1), we have the corresponding Dirac evolution equation

$$\partial \psi /\partial t=(⟨\alpha |\nabla ⟩+{\displaystyle \frac{i}{ħ}}\beta m)\psi +{\displaystyle \frac{ie}{ħ}}(\phi -⟨\alpha |A⟩\psi$$

(49)

on the operator spinor field $\psi :\mathsf{\Phi }\to {\mathsf{\Phi }}^4,$ where $\beta ,\alpha \in su\left(2\right)\times su\left(2\right)$ denotes the usual four-vector of Dirac matrices and $m\in {\mathbb{R}}_{+}$ is the electron mass. We can write down the related Hamiltonian operator expression

$$H_f^{\left(int\right)}={\int }_{{\mathbb{R}}^3}{d}^{3}x{\psi }^{+}(⟨\alpha |{\displaystyle \frac{ħ}{i}}\nabla ⟩+\beta m)\psi +{\int }_{{\mathbb{R}}^3}{d}^{3}x(e{\psi }^{+}\psi \phi -e{\psi }^{+}⟨\alpha |A⟩\psi )$$

(50)

for the electron operator field interacting with the electromagnetic operator field, where e denotes the elementary charge value. Thus, the evolution of the entire quantum field system, consisting of the fermionic and electromagnetic operator fields, is described by means of the joint Hamiltonian operator

$$H_{f-b}:=H_f^{\left(int\right)}+H_b$$

(51)

where the bosonic operator $H_b:\mathsf{\Phi }\to \mathsf{\Phi }$ is given by expression (7), via the Heisenberg commutator equations

$$\partial \psi /\partial t={\displaystyle \frac{i}{ħ}}[H_{f-b},\psi ],\partial {\psi }^{+}/\partial t={\displaystyle \frac{i}{ħ}}[H_{f-b},{\psi }^{+}]$$

(52)

and

$$\partial A/\partial t={\displaystyle \frac{i}{ħ}}[H_{f-b},A],\partial \phi /\partial t={\displaystyle \frac{i}{ħ}}[H_{f-b},\phi ]$$

(53)

with respect to the common temporal parameter $t\in {\mathbb{R}}_t.$ In this case, it is assumed that the corresponding proper temporal bosonic $\tilde{t}$$\in {\mathbb{R}}_{\tilde{t}}$ and proper temporal fermionic $\overline{t}$$\in {\mathbb{R}}_{\overline{t}}$ parameters coincide (i.e., $\tilde{t}=\overline{t}=t\in {\mathbb{R}}_t$). Furthermore, by definition, the operators satisfy $\psi (t,x):=\psi {(\overline{t},\tilde{t};x)|}_{\tilde{t}=\overline{t}=t},$ $\phi (t,x):=\phi (\tilde{t},x){|}_{\tilde{t}=\overline{t}=t},$ and $A(t,x)$ $:=$ $A(\tilde{t},x){|}_{\tilde{t}=t}$ at any point $t\in {\mathbb{R}}_t.$

Simultaneously, the parameters should satisfy both the electromagnetic field evolution derived above (Equation (4)) with respect to the reference frame $M_{\overline{t}}$, as well as the modified fermionic charged particle field equations

$$\partial \psi /\partial \overline{t}={\displaystyle \frac{i}{ħ}}[H_f^{\left(int\right)},\psi ],\partial {\psi }^{+}/\partial \overline{t}={\displaystyle \frac{i}{ħ}}[H_f^{\left(int\right)},{\psi }^{+}]$$

(54)

with respect to the reference frame $M_{\overline{t}}.$

As we are mostly interested in the evolution of the quantum particle fermionic field $\psi :\mathsf{\Phi }\to \mathsf{\Phi },$ we can get rid of the bosonic Hamiltonian’s impact on the evolution (54) by applying the following unitary canonical transformation to the Fock space $\mathsf{\Phi }$:

$$\mathsf{\Phi }\to \tilde{\mathsf{\Phi }}:=U\left(t\right)\mathsf{\Phi },$$

(55)

where we denote by $U\left(t\right):\mathsf{\Phi }\to \tilde{\mathsf{\Phi }}$ the unitary operator satisfying the determining equation

$$dU\left(t\right)/dt={\displaystyle \frac{i}{ħ}}{H}_{b}U\left(t\right)$$

(56)

generated by the bosonic Hamiltonian operator (7) with respect to the common temporal parameter $t\in \mathbb{R}.$ As a consequence of the transformation (55), we obtain the effective fermionic particle field interaction Hamiltonian operator

$$\begin{array}{cc}\hfill {\tilde{H}}_f^{\left(int\right)}& :=U\left(t\right)H_f^{\left(int\right)}{U}^{\ast }\left(t\right)={\int }_{{\mathbb{R}}^3}{d}^{3}x({\psi }^{+}⟨\alpha |{\displaystyle \frac{ħ}{i}}\nabla ⟩\psi +m{\psi }^{+}\beta \psi )+\hfill \\ \hfill & \hfill \\ \hfill & +{\int }_{{\mathbb{R}}^3}{d}^{3}x(e{\psi }^{+}\psi \tilde{\phi }-e{\psi }^{+}⟨\alpha |\tilde{A}⟩\psi ),\hfill \end{array}$$

(57)

where, by definition,

$$\begin{array}{cc}\hfill \tilde{A}& :=U\left(t\right)A{U}^{\ast }\left(t\right),\tilde{\phi }:=U\left(t\right)\phi U^{\ast }\left(t\right),\hfill \\ \hfill & \hfill \\ \hfill {\tilde{p}}_0& :=U\left(t\right)p_0{U}^{\ast }\left(t\right),\tilde{p}:=U\left(t\right)p{U}^{\ast }\left(t\right)\hfill \end{array}$$

(58)

We take into account that $\tilde{\psi }:=U\left(t\right)\psi U^{\ast }\left(t\right)=\psi$ and ${\tilde{\psi }}^{+}:=U\left(t\right){\psi }^{+}{U}^{\ast }\left(t\right)={\psi }^{+},$ subject to which the evolution in the transformed Fock space $\tilde{\mathsf{\Phi }},$ governed by the Hamiltonian operator (7) in the form

$${\tilde{H}}_b={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}_x^3}(|\tilde{p}{|}^2+|\nabla \times \tilde{A}{|}^2-{\tilde{p}}_0^2-{\tilde{p}}_0⟨\nabla |\tilde{A}⟩-⟨\tilde{p}|\nabla \tilde{\phi }⟩)d^{3}x$$

(59)

is completely eliminated. The electromagnetic operator field variables and their canonically adjoint ones (58) satisfy the following evolution equations, respectively:

$$\partial \tilde{A}/\partial \tilde{t}={\displaystyle \frac{i}{ħ}}[{\tilde{H}}_b,\tilde{A}],\partial \tilde{\phi }/\partial \tilde{t}={\displaystyle \frac{i}{ħ}}[{\tilde{H}}_b,\tilde{\phi }]$$

(60)

and

$$\partial {\tilde{p}}_0/\partial \tilde{t}={\displaystyle \frac{i}{ħ}}[{\tilde{H}}_b,{\tilde{p}}_0],\partial \tilde{p}/\partial \tilde{t}={\displaystyle \frac{i}{ħ}}[{\tilde{H}}_b,\tilde{p}]$$

(61)

with respect to the electromagnetic vector field proper time $\tilde{t}\in {\mathbb{R}}_{\tilde{t}}.$ From Equation (53), one obtains

$$\begin{array}{cc}\hfill \partial \tilde{A}/\partial t& ={\left.{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_{f-b},\tilde{A}]\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}={\left.{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_f^{\left(int\right)},\tilde{A}]\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}+{\left.{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_b,\tilde{A}]\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}.\hfill \\ \hfill \partial \tilde{\phi }/\partial t& ={\left.{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_{f-b},\tilde{\phi }]\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}={\left.{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_f^{\left(int\right)},\tilde{\phi }]\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}+{\left.{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_b,\tilde{\phi }]\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}.\hfill \end{array}$$

(62)

At the same time, one can write down equivalently that

$$\partial \tilde{A}/\partial t={\left.\partial \tilde{A}/\partial \tilde{t}\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}+{\left.{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_f^{\left(int\right)},\tilde{A}]\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}},\partial \tilde{\phi }/\partial t={\left.\partial \tilde{\phi }/\partial \tilde{t}\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}+{\left.{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_f^{\left(int\right)},\tilde{\phi }]\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}.$$

(63)

From the relationships (62) and (63), one can easily derive that

$${\left.\partial \tilde{A}/\partial \tilde{t}\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}={\left.{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_b,\tilde{A}]\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}{\left.\partial \tilde{\phi }/\partial \tilde{t}\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}={\left.{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_b,\tilde{\phi }]\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}},$$

(64)

coinciding with the system of Equation (60) above. Similarly, one obtains the system of equations

$${\left.{\displaystyle \frac{\partial {\tilde{p}}_0}{\partial \tilde{t}}}\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}={\left.{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_b,{\tilde{p}}_0]\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}{\left.{\displaystyle \frac{\partial \tilde{p}}{\partial \tilde{t}}}\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}}={\left.{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_b,\tilde{p}]\right|}_{\begin{array}{c}\tilde{t}=t,\\ \overline{t}=t\end{array}},$$

(65)

coinciding with the Equation (61).

Concerning the Hamiltonian operator (59), we need to mention that the related Dirac quantized commutation relationships for the operators $(\tilde{\phi },\tilde{A}):\tilde{\mathsf{\Phi }}\to {\tilde{\mathsf{\Phi }}}^4$ and their adjoints $({\tilde{p}}_0,\tilde{p}):\tilde{\mathsf{\Phi }}\to {\tilde{\mathsf{\Phi }}}^4$ remain the same; that is,

$$\begin{array}{cc}\hfill [\tilde{\phi }\left(x\right),\tilde{\phi }\left(y\right)]& =0=[\tilde{A}\left(x\right),\tilde{A}\left(y\right)],[{\tilde{p}}_0\left(x\right),{\tilde{p}}_0\left(y\right)]=0,[\tilde{p}\left(x\right),\tilde{p}\left(y\right)]=0,\hfill \\ \hfill [{\tilde{p}}_0\left(x\right),\tilde{\phi }\left(y\right)]& ={\displaystyle \frac{ħ}{i}}\delta (x-y),[\tilde{p}\left(x\right),\otimes \tilde{A}\left(y\right)]={\displaystyle \frac{ħ}{i}}I\delta (x-y),\hfill \end{array}$$

(66)

for all $x,y\in {\mathbb{R}}^3.$ Now, concerning the Hamiltonian operators (57) and (59), the Heisenberg evolution operator equations

$$\partial \psi /\partial \overline{t}={\displaystyle \frac{i}{ħ}}[{\tilde{H}}_f^{\left(int\right)},\psi ],\partial {\psi }^{+}/\partial \overline{t}={\displaystyle \frac{i}{ħ}}[{\tilde{H}}_f^{\left(int\right)},{\psi }^{+}]$$

(67)

with respect to the proper reference frame $M_{\overline{t}}$, together with the Heisenberg evolution operator Equations (60) and (61) with respect to the proper reference frame $M_{\tilde{t}}$, hold.

As we are further interested in the evolution Equations (52) and (53), suitably rewritten for the transformed Fock space $\tilde{\mathsf{\Phi }}$ with respect to the common temporal parameter $t\in \mathbb{R},$ we need to take into account [38,39,40] that the functional relationships

$$\begin{array}{cc}\hfill \psi \left(t\right)& :=\psi (\overline{t},\tilde{t}){|}_{\overline{t}=\tilde{t}=t},{\psi }^{+}\left(t\right):={\psi }^{+}(\overline{t},\tilde{t}){{|}_{\overline{t}=\tilde{t}=t},\tilde{\phi }\left(t\right):=\tilde{\phi }(\overline{t},\tilde{t})|}_{\overline{t}=\tilde{t}=t},\hfill \\ \hfill & \hfill \\ \hfill \tilde{A}\left(t\right)& :=\tilde{A}(\overline{t},\tilde{t}){|}_{\overline{t}=\tilde{t}=t},{\tilde{p}}_0\left(t\right):={\tilde{p}}_0(\overline{t},\tilde{t}){{|}_{\overline{t}=\tilde{t}=t},\tilde{p}\left(t\right):=\tilde{p}(\overline{t},\tilde{t})|}_{\overline{t}=\tilde{t}=t},\hfill \end{array}$$

(68)

hold. In particular, from (68), the evolution expressions satisfy

$$\begin{array}{cc}\hfill \partial \psi \left(t\right)/\partial t& =\partial \psi (\overline{t},\tilde{t})/\partial \overline{t}{{|}_{\overline{t}=\tilde{t}=t}+\partial \psi (\overline{t},\tilde{t})/\partial \tilde{t}|}_{\overline{t}=\tilde{t}=t},\hfill \\ \hfill & \hfill \\ \hfill \partial \tilde{\phi }\left(t\right)/\partial t& =\partial \tilde{\phi }(\overline{t},\tilde{t})/\partial \overline{t}{{|}_{\overline{t}=\tilde{t}=t}+\partial \tilde{\phi }(\overline{t},\tilde{t})/\partial \tilde{t}|}_{\overline{t}=\tilde{t}=t},\hfill \\ \hfill & \hfill \\ \hfill \partial \tilde{A}\left(t\right)/\partial t& =\partial \tilde{A}(\overline{t},\tilde{t})/\partial \overline{t}{{|}_{\overline{t}=\tilde{t}=t}+\partial \tilde{A}(\overline{t},\tilde{t})/\partial \tilde{t}|}_{\overline{t}=\tilde{t}=t},\hfill \\ \hfill & \hfill \\ \hfill \partial {\tilde{p}}_0\left(t\right)/\partial t& =\partial {\tilde{p}}_0(\overline{t},\tilde{t})/\partial \overline{t}{{|}_{\overline{t}=\tilde{t}=t}+\partial {\tilde{p}}_0(\overline{t},\tilde{t})/\partial \tilde{t}|}_{\overline{t}=\tilde{t}=t},\hfill \\ \hfill & \hfill \\ \hfill \partial \tilde{p}\left(t\right)/\partial t& =\partial \tilde{p}(\overline{t},\tilde{t})/\partial \overline{t}{{|}_{\overline{t}=\tilde{t}=t}+\partial \tilde{p}(\overline{t},\tilde{t})/\partial \tilde{t}|}_{\overline{t}=\tilde{t}=t},\hfill \end{array}$$

(69)

for all $t\in \mathbb{R}.$ The calculative expressions (69) prove to be useful when deriving the resulting quantum Maxwell electromagnetic equations. Before this, we need to check that the weak operator Lorenz constraints (25), rewritten in the case of the transformed Fock space $\tilde{\mathsf{\Phi }},$ are compatible with the evolution Equation (67) on the reduced Fock subspace

$${\tilde{\mathsf{\Phi }}}_0:=\{\tilde{f}\in \tilde{\mathsf{\Phi }}:{\tilde{p}}_0\left(x\right)|\tilde{f})=0\sim -\left(\partial \tilde{\phi }/\partial \tilde{t}+⟨\nabla |\tilde{A}⟩\right)|\tilde{f}⟩=0\}$$

(70)

for all $x\in {\mathbb{R}}^3$ with respect to $\tilde{t}\in \mathbb{R}:$

$$[{\tilde{p}}_{0,k}^{-},iħ\partial /\partial \tilde{t}-{\tilde{H}}_{b,0}]|\tilde{f})=0=[{\tilde{p}}_{0,k}^{+},iħ\partial /\partial \tilde{t}-{\tilde{H}}_{b,0}]|\tilde{f})$$

(71)

for all momenta $k\in {\mathbb{R}}^3$ and $|\tilde{f})\in {\tilde{\mathsf{\Phi }}}_0.$ However, they fail to be compatible with the evolution Equation (67), as

$$[{\tilde{p}}_{0,k}^{-},iħ\partial /\partial \overline{t}-{\tilde{H}}_f^{\left(int\right)}]|\tilde{f})\ne 0\ne [{\tilde{p}}_{0,k}^{+},iħ\partial /\partial \overline{t}-{\tilde{H}}_f^{\left(int\right)}]|\tilde{f})$$

(72)

for all $|\tilde{f})\in {\tilde{\mathsf{\Phi }}}_0$ with respect to the electron time $\overline{t}\in \mathbb{R}.$ This means that we cannot impose the constraint (70) on the Fock space ${\tilde{\mathsf{\Phi }}}_0$, as

$${\tilde{p}}_0\left(x\right)|\tilde{f}):=-\left(\partial \tilde{\phi }/\partial \tilde{t}+⟨\nabla |\tilde{A}⟩\right)|\tilde{f})\ne 0$$

(73)

for arbitrary $|\tilde{f})\in {\tilde{\mathsf{\Phi }}}_0$ and all $\overline{t}\in {\mathbb{R}}_{\overline{t}}$ and $\tilde{t}\in {\mathbb{R}}_{\tilde{t}}.$

Yet, it is easy enough to check [30,42,45] that the following deformed Lorenz-type operator

$${\tilde{p}}_0^{\left(def\right)}\left(x\right)={\tilde{p}}_0\left(x\right)+{\displaystyle \frac{e}{2\pi }}{\int }_{{\mathbb{R}}^3}{d}^{3}z{\int }_{{\mathbb{R}}^3}{d}^{3}y\tilde{\mathsf{\Theta }}(x;y,z){\tilde{p}}_0\left(z\right){\psi }^{+}\left(y\right)\psi \left(z\right),$$

(74)

where the kernel

$$\tilde{\mathsf{\Theta }}(x;y,z):={\displaystyle \frac{⟨(y-x)|{\nabla }_y{\delta }^3(y-z)⟩}{2iħ|y-x{|}^3}}$$

(75)

is the well-known Pauli–Jordan type [7] generalized function, guaranteeing that the compatibility conditions

$$\begin{array}{c}[{\tilde{p}}_0^{\left(def\right)}\left(x\right),{\tilde{p}}_0^{\left(def\right)}\left(y\right)]=0,[{\tilde{p}}_0^{\left(p\right)}\left(x\right),iħ\partial /\partial \overline{t}-{\tilde{H}}_f^{\left(int\right)}]|\tilde{f})=0,\\ [{\tilde{p}}_0^{\left(p\right)}\left(x\right),iħ\partial /\partial \tilde{t}-{\tilde{H}}_0\left]\right|\tilde{f})=0\end{array}$$

(76)

are satisfied for arbitrary temporal parameters $\overline{t}\in {\mathbb{R}}_{\overline{t}},$ $\tilde{t}\in {\mathbb{R}}_{\tilde{t}},$ all $x,y\in {\mathbb{R}}^3$, and any $|\tilde{f})\in {\tilde{\mathsf{\Phi }}}_0^{\left(def\right)}$ from the deformed reduced Fock subspace

$${\tilde{\mathsf{\Phi }}}_0^{\left(def\right)}:=\{|\tilde{f})\in \tilde{\mathsf{\Phi }}:{\tilde{p}}_0^{\left(def\right)}\left(x\right)|\tilde{f})=0,x\in {\mathbb{R}}^3\}.$$

(77)

Moreover, the operator expression

$$\widehat{\mathsf{\Theta }}(\tilde{t},\overline{t};x)={\displaystyle \frac{e}{2\pi }}{\int }_{{\mathbb{R}}^3}{d}^{3}z{\int }_{{\mathbb{R}}^3}{d}^{3}y\tilde{\mathsf{\Theta }}(x;y,z){\tilde{p}}_0\left(z\right){\psi }^{+}\left(y\right)\psi \left(z\right),$$

(78)

satisfies the determining differential operator relationship

$${\displaystyle \frac{\partial }{\partial \overline{t}}}\widehat{\mathsf{\Theta }}(\tilde{t},\overline{t};x)+{\displaystyle \frac{1}{ħ}}[H_f,\widehat{\mathsf{\Theta }}(\tilde{t},\overline{t};x)]={\displaystyle \frac{i}{ħ}}e{\psi }^{+}\left(x\right)\psi \left(x\right)$$

(79)

for all $x\in {\mathbb{R}}^3$ and $\tilde{t},\overline{t}\in \mathbb{R}$. This solves—at least, in part—Equation (47) of N.N. Bogolubov (Sr.). Thus, the related evolution flows (60), (61), and (67) on the reduced Fock subspace should be suitably considered, on the perturbed reduced Fock subspace (77), to be invariant with respect to the temporal parameters $\overline{t}\in {\mathbb{R}}_{\overline{t}}$ and $\tilde{t}\in {\mathbb{R}}_{\tilde{t}}.$

Now, we can proceed to deriving the corresponding quantum Maxwell equations. We start from the operator Equation (74) on the suitably transformed Fock space ${\tilde{\mathsf{\Phi }}}_0^{\left(def\right)}\subset \tilde{\mathsf{\Phi }}$ and the electromagnetic tension operator field definitions

$$\tilde{E}=-\partial \tilde{A}/\partial \tilde{t}-\nabla \tilde{\phi },\tilde{B}=\nabla \times \tilde{A}.$$

(80)

Applying the differential operators $\nabla \times$ and respectively ∇ to the operator expressions (80), one easily obtains that

$$\left(\nabla \times \tilde{B}-\partial \tilde{E}/\partial \tilde{t}\right){{|}_{{\tilde{\mathsf{\Phi }}}_0^{\left(def\right)}}={\displaystyle \frac{e}{2\pi }}\nabla {\int }_{{\mathbb{R}}^3}{d}^{3}y\mathsf{\Theta }(x;y,z){\tilde{p}}_0\left(z\right){\psi }^{+}\left(y\right)\psi \left(z\right)|}_{{\tilde{\mathsf{\Phi }}}_0^{\left(def\right)}}$$

(81)

and

$$⟨\nabla |\tilde{E}⟩{|}_{{\tilde{\mathsf{\Phi }}}_0^{\left(def\right)}}=-{\displaystyle \frac{e}{2\pi }}{\displaystyle \frac{\partial }{\partial \tilde{t}}}{\int }_{{\mathbb{R}}^3}{d}^{3}y\mathsf{\Theta }(x;y,z){\tilde{p}}_0\left(z\right){\psi }^{+}\left(y\right)\psi \left(z\right){{|}_{{\tilde{\mathsf{\Phi }}}_0^{\left(def\right)}},⟨\nabla |\tilde{B}⟩|}_{{\tilde{\mathsf{\Phi }}}_0^{\left(def\right)}}=0,$$

(82)

which are considered in the strong operator sense on the reduced Fock subspace ${\tilde{\mathsf{\Phi }}}_0^{\left(def\right)}\subset \tilde{\mathsf{\Phi }}$ with respect to the time reference frame $M_{\tilde{t}}^4$ of the electromagnetic vector field. It is also easy to calculate the strong operator relationship

$${\tilde{p}}_0{{|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}:=-\left(\partial \tilde{\phi }/\partial t+⟨\nabla |\tilde{A}⟩\right)|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}=0,$$

(83)

with respect to the limiting common reference frame $M_t^4,$ where we take into account that the space ${\tilde{\mathsf{\Phi }}}_0^{(lim)}:=lim{\tilde{\mathsf{\Phi }}}_0^{\left(def\right)}{|}_{\overline{t}=\tilde{t}\to t}$ at any $t\in {\mathbb{R}}_t.$

Now, we proceed to calculation of the strong Maxwell-type operator relationships (81) and (82) with respect to the common reference frame $M_t^4:$

$$\left(\nabla \times \tilde{B}-\partial \tilde{E}/\partial \tilde{t}\right){{|}_{{\tilde{\mathsf{\Phi }}}_0^{\left(def\right)}{|}_{\overline{t}=\tilde{t}\to t}}={\displaystyle \frac{e}{2\pi }}\nabla {\int }_{{\mathbb{R}}^3}{d}^{3}y\mathsf{\Theta }(x;y,z){\tilde{p}}_0\left(z\right){\psi }^{+}\left(y\right)\psi \left(z\right)|}_{{\tilde{\mathsf{\Phi }}}_0^{\left(def\right)}{|}_{\overline{t}=\tilde{t}\to t}}=0$$

(84)

and

$$\begin{array}{c}⟨\nabla |\tilde{E}⟩{|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}={\displaystyle \frac{i}{ħ}}[H_f,{\int }_{{\mathbb{R}}^3}{d}^{3}y\mathsf{\Theta }(x;y,z){\tilde{p}}_0\left(z\right){\psi }^{+}\left(y\right)\psi \left(z\right)]-\\ \\ -\frac{e}{2\pi }{\displaystyle \frac{\partial }{\partial \tilde{t}}}{\int }_{{\mathbb{R}}^3}{d}^{3}y\mathsf{\Theta }(x;y,z){\tilde{p}}_0\left(z\right){\psi }^{+}\left(y\right)\psi \left(z\right){{|}_{{\tilde{\mathsf{\Phi }}}_0^{\left(p\right)}{|}_{\overline{t}=\tilde{t}=t}}=e{\psi }^{+}\psi |}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}},\end{array}$$

(85)

where we used the corresponding evolution equations [7,46] for the operators ${\psi }^{+}\left(y\right),\psi \left(z\right):{\tilde{\mathsf{\Phi }}}_0^{(lim)}\to {\tilde{\mathsf{\Phi }}}_0^{(lim)},y,z\in {\mathbb{R}}^3.$ To further calculate the expression (84) with respect to the common reference frame $M_t^4,$ we need to make use of the strong operator relationships (69), from which we have that

$${\displaystyle \frac{\partial \tilde{E}}{\partial \tilde{t}}}{{|}_{\overline{t}=\tilde{t}\to t}={\displaystyle \frac{\partial \tilde{E}}{\partial t}}-{\displaystyle \frac{\partial \tilde{E}}{\partial \overline{t}}}|}_{\overline{t}=\tilde{t}\to t}={\displaystyle \frac{\partial \tilde{E}}{\partial t}}-{\displaystyle \frac{i}{ħ}}[{\tilde{H}}_f^{\left(int\right)},\tilde{E}]={\displaystyle \frac{\partial \tilde{E}}{\partial t}}+e{\psi }^{+}\alpha \psi .$$

(86)

Thus, from (86) and (84), one obtains

$$\left(\nabla \times \tilde{B}-\partial \tilde{E}/\partial t\right){{|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}=e{\psi }^{+}\alpha \psi |}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}$$

(87)

with respect to the common reference frame $M_t^4.$ Combining the strong operator relationships (85) and (87), we get

$$\left(\nabla \times \tilde{B}-\partial \tilde{E}/\partial t\right){|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}=e{\psi }^{+}{\alpha \psi |}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}},⟨\nabla |\tilde{E}⟩{{|}_{{\overline{\mathsf{\Phi }}}_0}=e{\psi }^{+}\psi |}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}$$

(88)

on the reduced Fock subspace ${\tilde{\mathsf{\Phi }}}_0^{(lim)}$. Together with the evident strong operator relationships

$$\nabla \times \tilde{E}+\partial \tilde{B}/\partial t=0,\nabla \times \tilde{B}=0$$

(89)

we can compile the complete system of quantum Maxwell equations with respect to the common reference frame $M_t^4.$ The reasoning presented above is reformulated as the final proposition, together with the described calculations.

Proposition 2. 

The electromagnetic vector field, interacting with an ambient electrically charged medium and quantized within the Dirac–Fock–Podolsky Lagrangian formalism, is a well-posed quantum field theory with a positive definite operator energy density defined on the strongly reduced Fock subspace, which is equal to the closure of the respectively quantized kernel of the auxiliary Lorentz operator condition. Moreover, the quantized electromagnetic field tensions satisfy the related quantum Maxwell equations defined on this strongly reduced Fock subspace, and are compatible with the corresponding quantized Dirac operator equations for the ambient charged medium.

Additionally, from the Heisenberg evolution Equation (66), an operator relationship for charge flux conservation can be easily obtained:

$$\partial \left(e{\psi }^{+}\psi \right)/\partial t+⟨\nabla |e{\psi }^{+}\alpha \psi ⟩=0,$$

(90)

in which the quantity $\rho :=e{\psi }^{+}\psi$ is interpreted as the operator charge density and the quantity $J:=e{\psi }^{+}\alpha \psi$ is naturally interpreted as the operator current density with respect to the common Minkowski reference frame $M_t^4.$ Hence, the strong operator Equation (88) can be rewritten, taking into account the above charge and current density definitions in the standard weak form of the Maxwell equations weak form, as follows:

$$\left(\nabla \times \tilde{B}-\partial \tilde{E}/\partial t\right){|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}{=J|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}},⟨\nabla |\tilde{E}⟩{{|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}=\rho |}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}},$$

(91)

considered on the Fock subspace ${\tilde{\mathsf{\Phi }}}_0^{(lim)},$ and determined by the time-invariant Lorenz operator constraint (83). Moreover, based on the strong Maxwell operator Equation (91) and the Lorenz constraint (83), one can easily derive the following linear strong operator wave equations:

$$\left({\partial }^2\tilde{\phi }/\partial t^2-\Delta \tilde{\phi }\right){{|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}=\rho |}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}$$

(92)

and

$$\left({\partial }^2\tilde{A}/\partial t^2-\Delta \tilde{A}\right){{|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}=J|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}$$

(93)

with respect to the common laboratory reference frame $M_t^4.$ This allows for calculation of the corresponding quantum electromagnetic potentials $(\tilde{\phi },\tilde{A}):{\tilde{\mathsf{\Phi }}}_0^{(lim)}\to {\left({\tilde{\mathsf{\Phi }}}_0^{(lim)}\right)}^4$ induced by the quantum charged fermionic field in the following analytical form:

$$\begin{array}{cc}\hfill \tilde{\phi }& ={\displaystyle \frac{1}{4\pi }}{\int }_{{\mathbb{R}}^3}{\displaystyle \frac{\rho (t^{\prime },y)d^{3}y}{|x-y|}}+{\displaystyle \frac{1}{4\pi }}{\int }_{{\mathbb{R}}^3}{\displaystyle \frac{{\rho }_0(t^{\prime },y)d^{3}y}{|x-y|}},\hfill \\ \hfill \tilde{A}& ={\displaystyle \frac{1}{4\pi }}{\int }_{{\mathbb{R}}^3}{\displaystyle \frac{J(t^{\prime },y)d^{3}y}{|x-y|}}+{\displaystyle \frac{1}{4\pi }}{\int }_{{\mathbb{R}}^3}{\displaystyle \frac{J_0(t^{\prime },y)d^{3}y}{|x-y|}},\hfill \end{array}$$

(94)

where $t^{\prime }:=t-|x-y|$$\in \mathbb{R}$ denotes the “retarded” temporal parameter, and the neutral density ${\rho }_0:{\tilde{\mathsf{\Phi }}}_0^{(lim)}\to {\tilde{\mathsf{\Phi }}}_0^{(lim)}$ and neutral current $J_0:{\tilde{\mathsf{\Phi }}}_0^{(lim)}\to {\tilde{\mathsf{\Phi }}}_0^{(lim)}$ operators are generated by the vacuum ${\tilde{\mathsf{\Phi }}}_0^{(lim)}$ and satisfy, by definition, the following strong operator constraints:

$${\rho }_0{{|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}=0,[{\tilde{H}}_f^{\left(int\right)}+iħ\partial /\partial t,{\rho }_0]|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}=0$$

(95)

on the reduced Fock space ${\tilde{\mathsf{\Phi }}}_0^{(lim)}.$ It is interesting to remark here that the neutral charge potential component

$${\tilde{\phi }}_0:={\displaystyle \frac{1}{4\pi }}{\int }_{{\mathbb{R}}^3}{\displaystyle \frac{{\rho }_0(t^{\prime },y)d^{3}y}{|x-y|}}$$

(96)

satisfies the quasi-stationary condition $\partial {\tilde{\phi }}_0(t,y)/\partial t\simeq 0,$ as due to the conditions (95) one can assume that $\partial {\rho }_0(t,y)/\partial t\simeq 0, ⟨\nabla \left|J_0\right(t,y)⟩\simeq 0$ for all $t\in \mathbb{R}$ and $y\in \mathbb{R},$ thus giving rise to the related Newton law

$$-\Delta {\tilde{\phi }}_0={\rho }_0,$$

(97)

which can be interpreted as a virtual gravity-type interaction source.

Considering the continuity relationship (90), the expressions (94) satisfy exactly the strong operator Lorenz subsidiary constraint

$$\left(\partial \tilde{\phi }/\partial t+⟨\nabla |\tilde{A}⟩\right){|}_{{\tilde{\mathsf{\Phi }}}_0^{(lim)}}=0,$$

(98)

coinciding with that of (83), regarding the common reference frame $M_t^4.$

Finally, we present an interesting observation derived from the reasoning presented above, as detailed in the Supplement, which relates to the characteristic strong wave operator relationship (92): if considered classically for the averaged potential $\overline{\phi }:=$$({\tilde{f}}_0|\phi |{\tilde{f}}_0)$ and charge density $\overline{\rho }:=({\tilde{f}}_0|\rho |{\tilde{f}}_0)$ quantities for $|{\tilde{f}}_0)\in {\tilde{\mathsf{\Phi }}}_0^{(lim)},$ it is possible to retrieve the complete system of classical Maxwell equations.

Conclusions

We demonstrated, within the Dirac and Fock multi-temporal paradigm, that the quantum electromagnetic field (both in vacuum and interacting with an ambient charged quantum medium), if considered on the time-invariant strongly reduced Fock subspace, presents a well-posed relativistic quantum field theory with a positive definite energy density. As a natural consequence from the presented reasoning and calculations, we re-derived the quantum Maxwell equations, which are strongly defined on the abovementioned reduced Fock subspace. Moreover, we confirmed the prediction of N.N. Bogolubov (Sr.) concerning a deformation of the strong Lorenz subsidiary constraint, imposed on quantum states of the related Fock space. It was also shown that the “weakened” Lorenz subsidiary condition—proposed by Gupta and Bleuler and widely accepted in modern quantum electrodynamics—is not time-invariant, and that the respectively reduced Hamiltonian operator is not both symmetric and positive definite; these issues have been strongly re-analyzed within the so-called BRST approach, eliminating many of the preceding physical inconsistencies. Moreover, we demonstrated that as the generally assumed averaged “weakened” Lorenz subsidiary condition is not time-invariant, it is thus insufficient to establish correspondence with the classical field, as claimed in [7], thus evidencing the poor physical motivation of this approach; in this context, the Hilbert space “indefinite” metric is similarly unsatisfactory.

4. Supplement: Spatial Electric Potential Energy Invariance and Retrieving the Maxwell Electromagnetic Equations

Consider, from the very beginning, a field-theoretic description of the classical electromagnetic vector field, based on a real smooth scalar charge density distribution $\rho \in C^1(M^4;\mathbb{R})$ on the Minkowski space $M^4\simeq$${\mathbb{R}}_{\tilde{t}}\times {\mathbb{R}}_x^3$ with coordinates $(\tilde{t},x)\in M^4.$ This generates, via the classical integral expression, the following electric retarded potential distribution:

$$\phi (\tilde{t},x)={\displaystyle \frac{1}{4\pi }}{\int }_{{\mathbb{R}}^3}{\displaystyle \frac{\rho (\tilde{t}-|x-y|,y)d^{3}y}{|x-y|}},$$

(99)

satisfying the well-known relativistic [6,7,8,10,11,47,48] Klein–Gordon–Fock wave propagation equation

$${\partial }^2\phi (\tilde{t},x)/\partial {\tilde{t}}^2-\Delta \phi (\tilde{t},x)=\rho (\tilde{t},x),$$

(100)

where we assume, for brevity, that $c=1.$ Consider the charge conservation property

$$dQ[\rho ;{\mathsf{\Omega }}_{\tilde{t}}]/d\tilde{t}=0$$

(101)

for any temporal parameter $\tilde{t}\in {\mathbb{R}}_{\tilde{t}},$ where

$$Q[\rho ;{\mathsf{\Omega }}_{\tilde{t}}]={\int }_{{\nleqq }_{\tilde{t}}}\rho (\tilde{t},x\left(\tilde{t}\right))d^{3}x\left(\tilde{t}\right)$$

(102)

denotes the charge inside any closed spatial region ${\mathsf{\Omega }}_{\tilde{t}}\subset {\mathbb{R}}_x^3$ with smooth boundary $\partial {\mathsf{\Omega }}_{\tilde{t}},$ whose spacetime pattern is formed by the evolution of the ambient charge. It is easy to calculate from the latter that the following partial differential relationship holds:

$$\partial \rho (\tilde{t},x)/\partial \tilde{t}+⟨\nabla |J(\tilde{t},x)⟩=0,$$

(103)

where $J(\tilde{t},x(\tilde{t})):=\rho (\tilde{t},x(\tilde{t}))u(\tilde{t},x(\tilde{t})$ denotes the corresponding electric current, generated by the charge velocity $u(\tilde{t},x(\tilde{t})):=dx(\tilde{t})/d\tilde{t}$ at internal points $(\tilde{t},x(\tilde{t}))\in {\mathsf{\Omega }}_{\tilde{t}}$ for all $\tilde{t}\in {\mathbb{R}}_{\tilde{t}}.$ Based on the charge conservation Equation (103) and the wave propagation Equation (100), we state the following instructive proposition.

Proposition 3. 

Consider the electric vector potential

$$\phi (\tilde{t},x)={\displaystyle \frac{e}{4\pi }}{\int }_{{\mathbb{R}}^3}{\displaystyle \frac{\rho (\tilde{t}-|x-y|,y\left)\right)d^{3}y}{|x-y|}}$$

(104)

at points $x=x(\tilde{t},x_0)$ $\in {\mathsf{\Omega }}_{\tilde{t}},$$x_0\in {\mathsf{\Omega }}_0,$ of an arbitrary closed domain ${\mathsf{\Omega }}_{\tilde{t}}\subset {\mathbb{R}}^3$ with smooth boundary $\partial {\mathsf{\Omega }}_{\tilde{t}}$ for all $\tilde{t}\in {\mathbb{R}}_{\tilde{t}}.$ Then, the following statements hold:

(a) 

the delayed temporal parameter ${\tilde{t}}^{\prime }=\tilde{t}-|x-y({\tilde{t}}^{\prime },y_0)|$ satisfies the wave propagation equation

$${\partial }^2{\tilde{t}}^{\prime }/\partial {\tilde{t}}^2-\Delta {\tilde{t}}^{\prime }=0$$

(105)

for any spatial evolution $y({\tilde{t}}^{\prime },y_0)\in {\mathbb{R}}_x^3,y_0\in {\mathbb{R}}^3,$ and all $\tilde{t}\in {\mathbb{R}}_{\tilde{t}};$

(b) 

the electric potential energy functional

$$I[\phi ;{\mathsf{\Omega }}_{\tilde{t}}]={\int }_{{\mathsf{\Omega }}_{\tilde{t}}}\phi (\tilde{t},x(\tilde{t}))d^{3}x(\tilde{t})$$

(106)

satisfies the temporal invariance condition

$$dI[\phi ;{\mathsf{\Omega }}_{\tilde{t}}]/d\tilde{t}=0$$

(107)

for all $\tilde{t}\in {\mathbb{R}}_{\tilde{t}},$ which is equivalent to the classical Lorenz subsidiary condition

$$\partial \phi (\tilde{t},x)/\partial \tilde{t}+⟨\nabla |A(\tilde{t},x)⟩=0,$$

(108)

where, by definition, $A(\tilde{t},x):=\phi (\tilde{t},x)dx(\tilde{t},x_0)/d\tilde{t},$$x(\tilde{t},x_0)\in {\mathsf{\Omega }}_{\tilde{t}},$ denotes the induced magnetic vector potential [49] for all temporal parameters $\tilde{t}\in {\mathbb{R}}_{\tilde{t}};$

(c) 

the classical Lorenz subsidiary condition (108), jointly with the differential relationships (100) and (103), reduces to the complete classical system of Maxwell equations:

$$\begin{array}{cc}\hfill \nabla \times B(\tilde{t},x)& =J(\tilde{t},x)+\partial E(\tilde{t},x)/\partial \tilde{t},⟨\nabla |E(\tilde{t},x)⟩=\rho (\tilde{t},x),\hfill \\ \hfill \partial B(\tilde{t},x)/\partial \tilde{t}& =-\nabla \times E(\tilde{t},x),⟨\nabla |B(\tilde{t},x)⟩=0,\hfill \end{array}$$

(109)

satisfied at all $(x,\tilde{t})\in M^4.$

Proof. 

(a) It is easy to observe that the first derivatives of the delayed parameter ${\tilde{t}}^{\prime }=\tilde{t}-|x-y({\tilde{t}}^{\prime },y_0)|$ are equal to the following expressions:

$${\displaystyle \frac{\partial {\tilde{t}}^{\prime }}{\partial \tilde{t}}}=U(\tilde{t},x),{\displaystyle \frac{\partial {\tilde{t}}^{\prime }}{\partial x}}=-U(\tilde{t},x){\displaystyle \frac{x-y({\tilde{t}}^{\prime },y_0)}{|x-y({\tilde{t}}^{\prime },y_0)|}},$$

(110)

where the scalar multiplier $U(\tilde{t},x):=(1-$$\frac{⟨x-y({\tilde{t}}^{\prime },y_0)|\dot{y}({\tilde{t}}^{\prime },y_0)⟩}{|x-y({\tilde{t}}^{\prime },y_0)|}{)}^{-1},\dot{y}({\tilde{t}}^{\prime },y_0):=\partial y({\tilde{t}}^{\prime },y_0)/\partial {\tilde{t}}^{\prime },$ is well-defined for all $(\tilde{t},x)\in M^4.$ Calculating the second temporal ${\partial }^2{\tilde{t}}^{\prime }/\partial {\tilde{t}}^2$ and spatial gradient $⟨\nabla |\nabla {\tilde{t}}^{\prime }⟩$ derivatives, one easily proves their equality, thus stating the wave propagation Equation (105).

(b) Taking into account both the charge invariance (103) and the electric potential relationships,

$$\phi (\tilde{t},x\left(t\right))={\displaystyle \frac{e}{4\pi }}{\int }_{{\mathbb{R}}^3}{\displaystyle \frac{\rho (\tilde{t}-|x(\tilde{t},x_0)-y|,y)d^{3}y}{|x(\tilde{t},x_0)-y|}}$$

(111)

one can obtain the following potential energy invariance equations by means of direct calculation:

$$\begin{array}{c}dI[\phi ;{\mathsf{\Omega }}_{\tilde{t}}]/d\tilde{t}=\frac{d}{d\tilde{t}}{\int }_{{\mathsf{\Omega }}_{\tilde{t}}}{d}^{3}x(t,x_0)\frac{e}{4\pi }{\int }_{{\mathbb{R}}^3}\frac{\rho (\tilde{t}-|x(t,x_0)-y|,y)d^{3}y}{\left|x\right(t)-y|}=\\ =\frac{e}{4\pi }{\int }_{{\mathbb{R}}^3}{d}^{3}z\frac{d}{d\tilde{t}}{\int }_{{\mathsf{\Omega }}_0}\frac{d^3{x}_0}{\left|z\right|}\rho (\tilde{t}-\left|z\right|,x(\tilde{t},x_0)-z\left)\mathrm{Jac}\right(x(t,x_0)=\\ =\frac{e}{4\pi }{\int }_{{\mathbb{R}}^3}{d}^{3}z{\int }_{{\mathsf{\Omega }}_0}\frac{d^3{x}_0}{\left|z\right|}\left[\partial \rho (\tilde{t}-\left|z\right|,x(\tilde{t},x_0)-z)/\partial \tilde{t}+\right.\\ +⟨\nabla \left|\rho \right(\tilde{t}-\left|z\right|,x(\tilde{t},x_0)-z)\dot{x}(\tilde{t},x_0)⟩\mathrm{Jac}\left(x(t,x_0)\right]=\\ =\frac{e}{4\pi }{\int }_{{\mathbb{R}}^3}{d}^{3}z{\int }_{{\mathsf{\Omega }}_t}\frac{d^{3}x(\tilde{t},x_0)}{\left|z\right|}\left[\partial \rho (\tilde{t}-\left|z\right|,x(\tilde{t},x_0)-z)/\partial \tilde{t}+\right.\\ +⟨\nabla \left|J\right(\tilde{t}-\left|z\right|,x(\tilde{t},x_0)-z)⟩]=0,\end{array}$$

(112)

where, by definition, we have $\mathrm{Jac}(x(\tilde{t},x_0)):=|det\partial x(\tilde{t},x_0)/\partial x_0|$—the usual Jacobian of the transformation ${\mathsf{\Omega }}_0\ni x_0\to x(\tilde{t},x_0)\in {\mathsf{\Omega }}_{\tilde{t}}$ for all $x_0\in {\mathsf{\Omega }}_0,\tilde{t}\in {\mathbb{R}}_{\tilde{t}}.$ The expression (112) can be re-calculated as

$$\begin{array}{c}dI[\phi ;{\mathsf{\Omega }}_{\tilde{t}}]/d\tilde{t}=\frac{d}{d\tilde{t}}{\int }_{{\mathsf{\Theta }}_{\tilde{t}}}\phi (\tilde{t},x(\tilde{t},x_0))d^{3}x(\tilde{t},x_0)=\\ =\frac{d}{d\tilde{t}}{\int }_{{\mathsf{\Omega }}_0}\phi (\tilde{t},x(\tilde{t},x_0))\mathrm{Jac}(x(\tilde{t},x_0))d^3{x}_0=\\ ={\int }_{{\mathsf{\Omega }}_0}\left[\partial \phi (\tilde{t},x(\tilde{t},x_0))/\partial \tilde{t}+⟨\nabla |u(\tilde{t},x)\phi (\tilde{t},x(\tilde{t},x_0))⟩\mathrm{Jac}(x(\tilde{t},x_0))\right]d^3{x}_0=\\ ={\int }_{{\mathsf{\Omega }}_{\tilde{t}}}\left[\partial \phi (\tilde{t},x(\tilde{t},x_0))/\partial \tilde{t}+⟨\nabla |u(\tilde{t},x)\phi (\tilde{t},x(\tilde{t},x_0))⟩\right]d^{3}x(\tilde{t},x_0)=0,\end{array}$$

(113)

which is exactly equivalent to the condition (108) at all points $(x,\tilde{t})\in M^4.$

(c) Having defined the electric field tension $E(\tilde{t},x):=-\nabla \phi (\tilde{t},x)-\partial A(\tilde{t},x)/\partial \tilde{t}$ and the magnetic field tension $B(\tilde{t},x):=\nabla \times A(\tilde{t},x)$ at all points $(\tilde{t},x)\in M^4,$ upon applying the curl operation $\nabla \times$, one easily obtains

$$\begin{array}{c}\nabla \times E(\tilde{t},x)=\nabla \times (-\nabla \phi (\tilde{t},x)-\partial A(\tilde{t},x)/\partial \tilde{t})=-\partial B(\tilde{t},x)/\partial \tilde{t},\\ \nabla \times B(\tilde{t},x)=\nabla \times (\nabla \times A(\tilde{t},x))=\nabla ⟨\nabla |A(\tilde{t},x)⟩-\Delta A.\end{array}$$

(114)

Taking the Lorenz condition (108) into account, one obtains

$$\begin{array}{c}\nabla \times E(\tilde{t},x)=-\partial B(\tilde{t},x)/\partial \tilde{t},\nabla \times B(\tilde{t},x):=\nabla \times (\nabla \times A(\tilde{t},x))=\\ =\partial E(\tilde{t},x)/\partial \tilde{t}+[{\partial }^{2}A(\tilde{t},x)/\partial {\tilde{t}}^2-\Delta A(\tilde{t},x)].\end{array}$$

(115)

Similarly, taking into account the wave propagation Equation (100) jointly with the charge continuity Equation (103), one obtains

$$⟨\nabla |{\partial }^{2}A(\tilde{t},x)/\partial {\tilde{t}}^2-\Delta A(\tilde{t},x)⟩=⟨\nabla |J(\tilde{t},x)⟩$$

(116)

which, upon substituting into (115), gives rise to the system

$$\begin{array}{c}\nabla \times E(\tilde{t},x)=-\partial B(\tilde{t},x)/\partial \tilde{t},\\ \nabla \times B(\tilde{t},x)=\partial E(\tilde{t},x)/\partial \tilde{t}+J(\tilde{t},x),\end{array}$$

(117)

that coincides exactly with the classical Faraday and Biot–Savart laws in the Minkowski spacetime $M^4.$ Similarly, having applied the gradient operation ∇ to the vector magnetic and electric potentials, from the wave propagation Equation (100) and the Lorenz condition (108), one simply derives the corresponding boundary conditions

$$⟨\nabla |B(\tilde{t},x)⟩=0,⟨\nabla |E(\tilde{t},x)⟩=\rho (\tilde{t},x)$$

(118)

for the above evolution Equation (117) at all points $(\tilde{t},x)\in M^4.$ Combining the differential relationships (117) with those of (118), we obtain the classical Maxwell equations, thus proving the proposition. □