Covariant Hamilton-Jacobi Formulation of Electrodynamics via the Polysymplectic Reduction and Its Relation to the Canonical Hamilton-Jacobi Theory

1. Introduction

The Hamilton-Jacobi (HJ) theory was originally formulated for classical mechanics and it has provided a fruitful framework for the analysis of classical equations of motion and evolved into a branch of applied mathematics [1]. It also played a heuristic role in the discovery of Schrödinger’s quantum mechanics and its relation with the classical mechanics. An extension of HJ theory to field theory or the calculus of variations of multiple integral functionals was pioneered by Vito Volterra [2]. His work has inspired multiple developments both in the theory of functionals [3,4] and geometric formulations of the variational problems, and the corresponding geometric Hamilton-Jacobi formulations, based on the notion of the Poincaré-Cartan form and its Lepage equivalents (see, e.g., [5,6,7] and the references therein). The earlier development of quantization of the electromagnetic field in 1925 has lead to the notion of fields as mechanical systems with infinitely many degrees of freedom and the corresponding canonical Hamiltonian formalism and canonical (infinite dimensional) HJ theory associated with it [8,9]. Unlike the former geometric approach to the generalization of HJ formulation to field theory, where all spacetime variables are treated equally as independent parameters of the variational problem and the resulting equations are formulated in terms of partial derivatives, the latter approach requires a distinction between the time variable and spacial variables, and it is formulated in terms of functionals of field configurations and their variational derivatives. The relation between these two approaches to the HJ formulation of field theories was not understood [10] until the clarifying result in [11] using the example of scalar field theory. In the end of this paper, it will be demonstrated how this result extends also to the case of the electromagnetic field.

Recently, the covariant HJ equation was derived for the teleparallel equivalent of General Relativity in the Palatini formulation [12]. This derivation employs the generalized Dirac-Bergmann algorithm developed in [13] to analyze second-class constraints within the covariant Hamiltonian framework of De Donder and Weyl [5,6,14,15,16], also referred to by various names such as polysymplectic, finite-dimensional, and multisymplectic. The analysis builds upon the extension of Poisson brackets to the De Donder–Weyl (DDW) formalism for field theories [17,18,19,20,21,22,23,24].

The HJ formulation is well known as the classical or geometric optics approximation of the quantum Schrödinger equation. In fact, the arguments grounded in the HJ theory historically led to the discovery of the Schrödinger equation in quantum mechanics [25] and they also underpin the interpretation of quantum mechanics known as "Bohmian mechanics" or the "causal interpretation" [26,27,28,29]. In General Relativity, the canonical HJ formulation [30] played a pivotal role in the heuristic development of the Wheeler-DeWitt equation in the canonical quantum gravity [31].

The covariant HJ framework in field theory introduced by De Donder and Weyl [5,6,14,15,16], which treats spatial and temporal variables on equal footing, has inspired the development of precanonical quantization in [32,33,34]. This approach has been applied to quantum gauge theories in [35,36,37] and to quantum general relativity, both in metric variables [38,39,40,41] and vielbein variables [42,43,44,45,46]. Other efforts to use the covariant HJ theory as a foundation of field quantization are also documented in [47,48]. The earlier work [12] was motivated by recent applications of precanonical quantization to the teleparallel equivalent of General Relativity in [49,50,51].

Both the covariant HJ theories and precanonical quantization treat spacetime variables equally. Consequently, they describe classical and quantum fields using eikonal and wave functions on a finite-dimensional configuration space comprising of spacetime and field variables. The spacetime variables serve as a multidimensional analogue of time in mechanics, while the field variables play the role of generalized coordinates. This raises a critical question regarding the relationship between the finite-dimensional descriptions of these approaches and the canonical description in the infinite-dimensional field configuration space. This relationship has been explored for HJ theories in [11,52] and for the precanonical quantization compared to the Schrödinger functional framework in quantum field theory of scalar fields in [53,54], gauge fields in [37] and scalar fields in curved spacetime in [55,56,57].

In this paper, we derive the covariant DDW HJ equation for classical Electrodynamics by following the approach of the previous work in [12]. One motivation is to test the method of the constraints analysis within the DDW Hamiltonian framework in the well understoon case of classical Electrodynamics. Another motivation is to formulate the covariant DDW HJ equation for the electromagnetic field, which can serve as a tool for numerical integration near caustics and as a consistency check for the classical limit of precanonical quantization of gauge fields. This study not only enhances the mathematical understanding of the Maxwell equations but it also provides a foundation for further investigations into the covariant HJ theories for other relativistic fields. Moreover, the connection between the polysymplectic formalism and the geometric aspects of field theory may also deepen our understanding of the interplay between symmetry, dynamics and topology in classical field theories and their quantum counterparts alike.

The paper is organized as follows. In Section 2, we present the Palatini-like Lagrangian formulation of Electrodynamics. Section 3 examines the DDW Hamiltonian framework for the Palatini-like Lagrangian that results in a singular DDW Hamiltonian system with second-class constraints. In Section 4, we compute the generalized Dirac brackets and demonstrate how the constrained DDW Hamiltonian system reduces to an unconstrained one on a reduced polymomentum phase space equipped with the corresponding reduced polysymplectic structure. Leveraging this reduced structure, we derive the covariant DDW HJ equation in Section 5, building on earlier geometric formulations of HJ theories in field theory in the multi-, poly-, and k-symplectic contexts. Then, in Section 6, we show how the canonical HJ equation in variational derivatives is derived from the covariant partial-derivative DDW HJ equation after the space+time splitting. Finally, Section 7 provides conclusions and discusses perspectives for future research.

2. The First-Order Formulation

In the conventional formulation of classical electrodynamics, the Lagrangian is expressed as follows:

$$L=-{\displaystyle \frac{1}{4}}{F}_{\mu \nu }{F}^{\mu \nu }-j^{\mu }{A}_{\mu },$$

(1)

where the field strength tensor $F_{\mu \nu }$ is defined in terms of the vector potential $A_{\mu }$ as

$$F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }.$$

(2)

Instead, in this paper, we adopt a Palatini-like Lagrangian

$$L={\displaystyle \frac{1}{4}}{P}^{\mu \nu }\left(P_{\mu \nu }-2F_{\mu \nu }\right)-j^{\mu }{A}_{\mu },$$

(3)

where the electromagnetic potential $A_{\mu }$ and and the antisymmetric tensor $P_{\mu \nu }=-P_{\nu \mu }$ are treated as independent field variables.

The variation of $P_{\mu \nu }$ yields

$$P_{\mu \nu }=F_{\mu \nu },$$

(4)

while the variation of $A_{\mu }$ leads to

$${\partial }_{\mu }{P}^{\mu \nu }=j^{\nu }.$$

(5)

Substituting $P_{\mu \nu }=F_{\mu \nu }$ into (2) we recover the standard Maxwell equations ${\partial }_{\mu }{F}^{\mu \nu }=j^{\nu }$ which are usually derived from the conventional Lagrangian (1).

Here we choose the Palatini-like formulation because it closely aligns with the formalism used in the context of the theories of gravity and their quantization in [12,43,46,50,51]. The primary motivation of this approach is to demonstrate that the analysis of the singular first-order formulation and its associated constraints, and by using the Poisson-Gerstenhaber brackets of differential forms and the generalized Dirac brackets of forms correctly reproduces the DDW HJ equation for the electromagnetic field which is known from earlier papers [6,16,58].

3. The De Donder–Weyl Hamiltonian Formulation

Using the procedure of the DDW Hamiltonian formulation we first obtain the expressions of the polymomenta associated with the independent field variables $A_{\mu },P^{\mu \nu }$

$$\begin{array}{cc}\hfill p_{A_{\nu }}^{\mu }& ={\displaystyle \frac{\partial L}{\partial {\partial }_{\mu }{A}_{\nu }}}=-P^{\mu \nu },\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill p_{P^{\mu \nu }}^{\alpha }& ={\displaystyle \frac{\partial L}{\partial {\partial }_{\alpha }{P}^{\mu \nu }}}=0.\hfill \end{array}$$

(7)

This results in a constrained DDW Hamiltonian system with the constraints

$$\begin{array}{cc}\hfill C_{A_{\nu }}^{\mu }& =p_{A_{\nu }}^{\mu }+P^{\mu \nu }\approx 0,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill C_{P^{\mu \nu }}^{\alpha }& =p_{P^{\mu \nu }}^{\alpha }\approx 0.\hfill \end{array}$$

(9)

Note that a consequence of (8) is that

$$p_{A_{\nu }}^{\mu }+p_{A_{\mu }}^{\nu }\approx 0.$$

(10)

We use here the standard Dirac notation ≈ for weak equalities on the surface of constraints (8), (9).

The DDW Hamiltonian function

$$\begin{array}{cc}\hfill H& =p_{A_{\nu }}^{\mu }{\partial }_{\mu }{A}_{\nu }+p_{P^{\mu \nu }}^{\alpha }{\partial }_{\alpha }{P}^{\mu \nu }-L\approx -{\displaystyle \frac{1}{2}}{P}^{\mu \nu }{F}_{\mu \nu }-{\displaystyle \frac{1}{4}}{P}^{\mu \nu }(P_{\mu \nu }-2F_{\mu \nu })+j^{\mu }{A}_{\mu }\hfill \\ \hfill & =-{\displaystyle \frac{1}{4}}{P}^{\mu \nu }{P}_{\mu \nu }+j^{\mu }{A}_{\mu }\approx -{\displaystyle \frac{1}{4}}{p}_{A_{\nu }}^{\mu }{p}_{\mu }^{A_{\nu }}+j^{\mu }{A}_{\mu }.\hfill \end{array}$$

(11)

Following the generalization of the Bergaman-Dirac analysis of constraints to the DDW Hamiltonian systems [13] and using the notation

$${\upsilon }_{\alpha }={\displaystyle \frac{\partial }{\partial x^{\alpha }}}⌟(d{x}^0\wedge d{x}^1\wedge ...\wedge d{x}^{n-1})={(-1)}^{\alpha }d{x}^0\wedge ...\widehat{d{x}^{\alpha }}...\wedge d{x}^{n-1}$$

(12)

for the basis of $(n-1)-$forms in n-dimensions, we introduce the $(n-1)-$forms of constraints

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_{A_{\nu }}& =C_{A_{\nu }}^{\alpha }{\upsilon }_{\alpha },\hfill \\ \hfill C_{P^{\mu \nu }}& =C_{P^{\mu \nu }}^{\alpha }{\upsilon }_{\alpha },\hfill \end{array}\end{array}$$

(13)

and calculate their brackets.

The brackets of forms are defined by the polysymplectic structure (first introduced in [17,19,20,21]) on the unconstrained polymomentum analogue of phase space, i.e. the space of field variables $(A,P)$ and their respective polymomenta $p_A^{\mu },p_P^{\mu }$, namely,

$$\mathsf{\Omega }=dA\wedge d{p}_A^{\alpha }\wedge {\upsilon }_{\alpha }+dP\wedge d{p}_P^{\alpha }\wedge {\upsilon }_{\alpha }.$$

(14)

As it is explained in [21], the polysymplectic structure in (14) represents the equivalence class of forms modulo the top horizontal n-forms $Hd{x}^0\wedge d{x}^1\wedge ...\wedge d{x}^{n-1}$ which depend on the specific Lagrangian L and the connection in the extended polymomentum phase space viewed as a bundle over the n–dimensional base of independent (spacetime) variables $x^{\mu }$. It was shown in [17,19,20] that this structure controls the dynamics of classical fields analogously to the symplectic structure controlling the phase space trajectories in classical mechanics.

The polysymplectic structure $\mathsf{\Omega }$ maps a form C of degree $(n-1)$ to the vector field ${\chi }_C$:

$${\chi }_C⌟\mathsf{\Omega }=dC,$$

(15)

and the bracket of two $(n-1)$-forms is defined as follows:

$$\left\{[C_1,C_2]\right\}:={\chi }_{C1}⌟d{C}_2.$$

(16)

It is easy to see that the bracket of two $(n-1)$-forms is a $(n-1)$-form. Note that this definition produces brackets for a limited class of forms called Hamiltonian forms. The Lie algebra structure defined by this bracket is embedded into a larger bi-graded (Gerstenhaber algebra) structure defined on forms of all degrees from 0 to $(n-1)$ [17,19,20,21,22,23,24].

By a direct calculation we obtain the following brackets of the forms of constraints

$$\begin{array}{cc}\hfill C_{P^{\mu \nu }{A}_{\sigma }}=\left\{[C_{P^{\mu \nu }},C_{A_{\sigma }}]\right\}& ={\upsilon }_{[\mu }{\delta }_{\nu ]}^{\sigma },\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill C_{A{A}^{\prime }}=\left\{[C_A,C_{A^{\prime }}]\right\}& =0,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill C_{P{P}^{\prime }}=\left\{[C_P,C_{P^{\prime }}]\right\}& =0.\hfill \end{array}$$

(19)

The right hand side of (17) does not vanish on the surface of constraints. Hence we have here the DDW analogue of the second-class constraints according to the Dirac classification [59,60,61].

4. The Dirac Brackets

A generalization of the Dirac bracket in the context of constrained DDW Hamiltonian systems with second-class constraints has been proposed in [13]. This approach has been used in precanonical quantization of Einstein gravity in  [42,43,44,45,46], in different models of gravity [71–73] and gauge theories [74], and in the previous work on the covariant HJ formulation of the teleparallel equivalent of general relativity [12] and its precanonical quantization in [49,50,51].

The generalized Dirac bracket of two $(n-1)$– forms A and B is defined as follows:

$${\left\{[A,B]\right\}}^{*}:=\left\{[A,B]\right\}-\sum _{U,V}\left\{[A,C_U]\right\}•C_{UV}^{\sim 1}\wedge \left\{[C_V,B]\right\},$$

(20)

where

$$A•B:={*}^{-1}(*A\wedge *B),$$

(21)

* is the Hodge star, the indices $U,V$ enumerate the primary constraints, i.e., in the present case, they run over all the indices of $A_{\mu }$ and $P^{\mu \nu }$. Here, $C^{\sim 1}=C_{\alpha }^{\sim 1}d{x}^{\alpha }$ denotes the pseudoinverse matrix defined by the relation

$$C•C^{\sim 1}\wedge C=C,$$

(22)

which generalizes the Moore-Penrose pseudoinverse [75] to the case of matrices whose components are exterior forms. Note that the distributive law for ∧ and • products is that the wedge product ∧ acts first.

For the matrix of constraints

$$C_{UV}=\left[\begin{array}{ccc}{C}_{A{A}^{\prime }}=0& C_{AP}& \\ -{\left(C_{AP}\right)}^T& C_{P{P}^{\prime }}=0\end{array}\right],$$

(23)

we obtain the pseudoinverse matrix

$$C_{UV}^{\sim 1}=\left[\begin{array}{ccc}{C}_{A{A}^{\prime }}^{\sim 1}=0& C_{AP}^{\sim 1}& \\ -{\left(C_{AP}^{\sim 1}\right)}^T& C_{P{P}^{\prime }}^{\sim 1}=0\end{array}\right],$$

(24)

whose components satisfy the relations

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sum _P{C^{\mu }}_{AP}{C}_{\mu P{A}^{\prime }}^{\sim 1}& ={\delta }_{A{A}^{\prime }},\hfill \\ \hfill \sum _A{C^{\mu }}_{PA}{C}_{\mu A{P}^{\prime }}^{\sim 1}& ={\delta }_{P{P}^{\prime }}.\hfill \end{array}\end{array}$$

(25)

The calculation of generalized Dirac brackets leads to the result for brackets between $(n-1)-$forms of polymomenta $p_A=p_A^{\alpha }{\upsilon }_{\alpha }$, $p_P=p_P^{\alpha }{\upsilon }_{\alpha }$ and $(n-1)-$forms of field variables $A{\upsilon }_{\alpha },P{\upsilon }_{\alpha }$

$$\begin{array}{cc}\hfill {\left\{[p_A,A^{\prime }{\upsilon }_{\alpha }]\right\}}^{*}& =\left\{[p_A,A^{\prime }{\upsilon }_{\alpha }]\right\}={\delta }_{A{A}^{\prime }}{\upsilon }_{\alpha },\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\left\{[p_P,P^{\prime }{\upsilon }_{\alpha }]\right\}}^{*}& ={\delta }_{P{P}^{\prime }}{\upsilon }_{\alpha }-\sum _{A,P^{″}}\left\{[p_P,C_A]\right\}•C_{A{P}^{″}}^{\sim 1}\wedge \left\{[C_{P^{\prime \prime }},P^{\prime }{\upsilon }_{\alpha }]\right\}=0,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\left\{[P^{\mu \nu }{\upsilon }_{\sigma },A_{\alpha }{\upsilon }_{\tau }]\right\}}^{*}& =-\sum _{P^{\prime },A^{\prime }}\left\{[P^{\mu \nu }{\upsilon }_{\sigma },C_{P^{\prime }}]\right\}•C_{P^{\prime }{A}^{\prime }}^{\sim 1}\wedge \left\{[C_{A^{\prime }},A_{\alpha }{\upsilon }_{\tau }]\right\}=C_{\tau P^{\mu \nu }{A}_{\alpha }}^{\sim 1}{\upsilon }_{\sigma },\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\left\{[p_P,A{\upsilon }_{\alpha }]\right\}}^{*}& =0.\hfill \end{array}$$

(29)

Hence the original polymomentum phase space of variables $(A,P,p_A,p_P)$ reduces to the space of A and $p_A$ equipped with the reduced polysymplectic structure given by

$${\mathsf{\Omega }}_R=d{A}_{\mu }\wedge d{p}_{A_{[\mu }}^{\alpha }\wedge {\upsilon }_{\alpha ]},$$

(30)

where the antisymmetrization appears as a consequence of the projection to the subspace of polymomenta which satisfy the constraint in (10).

The DDW Hamiltonian on this reduced polysymplectic space takes the form

$$H^{*}=-{\displaystyle \frac{1}{4}}{p}_{A_{\mu }}^{\alpha }{p_{\alpha }}^{A_{\mu }}+j^{\mu }{A}_{\mu }.$$

(31)

The DDW Hamiltonian field equations in terms of the Poisson bracket defined by the reduced polysymplectic structure take the form

$$\begin{array}{cc}\hfill d•p_{A_{[\mu }}^{\alpha }{\upsilon }_{\alpha ]}& =\left\{[H^{*},p_{A_{[\mu }}^{\alpha }{\upsilon }_{\alpha ]}]\right\}=-{\displaystyle \frac{\partial H^{*}}{\partial A_{\mu }}}=-j^{\mu },\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill d•A_{[\mu }{\upsilon }_{\alpha ]}& =\left\{[H^{*},A_{[\mu }{\upsilon }_{\alpha ]}]\right\}={\displaystyle \frac{\partial H^{*}}{\partial p_{A_{\mu }}^{\alpha }}}=-{\displaystyle \frac{1}{2}}{p_{\alpha }}^{A_{\mu }},\hfill \end{array}$$

(33)

where the operation $d•$ acts on $(n-1)-$forms $N=N^{\mu }{\upsilon }_{\mu }$ as follows

$$d•N={\partial }_{\alpha }{N}^{\mu }d{x}^{\alpha }•{\upsilon }_{\mu }={\partial }_{\alpha }{N}^{\alpha }.$$

(34)

Hence, equation (32) reproduces the Maxwell equation in the form

$${\partial }_{\alpha }{p}_{A_{\mu }}^{\alpha }=-j^{\mu },$$

(35)

and equation (33) reproduces the definition of the field strength represented by polymomenta ${p_{\alpha }}^{A_{\mu }}$ in terms of the vector potential (cf. (6))

$${\partial }_{\alpha }{A}_{\mu }-{\partial }_{\mu }{A}_{\alpha }=-{p_{\alpha }}^{A_{\mu }}.$$

(36)

Note that the symmetric part of $p_{\alpha }^{A_{\mu }}$ is vanishing according to the constraint in (8).

5. The Covariant Hamilton-Jacobi Equation

The geometrical formulation of the DDW HJ theory [62,63,64,65] allows us to write the corresponding equation based on the reduced polysymplectic structure (30) derived in the previous section

$$\begin{array}{c}\hfill {\partial }_{\mu }{S}^{\mu }+H^{*}\left(p_{A_{\mu }}^{\alpha }={\displaystyle \frac{\partial S^{\alpha }}{\partial A_{\mu }}},A_{\mu }\right)=0.\end{array}$$

(37)

Therefore, the covariant DDW HJ equation for the electromagnetic field takes the form

$${\partial }_{\mu }{S}^{\mu }-{\displaystyle \frac{1}{4}}{\displaystyle \frac{\partial S^{\mu }}{\partial A_{\nu }}}{\displaystyle \frac{\partial S_{\mu }}{\partial A^{\nu }}}+j^{\mu }{A}_{\mu }=0.$$

(38)

It is supplemented by the equation

$$p_{A_{\nu )}}^{(\mu }={\displaystyle \frac{\partial S^{(\mu }}{\partial A_{\nu )}}}=0,$$

(39)

which follows from the constraint (10) and the “embedding condition” (cf. eq. (36))

$$p_{A_{\nu ]}}^{[\mu }={\displaystyle \frac{\partial S^{[\mu }}{\partial A_{\nu ]}}}=-F^{\mu \nu }.$$

(40)

The result has been derived earlier in [16,58] using other methods based on the Lagrangian (1). In this case, the only constraint is (10) and the DDW Hamiltonian function is given by the last expression in (11). The covariant Hamilton-Jocobi equation can then be written in form (37), which reproduces solutions of Maxwell equation when the functions $S^{\mu }$ satisfy the constraint in (39) and the solutions are calculated from the solution of the HJ equation (38) using the embedding condition (40).

Thus we have shown that the formalism of the analysis of constraints in the DDW theory based on the generalization of Poisson and Dirac brackets and the generalization of Dirac–Bergmann analysis of constraints leads to the correct results in the well-understood and verifiable case of classical electromagnetic field.

6. The Relation with the Canonical Hamilton-Jacobi Formulation

6.1. Canonical Hamilton-Jacobi Equation for Electrodynamics

In the canonical Hamiltonian formalism we split the spacetime variables $x^{\mu }$ into the space variables $\mathbf{x}$ and the time variable $x^0=t$. Then, canonical momenta derived from the Lagrangian (1) are

$$p_{A_0}\left(\mathbf{x}\right)=0,p_{A_i}\left(\mathbf{x}\right)=-F^{0i}\left(\mathbf{x}\right),$$

(41)

and the canonical Hamiltonian has form

$$\mathbf{H}=\int d\mathbf{x}\left({\displaystyle \frac{1}{2}}{\left(F^{0i}\right)}^2+{\displaystyle \frac{1}{4}}{F}^{ij}{F}_{ij}+F^{0i}{\partial }_i{A}_0+j^{\mu }{A}_{\mu }\right).$$

(42)

The integration by parts in the last term leads to

$$-\int d\mathbf{x}{A}_0\left(\mathbf{x}\right){\partial }_i{F}^{i0}\left(\mathbf{x}\right),$$

(43)

so that the non-dynamical $A_0$ appears as the Lagrange multiplier and fixes the Gauss law constraint

$${\partial }_i{F}^{i0}\left(\mathbf{x}\right)=j^0.$$

(44)

The canonical HJ equation is formulated for a functional of field configurations at a fixed moment of time $\mathbf{S}(\left[A^i\left(\mathbf{x}\right)\right],t)$ such that

$$p_{A_i}\left(\mathbf{x}\right)={\displaystyle \frac{\delta \mathbf{S}}{\delta A_i\left(\mathbf{x}\right)}}$$

(45)

and it, thus, takes the form

$${\partial }_t\mathbf{S}=\int d\mathbf{x}\left({\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\delta \mathbf{S}}{\delta A_i\left(\mathbf{x}\right)}}\right)}^2+{\displaystyle \frac{1}{4}}{F}^{ij}{F}_{ij}+j^i{A}_i\right).$$

(46)

The Gauss law in HJ formulation has the form

$${\partial }_i{\displaystyle \frac{\delta \mathbf{S}}{\delta A_i\left(\mathbf{x}\right)}}=j^0.$$

(47)

The solutions of the Maxwell equations are related to the solutions of (46), (47) by means of the embedding condition

$${\displaystyle \frac{\delta \mathbf{S}}{\delta A_i\left(\mathbf{x}\right)}}=F^{i0}\left(\mathbf{x}\right).$$

(48)

An earlier appearence of this form of the HJ formulation is found in Max Born’s paper [66] and also in the context of the David Bohm’s causal interpretation in quantum field theory [67,68].

6.2. Canonical Hamilton-Jacobi from the Covariant Hamilton-Jacobi Equation

We have shown in Section 5 that the covariant DDW HJ equation for the electromagnetic field has the form (38) (see also [14,16,58,69,70]):

$${\partial }_{\mu }{S}^{\mu }-{\displaystyle \frac{1}{4}}{\displaystyle \frac{\partial S^{\mu }}{\partial A_{\nu }}}{\displaystyle \frac{\partial S_{\mu }}{\partial A^{\nu }}}+j^{\mu }{A}_{\mu }=0,$$

(49)

where $S^{\mu }$ are functions of the field components $A_{\nu }$ and spacetime coordinates $x^{\mu }$: $S^{\mu }=S^{\mu }(A,x)$. The solutions of the field equatons $A_{\mu }\left(x\right)$ are related to the solutions of the DDW HJ equation in (38) via the embedding condition [69]

$$F^{\mu \nu }\left(x\right)=-{\left.\left({\displaystyle \frac{\partial S^{[\mu }}{\partial A_{\nu ]}}}\right)\right|}_{\sigma },$$

(50)

where ${|}_{\sigma }$ means the restriction to a particular field configuration $\sigma$ which is a section $A_{\mu }\left(x\right)$ in the total space of the bundle of the electromagnetic potentials $A_{\mu }$ over the spacetime with the coordinates $x^{\mu }$. If the spacetime split is such that $x^{\mu }:=(t,\mathbf{x})$ then the configuration $\sigma$ is determined by the initial data $\mathsf{\Sigma }$ at an initial moment of time, namely, $A_{\mu }\left(\mathbf{x}\right)$.

We would like to understand the relation between the canonical HJ formulation in terms of the functional $\mathbf{S}\left(\right[A\left(\mathbf{x}\right)],t)$ and the covariant DDW HJ formulation that uses several functions $S^{\mu }(A,x)$. Let us start from the assumption that (cf. [11,52])

$$\mathbf{S}={\int }_{\mathsf{\Sigma }}{S}^{\mu }(A_{\mu },x^{\mu }){|}_{\mathsf{\Sigma }}{\upsilon }_{\mu }=\int d\mathbf{x}{S}^0(A_{\mu }\left(\mathbf{x}\right),\mathbf{x},t),$$

(51)

where $\mathsf{\Sigma }$ represents the surface of initial data $A_{\nu }=A_{\nu }\left(\mathbf{x}\right)$ at a fixed moment of time. In the following, we denote $S^{\mu }{|}_{\mathsf{\Sigma }}=S^{\mu }(A_{\nu }\left(\mathbf{x}\right),\mathbf{x},t)$ simply as $S^{\mu }$.

The time derivative of  $\mathbf{S}$ can be obtained from the covariant DDW HJ equation (38). Namely,

$$\begin{array}{c}\hfill {\partial }_t\mathbf{S}=\int d\mathbf{x}{\partial }_t{S}^0=\int d\mathbf{x}\left(-{\partial }_i{S}^i+{\displaystyle \frac{1}{4}}{\displaystyle \frac{\partial S^0}{\partial A_{\nu }}}{\displaystyle \frac{\partial S_0}{\partial A^{\nu }}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{\partial S^i}{\partial A_{\nu }}}{\displaystyle \frac{\partial S_i}{\partial A^{\nu }}}-j^{\mu }{A}_{\mu }\right).\end{array}$$

(52)

The first term can be rewritten as

$$\int d\mathbf{x}{\partial }_i{S}^i=\int d\mathbf{x}\left({\displaystyle \frac{d{S}^i}{d{x}^i}}-{\partial }_i{A}_{\mu }\left(\mathbf{x}\right){\displaystyle \frac{\partial S^i}{\partial A_{\mu }}}\right)=-\int d\mathbf{x}{\partial }_i{A}_{\mu }\left(\mathbf{x}\right){\displaystyle \frac{\partial S^i}{\partial A_{\mu }}},$$

(53)

where the total divergence whose integral is vanishing is given by

$${\displaystyle \frac{d{S}^i}{d{x}^i}}={\partial }_i{S}^i+{\partial }_i{A}_{\mu }\left(\mathbf{x}\right){\displaystyle \frac{\partial S^i}{\partial A_{\mu }}}.$$

(54)

Using the constraints, we can write the right hand side of (53) as

$$\int d\mathbf{x}\left(-{\partial }_i{A}_0\left(\mathbf{x}\right){\displaystyle \frac{\partial S^0}{\partial A_i}}+{\partial }_i{A}_j\left(\mathbf{x}\right){\displaystyle \frac{\partial S^{[i}}{\partial A_{j]}}}\right).$$

(55)

Integrating by parts the first term and using the embedding condition (50) in the second term we obtain

$$\int d\mathbf{x}\left(A_0\left(\mathbf{x}\right){\displaystyle \frac{d}{d{x}^i}}{\displaystyle \frac{\partial S^0}{\partial A_i}}-{\displaystyle \frac{1}{2}}{F}_{ij}{F}^{ij}\right).$$

(56)

In the spacetime with the signature $(+,-,-,-)$ which we use here, the second term in (52) transforms into

$$-{\displaystyle \frac{1}{4}}\int d\mathbf{x}{\displaystyle \frac{\partial S^0}{\partial A_i}}{\displaystyle \frac{\partial S_0}{\partial A_i}}.$$

(57)

Note that

$${\displaystyle \frac{\partial S^0}{\partial A_0}}=0$$

(58)

due to the constraint (39):

$$p_{A_{\nu )}}^{(\mu }={\displaystyle \frac{\partial S^{(\mu }}{\partial A_{\nu )}}}=0,$$

(59)

which follows from the definition of polymomenta from the Lagrangian (1) [69].

Using the constraints in (39) and the relation between polymomenta and field strengths, which follows from the Lagrangian in (1), namely,

$$p_{A_{\nu }}^{\mu }=-F^{\mu \nu },$$

(60)

and the embedding condition of the DDW HJ formulation in (50), the third term in (52) can be transformed as follows

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{4}}\int d\mathbf{x}\left(-{\left({\displaystyle \frac{\partial S^i}{\partial A_0}}\right)}^2+{\displaystyle \frac{\partial S^i}{\partial A_j}}{\displaystyle \frac{\partial S_i}{\partial A^j}}\right)\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill & =\int d\mathbf{x}\left(-{\displaystyle \frac{1}{4}}{\left({\displaystyle \frac{\partial S^0}{\partial A_i}}\right)}^2+{\displaystyle \frac{1}{4}}{F}^{ij}{F}_{ij}\right).\hfill \end{array}$$

(62)

By taking note of the fact that

$${\displaystyle \frac{\delta \mathbf{S}}{\delta A_i\left(\mathbf{x}\right)}}={\displaystyle \frac{\partial S^0}{\partial A_i}}(A\left(\mathbf{x}\right),\mathbf{x})$$

(63)

we obtain from (52)

$$\begin{array}{cc}\hfill {\partial }_t\mathbf{S}& =\int d\mathbf{x}\left(A_0\left(\mathbf{x}\right){\displaystyle \frac{d}{d{x}^i}}{\displaystyle \frac{\delta \mathbf{S}}{\delta A_i\left(\mathbf{x}\right)}}-{\displaystyle \frac{1}{2}}{\left(F_{ij}\right)}^2-{\displaystyle \frac{1}{4}}{\left({\displaystyle \frac{\delta \mathbf{S}}{\delta A_i\left(\mathbf{x}\right)}}\right)}^2-{\displaystyle \frac{1}{4}}{\left({\displaystyle \frac{\delta \mathbf{S}}{\delta A_i\left(\mathbf{x}\right)}}\right)}^2+{\displaystyle \frac{1}{4}}{\left(F_{ij}\right)}^2-j^{\mu }{A}_{\mu }\right)\hfill \\ \hfill & \\ \hfill & =-\int d\mathbf{x}\left({\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\delta \mathbf{S}}{\delta A_i\left(\mathbf{x}\right)}}\right)}^2+{\displaystyle \frac{1}{4}}{F}_{ij}\left(\mathbf{x}\right)F^{ij}\left(\mathbf{x}\right)+j^{\mu }{A}_{\mu }-A_0\left(\mathbf{x}\right){\displaystyle \frac{d}{d{x}^i}}{\displaystyle \frac{\delta \mathbf{S}}{\delta A_i\left(\mathbf{x}\right)}}\right).\hfill \end{array}$$

(64)

Thus, without any recourse to the procedures of canonical Hamiltonian formalism, we obtain the canonical HJ equation

$${\partial }_t\mathbf{S}+\int d\mathbf{x}\left({\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\delta \mathbf{S}}{\delta A_i\left(\mathbf{x}\right)}}\right)}^2+{\displaystyle \frac{1}{4}}{F}_{ij}{F}^{ij}+j^i{A}_i\right)=0$$

(65)

and the Gauss law in the HJ form

$${\displaystyle \frac{d}{d{x}^i}}{\displaystyle \frac{\delta \mathbf{S}}{\delta A_i\left(\mathbf{x}\right)}}=j^0$$

(66)

by applying the spacetime split and integration over the initial data to the covariant DDW HJ equation (38). The Gauss law constraint arises with the Lagrange multiplier $A_0\left(\mathbf{x}\right)$, and the latter decouples from the dynamics (cf. [37,61]).

7. Conclusions

We have shown how the covariant DDW HJ equation for classical electromagnetic field can be derived starting from the first-order Lagrangian formulation by using the formalism of the DDW Hamiltonian theory with constraints. We intentionally started from a more singular Palatini-like formulation in order to test the method of dealing with the singular DDW theories introduced in [13] which uses the generalized Dirac bracket on differential forms and effectively leads to what we call here the polysymplectic reduction from the polymomentum phase space with constraints to the reduced polymomentum phase space. The algorithm of the analysis of constraints within singular DDW field theories leads to the “reduced polysymplectic structure" that allows us to use the existing geometrical formulations of the covariant HJ theories in the context of closely related multisymplectic and k-symplectic formulations [62,63,64,65] in order to formulate the covariant DDW HJ equation for Maxwell’s electrodynamics. Note that the notion of the “polysymplectic reduction" in this paper is closer to the context of Dirac’s theory of constraints generalized to the DDW Hamiltonian formulation in [13], and its possible relations with other notions of the poly- or multi-symplectic reduction and poly- or multi-symplectic structures appearing in the mathematical literature (see e.g.  [76–78] and the references therein) deserves a separate study.

The DDW HJ equation in (38) could be an interesting reformulation of the electromagnetic field equations from the point of view of numerical integration near singular configurations of electromagnetic wave fields. In fact, the numerical integrator based on the DDW Hamiltonian formulation and the polysymplectic structure has proven to be considerably more efficient that other numerical schemes [79,80] and potentially extendable to numerical relativity [81].

Moreover, any quantization of the electromagnetic field which is based on the DDW Hamiltonian formalism is required to reproduce the DDW HJ equation in the classical limit. In [32], the DDW HJ equation for the scalar field theory is obtained in the classical limit of the analogue of the Schrödinger wave equation that was put forward as the foundation of the precanonical quantization [32]. It is interesting to understand how our DDW HJ Equation (38) emerges in the classical limit of the precanonically quantized Maxwell field. The latter is easy to obtain from the more general case of the quantum Yang-Mills theories considered in [34,35,36].

Another question to ask is: how the DDW HJ equation in (38), which uses the partial derivatives, is related to the canonical HJ equation for the electromagnetic field, which uses variational derivatives [67]? In [11] and [52] a similar question was discussed for the scalar field theory and gravity, respectively. Here we have shown that by properly applying the space+time split to the covariant DDW HJ equation for the Electromagnetic field we obtain the canonical HJ equation in variational derivatives and the HJ reformulation of the Gauss constraint without using the canonical Hamiltonian formalism per se. The quantum version of this correspondence in the case of quantum Yang-Mills theory was discussed in [36]. The construction of the correspondence between the DDW HJ formulation of Einstein’s equations [14,58] and Peres’ HJ formulation [30] without resorting to the temporal gauge as in [52] is still an open problem which we hope to address in the future.

Let us note that the considerations of this paper can be generalized to the case of non-abelian gauge fields in flat and curved spacetime rather straightforwardly. Moreover, the results of  [72–74] combined with the methods of this paper allow us to obtain the covariant DDW HJ equations for more general field theories than the electromagnetic field considered in this paper.