Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density

Abstract

We generalize Koopman–von Neumann classical mechanics to poly symplectic fields and recover De Donder–Weyl’s theory. Compared with Dirac’s Hamiltonian density, it inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz-covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman–von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize spacetime, energy–momentum, frequency–wavenumber, and the Fourier conjugate of energy–momentum. We clarify how first and second quantization can be found by simply mapping between operators in classical and quantum commutator algebras.

1. Introduction

Koopman–von Neumann (KvN) mechanics formulates classical mechanics (CM) with a complex wavefunction in a Hilbert space [1,2,3,4,5,6,7]. This formulation helps clarify the similarities and differences between CM and quantum mechanics (QM). Bondar et al. studied an algebra for KvN mechanics that can be quantized by mapping to the Heisenberg algebra [3]. The Koopman–von Neumann algebra contains the position operator ${\widehat{x}}^i$, its Fourier-conjugate wavenumber ${\widehat{k}}_j$, the momentum operator ${\widehat{p}}_k$, and its Fourier conjugate ${\widehat{q}}^l$. By recognizing that this KvN algebra has Fourier conjugate variables over phase space, quantization can be found by setting ${\widehat{p}}_i=\hslash {\widehat{k}}_i$. Our primary goal is to generalize this KvN quantization to relativistic field theories.

De Donder–Weyl (DDW) theory contains a covariant Hamiltonian density for relativistic field theories with DDW Equations [8,9,10,11]. DDW theory contains poly symplectic geometry, which introduces a conjugate poly momentum field of a higher tensor rank than the field. The prefix “poly” refers to the higher rank momentum not found in symplectic geometry. In classical mechanics, the Euler–Lagrange equations account for a time derivative of the momentum. In relativistic field theory, the analogous conjugate momentum is the poly momentum defined with a partial derivative, as found in DDW theory. DDW theory can be contrasted with Dirac’s canonical Hamiltonian density [12], which uses a partial time derivative ${\partial }_0$ for the canonical momentum despite the Euler–Lagrange equations containing all partial derivatives in ${\partial }_{\mathsf{\mu }}$.

Our initial goal was to recover the KvN quantization of DDW theory by generalizing KvN mechanics to poly symplectic fields. We find that the DDW equations can be found from a poly Liouville operator, which is related to previously studied poly symplectic Poisson brackets [13], but this obscures Dirac’s canonical quantization. However, new poly KvN commutator algebras allow for straightforward quantization by mapping commutators of classical operators to quantum operators. While this provides a new path towards DDW quantization, Kanatchikov has extensively discussed precanonical quantization as the (geometric) quantization of DDW theory by considering Gerstenhaber brackets as generalized Poisson brackets [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27].

The main result of this work is to find the KvN quantization of a new Hamiltonian density that is canonical, covariant, and symplectic. Canonical quantization is typically used with Dirac’s canonical Hamiltonian density, which is symplectic but not covariant [12]. Precanonical quantization can be applied to the DDW theory [19,20], whose Hamiltonian density is covariant but not symplectic, but rather poly symplectic. All known canonical formulations have some type of symplectic or poly syomplectic geometry. Symplectic geometry requires position and momentum to be of the same tensor rank, while poly symplectic geometry allows for the momentum to be of a higher rank. Comparing the Dirac and DDW Hamiltonian formulations inspires a new covariant, canonical, and symplectic Hamiltonian density, as shown in Figure 1. We introduce a generalized KvN algebra for these fields, whose second quantization leads to canonical commutation relations in terms of a covariant phase space of fields with symplectic geometry. Technically, for second quantization, the fields should be updated to operators.

While Witten and Crnkovic proved that covariant and symplectic structures for fields exist, this focused on identifying a symplectic charge that integrates over a hypersurface with time-like boundary [28,29,30,31,32], rather than considering a new type of Hamiltonian density found from the canonical structure of the fields. Covariant formulations of Hamiltonian dynamics have been previously discussed, but often do not present a Hamiltonian formulation, despite referring to the Hamiltonian dynamics [33,34]. In hindsight, Iyer and Wald’s formulation is close to ours, as a time-like vector $t^a$ was introduced in a similar manner to our ${\widehat{\tau }}^{\mathsf{\mu }}$ to find their Hamiltonian $H_X$ to study energy; however, no Hamiltonian density was found directly from the Lagrangian density in this formulation for gravitational fields [35,36,37]. Ashtekar found a symplectic Hamiltonian density that used a hypersurface based on null infinity, but this was restricted to the study of general relativity [38]. Covariant phase space methods have been recently explored in a wide class of gravitational theories [39,40,41,42]. However, a comprehensive formulation of arbitrary field theory with a covariant, symplectic, and canonical Hamiltonian density still appears to be lacking to the best of our knowldge.

To the best of our knowledge, this work includes the following results for the first time (or at least provides additional context and clarity):

• Construction of proper relativistic KvN mechanics and relativistic KvN algebra.
• Generalization of KvN mechanics to poly KvN fields as DDW theory with a new poly KvN algebra.
• A new covariant and symplectic Hamiltonian density formulation for relativistic fields.
• Second quantization via deformation of commutator algebras over fields.

This manuscript is organized as follows. Section 2 introduces the new Hamiltonian density, the analogous set of Hamilton’s equations, the generalized KvN algebra in terms of a Fourier-phase space of fields, and its KvN quantization. Appendix A focuses on a relativistic formulation of KvN mechanics and its quantization to give the relativistic generalization of the Heisenberg algebra. Appendix B introduces DDW theory, demonstrates that the poly symplectic generalization of KvN mechanics is equivalent to DDW theory, and presents the KvN quantization of DDW theory. Readers unfamiliar with KvN or DDW theory may prefer to start with the appropriate appendices, while experts may proceed directly to Section 2. While the appendices may contain some new results, they often contain rederivations of older results.

2. Quantization of a New Hamiltonian Density

In this section, we introduce a new Hamiltonian density inspired by Dirac’s canonical Hamiltonian density and the De Donder–Weyl Hamiltonian density. Our strategy is to develop a new Hamiltonian formulation by taking the local time-like component of a frame field and combining it with the DDW canonical momentum to obtain something more similar to the Dirac canonical momentum. By contracting a time-like unit vector ${\widehat{\tau }}_{\mathsf{\mu }}\left(x\right)=e_{\mathsf{\mu }}^0\left(x\right)$ with the poly momentum field to give $\pi \left(x\right)={\widehat{\tau }}_{\mathsf{\mu }}{\pi }^{\mathsf{\mu }}\left(x\right)$, the new Hamiltonian density limits to Dirac’s Hamiltonian density when the unit vector is ${\widehat{\tau }}^{\mathsf{\mu }}\left(x\right)=(1,0,0,0)$. In this manner, we can locally refer to different spacetime foliations with applications for dynamical foliation schemes. The new canonical momentum field is found to be covariant with respect to the global manifold, yet is symplectic with respect to the (position) field $\varphi \left(x\right)$. This new approach is well suited for canonical quantum relativistic evolution with manifestly covariant and symplectic fields.

A key result is the canonical and covariant commutation relations of a scalar field $\varphi$ with its conjugate field for classical, first, and second quantized fields. For classical fields $\varphi (x,p_x)$ over phase space (inspired by KvN mechanics), a Fourier-conjugate wavenumber field $\kappa (y,p_y)$ is found, which can be first quantized to give $\varphi \left(x\right)$ and $\kappa \left(y\right)$ as classical fields over spacetime instead of phase space.

Table 1. The field content for classical, first quantized, and second quantized scalar fields for the new covariant and symplectic approach are shown above. A field $A\left(a\right)$ with a noncommutative conjugate $B\left(b\right)$ satisfies $\left[A\right(a),B(b\left)\right]=i\delta (a-b)$. While KvN mechanics provides a wavefunction $\psi (x,p)$, the study of classical fields over phase space (zeroth quantized) such as $\varphi (x,p)$ has been relatively unexplored. Classical fields such as $\varphi \left(x\right)$ found in the Klein–Gordon equation are referred to as first quantized fields, since they are taken as a function of spacetime and have classical equations of motion with ℏ. Second quantized quantum fields such as $\mathsf{\Phi }\left(x\right)$ are found as deformations of the first quantized fields.

Quantization Level	Field	Noncommutative Conjugate
Classical phase space/zeroth quantized	$\varphi (x,p)$, $\pi (x,p)$	$\kappa (x,p)$, $\xi (x,p)$
Classical spacetime/first quantized	$\varphi \left(x\right)$, $\pi \left(x\right)$	$\kappa \left(x\right)$, $\xi \left(x\right)$
second quantized	$\mathsf{\Phi }\left(x\right)$	$\mathsf{\Pi }\left(x\right)$

summarizes the different possible phase space configurations of classical and quantum fields. Note that $\varphi \left(x\right)$ is typically referred to as a classical field, but in our context, the first quantization of KvN mechanics over fields implies that classical fields such as $\varphi \left(x\right)$ are actually first quantized, which is expanded upon in Appendix B.2.3. Quantum fields are depicted by capital letters, which lead to $\mathsf{\Phi }\left(x\right)$ and the conjugate momentum $\mathsf{\Pi }\left(y\right)$ as second quantized fields,

$$\begin{array}{ccc}& \mathrm{Zeroth} \mathrm{quantized} \left(\mathrm{classical}\right):& [\varphi (x,p_x),\kappa (y,p_y)]=i{\delta }^{\left(4\right)}(x-y){\delta }^{\left(4\right)}(p_x-p_y),\hfill \\ & \mathrm{First} \mathrm{quantized}:& [\varphi \left(x\right),\kappa \left(y\right)]=i{\delta }^{\left(4\right)}(x-y),\hfill \\ & \mathrm{Second} \mathrm{quantized}:& [\mathsf{\Phi }\left(x\right),\mathsf{\Pi }\left(y\right)]=i\hslash c\left|\tau \right|{\delta }^{\left(4\right)}(x-y),\hfill \end{array}$$

(1)

which includes a time-like vector ${\tau }^{\mathsf{\mu }}$ with dimensions of time to refer to the time-like displacement between spacetime foliations. An important insight was to realize that ${\widehat{\tau }}^{\mathsf{\mu }}$ also corresponds to the time-like component of a frame field. The simplest quantization procedure that we found is by setting $\mathsf{\Phi }\left(x\right)=\varphi \left(x\right)$ and $\mathsf{\Pi }\left(x\right)=\hslash c\left|\tau \right|\kappa \left(x\right)$, where the extra factor of length $c\left|\tau \right|$ in comparison to $\overrightarrow{p}=\hslash \overrightarrow{k}$ is included to account for our use of four-dimensional delta functions, while canonical quantization typically leads to three-dimensional delta functions.

2.1. A Covariant Hamiltonian Density Closer to Dirac’s

The Dirac canonical momentum of a field $\varphi \left(x\right)$ is given by ${\pi }_D\left(x\right)=\frac{\partial \mathcal{L}}{\partial \dot{\varphi }}$, while the DDW canonical momentum is ${\pi }^{\mathsf{\mu }}\left(x\right)=\frac{\partial \mathcal{L}}{\partial {\partial }_{\mathsf{\mu }}\varphi }$. One may also define ${\nu }_{\mathsf{\mu }}={\partial }_{\mathsf{\mu }}\varphi$ as the poly velocity field, while ${\nu }_D={\partial }_0\varphi$ is the Dirac or canonical velocity field. The canonical momentum has the advantage of intuitively being the same rank as the field itself with a symplectic structure, similar to spacetime and energy–momentum, while the DDW canonical momentum has the advantage of being Lorentz covariant. The canonical Hamiltonian theory requires a spacetime foliation with timelike separated hypersurfaces, while the DDW theory obscures the interpretation of time evolution for a more general notion of spacetime evolution. As it turns out, two different paths of exploration led to this symplectic Hamiltonian density by pushing the poly symplectic geometry of DDW closer to Dirac’s Hamiltonian formulation.

First, Koopman–von Neumann dynamics demonstrates that Poisson brackets and commutation relations provide different roles in classical theory, which becomes more apparent when generalizing to poly symplectic fields. The Liouville operator in KvN mechanics is analogous with the Hamiltonian in quantum mechanics, but the poly Liouville operator ${\widehat{\mathfrak{L}}}_{\mathsf{\mu }}$ found in Appendix B.2.2 is now a covector, while the Hamiltonian is a scalar. This motivated the search for an analogue of the poly Liouville operator as a type of energy–momentum density vector. The initial goal was to find a scheme in which Dirac’s Hamiltonian density could be found in the time-like component when choosing ${\widehat{\tau }}^{\mathsf{\mu }}=(1,0,0,0)$.

Taking inspiration from Koopman–von Neumann mechanics in a De Donder–Weyl formulation, we define a vectorial poly Hamiltonian density with the operator ${\widehat{\tau }}^{\mathsf{\mu }}$ which reduces to Dirac’s Hamiltonian in the zeroth component when ${\widehat{\tau }}^{\mathsf{\mu }}=(1,0,0,0)$ is chosen,

$${\mathcal{H}}^{\mathsf{\mu }}={\widehat{\tau }}^{\rho }{\nu }_{\rho }{\pi }^{\mathsf{\mu }}-{\widehat{\tau }}^{\mathsf{\mu }}\widehat{\mathcal{L}}$$

(2)

The timelike component of this operator contains the canonical momentum ${\pi }^0$. If ${\widehat{\tau }}^{\mathsf{\mu }}$ is a time-like constant; then, $\int d^{3}x{\mathcal{H}}^0$ is identical to the canonical Hamiltonian found within a particular inertial reference frame. One approach could be to multiply the DDW Hamiltonian density by ${\widehat{\tau }}^{\mathsf{\mu }}$, but this would not lead to the Dirac Hamiltonian density in the zeroth component in the same manner. However, the covector energy–momentum density functional above is no longer a Legendre transform of $\mathcal{L}$. Nevertheless, a scalar energy density functional can be defined by contracting ${\mathcal{H}}_{new}={\widehat{\tau }}_{\mathsf{\mu }}{\mathcal{H}}^{\mathsf{\mu }}$.

Second, consider a Legengre transformation with a velocity field $\nu$ and a new conjugate momentum field $\pi$ giving Dirac’s canonical momentum when ${\widehat{\tau }}^{\mathsf{\mu }}=(1,0,0,0)$,

$$\nu ={\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}\varphi ={\widehat{\tau }}^{\mathsf{\mu }}{\nu }_{\mathsf{\mu }},\pi =\frac{\partial \mathcal{L}}{\partial \nu }={\widehat{\tau }}_{\mathsf{\mu }}\frac{\partial \mathcal{L}}{\partial {\nu }_{\mathsf{\mu }}}.$$

(3)

The trade-off for introducing a symplectic canonical momentum field $\pi \left(x\right)$ is that the Lagrangian density $\mathcal{L}$ must be rewritten to isolate ${\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}\varphi$, which will be explicitly demonstrated later for the Klein–Gordon, Maxwell, and linearized gravity field theories. By taking inspiration from the covariant DDW Hamiltonian and striving for the simplicity of Dirac’s Hamiltonian with symplectic geometry, a new type of Hamiltonian density ${\mathcal{H}}_{new}$ can be found that is covariant, symplectic, and canonical

$${\mathcal{H}}_{new}=\nu \pi -\mathcal{L}={\widehat{\tau }}^{\mathsf{\mu }}{\nu }_{\mathsf{\mu }}{\widehat{\tau }}_{\nu }{\pi }^{\nu }-\mathcal{L}={\widehat{\tau }}_{\mathsf{\mu }}{\mathcal{H}}^{\mathsf{\mu }}$$

(4)

As mentioned, it turns out that ${\mathcal{H}}_{new}={\mathcal{H}}^{\mathsf{\mu }}{\widehat{\tau }}_{\mathsf{\mu }}$. The Hamiltonian density above is a Legendre transformation of the Lagrangian density since $\pi =\frac{\partial \mathcal{L}}{\partial {\widehat{\tau }}^{\mathsf{\mu }}{\nu }_{\mathsf{\mu }}}={\widehat{\tau }}_{\mathsf{\mu }}\frac{\partial \mathcal{L}}{\partial {\nu }_{\mathsf{\mu }}}$.

To obtain time evolution in a specific frame and to connect back to Dirac’s canonical quantization, consider a time-slice vector ${\widehat{\tau }}^{\mathsf{\mu }}=\frac{{\tau }^{\mathsf{\mu }}}{\left|\tau \right|}$ that is $(1,0,0,0)$ in the rest frame. This same concept of a time-like unit vector has been considered by Wald and Iyer as well as Rovelli and Vidotto [35,36,37,43]. The poly Liouville operator contracted with ${\widehat{\tau }}^{\mathsf{\mu }}$ gives time evolution for arbitrary inertial frames with coordinates $x^{\prime \mathsf{\mu }}=\left(c{t}^{\prime },{\overrightarrow{x}}^{\prime }\right)$,

$$i\frac{1}{c}\frac{\partial }{\partial t^{\prime }}\varphi \left(x\right)=i{\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}\varphi \left(x\right)={\widehat{\tau }}^{\mathsf{\mu }}{\widehat{\mathfrak{L}}}_{\mathsf{\mu }}\varphi \left(x\right)=i{\widehat{\tau }}^{\mathsf{\mu }}{\pi }_{\mathsf{\mu }}.$$

(5)

While this equation in a sense relates time evolution to the conjugate poly momentum, it does not dynamically evolve the scalar field with the Klein–Gordon field equations. This relates to the fact that Schrödinger and KvN dynamics are first-order in time, while Klein–Gordon is second-order and DDW contains two first-order equations. Before presenting the full equations of motion, we will demonstrate that $\varphi$ and $\pi$ are a symplectic phase space of fields.

The symplectic two-form is typically given by $\eta =dx\wedge dp$, motivating a $2d$-dimensional phase space for d-dimensional spacetime. To demonstrate that this new Hamiltonian formulation contains symplectic structure, recall how Poisson brackets for 2D phase space can be described as a dyadic differential operator in terms of a symplectic metric $\eta =\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$,

$$\{A,B\}=A\left(\begin{array}{cc}\overleftarrow{\frac{\partial }{\partial x}}& \overleftarrow{\frac{\partial }{\partial p}}\end{array}\right)\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{c}\frac{\partial }{\partial x}\\ \frac{\partial }{\partial p}\end{array}\right)B=\frac{\partial A}{\partial x}\frac{\partial B}{\partial p}-\frac{\partial A}{\partial p}\frac{\partial B}{\partial x}.$$

(6)

Generalizing to fields, the partial derivatives with respect to $\varphi$ and $\pi$ lead to field-theoretic Poisson brackets with symplectic structure,

$${\{A,B\}}_{new}=A\left(\begin{array}{cc}\overleftarrow{\frac{\partial }{\partial \varphi }}& \overleftarrow{\frac{\partial }{\partial \pi }}\end{array}\right)\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{c}\frac{\partial }{\partial \varphi }\\ \frac{\partial }{\partial \pi }\end{array}\right)B=\frac{\partial A}{\partial \varphi }\frac{\partial B}{\partial \pi }-\frac{\partial A}{\partial \pi }\frac{\partial B}{\partial \varphi }.$$

(7)

Mapping from the poly symplectic geometry of De Donder–Weyl theory to symplectic structure is found by contracting ${\widehat{\tau }}^{\mathsf{\mu }}$ with ${\{A,B\}}_{\mathsf{\mu }}$ shown in Equation (A87). Separate from the commutation relations for the poly KvN algebra, the Poisson brackets describe how these symplectic fields lead to canonical relations,

$${\{\varphi ,\pi \}}_{new}=1,{\{\varphi ,\varphi \}}_{new}={\{\pi ,\pi \}}_{new}=0.$$

(8)

Upon completing this work, we realized that Witten, Zuckerman, and Crnkovic have considered a symplectic current that is covariant [28,29,30]. Crnkovic clarified how a symplectic current was the variation in the field times the poly momentum, except the relationship to De Donder–Weyl theory was not realized [31,32]. Their symplectic form $\omega$ includes an integral over a spacelike hypersurface, similar to Dirac’s canonical quantization, but the Poincare invariance of $\omega$ was expressed. In this manner, our formulation is quite similar, as the choice of ${\widehat{\tau }}_{\mathsf{\mu }}$ relates to a choice of spacelike hypersurfaces. Our formulation seems to more easily provide a Hamiltonian density for general field theories.

The new set of Hamilton’s equations is found to be

$$\begin{array}{ccc}\hfill {\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}\varphi & =& {\{\varphi ,{\mathcal{H}}_{new}\}}_{new}=\frac{\partial {\mathcal{H}}_{new}}{\partial \pi },\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill {\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}\pi & =& {\{\pi ,{\mathcal{H}}_{new}\}}_{new}=-\frac{\partial {\mathcal{H}}_{new}}{\partial \varphi }.\hfill \end{array}$$

(10)

Note how these forms of Hamilton’s equations apply to arbitrary fields and their associated conjugate momentum fields, not just scalar fields. While these equations of motion are in terms of a new kind of Poisson brackets, the fields $\varphi$ and $\pi$ can be found in a commutator algebra that generalizes the KvN algebra to fields.

Since the KvN algebra contains Fourier conjugate variables of phase space such as $\overrightarrow{k}$ and $\overrightarrow{q}$ as shown in Equation (A11), there should also be generalized Fourier conjugate fields $\kappa \left(x\right)$ and $\xi \left(x\right)$ that are conjugate to $\varphi \left(x\right)$ and $\pi \left(x\right)$. Additionally, a field $\varphi \left(x\right)$ is in a sense a generalization of a quantum wavefunction $\psi \left(x\right)$, while the classical KvN wavefunction $\psi (x,p)$ depends both on spacetime and energy–momentum. For this reason, generalizing KvN mechanics to fields implies that the fully classical or zeroth quantized fields may be a function over all of phase space. From this perspective, classical field theory typically studies first quantized fields, which describes how the massive Klein–Gordon equation can be studied as a classical field theory, yet still contain ℏ, as the coordinates are quantized, but not the fields themselves.

The zeroth quantized KvN algebra associated with the fields $\varphi (x,p)$, $\pi (x,p)$, $\kappa (x,p)$, and $\xi (x,p)$ is given by the following commutation relations,

$$\begin{array}{ccc}\hfill [\varphi (x,p_x),\kappa (y,p_y)]& =& i{\delta }^{\left(4\right)}(x-y){\delta }^{\left(4\right)}(p_x-p_y),\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill [\xi (x,p_x),\pi (y,p_y)]& =& i{\delta }^{\left(4\right)}(x-y){\delta }^{\left(4\right)}(p_x-p_y).\hfill \end{array}$$

(12)

The “position field basis” for zeroth quantized fields are

$$\begin{array}{ccc}\hfill \kappa (x,p_x)& =& -i{\delta }^{\left(4\right)}(x-y){\delta }^{\left(4\right)}(p_x-p_y)\frac{\partial }{\partial \varphi (y,p_y)},\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill \xi (x,p_x)& =& i{\delta }^{\left(4\right)}(x-y){\delta }^{\left(4\right)}(p_x-p_y)\frac{\partial }{\partial \pi (y,p_y)}.\hfill \end{array}$$

(14)

The first quantized algebras associated with these fields $\varphi \left(x\right)$, $\pi \left(x\right)$, $\kappa \left(x\right)$, and $\xi \left(x\right)$ gives

$$\begin{array}{ccc}\hfill \left[\varphi \right(x),\kappa (y\left)\right]& =& i{\delta }^{\left(4\right)}(x-y),\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill \left[\xi \right(x),\pi (y\left)\right]& =& i{\delta }^{\left(4\right)}(x-y).\hfill \end{array}$$

(16)

The “position field basis” for zeroth and first quantized fields are

$$\begin{array}{ccc}\hfill \kappa \left(x\right)& =& -i{\delta }^{\left(4\right)}(x-y)\frac{\partial }{\partial \varphi \left(y\right)},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill \xi \left(x\right)& =& i{\delta }^{\left(4\right)}(x-y)\frac{\partial }{\partial \pi \left(y\right)}.\hfill \end{array}$$

(18)

Alternatively, a “wavenumber field basis” could be chosen that finds $\varphi \left(x\right)=i{\delta }^{\left(4\right)}\frac{\partial }{\partial \kappa \left(y\right)}$.

Next, we demonstrate that the interacting Klein–Gordon, Maxwell, and linearized gravity equations of motion can all be derived from this new Hamiltonian density.

2.1.1. Covariant Hamiltonian Density for Klein–Gordon Scalars

To derive the new Hamiltonian density, it is convenient to introduce a frame field $e_a^{\mathsf{\mu }}=\left({\widehat{\tau }}^{\mathsf{\mu }},{\widehat{x}}^{\mathsf{\mu }},{\widehat{y}}^{\mathsf{\mu }},{\widehat{z}}^{\mathsf{\mu }}\right)$. For non-gravitational theories, $e_a^{\mathsf{\mu }}$ as a constant encodes global Lorentz transformations between the global Minkowski manifold in some frame and local coordinates that can be in a different frame, such that $e_{\mathsf{\mu }}^a$ is independent of $x^{\mathsf{\mu }}$. The local and global frames are equivalent when $e_a^{\mathsf{\mu }}$ is given by the identity matrix, which leads to ${\widehat{\tau }}^{\mathsf{\mu }}=(1,0,0,0)$.

The Klein–Gordon action can be written as

$$S=\int d^{4}x\left[\frac{1}{2}{\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}\varphi {\widehat{\tau }}_{\nu }{\partial }^{\nu }\varphi +\frac{1}{2}{e}_i^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}\varphi e_{\nu }^i{\partial }^{\nu }\varphi -V\left(\varphi \right)\right],$$

(19)

where $i=1,2,3$. The velocity field and conjugate momentum field are

$$\nu ={\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}\varphi ,\pi =\frac{\partial \mathcal{L}}{\partial \nu }={\widehat{\tau }}_{\mathsf{\mu }}{\partial }^{\mathsf{\mu }}\varphi .$$

(20)

The new Hamiltonian density for an interacting Klein–Gordon field is

$${\mathcal{H}}_{new}=\frac{1}{2}\left({\pi }^2-e_i^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}\varphi e_{\nu }^i{\partial }^{\nu }\varphi \right)+V\left(\varphi \right).$$

(21)

Integrating by parts for the second term and dropping the boundary term allows for an easier evaluation of Hamilton’s equations,

$${\mathcal{H}}_{new}=\frac{1}{2}\left({\pi }^2+\varphi e_i^{\mathsf{\mu }}{e}_{\nu }^i{\partial }_{\mathsf{\mu }}{\partial }^{\nu }\varphi \right)+V\left(\varphi \right)+\mathrm{bdry}.$$

(22)

Applying the new set of Hamilton’s equations gives

$$\begin{array}{ccc}\hfill {\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}\varphi & =& {\{\varphi ,{\mathcal{H}}_{new}\}}_{new}=\pi ,\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}\pi & =& {\{\pi ,{\mathcal{H}}_{new}\}}_{new}=-e_i^{\mathsf{\mu }}{e}_{\nu }^i{\partial }_{\mathsf{\mu }}{\partial }^{\nu }\varphi -\frac{\partial V}{\partial \varphi }.\hfill \end{array}$$

(24)

Plugging the first equation into the second leads to the Klein–Gordon equation of motion. In this manner, the Hamiltonian dynamics are identical to Dirac’s Hamiltonian, except the frame fields are introduced to allow for arbitrary local frames. The new Hamiltonian density found is truly covariant. Understanding the role of $e_a^{\mathsf{\mu }}$ as a frame field also allows for a more dynamical realization of foliations. Despite ${\widehat{\tau }}^{\mathsf{\mu }}$ being the zeroth component of $e_a^{\mathsf{\mu }}$, the local structure is independent of the global manifold. In this manner, Dirac’s Hamiltonian is not covariant with respect to the global manifold since $\mathsf{\mu }=0$ is isolated. The frame field allows for $a=0$ to be chosen locally in a manner that the Hamiltonian density is still Lorentz covariant with respect to the global manifold.

2.1.2. Covariant Hamiltonian Density for Maxwell Theory

The Maxwell action is

$$S=\int d^{4}x\mathcal{L}=\int d^{4}x\left(-\frac{1}{4}{F}_{\mathsf{\mu }\nu }{F}^{\mathsf{\mu }\nu }-A^{\mathsf{\mu }}{J}_{\mathsf{\mu }}\right).$$

(25)

The velocity field of interest is

$${\nu }_{\mathsf{\mu }}={\widehat{\tau }}^{\nu }{\partial }_{\nu }{A}_{\mathsf{\mu }}.$$

(26)

The following relation allows for the Lagrangian to be expressed in terms of the velocity field,

$${\delta }_{\nu }^{\mathsf{\mu }}=e_a^{\mathsf{\mu }}{e}_{\nu }^a={\widehat{\tau }}^{\mathsf{\mu }}{\widehat{\tau }}_{\nu }+e_i^{\mathsf{\mu }}{e}_{\nu }^i.$$

(27)

The Lagrangian density can be expanded to give

$$\begin{array}{ccc}\hfill \mathcal{L}& =& -\frac{1}{4}{F}_{\mathsf{\mu }\nu }{F}_{\rho \sigma }\left({\widehat{\tau }}^{\mathsf{\mu }}{\widehat{\tau }}^{\nu }{\widehat{\tau }}^{\rho }{\widehat{\tau }}^{\sigma }+{\widehat{\tau }}^{\mathsf{\mu }}{e}_j^{\nu }{\widehat{\tau }}^{\rho }{e}^{\sigma j}+e_i^{\mathsf{\mu }}{\widehat{\tau }}^{\nu }{e}^{\rho i}{\widehat{\tau }}^{\sigma }+e_i^{\mathsf{\mu }}{e}_j^{\nu }{e}^{\rho i}{e}^{\sigma j}\right)\hfill \\ & =& -\frac{1}{2}{e}_j^{\nu }{\nu }^{\nu }{e}^{\sigma j}{\nu }_{\sigma }+e^{\sigma j}{\nu }_{\sigma }{e}^{\nu }{e}_j^{\nu }{\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\nu }{A}_{\mathsf{\mu }}-\frac{1}{2}{e}_j^{\nu }{\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\nu }{A}_{\mathsf{\mu }}{e}^{\sigma j}{\widehat{\tau }}^{\rho }{\partial }_{\sigma }{A}_{\mathsf{\mu }}-\frac{1}{4}{F}_{\mathsf{\mu }\nu }{F}^{\rho \sigma }{e}_i^{\mathsf{\mu }}{e}_j^{\nu }{e}_{\rho }^i{e}_{\sigma }^j.\hfill \end{array}$$

(28)

The conjugate momentum field is found to be

$${\pi }^{\sigma }=\frac{\partial \mathcal{L}}{\partial {\nu }_{\sigma }}=-e_j^{\nu }{e}^{\sigma j}\left({\nu }_{\nu }-{\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\nu }{A}_{\mathsf{\mu }}\right)=-e_j^{\nu }{e}^{\sigma j}{\widehat{\tau }}^{\mathsf{\mu }}{F}_{\mathsf{\mu }\nu }.$$

(29)

Another way to express the conjugate momentum field is given by

$${\widehat{\tau }}^{\nu }{F}_{\mathsf{\mu }\nu }=\left({\widehat{\tau }}^{\rho }{\widehat{\tau }}_{\mathsf{\mu }}+e_i^{\rho }{e}_{\mathsf{\mu }}^i\right)F_{\rho \nu }{\widehat{\tau }}^{\nu }={\widehat{\tau }}_{\mathsf{\mu }}{\widehat{\tau }}^{\rho }{\widehat{\tau }}^{\nu }{F}_{\rho \nu }+{\pi }_{\mathsf{\mu }}={\pi }_{\mathsf{\mu }},$$

(30)

where the antisymmetry of $F_{\rho \nu }$ led to the vanishing of the first term above. A background-independent derivation using differential forms would automatically impose this antisymmetry. Both forms of the conjugate momentum allow for the Lagrangian density to be rewritten as

$$\mathcal{L}=-\frac{1}{2}{\pi }^{\mathsf{\mu }}{\pi }_{\mathsf{\mu }}-\frac{1}{4}{F}_{\mathsf{\mu }\nu }{F}^{\rho \sigma }{e}_i^{\mathsf{\mu }}{e}_j^{\nu }{e}_{\rho }^i{e}_{\sigma }^j-A^{\mathsf{\mu }}{J}_{\mathsf{\mu }}.$$

(31)

The covariant Hamiltonian density is found as

$${\mathcal{H}}_{new}={\nu }^{\mathsf{\mu }}{\pi }_{\mathsf{\mu }}-\mathcal{L}={\pi }_{\mathsf{\mu }}\left(-\frac{1}{2}{\pi }^{\mathsf{\mu }}+{\widehat{\tau }}_{\nu }{\partial }^{\mathsf{\mu }}{A}^{\nu }\right)+\frac{1}{4}{F}_{\mathsf{\mu }\nu }{F}^{\rho \sigma }{e}_i^{\mathsf{\mu }}{e}_j^{\nu }{e}_{\rho }^i{e}_{\sigma }^j+A^{\mathsf{\mu }}{J}_{\mathsf{\mu }}.$$

(32)

Covariant Hamilton’s equations (after integrating by parts) lead to

$$\begin{array}{ccc}\hfill {\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}{A}_{\nu }& =& \frac{\partial {\mathcal{H}}_{new}}{\partial {\pi }^{\nu }}=-{\pi }_{\nu }+{\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\nu }{A}_{\mathsf{\mu }},\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill {\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}{\pi }_{\nu }& =& -\frac{\partial {\mathcal{H}}_{new}}{\partial A^{\nu }}={\partial }^{\mathsf{\mu }}{\pi }_{\mathsf{\mu }}{\widehat{\tau }}_{\nu }+{\partial }^{\mathsf{\mu }}{F}_{\rho \sigma }{e}_{\mathsf{\mu }}^i{e}_{\nu }^j{e}_i^{\rho }{e}_j^{\sigma }-J_{\nu }.\hfill \end{array}$$

(34)

Plugging the first equation into the second leads to

$$\begin{array}{ccc}\hfill J_{\nu }& =& -{\widehat{\tau }}^{\mathsf{\mu }}{\partial }_{\mathsf{\mu }}{\pi }_{\nu }+{\partial }^{\mathsf{\mu }}{\pi }_{\mathsf{\mu }}{\widehat{\tau }}_{\nu }+{\partial }^{\mathsf{\mu }}{F}_{\rho \sigma }{e}_{\mathsf{\mu }}^i{e}_{\nu }^j{e}_i^{\rho }{e}_j^{\sigma }\hfill \\ & =& e_{\nu }^i{e}_i^{\sigma }{\widehat{\tau }}_{\mathsf{\mu }}{\widehat{\tau }}^{\rho }{\partial }^{\mathsf{\mu }}{F}_{\rho \sigma }+{\widehat{\tau }}^{\rho }{\widehat{\tau }}_{\nu }{\partial }^{\mathsf{\mu }}{F}_{\mathsf{\mu }\rho }+{\partial }^{\mathsf{\mu }}{F}_{\rho \sigma }{e}_{\mathsf{\mu }}^i{e}_{\nu }^j{e}_i^{\rho }{e}_j^{\sigma }\hfill \\ & =& {\partial }^{\mathsf{\mu }}{F}_{\rho \sigma }\left({\widehat{\tau }}_{\mathsf{\mu }}{\widehat{\tau }}^{\rho }{e}_{\nu }^i{e}_i^{\sigma }+{\widehat{\tau }}^{\sigma }{\widehat{\tau }}_{\nu }{e}_i^{\sigma }{e}_{\mathsf{\mu }}^i+e_{\mathsf{\mu }}^i{e}_{\nu }^j{e}_i^{\rho }{e}_j^{\sigma }\right)\hfill \\ & =& {\partial }^{\mathsf{\mu }}{F}_{\mathsf{\mu }\nu }.\hfill \end{array}$$

(35)

where using Equation (27) and ${\widehat{\tau }}^{\rho }{\widehat{\tau }}^{\sigma }{F}_{\rho \sigma }=0$ were used to simplify the result above.

2.1.3. Covariant Hamiltonian Density for Linearized Gravity

Since general relativity would require a dynamical frame field $e_{\mathsf{\mu }}^i\left(x\right)$ that is position dependent, it is worthwhile to first establish the new Hamiltonian density for linearized gravity. The linearized gravity action with matter is

$$S=\int d^{4}x\left(\frac{1}{2}{\partial }^{\rho }{\overline{h}}^{\mathsf{\mu }\nu }{\partial }_{\rho }{\overline{h}}^{\mathsf{\mu }\nu }-\frac{1}{4}{\partial }_{\mathsf{\mu }}\overline{h}{\partial }^{\mathsf{\mu }}\overline{h}\right)$$

(36)

Treating ${\overline{h}}_{\mathsf{\mu }\nu }$ as the dynamical field leads to the velocity field ${\overline{\nu }}_{\mathsf{\mu }\nu }$,

$${\overline{\nu }}_{\mathsf{\mu }\nu }={\widehat{\tau }}^{\rho }{\partial }_{\rho }{\overline{h}}_{\mathsf{\mu }\nu }.$$

(37)

The Lagrangian can be rewritten as

$$\mathcal{L}=\frac{1}{2}\left({\overline{\nu }}_{\mathsf{\mu }\nu }{\overline{\nu }}^{\mathsf{\mu }\nu }-\frac{1}{2}\overline{\nu }\overline{\nu }+e_{\rho }^i{e}_i^{\sigma }\left({\partial }^{\rho }{\overline{h}}^{\mathsf{\mu }\nu }{\partial }_{\sigma }{\overline{h}}_{\mathsf{\mu }\nu }-\frac{1}{2}{\partial }^{\rho }\overline{h}{\partial }_{\sigma }\overline{h}\right)\right)-\frac{8\pi G}{c^4}\kappa {\widehat{T}}^{\mathsf{\mu }\nu }{\widehat{h}}_{\mathsf{\mu }\nu },$$

(38)

where $\overline{\nu }\equiv {\overline{\nu }}_{\mathsf{\mu }\nu }{\eta }^{\mathsf{\mu }\nu }$. The conjugate momentum field is found to be

$${\overline{\pi }}^{\mathsf{\mu }\nu }=\frac{\partial \mathcal{L}}{\partial {\overline{\nu }}_{\mathsf{\mu }\nu }}={\overline{\nu }}^{\mathsf{\mu }\nu }-\frac{1}{2}{\eta }^{\mathsf{\mu }\nu }\overline{\nu }={\nu }^{\mathsf{\mu }\nu }={\widehat{\tau }}_{\rho }{\partial }^{\rho }{h}^{\mathsf{\mu }\nu },$$

(39)

where a reciprocal relationship is found such that ${\nu }_{\mathsf{\mu }\nu }={\overline{\pi }}_{\mathsf{\mu }\nu }$, etc.

The covariant Hamiltonian density in terms of the gravitational field ${\overline{h}}_{\mathsf{\mu }\nu }$ and conjugate momentum field ${\overline{\pi }}^{\mathsf{\mu }\nu }$ is

$${\mathcal{H}}_{new}=\frac{1}{2}{\overline{\pi }}_{\mathsf{\mu }\nu }{\overline{\pi }}^{\mathsf{\mu }\nu }-\frac{1}{4}\overline{\pi }\overline{\pi }-\frac{1}{2}{e}_{\rho }^i{e}_i^{\sigma }\left({\partial }^{\rho }{\overline{h}}^{\mathsf{\mu }\nu }{\partial }_{\sigma }{\overline{h}}_{\mathsf{\mu }\nu }-\frac{1}{2}{\partial }^{\rho }\overline{h}{\partial }_{\sigma }\overline{h}\right)+\frac{8\pi G}{c^4}\kappa {\overline{T}}^{\mathsf{\mu }\nu }{\overline{h}}_{\mathsf{\mu }\nu }.$$

(40)

The covariant Hamilton’s equations give

$$\begin{array}{ccc}\hfill {\widehat{\tau }}^{\rho }{\partial }_{\rho }{\overline{h}}_{\mathsf{\mu }\nu }& =& \frac{\partial {\mathcal{H}}_{new}}{\partial {\overline{\pi }}^{\mathsf{\mu }\nu }}={\overline{\pi }}_{\mathsf{\mu }\nu }-\frac{1}{2}{\eta }_{\mathsf{\mu }\nu }\overline{\pi }={\pi }_{\mathsf{\mu }\nu },\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill {\widehat{\tau }}^{\rho }{\partial }_{\rho }{\overline{\pi }}_{\mathsf{\mu }\nu }& =& -\frac{\partial {\mathcal{H}}_{new}}{\partial {\overline{h}}^{\mathsf{\mu }\nu }}=-\left(e_{\rho }^i{e}_i^{\sigma }{\partial }^{\rho }{\partial }_{\sigma }\left({\overline{h}}_{\mathsf{\mu }\nu }-\frac{1}{2}{\eta }_{\mathsf{\mu }\nu }\overline{h}\right)+\frac{8\pi G}{c^4}\kappa {\overline{T}}^{\mathsf{\mu }\nu }\right).\hfill \end{array}$$

(42)

Plugging the first equation in the trace reversal of the second equation and rearranging leads to

$$\left({\widehat{\tau }}_{\rho }{\widehat{\tau }}^{\sigma }+e_{\rho }^i{e}_i^{\sigma }\right){\partial }^{\rho }{\partial }_{\sigma }{\overline{h}}_{\mathsf{\mu }\nu }={\partial }^{\rho }{\partial }_{\rho }{\overline{h}}_{\mathsf{\mu }\nu }=-\frac{8\pi G}{c^4}\kappa T^{\mathsf{\mu }\nu }.$$

(43)

This reproduces the known equations of motion for linearized gravity.

2.2. Quantization of the New Hamiltonian Theory

Using the covariant and canonical Hamiltonian density ${\mathcal{H}}_{new}$, four sets of classical fields $\varphi \left(x\right)$, $\kappa \left(x\right)$, $\pi \left(x\right)$, and $\xi \left(x\right)$ can be used to construct two quantum fields $\mathsf{\Phi }\left(x\right)$ and $\mathsf{\Pi }\left(x\right)$,

$$\begin{array}{ccc}\hfill \mathsf{\Phi }\left(x\right)& =& a\varphi \left(x\right)+b\xi \left(x\right),\hfill \end{array}$$

(44)

$$\begin{array}{ccc}\hfill \mathsf{\Pi }\left(x\right)& =& c\pi \left(x\right)+d\kappa \left(x\right).\hfill \end{array}$$

(45)

The commutation relations of the second quantized fields are therefore

$$[\mathsf{\Phi }\left(x\right),\mathsf{\Pi }\left(y\right)]=\left(ad+bc\right)i{\delta }^{\left(4\right)}(x-y).$$

(46)

Setting $a=1$, $d=\hslash c\left|\tau \right|$ and $b=c=0$ is the simplest quantization procedure, although other possibilities exist, as mentioned in Equation (A34).

The following commutation relations are found with $\mathsf{\Pi }\left(x\right)={\widehat{\tau }}^{\mathsf{\mu }}{\mathsf{\Pi }}_{\mathsf{\mu }}\left(x\right)$ as the symplectic, covariant, and canonical momentum,

$$\begin{array}{ccc}\hfill \left[\mathsf{\Phi }\right(x),\mathsf{\Pi }(y\left)\right]& =& i\hslash c\left|\tau \right|{\delta }^{\left(4\right)}(x-y),\hfill \\ \hfill \left[\mathsf{\Phi }\right(x),\mathsf{\Phi }(y\left)\right]& =& \left[\mathsf{\Pi }\right(x),\mathsf{\Pi }(y\left)\right]=0.\hfill \end{array}$$

(47)

Similar to how $p_{\mathsf{\mu }}=\hslash k_{\mathsf{\mu }}$, it is anticipated that $\mathsf{\Pi }\left(x\right)=\hslash c\left|\tau \right|\kappa \left(x\right)$.

If one were to model the universe as a quantum computer simulation, then this quantum computer must simulate the universe at a fast enough rate such that all possible observables lead to self-consistent results. Since the Planck time is thought to be the smallest possible duration of time that could be measured, it is sensible to assume that such a quantum computer simulation would use spacelike foliations separated by times no larger than the Planck time. Thiemann states that spin quantum numbers in LQG are created and destroyed in a Planck moment and Zizzi has discussed Planck-time foliations [44,45]. For our construction, this implies that $\left|\tau \right|=t_P=\sqrt{\frac{\hslash G}{c^5}}$. With this assumption, the new canonical, covariant, and symplectic commutation relations for fields in natural units with $c=\hslash =G=1$,

$$[\mathsf{\Phi }\left(x\right),\mathsf{\Pi }\left(y\right)]=i{\delta }^{\left(4\right)}(x-y).$$

(48)

This canonical commutation relation for second quantized fields appears to be the simplest possible that is manifestly covariant and symplectic. Building off of earlier work [46,47,48], Equation (48) was recently presented in Ref. [49] as a result of different motivations.

Four-dimensional delta functions are less common for commutation relations with quantum fields. However, the standard commutation relations for Dirac’s canonical momentum can be recovered. By taking ${\widehat{\tau }}^{\mathsf{\mu }}=(1,0,0,0)$, the new canonical momentum $\mathsf{\Pi }\left(x\right)$ becomes $c{\mathsf{\Pi }}_D$, where ${\mathsf{\Pi }}_D$ is Dirac’s canonical momentum. By continuing from Equation (47),

$$[\mathsf{\Phi }\left(x\right),\mathsf{\Pi }\left(y\right)]=[\mathsf{\Phi }\left(x\right),c{\mathsf{\Pi }}_D\left(y\right)]=i\hslash c{\delta }^{\left(3\right)}(x-y)\delta ((t_x-t_y)/\left|\tau \right|).$$

(49)

By integrating over $t_y$, one finds that $t_y=t_x$. Therefore, when Equation (47) is evaluated with $\mathsf{\Phi }(\overrightarrow{x},t)$ and $\mathsf{\Pi }(\overrightarrow{y},t)$ at the same time and ${\widehat{\tau }}^{\mathsf{\mu }}=(1,0,0,0)$, the standard commutation relations with Dirac’s canonical momenta are found.

3. Conclusions

In this work, we have demonstrated that a Hamiltonian density for relativistic field theory can be found that is canonical, covariant, and symplectic. This was found by taking the De Donder–Weyl poly momentum field and contracting with the time-like component of a frame field to obtain a symplectic momentum field. This momentum field can locally interpolate between different ADM foliations of spacetime, which makes it closer to Dirac’s canonical momentum. The generalized Hamilton’s equations for this new Hamiltonian density were found and demonstrated to give the correct equations of motion for Klein–Gordon, Maxwell, and linearized gravity theories. We also generalized the phase space of fields to a Fourier-phase space of fields as a generalization of Koopman–von Neumann mechanics, giving a classical commutator algebra of fields. KvN quantization of these new fields was provided, which bypasses the use of Poisson brackets to map from classical commutators to quantum commutators.

In Appendix A, a relativistic formulation of Koopman–von Neumann classical mechanics was constructed, which found a problem of time for the proper Liouville operator for the point particle action. The poly symplectic geometry of De Donder–Weyl theory was introduced in Appendix B to study the generalization of relativistic Koopman–von Neumann mechanics to field theory, which allows for the classical equations of motion to be derived quickly from the De Donder–Weyl Hamiltonian density. To our knowledge, classical, first, and second quantized poly Koopman–von Neumann algebras for fields were presented for the first time, which was shown to be compatible with De Donder–Weyl dynamics.

In future work, it would be worthwhile to derive the general Yang–Mills field equations and Einstein’s field equations from their respective covariant, canonical, and symplectic Hamiltonian densities. The introduction of the frame field for specifying local frames requires additional care in general relativity. Since the frame field is the gauge field of the translations, formulations of gauge gravity such as subsectors of metric-affine gauge gravity should be explored. Another avenue of exploration could use the relativistic Koopman–von Neumann algebra introduced here to extend the prequantum operator algebra defined in [7] and its possible category-theoretic realizations using pregeometric constructions discussed in [50,51].

Additionally, it would be worthwhile to explore singular Lagrangian densities, such as spinor fields. Other work has attempted to use a quadratic action with a constraint to describe spinors [52]. Revisiting the Dirac–Bergmann algorithm for singular Lagrangians would also be appropriate [53]. Also, Kanatchikov has introduced a generalization of the Dirac bracket formula to DDW theory for degenerate Lagrangian densities, such as the Dirac Lagrangian density discussed above [22]. Finally, our work may be inspirational for a covariant and canonical formulation of loop quantum gravity.