# Covariant, Canonical and Symplectic Quantization of Relativistic Field
Theories

David Chester<EMAIL_ADDRESS>Quantum Gravity Research,
Topanga, CA, USA  Xerxes D. Arsiwalla<EMAIL_ADDRESS>Pompeu Fabra
University, Barcelona, Spain  Wolfram Research, Champaign, IL, USA  Louis
Kauffman<EMAIL_ADDRESS>University of Illinois at Chicago, Chicago, IL, USA
Michel Planat<EMAIL_ADDRESS>Institut FEMTO-ST, Université de
Bourgogne/Franche-Comté, Besançon, France  Klee Irwin
<EMAIL_ADDRESS>Quantum Gravity Research, Topanga, CA, USA

###### Abstract

We generalize Koopman-von Neumann classical mechanics to relativistic field
theory. The manifestly covariant Koopman-von Neumann mechanics formulated over
polysympletic fields leads to De Donder-Weyl mechanics. Comparing this
polysymplectic formulation with Dirac’s quantization leads to a new
Hamiltonian density that is canonical and covariant with symplectic structure.
We provide the commutation relations for these classical and quantum fields
with a new type of canonical momentum that has the covariant structure of the
De Donder-Weyl poly-momentum, yet has the symplectic geometry of Dirac’s
conjugate momentum. For the first time, we clarify how 1st and 2nd
quantization relates to an algebraic deformation of commutation relations of
1st and 2nd Fourier-phase spaces over coordinates and fields, respectively.

###### Contents

  1. I Introduction
  2. II Manifestly covariant and canonical classical mechanics
    1. II.1 Classical relativistic Koopman-von Neumann mechanics
    2. II.2 Classical field theory with De Donder-Weyl mechanics
      1. II.2.1 Klein-Gordon scalar field
      2. II.2.2 Dirac spin-1/2 field
      3. II.2.3 Yang-Mills spin-1 field
      4. II.2.4 Linearized gravity with spin-2 field
    3. II.3 Poly-Koopman-von Neumann mechanics as De Donder-Weyl mechanics
      1. II.3.1 Sketch of assumptions for axiomatic interacting field theories
      2. II.3.2 Poly-Koopman-von Neumann mechanics
      3. II.3.3 0th and 1st quantized poly-Koopman-von Neumann algebras
    4. II.4 A covariant Hamiltonian density closer to Dirac’s
      1. II.4.1 Covariant Hamiltonian density for Klein-Gordon scalars
      2. II.4.2 Covariant Hamiltonian density for Dirac fields
      3. II.4.3 Covariant Hamiltonian density for Maxwell theory
      4. II.4.4 Covariant Hamiltonian density for linearized gravity
  3. III Covariant and canonical quantization of relativistic fields
    1. III.1 Relativistic Koopman-von Neumann quantization
    2. III.2 De Donder-Weyl quantization
    3. III.3 A new canonical and covariant quantization
  4. IV Conclusions
  5. V Acknowledgements

##  I Introduction

Reconciling canonical and covariant formulations of quantum gravity is
perceived as a challenge to this day [1]. While successful formulations have
been proposed by both string theorists and loop quantum gravity researchers,
relatively little attention has been given to Koopman-von Neumann (KvN) [2, 3,
4, 5, 6, 7, 8] and De Donder-Weyl (DDW) mechanics [9, 10, 11, 12]. While
Witten et al. have found canonical and covariant formulations with symplectic
structure, no Hamiltonian density was found and the relationship to the DDW
poly-momentum was overlooked [13, 14, 15, 16, 17]. Other researchers have
recently explored the use of covariant, canonical, and symplectic structures
in a wide range of gravitational theories [18, 19, 20, 21]. DDW mechanics
contains polysymplectic/multisymplectic geometry, which contains a Hamiltonian
density and a conjugate poly-momentum field. The symplectic approach refers to
the conjugate polymomentum, but the connection to DDW mechanics has been
largely overlooked by the covariant phase space approach. While DDW mechanics
has the more complicated polysymplectic geometry, the symplectic approach uses
a conjugate poly-momentum found in DDW mechanics with
polysymplectic/multisymplectic geometry. Thiemann acknowledges that
quantization of DDW mechanics is challenging and relatively unexplored [22],
in reference to progress from Kantachikov on DDW quantization [23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35].

By studying KvN mechanics, generalizing to relativistic field theory, and
comparing with DDW mechanics, we find a simpler form of polysymplectic Poisson
bracket. Additionally, new poly-KvN algebras are found, which allows for
straightforward 1st and 2nd quantization. These results motivate a new
covariant, canonical, and symplectic Hamiltonian density that is close to the
standard Hamiltonian formualtion of classical mechanics and Dirac’s canonical
quantization [36]. Covariant formulations of Hamiltonian dynamics have been
previously discussed, but are either overcomplicated [37] or do not present a
Hamiltonian formulation, despite referring to the Hamiltonian dynamics [38,
39]. 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 $\tau^{\mu}$ 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 [40, 41, 42].

Koopman-von Neumann mechanics formulates classical mechanics with a complex
wavefunction in a Hilbert space [2, 3, 4, 5, 6, 7, 8]. This formulation helps
clarify the similarities and differences between classical and (relativistic)
quantum mechanics, especially when recognizing that the (relativistic)
Koopman-von Neumann algebra operates with Fourier-phase space variables
$\hat{x}^{\mu}$, $\hat{p}_{\nu}$, $\hat{k}_{\rho}$, and $\hat{q}^{\sigma}$. To
the best of our knowledge, relativistic Koopman-von Neumann mechanics has not
been previously discussed. Here, we present a relativistic formulation in
terms of a proper Liouville operator that gives the proper time evolution. We
find that a problem of time occurs in this Koopman von-Neumann classical
mechanics for the relativistic point particle action. This suggests that
problems of time are not a problem with quantum mechanics (QM), but rather
stem from the choice of the appropriate Hamiltonian formulation.

A common path to quantization is Dirac’s $i\hbar$ prescription, which turns
classical Poisson brackets into quantum commutation relations. In general, it
is more straightforward to look at the KvN algebra of classical mechanics,
which includes commutators of operators acting on a complex Hilbert space,
similar to quantum mechanics [4]. It’s worth clarifying how the quantization
of KvN mechanics relates directly to setting
$\hat{\vec{p}}=\hbar\hat{\vec{k}}$ (see [8]), as the KvN algebra can be
expressed in terms of the position operator $\hat{\vec{x}}$, its Fourier-
conjugate wavenumber $\hat{\vec{k}}$, the momentum operator $\hat{\vec{p}}$,
and its Fourier conjugate $\hat{\vec{q}}$. Quantization from the KvN algebra
is similar to geometric quantization (which constructs a Hilbert space for
phase space), except no Poisson brackets are utilized. For the first time, we
generalize the KvN algebra of classical mechanics to study classical
relativistic fields that matches De Donder-Weyl mechanics, which provides a
straightforward path towards understanding canonical and covariant
quantization. Finally, a new Hamiltonian density that is canonical and
covariant is found to contain symplectic structure.

The De Donder-Weyl mechanics appropriately introduces polysymplectic geometry
that matches nicely with the conventional Euler-Lagrange equations used for
relativistic field theory [9, 10, 11, 12]. While the Euler-Lagrange equations
of classical mechanics consider time derivatives, relativistic field theory
considers a partial derivative, which motivates a conjugate poly-momentum with
an additional vector index. The De Donder-Weyl mechanics is one canonical and
covariant Hamiltonian approach for relativistic field theory. Here, we
consider the Koopman-von Neumann classical mechanics generalized to
polysymplectic fields, which leads to the De Donder-Weyl equations.
Multiplying this equation by Dirac matrices leads to Kanatchikov’s
precanonical formulation [23].

Figure 1: The Dirac, De Donder-Weyl (DDW), and the new covariant Hamiltonian
densities $\mathcal{H}_{D}$, $\mathcal{H}$, and $\mathcal{H}_{cov}$ are
compared and contrasted in a Venn diagram by depicting whether the
formulations are canonical, covariant, or symplectic.

Kanatchikov has extensively explored DDW mechanics, which discussed a
polysymplectic form $\Omega$ [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35]. Kanatchikov then generalizes the DDW Hamiltonian operator to act on a
Clifford algebra-valued wavefunction, which leads to an analogue of the
Schrodinger equation with Dirac’s gamma matrices included in the equation
$i\hbar\kappa\gamma^{\mu}\partial_{\mu}\Psi=\hat{H}\Psi$ for $\Psi$ in a
Clifford algebra. While this is often described as DDW mechanics with pre-
canonical quantization, our perspective is that this generalizes DDW
mechanics. The DDW equations for arbitrary fields need not require Clifford
algebras or gamma matrices.

Another natural generalization of the Schrodinger equation is to replace the
wavefunction with quantum fields. Up to now, the application of DDW mechanics
to classical fields has been relatively straightforward, while to the best of
our knowledge, only Kanatchikov has attempted 2nd quantization, but this
actually appears to be a generalization of DDW mechanics. While Kanatchikov’s
formulation seems to be relevant for Clifford relativity with applications to
membranes [43, 44, 45, 46, 47, 48, 49], introducing the Clifford-algebraic
realization of DDW mechanics may have caused some confusion for studying
classical and quantum fields. In this sense, our perspective is that the most
straightforward formulation of second quantization for DDW fields has not yet
been established.

A more straightforward formulation of DDW mechanics and quantization can be
found by removing the Clifford algebraic structure for the study of classical
or quantum fields. Further exploration of DDW mechanics has been explored by
other authors, but most of these works follow Kanatchikov’s approach or work
with an abstract formulation of poly-symplectic geometry [50, 51, 52, 53, 54,
55, 56, 57]. In this work, we find a new poly-Poisson bracket that most simply
leads to the DDW equations in a manner that mimics classical mechanics.

Our main result here is to develop a new canonical and covariant Hamiltonian
density with symplectic geometry, which is inspired by quantizing classical
Koopman-von Neumann mechanics generalized to relativistic field theory with De
Donder-Weyl mechanics with polysymplectic geometry. This quantization in
effect combines Dirac’s quantization procedure from arbitrary Lorentzian
frames. Dirac’s Hamiltonian (density) is canonical, symplectic, but not
covariant. The De Donder-Weyl Hamiltonian is canonical, polysymplectic, and
covariant. Our new Hamiltonian is canonical, symplectic, and covariant, as
shown in Fig. (1). While Witten and Crnkovic had proven that covariant and
symplectic structure for fields existed, this focused on identifying a
symplectic charge that integrates over a hypersurface with time-like boundary
[15], rather than considering a new type of Hamiltonian density found from the
canonical structure of the fields.

By introducing a time-like vector $\tau^{\mu}$, a new quantization procedure
is found that limits to Dirac’s canonical quantization when the unit vector is
$\hat{\tau}^{\mu}=(1,0,0,0)$. This new approach to quantum relativistic
evolution is manifestly covariant with canonical symplectic fields. A main
result is the canonical and covariant commutation relation of a scalar field
$\phi$ with its conjugate field for classical, 1st, and 2nd quantized fields.
For classical fields $\phi(x,p_{x})$ 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 $\phi(x)$ and $\kappa(y)$ as classical
fields over spacetime instead of phase space. Table (1) summarizes the
different possible phase space configurations of classical and quantum fields.
Note that $\phi(x)$ is typically referred to as a classical field, but in our
context, first quantization of KvN mechanics over fields implies that
classical fields such as $\phi(x)$ are actually first quantized. Quantum
fields are depicted by capital letters, which lead to $\Phi(x)$ and the
conjugate momentum $\Pi(y)$ as second quantized fields,

0th quantized (classical):
$\displaystyle[\phi(x,p_{x}),\kappa(y,p_{y})]=i\delta^{(4)}(x-y)\delta^{(4)}(p_{x}-p_{y}),$
1st quantized: $\displaystyle[\phi(x),\kappa(y)]=i\delta^{(4)}(x-y),$ 2nd
quantized: $\displaystyle[\Phi(x),\Pi(y)]=i\hbar c|\tau|\delta^{(4)}(x-y),$

which includes a time-like vector $\tau^{\mu}$ to refer to the time-like
displacement between spacetime foliations.

Quantization Level | Field | Noncommutative Conjugate
---|---|---
Classical phase space/0th quantized |  $\phi(x,p)$, $\pi(x,p)$ |  $\kappa(x,p)$, $\xi(x,p)$
Classical spacetime/1st quantized |  $\phi(x)$, $\pi(x)$ |  $\kappa(x)$, $\xi(x)$
2nd quantized | $\Phi(x)$ | $\Pi(x)$
Table 1: The field content for classical, 1st quantized, and 2nd quantized
scalar fields for the new covariant and symplectic approach are shown above. A
field $A(a)$ with a noncommutative conjugate $B(b)$ satisfies
$[A(a),B(b)]=i\delta(a-b)$. While KvN mechanics provides a wavefunction
$\psi(x,p)$, the study of classical fields over phase space (0th quantized)
such as $\phi(x,p)$ has been relatively unexplored. Classical fields such as
$\phi(x)$ found in the Klein-Gordon equation are referred to as 1st quantized
fields, since they are taken as a function of spacetime and have classical
equations of motion with $\hbar$. 2nd quantized quantum fields such as
$\Phi(x)$ are found as deformations of the 1st quantized fields.

To our knowledge, this work presents the following results for the first time
(or at least provided significant context and clarity):

  * •

Construction of proper relativistic KvN mechanics and relativistic KvN
algebra.

  * •

Generalization of KvN mechanics to poly-KvN relativistic fields as DDW
mechanics with poly-KvN algebra.

  * •

Covariant and symplectic Hamiltonian density for fields with corresponding
algebras.

  * •

2nd quantization via algebraic deformation of 2nd Fourier-phase space of
fields.

Section (II) introduces relativistic Koopman-von Neumann mechanics, gives a
review of De Donder-Weyl Hamiltonian densities for the Klein-Gordon, Dirac,
Yang-Mills, and linearized gravity theories, reconciles De Donder-Weyl
mechanics as a field theoretic generalization of Koopman-von Neumann
mechanics, and introduces a new covariant and canonical Hamiltonian (density)
with symplectic structure more similar to Dirac’s Hamiltonian. Section (III)
discusses quantization within the context of Koopman-von Neumann mechanics, De
Donder-Weyl mechanics, and the new canonical and covariant Hamiltonian
mechanics with symplectic structure. In particular, a new approach to 2nd
quantization is found by deforming the 2nd Fourier-phase space of fields.

##  II Manifestly covariant and canonical classical mechanics

To work towards manifestly canonical and covariant quantization of fields, the
classical Koopman-von Neumann mechanics is studied, as this contains a
wavefunction and a Hilbert space similar to quantum mechanics. For a
manifestly covariant formulation of relativistic phenomena, we update Koopman-
von Neumann mechanics to a relativistic formalism by focusing on the proper
time and a proper Hamiltonian. We demonstrate that the point particle action
in curved spacetime leads to a proper-relativistic Liouville operator that
vanishes, which is analogous to the so-called problem of time in quantum
gravity. This demonstrates that the problem of time has nothing to do with
quantum theory and relates more to the choice of Hamiltonian formulation for
relativistic dynamics. To bypass these issues, De Donder-Weyl mechanics is
introduced. To prepare for an understanding of canonical and covariant
quantization, the relativistic Koopman-von Neumann mechanics is improved by
considering relativistic fields with polysymplectic geometry of De Donder-Weyl
mechanics. Koopman-von Neumann-De Donder-Weyl algebras for classical fields
over spacetime and phase space are written down for the first time. A new
Hamiltonian density with symplectic geometry is introduced to give the
benefits of the covariant De Donder-Weyl Hamiltonian density with the
simplicity of Dirac’s covariant Hamiltonian density.

###  II.1 Classical relativistic Koopman-von Neumann mechanics

We start by introducing classical special relativity within Koopman-von
Neumann (KvN) mechanics. For an introduction to KvN mechanics, see Ref. [7].
The two-state-vector formalism interpretation of quantum mechanics has been
related to the entangled histories interpretation [58, 59, 60, 61, 62, 63, 64,
65]. Entangled histories allows for a Hilbert space for time, which has also
been discussed elsewhere [66, 67, 68, 69]. To the best of our knowledge, a
Hilbert space for time has never been discussed within KvN mechanics, which
helps provide a relativistic formulation of Koopman-von Neumann mechanics.

The standard Dirac bra-ket notation can be used for classical mechanics by
admitting a wavefunction $\psi(A)=\langle A|\psi\rangle$, and
$\psi^{*}(A)=\langle\psi|A\rangle$. A basis for phase space in classical
mechanics is provided with $|A\rangle=|x,p\rangle=|x\rangle\otimes|p\rangle$,
since position and momentum are independent. Unlike QM, the spacetime position
operator $\hat{x}^{\mu}$ commutes with the energy-momentum operator
$\hat{p}_{\nu}$ when acting on $\psi(x,p)$,

$\left[\hat{x}^{\mu},\hat{p}_{\nu}\right]\psi(x,p)=0.$ (2)

While classical non-relativistic phase space allows for
$\psi(\vec{x},\vec{p},t)$ as seven dimensions, the relativistic generalization
initially motivates 9 dimensions (8 independent) via
$\psi\left(x^{\mu},p_{\nu},\tau\right)$.

To provide proper time evolution of a Hilbert space in a classical setting, a
proper Hamiltonian $H$ is introduced from a proper Lagrangian $L$, such that a
canonical and covariant action is

$S=\int d\tau L=\int d\tau(v^{\mu}p_{\mu}-H),$ (3)

where $v^{\mu}=\frac{\partial x^{\mu}}{\partial\tau}$ is the relativistic
4-velocity and $p_{\mu}=\frac{\partial L}{\partial v^{\mu}}$ is the canonical
energy-momentum. Proper Hamilton’s equations are found by varying about the
classical path, giving

$\displaystyle\frac{\partial x^{\mu}}{\partial\tau}=\frac{\partial H}{\partial
p_{\mu}}=\\{x^{\mu},H\\}_{PB},$ (4) $\displaystyle\frac{\partial
p_{\mu}}{\partial\tau}=-\frac{\partial H}{\partial
x^{\mu}}=\\{p_{\mu},H\\}_{PB}$ (5)

where the relativistic Poisson brackets are given by

$\\{A,B\\}_{PB}=\frac{\partial A}{\partial x^{\mu}}\frac{\partial B}{\partial
p_{\mu}}-\frac{\partial A}{\partial p_{\mu}}\frac{\partial B}{\partial
x^{\mu}}.$ (6)

The proper Hamiltonian $H$ allows for a proper relativistic Liouville equation
in terms of the probability density $\rho(x,p,\tau)$,

$\frac{d\rho}{d\tau}=\frac{\partial\rho}{\partial\tau}+\frac{\partial
x^{\mu}}{\partial\tau}\frac{\partial\rho}{\partial x^{\mu}}+\frac{\partial
p_{\mu}}{\partial\tau}\frac{\partial\rho}{\partial
p_{\mu}}=\frac{\partial\rho}{\partial\tau}+\frac{\partial H}{\partial
p_{\mu}}\frac{\partial\rho}{\partial x^{\mu}}-\frac{\partial H}{\partial
x^{\mu}}\frac{\partial\rho}{\partial
p_{\mu}}=\frac{\partial\rho}{\partial\tau}+\\{\rho,H\\}_{PB}=\frac{\partial\rho}{\partial\tau}+i\hat{\mathfrak{L}}\rho=0.$
(7)

where $\hat{\mathfrak{L}}$ is the proper Liouville operator, which is
Hermitian in a manner similar to the Hamiltonian operator $\hat{H}$ in quantum
mechanics. The proper Liouville operator is found from the proper Liouville
equation after applying Hamilton’s equations. The proper Liouville operator
can be expressed with relativistic Poisson brackets, giving

$i\hat{\mathfrak{L}}A=\\{A,H\\}_{PB}=\frac{\partial A}{\partial
x^{\mu}}\frac{\partial H}{\partial p_{\mu}}-\frac{\partial A}{\partial
p_{\mu}}\frac{\partial H}{\partial x^{\mu}}.$ (8)

The proper Liouville operator acting on $x^{\mu}$ and $p_{\mu}$ gives proper
Hamilton’s equations, while acting on $\rho$ gives the proper Liouville
equation.

Koopman-von Neumann mechanics expresses probability densities
$\rho(x,p)=\psi^{*}(x,p)\psi(x,p)$ in terms of the Koopman-von Neumann
wavefunction $\psi(x,p)$. While relativistic field theory typically defines a
probability density of a quantum field by taking a time derivative after
generalizing the current to a 4-current, the Klein-Gordon field equation has
one additional time derivative not found in the Schrodinger equation. To
obtain a relativistic analogue of the KvN equations, we assume
$\rho(x,p,\tau)=\psi^{*}(x,p,\tau)\psi(x,p,\tau)=|\langle
x,p\mid\psi(\tau)\rangle|^{2}$. The proper Liouville operator is used to lead
to the proper Koopman-von Neumann equation,

$i\frac{\partial\psi}{\partial\tau}=\hat{\mathfrak{L}}\psi.$ (9)

This equation is a classical analogue of the Schrodinger equation (generalized
to a relativistic setting with proper time evolution).

In summary, classical KvN mechanics has four axioms

  1. 1.

The state of the system is representated by $|\psi\rangle$ in a complex
Hilbert space.

  2. 2.

An observable is a Hermitian operator $\hat{A}$ for an eigenvalue $A$ with
eigenstate $|A\rangle$ satisfying $\hat{A}|\psi\rangle=A|\psi\rangle$.

  3. 3.

The probability of $A$ is given by $P(A)=|\langle A\mid\psi(t)\rangle|^{2}$ as
the Born rule, which leads to instantaneous collapse of the wavefunction.

  4. 4.

The tensor product of subsystems leads to a description of the composite
system.

The third axiom should be questioned for a relativistic formulation of
classical mechanics, as retarded functions for information transfer of
measurements should be incorporated. For now, our interpretation is that the
wavefunction for observers infinitesimally close to the measurement should
observe the collapse of the wavefunction within an infinitesimal amount of
time.

The Poisson brackets of $x^{\mu}$ and $p_{\nu}$ describe the symplectic
structure of phase space, since

$\displaystyle\\{x^{\mu},p_{\nu}\\}_{PB}$ $\displaystyle=$
$\displaystyle\delta^{\mu}_{\nu},$ (10)
$\displaystyle\\{x^{\mu},x^{\nu}\\}_{PB}$ $\displaystyle=$
$\displaystyle\\{p_{\mu},p_{\nu}\\}_{PB}=0.$ (11)

While Dirac’s $i\hbar$ prescirption is coincidentally true, it is not general,
as shown by Groenewold [71]. The true quantization procedure relates more to
$p_{\mu}=\hbar k_{\mu}$, which can be understood more clearly from the
Koopman-von Neumann algebra. The KvN algebra is given by

$\displaystyle\left[\hat{x}^{\mu},\hat{k}_{\nu}\right]$ $\displaystyle=$
$\displaystyle i\delta_{\nu}^{\mu},$ (12)
$\displaystyle\left[\hat{q}^{\mu},\hat{p}_{\nu}\right]$ $\displaystyle=$
$\displaystyle i\delta_{\nu}^{\mu},$ (13)
$\displaystyle\left[\hat{x}^{\mu},\hat{p}_{\nu}\right]$ $\displaystyle=$
$\displaystyle\left[\hat{x}^{\mu},\hat{q}^{\nu}\right]=\left[\hat{p}_{\nu},\hat{k}_{\nu}\right]=\left[\hat{q}^{\mu},\hat{k}_{\nu}\right]=0,$
(14) $\displaystyle\left[\hat{x}^{\mu},\hat{x}^{\nu}\right]$ $\displaystyle=$
$\displaystyle\left[\hat{p}_{\mu},\hat{p}_{\nu}\right]=\left[\hat{k}_{\mu},\hat{k}_{\nu}\right]=\left[\hat{q}^{\mu},\hat{q}^{\nu}\right]=0.$
(15)

The KvN algebra establishes that we have both ${A,B}$ and $[A,B]$ in the
classical setting. Rather than applying
${A,B}\rightarrow\frac{1}{i\hbar}[A,B]$, quantization can be more rigorously
found by deforming the KvN algebra [4, 8].

Natural representations for the wavenumber operators $\hat{k}_{\nu}$ and
$\hat{q}^{\mu}$ in the phase space basis $|A\rangle=|x,p\rangle$ are

$\hat{k}_{\nu}=-i\partial_{\nu}\equiv-i\frac{\partial}{\partial
x^{\nu}},\quad\hat{q}^{\mu}=i\tilde{\partial}^{\mu}\equiv
i\frac{\partial}{\partial p_{\mu}}.$ (16)

The operators $\hat{x}^{\mu}$ and $\hat{p}_{\mu}$ lead to the following
eigenvalues when acting on a phase space eigenstate,

$\hat{x}^{\mu}|x,p\rangle=x^{\mu}|x,p\rangle,\quad\hat{p}_{\mu}|x,p\rangle=p_{\mu}|x,p\rangle.$
(17)

In the classical Koopman-von Neumann formulation, there is also a Fourier
space basis $|A\rangle=|k,q\rangle$. In this basis, the operators can be
represented with a bar, yet they must satisfy the same Koopman-von Neumann
algebra in Eq. (8). The position and momentum operators become differential
operators in the Fourier-conjugate basis,

$\hat{\bar{x}}^{\mu}=i\bar{\partial}^{\mu}=i\frac{\partial}{\partial
k_{\mu}},\quad\hat{\bar{p}}_{\mu}=-i\overline{\tilde{\partial}}_{\mu}=-i\frac{\partial}{\partial
q^{\mu}}.$ (18)

The wavenumber operators of phase space act on wavenumber eigenstates to
return wavenumber eigenvalues,

$\hat{\bar{k}}_{\mu}|k,q\rangle=k_{\mu}|k,q\rangle,\quad\hat{\bar{q}}^{\mu}|k,q\rangle=q^{\mu}|k,q\rangle.$
(19)

While this formulation leads to a manifestly relativistic KvN equation in
terms of proper time, it leads to a problem of time for the Lagrangian density
of a free point particle, as the proper Liouville operator leads to zero.
Consider the action for a point particle in curved spacetime given by the
proper length,

$S=-mc\int
d\tau\sqrt{-g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}}.$ (20)

The proper Lagrangian $L$ is given by $S=\int d\tau L$, which leads to a
proper covariant momentum,

$p_{\mu}=\frac{\delta L}{\delta
v^{\mu}}=mcg_{\mu\nu}v^{\nu}\frac{1}{\sqrt{-g_{\alpha\beta}v^{\alpha}v^{\beta}}}=mv_{\mu},$
(21)

where $v^{\mu}=\frac{dx^{\mu}}{d\tau}$ and $v^{\mu}v_{\mu}=-c^{2}$ was used to
find the result above. This allows for the solution of the proper Hamiltonian
$H$ to be found,

$H=v^{\mu}p_{\mu}-L=mv^{\mu}v_{\mu}+mc\sqrt{-g_{\mu\nu}v^{\mu}v^{\nu}}=-mc^{2}+mc^{2}=0.$
(22)

In the quantum theory, the proper Hamiltonian density operator would also be
zero.

A similar problem occurs with the Wheeler-DeWitt equation, except the zero
Hamiltonian is interpretted as a constraint [72, 73]. With relativistic
Koopman-von Neumann mechanics, the proper Hamiltonian would be zero, which
leads to a zero Liouville operator for time evolution of phase space. This may
suggest that the problem of time is independent of quantum theory, but rather
is a problem with some Hamiltonian systems for relativistic theories. To avoid
this problem and keep the manifest Lorentz invariance, De Donder-Weyl
mechanics is pursued next.

###  II.2 Classical field theory with De Donder-Weyl mechanics

Classical non-relativistic Lagrangian formulations lead to Euler-Lagrange
equations involving a time derivative $\frac{d}{dt}$. This naturally leads to
a covariant momentum in terms of a velocity that takes a time derivative,
giving a Hamiltonian. Relativistic field theory generalizes the Euler-Lagrange
equations to include a partial derivative $\partial_{\mu}$. However, the so-
called canonical Hamiltonian formulation from Dirac does not consider a poly-
momentum from $\partial_{\mu}$ and instead resorts to the time derivative
within some frame of reference, which breaks manifest Lorentz symmetry and is
not covariant [36]. As we saw for the relativistic point particle, using the
proper time with $\frac{d}{d\tau}$ instead of $\frac{d}{dt}$ also leads to a
vanishing proper Hamiltonian. A natural framework for obtaining a manifestly
covariant and canonical Hamiltonian density that generalizes the notion of the
relativistic Euler-Lagrange equations is to introduce a poly-momentum that
uses $\partial_{\mu}$ instead of $\frac{d}{dt}$ [9, 10, 11]. Relativistic
field theories motivate polysymplectic geometry, since the Lagrangian density
integrates over spacetime, not time. To summarize, consider the Euler-Lagrange
equations in classical mechanics vs classical field theory with a Klein-Gordon
scalar, which lead to the corresponding Hamiltonian formulations as shown in
Table (2).

| Euler-Lagrange | Hamiltonian
---|---|---
Newtonian particles | $\frac{d}{dt}\frac{dL}{dv^{i}}-\frac{dL}{dx^{i}}=0$ | $H=v^{i}p_{i}-L$
Relativistic fields | $\partial_{\mu}\frac{\delta\mathcal{L}}{\delta\nu_{\mu}^{A}}-\frac{\delta\mathcal{L}}{\delta\phi^{A}}=0$ | $\mathcal{H}=\nu_{\mu}^{A}\pi^{\mu}_{A}-\mathcal{L}$
Table 2: A comparison of the Euler-Lagrange equations with the Hamiltonian
(density) for Newtonian mechanics and relativistic field theory is shown
above. Since the Euler-Lagrange equations contain a functional derivative with
respect to the poly-velocity $\nu_{\mu}=\partial_{\mu}\phi^{A}$, the De
Donder-Weyl (DDW) Hamiltonian density $\mathcal{H}$ with poly-momentum
$\pi^{\mu A}$ conjugate to any field $\phi^{A}$ is natural for a manifestly
covariant and canonical Hamiltonian formulation.

A manifest covariant and canonical formulation motivates the De Donder-Weyl
Hamiltonian density $\mathcal{H}$ that is different than the canonical
Hamiltonian density $\mathcal{H}_{D}$. The poly-velocity and poly-momentum
densities of any field $\phi$ are given by

$\displaystyle\nu_{\mu}$ $\displaystyle=$ $\displaystyle\partial_{\mu}\phi$
(23) $\displaystyle\pi^{\mu}$ $\displaystyle=$
$\displaystyle\frac{\delta\mathcal{L}}{\delta\partial_{\mu}\phi}$ (24)

Since $\partial_{0}=\frac{1}{c}\frac{\partial}{\partial t}$, the standard
canonical momentum density
$\pi=\frac{\delta\mathcal{L}}{\delta\dot{\phi}}=\frac{1}{c}\pi^{0}$. Consider
a different type of Poisson bracket that contains a covector index to express
the polysymplectic geometry,

$\\{A,B\\}_{\mu}=\frac{\delta A}{\delta\phi(x)}\frac{\delta
B}{\delta\pi^{\mu}(x)}-\frac{\delta A}{\delta\pi^{\mu}(x)}\frac{\delta
B}{\delta\phi(x)}=\int d^{4}y\delta^{(4)}(x-y)\left[\frac{\delta
A}{\delta\phi(x)}\frac{\delta B}{\delta\pi^{\mu}(y)}-\frac{\delta
A}{\delta\pi^{\mu}(y)}\frac{\delta B}{\delta\phi(x)}\right].$ (25)

The polysympletic structure of phase space over fields is characterized by
this poly-Poisson bracket, since

$\displaystyle\\{\phi(x),\pi^{\mu}(x)\\}_{\nu}$ $\displaystyle=$
$\displaystyle\delta^{\mu}_{\nu},$ (26)
$\displaystyle\\{\phi(x),\phi(x)\\}_{\mu}$ $\displaystyle=$
$\displaystyle\\{\pi^{\mu}(x),\pi^{\nu}(x)\\}_{\rho}=0.$ (27)

It is not clear if the literature has explored this precise type of bracket
with an additional index, as various different brackets have been discussed in
the context of polysymplectic geometry [23, 24, 52]. Other Poisson brackets
over fields lead to Dirac delta functions over space [1]. Our conventions for
the poly-Poisson brackets allow for an easy identification of the De Donder-
Weyl equations in a manner that mimics classical mechanics. Note that Dirac’s
$i\hbar$ prescription no longer holds, since the polysymplectic geometry has a
conjugate poly-momentum of a different rank than the original field. This does
not obscure second quantization, especially once the generalization of the
Koopman-von Neumann algebra is articulated.

The De Donder-Weyl Hamiltonian density is

$\mathcal{H}=\pi^{\mu}\nu_{\mu}-\mathcal{L}.$ (28)

The De Donder-Weyl Hamiltonian is different than the canonical Hamiltonian
introduced by Dirac, as it contains additional Legendre transformations for
spatial components of the poly-momentum and poly-velocity. The canonical form
of the action is therefore

$S=\int d^{4}x\left(\nu_{\mu}\pi^{\mu}-\mathcal{H}\right).$ (29)

The variation of the action with respect to a classical orbit
$\phi=\phi_{\textrm{cl}}+\delta\phi$ and
$\pi^{\mu}=\pi^{\mu}_{\textrm{cl}}+\delta\pi^{\mu}$ leads to

$\displaystyle\delta S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\left[\delta\nu_{\mu}\pi^{\mu}+\nu_{\mu}\delta\pi^{\mu}-\frac{\delta\mathcal{H}}{\delta\phi}\delta\phi-\frac{\delta\mathcal{H}}{\delta\pi^{\mu}}\delta\pi^{\mu}\right],$
$\displaystyle=$ $\displaystyle\int
d^{4}x\left[\left(\nu_{\mu}-\frac{\delta\mathcal{H}}{\delta\pi^{\mu}}\right)\delta\pi^{\mu}-\left(\partial_{\mu}\partial^{\mu}+\frac{\delta\mathcal{H}}{\delta\phi}\right)\delta\phi+\partial_{\mu}\left(\pi^{\mu}\delta\phi\right)\right].$

Assuming that boundary terms vanish, the De Donder-Weyl equations are found to
give

$\displaystyle\partial_{\mu}\phi$ $\displaystyle=$
$\displaystyle\frac{\delta\mathcal{H}}{\delta\pi^{\mu}}=\\{\phi,\mathcal{H}\\}_{\mu},$
(31) $\displaystyle\partial_{\mu}\pi^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{\delta\mathcal{H}}{\delta\phi}=\frac{1}{d}\\{\pi^{\mu},\mathcal{H}\\}_{\mu},$
(32)

where $\frac{\delta\pi^{\mu}}{\delta\pi^{\nu}}=\delta^{\mu}_{\nu}$ and
$\delta^{\mu}_{\mu}=d$ for $d$-dimensional spacetime.

The canonical Hamiltonian density allows for the derivation of field
equations. Before discussing quantization, we demonstrate how derivations of
classical equations of motion are simpler with the De Donder-Weyl Hamiltonian
density in comparison to the canonical Hamiltonian density. The Hamilton-
Jacobi theory for De Donder-Weyl mechanics for Klein-Gordon, Dirac, and gauge
fields has been previously discussed [12]. Kanatchikov has discussed a wide
variety of field theories with De Donder-Weyl mechanics, including p-form
electrodynamics, but a slight modification was made that leads to a
complicated quantization procedure that latches onto the Dirac operator [23,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. Next, we provide simple and
straightforward derivations of the Klein-Gordon, Dirac, Maxwell, and Yang-
Mills fields using more standard notation familiar to quantum field theorists.
Later, we will also show that DDW quantization can be realized after
generalizing the KvN algebra.

####  II.2.1 Klein-Gordon scalar field

The action of a real and massive Klein-Gordon scalar field $\phi(x)$ with an
arbitrary potential $V(\phi)$ is

$S_{KG}=\int
d^{4}x\left(\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{m^{2}c^{2}}{2\hbar^{2}}\phi^{2}-V(\phi)\right),$
(33)

where the mostly-minus metric $\eta_{\mu\nu}$ is used. Throughout,
$\hbar=c=1$. The Klein-Gordon poly-velocity and poly-momentum are

$\nu_{\mu}=\partial_{\mu}\phi,\quad\pi^{\mu}=\frac{\delta\mathcal{L}}{\delta\nu_{\mu}}=\partial^{\mu}\phi.$
(34)

The De Donder-Weyl Hamiltonian density for the Klein-Gordon theory is

$\mathcal{H}=v_{\mu}\pi^{\mu}-\mathcal{L}=\frac{1}{2}\pi^{\mu}\pi_{\mu}+\frac{m^{2}}{2}\phi^{2}+V(\phi)$
(35)

The De Donder-Weyl equations give

$\displaystyle\partial_{\mu}\phi$ $\displaystyle=$
$\displaystyle\frac{\delta\mathcal{H}}{\delta\pi^{\mu}}=\pi_{\mu},$ (36)
$\displaystyle\partial_{\mu}\pi^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{\delta\mathcal{H}}{\delta\phi}=-m^{2}\phi-\frac{\delta
V}{\delta\phi}$ (37)

Plugging the first equation into the second equation gives the Klein-Gordon
equation,

$\left(\partial_{\mu}\partial^{\mu}+m^{2}\right)\phi=-\frac{\delta
V}{\delta\phi}.$ (38)

Since this Hamiltonian formulation is manifestly relativistic, it is more
straightforward than the canonical Hamiltonian approach. The tradeoff was to
introduce a poly-momentum vector field that is conjugate to the scalar field,
but the poly-velocity is already in the standard Euler-Lagrange equations for
fields.

Since it will be instructive for a later demonstration, we also consider a
complex Klein-Gordon field with the following action,

$S=\int
d^{4}x\left(\partial_{\mu}\phi\partial^{\mu}\phi^{*}-m^{2}|\phi|^{2}-V(\phi,\phi^{*})\right).$
(39)

Since $\phi=\phi_{1}+i\phi_{2}$, the poly-Poisson brackets must sum over both
$\phi$ and $\phi^{*}$. The following poly-velocities and poly-momenta are
found,

$\displaystyle\nu_{\mu}=\partial_{\mu}\phi,$
$\displaystyle\pi^{\mu}=\frac{\delta\mathcal{L}}{\delta\nu_{\mu}}=\partial^{\mu}\phi^{*},$
$\displaystyle\nu^{*}_{\mu}=\partial_{\mu}\phi^{*},$
$\displaystyle\pi^{*\mu}=\frac{\delta\mathcal{L}}{\delta\nu_{\mu}^{*}}=\partial^{\mu}\phi,$
(40)

where $\nu_{\mu}^{*}$ is defined as the poly-velocity of $\phi^{*}$, not
immediately as the complex conjugate of $\nu_{\mu}$. The poly-Poisson bracket
must consider both $\phi$ and $\phi^{*}$,

$\\{A,B\\}_{\mu}=\frac{\delta A}{\delta\phi}\frac{\delta
B}{\delta\pi^{\mu}}-\frac{\delta A}{\delta\pi^{\mu}}\frac{\delta
B}{\delta\phi}+\frac{\delta A}{\delta\phi^{*}}\frac{\delta
B}{\delta\pi^{*\mu}}-\frac{\delta A}{\delta\pi^{*\mu}}\frac{\delta
B}{\delta\phi^{*}}.$ (41)

For later convenience, the De Donder-Weyl Hamiltonian density is given by

$\mathcal{H}=\nu_{\mu}\pi^{\mu}+\nu_{\mu}^{*}\pi^{*\mu}-\mathcal{L}=\pi^{\mu}\pi_{\mu}^{*}+m^{2}|\phi|^{2}+V(\phi,\phi^{*}).$
(42)

####  II.2.2 Dirac spin-1/2 field

Next, the DDW Hamiltonian density for Dirac spinors is found. The spin-1/2
Dirac field $\psi^{\alpha}$ has the following action,

$S=\int
d^{4}x\bar{\psi}_{\alpha}\left(i\left(\gamma^{\mu}\right)^{\alpha}{}_{\beta}\partial_{\mu}-\delta_{\beta}^{\alpha}m\right)\psi^{\beta}=-\int
d^{4}x\left(\partial_{\mu}\bar{\psi}_{\alpha}i\left(\gamma^{\mu}\right)^{\alpha}{}_{\beta}+\delta_{\beta}^{\alpha}m\right)\psi^{\beta}$
(43)

where it is convenient to integrate by parts to identify the poly-momentum of
$\bar{\psi}_{\alpha}$,

$\displaystyle\nu_{\mu}^{\alpha}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\psi^{\alpha},\qquad\pi_{\alpha}^{\mu}=\frac{\delta\mathcal{L}}{\delta\nu_{\mu}^{\alpha}}=i\bar{\psi}_{\beta}\left(\gamma^{\mu}\right)^{\beta}{}_{\alpha},$
(44) $\displaystyle\bar{\nu}_{\mu\alpha}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\bar{\psi}_{\alpha},\qquad\bar{\pi}^{\mu\alpha}=\frac{\delta\mathcal{L}}{\delta\bar{\nu}_{\mu\alpha}}=-i\left(\gamma^{\mu}\right)^{\alpha}{}_{\beta}\psi^{\beta}.$
(45)

While the canonical Hamiltonian only considers a Legendre transformation with
$\psi$, here, we shift by both $\psi$ and $\bar{\psi}$, despite their
solutions being dependent, similar to the complex Klein-Gordon field. To find
the Hamiltonian, consider the Lagrangian that uses a mix of both terms, which
is what is typically done for Dirac spinors in curved spacetime [74],

$\mathcal{L}=\frac{1}{2}\bar{\psi}_{\alpha}i\left(\gamma^{\mu}\right)^{\alpha}{}_{\beta}\partial_{\mu}\psi^{\beta}-\frac{1}{2}\partial_{\mu}\bar{\psi}_{\alpha}i\left(\gamma^{\mu}\right)^{\alpha}{}_{\beta}\psi^{\beta}-m\bar{\psi}_{\alpha}\psi^{\alpha}.$
(46)

The De Donder-Weyl Hamiltonian for Dirac spinors is therefore

$\mathcal{H}=\pi_{\alpha}^{\mu}\nu_{\mu}^{\alpha}+\bar{\nu}_{\mu\alpha}\bar{\pi}^{\mu\alpha}-\mathcal{L}=\frac{1}{2}\pi_{\alpha}^{\mu}\partial_{\mu}\psi^{\alpha}+\frac{1}{2}\partial_{\mu}\bar{\psi}_{\alpha}\bar{\pi}^{\mu\alpha}+m\bar{\psi}_{\alpha}\psi^{\alpha}=\partial_{\mu}\bar{\psi}_{\alpha}\bar{\pi}^{\mu\alpha}+m\bar{\psi}_{\alpha}\psi^{\alpha},$
(47)

where the last term integrated by parts and dropped the boundary term. The De
Donder-Weyl equations of the conjugate spinor give equations of motion for the
Dirac spinor,

$\displaystyle\partial_{\mu}\bar{\psi}_{\alpha}$ $\displaystyle=$
$\displaystyle\frac{\delta\mathcal{H}}{\partial\bar{\pi}^{\mu\alpha}}=\partial_{\mu}\bar{\psi}_{\alpha},$
(48) $\displaystyle\partial_{\mu}\bar{\pi}^{\mu\alpha}$ $\displaystyle=$
$\displaystyle-\frac{\delta\mathcal{H}}{\delta\bar{\psi}_{\alpha}}=-m\psi^{\alpha}.$
(49)

Here, the first-order action leads to a trivial first equation that does not
give the definition of the conjugate poly-momentum. Nevertheless, plugging in
the known poly-momentum leads to the Dirac equation,

$i\left(\gamma^{\mu}\right)^{\alpha}{}_{\beta}\partial_{\mu}\psi^{\beta}=m\psi^{\alpha}.$
(50)

####  II.2.3 Yang-Mills spin-1 field

Next, the DDW Hamiltonian for Yang-Mills theory is derived. The spin-1 Yang-
Mills field has the following action,

$S=\int d^{4}x\left(-\frac{1}{4}F_{\mu\nu}^{A}F^{\mu\nu A}-J^{\mu
A}A_{\mu}^{A}\right),$ (51)

where the Yang-Mills field strength $F_{\mu\nu}^{A}$ is found in terms of the
gauge potential $A_{\mu}^{A}$,

$F_{\mu\nu}^{A}=\partial_{\mu}A_{\nu}^{A}-\partial_{\nu}A_{\mu}^{A}-gf^{ABC}A_{\mu}^{b}A_{\nu}^{c}.$
(52)

The Yang-Mills poly-velocity and poly-momentum are

$\nu_{\mu\nu}{}^{A}=\partial_{\mu}A_{\nu}^{A},\quad\pi^{\mu\nu A}=-F^{\mu\nu
A}.$ (53)

The De Donder-Weyl Hamiltonian density for Yang-Mills theory is

$\mathcal{H}=-\frac{1}{4}\pi_{\mu\nu}^{A}\pi^{\mu\nu A}+\frac{g}{2}\pi^{\mu\nu
A}f^{ABC}A_{\mu}^{B}A_{\nu}^{C}+J^{\mu A}A_{\mu}^{A}.$ (54)

The De Donder-Weyl equations give the equations of motion for Yang-Mills
theory when asserting that $\pi^{\mu\nu A}$ is antisymmetric,

$\displaystyle\frac{1}{2}\left(\partial_{\mu}A_{\nu}^{A}-\partial_{\nu}A_{\mu}^{A}\right)$
$\displaystyle=$ $\displaystyle\frac{\delta\mathcal{H}}{\delta\pi^{\mu\nu
A}}=-\frac{1}{2}\pi_{\mu\nu}{}^{A}+\frac{g}{2}f^{ABC}A_{\mu}^{B}A_{\nu}^{C}$
(55) $\displaystyle\partial_{\mu}\pi^{\mu\nu A}$ $\displaystyle=$
$\displaystyle-\frac{\delta\mathcal{H}}{\delta
A_{\nu}^{A}}=-\frac{g}{2}\pi^{\nu\mu
B}f^{BAC}A_{\mu}^{C}-\frac{g}{2}\pi^{\mu\nu C}f^{CBA}A_{\mu}^{B}-J^{\nu
A}=gf^{ABC}A_{\mu}^{B}\pi^{\mu\nu C}-J^{\nu A}$ (56)

The first equation leads to $\pi^{\mu\nu A}=-F^{\mu\nu A}$, which when
plugging this into the second equation gives the Yang-Mills equations of
motion,

$\partial_{\mu}F^{\mu\nu A}=J^{\nu A}+gf^{ABC}A_{\mu}^{B}F^{\mu\nu C}=j^{\nu
A},$ (57)

where $j^{\nu A}$ is the gauge-dependent current. The equations of motion are
also expressed as $D_{\mu}F^{\mu\nu A}=J^{\nu A}$, where $D_{\mu}$ is the
covariant derivative. Choosing the Abelian gauge group $U(1)$ would lead to
Maxwell’s electrodynamics. This completes the derivation of the equations of
motion for all of the field content in the standard model within De Donder-
Weyl mechanics.

Since the gauge-dependent Yang-Mills current will be inspirational for
generalizing the Liouville equation in the relativistic setting, consider how
charge conservation can be stated with the covariant or partial derivative,

$D_{\mu}J^{\mu A}=\partial_{\mu}J^{\mu A}-gf^{ABC}A_{\mu}^{B}J^{\mu
C}=0,\qquad\partial_{\mu}j^{\mu A}=0,$ (58)

where $J^{\mu A}$ is the gauge-covariant current, while $j^{\mu A}$ is the
conserved Noether current. Later, the poly-Liouville equation for probability
4-current conservation of relativistic fields will be derived in analogy with
$D_{\mu}J^{\mu A}=0$.

####  II.2.4 Linearized gravity with spin-2 field

In linearized gravity, the spacetime metric $g_{\mu\nu}$ is perturbed with
respect to a background Minkowski metric $\eta_{\mu\nu}$,

$\displaystyle g_{\mu\nu}$ $\displaystyle=$ $\displaystyle\eta_{\mu\nu}+\kappa
h_{\mu\nu},$ (59) $\displaystyle g^{\mu\nu}$ $\displaystyle=$
$\displaystyle\eta^{\mu\nu}-\kappa h^{\mu\nu}+\mathcal{O}(\kappa^{2}),$ (60)

where $h_{\mu\nu}(x)$ is the linearized gravitational field and
$\kappa=\sqrt{\frac{32\pi G}{c^{3}}}$. The linearized gravity action is

$S=\int
d^{4}x\left(\frac{1}{2}\partial^{\rho}h^{\mu\nu}\partial_{\rho}h_{\mu\nu}-\partial_{\mu}h^{\mu\rho}\partial^{\nu}h_{\nu\rho}+\partial^{\mu}h_{\mu\nu}\partial^{\nu}h-\frac{1}{2}\partial_{\mu}h\partial^{\mu}h\right),$
(61)

where $h=h_{\mu\nu}\eta^{\mu\nu}$. The trace-reversed metric
$\bar{h}_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h$ is the field mapped
from the Yang-Mills gauge field with respect to the radiative double copy
[75]. This allows for a simplification in the action and the field equations
as well, since

$S=\int
d^{4}x\left(\frac{1}{2}\partial^{\rho}\bar{h}^{\mu\nu}\partial_{\rho}\bar{h}_{\mu\nu}-\frac{1}{4}\partial_{\mu}\bar{h}\partial^{\mu}\bar{h}\right).$
(62)

Plugging $\bar{h}_{\mu\nu}$ in terms of $h_{\mu\nu}$ of the action above leads
to

$S=\int
d^{4}x\left(\frac{1}{2}\partial^{\rho}h^{\mu\nu}\partial_{\rho}h_{\mu\nu}-\frac{1}{4}\partial_{\mu}h\partial^{\mu}h\right).$
(63)

Consider the poly-velocity and poly-momentum of the trace-reversed metric,

$\displaystyle\bar{\nu}_{\rho\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\rho}\bar{h}_{\mu\nu},$ (64)
$\displaystyle\bar{\pi}^{\rho\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\delta\mathcal{L}}{\delta\bar{\nu}_{\rho\mu\nu}}=\partial^{\rho}\bar{h}^{\mu\nu}-\frac{1}{2}\eta^{\mu\nu}\partial^{\rho}\bar{h}=\partial^{\rho}h^{\mu\nu},$
(65)

where the poly-momentum of the trace-reversed metric is the partial derivative
of the metric. The De Donder-Weyl Hamiltonian density for linearized gravity
is

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\bar{\nu}_{\rho\mu\nu}\bar{\pi}^{\rho\mu\nu}-\mathcal{L}=\frac{1}{2}\partial^{\rho}\bar{h}^{\mu\nu}\partial_{\rho}\bar{h}_{\mu\nu}-\frac{1}{4}\partial_{\rho}\bar{h}\partial^{\rho}\bar{h}=\mathcal{L}$
(66) $\displaystyle=$
$\displaystyle\frac{1}{2}\bar{\pi}^{\rho\mu\nu}\bar{\pi}_{\rho\mu\nu}-\bar{\pi}_{\mu}^{\,\,\,\mu\rho}\bar{\pi}^{\nu}_{\,\,\,\nu\rho}+\bar{\pi}^{\mu}_{\,\,\,\mu\nu}\eta^{\alpha\beta}\bar{\pi}^{\nu}_{\,\,\,\alpha\beta}-\frac{1}{2}\eta^{\alpha\beta}\eta^{\rho\sigma}\bar{\pi}_{\mu\alpha\beta}\bar{\pi}^{\mu}_{\,\,\,\rho\sigma},$

where the massless linearized graviton’s De Donder-Weyl Hamiltonian density is
the same as the Lagrangian density, which is encouraging, as there is only
gravitational kinetic energy present. The De Donder-Weyl equations give

$\displaystyle\partial_{\mu}\bar{h}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{\delta\mathcal{H}}{\delta\bar{\pi}^{\mu\alpha\beta}}=\bar{\pi}_{\mu\alpha\beta}-2\eta_{\mu\alpha}\bar{\pi}^{\nu}_{\,\,\,\nu\beta}+\eta_{\mu\alpha}\eta^{\rho\sigma}\bar{\pi}_{\beta\rho\sigma}+\bar{\pi}^{\rho}_{\,\,\,\rho\mu}\eta_{\alpha\beta}-\eta_{\alpha\beta}\eta^{\rho\sigma}\bar{\pi}_{\mu\rho\sigma},$
(67) $\displaystyle\partial_{\mu}\bar{\pi}^{\mu\alpha\beta}$ $\displaystyle=$
$\displaystyle-\frac{\delta\mathcal{H}}{\delta\bar{h}_{\alpha\beta}}=0.$ (68)

The equation of motion can be found by solving the first equation for
$\pi^{\mu\alpha\beta}$ and plugging into the second equation.

The first equation of motion differs from the assignment of the poly-momentum
specifically unless the De Donder gauge condition is taken,

$\partial_{\mu}\bar{h}^{\mu\nu}=0.$ (69)

This “gauge condition” is sometimes called the harmonic, Hilbert, Lorenz,
Lorentz, or Fock gauge. Specifically when this gauge is taken, then the
definition of the poly-momentum found from the Lagrangian density can be
found, which leads to the following vacuum equations of motion,

$\partial_{\mu}\partial^{\mu}h^{\alpha\beta}=0.$ (70)

Since the stress-energy-momentum tensor $T^{\mu\nu}$ is given by

$T^{\mu\nu}=-\frac{2c}{\sqrt{-g}}\frac{\delta\sqrt{-g}\mathcal{L}_{M}}{\delta
g_{\mu\nu}}=-2c\frac{\delta\mathcal{L}_{M}}{\delta
g_{\mu\nu}}+c\mathcal{L}g^{\mu\nu},$ (71)

it is understood that $\kappa\bar{h}_{\mu\nu}$ will couple to
$\bar{T}^{\mu\nu}$ at lowest order. This implies that the matter Lagrangian is

$S_{M}=-\frac{\kappa}{2c}\int
d^{4}x\sqrt{-g}\bar{T}^{\mu\nu}\bar{h}_{\mu\nu}.$ (72)

To derive the De Donder-Weyl Hamiltonian for linearized gravity in another
manner, consider the solution for $\partial_{\mu}\bar{h}_{\alpha\beta}$ in
terms of $\pi_{\mu\alpha\beta}$ from Eq. (67),

$\bar{\nu}_{\mu\alpha\beta}=\partial_{\mu}\bar{h}_{\alpha\beta}=\bar{\pi}_{\mu\alpha\beta}-\frac{1}{2}\eta_{\alpha\beta}\bar{\pi}_{\mu}\equiv\pi_{\mu\alpha\beta},$
(73)

A reciprocal relationship is found between the trace-reversed poly-velocity
$\bar{\nu}_{\mu\alpha\beta}$ with the trace reverse of the trace-reversed
poly-momentum $\pi_{\mu\alpha\beta}$. This allows for the De Donder-Weyl
Hamiltonian density to be written as

$\mathcal{H}=\frac{1}{2}\pi_{\mu\alpha\beta}\pi^{\mu\alpha\beta}-\frac{1}{4}\pi_{\mu}\pi^{\mu}+\frac{8\pi
G}{c^{4}}\kappa\bar{T}^{\mu\nu}\bar{h}_{\mu\nu}=\frac{1}{2}\bar{\pi}_{\mu\alpha\beta}\bar{\pi}^{\mu\alpha\beta}-\frac{1}{4}\bar{\pi}_{\mu}\bar{\pi}^{\mu}+\frac{8\pi
G}{c^{4}}\kappa\bar{T}^{\mu\nu}\bar{h}_{\mu\nu},$ (74)

where $\bar{\pi}_{\mu}=-\pi_{\mu}=\partial_{\mu}h=-\partial_{\mu}\bar{h}$. For
this Hamiltonian density, the De Donder gauge was not explicitly solved for,
but the reciprocal relationship of the polymomenta and the polyvelocities
implies the De Donder gauge, as the De Donder-Weyl-Hamilton equations from the
Hamiltonian density above lead to

$\displaystyle\partial_{\mu}\bar{h}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{\delta\mathcal{H}}{\delta\bar{\pi}^{\mu\alpha\beta}}=\bar{\pi}_{\mu\alpha\beta}-\frac{1}{2}\eta_{\alpha\beta}\bar{\pi}_{\mu}=\pi_{\mu\alpha\beta},$
(75) $\displaystyle\partial_{\mu}\bar{\pi}^{\mu\alpha\beta}$ $\displaystyle=$
$\displaystyle-\frac{\delta\mathcal{H}}{\delta\bar{h}_{\alpha\beta}}=-\kappa\frac{8\pi
G}{c^{4}}\bar{T}^{\alpha\beta}.$ (76)

The trace-reversal of the second equation above after plugging in the first
equation leads to the equations of motion for linearized gravity sourced by
matter in the De Donder gauge,

$\partial_{\mu}\pi^{\mu\alpha\beta}=\partial_{\mu}\partial^{\mu}\bar{h}^{\alpha\beta}=-\frac{8\pi
G\kappa}{c^{4}}T^{\alpha\beta}.$ (77)

The De Donder gauge is therefore a type of Hamiltonian constraint. Just as
Dirac’s canonical Hamiltonian formulation of general relativity reduces the
metricial degrees of freedom from 10 to 6, so does imposing the De Donder
gauge condition [70]. For general relativity, $\bar{h}_{\mu\nu}$ would be
generalized to $\sqrt{-g}g_{\mu\nu}$, which has been discussed previously [76,
77]

###  II.3 Poly-Koopman-von Neumann mechanics as De Donder-Weyl mechanics

Next, we generalize relativistic Koopman-von Neumann mechanics to classical
field theory with polysymplectic fields and recover De Donder-Weyl mechanics.
First, the Koopman-von Neumann mechanics must be generalized to work on Fock
spaces to describe classical fields. Second, a poly-Liouville operator
$\hat{\mathfrak{L}}_{\mu}$ is constructed, which leads to a poly-Liouville
equation for probability 4-current conservation. The classical analogue of a
poly-Schrodinger equation with $\partial_{\mu}$ instead of
$\frac{\partial}{\partial t}$ is also found, which corresponds to the first De
Donder-Weyl equation, while the second De Donder-Weyl equation of motion is
interpreted as a poly-Schrodinger equation for the conjugate poly-momentum.
Finally, we formulate two types of poly-Koopman-von Neumann algebras with
covariant and canonical commutation relations over a classical Fourier-phase
space of functionals with polysymplectic geometry for classical and 1st
quantized fields.

####  II.3.1 Sketch of assumptions for axiomatic interacting field theories

A Fock space is defined as the tensor product of $N$-particle Hilbert spaces.
In quantum field theory, a non-interacting bare vacuum $|0\rangle$ is
typically presumed to be embedded in Minkowski space with the Wightman or
Haag-Kastler axioms [78, 79]. Hall and Wightman concluded from Haag’s theorem
that there is not a single Hilbert space representation for the free and
interacting fields [80]. We embrace this notion by considering a dynamical
discrete spectrum of the renormalized (physical) vacuum state $|\Omega\rangle$
that is uniquely determined by the phase space geometry, which is not found in
$|0\rangle$. The renormalized vacuum $|\Omega\rangle$ need not have the same
spectrum as $|0\rangle$, since the setting of the energy scale relates to the
types of measurements that can be performed. We suppose that algebraic quantum
field theory can be constructed via operator algebras that need not be related
to Minkowski space in the interacting field theory. From a philosophical
perspective, QFTs should include gravity, since all fields carry energy and
momentum as a source of gravitation.

While quantum mechanics contains a measurement problem, quantum field theory
does not in the same sense, as the renormalized measurement apparatus
$\langle\Omega|$ contains additional information not found in the non-
interacting bare template of abstract measurements $\langle 0|$. The true
state of measurement is defined by a destruction operator acting on a
renormalized vacuum $\langle\chi|=\langle\Omega|\hat{a}$, which denotes a
notion of measurement via absorption of quantum information. Quantum mechanics
has no destruction operators without Fock spaces.

Since Fock spaces are used in field theory rather than Hilbert spaces, the
wavefunction must be replaced with raising and lowering operators associated
with quantum fields. If a field $\phi$ has $a^{\dagger}$ and $a$ as creation
and annihilation operators that act on a renormalized vacuum state
$|\Omega\rangle$, then projection operators are replaced via
$\left|s^{\prime}\right\rangle\langle s|\rightarrow
a_{s^{\prime}}^{\dagger}a_{s}$. Without rigorously developing axiomatic
interacting QFT, the progression from wavefunctions $\psi(x)$ to (classical or
quantum) fields $\phi(x)$ is aided by the Dirac bra-ket notation,

$\hat{\mathcal{O}}\psi(x)\equiv\langle
x|\hat{\mathcal{O}}|\psi\rangle\rightarrow\mathcal{O}\phi(x)\left\langle
x\left|\hat{\mathcal{O}}a^{\dagger}\right|\Omega\right\rangle=\langle
x|\hat{\mathcal{O}}|\phi\rangle,$ (78)

where a single-particle state $|\phi\rangle=a^{\dagger}|\Omega\rangle$ was
constructed above. Without loss of generality, $|\phi\rangle$ can contain an
arbitrary number of creation operators acting on $|\Omega\rangle$. In this
manner, the classical Koopman-von Neumann wavefunction can be appropriately
generalized to classical Koopman-von Neumann fields, including the
relativistic Klein-Gordon field.

####  II.3.2 Poly-Koopman-von Neumann mechanics

Manifestly covariant and canonical evolution can be found from the De Donder-
Weyl equations, which motivates the introduction of a covector poly-Liouville
operator in poly-Koopman-von Neumann mechanics. The poly-Liouville operator is
introduced as

$i\hat{\mathfrak{L}}_{\mu}A=\\{A,\mathcal{H}\\}_{\mu}=\sum_{\phi}\left(\frac{\delta\mathcal{H}}{\delta\pi^{\mu}}\frac{\delta}{\delta\phi}-\frac{\delta\mathcal{H}}{\delta\phi}\frac{\delta}{\delta\pi^{\mu}}\right)A.$
(79)

If a wavefunction was evolved, then the classical poly-Koopman-von Neumann
equation would be

$i\partial_{\mu}\psi=\hat{\mathfrak{L}}_{\mu}\psi.$ (80)

However, we desire fields, not wavefunctions. As mentioned, since Fock spaces
are constructed from Hilbert spaces, it is anticipated that a similar equation
should hold for classical fields over Fock space.

Consider the poly-Liouville operator acting on the polysymplectic fields
$\phi$ and $\pi^{\mu}$. The poly-Koopman-von Neumann equations for fields are
defined as the poly-Liouville operator acting on classical fields of
polysymplectic phase space, which are found by applying the De Donder-Weyl
equations,

$\displaystyle i\hat{\mathfrak{L}}_{\mu}\phi$ $\displaystyle=$
$\displaystyle\\{\phi,\mathcal{H}\\}_{\mu}=\frac{\delta\mathcal{H}}{\delta\pi^{\mu}}=\partial_{\mu}\phi,$
(81) $\displaystyle i\hat{\mathfrak{L}}_{\mu}\pi^{\nu}$ $\displaystyle=$
$\displaystyle\\{\pi^{\nu},\mathcal{H}\\}_{\mu}=-\frac{\delta\mathcal{H}}{\delta\phi}\frac{\delta\pi^{\nu}}{\delta\pi^{\mu}}=-\frac{\delta\mathcal{H}}{\delta\phi}\delta_{\mu}^{\nu}=\partial_{\rho}\pi^{\rho}\delta_{\mu}^{\nu}.$
(82)

These poly-Koopman-von Neumann equations replace the classical wavefunction
$\psi(x,p)$ with the classical fields $\phi$ and $\pi^{\mu}$. The poly-
Liouville operator $\hat{\mathfrak{L}}_{\mu}$ acting on $\pi^{\nu}$ does not
simply take the partial derivative $i\partial_{\mu}$, which is different than
how the Liouville operator gives time evolution of the Koopman-von Neumann
wavefunction similar to the Hamiltonian operator for the quantum wavefunction.
Nevertheless, $i\partial_{\mu}\pi^{\mu}$ gives De Donder-Weyl equations,

$\displaystyle i\partial_{\mu}\phi$ $\displaystyle=$
$\displaystyle-\hat{\mathfrak{L}}_{\mu}\phi=i\left\\{\phi,\mathcal{H}\right\\}_{\mu}=i\frac{\delta\mathcal{H}}{\delta\pi^{\mu}},$
(83) $\displaystyle i\partial_{\mu}\pi^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{1}{d}\hat{\mathfrak{L}}_{\mu}\pi^{\mu}=\frac{i}{d}\left\\{\pi^{\mu},\mathcal{H}\right\\}_{\mu}=-i\frac{\delta\mathcal{H}}{\delta\phi}.$
(84)

This demonstrates for the first time that the De Donder-Weyl equations can be
rewritten with a poly-Liouville operator, since
$\\{\pi^{\mu},\mathcal{H}\\}_{\mu}=-i\hat{\mathfrak{L}}_{\mu}\pi^{\mu}$.

The relationship of the Liouville equation and Hamilton’s equations motivates
a poly-Liouville equation from De Donder-Weyl equations, which replaces the
partial time derivative with the partial derivative of spacetime
$\partial_{\mu}$. This implies that the poly-Liouville equation should involve
a probability 4-current $J^{\mu}(\phi(x),\pi(x))$, rather than the probabilty
density $\rho$. The probability 3-current of a wavefunction generalized to a
4-current leads to the following probability density for $\phi$ in quantum
field theory,

$J^{\mu}_{QFT}=\frac{i\hbar}{2m}\left(\phi^{*}\partial^{\mu}\phi-(\partial^{\mu}\phi^{*})\phi\right).$
(85)

Classical theory does not have $\hbar$. Also, massless Klein-Gordon fields can
be studied. Taking inspiration from above, consider the following canonical
classical Klein-Gordon probability current for a complex scalar field,

$J^{\mu}(\phi(x),\pi(x))=\frac{i}{2}\left(\phi^{*}\pi^{*\mu}-\pi^{\mu}\phi\right).$
(86)

While the Liouville equation typically implements time evolution of global
phase space with Poisson brackets of a Hamiltonian, the poly-Liouville
equation implements local spacetime evolution with a Hamiltonian density. This
analogy implies for the existence of a total spacetime derivative
$\frac{d}{dx^{\mu}}$, which contains the partial derivative plus functional
derivatives with respect to the fields and their conjugate polymomenta. It
turns out to be trivial for real Klein-Gordon fields $\phi$, as
$\partial_{\mu}J^{\mu}=0$ and the Liouville operator on $J^{\mu}$ would
vanish. For complex Klein-Gordon fields, the poly-Liouville equation can be
derived by considering the following,

$\displaystyle\partial_{\mu}J^{\mu}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\left(\partial_{\mu}\phi^{*}\pi^{*\mu}+\phi^{*}\partial_{\mu}\pi^{*\mu}-\partial_{\mu}\pi^{\mu}-\pi^{\mu}\partial_{\mu}\phi\right)=\frac{i}{2}\left(\phi^{*}\partial_{\mu}\pi^{*\mu}-\partial_{\mu}\pi^{\mu}\phi\right),$
(87) $\displaystyle i\hat{\mathfrak{L}}_{\mu}J^{\mu}$ $\displaystyle=$
$\displaystyle\\{J^{\mu},\mathcal{H}\\}_{\mu}=\frac{\delta
J^{\mu}}{\delta\phi}\frac{\delta\mathcal{H}}{\delta\pi^{\mu}}-\frac{\delta
J^{\mu}}{\delta\pi^{\mu}}\frac{\delta\mathcal{H}}{\delta\phi}+\frac{\delta
J^{\mu}}{\delta\phi^{*}}\frac{\delta\mathcal{H}}{\delta\pi^{*\mu}}-\frac{\delta
J^{\mu}}{\delta\pi^{*\mu}}\frac{\delta\mathcal{H}}{\delta\phi^{*}}$ (88)
$\displaystyle=$
$\displaystyle\frac{id}{2}\left(\phi^{*}\partial_{\rho}\pi^{*\rho}-\phi\partial_{\rho}\pi^{\rho}\right).$
(89)

Putting these two terms together, the poly-Liouville equation for arbitrary
field theories is found to be

$\frac{dJ^{\mu}}{dx^{\mu}}\equiv\partial_{\mu}J^{\mu}-\frac{i}{d}\hat{\mathfrak{L}}_{\mu}J^{\mu}=\partial_{\mu}J^{\mu}-\frac{1}{d}\\{J^{\mu},\mathcal{H}\\}_{\mu}=\partial_{\mu}J^{\mu}-\frac{1}{d}\sum_{i}\partial_{\mu}\phi_{i}\frac{\delta
J^{\mu}}{\delta\phi_{i}}+\partial_{\nu}\pi_{i}^{\nu}\frac{\delta
J^{\mu}}{\delta\pi^{\mu}_{i}}=0,$ (90)

where the complex Klein-Gordon scalar field theory has
$\phi_{i}=(\phi,\phi^{*})$. The polysymplectic geometry leads to a contraction
between $\partial_{\nu}\pi^{\nu}$ in the last term above. Curiously, if
$d_{\mu}\equiv\frac{d}{dx^{\mu}}$ is thought of as a covariant derivative, the
poly-Liouville operator $\hat{\mathfrak{L}}_{\mu}$ is like a gauge field.

The poly-Liouville equation describes the conservation of a 4-current density,
whose structure is more similar to the second De Donder-Weyl equation. In this
sense, the equations for the 4-current and the conjugate poly-momentum are
similar, while the non-relativistic Liouville equation is more similar to the
first De Donder-Weyl equation. The poly-Liouville equation acting on classical
fields cannot be found by applying the chain rule due to the structure of
polysymplectic geometry. While this equation does give a partial derivative
that includes a time derivative as used in time evolution, the poly-Liouville
operator acting on $\phi$ gives $\pi^{\mu}$ rather than the evolution of
$\phi$, as the poly-Liouville operator acts on $\pi^{\mu}$ to give the
equations of motion for $\phi$ after plugging in the first equation. In
summary, poly-KvN mechanics is equivalent to DDW mechanics.

####  II.3.3 0th and 1st quantized poly-Koopman-von Neumann algebras

Finally, we consider poly-Koopman-von Neumann algebras that generalize the
Koopman-von Neumann algebra. Since phase space coordinates are replaced with
polysymplectic fields, Fourier-conjugate fields $\kappa$ and $\xi^{\mu}$ are
introduced,

$\begin{array}[]{ccccccc}x^{\mu}&&&\rightarrow&\phi&&\\\
k_{\nu}&=&-i\frac{\partial}{\partial
x^{\nu}}&\rightarrow&\kappa&\propto&-i\frac{\delta}{\delta\phi}\\\
p_{\rho}&&&\rightarrow&\pi^{\rho}&&\\\ q^{\sigma}&=&i\frac{\partial}{\partial
p_{\sigma}}&\rightarrow&\xi_{\sigma}&\propto&i\frac{\delta}{\delta\pi^{\sigma}}.\end{array}$
(91)

While it is common to derive the classical equations of motion for the Klein-
Gordon equation in terms of a scalar field $\phi(x)$, the poly-Koopman-von
Neumann approach to field theory leads to a different type of classical field,
as it may depend both on position and momentum. Since the polysymplectic
fields are generalizations of the wavefunction, the fully classical poly-
Koopman-von Neumann fields should depend on position and momentum such as
$\phi(x,p)$ or $\bar{\phi}(k,q)$ as Fourier-phase-space conjugates. First
quantization is realized as quantizing phase space coordinates with
$p_{\mu}=\hbar k_{\mu}$ without quantizing fields, which leads to $\phi(x)$
and $\phi(p)$ as Fourier conjugates. Second quantization applies quantization
of the fields. As such, classical field theory admits 0th quantized
(classical) and 1st quantized poly-Koopman-von Neumann algebras.

It is also worth clarifying that the study of KvN fields introduces the
concept of 1st and 2nd Fourier conjugation. At the classical phase space
level, a field $\phi(x,p)$ admits a 1st Fourier transformation to give
$\tilde{\phi}(k,q)$. Half-Fourier transformations giving $\tilde{\phi}(k,p)$
or $\tilde{\phi}(x,q)$ can be found. In another sense, a functional such as
$\mathcal{H}(\phi,\pi)$ may admit a 2nd Fourier transformation, leading to
$\tilde{\mathcal{H}}(\kappa,\xi)$. In general, $\kappa(k,q)$ is not the 1st
Fourier transform of $\phi(x,p)$, but rather $\kappa(x,p)$ is the 2nd Fourier
conjugate field to $\phi(x,p)$. This matches how $x^{\mu}$ and $p_{\nu}$
denote coordinates for (1st) phase space, while the covariant phase space
description refers to phase space over fields (2nd phase space). Kvn fields
require the articulation of of 1st and 2nd Fourier-phase space over
coordinates and fields as shown in Eq. (91). Throughout, we refer to 1st
quantization as the reduction of 1st Fourier-phase space, while 2nd
quantization is the reduction of 2nd Fourier-phase space.

While classical field theory is typically over spacetime, poly-KvN mechanics
motivates classical fields over phase space. The 0th quantized poly-KvN
algebra is therefore

$\displaystyle{[}\phi(x,p_{x}),\kappa(y,p_{y})]$ $\displaystyle=$
$\displaystyle i\delta^{(4)}(x-y)\delta^{(4)}(p_{x}-p_{y}),$ (92)
$\displaystyle{[}\xi_{\mu}(x,p_{x}),\pi^{\nu}(y,p_{y})]$ $\displaystyle=$
$\displaystyle i\delta_{\mu}^{\nu}\delta^{(4)}(x-y)\delta^{(4)}(p_{x}-p_{y}),$
(93) $\displaystyle{[}\phi(x,p_{x}),\pi^{\mu}(y,p_{y})]$ $\displaystyle=$
$\displaystyle 0,$ (94) $\displaystyle{[}\xi_{\mu}(x,p_{x}),\kappa(y,p_{y})]$
$\displaystyle=$ $\displaystyle 0,$ (95)

where all other commutation relations vanish. This algebra contains fields
that differ from standard classical field theory, as the Klein-Gordon scalar
field typically is $\phi(x)$, not $\phi(x,p)$.

Assuming $\phi(x)$ is a 1st-quantized field, the introduction of fields
$\kappa(x)$ and $\xi_{\sigma}(x)$ over spacetime implies that the following
covariant and canonical commutation relations should hold

$\displaystyle{[}\phi(x),\kappa(y)]$ $\displaystyle=$ $\displaystyle
i\delta^{4}(x-y),$ (96) $\displaystyle{[}\xi_{\mu}(x),\pi^{\nu}(y)]$
$\displaystyle=$ $\displaystyle i\delta_{\mu}^{\nu}\delta^{4}(x-y),$ (97)
$\displaystyle{[}\phi(x),\pi^{\mu}(y)]$ $\displaystyle=$ $\displaystyle 0,$
(98) $\displaystyle{[}\xi_{\mu}(x),\kappa(y)]$ $\displaystyle=$ $\displaystyle
0.$ (99)

The units of $\kappa(x,p)$ must differ from $\kappa(x)$ if we assume that the
units of $\phi(x)$ and $\phi(x,p)$ are the same. To understand second
quantization of fields, the classical fields satisfying the first-quantized
classical commutation relations are the most helpful.

The dimensional analysis of fields over phase space differs from fields over
spacetime. Consider a 0th quantized Klein-Gordon scalar field $\phi(x,p)$. If
such a theory is to exhibit phase space symmetry with Born’s reciprocity, then
the Klein-Gordon scalar would be massless if we consider that $Spin(4,4)$ or
$Spin(6,2)$ for relativistic phase space both contain the conformal group
$Spin(4,2)$. The conformal group has been studied as a shadow of phase space
[81] and conformal theories are massless. This ensures that the action
principle can be considered without $\hbar$, since the kinetic term is void of
$\hbar$, while the mass term is
$-\frac{1}{2}\left(\frac{mc}{\hbar}\right)^{2}\phi^{2}$. This gives
$\phi(x,p)$ the dimensions of $\sqrt{\frac{m}{t}}$. The conjugate poly-
momentum $\pi^{\mu}(x,p)$ has dimensions of $\frac{1}{l}\sqrt{\frac{m}{t}}$.
The dimensional analysis of 0th and 1st quantized fields are shown in Table
(3).

Field | 1st Quantized | Dimensions | Mass Dimensions
---|---|---|---
$\phi(x,p)$ | False | $\sqrt{\frac{m}{t}}$ | 1
$\pi^{\mu}(x,p)$ | False | $\frac{1}{l}\sqrt{\frac{m}{t}}$ | 2
$\kappa(x,p)$ | False | $\frac{1}{l^{8}}\left(\frac{t}{m}\right)^{\frac{9}{2}}$ | -1
$\xi_{\nu}(x,p)$ | False | $\frac{1}{l^{7}}\left(\frac{t}{m}\right)^{\frac{9}{2}}$ | -2
$\phi(x)$ | True | $\sqrt{\frac{m}{t}}$ | 1
$\pi^{\mu}(x)$ | True | $\frac{1}{l}\sqrt{\frac{m}{t}}$ | 2
$\kappa(x)$ | True | $\frac{1}{l^{4}}\sqrt{\frac{t}{m}}$ | 3
$\xi_{\nu}(x)$ | True | $\frac{1}{l^{3}}\sqrt{\frac{t}{m}}$ | 2
Table 3: The dimensional analysis of 0th and 1st quantized fields in KvNdDW
mechanics are depicted above.

Assuming that the dimensions of $\phi(x,p)$ and $\phi(x)$ are the same, then
the dimensions of $\int d^{4}p\kappa(x,p)$ are equal to $\kappa(x)$. First
quantization of fields over phase space may involve integrating out energy-
momentum, as

$\int d^{4}p_{y}[\phi(x,p_{x}),\kappa(y,p_{y})]=i\delta^{4}(x-y)\int
d^{4}p_{y}\delta^{4}(p_{x}-p_{y})=[\phi(x),\kappa(y)].$ (100)

In general, for the “phase space field basis,” the conjugate fields as
operators for 0th quantized fields are given by

$\displaystyle\kappa(x,p_{x})$ $\displaystyle=$
$\displaystyle-i\delta^{(4)}(x-y)\delta^{(4)}(p_{x}-p_{y})\frac{\delta}{\delta\phi(y,p_{y})},$
(101) $\displaystyle\xi_{\mu}(x,p_{x})$ $\displaystyle=$ $\displaystyle
i\delta^{(4)}(x-y)\delta^{(4)}(p_{x}-p_{y})\frac{\delta}{\delta\pi^{\mu}(y,p_{y})},$
(102)

while for 1st quantized fields,

$\displaystyle\kappa(x)$ $\displaystyle=$
$\displaystyle-i\delta^{(4)}(x-y)\frac{\delta}{\delta\phi(y)},$ (103)
$\displaystyle\xi_{\mu}(x)$ $\displaystyle=$ $\displaystyle
i\delta^{(4)}(x-y)\frac{\delta}{\delta\pi^{\mu}(y)}.$ (104)

###  II.4 A covariant Hamiltonian density closer to Dirac’s

Dirac introduced the so-called canonical quantization procedure and a
symplectic Hamiltonian density that is not manifestly Lorentz covariant. While
De Donder-Weyl mechanics allows for a canonical and covariant procedure, the
poly-symplectic geometry leads to complications. For instance, one trick is to
apply Dirac’s $i\hbar$ prescription to switch out Poisson brackets for
commutators. This cannot be done with poly-symplectic geometry, as the poly-
Poisson bracket has an index not contained in a commutator. Additionally,
study of Koopman-von Neumann dynamics demonstrates that the Poisson brackets
and commutation relations can be found separately in classical theory.
Alternatively, the Liouville operator in KvN mechanics is analogous with the
Hamiltonian in quantum mechanics, but the poly-Liouville operator
$\hat{\mathfrak{L}}_{\mu}$ is now a covector, while the Hamiltonian is a
scalar. This motivated the search for an analogue of the poly-Liouville
operator in quantum theory as a type of Hamiltonian covector density that
could be found in classical or quantum theory. Finding such an operator in
terms of $\tau^{\mu}$ will be shown to give a new covariant and canonical
Hamiltonian density with symplectic geometry. This combines the elegance of a
covariant formulation found with the De Donder-Weyl Hamiltonian and the
simplicity of the symplectic geometry found in Dirac’s Hamiltonian.

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

$\mathcal{H}^{\mu}=\hat{\tau}^{\rho}\nu_{\rho}\pi^{\mu}-\hat{\tau}^{\mu}\hat{\mathcal{L}}$
(105)

The timelike component of this operator contains the canonical momentum
$\pi^{0}$. When $\hat{\tau}^{\mu}$ is taken as purely time-like, $\int
d^{3}x\mathcal{H}^{0}$ is identical to the canonical Hamiltonian found within
a particular frame. For this reason, we call this the canonical poly-
Hamiltonian density.

To get time evolution in a specific frame and to connect back to Dirac’s
canonical quantization, consider a time-slice vector
$\hat{\tau}^{\mu}=\frac{\tau^{\mu}}{|\tau|}$ that is $(1,0,0,0)$ in the rest
frame. This same concept of a time-like unit vector has been considered by
Rovelli and Vidotto [82]. The poly-Liouville operator contracted with
$\hat{\tau}^{\mu}$ gives time evolution for arbitrary coordinates
$x^{\prime\mu}=\left(ct^{\prime},\vec{x}^{\prime}\right)$,

$i\frac{1}{c}\frac{\partial}{\partial
t^{\prime}}\phi(x)=i\hat{\tau}^{\mu}\partial_{\mu}\phi(x)=\hat{\tau}^{\mu}\hat{\mathfrak{L}}_{\mu}\phi(x)=i\hat{\tau}^{\mu}\pi_{\mu}.$
(106)

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 Schrodinger and KvN
dynamics are first-order in time, while Klein-Gordon is second order and DDW
contains two first-order equations.

Consider another Legengre transformation with a “velocity field” $\nu$ with
the new conjugate momentum field $\pi$ giving Dirac’s canonical momentum when
$\hat{\tau}^{\mu}=(1,0,0,0)$,

$\nu=\hat{\tau}^{\mu}\partial_{\mu}\phi=\hat{\tau}^{\mu}\nu_{\mu},\qquad\pi=\frac{\delta\mathcal{L}}{\delta\nu}=\hat{\tau}_{\mu}\frac{\delta\mathcal{L}}{\delta\nu_{\mu}}.$
(107)

By taking inspiration from the De Donder-Weyl Hamiltonian and striving for the
simplicity of Dirac (canonical) Hamiltonians symplectic geometry rather than
using polysymplectic geometry, a new type of Hamiltonian density
$\mathcal{H}_{cov}$ can be found that is covariant and in a sense canonical,

$\mathcal{H}_{cov}=\nu\pi-\mathcal{L}=\hat{\tau}^{\mu}\nu_{\mu}\hat{\tau}_{\nu}\pi^{\nu}-\mathcal{L}=\hat{\tau}_{\mu}\mathcal{H}^{\mu}$
(108)

While this new covariant Hamiltonian density was initially constructed
independently, it turns out that
$\mathcal{H}_{cov}=\mathcal{H}^{\mu}\hat{\tau}_{\mu}$. The Hamiltonian density
above can be considered canonical as a Legendre transformation of the
Lagrangian if
$\frac{\delta\mathcal{L}}{\delta\hat{\tau}^{\mu}\nu_{\mu}}=\hat{\tau}_{\mu}\frac{\delta\mathcal{L}}{\delta\nu_{\mu}}$.
For a time-like vector $\hat{\tau}^{\mu}$, a Lorentz transformation can always
be peformed to rotate $\hat{\tau}^{\mu}$ to $(1,0,0,0)$, which ensures that
the Hamiltonian density is canonical. While $\mathcal{H}^{\mu}$ is technically
not a Legendre transformation, $\mathcal{H}_{cov}$ is.

The 0th quantized algebras associated with the fields $\phi(x,p)$, $\pi(x,p)$,
$\kappa(x,p)$, and $\xi(x,p)$ motivates the following commutation relations,

$\displaystyle[\phi(x,p_{x}),\kappa(y,p_{y})]$ $\displaystyle=$ $\displaystyle
i\delta^{(4)}(x-y)\delta^{(4)}(p_{x}-p_{y}),$ (109)
$\displaystyle{[}\xi(x,p_{x}),\pi(y,p_{y})]$ $\displaystyle=$ $\displaystyle
i\delta^{(4)}(x-y)\delta^{(4)}(p_{x}-p_{y}).$ (110)

The “position field basis” for 0th quantized fields are

$\displaystyle\kappa(x,p_{x})$ $\displaystyle=$
$\displaystyle-i\delta^{(4)}(x-y)\delta^{(4)}(p_{x}-p_{y})\frac{\delta}{\delta\phi(y,p_{y})},$
(111) $\displaystyle\xi(x,p_{x})$ $\displaystyle=$ $\displaystyle
i\delta^{(4)}(x-y)\delta^{(4)}(p_{x}-p_{y})\frac{\delta}{\delta\pi(y,p_{y})}.$
(112)

The 1st quantized algebras associated with these fields $\phi(x)$, $\pi(x)$,
$\kappa(x)$, and $\xi(x)$ gives

$\displaystyle[\phi(x),\kappa(y)]$ $\displaystyle=$ $\displaystyle
i\delta^{(4)}(x-y),$ (113) $\displaystyle{[}\xi(x),\pi(y)]$ $\displaystyle=$
$\displaystyle i\delta^{(4)}(x-y).$ (114)

The “position field basis” for 0th and 1st quantized fields are

$\displaystyle\kappa(x)$ $\displaystyle=$
$\displaystyle-i\delta^{(4)}(x-y)\frac{\delta}{\delta\phi(y)},$ (115)
$\displaystyle\xi(x)$ $\displaystyle=$ $\displaystyle
i\delta^{(4)}(x-y)\frac{\delta}{\delta\pi(y)}.$ (116)

The symplectic two-form is typically given by $\eta=dx\wedge dp$, motivating a
$2d$-dimensional phase space for $d$-dimensional spacetime. To justify that
this new Hamiltonian 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 $\eta=\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)$,

$\\{A,B\\}=A\left(\begin{array}[]{cc}\overleftarrow{\frac{\partial}{\partial
x}}&\overleftarrow{\frac{\partial}{\partial
p}}\end{array}\right)\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)\left(\begin{array}[]{c}\frac{\partial}{\partial x}\\\
\frac{\partial}{\partial p}\end{array}\right)B=\frac{\partial A}{\partial
x}\frac{\partial B}{\partial p}-\frac{\partial A}{\partial p}\frac{\partial
B}{\partial x}.$ (117)

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

$\\{A,B\\}_{cov}=A\left(\begin{array}[]{cc}\overleftarrow{\frac{\delta}{\delta\phi}}&\overleftarrow{\frac{\delta}{\delta\pi}}\end{array}\right)\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)\left(\begin{array}[]{c}\frac{\delta}{\delta\phi}\\\
\frac{\delta}{\delta\pi}\end{array}\right)B=\frac{\delta
A}{\delta\phi}\frac{\delta B}{\delta\pi}-\frac{\delta
A}{\delta\pi}\frac{\delta B}{\delta\phi}.$ (118)

Mapping from the polysymplectic geometry of De Donder-Weyl mechanics to
symplectic structure is found simply by contracting $\hat{\tau}^{\mu}$ with
$\\{A,B\\}_{\mu}$. Separate from the commutation relations for the poly-KvN
algebra, the Poisson brackets describe how these symplectic fields lead to
canonical relations,

$\\{\phi,\pi\\}_{cov}=1,\qquad\\{\phi,\phi\\}_{cov}=\\{\pi,\pi\\}_{cov}=0.$
(119)

After completing this work, we realized that Witten, Zuckerman, and Crnkovic
have considered a symplectic current that is covariant [13, 14, 15]. Crnkovic
clarified how a symplectic current was the variation of the field times the
polymomentum, except the relationship to De Donder-Weyl mechanics was not
realized [16, 17]. 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 $\hat{\tau}_{\mu}$ relates to a choice of
spacelike hypersurfaces. Our formulation more easily provides a Hamiltonian
density that can be integrated over all of spacetime, rather than a spacelike
hypersurface.

A new set of Hamilton’s equations are found to be

$\displaystyle\hat{\tau}^{\mu}\partial_{\mu}\phi$ $\displaystyle=$
$\displaystyle\\{\phi,\mathcal{H}_{cov}\\}_{cov}=\frac{\delta\mathcal{H}_{cov}}{\delta\pi},$
(120) $\displaystyle\hat{\tau}^{\mu}\partial_{\mu}\pi$ $\displaystyle=$
$\displaystyle\\{\pi,\mathcal{H}_{cov}\\}_{cov}=-\frac{\delta\mathcal{H}_{cov}}{\delta\phi}.$
(121)

Note how these form of Hamilton’s equations apply to arbitrary fields and
their associated conjugate momentum fields. Kanatchikov introduced
$\gamma^{\mu}$ instead of $\hat{\tau}^{\mu}$, which then must act on elements
in a Clifford algebra. Kanatchikov discussed a symplectic form
$\Omega=-d\phi^{a}\wedge d\pi_{a}^{\mu}\wedge\omega_{\mu}$, where
$\omega=dx^{1}\wedge\dots dx^{n}$ and
$\omega_{\mu}=\partial_{\mu}\lrcorner\omega$. Kanatchikov’s $\omega_{\mu}$ is
defnied as the interior product of $\partial_{\mu}$ with $dx^{1}\wedge\dots
dx^{n}$ [23]. Note that Kanatchikov defines momenta
$\pi_{a}=\pi_{a}^{\mu}\omega_{\mu}$, which introduces a 3 volume, while we
focus on isolating the time-like component of a frame field to obtain results
more similar to Dirac’s canonical quantization.

####  II.4.1 Covariant Hamiltonian density for Klein-Gordon scalars

Due to the similarity with Dirac’s Hamiltonian, we will only review the
derivation of the Klein-Gordon equation with this new Hamiltonian. To work
with the interacting Klein-Gordon field, it is convenient to introduce an
orthogonal frame of vectors or frame field
$e_{a}^{\mu}=\left(\hat{\tau}^{\mu},\hat{x}^{\mu},\hat{y}^{\mu},\hat{z}^{\mu}\right)$.
For non-gravitational theories, $e_{a}^{\mu}$ simply encodes Lorentz
transformations between the global Minkowski manifold in some frame and local
coordinates that can be in a different frame. The local and global frames are
equivalent when $e_{a}^{\mu}$ is given by the identity matrix, which leads to
$\hat{\tau}^{\mu}=(1,0,0,0)$.

The Klein-Gordon action can be written as

$S=\int
d^{4}x\left[\frac{1}{2}\hat{\tau}^{\mu}\partial_{\mu}\phi\hat{\tau}_{\nu}\partial^{\nu}\phi+\frac{1}{2}e_{i}^{\mu}\partial_{\mu}\phi
e^{i}_{\nu}\partial^{\nu}\phi-V(\phi)\right],$ (122)

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

$\nu=\hat{\tau}^{\mu}\partial_{\mu}\phi,\qquad\pi=\frac{\delta\mathcal{L}}{\delta\nu}=\hat{\tau}_{\mu}\partial^{\mu}\phi.$
(123)

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

$\mathcal{H}_{cov}=\frac{1}{2}\left(\pi^{2}-e_{i}^{\mu}\partial_{\mu}\phi
e_{\nu}^{i}\partial^{\nu}\phi\right)+V(\phi).$ (124)

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

$\mathcal{H}_{cov}=\frac{1}{2}\left(\pi^{2}+\phi
e_{i}^{\mu}e_{\nu}^{i}\partial_{\mu}\partial^{\nu}\phi\right)+V(\phi)+\mbox{bdry},$
(125)

where we assume that $e_{i}^{\mu}$ only can specify local inertial frames that
correspond to global Lorentz transformations on Minkowski spacetime. Applying
the new set of Hamilton’s equations gives

$\displaystyle\hat{\tau}^{\mu}\partial_{\mu}\phi$ $\displaystyle=$
$\displaystyle\\{\phi,\mathcal{H}_{cov}\\}_{cov}=\pi,$ (126)
$\displaystyle\hat{\tau}^{\mu}\partial_{\mu}\pi$ $\displaystyle=$
$\displaystyle\\{\pi,\mathcal{H}_{cov}\\}_{cov}=-e_{i}^{\mu}e^{i}_{\nu}\partial_{\mu}\partial^{\nu}\phi-\frac{\delta
V}{\delta\phi}.$ (127)

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 $e_{a}^{\mu}$ as a frame field also allows for a
more dynamical realization of foliations. Despite $\hat{\tau}^{\mu}$ being the
zeroth component of $e_{a}^{\mu}$, 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 $\mu=0$ 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.

####  II.4.2 Covariant Hamiltonian density for Dirac fields

The Dirac action can be written as

$S=\int
d^{4}x\left[\bar{\psi}_{\alpha}i\left(\gamma^{0}\right)^{\alpha}_{\,\,\,\beta}\hat{\tau}^{\mu}\partial_{\mu}\psi^{\beta}+\bar{\psi}_{\alpha}i\left(\gamma^{i}\right)^{\alpha}_{\,\,\,\beta}e^{\mu}_{i}\partial_{\mu}\psi^{\beta}-m\bar{\psi}_{\alpha}\psi^{\alpha}\right].$
(128)

The new covariant velocity and momentum fields are

$\displaystyle\nu^{\beta}=\hat{\tau}^{\mu}\partial_{\mu}\psi^{\beta},$
$\displaystyle\pi_{\beta}=\frac{\delta\mathcal{L}}{\delta\nu^{\beta}}=\bar{\psi}_{\alpha}i(\gamma^{0})^{\alpha}_{\,\,\,\beta},$
$\displaystyle\bar{\nu}_{\alpha}=\hat{\tau}^{\mu}\partial_{\mu}\bar{\psi}_{\alpha},$
$\displaystyle\bar{\pi}^{\alpha}=\frac{\delta\mathcal{L}}{\delta\bar{\nu}_{\alpha}}=-i(\gamma^{0})^{\alpha}_{\,\,\,\beta}\psi^{\beta}.$
(129)

After splitting the Lagrangian density into two terms with a derivative acting
on both the spinor $\psi^{\alpha}$ and its conjugate $\bar{\psi}_{\alpha}$,
the covariant Dirac Hamiltonian density is given by

$\mathcal{H}_{cov}=\frac{1}{2}\left[\pi_{\alpha}\hat{\tau}^{\mu}\partial_{\mu}\psi^{\alpha}+\hat{\tau}^{\mu}\partial_{\mu}\bar{\psi}_{\alpha}\bar{\pi}^{\alpha}-\bar{\psi}_{\alpha}i\left(\gamma^{j}\right)^{\alpha}_{\,\,\,\beta}e^{\mu}_{j}\partial_{\mu}\psi^{\beta}+\partial_{\mu}\bar{\psi}_{\alpha}i\left(\gamma^{j}\right)^{\alpha}_{\,\,\,\beta}e_{j}^{\mu}\psi^{\beta}\right]+m\bar{\psi}_{\alpha}\psi^{\alpha}$
(130)

Since the Dirac equation is first order, the Hamiltonian contains kinetic
terms as $1/2$ the velocity field times the conjugate momentum field, rather
than being quadratic in the momentum field. When integrating by partrs and
dropping the boundary term, the Hamiltonian density can be found as

$\mathcal{H}_{cov}=\hat{\tau}^{\mu}\partial_{\mu}\bar{\psi}_{\alpha}\bar{\pi}^{\alpha}-\bar{\psi}_{\alpha}i\left(\gamma^{j}\right)^{\alpha}_{\,\,\,\beta}e^{\mu}_{j}\partial_{\mu}\psi^{\beta}+m\bar{\psi}_{\alpha}\psi^{\alpha}$
(131)

The set of Hamilton’s equations are given by

$\displaystyle\hat{\tau}^{\mu}\partial_{\mu}\bar{\psi}_{\alpha}$
$\displaystyle=$
$\displaystyle\frac{\delta\mathcal{H}_{cov}}{\delta\bar{\pi}^{\alpha}}=\hat{\tau}^{\mu}\partial_{\mu}\bar{\psi}_{\alpha},$
(132) $\displaystyle\hat{\tau}^{\mu}\partial_{\mu}\bar{\pi}^{\alpha}$
$\displaystyle=$
$\displaystyle-\frac{\delta\mathcal{H}_{cov}}{\delta\bar{\psi}_{\alpha}}=i\left(\gamma^{j}\right)^{\alpha}_{\,\,\,\beta}e^{\mu}_{j}\partial_{\mu}\psi^{\beta}-m\psi^{\alpha}.$
(133)

Once again, the first equation is trivial due to the first-order nature of the
Dirac action. Nevertheless, plugging in the known definition of the conjugate
momentum into the second equation gives

$\hat{\tau}^{\mu}\partial_{\mu}\left(-i\left(\gamma^{0}\right)^{\alpha}_{\,\,\,\beta}\psi^{\beta}\right)=i\left(\gamma^{j}\right)^{\alpha}_{\,\,\,\beta}e^{\mu}_{j}\partial_{\mu}\psi^{\beta}-m\psi^{\alpha},$
(134)

which leads directly to the Dirac equation,

$i\left(\gamma^{a}\right)^{\alpha}_{\,\,\,\beta}e^{\mu}_{a}\partial_{\mu}\psi^{\beta}=m\psi^{\alpha}.$
(135)

####  II.4.3 Covariant Hamiltonian density for Maxwell theory

Due to complications of an antisymmetric field strength, we consider Maxwell’s
electrodynamics before Yang-Mills theory. The Maxwell action is

$S=\int d^{4}x\mathcal{L}=\int
d^{4}x\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-A^{\mu}J_{\mu}\right).$ (136)

The velocity field of interest is

$\nu_{\mu}=\hat{\tau}^{\nu}\partial_{\nu}A_{\mu}.$ (137)

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

$\delta^{\mu}_{\nu}=e^{\mu}_{a}e^{a}_{\nu}=\hat{\tau}^{\mu}\hat{\tau}_{\nu}+e^{\mu}_{i}e^{i}_{\nu}.$
(138)

The Lagrangian density can be expanded to give

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F_{\rho\sigma}\left(\hat{\tau}^{\mu}\hat{\tau}^{\nu}\hat{\tau}^{\rho}\hat{\tau}^{\sigma}+\hat{\tau}^{\mu}e^{\nu}_{j}\hat{\tau}^{\rho}e^{\sigma
j}+e^{\mu}_{i}\hat{\tau}^{\nu}e^{\rho
i}\hat{\tau}^{\sigma}+e^{\mu}_{i}e^{\nu}_{j}e^{\rho i}e^{\sigma j}\right)$
$\displaystyle=$ $\displaystyle-\frac{1}{2}e^{\nu}_{j}\nu^{\nu}e^{\sigma
j}\nu_{\sigma}+e^{\sigma
j}\nu_{\sigma}e^{\nu}e^{\nu}_{j}\hat{\tau}^{\mu}\partial_{\nu}A_{\mu}-\frac{1}{2}e^{\nu}_{j}\hat{\tau}^{\mu}\partial_{\nu}A_{\mu}e^{\sigma
j}\hat{\tau}^{\rho}\partial_{\sigma}A_{\mu}-\frac{1}{4}F_{\mu\nu}F^{\rho\sigma}e^{\mu}_{i}e^{\nu}_{j}e_{\rho}^{i}e_{\sigma}^{j}.$

The conjugate momentum field is found to be

$\pi^{\sigma}=\frac{\delta\mathcal{L}}{\delta\nu_{\sigma}}=-e^{\nu}_{j}e^{\sigma
j}\left(\nu_{\nu}-\hat{\tau}^{\mu}\partial_{\nu}A_{\mu}\right)=-e^{\nu}_{j}e^{\sigma
j}\hat{\tau}^{\mu}F_{\mu\nu}.$ (140)

Another way to express the conjugate momentum field is given by

$\hat{\tau}^{\nu}F_{\mu\nu}=\left(\hat{\tau}^{\rho}\hat{\tau}_{\mu}+e^{\rho}_{i}e_{\mu}^{i}\right)F_{\rho\nu}\hat{\tau}^{\nu}=\hat{\tau}_{\mu}\hat{\tau}^{\rho}\hat{\tau}^{\nu}F_{\rho\nu}+\pi_{\mu}=\pi_{\mu},$
(141)

where the antisymmetry of $F_{\rho\nu}$ led to the vanishing of the first term
above. Both forms of the conjugate momentum allow for the Lagrangian density
to be rewritten as

$\mathcal{L}=-\frac{1}{2}\pi^{\mu}\pi_{\mu}-\frac{1}{4}F_{\mu\nu}F^{\rho\sigma}e^{\mu}_{i}e^{\nu}_{j}e_{\rho}^{i}e_{\sigma}^{j}-A^{\mu}J_{\mu}.$
(142)

The covariant Hamiltonian density is found as

$\mathcal{H}_{cov}=\nu^{\mu}\pi_{\mu}-\mathcal{L}=\pi_{\mu}\left(-\frac{1}{2}\pi^{\mu}+\hat{\tau}_{\nu}\partial^{\mu}A^{\nu}\right)+\frac{1}{4}F_{\mu\nu}F^{\rho\sigma}e^{\mu}_{i}e^{\nu}_{j}e_{\rho}^{i}e_{\sigma}^{j}+A^{\mu}J_{\mu}.$
(143)

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

$\displaystyle\hat{\tau}^{\mu}\partial_{\mu}A_{\nu}$ $\displaystyle=$
$\displaystyle\frac{\delta\mathcal{H}_{cov}}{\delta\pi^{\nu}}=-\pi_{\nu}+\hat{\tau}^{\mu}\partial_{\nu}A_{\mu},$
(144) $\displaystyle\hat{\tau}^{\mu}\partial_{\mu}\pi_{\nu}$ $\displaystyle=$
$\displaystyle-\frac{\delta\mathcal{H}_{cov}}{\delta
A^{\nu}}=\partial^{\mu}\pi_{\mu}\hat{\tau}_{\nu}+\partial^{\mu}F_{\rho\sigma}e_{\mu}^{i}e_{\nu}^{j}e^{\rho}_{i}e^{\sigma}_{j}-J_{\nu}.$
(145)

Plugging the first equation into the second leads to

$\displaystyle J_{\nu}$ $\displaystyle=$
$\displaystyle-\hat{\tau}^{\mu}\partial_{\mu}\pi_{\nu}+\partial^{\mu}\pi_{\mu}\hat{\tau}_{\nu}+\partial^{\mu}F_{\rho\sigma}e_{\mu}^{i}e_{\nu}^{j}e^{\rho}_{i}e^{\sigma}_{j}$
(146) $\displaystyle=$ $\displaystyle
e_{\nu}^{i}e^{\sigma}_{i}\hat{\tau}_{\mu}\hat{\tau}^{\rho}\partial^{\mu}F_{\rho\sigma}+\hat{\tau}^{\rho}\hat{\tau}_{\nu}\partial^{\mu}F_{\mu\rho}+\partial^{\mu}F_{\rho\sigma}e_{\mu}^{i}e_{\nu}^{j}e^{\rho}_{i}e^{\sigma}_{j}$
$\displaystyle=$
$\displaystyle\partial^{\mu}F_{\rho\sigma}\left(\hat{\tau}_{\mu}\hat{\tau}^{\rho}e_{\nu}^{i}e^{\sigma}_{i}+\hat{\tau}^{\sigma}\hat{\tau}_{\nu}e^{\sigma}_{i}e_{\mu}^{i}+e_{\mu}^{i}e_{\nu}^{j}e^{\rho}_{i}e^{\sigma}_{j}\right)$
$\displaystyle=$ $\displaystyle\partial^{\mu}F_{\mu\nu}.$

where using Eq. (138) and
$\hat{\tau}^{\rho}\hat{\tau}^{\sigma}F_{\rho\sigma}=0$ were used to simplify
the result above.

####  II.4.4 Covariant Hamiltonian density for linearized gravity

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

$S=\int
d^{4}x\left(\frac{1}{2}\partial^{\rho}\bar{h}^{\mu\nu}\partial_{\rho}\bar{h}^{\mu\nu}-\frac{1}{4}\partial_{\mu}\bar{h}\partial^{\mu}\bar{h}\right)$
(147)

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

$\bar{\nu}_{\mu\nu}=\hat{\tau}^{\rho}\partial_{\rho}\bar{h}_{\mu\nu}.$ (148)

The Lagrangian can be rewritten as

$\mathcal{L}=\frac{1}{2}\left(\bar{\nu}_{\mu\nu}\bar{\nu}^{\mu\nu}-\frac{1}{2}\bar{\nu}\bar{\nu}+e_{\rho}^{i}e^{\sigma}_{i}\left(\partial^{\rho}\bar{h}^{\mu\nu}\partial_{\sigma}\bar{h}_{\mu\nu}-\frac{1}{2}\partial^{\rho}\bar{h}\partial_{\sigma}\bar{h}\right)\right)-\frac{8\pi
G}{c^{4}}\kappa\hat{T}^{\mu\nu}\hat{h}_{\mu\nu},$ (149)

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

$\bar{\pi}^{\mu\nu}=\frac{\delta\mathcal{L}}{\delta\bar{\nu}_{\mu\nu}}=\bar{\nu}^{\mu\nu}-\frac{1}{2}\eta^{\mu\nu}\bar{\nu}=\nu^{\mu\nu}=\hat{\tau}_{\rho}\partial^{\rho}h^{\mu\nu},$
(150)

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

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

$\mathcal{H}_{cov}=\frac{1}{2}\bar{\pi}_{\mu\nu}\bar{\pi}^{\mu\nu}-\frac{1}{4}\bar{\pi}\bar{\pi}-\frac{1}{2}e_{\rho}^{i}e^{\sigma}_{i}\left(\partial^{\rho}\bar{h}^{\mu\nu}\partial_{\sigma}\bar{h}_{\mu\nu}-\frac{1}{2}\partial^{\rho}\bar{h}\partial_{\sigma}\bar{h}\right)+\frac{8\pi
G}{c^{4}}\kappa\bar{T}^{\mu\nu}\bar{h}_{\mu\nu}.$ (151)

The covariant Hamilton’s equations give

$\displaystyle\hat{\tau}^{\rho}\partial_{\rho}\bar{h}_{\mu\nu}$
$\displaystyle=$
$\displaystyle\frac{\delta\mathcal{H}_{cov}}{\delta\bar{\pi}^{\mu\nu}}=\bar{\pi}_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}\bar{\pi}=\pi_{\mu\nu},$
(152) $\displaystyle\hat{\tau}^{\rho}\partial_{\rho}\bar{\pi}_{\mu\nu}$
$\displaystyle=$
$\displaystyle-\frac{\delta\mathcal{H}_{cov}}{\delta\bar{h}^{\mu\nu}}=-\left(e_{\rho}^{i}e^{\sigma}_{i}\partial^{\rho}\partial_{\sigma}\left(\bar{h}_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}\bar{h}\right)+\frac{8\pi
G}{c^{4}}\kappa\bar{T}^{\mu\nu}\right).$ (153)

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

$\left(\hat{\tau}_{\rho}\hat{\tau}^{\sigma}+e_{\rho}^{i}e^{\sigma}_{i}\right)\partial^{\rho}\partial_{\sigma}\bar{h}_{\mu\nu}=\partial^{\rho}\partial_{\rho}\bar{h}_{\mu\nu}=-\frac{8\pi
G}{c^{4}}\kappa T^{\mu\nu}.$ (154)

This reproduces the known equations of motion for linearized gravity.

##  III Covariant and canonical quantization of relativistic fields

Next, quantization of relativistic KvN mechanics, 2nd quantization of DDW or
poly-KvN mechanics, and 2nd quantization of the new Hamiltonian mechanics is
presented. Throughout, lower-case letters represent classical operators and
fields, while upper-case letters denote quantum operators and fields. Latin
letters are given for operators, while greek letters are used for fields.

###  III.1 Relativistic Koopman-von Neumann quantization

While classical theory treats $p_{\mu}$ and $k_{\mu}$ as independent, quantum
mechanics treats the operators as linearly dependent via
$\hat{P}_{\mu}=\hbar\hat{K}_{\mu}$ and $\hat{Q}^{\mu}=\hbar\hat{X}^{\mu}$.
This relates to the Heisenberg uncertainty relations of spacetime and energy-
momentum in the relativistic setting. From the operator algebra perspective,
quantization of classical theory can be seen in a straightforward manner with
Koopman-von Neumann mechanics, as both utilize a Hilbert space. A quantum
deformation of classical phase space can be implemented as a change of
variables from $\hat{x}^{\mu}\rightarrow\hat{X}^{\mu}$ and
$\hat{p}_{\nu}\rightarrow\hat{P}_{\nu}$

$\left[\hat{x}^{\mu},\hat{p}_{\nu}\right]=0\rightarrow\left[\hat{X}^{\mu},\hat{P}_{\nu}\right]=i\hbar\delta_{\nu}^{\mu}.$
(155)

The 16 independent variables of classical Fourier-phase space
$x^{\mu},p_{\nu},k_{\rho}$, and $q^{\sigma}$ can be replaced by the 8
independent variables of quantum Fourier space (or quantum phase space)
$\hat{X}^{\mu}$ and $\hat{P}_{\nu}$. This results in the the Heisenberg
algebra with
$\left[\hat{X}^{\mu},\hat{X}^{\nu}\right]=\left[\hat{P}_{\mu},\hat{P}_{\nu}\right]=0$.

The Heisenberg algebra can be found from the Koopman-von Neumann algebra by
shifting coordinates [4]. However, the relationship between wavenumber and the
variable conjugate to position in $\mathrm{KvN}$ was not mentioned, which
overlooked the notion of $\hat{P}_{\mu}=\hbar\hat{K}_{\mu}$. An important
property of quantum mechanics is that the eigenvalues of $\hat{X}^{\mu}$
should lead to coordinates of $\hat{x}^{\mu}$, assuming the quantum
eigenvalues relate to a notion of classical reality. To understand what shifts
in coordinates can be made, consider

$\displaystyle\hat{X}^{\mu}$ $\displaystyle=$ $\displaystyle
a\hat{x}^{\mu}+b\hbar\hat{q}^{\mu},$ (156) $\displaystyle\hat{P}_{\nu}$
$\displaystyle=$ $\displaystyle c\hat{p}_{\nu}+d\hbar\hat{k}_{\nu},$ (157)

where $a,b,c$, and $d$ are constants to be determined partially by the
following constraint,

$\left[\hat{X}^{\mu},\hat{P}_{\nu}\right]=\left[a\hat{x}^{\mu},c\hat{p}_{\nu}\right]+\left[a\hat{x}^{\mu},d\hbar\hat{k}_{\nu}\right]+\left[b\hbar\hat{q}^{\mu},c\hat{p}_{\nu}\right]+\left[b\hbar\hat{q}^{\mu},d\hbar\hat{k}_{\nu}\right]=i\hbar(ad+bc)\quad\rightarrow\quad
ad+bc=1.$ (158)

The quantum theory should have operators $\hat{K}_{\rho}$ and
$\hat{Q}^{\sigma}$ which satisfy the following relations,

$\hat{P}_{\rho}=\hbar\hat{K}_{\rho},\quad\hat{X}^{\sigma}=\hbar\hat{Q}^{\sigma},$
(159)

which results in

$\displaystyle\hat{K}_{\rho}$ $\displaystyle=$ $\displaystyle
d\hat{k}_{\rho}+\frac{c}{\hbar}\hat{p}_{\rho},$ (160)
$\displaystyle\hat{Q}^{\sigma}$ $\displaystyle=$ $\displaystyle
b\hat{q}^{\sigma}+\frac{a}{\hbar}\hat{x}^{\sigma}.$ (161)

In order to have a quantum theory whose operators
$\hat{X}^{\mu},\hat{P}_{\nu},\hat{K}_{\rho}$, and $\hat{Q}^{\sigma}$ lead to
the same eigenvalues as found in classical theory with
$\hat{x}^{\mu},\hat{p}_{\nu},\hat{k}_{\rho}$, and $\hat{q}^{\sigma}$, the
following operator relations are found when $\hat{x}^{\mu}=\hbar\hat{q}^{\mu}$
and $\hat{p}_{\nu}=\hbar\hat{k}_{\nu}$,

$\displaystyle\hat{X}^{\mu}$ $\displaystyle=$
$\displaystyle(a+b)\hat{x}^{\mu}\quad\rightarrow\quad a+b=1\text{, }$ (162)
$\displaystyle\hat{P}_{\nu}$ $\displaystyle=$
$\displaystyle(c+d)\hat{p}_{\nu}\quad\rightarrow\quad c+d=1,$ (163)
$\displaystyle\hat{K}_{\rho}$ $\displaystyle=$
$\displaystyle(b+a)\hat{k}_{\rho}\quad\rightarrow\quad a+b=1,$ (164)
$\displaystyle\hat{Q}^{\sigma}$ $\displaystyle=$
$\displaystyle(d+c)\hat{q}^{\sigma}\quad\rightarrow\quad c+d=1\text{. }$ (165)

When the following relations are satisfied, the quantum theory satisfies
$\hat{P}_{\nu}=\hbar\hat{K}_{\nu}=\hat{p}_{\nu}=\hbar\hat{k}_{\nu}$, etc. This
assures that the quantum and classical Fourier phase space operators all agree
such that the quantum eigenvalues correspond to classical variables as
introduced in the Copenhagen interpretation. The solutions are not fully
constrained, which allows for $b,c$, and $d$ to be found in terms of $a$,

$b=1-a,\quad c=\frac{a-1}{2a-1},\quad d=\frac{a}{2a-1}.$ (166)

Note that Ref. [4] chose $a=1,b=1/2,c=1$, and $d=1/2$ (since their
$\hat{\lambda}_{p}$ is our $-\hat{q}$). What this suggests is that the quantum
theory in their formalism leads to quantum position eigenvalues that do not
correspond to the position eigenvalues in the classical theory. For instance,
if their $\hat{p}_{q}=\hbar\hat{k}_{q}$ and $\hat{p}=\hbar\hat{k}$, then
$\hat{p}_{q}=\frac{3}{2}\hat{p}$. This was not considered problematic, as
taking $\alpha$ (or $\hbar$ ) to zero leads to $\hat{p}_{q}=\hat{p}$. However,
quantum theory leads to measurements of classical observables with real
eigenvalues, so it seems to be an improvement to select only the classical
formulations that admit a correspondence between the operators. While
classical mechanics does not satisfy $\hat{p}=\hbar\hat{k}$, our assertion is
that quantum theory must use operators $\hat{P}=\hbar\hat{K}$ that contain
eigenvalues that match $\hat{P}=\hat{p}$ and $\hat{K}=\hat{k}$ in some
classical framework, which implies that $\hat{p}=\hbar\hat{k}$ within the
quantum theory.

A few simple solutions for the coefficients include

$\begin{array}[]{cccc}\mbox{Solution 1: }a=1,&b=0,&c=0,&d=1,\\\ \mbox{Solution
2: }a=2,&b=-1,&c=\frac{1}{3},&d=\frac{2}{3},\\\ \mbox{Solution 3:
}a=\frac{3}{2},&b=-\frac{1}{2},&c=\frac{1}{4},&d=\frac{3}{4},\\\
\mbox{Solution 4:
}a=\frac{1}{\sqrt{2}},&b=1-\frac{1}{\sqrt{2}},&c=-\frac{1}{\sqrt{2}},&d=1+\frac{1}{\sqrt{2}},\end{array}$
(167)

where all of these satisfy $ad+bc=1,a+b=1$, and $c+d=1$. The solutions for
$a,b,c$, and $d$ are shown in Fig. (2). The first solution above is clearly
the simplest, as it leads to

$\displaystyle\hat{X}^{\mu}$ $\displaystyle=$ $\displaystyle\hat{x}^{\mu},$
(168) $\displaystyle\hat{P}_{\nu}$ $\displaystyle=$
$\displaystyle\hbar\hat{k}_{\nu},$ (169) $\displaystyle\hat{K}_{\rho}$
$\displaystyle=$ $\displaystyle\hat{k}_{\rho},$ (170)
$\displaystyle\hat{Q}^{\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{\hbar}\hat{x}^{\sigma}.$ (171)

This change of variables from the Koopman-von Neumann classical mechanics to
quantum mechanics is the simplest and most unambiguous, as it clearly states
that the quantum momentum is equal to $\hbar$ times the classical wavenumber.
For this solution, the classical momentum $\hat{p}_{\nu}$ and its conjugate
momentum-wavenumber $\hat{q}^{\sigma}$ are not expressed in the quantum
theory.

Figure 2: The solution for coefficients $a,b,c$, and $d$ are shown as a
function of $a$ such that both the classical and quantum phase space
coordinates equal to Planck’s constant times their corresponding wavenumbers.

###  III.2 De Donder-Weyl quantization

Next, quantum fields in the De Donder-Weyl formalism are discussed. Rather
than referring to a particle’s position and momentum, a quantum field
$\Phi(x)$ replaces position, which admits a canonical poly-momentum
$\Pi_{\mu}(x)$. In principle, one may consider a mixture of $\phi$ and
$\xi^{\sigma}$ as well as $\pi^{\mu}$ and $\kappa$, but the mismatch in tensor
rank implies the following quantization schemes may be considered,

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle a\phi+b^{\mu}\xi_{\mu},$
(172) $\displaystyle\Pi^{\mu}$ $\displaystyle=$ $\displaystyle
c\pi^{\mu}+d^{\mu}\kappa.$ (173)

The commutator of the second-quantized fields $\Phi(x)$ and $\Pi^{\mu}(y)$ for
arbitrary coefficients is found from Eq. (99)

$[\Phi(x),\Pi^{\mu}(y)]=\left(ad^{\mu}+b^{\mu}c\right)i\delta^{(4)}(x-y).$
(174)

The simplest scheme is $a=1$, $d^{\mu}\propto\hat{\tau}^{\mu}$, $b^{\mu}=c=0$.
Next, we seek a solution for $d^{\mu}$ given $a=1$. If the dimensions of 1st
and 2nd quantized fields $\phi(x)$ and $\Phi(x)$ are the same, this is
analogous to the dimensions of $\hat{x}^{\mu}$ matching the dimensions of
$\hat{X}^{\mu}$. However, $\kappa(x)$ and $\Pi^{\mu}(x)$ do not differ by
dimensions of $\hbar$. The dimensions of $d^{\mu}$ must be the same as $\hbar
c\tau^{\mu}$. One cannot set $\Pi^{\mu}(x)=\hbar\kappa(x)$ due to dimensional
analysis and the polysymplectic geometry, but $\Pi^{\mu}(x)=\hbar
c\tau^{\mu}\kappa(x)$ is possible.

The commutation relations of a field $\Phi$ and its conjugate poly-momentum
$\Pi^{\mu}$ can be found by recognizing that
$\Pi_{D}=\frac{\delta\mathcal{L}}{\delta\frac{\partial\Phi}{\partial
t}}=\frac{1}{c}\Pi^{0}$ is Dirac’s canonical momentum, where

$[\Phi(x),\Pi_{D}(y)]_{x^{0}=y^{0}}=i\frac{\hbar}{c}\delta^{(3)}(x-y).$ (175)

While this reference to $x^{0}$ and $y^{0}$ implies a specific frame, the
canonical commutation relations in with the poly-momentum can utilize
$\hat{\tau}^{\mu}$, giving

$\left[\Phi(x),\Pi^{\mu}(y)\right]=i\hbar c\tau^{\mu}\delta^{(4)}(x-y)=i\hbar
c|\tau|\hat{\tau}^{\mu}\delta^{(4)}(x-y),$ (176)

where $[\Phi(x),\Phi(y)]=\left[\Pi^{\mu}(x),\Pi^{\nu}(y)\right]=0$. These
commutation relations are manifestly Lorentz invariant. Since $\tau^{\mu}$ is
not meant to encode the time coordinate, but rather the direction of a time-
like curve in relation to another frame, $|\tau|$ should never be zero. The
magnitude relates to differences in time between spacetime foliations, which
may motivate Planck’s time as the interval to be used in Feynman’s discrete
path integral. The discrete calculus of time evolution has been discussed
previously by one of the authors [83, 84]. The canonical commutation relations
are recovered in the frame where $\hat{\tau}^{\mu}=(1,0,0,0)$,

$\displaystyle{\left[\Phi(x),\frac{1}{c}\Pi^{0}(y)\right]}$ $\displaystyle=$
$\displaystyle
i\hbar|\tau|\delta^{(4)}(x-y)=i\hbar|\tau|\delta^{(3)}(x-y)\delta\left(t_{x}-t_{y}\right)$
(177) $\displaystyle=$ $\displaystyle
i\hbar\delta^{(3)}(x-y)|\tau|\delta\left(c\left(t_{x}-t_{y}\right)\right)=i\hbar\delta^{(3)}(x-y)\delta\left(\left(t_{x}-t_{y}\right)/|\tau|\right).$
(178)

Integrating over $t_{y}$ leads to $t_{y}=t_{x}$ and recovers Eq. (175). In
conclusion, the second-quantized De Donder-Weyl algebra leads to Dirac’s
canonical quantization when the zeroth component of the polymomentum is
considered and $\Phi(x)$ and $\Pi^{0}(y)$ are evaluated at the same time.

###  III.3 A new canonical and covariant quantization

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

$\displaystyle\Phi(x)$ $\displaystyle=$ $\displaystyle a\phi(x)+b\xi(x),$
(179) $\displaystyle\Pi(x)$ $\displaystyle=$ $\displaystyle
c\pi(x)+d\kappa(x).$ (180)

The commutation relations of the second-quantized fields are therefore

$[\Phi(x),\Pi(y)]=\left(ad+bc\right)i\delta^{(4)}(x-y).$ (181)

Setting $a=1$, $d=\hbar c|\tau|$ and $b=c=0$ is once again the simplest
quantization procedure, although other possibilities exist.

The following commutation relations are anticipated with
$\Pi(x)=\hat{\tau}^{\mu}\Pi_{\mu}(x)$ as the symplectic covariant canonical
momentum,

$[\Phi(x),\Pi(y)]=i\hbar c|\tau|\delta^{(4)}(x-y).$ (182)

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

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 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 [22, 87]. For our construction,
this implies that $|\tau|=t_{P}=\sqrt{\frac{\hbar G}{c^{5}}}$. With this
assumption, the new canonical, covariant, and symplectic commutation relations
for fields in natural units

$[\Phi(x),\Pi(y)]=i\delta^{(4)}(x-y).$ (183)

This canonical commutation relation for second quantized fields appears to be
the simplest possible that are manifestly covariant and symplectic.

##  IV Conclusions

In this work, 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 mechanics was introduced 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, 1st, and 2nd quantized
poly-Koopman-von Neumann algebras for fields were presented for the first
time, which is compatible with De Donder-Weyl dynamics. Comparing the
covariant poly-symplectic geometry of the De Donder-Weyl Hamiltonian with the
non-covariant symplectic geometry of the Dirac Hamiltonian led to the
development of a new type of Hamiltonian density with canonical, covariant,
and symplectic structure that allows for $\tau^{\mu}$ to be modified to encode
Dirac’s Hamiltonian for different foliations. While covariant, canonical, and
symplectic structures had been previously discussed, generalizing KvN
mechanics to DDW mechanics provides a more comprehensive framework, including
explicit Hamiltonian densities, which were demonstrated to give the correct
equations of motion for Klein-Gordon, Dirac, Maxwell, and linearized gravity
theories.

The first quantization of relativistic Koopman-von Neumann mechanics, second
quantization of (poly-Koopman-von Neumann) De Donder-Weyl mechanics, and
second quantization of the new covariant and symplectic approach were found
from an algebraic perspective that uses commutators for operators on Hilbert
spaces from classical to quantum theory. Loop quantum gravity has mostly
avoided De Donder-Weyl mechanics, partially due to its obscurity and partially
due to its complicated poly-symplectic structure [1, 22]. This has led to
canonical and covariant quantization as separate procedures [82, 1], although
recent work has explored approaches stemming from Witten [13, 14, 15, 16, 17,
18, 19, 20, 21]. Similar to how Witten discussed Poincare transformations
between space-like hypersurfaces, the time-like component of the frame field
was identified as $\hat{\tau}^{\mu}$, which allows for dynamics similar to
Dirac’s quantization procedure, yet no specific foliation was chosen.

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 and
recent work has explored Palatini or metric-affine spacetime structures,
exploring Hamiltonian dynamics from metric-affine or metric-spin-frame
formulations of gauge gravitational theories would also be relevant. Another
avenue of exploration involves using the relativistic Koopman-von Neumann
algebra introduced here to extend the prequantum operator algebra defined in
[8] and its possible category-theoretic realizations using pregeometric
constructions discussed in [85, 86].

##  V Acknowledgements

Thanks to Peter Morgan for making us aware of Koopman-von Neumann mechanics
[5, 6]. Thanks to Marcelo Amaral and Ray Aschheim for discussions related to
canonical quantization and De Donder-Weyl mechanics.

## References

  * [1] R. Loll, G. Fabiano, D. Frattulillo and F. Wagner, “Quantum Gravity in 30 Questions,” PoS CORFU2021, 316 (2022), doi:10.22323/1.406.0316, arXiv:2206.06762 [hep-th].
  * [2] B. O. Koopman, “Hamiltonian Systems and Transformation in Hilbert Space,” Proc. Natl. Acad. Sci. 17(5): 315-318 (1931), doi:10.1073%2Fpnas.17.5.315.
  * [3] J. von Neumann, “Zur operatorenmethode in der Klassi sehen Mechanik,” Ann. Math 33 (3), 587-642 (1932), doi:10.2307%2F1968537.
  * [4] D. I. Bondar, R. Cabrera, R. R. Lompay, M. Y. Ivanov, and H. A. Rabitz, “Operational Dynamic Modeling Transcending Quantum and Classical Mechanics,” Phys. Rev. Lett. 109, 190403 (2012), doi:10.1103/PhysRevLett.109.190403, arXiv:1105.4014 [quant-ph].
  * [5] P. Morgan, “An algebraic approach to Koopman classical mechanics,” Annals Phys. 414, 168090 (2020), doi:10.1016/j.aop.2020.168090, arXiv:1901.00526 [quant-ph].
  * [6] P. Morgan, “The collapse of a quantum state as a joint probability construction,” J. Phys. A 55, 25, 254006 (2022), doi:10.1088/1751-8121/ac6f2f, arXiv:2101.10931 [quant-ph].
  * [7] D. Piasecki, “Introduction to Koopman-von Neumann Mechanics,” BS Thesis, Tulane University, arXiv:2112.05619 [quant-ph].
  * [8] X. D. Arsiwalla, D. Chester and L. H. Kauffman, “On the Operator Origins of Classical and Quantum Wave Functions", (2022), arXiv:2211.01838 [math-ph].
  * [9] T. De Donder, “Théorie invariantive du calcul des variations,” The Mathematical Gazette, 20(240), 287-288 (1935), doi:10.2307/3607791.
  * [10] H. Weyl, “Observations on Hilbert’s Independence Theorem and Born’s Quantization of Field Equations,” Phys. Rev. 46, 505-508 (1934), doi:10.1103/PhysRev.46.505
  * [11] H. Weyl, “Geodesic Fields in the Calculus of Variation for Multiple Integrals,” Annals of Math. 36 (3), 607-629 (1935).
  * [12] J. von Rieth, “The Hamilton–Jacobi theory of De Donder and Weyl applied to some relativistic field theories,” J. Math. Phys. 25, 1102 (1984), doi:10.1063/1.526253.
  * [13] E. Witten, “Interacting field theory of open superstrings,” Nucl. Phys. B 276 2, 291-324 (1986), doi:10.1016/0550-3213(86)90298-1.
  * [14] G. J. Zuckerman, “Action principles and global geometry,” Math. Aspects of Str. Th., 259-284 (1987). doi:10.1142/9789812798411_0013.
  * [15] C. Crnkovic and E. Witten, “Covariant description of canonical formalism in geometrical theories,” in: “Three Hundred Years of Gravitation,” 676 - 684, Cambridge University Press (1987).
  * [16] C. Crnkovic, “Symplectic geometry and (super-)Poincaré algebra in geometrical theories,” Nucl. Phys. B 288, 419-430 (1987), doi:10.1016/0550-3213(87)90221-5.
  * [17] C. Crnkovic, “Symplectic geometry of the covariant phase space,” Class. and Quant. Grav. 5(12), 1557-1575, doi:10.1088/0264-9381/5/12/008.
  * [18] J. Margalef-Bentabol and E. J. S. Villaseñor, “Geometric formulation of the Covariant Phase Space methods with boundaries,” Phys. Rev. D 103 2, 025011 (2021), doi:10.1103/PhysRevD.103.025011, arXiv:2008.01842 [math-ph].
  * [19] J. F. Barbero G., J. Margalef-Bentabol, V. Varo and E. J. S. Villaseñor, “Covariant phase space for gravity with boundaries: Metric versus tetrad formulations,” Phys. Rev. D 104 4, 044048 (2021), doi:10.1103/PhysRevD.104.044048, arXiv:2103.06362 [gr-qc].
  * [20] J. F. Barbero G., J. Margalef-Bentabol, V. Varo and E. J. S. Villaseñor, “Palatini gravity with nonmetricity, torsion, and boundaries in metric and connection variables,” Phys. Rev. D 104 4, 044046 (2021), doi:10.1103/PhysRevD.104.044046, arXiv:2105.07053 [gr-qc].
  * [21] J. F. Barbero G., J. Margalef-Bentabol, V. Varo and E. J. S. Villaseñor, “On-shell equivalence of general relativity and Holst theories with nonmetricity, torsion, and boundaries,” Phys. Rev. D 105, no.6, 064066 (2022), doi:10.1103/PhysRevD.105.064066, arXiv:2201.12141 [gr-qc].
  * [22] T. Thiemann, “Modern Canonical Quantum General Relativity,” Cambridge Monographs on Mathematical Physics, Cambridge University Press, (2007), doi:10.1017/CBO9780511755682.
  * [23] I. V. Kanatchikov, “Towards the Born-Weyl quantization of fields,” Int. J. Theor. Phys. 37, 333-342 (1998), doi:10.1023/A:1026679104397, arXiv:quant-ph/9712058 [quant-ph].
  * [24] I. V. Kanatchikov, “On field theoretic generalizations of a Poisson algebra,” Rept. Math. Phys. 40, 225 (1997), doi:10.1016/S0034-4877(97)85919-8, arXiv:hep-th/9710069 [hep-th].
  * [25] I. V. Kanatchikov, “Canonical structure of classical field theory in the polymomentum phase space,” Rept. Math. Phys. 41, 49-90 (1998), doi:10.1016/S0034-4877(98)80182-1, arXiv:hep-th/9709229 [hep-th].
  * [26] I. V. Kanatchikov, “From the De Donder-Weyl Hamiltonian formalism to quantization of gravity,” International Seminar on Mathematical Cosmology (ISMC 98), 472-482 (1998), arXiv:gr-qc/9810076 [gr-qc].
  * [27] I. V. Kanatchikov, “De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory,” Rept. Math. Phys. 43, 157-170 (1999), doi:10.1016/S0034-4877(99)80024-X, arXiv:hep-th/9810165 [hep-th].
  * [28] I. V. Kanatchikov, “On quantization of field theories in polymomentum variables,” AIP Conf. Proc. 453, no.1, 356-367 (1998), doi:0.1063/1.57105, arXiv:hep-th/9811016 [hep-th].
  * [29] I. V. Kanatchikov, “Geometric (pre)quantization in the polysymplectic approach to field theory,” 8th International Conference on Differential Geometry and Its Applications, (2001), arXiv:hep-th/0112263 [hep-th].
  * [30] I. V. Kanatchikov, “Precanonical quantization of Yang-Mills fields and the functional Schrodinger representation,” Rept. Math. Phys. 53, 181-193 (2004), doi:10.1016/S0034-4877(04)90011-0, arXiv:hep-th/0301001 [hep-th].
  * [31] I. V. Kanatchikov, “On precanonical quantization of gravity in spin connection variables,” AIP Conf. Proc. 1514 1, 73-76 (2013), doi:10.1063/1.4791728, arXiv:1212.6963 [gr-qc].
  * [32] I. V. Kanatchikov, “De Donder-Weyl Hamiltonian formulation and precanonical quantization of vielbein gravity,” J. Phys. Conf. Ser. 442, 012041 (2013), doi:10.1088/1742-6596/442/1/012041, arXiv:1302.2610 [gr-qc].
  * [33] I. V. Kanatchikov, “Ehrenfest Theorem in Precanonical Quantization,” J. Geom. Symmetry Phys. 37, 43-66 (2015), doi:10.7546/jgsp-37-2015-43-66, arXiv:1501.00480 [hep-th].
  * [34] I. V. Kanatchikov, “On the spectrum of DW Hamiltonian of quantum SU(2) gauge field,” Int. J. Geom. Meth. Mod. Phys. 14, 1750123 (2017), doi:10.1142/S0219887817501237, arXiv:1706.01766 [hep-th].
  * [35] I. V. Kanatchikov, “Schrödinger Wave Functional in Quantum Yang–Mills Theory from Precanonical Quantization,” Rept. Math. Phys. 82, no.3, 373-388 (2018), doi:10.1016/S0034-4877(19)30008-4, arXiv:1805.05279 [hep-th].
  * [36] P. A. M. Dirac, “Generalized Hamiltonian Dynamics,” Can. J. Math. 2, 129-148 (1950), doi:10.4153/CJM-1950-012-1.
  * [37] A. Ashtekar and A. Magnon-Ashtekar, “On the symplectic structure of general relativity,” Comm. Math. Phys. 86, 55–68 (1982), doi:10.1007/BF01205661.
  * [38] Jorg Frauendiener and George A. J. Sparling, “On the Symplectic Formalism for General Relativity,” Proc. R. Soc. Lond. A 436, 141-153 (1992), doi:10.1098/rspa.1992.0010.
  * [39] D. Harlow and J. Q. Wu, “Covariant phase space with boundaries,” JHEP 10, 146 (2020), doi:10.1007/JHEP10(2020)146, arXiv:1906.08616 [hep-th].
  * [40] V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamical black hole entropy,” Phys. Rev. D 50, 846-864 (1994), doi:10.1103/PhysRevD.50.846, arXiv:gr-qc/9403028 [gr-qc].
  * [41] V. Iyer and R. M. Wald, “A Comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes,” Phys. Rev. D 52, 4430-4439 (1995), doi:10.1103/PhysRevD.52.4430, arXiv:gr-qc/9503052 [gr-qc].
  * [42] R. M. Wald and A. Zoupas, “A General definition of ’conserved quantities’ in general relativity and other theories of gravity,” Phys. Rev. D 61, 084027 (2000), doi:10.1103/PhysRevD.61.084027, arXiv:gr-qc/9911095 [gr-qc].
  * [43] C. Castro, “On n-ary algebras, branes and poly-vector gauge theories in noncommutative Clifford spaces,” J. Phys. A: Math. Theor. 43, 365201 (2010), doi:10.1088/1751-8113/43/36/365201.
  * [44] C. Castro, “A Clifford algebra-based grand unification program of gravity and the Standard Model: a review study,” Can. J. Phys. 92, 12 (2014), doi:10.1139/cjp-2013-0686.
  * [45] C. Castro, “On Clifford Space Relativity, Black Hole Entropy, Rainbow Metrics, Generalized Dispersion and Uncertainty Relations,”, Found. Phys. 44, 990-1008 (2014) doi:10.1007/s10701-014-9825-x.
  * [46] C. Castro, “Extended Lorentz Transformations in Clifford Space Relativity Theory,” Adv. Appl. Cliff. Alg. 25, 553-5676 (2015), doi:10.1007/s00006-015-0529-x.
  * [47] C. Castro, “A unified description of particles, strings and branes in Clifford spaces and p-brane/polyparticle duality,” Int. J. Geom. Met. Mod. Phys. 13 01, 1550133 (2016), doi:10.1142/S0219887815501339.
  * [48] C. Castro, “On Some Novel Consequences of Clifford Space Relativity Theory,” Adv. Appl. Cliff. Alg. 27, 255-266 (2017), doi:10.1007/s00006-015-0553-x.
  * [49] C. Castro Perelman, “The Clifford-Yang algebra, noncommutative Clifford phase spaces and the deformed quantum oscillator,” Int. J. Geom. Met. Mod. Phys. 0, (2022), doi:10.1142/S0219887823500780.
  * [50] M. McLean and L. K. Norris, “Covariant field theory on frame bundles of fibered manifolds,” J. Math. Phys. 41, 6808 (2000), doi:10.1063/1.1288797.
  * [51] F. Hélein, “Multisymplectic formalism and the covariant phase,” (2011), arXiv:1106.2086 [math-ph].
  * [52] M. Forger and M. O. Salles, “On Covariant Poisson Brackets in Classical Field Theory,” J. Math. Phys. 56, 102901 (2015). doi:10.1063/1.4932011, arXiv:1501.03780 [math-ph].
  * [53] H. Nikolic, “Covariant canonical quantization of fields and Bohmian mechanics,” Eur. Phys. J. C 42, 365-374 (2005), doi:10.1140/epjc/s2005-02296-7.
  * [54] N. Riahi and M. E. Pietrzyk, “On the Relation Between the Canonical Hamilton–Jacobi Equation and the De Donder–Weyl Hamilton–Jacobi Formulation in General Relativity,” Acta Phys. Polon. Supp. 13, 213 (2020), doi:10.5506/APhysPolBSupp.13.213, arXiv:1912.13363 [gr-qc].
  * [55] J. Berra-Montiel, A. Molgado and D. Serrano-Blanco, “De Donder–Weyl Hamiltonian formalism of MacDowell–Mansouri gravity,” Class. Quant. Grav. 34 23, 235002 (2017), doi:10.1088/1361-6382/aa924a, arXiv:1703.09755 [gr-qc].
  * [56] J. Berra-Montiel, A. Molgado and Á. Rodríguez-López, “Polysymplectic formulation for BF gravity with Immirzi parameter,” Class. Quant. Grav. 36 11, 115003 (2019), doi:10.1088/1361-6382/ab1365, arXiv:1901.11532 [gr-qc].
  * [57] J. Berra-Montiel, A. Molgado and Á. Rodríguez-López, “A review on geometric formulations for classical field theory: the Bonzom–Livine model for gravity,” Class. Quant. Grav. 38 13, 135012 (2021), doi:10.1088/1361-6382/abf711, arXiv:2101.08960 [gr-qc].
  * [58] S. Watanabe, “Symmetry of physical laws. Part III. Prediction and retrodiction,” Rev. Mod. Phys. 27 (2), 179-186, doi:10.1103/RevModPhys.27.179.
  * [59] Y. Aharonov, P. G. Bergmann, and J. L. Lebowitz, “Time Symmetry in the Quantum Process of Measurement,” Phys. Rev. 134, B1410 (1964), doi:10.1103/PhysRev.134.B1410.
  * [60] Y. Aharonov and L. Vaidman, “Complete description of a quantum system at a given time,” J. Phys. A: Math. Gen. 24, 2315 (1991), doi:10.1088/0305-4470/24/10/018.
  * [61] Y. Aharonov, S. Popescu, J. Tollaksen, and L. Vaidman, “Multiple-time states and multiple-time measurements in quantum mechanics,” Phys. Rev. A 79, 052110 (2009), doi:10.1103/PhysRevA.79.052110, arXiv:0712.0320 [quant-ph].
  * [62] J. Cotler and F. Wilczek, “Entangled histories,” Physica Scripta T168, 014004 (2016), doi:10.1088/0031-8949/2016/T168/014004, arXiv:1502.02480 [quant-ph].
  * [63] M. Nowakowski, “Monogamy of quantum entanglement in time,” arXiv:1604.03976 [quant-ph].
  * [64] M. Nowakowski, “Quantum Entanglement in Time,” AIP Conf. Proc. 1841, 020007 (2017), doi:10.1063/1.4982771, arXiv:1701.08116 [quant-ph].
  * [65] M. Nowakowski, E. Cohen and P. Horodecki, “Entangled histories versus the two-state-vector formalism: Towards a better understanding of quantum temporal correlations,” Phys. Rev. A 98 3, 032312 (2018), doi:10.1103/PhysRevA.98.032312, arXiv:1803.11267 [quant-ph].
  * [66] V. Giovannetti, S. Lloyd and L. Maccone, “Quantum Time,” Phys. Rev. D 92 4, 045033 (2015), doi:10.1103/PhysRevD.92.045033, arXiv:1504.04215 [quant-ph].
  * [67] M. Fadel and L. Maccone, “Time-energy uncertainty relation for quantum events,” Phys. Rev. A 104, L050204 (2021), doi:10.1103/PhysRevA.104.L050204, arXiv:2108.13974 [quant-ph].
  * [68] V. Giovannetti, S. Lloyd and L. Maccone, “Geometric Event-Based Relativistic Quantum Mechanics,” New J. Phys. 25, 023027 (2023), doi:10.1088/1367-2630/acb793, arXiv:2206.08359 [quant-ph].
  * [69] J. A. Vaccaro, “The quantum theory of time, the block universe and human experience,” Phil. Trans. Roy. Soc. Lond. A 376, 0316 (2018), doi:10.1098/rsta.2017.0316, arXiv:1804.05177 [quant-ph].
  * [70] P. A. M. Dirac, “The theory of gravitation in Hamiltonian form,” Proc. Roy. Soc. A 246, 1246 (1958), 10.1098/rspa.1958.0142. Published:19 August 1958https://doi.org/10.1098/rspa.1958.0142
  * [71] H. J. Groenewold, “On the principles of elementary quantum mechanics,” Physica 12 7, 405-460 (1946), doi:10.1016/S0031-8914(46)80059-4.
  * [72] B. S. DeWitt, “Quantum Theory of Gravity. 1. The Canonical Theory,” Phys. Rev. 160, 1113-1148 (1967), doi:10.1103/PhysRev.160.1113.
  * [73] C. J. Isham, “Canonical quantum gravity and the problem of time,” NATO Sci. Ser. C 409, 157-287 (1993), arXiv:gr-qc/9210011 [gr-qc].
  * [74] F. W. Hehl and B. K. Datta, “Nonlinear Spinor Equation and Asymmetric Connection in General Relativity,” J. Math. Phys. 12, 1334 (1971), 10.1063/1.1665738.
  * [75] D. Chester, “Radiative double copy for Einstein-Yang-Mills theory,” Phys. Rev. D 97 (8), 084025 (2018), doi:10.1103/PhysRevD.97.084025, arXiv:1712.08684 [hep-th].
  * [76] P. Horava, “On a covariant Hamilton-Jacobi framework for the Einstein-Maxwell theory,” Class. and Quant. Grav., 8(11), 2069–2084 (1991), doi:10.1088/0264-9381/8/11/016.
  * [77] C. Cheung and G. N. Remmen, “Twofold Symmetries of the Pure Gravity Action,” JHEP 01, 104 (2017), doi:10.1007/JHEP01(2017)104, arXiv:1612.03927 [hep-th].
  * [78] A. S. Wightman, “Quantum Field Theory in Terms of Vacuum Expectation Values,” Phys. Rev. 101, 860, (1956), doi:10.1103/PhysRev.101.860.
  * [79] R. Haag and D. Kastler, “An Algebraic Approach to Quantum Field Theory,” J. Math. Phys. 5, 848 (1964), doi:10.1063/1.1704187.
  * [80] D. Hall and A. S. Wightman, “A Theorem on Invariant Analytic Functions with Applications to Relativistic Quantum Field Theory,” Mat. Fys. Medd. Dan. Vid. Seisk. 31, 5 (1957).
  * [81] I. Bars, “Gauge Symmetry in Phase Space, Consequences for Physics and Spacetime,” Int. J. Mod. Phys. A 25, 5235-5252 (2010), doi:10.1142/S0217751X10051128, arXiv:1004.0688 [hep-th].
  * [82] C. Rovelli and F. Vidotto, “Covariant Loop Quantum Gravity,” Cambridge University Press, (2015), doi:10.1017/CBO9781107706910.
  * [83] L. H. Kauffman, “Non-commutative Calculus and Discrete Physics,” Proc. ANPA Conf., (2002), arXiv:quant-ph/0303058.
  * [84] L. H. Kauffman, “Calculus, Gauge Theory and Noncommutative Worlds,” Symmetry 14 (3), 430 (2022), doi:10.3390/sym14030430, arXiv:2108.03007 [math.DG].
  * [85] X. D. Arsiwalla and J. Gorard, “Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types", (2021), arXiv:2111.03460 [math.CT].
  * [86] X. D. Arsiwalla, J. Gorard and H. Elshatlawy, “Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories", (2021), arXiv:2105.10822 [math.CT].
  * [87] P. Zizzi, “Entangled spacetime,” Mod. Phys. Lett. A 33 (29), 1850168 (2018) doi:10.1142/S0217732318501687, arXiv:1807.06433 [physics.gen-ph].