Skip to main content# Diffeomorphisms and gravity

# Quantum gravity

# Classical Hamiltonian

# Quantum Hamiltonian

Definitions of Quantum Yang-Mills, Quantum Gravity and Classical Statistical Field Theory, with gauge symmetries through algebraic ideals.

Published onMay 29, 2022

Diffeomorphisms and gravity

The first step towards Quantum Gravity, is to choose which version of (classical) General Relativity we want to quantize and the corresponding definition of space-time. For that purpose, it is useful to review the case of the Yang-Mills theory exposed in the previous chapters: in a non-abelian gauge-theory, the Gribov ambiguity forces us to consider a phase-space formed by fields defined on not only space but also time. This is related to the fact that in a fibre bundle (the mathematical formulation of a classical gauge theory) the time cannot be factored out from the total space because the topology of the total space is not a product of the base-space (time) and the fibre-space, despite that the total space is *locally* a product space. Thus, the Hamiltonian constraints are in fact a tool to define a probability measure for a manifold with a non-trivial topology (a principal fibre bundle for the gauge group) such that the only measurable functions are the gauge-invariant functions[1], because a phase-space of gauge fields defined *globally* on a 4-dimensional space-time (i.e. a fibre bundle with a trivial topology, when the base space is the Minkowski space-time) produces well-defined expectation functionals for the gauge-invariant operators acting on a fibre bundle with a non-trivial topology[1].

Since only gauge-invariant operators are allowed, we must distinguish between the concrete manifold appearing in the phase-space and the family of manifolds (obtained from the concrete manifold through different choices of transition maps between local charts) to which the expectation values correspond. Regarding General Relativity, the tetrad fields can be reconstructed from diffeomorphism- and local Lorentz-invariant operators [2], as in the reconstruction theorem of Yang-Mills theory [3]. The concrete space-time has a spinor structure, otherwise it would be necessary to define spinor fields which transform non-linearly under the diffeomorphism group [4]. Also, a noncompact (4 dimensional) space-time admits a spinor structure if and only if it has a global field of orthonormal tetrads [5]; meaning that the concrete space-time is parallelizable [5] and so the Levi-Civita and the Weizenbock connections coexist [6] and General Relativity may be equivalently defined as Teleparallel Gravity [7]. Many well-known solutions of the Einstein equations do have a spinor structure [5]. However, if only diffeomorphism- and local Lorentz-invariant operators are allowed, then the expectation values would correspond to any space-time with a Lorentz metric as if we would have defined the tetrads only *locally*.

In fact, the space-time in the Hamiltonian formalism is time-orientable since it requires a global time-like vector field giving the direction of the time-evolution [8]; the space-time also requires a volume form (to define the Hamiltonian) which is an orientation [9]; the space-time is also space-orientable, because it is orientable and time-orientable [8]. Both the volume form and the global time-like vector field are invariant under local Lorentz transformations. Crucially, the diffeomorphism invariance of the Hamiltonian formalism remains intact.

Moreover, the Einstein-Hilbert action with the metric as a dynamical variable has second-order derivatives of the metric, which requires adding extra terms to the action in the presence of some boundary conditions [10].

We stress that a parallelizable space-time doesn’t need to have a Cauchy surface because the derivatives corresponding to different tetrads do not need to commute, thus the parallelizable space-time doesn’t need to be globally hyperbolic (i.e. topologically a product space of time and a 3-dimensional manifold)[11][12][5][13].

If we assume that there is also a global scalar field corresponding to time (which implies that the space-time has stable causality[8]), then we can define a family of 3-dimensional achronal (but not necessarily space-like) hypersurfaces corresponding to constant times[8]. However, the step from local existence of solutions to the Einstein equations to the existence of a maximal development is non-trivial, due to the diffeomorphism invariance of the equations [14][15]. It turns out that given initial data (that is, one 3-dimensional hypersurface corresponding to a specific time), there is a unique maximal globally hyperbolic development [14]. Despite that the maximal development is only a subset of the total space-time, it is nevertheless a space-time in itself which can be defined by an Hamiltonian formalism. This justifies the success of the predictions of numerical relativity based on the ADM formalism [16][17]) which assumes an orientable globally hyperbolic spacetime (that is, all of the 3-dimensional achronal hypersurfaces are space-like).

Thus, there is no guarantee “a priori” that in Quantum Gravity the space-time is globally hyperbolic as it is assumed in the quantum ADM formalism. Moreover, the classical ADM formalism is a weakly hyperbolic system of partial differential equations and thus it is divergent and not well-posed[15]. Thus, there is no classical Hamiltonian formalism which is well-posed. Note that the “Hamiltonian formalism” proposed in reference[15] is admittedly only defined locally and thus it is not a solution to the Cauchy problem in General Relativity as a complete Hamiltonian formalism should be.

The second step towards Quantum Gravity is to choose the mathematical tools needed to define the theory. There is a widespread belief that the sequence of generalizations on the descriptions of space and time 1) Galilean invariance, 2) special relativity, 3) general relativity; which happened for deterministic theories should also happen for quantum theories [18] (the present author also shared this belief in the past [19]). Following this belief, the special role of the little group of rotations and the time evolution in our definition of a (special relativistic) Quantum Field Theory seems to be a step back in the road towards a general relativistic quantum theory. Thus, our definition would not be of much value since a rigorous definition of a Quantum Yang-Mills theory should mark *“a turning point in the mathematical understanding of quantum field theory, with a chance of opening new horizons for its applications”*, as stated in reference [20].

But in the Hamiltonian formalism (and so in any quantum theory), the diffeomorphisms are generated by constraints [21] while the Poincare transformations act in a non-trivial way in the algebra of operators. Hence, the diffeomorphisms cannot be a generalization of a non-trivial Poincare symmetry and so, *a priori* diffeomorphisms are not incompatible with the special role of the little group of rotations and the time evolution[21] . *A priori*, the formalism of gauge-invariant Quantum Field Theory may be enough to define quantum gravity.

The crucial fact that allows us to distinguish the Poincare symmetry from diffeomorphisms and gauge symmetry is the fact that the time-evolution generated by the Hamiltonian is *not* a diffeomorphism or a gauge symmetry. There are indeed diffeomorphisms which advance the time coordinate of the phase-space (as we would expect from the time-evolution), however they differ from the time-evolution in how they affect the wave-function as a whole. Thus, the Poincare symmetry is not a diffeomorphism or a gauge symmetry and it is not generated by constraints. The algebra of operators is a non-trivial representation of the Poincare group, while it is diffeomorphism invariant and gauge invariant. The generators of the (global) Poincare group are also diffeomorphism invariant and gauge invariant.

We define mathematically a general relativistic Quantum Field Theory, verifying all the properties we expect from Quantum Gravity, without adding complexity with respect to the mathematical definition of gauge-invariant Quantum Field Theory presented in the previous sections.

Unlike in Yang-Mills theory, the Hamiltonian of the Einstein-Cartan theory involves canonical commutation relations of analytic functions of operators, which is no challenge by itself when there is a non-polynomial function of only one of the operators involved [22]. With the Weyl ordering of products of operators and the measure defined in the previous section, there is no challenge or ambiguity if the kinetic part of the Hamiltonian is polynomial both in the fields and in its canonical conjugates while the potential is rational in the fields only (with no dependence on the canonical conjugates), just like in the Hydrogen atom in non-relativistic Quantum Mechanics. Note that since the Hamiltonian is defined in a 4-dimensional space-time, the diffeomorphisms (and all other constraints) are polynomial in the fields and its canonical conjugates, unlike in Loop Quantum Gravity [23][24].

Moreover, no vaccuum state needs to be defined (the Hamiltonian needs not to be bounded from below), thus there is no need for reguralization and so renormalization plays no essential role (at the non-perturbative level).

Moreover, the operator ordering ambiguity in Loop Quantum Gravity may limit it to the same peculiar notion of “continuum” through renormalization as in (Wilson’s) Quantum Field Theory, as suggested in reference [24]:

“[...] the ultimate goal is to use Hamiltonian renormalisation to find a continuum theory for canonical Quantum Gravity. Here we can use the LQG [Loop Quantum Gravity] candidate as a starting point because it is rather far developed, but of course the flow scheme developed can be applied to any other canonical programme”.

We propose instead that Quantum Gravity at all stages of the quantization programme, should be seen as a particular case of a (special relativistic) Quantum Field Theory defined in the continuum, which is a major simplification with respect to Loop Quantum Gravity.

Therefore, in the quantization due to time-evolution, there is no obvious advantage of trying to rewrite the Hamiltonian as a polynomial in the canonical operators, in contrast with Loop Quantum Gravity. We can still do it after the definition of the quantum theory[25], for the purpose of numerical computations in perturbation theory for instance.

We will start by defining the Hamiltonian for Classical General Relativity within the formalism of Quantization due to the time-evolution. Just for the purpose of solving initial-value problems within the classical theory, it is already advantageous since it allows us to define an Hamiltonian which is diffeomorphism invariant (unlike in the ADM formalism[16][17]), avoiding the technical complexity of the definition of chains of constraints[15]. Moreover, the low-energy limit of Quantum Gravity must anyway be studied and it should match the classical theory.

The classical Action for the Einstein–Cartan theory of gravitation is:

$\begin{aligned}
S=&\int d^4 x \mathcal{L}\\
\mathcal{L}=&e R\\
e=&\det{e_{\mu}^{a}}\\
R=&g^{\mu \nu} R_{\mu \nu}\\
R_{\mu \nu}=&R^{\alpha}_{\mu \alpha \nu}\\
R^{\alpha}_{\beta \mu \nu}=&e^{\alpha}_{a} e_{\beta}^{b}\partial_{\mu}{\omega^{a}_{\nu b}}
-e^{\alpha}_{a} e_{\beta}^{b}\partial_{\nu}{\omega^{a}_{\mu b}}
+e^{\alpha}_{a} e_{\beta}^{b}\omega^{a}_{\mu c}\omega^{c}_{\nu b}
-e^{\alpha}_{a} e_{\beta}^{b}\omega^{a}_{\nu c}\omega^{c}_{\mu b}\\
\omega^{a}_{\mu b}=&1/2 (\eta^{a c} e^{\nu}_{c} \eta_{b k}\delta_{\mu}^{\alpha}\delta_{\nu}^{\beta}-e^{\nu}_{b}\delta_{\mu}^{\alpha}\delta_{\nu}^{\beta}\delta^{a}_{k}-\eta^{a c} e^{\alpha}_{c}e_{\mu}^{j}\eta_{j k}
e^{\beta}_{b})(\partial_{\alpha}{e_{\beta}^{k}}-\partial_{\beta}{e_{\alpha}^{k}})\\
g^{\mu \nu}=&e^{\mu}_{a} e^{\nu}_{b} \eta^{a b}\end{aligned}$

Where $a, b, c, j, k\in\{0,1,2,3\}$.

Then we integrate by parts in the variable $\partial_{\nu}{e_{\alpha}^{a}}$ and replace:

$\begin{aligned}
\partial_{\alpha}{e}=&-e e_{\mu}^{a}\partial_{\alpha}{e^{\mu}_{a}}\\
\partial_{\alpha}{e_{\mu}^{a}}=&-e_{\nu}^{a}\partial_{\alpha}{e^{\nu}_{b}} e_{\mu}^{b}\end{aligned}$

We also define:

$\begin{aligned}
\chi_{a}^{\ b}=&\delta_{a}^{\ b}+v_{a} v^{b}\\
e_{a b c}=&\partial_{\alpha}{e^{\beta}\,_{b}} e^{\alpha}\,_{a} e_{\beta}\,^{k}\eta_{k c}\\
E_{a b}=& \chi_{a}^{\ a_1} v^{c} \chi_{b}^{\ a_2} e_{a_1 c b}\\
E_{a}=& \chi_{a}^{\ a_1} v^{c} v^{b} e_{a_1 c b}\\
T_{a b c}=&e_{a b c}-e_{b a c}\\
T_{a b}=& v^{c} \chi_{a}^{\ a_1} T_{c a_1 b}\\
A_{a b}=& \chi_{a}^{\ a_1} \chi_{b}^{\ a_2} (T_{a_1 a_2}-T_{a_2 a_1})\\
T=&\chi_{a}^{\ a_1} \chi_{b}^{\ a_2} T_{a_1 a_2} \eta^{a b}\\
S_{a b}=&\chi_{a}^{\ a_1}\chi_{b}^{\ a_2}(T_{a_1 a_2}+T_{a_2 a_1}-\frac{2}{3}\eta_{a_1 a_2} T)\\
T_{a}=&v^c \chi_{a}^{\ a_2} v^{b} T_{c a_2 b}\\
\mathcal{T}_{a b c}=& \chi_{a}^{\ a_1} \chi_{b}^{\ a_2} \chi_{c}^{\ a_3} T_{a_1 a_2 a_3}\\
\mathcal{T}_{a b}=& \chi_{a}^{\ a_1} \chi_{b}^{\ a_2} v^{c} T_{a_1 a_2 c}\end{aligned}$

Where $v^a=v^\mu e_\mu^a$ are the tetrad components of the globally defined time-like vector verifying $v^\mu v_\mu=-1$, $a,b,c,a_1,a_2,a_3\in\{0,1,2,3\}$ and $T_{c b a}=-T_{c a b}$.

Up to a divergence term, the Lagrangian density is given by:

$\begin{aligned}
&\mathcal{L}\approx e\left(T_{a b}^{\ \ b} T^{a c}_{\ \ c} - \frac{1}{2} T_{a b c} T^{a c b} - \frac{1}{4}T_{a b c} T^{a b c}\right)=\\
% &- e T_{a b c} T_{j k l} \eta^{a j}\left(-\eta^{b c} \eta^{k l} + \frac{1}{8}\eta^{b l} \eta^{c k}-\frac{1}{8}\eta^{b k} \eta^{c l} + \frac{3}{8}\eta^{b l} \eta^{c k} + \frac{3}{8}\eta^{b k} \eta^{c l}\right)=\\
% &- e T_{a b c} T_{j k l} \eta^{a j}\left(-\eta^{b c} \eta^{k l} + \frac{1}{8}\eta^{b l} \eta^{c k}-\frac{1}{8}\eta^{b k} \eta^{c l} + \frac{3}{8}\eta^{b l} \eta^{c k} + \frac{3}{8}\eta^{b k} \eta^{c l}\right)=\\
% &=+\frac{13}{16}e (T)^2+\frac{1}{16} e A_{a c} S_{k l}\eta^{a k} \eta^{c l}-e\frac{3}{16} S_{a c} S_{k l}\eta^{a k} \eta^{c l} + e T_{b a c} \chi^{b j} T_{j k l}\left(\eta^{a c}\eta^{k l} - \frac{1}{2}\eta^{a l} \eta^{c k} - \frac{1}{4}\eta^{a k} \eta^{l c}\right)\\
&=\frac{1}{4}S_{a b} S^{a b} e - \frac{2}{3}(T)^2 e - \frac{1}{2}A_{a b} \mathcal{T}^{a b} e
-2 \mathcal{T}^{a b}_{\ \ a} T_{b} e+\\
&- \frac{1}{2}\mathcal{T}_{a b c} \mathcal{T}^{a c b} e - \frac{1}{4} \mathcal{T}_{a b c} T^{a b c}e+\mathcal{T}_{b a}^{\ \ a} \mathcal{T}^{b c}_{\ \ c} e+\frac{1}{4}\mathcal{T}_{a b} \mathcal{T}^{a b} e \end{aligned}$

This is in agreement with reference [7], up to a global sign of the Lagrangian density (which does not affect the classical equations of motion). Then we vary the resulting Lagrangian density in the variables $\partial_{\alpha}{e^{\beta}_{a}}$, to obtain the polymomentum [26]:

$\begin{aligned}
&p^{a b}=\frac{\delta\mathcal{L}}{\delta \partial_{\alpha}{e^{\beta}_{a}}}v_{\alpha} e^{\beta}_{c}\eta^{c b}=\\
&=\bigl(\frac{\delta\mathcal{L}}{\delta S_{a_3 a_4}}\frac{\delta S_{a_3 a_4}}{\delta T_{a_1 a_2}}
+\frac{\delta\mathcal{L}}{\delta T}\frac{\delta T}{\delta T_{a_1 a_2}}
+\frac{\delta\mathcal{L}}{\delta A_{a_3 a_4}}\frac{\delta A_{a_3 a_4}}{\delta T_{a_1 a_2}}
+\frac{\delta\mathcal{L}}{\delta T_{a_3}}\frac{\delta T_{a_3}}{\delta T_{a_1 a_2}}\bigr)
\frac{\delta T_{a_1 a_2}}{\delta \partial_{\alpha}{e^{\beta}_{a}}} v_{\alpha}e^{\beta}_{c}\eta^{c b}\\
&=e\biggl(\frac{1}{2}S_{a_3 a_4}(\eta^{a_3 a_1}\eta^{a_4 a_2}+\eta^{a_4 a_1}\eta^{a_3 a_2}-\frac{2}{3}\eta^{a_3 a_4}\eta^{a_1 a_2})
-\frac{4}{3}T \eta^{a_1 a_2}+\\
&-\frac{1}{2} \mathcal{T}_{a_3 a_4} (\eta^{a_3 a_1}\eta^{a_4 a_2}
-\eta^{a_4 a_1}\eta^{a_3 a_2})+2 \mathcal{T}_{a_3 a_4}\,^{a_4} \eta^{a_3 a_1} v^{a_2}\biggr)
\delta_{a_1}^{\ a}\delta_{a_2}^{\ b}\\
&=e\biggl(S^{a b}
-\frac{4}{3}T \eta^{a b}-\mathcal{T}^{a b}+2 \mathcal{T}^{a c}_{\ \ c} v^{b}\biggr)\\
&\mathcal{A}^{a b}=\chi^{a}_{\ a_1} \chi^{b}_{\ a_2}(p^{a_1 a_2}-p^{a_2 a_1})=
-2 e \mathcal{T}^{a b}\\
&\mathcal{P}=\eta_{a b}\chi^{a}_{\ a_1} \chi^{b}_{\ a_2} p^{a_1 a_2}=-4 e T\\
&\mathcal{S}^{a b}=\chi^{a}_{\ a_1} \chi^{b}_{\ a_2}
(p^{a_1 a_2}+p^{a_2 a_1}-\frac{2}{3}\eta^{a_1 a_2} \mathcal{P})=2 e S^{a b}\\
&p^{a}=v_b p^{a b}=2 e \mathcal{T}^{a c}_{\ \ c}\end{aligned}$

The Hamiltonian density in 3-dimensional space is:

$\begin{aligned}
&\mathcal{H}=v_{\alpha}\frac{\delta\mathcal{L}}{\delta \partial_{\alpha}{e^{\beta}_{a}}}v^{\mu}\partial_{\mu} e^{\beta}_{a}-\mathcal{L}=
p^{a b}T_{a b}+p^{a b}E_{a b}-\mathcal{L}=-\mathcal{L}+\frac{1}{4}\mathcal{A}^{a b} A_{a b}+\frac{1}{4}\mathcal{S}^{a b}S_{a b}+\\
&+\frac{1}{3}\mathcal{P} T+p^{a} T_{a}+\frac{1}{2}\mathcal{A}^{a b} E_{a b}+\frac{1}{2}\mathcal{S}^{a b}E_{a b}
+\frac{1}{3}\mathcal{P} E_{a}\,^{a}+p^{a} E_{a}\\
&\approx e\biggl(\frac{1}{4}S^{a b} S_{a b} - \frac{2}{3}(T)^2
+S^{a b}E_{a b}-\frac{4}{3}T E_{a}\,^{a}-\mathcal{T}^{a b} E_{a b} +2 \mathcal{T}^{a b}\,_b E_{a}+\\
&+ \frac{1}{2}\mathcal{T}_{a b c} \mathcal{T}^{a c b} + \frac{1}{4} \mathcal{T}_{a b c} T^{a b c}-\mathcal{T}_{b a}^{\ \ a} \mathcal{T}^{b c}_{\ \ c}-\frac{1}{4}\mathcal{T}_{a b} \mathcal{T}^{a b}\biggr)\end{aligned}$

As expected, the Hamiltonian density in 3-dimensional space only depends on the momenta $\mathcal{S}^{m n}$ and $\mathcal{P}$ through $S^{m n}$ and $T_{0}$:

$\begin{aligned}
&\mathcal{H}=\frac{1}{16 e}\mathcal{S}^{a b} \mathcal{S}_{a b} - \frac{1}{24 e} (\mathcal{P})^2
+\frac{1}{2}\mathcal{S}^{a b} E_{a b}
+\frac{1}{3}\mathcal{P} E_{a}\,^{a}-e\biggl(\mathcal{T}^{a b} E_{a b}+2 \mathcal{T}^{a b}\,_{b} E_{a}+\\
&+\frac{1}{2}\mathcal{T}_{a b c} \mathcal{T}^{a c b} + \frac{1}{4} \mathcal{T}_{a b c} T^{a b c}-\mathcal{T}_{b a}^{\ \ a} \mathcal{T}^{b c}_{\ \ c}-\frac{1}{4}\mathcal{T}_{a b} \mathcal{T}^{a b}\biggr)\end{aligned}$

The Hamiltonian density in 4-dimensional space is given by:

$\begin{aligned}
&\mathcal{H}_4=-p^{a b}T_{a b}-p^{a b}E_{a b}+\mathcal{H}=\\
&=\frac{1}{4}\mathcal{A}^{a b} A_{a b}+\frac{1}{4}\mathcal{S}^{a b} S_{a b}+\frac{1}{3} \mathcal{P}T+ p^{a} T_{a}+\\
&- \frac{1}{16 e}\mathcal{S}^{a b} \mathcal{S}_{a b}+\frac{1}{24 e}(\mathcal{P})^2- \frac{1}{2}\mathcal{T}^{a c b} \mathcal{T}_{a b c} e
-\mathcal{T}^{a c}\,_{a} \mathcal{T}_{c b}\,^{b} e+\\
&+\frac{1}{4} \mathcal{T}^{a b} \mathcal{T}_{a b} e - \frac{1}{4} \mathcal{T}^{a b c} \mathcal{T}_{a b c} e\end{aligned}$

The constraints are the diffeomorphisms, local Lorentz transformations and global translations:

$\begin{aligned}
% -G_\mu= e_{\nu}^{a} \partial_{\mu} p^{\nu}_{a} +\partial_{\nu}(e_{\mu}^{a}p^{\nu}_{a})
G_\mu=&p^{\nu}_{a} \partial_{\mu} e_{\nu}^{a}-\partial_{\nu}(p^{\nu}_{a} e_{\mu}^{a})+p^{\nu}_{v} \partial_{\mu} v_{\nu}-\partial_{\nu}(p^{\nu}_{v} v_{\mu})\\
J_{a b}=&p^{\mu}_{a} e_{\mu}^{c}\eta_{c b}-p^{\mu}_{b} e_{\mu}^{c}\eta_{c a}\\
P_\mu=&\int d^4x \biggl(p^{\nu}_{a} \partial_{\mu} e_{\nu}^{a}+p^{\nu}_{v} \partial_{\mu} v_{\nu}\biggr)\end{aligned}$

Note that the diffeomorphisms and global translations are consistent with each other, but they are two independent constraints. The BRST charge is:

$\begin{aligned}
% [G,b_0]=p^{\nu}_{a} \partial_{0} e_{\nu}^{a}-\partial_{0}(p^{0}_{a} e_{0}^{a})-\partial_{j}(p^{j}_{a} e_{0}^{a})
G=& \int d^4x \biggl(p_{\nu}^{a}c^{\mu}\partial_{\mu}e^{\nu}_{a}-p_{\nu}^{a}e^{\mu}_{a}\partial_{\mu}c^{\nu}
+\pi_{\nu}c^{\mu}\partial_{\mu}v^{\nu}-\pi_{\nu}v^{\mu}\partial_{\mu}c^{\nu}
+i \partial_{\beta}c^{\alpha} c^{\beta}b_{\alpha}\biggr)\end{aligned}$

When the globally-defined time-like vector is constant and given by $v^\mu=\delta^\mu_0$, the 4-dimensional Hamiltonian formalism can be reduced to a 3-dimensional Hamiltonian formalism. Such formalism is different from the ADM formalism since the constraints are different, which is in fact a good property. The standard ADM evolution system is only weakly hyperbolic and so it is not well-posed[27]. Well-posed versions of the Einstein equations have in fact been known since the 1950s, but they were not based on a 3+1 decomposition and they are not suitable for numerical calculations[27]. Thus a strongly hyperbolic reformulation of the 3+1 evolution equations is required anyway[27].

The diffeomorphisms must now conserve the condition $v^\mu=\delta^\mu_0$, so the ghosts appearing in the BRST charge for the 3-dimensional Hamiltonian formalism are constant in the timepiece, with the BRST charge having the same functional form as in the 4-dimensional formalism. Such a BRST charge is significantly different from the BRST charge of the ADM formalism.

The Hilbert space (in the particular case of an empty spacetime) is the tensor product of the symmetric and antisymmetric Fock spaces $\Gamma^s(L^2(\mathbb{R}^{84}\times \mathbb{Z}_2^{19}))\otimes \Gamma^a(L^2(\mathbb{R}^{84}\times\mathbb{Z}_2^{19}))$ [28][29]. This gives us a graded Lie superalgebra of creation and annihilation operators, of both bosonic and fermionic types. The total fermionic number is defined as the sum of all number operators corresponding to fermionic degrees of freedom. The fermionic creation and annihilation operators correspond to a field configuration where the total fermionic number is odd, while the bosonic creation and annihilation operators correspond to a field configuration where the total fermionic number is even.

The $\mathbb{R}^{84}$ degrees of freedom correspond to 4 space coordinates (x), 4x4=16 fields ($e_{\mu}^{a}$) and its corresponding space-derivatives ($\partial_{\mu}e_{\nu}^{a}$). The $\mathbb{Z}_2^{19}$ degrees of freedom correspond to 4 ghosts corresponding to each generator of diffeomorphisms ($\psi_{\mu}$) and its corresponding spacetime-derivatives ($\partial_{\mu}\psi_{\nu}$), minus one because we are taking the tensor product of two Fock-spaces (bosonic and fermionic) which introduces an extra $\mathbb{Z}_2$ degree of freedom.

We define:

$\begin{aligned}
& a(x, ...) = a(x,\oplus_{a} (\alpha_{a},(\oplus_{\mu}A_{\mu,a}),\oplus_{\nu}(\alpha_{a \nu},(\oplus_{\mu}A_{\mu a,\nu}))))\\
& \int d^4 x ... = \int d^4 x \prod_{a}((\prod_{\mu}dA_{\mu a})\sum_{\alpha_{a}=0}^1\prod_{\nu}((\prod_{\mu}d A_{\mu a,\nu})\sum_{\alpha_{a \nu}=0}^1)) \\
&\{[a(x, ..., \mu),a^\dagger(y, ...,\nu)]\}=a(x, ...,\alpha) a^\dagger(y, ...,\beta)+(-1)^{(\alpha \mathrm{mod} 2)(\beta \mathrm{mod} 2)}a^\dagger(y, ...,\alpha) a(x, ...,\beta)=\\
&=\delta((x,...)-(y,...))\\
&[A_{\mu a},\pi^{\nu}_{b}]=A_{\mu a}\pi^{\nu}_{b}-\pi^{\nu}_{b} A_{\mu a}= i\delta^{\nu}_{\mu} \delta_{a b}\\
&[ A_{\nu a,\mu},\pi^{\alpha\beta}_{b}]=A_{\nu a,\mu}\pi^{\alpha \beta}_{b}-\pi^{\alpha\beta}_{b} A_{\nu a,\mu}= i\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}\delta_{a b}\\
&\{\psi_{a},\psi^\dagger_{b}\}=\psi_{a} \psi^\dagger_{b}(y)+\psi^\dagger_{b}\psi_{a}=\delta_{a b}\\
&\{\psi_{a \mu},\psi^\dagger_{b \nu}\}=\psi_{a \mu} \psi^\dagger_{b \nu}+\psi^\dagger_{b \nu}\psi_{a \mu}=\delta_{a b}\delta_{\mu \nu}
\end{aligned}$

where

$\begin{aligned}
& \psi^\dagger\{a\}(x, ...,j)= a(x, ...,1) \delta_{j 0}\\
& \psi\{a\}(x, ...,j)= a(x, ...,0) \delta_{j 1}
\end{aligned}$

The Hamiltonian for the Yang-Mills theory in our formalism is related to the classical Hamiltonian Action[21]:

$\begin{aligned}
&H=\int d^4 x ... a^\dagger(x, ...)H(x, ...) a(x, ...)\\
&H(x, ...)=i\partial_0-\pi^i_a A_{i a, 0}+ \pi^i_a (D_i A_{0 a})-\frac{1}{2}\pi^i_a\pi^i_a
-\frac{1}{2} B_{i a} B_{i a}\\
&\Omega=\int d^4 x ... a^\dagger(x, ...) \Omega(x,...) a(x, ...)\\
&\Omega(x,..)=\left[ \pi^{\mu}_{a} \partial_{\mu} \psi^\dagger_{a}-\pi^{\mu}_{a} f_{a b c} A_{\mu b} \psi^\dagger_{c} -i \frac{1}{2}f_{a b c} \psi^\dagger_{a} \psi^\dagger_{b} \psi_{c}\right]
\end{aligned}$