Quantum Yang-Mills theory, Classical Statistical Field Theory (for Hamiltonians which are non-polynomial in the fields, e.g. General relativistic statistical mechanics) and Quantum Gravity all suffer from severe mathematical inconsistencies and produce unreliable predictions at best.

We define a class of statistical field theories where the probability distribution of the infinite-dimensional phase-space is mathematically defined by a wave-function in a Fock-space, allowing Hamiltonians which are non-polynomial in the fields. In this formalism, the wave-function is crucial already for classical statistical field theory due to mathematical reasons.

By defining gauge symmetries through algebraic ideals, we propose definitions for Quantum Yang-Mills and Classical Statistical Field Theory. Finally, we test the consistency of our formalism with the quantization of the free Electromagnetic field.

Lorentz covariance

Concerning the relation between the Hamiltonian formalism and special relativity, in this section we will just define the theory and show that it is Poincare invariant, postponing foundational questions to when a diffeomorphism-invariant (which includes Poincare invariance) theory is defined1. In short, the fact that time evolution plays a special role allows us to use only Poincare representations with positive squared mass. Considering Poincare representations with positive squared mass is self-consistent and it is in no way in conflict with Lorentz symmetry.

A complete physical system is a free system. If we neglect gravity, the wave-function associated to the free system is a unitary representation of the Poincare group, regardless of the interactions occurring within the free system [1].

When the Hilbert space is the direct sum of irreducible representations of a symmetry group, then these representations will be defined by numbers (e.g. mass and spin) which are invariant under the symmetry group. Thus there will be a set of operators whose diagonal form corresponds to those invariants, we will call them Casimir operators. When the symmetry group is abelian and continuous (e.g. translation in time), then the generator of the group (e.g. the Hamiltonian) is a Casimir operator, and the invariant numbers defining the representations are called the constants of motion. Certainly, when we move from non-relativistic quantum mechanics and consider instead the Poincare group, then the Hamiltonian is no longer a Casimir operator and the notion of constants of motion needs to be reviewed. Nevertheless, the Casimir operators can be chosen arbitrarily (just like the Hamiltonian in non-relativistic quantum mechanics).

For a positive squared mass, the spin and the sign of the Energy are also Poincare invariants. The sign of the Energy times the modulus of the mass is the center-of-mass Energy, while the spin is the center-of-mass angular momentum. Thus, the Casimir operator whose eigenvalues are the center-of-mass Energy may have negative eigenvalues and it will be the analogous operator to the Hamiltonian of the non-relativistic formalism2. The Casimir operators necessarily commute with the momentum operator and thus they do not change the 3-momentum eigenstate. Thus we can solve the problem in a basis where the 3-momentum operator is diagonal.

In such a basis, the translations in space-time can be written as:

T(x)\Psi(\gamma\vec{v})=e^{i M \tau(\gamma\vec{v}, x)}\Psi(\gamma\vec{v})

Where $\tau(\gamma\vec{v}, x)=\gamma x_0-\gamma \vec{v}\cdot \vec{x}$ .

Note that $\gamma=\sqrt{1+\gamma^2\vec{v}^2}$ is a function of $\gamma\vec{v}$ . Thus in a basis where the 3-momentum is diagonal, the translations in space-time have the same structure as the time-evolution in non-relativistic space-time, with $M$ playing the role of the non-relativistic Hamiltonian and the numerical factor $\tau(\gamma\vec{v}, x)$ playing the role of the time (it is indeed the proper-time).

For each 3-momentum eigenstate, there is a corresponding inertial referential where the 3-momentum is null, i.e. the referential of the center-of-mass. In such referential, the modulus of the energy is the invariant mass, the signal of the energy is also a Lorentz invariant and the angular momentum is the spin. Thus, the eigenvalues of the Hamiltonian and angular momentum operator in the center-of-mass define three Lorentz invariants which define the Poincare representation completely.

Despite we do not know a priori the diagonal form of the Hamiltonian, we know that it is either continuous or discrete in the neighborhood of the eigenvalue 0 (in the referential of the center-of-mass). If it is continuous then the zero energy has null measure. If it is discrete, we can modify the Hamiltonian adding an appropriate constant such that the zero energy is not one of the eigenvalues. Note that this is only possible in a complex Hilbert space and this is equivalent to adding to the system a free massive particle with null 3-momentum relative to the system. In any case, we can assume without loss of generality that our system is a quantum superposition of massive free systems with null 3-momentum. Then, the Lorentz transformations become known and are given by the Wigner irreducible massive representations of the Poincare group[1]. If the Hamiltonian is bounded from below then the vacuum state is not Lorentz invariant, as it was already suggested[2].

In the center-of-mass, the relevant group is not the Poincare group, but the little group of spatial rotations and the translation in time[1]. Thus the spatial and time coordinates of space-time, become separated. The fields are no longer representations of the Lorentz group, but only of the rotation group and the canonical commutation relations are not in conflict with the little group of spatial rotations.

Note that we use Wigner’s convention for the definition of the 3-momentum of the free complete system: it is the eigenvalue of the generator of the translations in space for the complete system (i.e. all fields defining the phase-space are translated in space). Thus in the center-of-mass, the algebra of operators has a constraint imposing that the operators are translation invariant. This constraint is the same needed to define the free field parametrization, the constraint is $iD_x=0$ as defined in chapter /pub/field, so no generality is lost and the free field parametrization is consistent with Poincare covariance.

The cyclic vector defining the Hilbert space needs not be translation-invariant (in the center-of-mass), just the operators need to be translation-invariant in the center-of-mass3. This gives us a big freedom to choose the cyclic vector defining the Hilbert space (which is related with the initial state of the system).

We assume that the translations in space of the complete system conserve the Hamiltonian and the constraints equations, such that setting the total 3-momentum to zero in no way conflicts with the constrained Hamiltonian system. Nevertheless, the restriction that there is a referential where the total 3-momentum is null, excludes the free complete system from traveling at the speed of light (e.g. a photon with non-null energy). Then the dynamics determined by the Hamiltonian becomes linked with the time coordinate (for a photon this would not be the case[3]).

Therefore and unlike what it is often claimed in the literature, it is false that (canonical) quantization is incompatible with Lorentz covariance. Note that the phenomenologically successful (but ill defined) path integral formalism based on the Lagrangian is in fact equivalent to a path integral based on the Hamiltonian[4]. In our formalism, the only restriction is that we need to consider representations with positive squared mass, then the dynamics determined by the Hamiltonian becomes linked with the time coordinate[3]. The question why only positive squared masses are relevant is a reformulation of question 2) which will be left open in this paper. Similar assumptions concerning the energy-momentum of the full system are also done in the Källén-Lehmann representation of a non-perturbative two-point correlation function, where it is assumed that the eigenvalues of the 3-momentum squared are not larger than those of the squared energy[5].

Gauge-variant gaussian measure and the phase diagram

Unlike in non-relativistic Quantum Mechanics, there is no rigorous definition of what is a Quantum Yang-Mills theory to compare our results with, then our approach in field theory is different than in the non-relativistic case. Our goal is to build a self-consistent rigorous theory which after some approximations (e.g. perturbative expansion, or a ultra-violet cutoff) reproduces the successful predictions of the Standard Model of Particle Physics.

The Feynman’s path-integral produces the notion of a gauge-invariant vacuum state, when the canonical formalism is “derived” from the path-integral formalism. However, the Feynman’s path integral assumes the existence of a translation-invariant $\sigma$ -finite (i.e. Lebesgue like) measure which is therefore gauge-invariant. Yet, it is proved that in rigor such infinite-dimensional Lebesgue measure cannot exist. As a consequence, the notion of a gauge-invariant vacuum state is inconsistent. On the other hand, a gauge-variant probability measure for an infinite-dimensional phase-space is mathematically consistent (e.g. a gaussian measure).

Some symmetries of the algebra of operators cannot be symmetries of the cyclic state defining the Hilbert space4. In the case of gauge symmetry, the gauge potentials can be fully reconstructed from the algebra of gauge-invariant operators [6]. Moreover, the Fock space (defined on a 3-dimensional space) produces well-defined expectation functionals for the gauge-invariant operators [7]. The expectation-values of the gauge-invariant operators fully define the statistical gauge field theory (since the gauge potentials can be fully reconstructed [6]), thus the gauge-variant operators can be neglected. Of course, gauge-variant operators can act on the Fock-space, but the link between these operators and the underlying manifold of gauge potentials is destroyed since the expectation-value is not gauge-invariant.

Since only (fully) gauge-invariant operators are allowed and the wave-functions necessarily break the gauge-symmetry, in scattering theory we always need to work in the in-in formalism. Of course, we can use the more common in-out formalism in intermediate steps. Explicitly, any bounded normal operator can be expressed in diagonal form using projection-valued measures. These projections are built using a basis of the Hilbert space, which can be expressed using gauge-invariant operators acting on the cyclic state (initial state). Thus, the complex amplitudes of the in-out formalism can be expressed as expectation values of the in-in formalism. The amplitudes are complex to allow that a constant can be added to the Hamiltonian without observable consequences, which is crucial to ensure the Lorentz covariance of the theory as it was discussed in the previous section.

Thus the gauge-variant initial (cyclic) states are perfectly fine, even at the non-perturbative level. In the canonical formalism the same gauge-variant initial states are allowed due to the adiabatic approximation and the Gell-Mann and Low’s theorem[8][9], although this issue is more subtle than in our formalism[10].

This does not imply that the mean-field approximation (that is, the usual choice of exact wave-function around which perturbative corrections are applied) always works, but the non-perturbative problems of the mean-field approximation are not exclusive to the gauge-symmetry (e.g. the mean-field approximation also breaks global symmetries in the two-Higgs-doublet model). So it can happen that within our (non-perturbative) mathematical definition of Quantum Gauge Field Theory, some exact predictions differ substantially from the corresponding perturbative approximation, being the predictions involving non-null non-abelian global charges obvious candidates for this difference to show up [11][12][13][14]. However, the non-perturbative validity of our definition also shows that there is no fundamental reason why all non-abelian global charges must be null as claimed recently [15].

The global charge operator is different from a linear combination of gauge generators which are constrained to be zero (due to surface terms at infinity). In the in-in formalism, we can characterize an initial state with a non-null global charge using only gauge-invariant operators (e.g. the Casimir operators of the algebra generated by the global charge of an abelian or non-abelian gauge-theory, are gauge-invariant operators).

Spontaneous symmetry breaking

There are several definitions of spontaneous symmetry breaking in the context of statistical mechanics, based on: a long-range order parameter which is the expectation value of a function $f(A)$ invariant under a group $G$ (e.g. the modulus $f(A)=|A|$ ) of an operator $A$ which is translation invariant and breaks $G$ ; or a conditional expectation value of some operator $A$ given some condition $C=0$ that breaks the symmetry; or a two-point correlation function with the points at an infinite distance from each other (related with boundary conditions)[16].

In the translation-invariant time-evolution discussed in Chapter /pub/field, the probability distribution is asymptotically constant in the continuum space so that spontaneous symmetry breaking is possible.

As it was discussed in the previous section, the wave-function necessarily breaks the full gauge symmetry. But it may or not break the global gauge symmetry (remnant of the full gauge symmetry). However, there is no spontaneous symmetry breaking of the (full or global) gauge symmetry, since the expectation-values of the gauge-variant operators are null due to the constraints5. Note that the gauge-invariant operators are not sensitive to the long-range correlations which could signal spontaneous symmetry breaking[17].

Still the phase diagram of the theory may be sensitive to whether or not the field configuration at infinity conserves the global gauge symmetry. Since only gauge-invariant operators are allowed6, 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. Thus, the concrete manifold appearing in the phase-space may feature spontaneous symmetry breaking of the global gauge symmetry and this may cause effects in the phase-diagram (such as the existence of a phase transition), despite that there is no spontaneous symmetry breaking of the (full or global) gauge symmetry; the Higgs mechanism is an example of such a case.

Local operators and the momentum constraint

The momentum constraint also generates a gauge symmetry, once we consider a spectral measure where the fields are functions of space. Then the momentum constraint always modifies the spectral measure and so we have a complete unconstrained gauge-fixing.

Since all operators must be invariant under a translation in space, how can we define local operators? In rigor we can’t, but we can define operators which are linear combinations of local operators, which behave effectively as local operators. Consider the operator $\int d\vec{x}\ l(\vec{x})$ which is translation invariant, where $l(\vec{x})$ is a local operator. The translation-invariant operator behaves effectively as a local operator when the wave-function is concentrated around one point in the 3-dimensional space. Note that then the wave-function is not translation invariant which is fine, as long as the operators are translation invariant.

Quantization due to time evolution

We introduce now the procedure of quantization due to unitary time evolution. The idea is that every operator corresponding to an observable is the time evolution of an operator belonging to a common commutative algebra of observables. The time-evolution transforms a sequence of time-ordered operators [18] (which commute algebraically but the time-ordering is non-commutative) into a sequence of (algebraically) non-commuting operators acting on a single slice of time of the wave-function. For a class of time evolutions (of the type of non-relativistic Quantum Mechanics), the canonical commutation relation of position and momentum are reproduced (strictly speaking, it is the Weyl relation that is reproduced, i.e. the exponenciated version of the canonical commutation relation).

We use again the Trotter exponential product approximation [19], verifying for small $\epsilon$ :

\begin{aligned} e^{i\epsilon A}e^{i\epsilon B}= e^{i\epsilon (A+B)-\frac{\epsilon^2}{2}[A,B]+i\mathcal{O}(\epsilon^3)}\end{aligned}

This is a good approximation since it works for unbounded self-adjoint operators $A,B$ .

Let now $\epsilon=\frac{1}{n}$ with $n$ arbitrarily large. Then,

\begin{aligned} e^{i\epsilon A}e^{iB}e^{-i\epsilon A}=(e^{i\epsilon A}e^{i\epsilon B}e^{-i\epsilon A})^n=e^{iB-\epsilon[A,B]+i\mathcal{O}(\epsilon^2)}\end{aligned}

Therefore, for small enough $\epsilon$

\begin{aligned} &e^{i\epsilon p^2}e^{i x}e^{-i\epsilon p^2} =e^{i (x+\epsilon p)}\end{aligned}

Now we need a definition of covariant derivative in time of the position operator $x$ , consistent with the fact that only $F(a)=U(a)e^{i x}U^\dagger (a)$ (where $U(a)$ is the time-evolution) is bounded while $x$ is unbounded. If we would be dealing with a commutative algebra, then the natural definition would be:

\begin{aligned} \lim_{\epsilon \to 0} F^{\frac{1}{\epsilon}}(0)(F^{\frac{1}{\epsilon}}(\epsilon))^{\dagger}\end{aligned}

For a trivial parallel transport, we would get as required:

\begin{aligned} \lim_{\epsilon \to 0} F^{\frac{1}{\epsilon}}(0)(F^{\frac{1}{\epsilon}}(\epsilon))^{\dagger}=\lim_{\epsilon \to 0}e^{i \frac{x(0)-x(-\epsilon)}{\epsilon}}\end{aligned}

But since we are dealing with a non-commutative algebra, we need to use the Trotter exponential product approximation formula, to define the exponential version of the time derivative:

\begin{aligned} &\lim_{\epsilon \to 0}\lim_{n \to \infty}(F^{\frac{1}{n\epsilon}}(0)(F^{\frac{1}{n\epsilon}}(\epsilon))^{\dagger})^n\end{aligned}

And so for the parallel transport $U(\epsilon)=e^{i\epsilon (p^2+V(x))}$ where $V(x)$ is a potential only dependent on the position operator, the exponential version of the time derivative of the position operator $x$ is:

\begin{aligned} &\lim_{\epsilon \to 0}\lim_{n \to \infty}(F^{\frac{1}{n\epsilon}}(0)(F^{\frac{1}{n\epsilon}}(\epsilon))^{\dagger})^n=\\ &\lim_{\epsilon \to 0}\lim_{n \to \infty} (e^{i\frac{1}{n\epsilon} x} U(\epsilon) e^{-i\frac{1}{n \epsilon} x} U^\dagger (\epsilon))^n= e^{i p} \end{aligned}

The result is the exponential of the momentum operator, which verifies the Weyl relations with respect to the exponential of the position operator. With some abuse of language, we can say that for this type of time-evolution (and thus for this type of Hamiltonian $p^2+V(x)$ , which is most common in non-relativistic Quantum Mechanics), the time derivative of the position operator is the momentum operator. Thus the quantization (i.e. the Weyl relations) may appear in a statistical field theory due to a particular non-deterministic time-evolution.

One major advantage of the quantization due to time-evolution is that it applies not only to the variables of position and momentum which have an obvious correspondence in classical statistical mechanics, but it also applies to variables without an obvious correspondence in classical statistical mechanics such as fermions and ghosts (we just apply the time-evolution to the variables) and also in the presence of gauge symmetries as we will see in the remaining of this work.

Relation with the BRST formalism

The results of reference [20], are consistent with our formalism:
1) there is no fundamental reason why BRST-like gauge-fixing should be Lorentz invariant;
2) but it is crucial that the BRST-like gauge-fixing term involves the time derivative of the fields which have an arbitrary time-evolution, otherwise the theory becomes non-local and the perturbative expansion becomes inconsistent (as it happens in the Coulomb gauge, see also [21][22]);
3) for technical reasons that (apparently) are not related with Lorentz covariance, the $R_\xi$ gauges are better suited for perturbation theory than any other gauge (Lorentz covariant or non-covariant).

The result 1) is consistent with our formalism where the fields are representations of the little group of rotations and not of the Lorentz group. In our formalism, we do not need the BRST-like gauge-fixing to define the theory, however to go from a phase-space defined in space-time to a phase-space defined in space only we need to fix the time-evolution of all fields. As we will see in this Section, this is done using a BRST-like gauge fixing in agreement with result 2). Finally, there does not seem to exist an obvious reason for result 3), but in any case our formalism is not incompatible with the result 3).

We are working from the start with a self-consistent statistical field theory.

The BRST charge is useful for a non-commutative algebra, because when we multiply the right and left ideal the result would not be an ideal if the BRST charge would not square to 0. The alternatives would be to use (standard) gauge-fixing, which suffers from nonlocality in general and suffers from the Gribov problem in non-abelian gauge theories; or to work only with a commutative sub-algebra of the algebra of operators satisfying the constraints, which is challenging for a non-deterministic time-evolution.

The algebra of operators is then enlarged from gauge-invariant operators to BRST-invariant (not necessarily gauge-invariant) operators. These BRST-invariant operators are divided into equivalence classes. The BRST cohomology maps (in a bijective way) each equivalence class with a corresponding gauge-invariant operator satisfying the constraints. When performing algebraic manipulations in a gauge-invariant operator, we can convert the gauge-invariant operator into BRST-invariant, then do the manipulations in the space of BRST-invariant operators and then convert the resulting BRST-invariant operator again into a gauge-invariant operator.

Note that there are two possible inner-products: one degenerate where the ghost fields are self-adjoint and another non-degenerate where the ghost fields are not self-adjoint and behave like standard fermioninc creation and annihilation operators [23]. We use the non-degenerate inner-product, so that we are always working within an Hilbert space formalism. We can do this, because the vacuum state in our formalism is not Lorentz invariant if it exists and the existence of such vacuum state is not mandatory for our formalism. This would not possible in the covariant operator formalism [24] where the vaccuum is Lorentz invariant. Since our vacuum breaks Lorentz invariance, then the spin-statistics theorem does not hold and the ghosts are as consistent as a Schrödinger field. Moreover, since the ghosts only appear in intermediate stepts and the ghosts never appear in the definition of expectation values of observables (neither in the operators nor in the wave-function), then the complete system when treated as a free particle respects the spin-statistics theorem.

Crucially, the BRST cohomology merely simplifies the expression defining a gauge-invariant operator into another equivalent expression, it does not affect the gauge-variant wave-function and therefore the Gribov problem does not arise.

There is a well-known subtlety with the BRST cohomology that we will address in chapter /pub/timepiece: the BRST cohomology is itself gauge-invariant and mathematically well-defined, but it is merely a dispensable auxiliary step in a calculation performed in the context of a quantum formalism. If the quantum formalism is mathematically inconsistent, if the formulation of the calculation crucially depends on the BRST-invariant algebra (not our case in chapter /pub/timepiece, but it is the case of the path integral), surprises are possible. In particular, if all the details of the calculation are only known for the BRST-invariant algebra and not before for the gauge-invariant operators (not our case in chapter /pub/timepiece, but it is the case of the quantum BRST formalism), then the gauge-invariance of the BRST cohomology does not imply that the calculation would be the same in all gauges or that the quantum formalism is logically consistent (and in fact it is not due to the Gribov problem). This is discussed in reference [25]:

“Being gauge invariant, the BFV-PI necessarily reduces to an integral over modular space, irrespective of the gauge fixing choice. Nevertheless, which domain and integration measure over modular space are thereby induced are function of the choice of gauge fixing conditions. The BFV-PI is not totally independent of the choice of gauge fixing fermion $\Psi$ .”

In the (our) case of the quantization due to time-evolution (see chapter /pub/timepiece), the quantum formalism is mathematically well-defined and all details of the calculations are known, regardless of whether we apply the BRST cohomology or not. Since we use the BRST cohomology to merely simplify the expression defining a gauge-invariant operator into another equivalent expression, the Gribov problem does not affect us.

Pure SU(3) Yang-Mills theory

In the following, we follow reference [26] where it is possible. The structure constants $f_{a b c}$ are related with the $SU(N)$ generators $T_{a}$ by:

\begin{aligned} =i f_{a b c} T_{c}\\ \mathrm{tr}(T_a T_b)&=\frac{1}{2}\delta_{a b}\end{aligned}

The $SU(N)$ index $\{a,b,c\}$ allows us to cover also the electromagnetism by eliminating the $SU(N)$ index and thus setting the structure constants $f_{a b c}$ to zero. The covariant derivative is given by:

\begin{aligned} D_{j}&=\partial_{j} -i g T_a A_{j a}\\ [D_j,D_k]&=-ig T_a F_{j k a}\end{aligned}

The magnetic components of the gauge field strength tensor are given by:

\begin{aligned} B_{i a}&=\frac{1}{2}\epsilon_{i j k} (\partial_j A_{k a}-\partial_k A_{j a}-g f_{a b c} A_{j b} A_{k c})\\ B_{i}&=i\frac{1}{2 g}\epsilon_{i j k} [D_j, D_k]=B_{i a} T_a\end{aligned}

Where the indices i,j,k correspond to the spatial dimensions only, i.e. from 1 to 3. It verifies the Jacobi relation

\begin{aligned} =i\frac{1}{2 g}\epsilon_{i j k}[D_i,[D_j, D_k]]=0\end{aligned}

The Hilbert space is the tensor product of the symmetric and antisymmetric Fock spaces $\Gamma^s(L^2(\mathbb{R}^{99}\times \mathbb{Z}_2^{31}))\otimes \Gamma^a(L^2(\mathbb{R}^{99}\times\mathbb{Z}_2^{31}))$ [27][28]. 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 $\mathbb{R}^{99}$ degrees of freedom correspond to 3 space coordinates (x), 3x8=24 fields ( $A_{k a}$ ) and its corresponding space-derivatives ( $\partial_{j}A_{k a}$ ). The $\mathbb{Z}_2^{31}$ degrees of freedom correspond to 8 ghosts corresponding to each SU(3) generator ( $\psi_{a}$ ) and its corresponding space-derivatives ( $\partial_{j}\psi_{a}$ ), minus one because we are taking the tensor product of two Fock-spaces which introduces an extra $\mathbb{Z}_2$ degree of freedom.

We define:

\begin{aligned} & a(\vec{x}, ...) = a(\vec{x},\oplus_{a}(\vec{A}_{a},\mu_{a},\oplus_{k}(\vec{A}_{a,k},\mu_{a k})))\\ & \int d^3\vec{x} ... = \int d^3\vec{x}\prod_{a}(d^3\vec{A}_{a}\sum_{\mu_{a}=0}^1\prod_{k}(d^3 \vec{A}_{a,k}\sum_{\mu_{a k}=0}^1)) \\ &\{[a(\vec{x}, ..., \mu),a^\dagger(\vec{y}, ...,\nu)]\}=a(\vec{x}, ...,\mu) a^\dagger(\vec{y}, ...,\nu)+(-1)^{(\mu \mathrm{mod} 2)(\nu \mathrm{mod} 2)}a^\dagger(\vec{y}, ...,\nu) a(\vec{x}, ...,\mu)=\\ &=\delta((\vec{x},...)-(\vec{y},...))\\ &[A_{j a},\pi^{k}_{b}]=A_{j a}\pi^{k}_{b}-\pi^{k}_{b} A_{j a}= i\delta^{k}_{j} \delta_{a b}\\ &[ A_{j a,k},\pi^{m n}_{b}]=A_{j a,k}\pi^{m n}_{b}-\pi^{m n}_{b} A_{j a,k}= i\delta^{n}_{j}\delta^{m}_{k}\delta_{a b}\\ &\{\psi_{a},\psi^\dagger_{b}\}=\psi_{a} \psi^\dagger_{b}(y)+\psi^\dagger_{b}\psi_{a}=\delta_{a b}\\ &\{\psi_{a j},\psi^\dagger_{b k}\}=\psi_{a j} \psi^\dagger_{b k}(y)+\psi^\dagger_{b k}\psi_{a j}=\delta_{a b}\delta_{j k} \end{aligned}

where

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

The Hamiltonian for the Yang-Mills theory in the Weyl gauge is defined as:

\begin{aligned} &H=\int d^3\vec{x} ... a^\dagger(\vec{x}, ...)H(\vec{x}, ...) a(\vec{x}, ...)\\ &H(\vec{x}, ...)=-\frac{1}{2}\pi^i_a\pi^i_a -\frac{1}{2} B_{i a} B_{i a}\\ &\Omega=\int d^3\vec{x} ... a^\dagger(\vec{x}, ...) \Omega(\vec{x},...) a(\vec{x}, ...)\\ &\Omega(\vec{x},..)=\left[ \pi^{k}_{a} \partial_{k} \psi^\dagger_{a}-\pi^{k}_{a} f_{a b c} A_{k b} \psi^\dagger_{c} -i \frac{1}{2}f_{a b c} \psi^\dagger_{a} \psi^\dagger_{b} \psi_{c}\right] \end{aligned}

where $(\phi,\pi)$ is a complex field/conjugate momentum field over timepiece $t$ (correspondingly). $V(|\phi|^2)$ is a functional (a polynomial or other) on $|\phi|^2$ . The BRST charge is $\Omega$ , where $\psi$ is the ghost field, $\lambda$ is the (real) gauge field and $p_\lambda$ its conjugate momentum.

We need to separate the ideal (gauge generator) from the gauge-invariant algebra. That is, not only the gauge-invariant algebra must commute with the ideal, but also the ideal cannot be included in the gauge-invariant algebra. This is guaranteed by (unconstrained) gauge-fixing as it was discussed in Chapter /pub/parametrization. For instance, in the case of the free Electromagnetic field, while the gauge generator commutes with the ideal (because the ideal is the gauge generator itself) it cannot be included in the gauge-invariant algebra; thus, the Electromagnetic field as a whole is not part of the gauge-invariant algebra. And that is a good thing, since then we always work with local field operators, unlike those of the Coulomb gauge.

Since the Weyl gauge is an incomplete gauge-fixing, this Hamiltonian is invariant under a remnant gauge symmetry generated by $\theta_a (D_i \pi^i)_{a}$ , where the function $\theta$ is constant in time (the remnant gauge transformations are constant in time). We can now apply the BRST cohomology for the remnant gauge symmetry (as discussed in reference [29]) and choose the Lorentz gauge.

The first step is to introduce a new field $\Phi$ and a new constraint $\Pi=0$ where $\Pi$ is the conjugate field of $\Phi$ . Such a field is completely spurious because it allows a complete and unconstrained gauge-fixing, thus we can remove it from the gauge-invariant operators (as expected, otherwise we would have a different theory from the one we started before introducing the new field).

Quantization due to time-evolution: Yang-Mills and Classical Statistical Field Theory