Learn Gauge Fixing in Mathematical Quantum Field Theory

The following is one chapter in a series of Mathematical Quantum Field Theory.

The previous chapter is 11. Reduced phase space.

The next chapter is 13. Quantization.

12. Gauge fixing

While in the previous chapter we had constructed the reduced phase space of a Lagrangian field theory, embodied by the local BV-BRST complex (example 11.21), as the homotopy quotient by the infinitesimal gauge symmetries of the homotopy intersection with the shell, this in general still does not yield a covariant phase space of on-shell field histories (prop. 8.6), since Cauchy surfaces for the equations of motion may still not exist (def. 8.1).

However, with the homological resolution constituted by the BV-BRST complex in hand, we now have the freedom to adjust the field-content of the theory without changing its would-be reduced phase space, namely without changing its BV-BRST cohomology. In particular, we may adjoin further “auxiliary fields” in various degrees, as long as they contribute only a contractible cochain complex to the BV-BRST complex. If such a quasi-isomorphism of BV-BRST complexes brings the Lagrangian field theory into a form such that the equations of motion of the combined fields, ghost fields, and potential further auxiliary fields are Green hyperbolic differential equations after all, and thus admit a covariant phase space, then this is called a gauge fixing (def. 12.1 below), since it is the infinitesimal gauge symmetries which obstruct the existence of Cauchy surfaces (by prop. 10.1 and remark 10.6).

The archetypical example is the Gaussian-averaged Lorenz gauge fixing of the electromagnetic field (example 12.8 below) which reveals that the gauge-invariant content of electromagnetic waves is only in their transversal wave polarization (prop. 12.13 below).

The tool of gauge fixing via quasi-isomorphisms of BV-BRST complexes finally brings us in position to consider, in the following chapters, the quantization also of gauge theories: We use gauge fixing quasi-isomorphisms to bring the BV-BRST complexes of the given Lagrangian field theories into a form that admits degreewise quantization of a graded covariant phase space of fields, ghost fields and possibly further auxiliary fields, compatible with the gauge-fixed BV-BRST differential:

$$
\array{
\underline{\mathbf{\text{pre-quantum geometry}}}
&&
\underline{\mathbf{\text{higher pre-quantum geometry}}}
\\
\,
\\
\left\{
\array{
\text{Lagrangian field theory with}
\\
\text{infinitesimal gauge transformations}
}
\right\}
&\overset{ \text{homotopy quotient by} \atop \text{gauge transformations} }{\longrightarrow}&
\left\{
\array{
\text{dg-Lagrangian field theory with}
\\
\text{quotiented by gauge transformations}
\\
\text{embodied by BRST complex }
}
\right\}
\\
&& \Big\downarrow{}^{\substack{ \text{pass to} \atop \text{derived critical locus} }}
\\
&&
\left\{
\array{
\text{dg-reduced phase space}
\\
\text{ embodied by BV-BRST complex }
}
\right\}
\\
&& {}^{\substack{\simeq}}\Big\downarrow{}^{\substack{\text{fix gauge} }}
\\
\left\{
\array{
\text{ decategorified }
\\
\text{ covariant }
\\
\text{ reduced phase space }
}
\right\}
&\underset{\text{pass to cohomology}}{\longleftarrow}&
\left\{
\array{
\text{ dg-covariant}
\\
\text{reduced phase space }
}
\right\}
\\
&& \Big\downarrow{}^{\substack{
\array{
\text{ quantize }
\\
\text{degreewise}
}
}}
\\
\left\{
\array{
\text{gauge invariant}
\\
\text{quantum observables}
}
\right\}
&\underset{\text{pass to cohomology}}{\longleftarrow}&
\left\{
\array{
\text{quantum}
\\
\text{BV-BRST complex}
}
\right\}
}
$$

Here:

term	meaning
“phase space”	derived critical locus of Lagrangian equipped with Poisson bracket
“reduced”	gauge transformations have been homotopy-quotiented out
“covariant”	Cauchy surfaces exist degreewise

We now discuss these topics:

1. Quasi-isomorphisms between local BV-BRST complexes
   1. gauge fixing chain maps;
   2. adjoining contractible complexes of auxiliary fields
2. Example: gauge fixed electromagnetic field

quasi-isomorphisms between local BV-BRST complexes

Recall (prop. 11.10) that given a local BV-BRST complex (example 11.21) with BV-BRST differential $s$, then the space of local observables which are on-shell and gauge-invariant is the cochain cohomology of $s$ in degree zero:

$$
H^0(s \vert d)
\;=\;
\left\{
\array{
\text{gauge invariant on-shell}
\\
\text{local observables}
}
\right\}
$$

The key point of having resolved (in chapter Reduced phase space) the naive quotient by infinitesimal gauge symmetries of the naive intersection with the shell by the L-infinity algebroid whose Chevalley-Eilenberg algebra is called the local BV-BRST complex, is that placing the reduced phase space into the context of homotopy theory/homological algebra this way provides the freedom of changing the choice of field bundle and of Lagrangian density without actually changing the Lagrangian field theory up to equivalence, namely without changing the cochain cohomology of the BV-BRST complex.

A homomorphism of differential graded-commutative superalgebras (such as BV-BRST complexes) that induces an isomorphism in cochain cohomology is called a quasi-isomorphism. We now discuss two classes of quasi-isomorphisms between BV-BRST complexes:

1. gauge fixing (def. 12.1 below)
2. adjoining auxiliary fields (def. 12.4 below).

gauge fixing chain maps

Let

$$
CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)_{\delta_{EL} L \simeq 0} \right)
\;=\;
\left(
\Omega^{0,0}_\Sigma\left(
T^\ast_{\Sigma,inf}[-1]\left(E \times_\Sigma \mathcal{G}[1]\right) \times_\Sigma T \Sigma[1]
\right)
\;,\;
d_{CE}
=
\underset{s}{
\underbrace{
\left\{
-\mathbf{L} + \mathbf{L}_{BRST}
\,,\,
–
\right\}
}
}
\;+\;
d
\right)
$$

be a local BV-BRST complex of a Lagrangian field theory $(E,\mathbf{L})$ (example 11.21).

Then for

$$
\mathbf{L}_{gf}
\;\in\;
\Omega^{p+1,0}
\left(
T^\ast_{\Sigma,inf}\left(E \times_\Sigma \mathcal{G}\right) \times_\Sigma T \Sigma[1]
\right)
$$

a Lagrangian density (def. 5.1) on the graded field bundle

$$
\mathbf{L}_{gf} \;=\; L_{gf} \ dvol_\Sigma
$$

of degree

$$
deg(L) = (-1, even)
$$

then the exponential of forming the local antibracket (def. 11.15) with $\mathbf{L}_{gf}$

$$
\array{
\Omega^{p+1,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma \mathcal{G}[1]\right) \right)
&
\overset{
e^{\left\{ \mathbf{L}_{gf} \,,\, -\right\}}(-)
}{\longrightarrow}
&
\Omega^{p+1,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma \mathcal{G}[1]\right) \right)
\\
\mathbf{K}
&\mapsto&
\left\{ \mathbf{L}_{gf} , \mathbf{K} \right\}
+
\tfrac{1}{2}
\left\{ \mathbf{L}_{gf} \,,\, \left\{ \mathbf{L}_{gf} \,,\, \mathbf{K} \right\} \right\}
+
\tfrac{1}{6}
\left\{ \mathbf{L}_{gf} \,,\,\left\{ \mathbf{L}_{gf} \,,\, \left\{ \mathbf{L}_{gf} \,,\,\mathbf{K} \right\} \right\} \right\}
+
\cdots
}
$$

is an endomorphism of the local antibracket (def. 11.15) in that

$$
e^{
\left\{ \mathbf{\psi} \,,\, – \right\}
}
\left(
\left\{ \mathbf{A} \,,\, \mathbf{B} \right\}
\right)
\;=\;
\left\{
e^{
\left\{ \mathbf{\psi} \,,\, – \right\}
}
\left(\mathbf{A}\right)
\,,\,
e^{
\left\{ \psi \,,\, – \right\}
}
\left(\mathbf{B}\right)
\right\}
$$

and in fact an automorphism, with inverse morphism given by

$$
\left(e^{\left\{ \psi \,,\, -\right\}}(-)\right)^{-1}
\;=\;
e^{\left\{ -\psi \,,\, -\right\}}(-)
\,.
$$

We may think of this as the Hamiltonian flow of $\mathbf{L}_{gf}$ under the local antibracket.

In particular when applied to the BV-Lagrangian density

$$
s_{gf}
\;:=\;
\left\{
e^{\left\{ \mathbf{L}_{gf},-\right\}}\left(- \mathbf{L} + \mathbf{L}_{BRST}\right)
\,,\,
–
\right\}
$$

this yields another differential

$$
\left( s_{gf}\right)^2
\;=\;
0
$$

and hence another differential graded-commutative superalgebra (def. 10.17)

$$
CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)^{gf}_{\delta_{EL} L \simeq 0} \right)
\;=\;
\left(
\Omega^{0,0}_\Sigma\left(
T^\ast_{\Sigma,inf}[-1]\left(E \times_\Sigma \mathcal{G}[1]\right) \times_\Sigma T \Sigma[1]
\right)
\;,\;
d_{CE}
=
\underset{s_{gf}}{
\underbrace{
\left\{
e^{\left\{ \mathbf{L}_{gf}, – \right\}}\left( – \mathbf{L} + \mathbf{L}_{BRST} \right)
\,,\,
–
\right\}
}
}
\;+\;
d
\right)
$$

Finally, $e^{\left\{\mathbf{L}_{gf},-\right\}}$ constitutes a chain map from the local BV-BRST complex to this deformed version, in fact a homomorphism of differential graded-commutative superalgebras, in that

$$
s_{gf} \circ e^{ \left\{ \mathbf{L}_{gf}\,,\, – \right\} }
\;=\;
e^{ \left\{ \mathbf{L}_{gf}\,,\, – \right\} } \circ s
\,.
$$

Proof. By prop. 11.16
the local antibracket $\left\{ -,-\right\}$ is a graded derivation in its second argument, of degree one more than the degree of its first argument (187). Hence for the first argument of degree -1 this implies that $e^{\{\mathbf{L}_{gf}, – \}}$ is an automorphism of the local antibracket. Moreover, it is clear from the definition that $\left\{ \mathbf{L}_{gf},-\right\}$ is a derivation with respect to the pointwise product of smooth functions, so that $e^{\{\mathbf{L}_{gf},-\}}$ is also a homomorphism of graded algebras.

Since $e^{\{\mathbf{L}_{gf}, -\}}$ is an automorphism of the local antibracket, and since $s$ and $s_{gf}$ are themselves given by applying the local antibracket in the second argument, this implies that $e^{\{\mathbf{L}_{gf},-\}}$ respects the differentials:

$$
\array{
\mathbf{A}
&\overset{e^{\{\mathbf{L}_{gf},-\}}}{\longrightarrow}&
e^{\{\mathbf{L}_{gf},-\}}\left( \mathbf{A} \right)
\\
{}^{\substack{s}}\downarrow && \downarrow^{\substack{s_{gf}}}
\\
\left\{ \left(-\mathbf{L} + \mathbf{L}_{BRST}\right)\,,\, \mathbf{A}\right\}
&\underset{ e^{\{\mathbf{L}_{gf}\,,\,-\}} }{\longrightarrow}&
\left\{ e^{\{\mathbf{L}_{gf},-\}}\left(-\mathbf{L} + \mathbf{L}_{BRST}\right)
\,,\,
e^{\{\mathbf{L}_{gf},-\}}(\mathbf{A})
\right\}
}
$$

Definition 12.1. (gauge fixing Lagrangian density)

Let

$$
CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)_{\delta_{EL} L \simeq 0} \right)
\;=\;
\left(
\Omega^{0,0}_\Sigma\left(
T^\ast_{\Sigma,inf}[-1]\left(E \times_\Sigma \mathcal{G}[1]\right) \times_\Sigma T \Sigma[1]
\right)
\;,\;
d_{CE}
=
\underset{s}{
\underbrace{
\left\{
-\mathbf{L} + \mathbf{L}_{BRST}
\,,\,
–
\right\}
}
}
\;+\;
d
\right)
$$

be a local BV-BRST complex of a Lagrangian field theory $(E,\mathbf{L})$ (example 11.21) and let

$$
\mathbf{L}_{gf}
\;\in\;
\Omega^{p+1,0}
\left(
T^\ast_{\Sigma,inf}\left(E \times_\Sigma \mathcal{G}\right) \times_\Sigma T \Sigma[1]
\right)
$$

be a Lagrangian density (def. 5.1) on the graded field bundle such that

$$
deg(L_{gf}) = -1
\,.
$$

If the quasi-isomorphism of BV-BRST complexes given by the local anti-Hamiltonian flow $\mathbf{L}_{gf}$ via prop. link

$$
e^{\left\{ \mathbf{L}_{gf},-\right\}}
\;\colon\;
CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)^{gf}_{\delta_{EL} L \simeq 0} \right)
\overset{\phantom{A}\simeq_{qi}\phantom{A}}{\longrightarrow}
CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)^{gf}_{\delta_{EL} L \simeq 0} \right)
$$

is such that for the transformed graded Lagrangian field theory

$$
\label{GaugeFixedLagrangianDensity}
-\underset{deg_{af} = 0}{\underbrace{\mathbf{L}’ }} + \mathbf{L}’_{BRST}
\;:=\;
e^{\{\mathbf{L}_{gf},-\}}(-\mathbf{L} + \mathbf{L}_{BRST})
$$ (195)

(with Lagrangian density $\mathbf{L}’$ the part independent of antifields) the Euler-Lagrange equations of motion (def. 5.24) admit Cauchy surfaces (def. 8.1), then we call $\mathbf{L}_{gf}$ a gauge fixing Lagrangian density for the original Lagrangian field theory, and $\mathbf{L}’$ the corresponding gauge fixed form of the original Lagrangian density $\mathbf{L}$.

Remark 12.2. (warning on terminology)

What we call a gauge fixing Lagrangian density $\mathbf{L}_{gf}$ in def. 12.1 is traditionally called a gauge fixing fermion and denoted by “$\psi$” (Henneaux 90, section 8.3, 8.4).

Here “fermion” is meant as a reference to the fact that the cohomological degree $deg(L_{gf}) = -1$, which is reminiscent of the odd super-degree of fermion fields such as the Dirac field (example 3.50); see at signs in supergeometry the section The super odd sign rule.

Example 12.3. (gauge fixing via anti-Lagrangian subspaces)

Let $\mathbf{L}_{gf}$ be a gauge fixing Lagrangian density as in def. 12.1
such that

1. its local antibracket-square vanishes$$
   \left\{ \mathbf{L}_{gf},\, \left\{ \mathbf{L}_{gf}, \, -\right\} \right\} = 0
   $$hence its anti-Hamiltonian flow has at most a linear component in its argument $\mathbf{A}$:$$
   e^{\left\{ \mathbf{L}_{gf} \,,\, \mathbf{A} \right\}}
   \;=\;
   \mathbf{A} + \left\{ \mathbf{L}_{gf} \,,\, \mathbf{A} \right\}
   $$
2. it is independent of the antifields$$
   deg_{af}\left( L_{gf} \right) \;=\; 0
   \,.
   $$

Then with

• $(\phi^A)$ collectively denoting all the field coordinates(including the actual fields $\phi^a$, the ghost fields $c^\alpha$ as well as possibly further auxiliary fields)
• $(\phi^\ddagger_A)$ collectively denoting all the antifield coordinates(includion the antifields $\phi^\ddagger_a$ of the actual fields, the antifields $c^\ddagger_\alpha$ of the ghost fields as well as those of possibly further auxiliary fields )

we have

$$
\begin{aligned}
(\phi’)^A
& :=
e^{\left\{ \mathbf{L}_{gf}\,,\, – \right\}}(\phi^A)
\\
& = \phi^A
\\
\phantom{A}
\\
(\phi’)^\ddagger_A
& :=
e^{\left\{ \mathbf{L}_{gf}\,,\, – \right\}}
\left( \phi^\ddagger_A \right)
\\
& =
\phi^\ddagger_A – \frac{\overset{\leftarrow}{\delta}_{EL} \mathbf{L}_{gf}}{\delta \phi^a}
\end{aligned}
$$

(and similarly for the higher jets); and the corresponding transformed Lagrangian density (195) may be written as

$$
\begin{aligned}
-\mathbf{L}’ + \mathbf{L}’_{BRST}
& :=
e^{\left\{ \mathbf{L}_{gf}\,,\, – \right\}}\left( -\mathbf{L} + \mathbf{L}_{BRST} \right)
\\
& =
\left(
-\mathbf{L}
+
\mathbf{L}_{BRST}
\right)
\left( \phi’, (\phi’)^\ddagger \right)
\end{aligned}
\,,
$$

where the notation on the right denotes that $\phi’$ is substituted for $\phi$ and $\phi’_\ddagger$ for $\phi_\ddagger$.

This means that the defining condition that $\mathbf{L}’$ be the antifield-independent summand (195), which we may write as

$$
\mathbf{L}’
:=
\left(
-\mathbf{L}
+
\mathbf{L}_{BRST}
\right)
\left( \phi'(\phi), \phi_\ddagger = 0 \right)
$$

translates into

$$
\mathbf{L}’
:=
\left(
-\mathbf{L}
+
\mathbf{L}_{BRST}
\right)
\left( \phi’, (\phi’)^\ddagger_A = -\frac{\overset{\leftarrow}{\delta}_{EL} L_{gf}}{\delta \phi^A} \right)
\,.
$$

In this form BV-gauge fixing is considered traditionally (e.g. Hennaux 90, section 8.3, page 83, equation (76b) and item (iii)).

adjoining contractible cochain complexes of auxiliary fields

Typically a Lagrangian field theory $(E,\mathbf{L})$ for a given choice of field bundle, even after finding appropriate gauge parameter bundles $\mathcal{G}$, does not yet admit a gauge fixing Lagrangian density (def. 12.1). But if the gauge parameter bundle has been chosen suitably, then the remaining obstruction vanishes “up to homotopy” in that a gauge fixing Lagrangian density does exist if only one adjoins sufficiently many auxiliary fields forming a contractible complex, hence without changing the cochain cohomology of the BV-BRST complex:

Definition 12.4. (auxiliary fields and antighost fields)

Over Minkowski spacetime $\Sigma$, let

$$
A \overset{aux}{\longrightarrow} \Sigma
$$

be any graded vector bundle (remark 10.23), to be regarded as a field bundle (def. 3.1) for auxiliary fields. If this is a trivial vector bundle (example 3.4) we denote its field coordinates by $(b^i)$. On the corresponding graded bundle with degrees shifted down by one

$$
A[-1] \overset{aux[-1]}{\longrightarrow} \Sigma
$$

we write $(\overline{c}^i)$ for the induced field coordinates.

Accordingly, the shifted infinitesimal vertical cotangent bundle (def. 11.14) of the fiber product of these bundles

$$
T^\ast_{\Sigma,inf}[-1]\left( A \times_\Sigma A^\ast[-1] \right)
$$

has the following coordinates:

$$
\array{
\text{name:}
&
\array{
\text{antifield of}
\\
\text{antighost field}
}
&
\array{
\text{antifield of}
\\
\text{auxiliary field}
}
&
\text{antighost field}
&
\text{auxiliary field}
\\
\text{symbol:} & \overline{c}^\ddagger_i & b^\ddagger_i & \overline c^i & b^i
\\
deg = & -(deg(b^i)-1)-1 & -deg(b^i)-1 & deg(b^i)-1 & deg(b^i)
\\
& = -deg(b^i)
}
$$

On this fiber bundle consider the Lagrangian density (def. 5.1)

$$
\label{LagrangianDensityForAuxiliaryFields}
\mathbf{L}_{aux}
\;\in\;
\Omega^{p+1,0}_\Sigma( T^\ast_{\Sigma,inf}[-1]\left( A \times_\Sigma A[-1] \right) )
$$ (196)

given in local coordinates by

$$
\mathbf{L}_{aux}
\;:=\;
\overline{c}^\ddagger_i b^i \, dvol_\Sigma
\,.
$$

This is such that the local antibracket (def. 11.15) with this Lagrangian acts on generators as follows:

$$
\label{BVDifferentialOnauxiliaryFields}
\array{
&& \left\{ \mathbf{L}_{aux},- \right\}
\\
\text{auxiliary field}
&
b^i &\mapsto& 0
\\
\text{antighost field}
&
\overline{c}^i &\mapsto& b^i
\\
\text{antifield of auxiliary field}
&
b^\ddagger_i &\mapsto& – \overline{c}^\ddagger_i
\\
\text{antifield of antighost field}
&
\overline{c}^\ddagger_i &\mapsto& 0
}
$$ (197)

Remark 12.5. (warning on terminology)

Beware that when adjoining antifields as in def. 12.4 to a Lagrangian field theory which also has ghost fields $(c^\alpha)$ adjoined (example 10.28) then there is no relation, a priori, between

• the “antighost field” $\overline{c}^i$

and

• the “antifield of the ghost field” $c^\ddagger_\alpha$

In particular, there is also the

• “antifield of the antighost field” $\overline{c}^\ddagger_i$

The terminology and notation are maybe unfortunate but entirely established.

The following is immediate from def. 12.4, in fact, this is the purpose of the definition:

Proposition 12.6. (adjoining auxiliary fields is quasi-isomorphism of BV-BRST complexes)

Let

$$
CE\left( E/(\mathcal{G} \times_\Sigma T\Sigma)_{\delta_{EL} L \simeq 0} \right)
\;=\;
\left(
\Omega^{0,0}_\Sigma\left(
T^\ast_{\Sigma,inf}[-1]\left(E \times_\Sigma \mathcal{G}[1]\right) \times_\Sigma T \Sigma[1]
\right)
\;,\;
d_{CE}
=
\underset{s}{
\underbrace{
\left\{
-\mathbf{L} + \mathbf{L}_{BRST}
\,,\,
–
\right\}
}
}
\;+\;
d
\right)
$$

be a local BV-BRST complex of a Lagrangian field theory $(E,\mathbf{L})$ (example 11.21).

Let moreover $A \overset{aux}{\longrightarrow} \Sigma$ be any auxiliary field bundle (def. 12.4). Then on the fiber product of the original field bundle $E$ and the shifted gauge parameter bundle $\mathcal{G}[1]$ with the auxiliary field bundle $A$ the sum of the original BV-Lagrangian density $-\mathbf{L} + \mathbf{L}_{BRST}$ with the auxiliary Lagrangian density $\mathbf{L}_{aux}$ (196) induce a new differential graded-commutative superalgebra:

$$
\begin{aligned}
& CE\left( E/(\mathcal{G} \times_\Sigma (A \times_\Sigma A[-1]) \times_\Sigma T \Sigma)^{aux}_{\delta_{EL} L \simeq 0} \right)
\\
& :=\;
\left(
\Omega^{0,0}_\Sigma\left(
T^\ast_{\Sigma,inf}[-1]
\left(
E \times_\Sigma \mathcal{G}[1] \times_\Sigma \left( A \times_\Sigma A[-1]\right) \right) \times_\Sigma T \Sigma[1]
\right)
\;,\;
d_{CE}
=
\underset{s}{
\underbrace{
\left\{
\left( – L + L_{BRST} + \mathbf{L}_{aux} \right) dvol_\Sigma
\,,\,
–
\right\}
}
}
\;+\;
d
\right)
\end{aligned}
$$

with generators

$$
\array{
\text{fields} & \phi^a & E & \phi^\ddagger_a & \text{antifields}
\\
\\
\text{ghost fields} & c^\alpha & \mathcal{G}[1] & c^\ddagger_\alpha & \array{ \text{antifields of} \\ \text{ghost fields} }
\\
\\
\text{ auxiliary fields } & b^i & A & b^\ddagger_i & \array{ \text{antifields of} \\ \text{auxiliary fields} }
\\
\\
\text{ antighost fields } & \overline{c}^i & A[-1] & \overline{c}^{\ddagger}_i & \array{ \text{antifields of} \\ \text{antighost fields} }
}
$$

Moreover, the differential graded-commutative superalgebra of auxiliary fields and their antighost fields is a contractible chain complex

$$
\left(
\Omega^{0,0}_\Sigma( A \times_{\Sigma} A[-1] )
\,,\,
d_{CE} = \left\{ \overline{c}^\ddagger_i b^i \, dvol_\Sigma \,,\, – \right\}
\right)
\overset{\simeq_{qi}}{\longrightarrow}
0
$$

and thus the canonical inclusion map

$$
CE\left( E/(\mathcal{G} \times_\Sigma \times_\Sigma T \Sigma)_{\delta_{EL} L \simeq 0} \right)
\overset{\phantom{AA} \simeq_{qi} \phantom{aa}}{\rightarrow}
CE\left( E/(\mathcal{G} \times_\Sigma (A \times_\Sigma A[-1]) \times_\Sigma T \Sigma)^{aux}_{\delta_{EL} L \simeq 0} \right)
$$

(of the original BV-BRST complex into its tensor product with that for the auxiliary fields and their antighost fields) is a quasi-isomorphism.

Proof. From (197) we read off that

1. the map $s_{aux} := \left\{ \mathbf{L}_{aux},- \right\}$ is a differential (squares to zero), and the auxiliary Lagrangian density satisfies its classical master equation (remark 11.17) strictly$$
   \{\mathbf{L}_{aux}, \mathbf{L}_{aux}\} = 0
   $$
2. the cochain cohomology of this differential is trivial:$$
   H^\bullet( s_{aux} )\;=\;0
   $$
3. The local antibracket of the BV-Lagrangian density with the auxiliary Lagrangian density vanishes:$$
   \left\{
   – \mathbf{L} + \mathbf{L}_{BRST}
   \,,\,
   \mathbf{L}_{aux}
   \right\}
   \;=\;
   0
   $$

Together this implies that the sum $-\mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}$ satisfies the classical master equation (remark 11.17)

$$
\left\{
\left(
– \mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}
\right)
\,,\,
\left(
– \mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}
\right)
\right\}
\;=\;
0
$$

and hence that

$$
s + s_{aux}
\;:=\;
\left\{
– \mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}
\,,\,
–
\right\}
$$

is indeed a differential; such that its cochain cohomology is identified with that of $s = \left\{-\mathbf{L} + \mathbf{L}_{BRST},-\right\}$ under the canonical inclusion map.

Remark 12.7. (gauge fixed BV-BRST field bundle)

In conclusion, we have that, given

1. $(E,\mathbf{L})$ a Lagrangian field theory (def. 5.1), with field bundle $E$ (def. 3.1);
2. $\mathcal{G}$ a choice of gauge parameters (def. 10.5),hence$\mathcal{G}[1]$ a choice of ghost fields (example 10.28);
3. $A$ a choice of auxiliary fields (def. 12.4),hence$A[-1]$ a choice of antighost fields (def. 12.4)
4. $T^\ast_{\Sigma,inf}[-1](\cdots)$ the corresponding antifields (def. 11.14)
5. a gauge fixing Lagrangian density $\mathbf{L}_{gf}$ (def. 12.1)

then the result is a new Lagrangian field theory

$$
\left( E_{\text{BV-BRST}}, \mathbf{L}’ \right)
$$

now with graded field bundle (remark 10.23) the fiber product

$$
E_{\text{BV-BRST}}
\;:=\;
\underset{
\array{ \text{anti-} \\ \text{fields} }
}{
\underbrace{
T^\ast_{\Sigma,inf}
}
}
\left(
\underset{\text{fields}}{\underbrace{E}}
\times_\Sigma
\underset{ \array{ \text{ghost} \\ \text{fields} }}{\underbrace{\mathcal{G}[1]}}
\times_\Sigma
\underset{\array{ \text{auxiliary} \\ \text{fields} }}{\underbrace{A}}
\times_{\Sigma}
\underset{ \array{ \text{antighost} \\ \text{fields} } }{\underbrace{A[-1]}}
\right)
$$

and with Lagrangian density $\mathbf{L’}$ independent of the antifields, but complemented by an auxiliary Lagrangian density $\mathbf{L}’_{BRST}$.

The key point is that $\mathbf{L}’$ admits a covariant phase space (while $\mathbf{L}$ may not), while in BV-BRST cohomology both theories still have the same gauge-invariant on-shell observables.

Gauge fixed electromagnetic field

As an example of the general theory of BV-BRST gauge fixing above we now discuss the gauge fixing of the electromagnetic field.

Example 12.8. (Gaussian-averaged Lorenz gauge fixing of vacuum electromagnetism)

Consider the local BV-BRST complex for the free electromagnetic field on Minkowski spacetime from example 11.23:

The field bundle is $E := T^\ast \Sigma$ and the gauge parameter bundle is $\mathcal{G} := \Sigma \times \mathbb{R}$. The 0-jet field coordinates are

$$
\array{
& c^\ddagger & (a^\ddagger)^\mu & a_\mu & c
\\
deg =
&
-2 & -1 & 0 & 1
}
$$

the Lagrangian density is (43)

$$
\label{VacuumEMLagrangianDensityRecalledForNLFields}
\mathbf{L}_{EM} := \tfrac{1}{2} f_{\mu \nu} f^{\mu \nu}
$$ (198)

and the BV-BRST differential acts as:

$$
\array{
& &\array{ \text{BV-BRST} \\ \text{differential} }&
\\
\array{
\text{ electromagnetic field }
\\
\text{ (“vector potential”) }
}
& a_\mu &\mapsto& c_{,\mu} & \text{gauge transformation}
\\
\phantom{A}
\\
\text{ ghost field }
& c &\mapsto& 0 & \text{abelian Lie algebra}
\\
\phantom{A}
\\
\array{
\text{antifield of}
\\
\text{electromagnetic field}
}
& (a^\ddagger)^\mu &\mapsto& f^{\nu \mu}_{,\nu} & \text{equations of motion}
\\
\phantom{A}
\\
\array{
\text{antifield of}
\\
\text{ghostfield}
}
& c^\ddagger &\mapsto& (a^\ddagger)^\mu_{,\mu} & \text{Noether identity}
\\
\phantom{A}
\\
\text{Nakanishi-Lautrup field}
&
b &\mapsto& 0 & \text{vanishing of auxiliary fields…}
\\
\phantom{A}
\\
\text{antighost field}
&
\overline{c} &\mapsto& b & \text{… in cohomology}
\\
\phantom{A}
\\
\array{
\text{antifield of}
\\
\text{ Nakanishi-Lautrup field }
}
&
b^\ddagger &\mapsto& -\overline{c}^\ddagger
&
\text{vanishing of antifields of auxiliary fields…}
\\
\phantom{A}
\\
\array{
\text{antifield of}
\\
\text{antighost field}
}
&
\overline{c}^\ddagger &\mapsto& 0
&
\text{… in cohomology}
}
$$

Introduce a trivial real line bundle for auxiliary fields $b$ in degree 0 and their antighost fields $\overline{c}$ (def. 12.4) in degree -1:

$$
\array{
&
\Sigma \times \langle \overline{c}\rangle
&\overset{ \overline{c} \mapsto b}{\longrightarrow}&
\Sigma \times\langle b\rangle
\\
deg = & -1 && 0
}
\,.
$$

In the present context, the auxiliary field $b$ is called the abelian Nakanishi-Lautrup field.

The corresponding BV-BRST complex with auxiliary fields adjoined, which, by prop. 12.6, is quasi-isomorphic to the original one above, has coordinate generators

$$
\array{
& c^\ddagger & (a^\ddagger)^\mu & a_\mu & c
\\
& & \overline{c} & b
\\
& & b^{\ddagger} & \overline{c}^\ddagger
\\
deg =
&
-2 & -1 & 0 & 1
}
\,.
$$

and BV-BRST differential given by the local antibracket (def. 11.15) with $-\mathbf{L}_{EM} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}$:

$$
s
\;=\;
\left\{
\left(
– \underset{ = L_{EM}}{\underbrace{\tfrac{1}{2}f_{\mu \nu} f^{\mu \nu}}}
+
\underset{ = L_{BRST} }{\underbrace{ c_{,\mu} (a^\ddagger)^\mu }}
+
\underset{ = L_{aux} }{\underbrace{ b \overline{c}^{\ddagger} }}
\right)
dvol_\Sigma
\,,\,
(-)
\right\}
$$

We say that the gauge fixing Lagrangian (def. 12.1) for Gaussian-averaged Lorenz gauge_ for the electromagnetic field

$$
\mathbf{L}_{gf}
\;\in\;
\Omega^{p+1}_\Sigma\left(
E \times_\Sigma \mathcal{G}[1] \times_\Sigma A \times_\Sigma A[-1]
\right)
\,.
$$

is given by (Henneaux 90 (103a))

$$
\label{GaugeFixingLagrangianForGaussianAveragedLorentzGauge}
\mathbf{L}_{gf}
\;:=\;
\underset{deg = -1}{
\underbrace{
\phantom{A}\overline{c}\phantom{A}
}}
\underset{deg = 0}{\underbrace{( b – a^{\mu}_{,\mu} )}} \, dvol_\Sigma
\,.
$$ (199)

We check that this really is a gauge fixing Lagrangian density according to def. 12.1:

From (198) and (199) we find the local antibrackets (def. 11.15) with this gauge fixing Lagrangian density to be

$$
\begin{aligned}
\left\{\mathbf{L}_{gf}\,,\,\left( – \mathbf{L}_{EM} + \mathbf{L}_{BRST} + \mathbf{L}_{aux} \right) \right\}
& =
\left\{
\overline{c}\left( b – a^\mu_{,\mu}\right) \, dvol_\Sigma
\,,\,
\left(
-\tfrac{1}{2}f_{\mu \nu}f^{\mu \nu}
+
c_{,\mu} (a^\ddagger)^\mu
+
b \overline{c}^\ddagger
\right)
dvol_\Sigma
\right\}
\\
& =
\left\{
\overline{c}\left( b – a^\mu_{,\mu}\right) \, dvol_\Sigma
\,,\,
b \overline{c}^{\ddagger} \, dvol_\Sigma
\right\}
+
\left\{
\overline{c}\left( b – a^\mu_{,\mu}\right) \, dvol_\Sigma
\,,\,
c_{,\mu} (a^{\ddagger})^\mu \, dvol_\Sigma
\right\}
\\
& =
–
\left(
b ( b – a^{\mu}_{,\mu} ) + \overline{c}_{,\mu} c^{,\mu}
\right)
\, dvol_\Sigma
\\
\phantom{A}
\\
\{ \mathbf{L}_{gf}, \{ \mathbf{L}_{gf} , (-\mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux} )\}\}
& = 0
\end{aligned}
$$

(So we are in the traditional situation of example 12.3.)

Therefore the corresponding gauge fixed Lagrangian density (195) is (see also Henneaux 90 (103b)):

$$
\label{GaussianAveragedLorentzianGaugeFixOfElectromagneticFieldOnMinkowskiSpacetime}
\begin{aligned}
-\mathbf{L}’ + \mathbf{L}’_{BRST}
& :=
e^{\left\{ \mathbf{L}_{gf} ,-\right\}}\left( -\mathbf{L}_{EM} + \mathbf{L}_{BRST} + \mathbf{L}_{aux} \right)
\\
& =
–
\underset{
= \mathbf{L}’
}{
\underbrace{
\left(
\underset{ = L_{EM} }{
\underbrace{
\tfrac{1}{2} f_{\mu \nu} f^{\mu \nu}
}
}
+
\underset{ = -\left\{ L_{gf}, L_{BRST} + L_{aux} \right\} }{
\underbrace{
b ( b – a^{\mu}_{,\mu} )
+
\overline{c}_{,\mu} c^{,\mu}
}
}
\right) dvol_\Sigma
}
}
\;+\;
\underset{ = \mathbf{L}’_{BRST} }{
\underbrace{
\left(
\underset{ = L_{BRST} }{
\underbrace{
c_{,\mu} (a^\ddagger)^\mu
}
}
+
\underset{ = L_{aux} }{
\underbrace{
b \overline{c}^\ddagger
}
}
\right)
dvol_\Sigma
}
}
\end{aligned}
\,.
$$ (200)

The Euler-Lagrange equation of motion (def. 5.24) induced by the gauge fixed Lagrangian density $\mathbf{L}’$ at antifield degree 0 are (using (62)):

$$
\label{LorenzGaugeFixedEOMForVacuumElectromagnetism}
\delta_{EL} \mathbf{L}’ \;=\; 0
\phantom{AAA}
\Leftrightarrow
\phantom{AAA}
\left\{
\begin{aligned}
-\frac{d}{d x^\mu} f^{\mu \nu} & = b^{,\nu}
\\
b & = \tfrac{1}{2} a^\mu_{,\mu}
\\
c_{,\mu}{}^{,\mu} & = 0
\\
\overline{c}_{,\mu}{}^{,\mu} & = 0
\end{aligned}
\right.
\phantom{AAA}
\Leftrightarrow
\phantom{AAA}
\left\{
\begin{aligned}
\Box a^\mu & = 0
\\
b & = \tfrac{1}{2} div a
\\
\Box c & = 0
\\
\Box \overline{c} & = 0
\end{aligned}
\right.
$$ (201)

(e.g. Rejzner 16 (7.15) and (7.16)).

(Here in the middle we show the equations as the appear directly from the Euler-Lagrange variational derivative (prop. 5.12). The differential operator $\Box = \eta^{\mu \nu} \frac{d}{d x^\mu} \frac{d}{d x^\nu}$ on the right is the wave operator (example 5.27) and $div$ denotes the divergence. The equivalence to the equations on the right follows from using in the first equation the derivative of the second equation on the left, which is $b^{,\nu} = \tfrac{1}{2} a^{\mu,\nu}{}_{,\mu}$ and recalling the definition of the universal Faraday tensor (30): $\frac{d}{d x^\mu} f^{\mu \nu} = \tfrac{1}{2} \left( a^{\nu,\mu}{}_{,\mu} – a^{\mu,\nu}{}_{,\mu} \right)$.)

Now the differential equations for gauge-fixed electromagnetism on the right in (201) are nothing but the wave equations of motion of $(p+1) + 1 + 1$ free massless scalar fields (example 5.27).

As such, by example 7.19 they are a system of Green hyperbolic differential equations (def. 7.18), hence admit Cauchy surfaces (def. 8.1).

Therefore (200) indeed is a gauge fixing of the Lagrangian density of the electromagnetic field on Minkowski spacetime according to def. 12.1.

The gauge-fixed BRST operator-induced from the gauge fixed Lagrangian density (200) acts as

$$
\label{GaussianAveragedLorentzGaugeFixedBRSTOperator}
\array{
& \array{ s’_{BRST} = \\ \left\{ \left( c_{,\mu} (a^\ddagger)^\mu + b \overline{c}^{\ddagger}\right) dvol_\Sigma, (-) \right\} }
\\
a_\mu &\mapsto& c_{,\mu}
\\
b &\mapsto& 0
\\
\overline{c} &\mapsto & b
}
$$ (202)

From this we immediately obtain the propagators for the gauge-fixed electromagnetic field:

Proposition 12.9. (photon propagator in Gaussian-averaged Lorenz gauge)

After fixing Gaussian-averaged Lorenz gauge (example 12.8) of the electromagnetic field on Minkowski spacetime, the causal propagator (prop. 7.23) of the combined electromagnetic field and Nakanishi-Lautrup field is of the form

$$
\Delta^{EM, EL}
\;=\;
\left(
\array{
\Delta^{photon} & \ast
\\
\ast & \ast
}
\right)
$$

with

$$
\Delta^{photon}_{\mu \nu}(x,y)
\;=\;
\eta_{\mu \nu} \Delta(x,y)
\,,
$$

where

1. $\eta_{\mu \nu}$ is the Minkowski metric tensor (def. 2.17);
2. $\Delta(x,y)$ is the causal propagator of the free field theory massless real scalar field (prop. 9.54).

Accordingly, the Feynman propagator of the electromagnetic field in Gaussian-averaged Lorenz gauge is

$$
(\Delta^{photon}_F)_{\mu \nu}(x,y)
\;=\;
\eta_{\mu \nu} \Delta_F(x,y)
\,,
$$

where on the right $\Delta_F(x,y)$ is the Feynman propagator of the free massless real scalar field (def. 9.61).

This is also called the photon propagator.

Hence by prop. 9.64 the distributional Fourier transform of the photon propagator is

$$
\widehat{\Delta^{photon}_F}_{\mu \nu}(k)
\;=\;
\frac{1}{- k^\mu k_\mu + i 0^+}
\,.
$$

(this is a special case of Khavkine 14 (99), see also Rejzner 16, (7.20))

Proof. The Gaussian-averaged Lorenz gauge-fixed equations of motion (201) of the electromagnetic field are just $(p+1)$ uncoupled massless Klein-Gordon equations, hence wave equations (example 5.27) for the $(p+1)$ real components of the electromagnetic field (“vector potential“)

$$
\Box A_\mu = 0\phantom{AAAA} \mu \in \{0,1,\cdots, p\}
\,.
$$

This shows that the propagator is proportional to that of the real scalar field.

To see that the index structure is as claimed, recall that the domain and codomain of the advanced and retarded propagators in def. 7.17 is

$$
\array{
\Gamma_\Sigma(T\Sigma)
&\overset{\left( (\mathrm{G}_{\pm})_{\mu \nu} \right)}{\longrightarrow}&
\Gamma_\Sigma(T^\ast \Sigma)
}
$$

corresponding to a differential operator for the equations of motion which by (62) and (201) is given by

$$
\array{
\Gamma_\Sigma(T^\ast \Sigma)
&\overset{ \eta^{-1} \circ \Box }{\longrightarrow}&
\Gamma_\Sigma(T \Sigma)
\\
A_\mu &\mapsto& \eta^{\mu \nu} \Box A_\nu
}
$$

Then the defining equation (90) for the advanced and retarded Green functions is, in terms of their integral kernels, the advanced and retarded propagators $\Delta_{\pm}$

$$
\eta^{\mu’ \mu} \Box \underset{y \in X}{\int} (\Delta_{\pm})_{\mu \nu}((-),y) A^{\nu}(y) \, dvol_\Sigma(x)
=
A^\nu(x)
\,.
$$

This shows that

$$
(\Delta_{\pm})_{\mu \nu}
\;=\;
\eta_{\mu\nu} \Delta_{\pm}
$$

with $\Delta_{\pm}$ the advanced and retarded propagator of the free real scalar field on Minkowski spacetime (prop. 9.52), and hence

$$
\begin{aligned}
\Delta_{\mu \nu}
&=
(\Delta_+)_{\mu \nu}
–
(\Delta_-)_{\mu \nu}
\\
& =
\eta_{\mu \nu} (\Delta_+ – \Delta_-)
\\
& =
\eta_{\mu \nu} \Delta
\end{aligned}
$$

Next we compute the gauge-invariant on-shell polynomial observables of the electromagnetic field. The result will involve the following concept:

Definition 12.10. (wave polarization of linear observables of the electromagnetic field)

Consider the electromagnetic field on Minkowski spacetime $\Sigma$, with field bundle the cotangent bundle

The space of off-shell linear observables is spanned by the point evaluation observables

$$
e^\mu \mathbf{A}_\mu(x)
\;\in\;
LinObs(T^\ast \Sigma)
$$

where

1. $e = (e^\mu) \in \mathbb{R}^{p,1}$ is some vector;
2. $x \in \mathbb{R}^{p,1}$ is some point in Minkowski spacetime
3. $\mathbf{A}_\mu(x) \;\colon\; A \mapsto A_\mu(x)$is the functional which sends a section $A \in \Gamma_\Sigma(E) = \Omega^1(\Sigma)$ to its $\mu$-component at $x$.

After Fourier transform of distributions, this is

$$
e^\mu \widehat{\mathbf{A}}_\mu(k)
\;\in\;
LinObs(T^\ast \Sigma)
$$

for $k = (k_\mu) \in (\mathbb{R}^{p,1})^\ast$ the wave vector

for $e = (e^\mu) \in \mathbb{R}^{p,1}$ the wave polarization

The linear on-shell observables are spanned by the same expressions, but subject to the condition that

$$
{\vert k\vert}_\eta^2 = k^\mu k_\mu = 0
$$

hence

$$
LinObs(T^\ast \Sigma,\mathbf{L}_{EM})
\;=\;
\left\langle
e^\mu \widehat{\mathbf{A}}_\mu(k)
\;\vert\;
k^\mu k_\mu = 0
\right\rangle
$$

We say that the space of transversally polarized linear on-shell observables is the quotient vector space

$$
\label{ElectromagneticFieldLinearObservablesTransversallyPolarized}
LinObs(T^\ast \Sigma,\mathbf{L}_{EM})_{trans}
\;:=\;
\frac{
\langle
e^\mu \widehat{\mathbf{A}}_\mu(k)
\;\vert\;
k^\mu k_\mu = 0 \,\, \text{and} \,\, e^\mu k_\mu = 0
\rangle
}{
\langle
e^\mu \widehat{\mathbf{A}}_\mu(k)
\;\vert\;
k^\mu k_\mu = 0 \,\, \text{and} \,\, e_\mu \propto k_\mu
\rangle
}
$$ (203)

of those observables whose Fourier modes involve wave polarization vectors $e$ that vanish when contracted with the wave vector $k$, modulo those whose wave polarization vector $e$ is proportional to the wave vector.

For example if $k = (\kappa, 0, \cdots, \kappa)$, then the corresponding space of transversal polarization vectors may be identified with $\left\{e \,\vert\, e = (0,e_1, e_2, \cdots, e_{p-1}, 0) \right\}$.

Proposition 12.11. (BRST cohomology on linear on-shell observables of the Gaussian-averaged Lorenz gauge fixed electromagnetic field)

After fixing the Gaussian-averaged Lorenz gauge (example 12.8) of the electromagnetic field on Minkowski spacetime, the global BRST cohomology (def. 11.28) on the Gaussian-averaged Lorenz gauge fixed (def. 12.8) on-shell linear observables (def. 7.3) at $deg_{gh} = 0$ (prop. 11.9) is isomorphic to the space of transversally polarized linear observables, def. 12.10:

$$
H^0( LinObs( T^\ast \Sigma \times_\Sigma A \times_\Sigma A[-1] \times_\Sigma \mathcal{G}[1], \mathbf{L}’ ), s’_{BRST} )
\;\simeq\;
LinObs( T^\ast \Sigma, \mathbf{L}_{EM})_{trans}
\,.
$$

(e.g. Dermisek 09 II-5, p. 325)

Proof. The gauge fixed BRST differential (202) acts on the Fourier modes of the linear observables (def. 7.3) as follows

$$
\array{
& & s’_{BRST}
\\
\array{ \text{antighost} \\ \text{field} }
&
\widehat{\overline{\mathbf{C}}}(k)
&\mapsto&
\widehat{\mathbf{B}}(k) & \array{ \text{Nakanishi-Lautrup} \\ \text{field} }
\\
\phantom{a}
\\
&&& \underset{\text{on-shell}}{=}
\tfrac{i}{2} k^\mu \widehat{\mathbf{A}}_\mu(k)
&
\array{ \text{Lorenz gauge} \\ \text{condition} }
\\
\phantom{A}
\\
\array{ \text{electromagnetic} \\ \text{field} }
&
e^\mu \widehat{\mathbf{A}}_\mu(k)
&\mapsto&
i \left(e^\mu k_\mu\right) \widehat{\mathbf{C}}(k)
&
\array{
\text{polarization contracted}
\\
\text{with wave vector}
\\
\text{times ghost field}
}
\\
\phantom{A}
\\
\array{ \text{Nakanishi-Lautrup} \\ \text{field} }
& \widehat{\mathbf{B}}
&\mapsto&
0
}
$$

This implies that the gauge fixed BRST cohomology on linear on-shell observables at $deg_{gh} = 0$ is the space of transversally polarized linear observables (def. 12.10):

$$
\label{LinearOnShellObservablesGaugeFixedBRSTCohomologyForEMField}
\begin{aligned}
H^0(LinObs(E,\mathbf{L}_{EM}), s’_{BRST})
& =
\left\langle
\frac{
\left\{
e^\mu \widehat{\mathbf{A}}_{\mu}(k)
\,\vert\, k^\mu k_\mu = 0 \,\,\text{and}\,\,0 = d_{BRST}\left( e^\mu \widehat{\mathbf{A}}_\mu(k)
\right) = i (e^\mu k_\mu) \widehat{\mathbf{C}}(k)
\right\}
}{
\left\{
e^\mu \widehat{\mathbf{A}}_\mu(k)
\,\vert\, k^\mu k_\mu = 0 \,\,\text{and}\,\, e^\mu \widehat{\mathbf{A}}_\mu(k) \propto s’_{BRST}( \widehat{\overline{\mathbf{C}}}(k) ) = \tfrac{i}{2} k^\mu \widehat{ \mathbf{A} }_\mu(k)
\right\}
}
\right\rangle
\\
& =
\left\langle
\frac{
\left\{
e^\mu \widehat{\mathbf{A}}_\mu(k) \,\vert \, k^\mu k_\mu = 0 \,\, \text{and} \,\, e^\mu k_\mu = 0
\right\}
}
{
\left\{
e^\mu \widehat{\mathbf{A}}_\mu(k) \,\vert \, k^\mu k_\mu = 0 \,\, \text{and} \,\, e^\mu \propto k^\mu
\right\}
}
\right\rangle
\\
& =
LinObs(T^\ast \Sigma,\mathbf{L}_{EM})_{trans}
\end{aligned}
$$ (204)

Here the first line is the definition of cochain cohomology (using that both $\widehat{\mathbf{B}}$ and $\widehat{\overline{\mathbf{C}}}$ are immediately seen to vanish in cohomology), the second line is spelling out the action of the BRST operator and using the on-shell relations (201) for $\widehat{\mathbf{B}}$ and the last line is by def. 12.10.

As a corollary we obtain:

Proposition 12.12. (BRST cohomology on polynomial on-shell observables of the Gaussian-averaged Lorenz gauge fixed electromagnetic field)

After fixing Gaussian-averaged Lorenz gauge (example 12.8) of the electromagnetic field on Minkowski spacetime, the global BRST cohomology (def. 11.28) on the Gaussian-averaged Lorenz gauge fixed (def. 12.8) polynomial on-shell observables (def. 7.13) at $deg_{gh} = 0$ (prop. 11.9) is isomorphic to the distributional polynomial algebra on transversally polarized linear observables, def. 12.10:

$$
\label{EMBRSTCohomologyOnPolynomialOnShellObservables}
H^0(PolyObs( T^\ast \Sigma \times_\Sigma \mathcal{G}[1] \times_\Sigma A \times_\Sigma A[-1] ,\mathbf{L}), s’_{BRST})
\;\simeq\;
Sym\left(
LinObs(T^\ast \Sigma,\mathbf{L}_{EM})_{trans}
\right)
$$ (205)

Proof. Generally, if $(V^\bullet,d)$ is a cochain complex over a ground field of characteristic zero (such as the real numbers in the present case) and $Sym(V^\bullet,d)$ the differential graded-symmetric algebra that it induces (this example), then

$$
H^\bullet(Sym(V,d)) = Sym(H^\bullet(V,d))
\,.
$$

(by this prop.).

In conclusion, we finally obtain:

Proposition 12.13. (gauge-invariant polynomial on-shell observables of the free field theory electromagnetic field)

The BV-BRST cohomology on infinitesimal observables (def. 7.41) of the free electromagnetic field on Minkowski spacetime (example 11.23) at $deg_{gh} = 0$ is the distributional polynomial algebra in the transversally polarized linear on-shell observables, def. 12.10, as in prop. 12.12.

Proof. By the classes of quasi-isomorphisms of prop. link and prop. 12.6 we may equivalently compute the cohomology if the BV-BRST complex with differential $s’$, obtained after Gaussian-averaged Lorenz gauge fixing from example 12.8. Since the equations of motion (201) are manifestly Green hyperbolic differential equations after this gauge fixing Cauchy surfaces for the equations of motion exist and hence prop. 10.1 together with prop. 10.4 implies that the gauge fixed BV-complex $s’_{BV}$ has its cohomology concentrated in degree zero on the on-shell observables. Therefore prop. 11.10 (i.e. the collapsing of the spectral sequence for the BV/BRST bi-complex) implies that the gauge fixed BV-BRST cohomology at ghost number zero is given by the on-shell BRST-cohomology. This is characterized by prop. 12.12.

This concludes our discussion of gauge fixing. With the covariant phase space for gauge theories obtained thereby, we may finally pass to the quantization of field theory to quantum field theory proper, in the next chapter.