In this chapter we discuss the following topics:
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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 ), 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. ), since Cauchy surfaces for the equations of motion may still not exist (def. ).
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. below), since it is the infinitesimal gauge symmetries which obstruct the existence of Cauchy surfaces (by prop. and remark ).
The archetypical example is the Gaussian-averaged Lorenz gauge fixing of the electromagnetic field (example below) which reveals that the gauge-invariant content of electromagnetic waves is only in their transversal wave polarization (prop. 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:
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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 |
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quasi-isomorphisms between local BV-BRST complexes
Recall (prop. ) that given a local BV-BRST complex (example ) 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:
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) which induces an isomorphism in cochain cohomology is called a quasi-isomorphism. We now discuss two classes of quasi-isomorphisms between BV-BRST complexes:
gauge fixing (def. below)
adjoining auxiliary fields (def. below).
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(local anti-Hamiltonian flow is automorphism of local antibracket)
Let
be a local BV-BRST complex of a Lagrangian field theory $(E,\mathbf{L})$ (example ).
Then for
a Lagrangian density (def. ) on the graded field bundle
of degree
then the exponential of forming the local antibracket (def. ) with $\mathbf{L}_{gf}$
is an endomorphism of the local antibracket (def. ) in that
and in fact an automorphism, with inverse morphism given by
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
this yields another differential
and hence another differential graded-commutative superalgebra (def. )
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
By prop. the local antibracket $\left\{ -,-\right\}$ is a graded derivation in its second argument, of degree one more than the degree of its first argument (eq:LocalAntibracketGradedDerivationInSecondArgument). 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 homomorpism 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:
(gauge fixing Lagrangian density)
Let
be a local BV-BRST complex of a Lagrangian field theory $(E,\mathbf{L})$ (example ) and let
be a Lagrangian density (def. ) on the graded field bundle such that
If the quasi-isomorphism of BV-BRST complexes given by the local anti-Hamiltonian flow $\mathbf{L}_{gf}$ via prop.
is such that for the transformed graded Lagrangian field theory
(with Lagrangian density $\mathbf{L}'$ the part independent of antifields) the Euler-Lagrange equations of motion (def. ) admit Cauchy surfaces (def. ), 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 $\mathbf{L}$.
(warning on terminology)
What we call a gauge fixing Lagrangian density $\mathbf{L}_{gf}$ in def. 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 ); see at signs in supergeometry the section The super odd sign rule.
(gauge fixing via anti-Lagrangian subspaces)
Let $\mathbf{L}_{gf}$ be a gauge fixing Lagrangian density as in def. such that
its local antibracket-square vanishes
hence its anti-Hamiltonian flow has at most a linear component in its argument $\mathbf{A}$:
it is independent of the antifields
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
(and similarly for the higher jets); and the corresponding transformed Lagrangian density (1) may be written as
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 (1), which we may write as
translates into
In this form BV-gauge fixing is considered traditionally (e.g. Hennaux 90, section 8.3, page 83, equation (76b) and item (iii)).
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adjoining contractible cochain complexes of auxiliary fields
Typically a Lagrangian field theory $(E,\mathbf{L})$ for given choice of field bundle, even after finding appropriate gauge parameter bundles $\mathcal{G}$, does not yet admit a gauge fixing Lagrangian density (def. ). 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:
(auxiliary fields and antighost fields)
Over Minkowski spacetime $\Sigma$, let
be any graded vector bundle (remark ), to be regarded as a field bundle (def. ) for auxiliary fields. If this is a trivial vector bundle (example ) we denote its field coordinates by $(b^i)$. On the corresponding graded bundle with degrees shifted down by one
we write $(\overline{c}^i)$ for the induced field coordinates.
Accordingly, the shifted infinitesimal vertical cotangent bundle (def. ) of the fiber product of these bundles
has the following coordinates:
On this fiber bundle consider the Lagrangian density (def. )
given in local coordinates by
This is such that the local antibracket (def. ) with this Lagrangian acts on generators as follows:
(warning on terminology)
Beware that when adjoining antifields as in def. to a Lagrangian field theory which also has ghost fields $(c^\alpha)$ adjoined (example ) then there is no relation, a priori, between
and
In particular there is also the
The terminology and notation is maybe unfortunate but entirely established.
The following is immediate from def. , in fact this is the purpose of the definition:
(adjoining auxiliary fields is quasi-isomorphism of BV-BRST complexes)
Let
be a local BV-BRST complex of a Lagrangian field theory $(E,\mathbf{L})$ (example ).
Let moreover $A \overset{aux}{\longrightarrow} \Sigma$ be any auxiliary field bundle (def. ). 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 $\mathbf{L}_{aux}$ (2) induce a new differential graded-commutative superalgebra:
with generators
Moreover, the differential graded-commutative superalgebra of auxiliary fields and their antighost fields is a contractible chain complex
and thus the canonical inclusion map
(of the original BV-BRST complex into its tensor product with that for the auxiliary fields and their antighost fields) is a quasi-isomorphism.
From (3) we read off that
the map $s_{aux} \coloneqq \left\{ \mathbf{L}_{aux},- \right\}$ is a differential (squares to zero), and the auxiliary Lagrangian density satisfies its classical master equation (remark ) strictly
the cochain cohomology of this differential is trivial:
The local antibracket of the BV-Lagrangian density with the auxiliary Lagrangian density vanishes:
Together this implies that the sum $-\mathbf{L} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}$ satisfies the classical master equation (remark )
and hence that
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.
(gauge fixed BV-BRST field bundle)
In conclusion, we have that, given
$(E,\mathbf{L})$ a Lagrangian field theory (def. ), with field bundle $E$ (def. );
$\mathcal{G}$ a choice of gauge parameters (def. ),
hence
$\mathcal{G}[1]$ a choice of ghost fields (example );
$A$ a choice of auxiliary fields (def. ),
hence
$A[-1]$ a choice of antighost fields (def. )
$T^\ast_{\Sigma,inf}[-1](\cdots)$ the corresponding antifields (def. )
a gauge fixing Lagrangian density $\mathbf{L}_{gf}$ (def. )
then the result is a new Lagrangian field theory
now with graded field bundle (remark ) the fiber product
and with Lagrangian density $\mathbf{L'}$ independent of the antifields, but complemented by an auxiliary Lagrangian density $\mathbf{L}'_{BRST}$.
The key point being 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.
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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.
(Gaussian-averaged Lorenz gauge fixing of vacuum electromagnetism)
Consider the local BV-BRST complex for the free electromagnetic field on Minkowski spacetime from example :
The field bundle is $E \coloneqq T^\ast \Sigma$ and the gauge parameter bundle is $\mathcal{G} \coloneqq \Sigma \times \mathbb{R}$. The 0-jet field coordinates are
the Lagrangian density is (eq:ElectromagnetismLagrangian)
and the BV-BRST differential acts as:
Introduce a trivial real line bundle for auxiliary fields $b$ in degree 0 and their antighost fields $\overline{c}$ (def. ) in degree -1:
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. , is quasi-isomorphic to the original one above, has coordinate generators
and BV-BRST differential given by the local antibracket (def. ) with $-\mathbf{L}_{EM} + \mathbf{L}_{BRST} + \mathbf{L}_{aux}$:
We say that the gauge fixing Lagrangian (def. ) for Gaussian-averaged Lorenz gauge_ for the electromagnetic field
is given by (Henneaux 90 (103a))
We check that this really is a gauge fixing Lagrangian density according to def. :
From (4) and (5) we find the local antibrackets (def. ) with this gauge fixing Lagrangian density to be
(So we are in the traditional situation of example .)
Therefore the corresponding gauge fixed Lagrangian density (1) is (see also Henneaux 90 (103b)):
The Euler-Lagrange equation of motion (def. ) induced by the gauge fixed Lagrangian density $\mathbf{L}'$ at antifield degree 0 are (using (eq:ElectromagneticFieldEulerLagrangeForm)):
(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. ). 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 ) 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 (eq:FaradayTensorJet): $\frac{d}{d x^\mu} f^{\mu \nu} = \tfrac{1}{2} \left( a^{\nu,\mu}{}_{,\mu} - a^{\mu,\nu}{}_{,\mu} \right)$.)
Now the differential equations for gauge-fixed electromagnetism on the right in (7) are nothing but the wave equations of motion of $(p+1) + 1 + 1$ free massless scalar fields (example ).
As such, by example they are a system of Green hyperbolic differential equations (def. ), hence admit Cauchy surfaces (def. ).
Therefore (6) indeed is a gauge fixing of the Lagrangian density of the electromagnetic field on Minkowski spacetime according to def. .
The gauge-fixed BRST operator induced from the gauge fixed Lagrangian density (6) acts as
From this we immediately obtain the propagators for the gauge-fixed electromagnetic field:
(photon propagator in Gaussian-averaged Lorenz gauge)
After fixing Gaussian-averaged Lorenz gauge (example ) of the electromagnetic field on Minkowski spacetime, the causal propagator (prop. ) of the combined electromagnetic field and Nakanishi-Lautrup field is of the form
with
where
$\eta_{\mu \nu}$ is the Minkowski metric tensor (def. );
$\Delta(x,y)$ is the causal propagator of the free field theory massless real scalar field (prop. ).
Accordingly the Feynman propagator of the electromagnetic field in Gaussian-averaged Lorenz gauge is
where on the right $\Delta_F(x,y)$ is the Feynman propagator of the free massless real scalar field (def. ).
This is also called the photon propagator.
Hence by prop. the distributional Fourier transform of the photon propagator is
(this is a special case of Khavkine 14 (99), see also Rejzner 16, (7.20))
The Gaussian-averaged Lorenz gauge-fixed equations of motion (7) of the electromagnetic field are just $(p+1)$ uncoupled massless Klein-Gordon equations, hence wave equations (example ) for the $(p+1)$ real components of the electromagnetic field (“vector potential”)
This shows that the propoagator 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. is
corresponding to a differential operator for the equations of motion which by (eq:ElectromagneticFieldEulerLagrangeForm) and (7) is given by
Then the defining equation (eq:AdvancedRetardedGreenFunctionIsRightInverseToDiffOperator) for the advanced and retarded Green functions is, in terms of their integral kernels, the advanced and retarded propagators $\Delta_{\pm}$
This shows that
with $\Delta_{\pm}$ the advanced and retarded propagator of the free real scalar field on Minkowski spacetime (prop. ), and hence
Next we compute the gauge-invariant on-shell polynomial observables of the electromagnetic field. The result will involve the following concept:
(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
where
$e = (e^\mu) \in \mathbb{R}^{p,1}$ is some vector;
$x \in \mathbb{R}^{p,1}$ is some point in Minkowski spacetime
$\mathbf{A}_\mu(x) \;\colon\; A \mapsto A_\mu(x)$
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
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
hence
We say that the space of transversally polarized linear on-shell observables is the quotient vector space
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 $\left\{e \,\vert\, e = (0,e_1, e_2, \cdots, e_{p-1}, 0) \right\}$.
(BRST cohomology on linear on-shell observables of the Gaussian-averaged Lorenz gauge fixed electromagnetic field)
After fixing Gaussian-averaged Lorenz gauge (example ) of the electromagnetic field on Minkowski spacetime, the global BRST cohomology (def. ) on the Gaussian-averaged Lorenz gauge fixed (def. ) on-shell linear observables (def. ) at $deg_{gh} = 0$ (prop. ) is isomorphic to the space of transversally polarized linear observables, def. :
(e.g. Dermisek 09 II-5, p. 325)
The gauge fixed BRST differential (8) acts on the Fourier modes of the linear observables (def. ) as follows
This impies that the gauge fixed BRST cohomology on linear on-shell observables at $deg_{gh} = 0$ is the space of transversally polarized linear observables (def. ):
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 (7) for $\widehat{\mathbf{B}}$ and the last line is by def. .
As a corollary we obtain:
(BRST cohomology on polynomial on-shell observables of the Gaussian-averaged Lorenz gauge fixed electromagnetic field)
After fixing Gaussian-averaged Lorenz gauge (example ) of the electromagnetic field on Minkowski spacetime, the global BRST cohomology (def. ) on the Gaussian-averaged Lorenz gauge fixed (def. ) polynomial on-shell observables (def. ) at $deg_{gh} = 0$ (prop. ) is isomorphic to the distributional polynomial algebra on transversally polarized linear observables, def. :
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
(by this prop.).
In conclusion we finally obtain:
(gauge-invariant polynomial on-shell observables of the free field theory electromagnetic field)
The BV-BRST cohomology on infinitesimal observables (def. ) of the free electromagnetic field on Minkowski spacetime (example ) at $deg_{gh} = 0$ is the distributional polynomial algebra in the transversally polarized linear on-shell observables, def. , as in prop. .
By the classes of quasi-isomorphisms of prop. and prop. we may equivalently compute the cohomology if the BV-BRST complex with differential $s'$, obtained after Gaussian-averaged Lorenz gauge fixing from example . Since the equations of motion (7) are manifestly Green hyperbolic differential equations after this gauge fixing Cauchy surfaces for the equations of motion exist and hence prop. together with prop. implies that the gauge fixed BV-complex $s'_{BV}$ has its cohomology concentrated in degree zero on the on-shell observables. Therefore prop. (i.e. the collapsing of the spectral sequence for the BV/BRST bicomplex) 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. .
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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.
Last revised on August 2, 2018 at 03:37:43. See the history of this page for a list of all contributions to it.