BRST quantization
In theoretical physics, the BRST formalism, or BRST quantization (where the BRST refers to the last
names of Carlo Becchi, Alain Rouet, Raymond Stora and Igor Tyutin) denotes a relatively rigorous
mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier
quantum field theory (QFT) frameworks resembled "prescriptions" or "heuristics" more than proofs,
especially in non-abelian QFT, where the use of "ghost fields" with superficially bizarre properties is almost
unavoidable for technical reasons related to renormalization and anomaly cancellation.
The BRST global supersymmetry introduced in the mid-1970s was quickly understood to rationalize the
introduction of these Faddeev–Popov ghosts and their exclusion from "physical" asymptotic states when
performing QFT calculations. Crucially, this symmetry of the path integral is preserved in loop order, and
thus prevents introduction of counterterms which might spoil renormalizability of gauge theories. Work by
other authors a few years later related the BRST operator to the existence of a rigorous alternative to path
integrals when quantizing a gauge theory.
Only in the late 1980s, when QFT was reformulated in fiber bundle language for application to problems in
the topology of low-dimensional manifolds (topological quantum field theory), did it become apparent that
the BRST "transformation" is fundamentally geometrical in character. In this light, "BRST quantization"
becomes more than an alternate way to arrive at anomaly-cancelling ghosts. It is a different perspective on
what the ghost fields represent, why the Faddeev–Popov method works, and how it is related to the use of
Hamiltonian mechanics to construct a perturbative framework. The relationship between gauge invariance
and "BRST invariance" forces the choice of a Hamiltonian system whose states are composed of
"particles" according to the rules familiar from the canonical quantization formalism. This esoteric
consistency condition therefore comes quite close to explaining how quanta and fermions arise in physics to
begin with.
In certain cases, notably gravity and supergravity, BRST must be superseded by a more general formalism,
the Batalin–Vilkovisky formalism.
Contents
Technical summary
Classical BRST
Gauge transformations in QFT
Gauge fixing and perturbation theory
Pre-BRST approaches to gauge fixing
Mathematical approach to BRST
The BRST operator and asymptotic Fock space
The Kugo–Ojima answer to unitarity questions
Gauge bundles and the vertical ideal
BRST formalism
Quantum version
Example
See also
References
Citations
Textbook treatments
Mathematical treatment
Primary literature
Alternate perspectives
External links
Technical summary
BRST quantization is a differential geometric approach to performing consistent, anomaly-free perturbative
calculations in a non-abelian gauge theory. The analytical form of the BRST "transformation" and its
relevance to renormalization and anomaly cancellation were described by Carlo Maria Becchi, Alain
Rouet, and Raymond Stora in a series of papers culminating in the 1976 "Renormalization of gauge
theories". The equivalent transformation and many of its properties were independently discovered by Igor
Viktorovich Tyutin. Its significance for rigorous canonical quantization of a Yang–Mills theory and its
correct application to the Fock space of instantaneous field configurations were elucidated by Taichiro
Kugo and Izumi Ojima. Later work by many authors, notably Thomas Schücker and Edward Witten, has
clarified the geometric significance of the BRST operator and related fields and emphasized its importance
to topological quantum field theory and string theory.
In the BRST approach, one selects a perturbation-friendly gauge fixing procedure for the action principle of
a gauge theory using the differential geometry of the gauge bundle on which the field theory lives. One
then quantizes the theory to obtain a Hamiltonian system in the interaction picture in such a way that the
"unphysical" fields introduced by the gauge fixing procedure resolve gauge anomalies without appearing in
the asymptotic states of the theory. The result is a set of Feynman rules for use in a Dyson series
perturbative expansion of the S-matrix which guarantee that it is unitary and renormalizable at each loop
order—in short, a coherent approximation technique for making physical predictions about the results of
scattering experiments.
Classical BRST
This is related to a supersymplectic manifold where pure operators are graded by integral ghost numbers
and we have a BRST cohomology.
Gauge transformations in QFT
From a practical perspective, a quantum field theory consists of an action principle and a set of procedures
for performing perturbative calculations. There are other kinds of "sanity checks" that can be performed on
a quantum field theory to determine whether it fits qualitative phenomena such as quark confinement and
asymptotic freedom. However, most of the predictive successes of quantum field theory, from quantum
electrodynamics to the present day, have been quantified by matching S-matrix calculations against the
results of scattering experiments.
In the early days of QFT, one would have to have said that the quantization and renormalization
prescriptions were as much part of the model as the Lagrangian density, especially when they relied on the
powerful but mathematically ill-defined path integral formalism. It quickly became clear that QED was
almost "magical" in its relative tractability, and that most of the ways that one might imagine extending it
would not produce rational calculations. However, one class of field theories remained promising: gauge
theories, in which the objects in the theory represent equivalence classes of physically indistinguishable
field configurations, any two of which are related by a gauge transformation. This generalizes the QED
idea of a local change of phase to a more complicated Lie group.
QED itself is a gauge theory, as is general relativity, although the latter has proven resistant to quantization
so far, for reasons related to renormalization. Another class of gauge theories with a non-Abelian gauge
group, beginning with Yang–Mills theory, became amenable to quantization in the late 1960s and early
1970s, largely due to the work of Ludwig D. Faddeev, Victor Popov, Bryce DeWitt, and Gerardus 't Hooft.
However, they remained very difficult to work with until the introduction of the BRST method. The BRST
method provided the calculation techniques and renormalizability proofs needed to extract accurate results
from both "unbroken" Yang–Mills theories and those in which the Higgs mechanism leads to spontaneous
symmetry breaking. Representatives of these two types of Yang–Mills systems—quantum chromodynamics
and electroweak theory—appear in the Standard Model of particle physics.
It has proven rather more difficult to prove the existence of non-Abelian quantum field theory in a rigorous
sense than to obtain accurate predictions using semi-heuristic calculation schemes. This is because
analyzing a quantum field theory requires two mathematically interlocked perspectives: a Lagrangian
system based on the action functional, composed of fields with distinct values at each point in spacetime
and local operators which act on them, and a Hamiltonian system in the Dirac picture, composed of states
which characterize the entire system at a given time and field operators which act on them. What makes this
so difficult in a gauge theory is that the objects of the theory are not really local fields on spacetime; they
are right-invariant local fields on the principal gauge bundle, and different local sections through a portion
of the gauge bundle, related by passive transformations, produce different Dirac pictures.
What is more, a description of the system as a whole in terms of a set of fields contains many redundant
degrees of freedom; the distinct configurations of the theory are equivalence classes of field configurations,
so that two descriptions which are related to one another by a gauge transformation are also really the same
physical configuration. The "solutions" of a quantized gauge theory exist not in a straightforward space of
fields with values at every point in spacetime but in a quotient space (or cohomology) whose elements are
equivalence classes of field configurations. Hiding in the BRST formalism is a system for parameterizing
the variations associated with all possible active gauge transformations and correctly accounting for their
physical irrelevance during the conversion of a Lagrangian system to a Hamiltonian system.
Gauge fixing and perturbation theory
The principle of gauge invariance is essential to constructing a workable quantum field theory. But it is
generally not feasible to perform a perturbative calculation in a gauge theory without first "fixing the
gauge"—adding terms to the Lagrangian density of the action principle which "break the gauge symmetry"
to suppress these "unphysical" degrees of freedom. The idea of gauge fixing goes back to the Lorenz gauge
approach to electromagnetism, which suppresses most of the excess degrees of freedom in the fourpotential while retaining manifest Lorentz invariance. The Lorenz gauge is a great simplification relative to
Maxwell's field-strength approach to classical electrodynamics, and illustrates why it is useful to deal with
excess degrees of freedom in the representation of the objects in a theory at the Lagrangian stage, before
passing over to Hamiltonian mechanics via the Legendre transformation.
The Hamiltonian density is related to the Lie derivative of the Lagrangian density with respect to a unit
timelike horizontal vector field on the gauge bundle. In a quantum mechanical context it is conventionally
rescaled by a factor . Integrating it by parts over a spacelike cross section recovers the form of the
integrand familiar from canonical quantization. Because the definition of the Hamiltonian involves a unit
time vector field on the base space, a horizontal lift to the bundle space, and a spacelike surface "normal"
(in the Minkowski metric) to the unit time vector field at each point on the base manifold, it is dependent
both on the connection and the choice of Lorentz frame, and is far from being globally defined. But it is an
essential ingredient in the perturbative framework of quantum field theory, into which the quantized
Hamiltonian enters via the Dyson series.
For perturbative purposes, we gather the configuration of all the fields of our theory on an entire threedimensional horizontal spacelike cross section of P into one object (a Fock state), and then describe the
"evolution" of this state over time using the interaction picture. The Fock space is spanned by the multiparticle eigenstates of the "unperturbed" or "non-interaction" portion
of the Hamiltonian . Hence the
instantaneous description of any Fock state is a complex-amplitude-weighted sum of eigenstates of
. In
the interaction picture, we relate Fock states at different times by prescribing that each eigenstate of the
unperturbed Hamiltonian experiences a constant rate of phase rotation proportional to its energy (the
corresponding eigenvalue of the unperturbed Hamiltonian).
Hence, in the zero-order approximation, the set of weights characterizing a Fock state does not change over
time, but the corresponding field configuration does. In higher approximations, the weights also change;
collider experiments in high-energy physics amount to measurements of the rate of change in these weights
(or rather integrals of them over distributions representing uncertainty in the initial and final conditions of a
scattering event). The Dyson series captures the effect of the discrepancy between
and the true
Hamiltonian , in the form of a power series in the coupling constant g; it is the principal tool for making
quantitative predictions from a quantum field theory.
To use the Dyson series to calculate anything, one needs more than a gauge-invariant Lagrangian density;
one also needs the quantization and gauge fixing prescriptions that enter into the Feynman rules of the
theory. The Dyson series produces infinite integrals of various kinds when applied to the Hamiltonian of a
particular QFT. This is partly because all usable quantum field theories to date must be considered effective
field theories, describing only interactions on a certain range of energy scales that we can experimentally
probe and therefore vulnerable to ultraviolet divergences. These are tolerable as long as they can be
handled via standard techniques of renormalization; they are not so tolerable when they result in an infinite
series of infinite renormalizations or, worse, in an obviously unphysical prediction such as an uncancelled
gauge anomaly. There is a deep relationship between renormalizability and gauge invariance, which is
easily lost in the course of attempts to obtain tractable Feynman rules by fixing the gauge.
Pre-BRST approaches to gauge fixing
The traditional gauge fixing prescriptions of continuum electrodynamics select a unique representative from
each gauge-transformation-related equivalence class using a constraint equation such as the Lorenz gauge
. This sort of prescription can be applied to an Abelian gauge theory such as QED, although it
results in some difficulty in explaining why the Ward identities of the classical theory carry over to the
quantum theory—in other words, why Feynman diagrams containing internal longitudinally polarized
virtual photons do not contribute to S-matrix calculations. This approach also does not generalize well to
non-Abelian gauge groups such as the U(2) of Yang–Mills and electroweak theory and the SU(3) of
quantum chromodynamics. It suffers from Gribov ambiguities and from the difficulty of defining a gauge
fixing constraint that is in some sense "orthogonal" to physically significant changes in the field
configuration.
More sophisticated approaches do not attempt to apply a delta function constraint to the gauge
transformation degrees of freedom. Instead of "fixing" the gauge to a particular "constraint surface" in
configuration space, one can break the gauge freedom with an additional, non-gauge-invariant term added
to the Lagrangian density. In order to reproduce the successes of gauge fixing, this term is chosen to be
minimal for the choice of gauge that corresponds to the desired constraint and to depend quadratically on
the deviation of the gauge from the constraint surface. By the stationary phase approximation on which the
Feynman path integral is based, the dominant contribution to perturbative calculations will come from field
configurations in the neighborhood of the constraint surface.
The perturbative expansion associated with this Lagrangian, using the method of functional quantization, is
generally referred to as the Rξ gauge. It reduces in the case of an Abelian U(1) gauge to the same set of
Feynman rules that one obtains in the method of canonical quantization. But there is an important
difference: the broken gauge freedom appears in the functional integral as an additional factor in the overall
normalization. This factor can only be pulled out of the perturbative expansion (and ignored) when the
contribution to the Lagrangian of a perturbation along the gauge degrees of freedom is independent of the
particular "physical" field configuration. This is the condition that fails to hold for non-Abelian gauge
groups. If one ignores the problem and attempts to use the Feynman rules obtained from "naive" functional
quantization, one finds that one's calculations contain unremovable anomalies.
The problem of perturbative calculations in QCD was solved by introducing additional fields known as
Faddeev–Popov ghosts, whose contribution to the gauge-fixed Lagrangian offsets the anomaly introduced
by the coupling of "physical" and "unphysical" perturbations of the non-Abelian gauge field. From the
functional quantization perspective, the "unphysical" perturbations of the field configuration (the gauge
transformations) form a subspace of the space of all (infinitesimal) perturbations; in the non-Abelian case,
the embedding of this subspace in the larger space depends on the configuration around which the
perturbation takes place. The ghost term in the Lagrangian represents the functional determinant of the
Jacobian of this embedding, and the properties of the ghost field are dictated by the exponent desired on the
determinant in order to correct the functional measure on the remaining "physical" perturbation axes.
Mathematical approach to BRST
BRST construction applies to a situation of a hamiltonian action of a compact, connected Lie group G on a
phase space M.[1][2] Let be the Lie algebra of G and
a regular value of the moment map
. Let
. Assume the G-action on M0 is free and proper, and consider the space
of G-orbits on M0 , which is also known as a Symplectic reduction quotient
.
First, using the regular sequence of functions defining M0 inside M, construct a Koszul complex
The differential, δ, on this complex is an odd C∞(M)-linear derivation of the graded C∞(M)-algebra
. This odd derivation is defined by extending the Lie algebra homomorphim
of the hamiltonian action. The resulting Koszul complex is the Koszul complex of the
∞
-module C (M), where
is the symmetric algebra of , and the module structure comes from a ring
homomorphism
induced by the hamiltonian action
.
This Koszul complex is a resolution of the
-module
, i.e.,
Then, consider the Chevalley-Eilenberg cochain complex for the Koszul complex
considered as a dg module over the Lie algebra :
The "horizontal" differential
is defined on the coefficients
by the action of and on
as the exterior derivative of right-invariant differential forms on the Lie
group G, whose Lie algebra is .
Let Tot(K) be a complex such that
with a differential D = d + δ. The cohomology groups of (Tot(K), D) are computed using a spectral
sequence associated to the double complex
.
The first term of the spectral sequence computes the cohomology of the "vertical" differential δ:
, if j = 0 and zero otherwise.
The first term of the spectral sequence may be interpreted as the complex of vertical differential forms
for the fiber bundle
.
The second term of the spectral sequence computes the cohomology of the "horizontal" differential d on
:
, if
The spectral sequence collapses at the second term, so that
zero.
and zero otherwise.
, which is concentrated in degree
Therefore,
, if p = 0 and 0 otherwise.
The BRST operator and asymptotic Fock space
Two important remarks about the BRST operator are due. First, instead of working with the gauge group G
one can use only the action of the gauge algebra on the fields (functions on the phase space).
Second, the variation of any "BRST exact form" sBX with respect to a local gauge transformation dλ is
which is itself an exact form.
More importantly for the Hamiltonian perturbative formalism (which is carried out not on the fiber bundle
but on a local section), adding a BRST exact term to a gauge invariant Lagrangian density preserves the
relation sBX = 0. As we shall see, this implies that there is a related operator QB on the state space for which
—i. e., the BRST operator on Fock states is a conserved charge of the Hamiltonian system.
This implies that the time evolution operator in a Dyson series calculation will not evolve a field
configuration obeying
into a later configuration with
(or vice versa).
Another way of looking at the nilpotence of the BRST operator is to say that its image (the space of BRST
exact forms) lies entirely within its kernel (the space of BRST closed forms). (The "true" Lagrangian,
presumed to be invariant under local gauge transformations, is in the kernel of the BRST operator but not in
its image.) The preceding argument says that we can limit our universe of initial and final conditions to
asymptotic "states"—field configurations at timelike infinity, where the interaction Lagrangian is "turned
off"—that lie in the kernel of QB and still obtain a unitary scattering matrix. (BRST closed and exact states
are defined similarly to BRST closed and exact fields; closed states are annihilated by QB, while exact
states are those obtainable by applying QB to some arbitrary field configuration.)
We can also suppress states that lie inside the image of QB when defining the asymptotic states of our
theory—but the reasoning is a bit subtler. Since we have postulated that the "true" Lagrangian of our theory
is gauge invariant, the true "states" of our Hamiltonian system are equivalence classes under local gauge
transformation; in other words, two initial or final states in the Hamiltonian picture that differ only by a
BRST exact state are physically equivalent. However, the use of a BRST exact gauge breaking prescription
does not guarantee that the interaction Hamiltonian will preserve any particular subspace of closed field
configurations that we can call "orthogonal" to the space of exact configurations. (This is a crucial point,
often mishandled in QFT textbooks. There is no a priori inner product on field configurations built into the
action principle; we construct such an inner product as part of our Hamiltonian perturbative apparatus.)
We therefore focus on the vector space of BRST closed configurations at a particular time with the
intention of converting it into a Fock space of intermediate states suitable for Hamiltonian perturbation. To
this end, we shall endow it with ladder operators for the energy-momentum eigenconfigurations (particles)
of each field, complete with appropriate (anti-)commutation rules, as well as a positive semi-definite inner
product. We require that the inner product be singular exclusively along directions that correspond to BRST
exact eigenstates of the unperturbed Hamiltonian. This ensures that one can freely choose, from within the
two equivalence classes of asymptotic field configurations corresponding to particular initial and final
eigenstates of the (unbroken) free-field Hamiltonian, any pair of BRST closed Fock states that we like.
The desired quantization prescriptions will also provide a quotient Fock space isomorphic to the BRST
cohomology, in which each BRST closed equivalence class of intermediate states (differing only by an
exact state) is represented by exactly one state that contains no quanta of the BRST exact fields. This is the
Fock space we want for asymptotic states of the theory; even though we will not generally succeed in
choosing the particular final field configuration to which the gauge-fixed Lagrangian dynamics would have
evolved that initial configuration, the singularity of the inner product along BRST exact degrees of freedom
ensures that we will get the right entries for the physical scattering matrix.
(Actually, we should probably be constructing a Krein space for the BRST-closed intermediate Fock states,
with the time reversal operator playing the role of the "fundamental symmetry" relating the Lorentzinvariant and positive semi-definite inner products. The asymptotic state space is presumably the Hilbert
space obtained by quotienting BRST exact states out of this Krein space.)
In sum, no field introduced as part of a BRST gauge fixing procedure will appear in asymptotic states of
the gauge-fixed theory. However, this does not imply that we can do without these "unphysical" fields in
the intermediate states of a perturbative calculation! This is because perturbative calculations are done in the
interaction picture. They implicitly involve initial and final states of the non-interaction Hamiltonian
,
gradually transformed into states of the full Hamiltonian in accordance with the adiabatic theorem by
"turning on" the interaction Hamiltonian (the gauge coupling). The expansion of the Dyson series in terms
of Feynman diagrams will include vertices that couple "physical" particles (those that can appear in
asymptotic states of the free Hamiltonian) to "unphysical" particles (states of fields that live outside the
kernel of sB or inside the image of sB) and vertices that couple "unphysical" particles to one another.
The Kugo–Ojima answer to unitarity questions
T. Kugo and I. Ojima are commonly credited with the discovery of the principal QCD color confinement
criterion. Their role in obtaining a correct version of the BRST formalism in the Lagrangian framework
seems to be less widely appreciated. It is enlightening to inspect their variant of the BRST transformation,
which emphasizes the hermitian properties of the newly introduced fields, before proceeding from an
entirely geometrical angle. The gauge fixed Lagrangian density is below; the two terms in parentheses form
the coupling between the gauge and ghost sectors, and the final term becomes a Gaussian weighting for the
functional measure on the auxiliary field B.
The Faddeev–Popov ghost field c is unique among the new fields of our gauge-fixed theory in having a
geometrical meaning beyond the formal requirements of the BRST procedure. It is a version of the Maurer–
Cartan form on
, which relates each right-invariant vertical vector field
to its representation
(up to a phase) as a -valued field. This field must enter into the formulas for infinitesimal gauge
transformations on objects (such as fermions ψ, gauge bosons Aμ, and the ghost c itself) which carry a nontrivial representation of the gauge group. The BRST transformation with respect to δλ is therefore:
Here we have omitted the details of the matter sector ψ and left the form of the Ward operator on it
unspecified; these are unimportant so long as the representation of the gauge algebra on the matter fields is
consistent with their coupling to δAμ. The properties of the other fields we have added are fundamentally
analytical rather than geometric. The bias we have introduced towards connections with
is
gauge-dependent and has no particular geometrical significance. The anti-ghost is nothing but a Lagrange
multiplier for the gauge fixing term, and the properties of the scalar field B are entirely dictated by the
relationship
. (The new fields are all Hermitian in Kugo–Ojima conventions, but the parameter
δλ is an anti-Hermitian "anti-commuting c-number". This results in some unnecessary awkwardness with
regard to phases and passing infinitesimal parameters through operators; this will be resolved with a change
of conventions in the geometric treatment below.)
We already know, from the relation of the BRST operator to the exterior derivative and the Faddeev–Popov
ghost to the Maurer–Cartan form, that the ghost c corresponds (up to a phase) to a -valued 1-form on
.
In order for integration of a term like
to be meaningful, the anti-ghost
must carry
representations of these two Lie algebras—the vertical ideal
and the gauge algebra —dual to those
carried by the ghost. In geometric terms, must be fiberwise dual to and one rank short of being a top
form on
. Likewise, the auxiliary field B must carry the same representation of (up to a phase) as , as
well as the representation of
dual to its trivial representation on Aμ—i. e., B is a fiberwise -dual top
form on
.
Let us focus briefly on the one-particle states of the theory, in the adiabatically decoupled limit g → 0.
There are two kinds of quanta in the Fock space of the gauge-fixed Hamiltonian that we expect to lie
entirely outside the kernel of the BRST operator: those of the Faddeev–Popov anti-ghost and the forward
polarized gauge boson. This is because no combination of fields containing is annihilated by sB and we
have added to the Lagrangian a gauge breaking term that is equal up to a divergence to
Likewise, there are two kinds of quanta that will lie entirely in the image of the BRST operator: those of the
Faddeev–Popov ghost c and the scalar field B, which is "eaten" by completing the square in the functional
integral to become the backward polarized gauge boson. These are the four types of "unphysical" quanta
which will not appear in the asymptotic states of a perturbative calculation—if we get our quantization rules
right.
The anti-ghost is taken to be a Lorentz scalar for the sake of Poincaré invariance in
.
However, its (anti-)commutation law relative to c—i. e., its quantization prescription, which ignores the
spin–statistics theorem by giving Fermi–Dirac statistics to a spin-0 particle—will be given by the
requirement that the inner product on our Fock space of asymptotic states be singular along directions
corresponding to the raising and lowering operators of some combination of non-BRST-closed and BRSTexact fields. This last statement is the key to "BRST quantization", as opposed to mere "BRST symmetry"
or "BRST transformation".
(Needs to be completed in the language of BRST cohomology, with reference to the Kugo–Ojima
treatment of asymptotic Fock space.)
Gauge bundles and the vertical ideal
In order to do the BRST method justice, we must switch from the "algebra-valued fields on Minkowski
space" picture typical of quantum field theory texts (and of the above exposition) to the language of fiber
bundles, in which there are two quite different ways to look at a gauge transformation: as a change of local
section (also known in general relativity as a passive transformation) or as the pullback of the field
configuration along a vertical diffeomorphism of the principal bundle. It is the latter sort of gauge
transformation that enters into the BRST method. Unlike a passive transformation, it is well-defined
globally on a principal bundle with any structure group over an arbitrary manifold. (However, for
concreteness and relevance to conventional QFT, this article will stick to the case of a principal gauge
bundle with compact fiber over 4-dimensional Minkowski space.)
A principal gauge bundle P over a 4-manifold M is locally isomorphic to U × F, where U ⊂ R4 and the
fiber F is isomorphic to a Lie group G, the gauge group of the field theory (this is an isomorphism of
manifold structures, not of group structures; there is no special surface in P corresponding to 1 in G, so it is
more proper to say that the fiber F is a G-torsor). Thus, the (physical) principal gauge bundle is related to
the (mathematical) principal G-bundle but has more structure. Its most basic property as a fiber bundle is the
"projection to the base space" π : P → M, which defines the "vertical" directions on P (those lying within
the fiber π−1 (p) over each point p in M). As a gauge bundle it has a left action of G on P which respects the
fiber structure, and as a principal bundle it also has a right action of G on P which also respects the fiber
structure and commutes with the left action.
The left action of the structure group G on P corresponds to a mere change of coordinate system on an
individual fiber. The (global) right action Rg : P → P for a fixed g in G corresponds to an actual
automorphism of each fiber and hence to a map of P to itself. In order for P to qualify as a principal Gbundle, the global right action of each g in G must be an automorphism with respect to the manifold
structure of P with a smooth dependence on g—i. e., a diffeomorphism P × G → P.
The existence of the global right action of the structure group picks out a special class of right invariant
geometric objects on P—those which do not change when they are pulled back along Rg for all values of g
in G. The most important right invariant objects on a principal bundle are the right invariant vector fields,
which form an ideal of the Lie algebra of infinitesimal diffeomorphisms on P. Those vector fields on P
which are both right invariant and vertical form an ideal
of , which has a relationship to the entire
bundle P analogous to that of the Lie algebra of the gauge group G to the individual G-torsor fiber F.
The "field theory" of interest is defined in terms of a set of "fields" (smooth maps into various vector
spaces) defined on a principal gauge bundle P. Different fields carry different representations of the gauge
group G, and perhaps of other symmetry groups of the manifold such as the Poincaré group. One may
define the space Pl of local polynomials in these fields and their derivatives. The fundamental Lagrangian
density of one's theory is presumed to lie in the subspace Pl0 of polynomials which are real-valued and
invariant under any unbroken non-gauge symmetry groups. It is also presumed to be invariant not only
under the left action (passive coordinate transformations) and the global right action of the gauge group but
also under local gauge transformations—pullback along the infinitesimal diffeomorphism associated with
an arbitrary choice of right invariant vertical vector field
.
Identifying local gauge transformations with a particular subspace of vector fields on the manifold P equips
us with a better framework for dealing with infinite-dimensional infinitesimals: differential geometry and
the exterior calculus. The change in a scalar field under pullback along an infinitesimal automorphism is
captured in the Lie derivative, and the notion of retaining only the term linear in the scale of the vector field
is implemented by separating it into the inner derivative and the exterior derivative. (In this context, "forms"
and the exterior calculus refer exclusively to degrees of freedom which are dual to vector fields on the
gauge bundle, not to degrees of freedom expressed in (Greek) tensor indices on the base manifold or
(Roman) matrix indices on the gauge algebra.)
The Lie derivative on a manifold is a globally well-defined operation in a way that the partial derivative is
not. The proper generalization of Clairaut's theorem to the non-trivial manifold structure of P is given by
the Lie bracket of vector fields and the nilpotence of the exterior derivative. And we obtain an essential tool
for computation: the generalized Stokes theorem, which allows us to integrate by parts and drop the surface
term as long as the integrand drops off rapidly enough in directions where there is an open boundary. (This
is not a trivial assumption, but can be dealt with by renormalization techniques such as dimensional
regularization as long as the surface term can be made gauge invariant.)
BRST formalism
In theoretical physics, the BRST formalism is a method of implementing first class constraints. The letters
BRST stand for Becchi, Rouet, Stora, and (independently) Tyutin who discovered this formalism. It is a
sophisticated method to deal with quantum physical theories with gauge invariance. For example, the
BRST methods are often applied to gauge theory and quantized general relativity.
Quantum version
The space of states is not a Hilbert space (see below). This vector space is both Z2 -graded and R-graded. If
you wish, you may think of it as a Z2 × R-graded vector space. The former grading is the parity, which can
either be even or odd. The latter grading is the ghost number. Note that it is R and not Z because unlike the
classical case, we can have nonintegral ghost numbers. Operators acting upon this space are also Z2 × Rgraded in the obvious manner. In particular, Q is odd and has a ghost number of 1.
Let Hn be the subspace of all states with ghost number n. Then, Q restricted to Hn maps Hn to Hn+1 . Since
Q2 = 0, we have a cochain complex describing a cohomology.
The physical states are identified as elements of cohomology of the operator Q, i.e. as vectors in
Ker(Qn+1 )/Im(Qn ). The BRST theory is in fact linked to the standard resolution in Lie algebra
cohomology.
Recall that the space of states is Z2 -graded. If A is a pure graded operator, then the BRST transformation
maps A to [Q, A) where [ , ) is the supercommutator. BRST-invariant operators are operators for which
[Q, A) = 0. Since the operators are also graded by ghost numbers, this BRST transformation also forms a
cohomology for the operators since [Q, [Q, A)) = 0.
Although the BRST formalism is more general than the Faddeev-Popov gauge fixing, in the special case
where it is derived from it, the BRST operator is also useful to obtain the right Jacobian associated with
constraints that gauge-fix the symmetry.
The BRST operator is a supersymmetry. It generates the Lie superalgebra with a zero-dimensional even
part and a one-dimensional odd part spanned by Q. [Q, Q) = {Q, Q} = 0 where [ , ) is the Lie superbracket
(i.e. Q2 = 0). This means Q acts as an antiderivation.
Because Q is Hermitian and its square is zero but Q itself is nonzero, this means the vector space of all
states prior to the cohomological reduction has an indefinite norm! This means it is not a Hilbert space.
For more general flows which can't be described by first class constraints, see Batalin–Vilkovisky
formalism.
Example
For the special case of gauge theories (of the usual kind described by sections of a principal G-bundle) with
a quantum connection form A, a BRST charge (sometimes also a BRS charge) is an operator usually
denoted Q.
Let the -valued gauge fixing conditions be
where ξ is a positive number determining the
gauge. There are many other possible gauge fixings, but they will not be covered here. The fields are the valued connection form A, -valued scalar field with fermionic statistics, b and c and a -valued scalar field
with bosonic statistics B. c deals with the gauge transformations whereas b and B deal with the gauge
fixings. There actually are some subtleties associated with the gauge fixing due to Gribov ambiguities but
they will not be covered here.
where D is the covariant derivative.
where [ , ]L is the Lie bracket.
Q is an antiderivation.
The BRST Lagrangian density
While the Lagrangian density is not BRST invariant, its integral over all of spacetime, the action, is.
The operator Q is defined as
where
are the Faddeev–Popov ghosts and antighosts (fields with a negative ghost number),
respectively, Li are the infinitesimal generators of the Lie group, and
are its structure constants.
See also
Batalin–Vilkovisky formalism
Quantum chromodynamics
References
Citations
1. Figueroa-O'Farrill & Kimura 1991, pp. 209–229
2. Kostant & Sternberg 1987, pp. 49–113
Textbook treatments
Chapter 16 of Peskin & Schroeder (ISBN 0-201-50397-2 or ISBN 0-201-50934-2) applies
the "BRST symmetry" to reason about anomaly cancellation in the Faddeev–Popov
Lagrangian. This is a good start for QFT non-experts, although the connections to geometry
are omitted and the treatment of asymptotic Fock space is only a sketch.
Chapter 12 of M. Göckeler and T. Schücker (ISBN 0-521-37821-4 or ISBN 0-521-32960-4)
discusses the relationship between the BRST formalism and the geometry of gauge
bundles. It is substantially similar to Schücker's 1987 paper (http://projecteuclid.org/Dienst/U
I/1.0/Summarize/euclid.cmp/1104116716).
Mathematical treatment
Figueroa-O'Farrill, J. M.; Kimura, T. (1991). "Geometric BRST Quantization I.
Prequantization" (http://projecteuclid.org/euclid.cmp/1104202348). Commun. Math. Phys.
Springer-Verlag. 136 (2): 209–229. Bibcode:1991CMaPh.136..209F (https://ui.adsabs.harvar
d.edu/abs/1991CMaPh.136..209F). doi:10.1007/BF02100022 (https://doi.org/10.1007%2FB
F02100022). ISSN 0010-3616 (https://www.worldcat.org/issn/0010-3616). MR 1096113 (http
s://www.ams.org/mathscinet-getitem?mr=1096113). S2CID 120119621 (https://api.semantic
scholar.org/CorpusID:120119621).
Kostant, B.; Sternberg, S. (1987). "Symplectic reduction, BRS cohomology, and infinitedimensional Clifford algebras". Ann. Phys. Elsevier. 176 (1): 49–113.
Bibcode:1987AnPhy.176...49K (https://ui.adsabs.harvard.edu/abs/1987AnPhy.176...49K).
doi:10.1016/0003-4916(87)90178-3 (https://doi.org/10.1016%2F0003-4916%2887%299017
8-3).
Primary literature
Original BRST papers:
Brandt, Friedemann; Barnich, Glenn; Henneaux, Marc (2000), "Local BRST cohomology in
gauge theories", Physics Reports, 338 (5): 439–569, arXiv:hep-th/0002245 (https://arxiv.org/
abs/hep-th/0002245), Bibcode:2000PhR...338..439B (https://ui.adsabs.harvard.edu/abs/200
0PhR...338..439B), doi:10.1016/S0370-1573(00)00049-1 (https://doi.org/10.1016%2FS0370
-1573%2800%2900049-1), ISSN 0370-1573 (https://www.worldcat.org/issn/0370-1573),
MR 1792979 (https://www.ams.org/mathscinet-getitem?mr=1792979), S2CID 119420167 (ht
tps://api.semanticscholar.org/CorpusID:119420167)
Becchi, C.; Rouet, A.; Stora, R. (1974). "The abelian Higgs Kibble model, unitarity of the Soperator". Physics Letters B. Elsevier BV. 52 (3): 344–346. doi:10.1016/03702693(74)90058-6 (https://doi.org/10.1016%2F0370-2693%2874%2990058-6). ISSN 03702693 (https://www.worldcat.org/issn/0370-2693).
Becchi, C.; Rouet, A.; Stora, R. (1975). "Renormalization of the abelian Higgs-Kibble model"
(http://www.numdam.org/item/RCP25_1975__22__A9_0/). Communications in
Mathematical Physics. Springer Science and Business Media LLC. 42 (2): 127–162.
doi:10.1007/bf01614158 (https://doi.org/10.1007%2Fbf01614158). ISSN 0010-3616 (https://
www.worldcat.org/issn/0010-3616). S2CID 120552882 (https://api.semanticscholar.org/Corp
usID:120552882).
Becchi, C; Rouet, A; Stora, R (1976). "Renormalization of gauge theories" (http://www.numd
am.org/item/RCP25_1975__22__A10_0/). Annals of Physics. Elsevier BV. 98 (2): 287–321.
doi:10.1016/0003-4916(76)90156-1 (https://doi.org/10.1016%2F0003-4916%2876%299015
6-1). ISSN 0003-4916 (https://www.worldcat.org/issn/0003-4916).
I.V. Tyutin, "Gauge Invariance in Field Theory and Statistical Physics in Operator
Formalism" (https://arxiv.org/abs/0812.0580), Lebedev Physics Institute preprint 39 (1975),
arXiv:0812.0580.
Kugo, Taichiro; Ojima, Izumi (1979). "Local Covariant Operator Formalism of Non-Abelian
Gauge Theories and Quark Confinement Problem" (https://doi.org/10.1143%2Fptps.66.1).
Progress of Theoretical Physics Supplement. Oxford University Press (OUP). 66: 1–130.
doi:10.1143/ptps.66.1 (https://doi.org/10.1143%2Fptps.66.1). ISSN 0375-9687 (https://www.
worldcat.org/issn/0375-9687).
A more accessible version of Kugo–Ojima is available online in a series of papers, starting
with: Kugo, T.; Ojima, I. (1978-12-01). "Manifestly Covariant Canonical Formulation of the
Yang-Mills Field Theories. I: -- General Formalism --" (https://doi.org/10.1143%2Fptp.60.186
9). Progress of Theoretical Physics. Oxford University Press (OUP). 60 (6): 1869–1889.
doi:10.1143/ptp.60.1869 (https://doi.org/10.1143%2Fptp.60.1869). ISSN 0033-068X (https://
www.worldcat.org/issn/0033-068X). This is probably the single best reference for BRST
quantization in quantum mechanical (as opposed to geometrical) language.
Much insight about the relationship between topological invariants and the BRST operator
may be found in: E. Witten, "Topological quantum field theory" (http://projecteuclid.org/Diens
t/UI/1.0/Summarize/euclid.cmp/1104161738), Commun. Math. Phys. 117, 3 (1988), pp. 353–
386
Alternate perspectives
BRST systems are briefly analyzed from an operator theory perspective in: S. S. Horuzhy
and A. V. Voronin, "Remarks on Mathematical Structure of BRST Theories" (http://projecteuc
lid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1104178989), Comm. Math. Phys. 123, 4 (1989)
pp. 677–685
A measure-theoretic perspective on the BRST method may be found in Carlo Becchi's 1996
lecture notes (https://arxiv.org/abs/hep-th/9607181).
External links
Brst cohomology on arxiv.org (https://arxiv.org/search/?query=brst+cohomology&searchtype
=all&source=header)
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