BRST quantization
Encyclopedia
In theoretical physics
, BRST quantization (where the BRST refers to Becchi, Rouet, Stora and Tyutin) is a relatively rigorous mathematical approach to quantizing
a field theory
with a gauge symmetry. Quantization
rules in earlier 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 supersymmetry
was introduced in the mid-1970s and was quickly understood to justify the introduction of these Faddeev–Popov ghosts and their exclusion from "physical" asymptotic states when performing QFT calculations. 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
, 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.
-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 Kugo Taichiro and Ojima Izumi. 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.
where pure operators are graded by integral ghost numbers and we have a BRST cohomology
.
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 an active 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.
goes back to the Lorenz gauge approach to electromagnetism, which suppresses most of the excess degrees of freedom in the four-potential 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 transform.
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 connexion
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 three-dimensional horizontal spacelike cross section of
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 multi-particle 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
; 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.
. 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 SU(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
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.
on a phase space
. Let 
be the Lie algebra of 
and 
a regular value of the moment map
. Let 
. Assume the 
-action on 
is free and proper, and consider the space 
of 
-orbits on 
, which is also known as a Symplectic Reduction quotient 
.
First, using the regular sequence of functions defining
inside 
, construct a Koszul complex
. The differential, 
, on this complex is an odd 
-linear derivation of the graded 
-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 
, 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.,
, if 
and zero otherwise.
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 
, whose Lie algebra is 
.
Let
be a complex such that 
with a differential 
. The cohomology groups of 
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
:
, if 
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
on 
:
, if 
and zero otherwise.
The spectral sequence collapses at the second term, so that
, which is concentrated in degree zero.
Therefore,
, if p = 0 and 0 otherwise.
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"
with respect to a local gauge transformation 
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
. As we shall see, this implies that there is a related operator 
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
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 
, while exact states are those obtainable by applying 
to some arbitrary field configuration.)
We can also suppress states that lie inside the image of
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 Lorentz-invariant 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
or inside the image
of
) and vertices that couple "unphysical" particles to one another.
.
The Faddeev–Popov ghost field
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 
, gauge bosons 
, and the ghost 
itself) which carry a non-trivial representation of the gauge group. The BRST transformation with respect to 
is therefore:
Here we have omitted the details of the matter sector
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 
. 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 
are entirely dictated by the relationship 
. (The new fields are all Hermitian in Kugo–Ojima conventions, but the parameter 
is an anti-Hermitian "anti-commuting 
-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
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 
must carry the same representation of 
(up to a phase) as 
, as well as the representation of 
dual to its trivial representation on 
—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
. 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 
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 
and the scalar field 
, 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 
—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 BRST-exact fields. This last statement is the key to "BRST quantization", as opposed to mere "BRST symmetry" or "BRST transformation".
s, 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; this is important in several approaches to a Theory of Everything
. (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
over a 4-manifold 
is locally isomorphic to 
, where the fiber
is isomorphic to a Lie group
, the gauge group of the field theory. (This is an isomorphism of manifold structures, not of group structures; there is no special surface in 
corresponding to 
, so it is more proper to say that the fiber 
is a 
-torsor.) Its most basic property as a fiber bundle
is the "projection to the base space"
, which defines the "vertical" directions on 
(those lying within the fiber 
over each point 
). As a gauge bundle it has a left action
of
on 
which respects the fiber structure, and as a principal bundle
it also has a right action
of
on 
which also respects the fiber structure and commutes with the left action.
The left action of the structure group
on 
corresponds to a mere change of coordinate system
on an individual fiber. The (global) right action
of a (fixed) 
corresponds to an actual automorphism
of each fiber and hence to a map of
to itself. In order for 
to qualify as a principal 
-bundle, the global right action of each 
must be an automorphism with respect to the manifold structure of 
with a smooth dependence on 
—i. e., a diffeomorphism from 
to 
.
The existence of the global right action of the structure group picks out a special class of right invariant geometric objects on
—those which do not change when they are pulled back along 
for all values of 
. 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
. Those vector fields on 
which are both right invariant and vertical form an ideal 
of 
, which has a relationship to the entire bundle 
analogous to that of the Lie algebra
of the gauge group 
to the individual 
-torsor fiber 
.
We suppose that 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
. Different fields carry different representations
of the gauge group
, and perhaps of other symmetry group
s of the manifold such as the Poincaré group
. One may define the space
of local polynomials in these fields and their derivatives. The fundamental Lagrangian density of one's theory is presumed to lie in the subspace 
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
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
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.)
, the BRST formalism is a method of implementing first class constraint
s. 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
.
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×R-graded
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
, 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 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.
(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
.
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 wheareas 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
, NOT the commutator
.
Q is an antiderivation.
The BRST Lagrangian density
While the Lagrangian density isn't BRST invariant, its integral over all of spacetime, the action is.
The operator
is defined as
where
are the Faddeev–Popov ghosts and antighosts (fields with a negative ghost number), respectively, 
are the infinitesimal generators of the Lie group
, and
are its structure constants.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, BRST quantization (where the BRST refers to Becchi, Rouet, Stora and Tyutin) is a relatively rigorous mathematical approach to quantizing
Quantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
a field theory
Field theory (mathematics)
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
with a gauge symmetry. Quantization
Quantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
rules in earlier 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
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
and anomaly cancellation. The BRST supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
was introduced in the mid-1970s and was quickly understood to justify the introduction of these Faddeev–Popov ghosts and their exclusion from "physical" asymptotic states when performing QFT calculations. Work by other authors a few years later related the BRST operator to the existence of a rigorous alternative to path integrals
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
when quantizing a gauge theory.
Only in the late 1980s, when QFT was reformulated in fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
language for application to problems in the topology of low-dimensional manifolds
Donaldson theory
Donaldson theory is the study of smooth 4-manifolds using gauge theory. It was started by Simon Donaldson who proved Donaldson's theorem restricting the possible quadratic forms on the second cohomology group of a compact simply connected 4-manifold....
, 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
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
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
Canonical quantization
In physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...
formalism. This esoteric consistency condition therefore comes quite close to explaining how quanta
Quantum
In physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...
and fermions arise in physics to begin with.
Technical summary
BRST quantization (or the BRST formalism) is a differential geometric approach to performing consistent, anomalyAnomaly (physics)
In quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics an anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking...
-free perturbative calculations in a non-abelian gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
. The analytical form of the BRST "transformation" and its relevance to renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
and anomaly cancellation were described by Carlo Maria Becchi
Carlo Becchi
Carlo Maria Becchi is an Italian theoretical physicist.Becchi studied at the University of Genoa, where he received his university degree in physics in 1962. He is since 1976 full professor for theoretical physics at the University of Genoa. Twice he was chairman of the physics faculty there...
, 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
Igor Tyutin
Igor Viktorovich Tyutin is a Russian theoretical physicist, who works on quantum field theory.Tyutin is a professor at the Lebedev Institute in Moscow...
. Its significance for rigorous canonical quantization
Canonical quantization
In physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...
of a Yang–Mills theory and its correct application to the Fock space
Fock space
The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...
of instantaneous field configurations were elucidated by Kugo Taichiro and Ojima Izumi. Later work by many authors, notably Thomas Schücker and Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
, has clarified the geometric significance of the BRST operator and related fields and emphasized its importance to topological quantum field theory
Topological quantum field theory
A topological quantum field theory is a quantum field theory which computes topological invariants....
and string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
.
In the BRST approach, one selects a perturbation-friendly gauge fixing
Gauge fixing
In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field...
procedure for the action principle of a gauge theory using the differential geometry of the gauge bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
on which the field theory lives. One then quantizes
Quantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
the theory to obtain a Hamiltonian system
Hamiltonian system
In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
in the interaction picture
Interaction picture
In quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...
in such a way that the "unphysical" fields introduced by the gauge fixing procedure resolve gauge anomalies
Gauge anomaly
In theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e...
without appearing in the asymptotic states of the theory. The result is a set of Feynman rules for use in a Dyson series
Dyson series
In scattering theory, the Dyson series, formulated by British-born American physicist Freeman Dyson, is a perturbative series, and each term is represented by Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics at the second order the difference from...
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 manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
where pure operators are graded by integral ghost numbers and we have a BRST cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
.
Gauge transformations in QFT
From a practical perspective, a quantum field theoryQuantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
consists of an action principle and a set of procedures for performing perturbative calculations
Perturbation theory (quantum mechanics)
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...
. 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
Asymptotic freedom
In physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become arbitrarily weak at energy scales that become arbitrarily large, or, equivalently, at length scales that become arbitrarily small .Asymptotic freedom is a feature of quantum...
. However, most of the predictive successes of quantum field theory, from quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
to the present day, have been quantified by matching S-matrix calculations against the results of scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
experiments.
In the early days of QFT, one would have to have said that the quantization
Quantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
and renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
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
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
. 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
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
.
QED itself is a gauge theory, as is general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, 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
Victor Popov
Victor Nikolaevich Popov was a Russian theoretical physicist known for his contribution to the quantization of non-abelian gauge fields. His work with Ludvig Faddeev on that subject introduced the fundamental objects now known as Faddeev–Popov ghosts....
, Bryce DeWitt
Bryce DeWitt
Bryce Seligman DeWitt was a theoretical physicist renowned for advancing gravity and field theories.-Biography:...
, and Gerardus 't Hooft
Gerardus 't Hooft
Gerardus 't Hooft is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G...
. 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
Higgs mechanism
In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
leads to spontaneous symmetry breaking
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
. Representatives of these two types of Yang–Mills systems—quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
and electroweak theory—appear in the Standard Model
Standard Model
The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
of particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
.
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
Lagrangian system
In mathematics, a Lagrangian system is a pair of a smoothfiber bundle Y\to X and a Lagrangian density L which yields the Euler-Lagrange differential operator acting on sections of Y\to X.In classical mechanics, many dynamical systems are...
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
Hamiltonian system
In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
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 an active 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
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
(or cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
) 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 fixingGauge fixing
In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field...
goes back to the Lorenz gauge approach to electromagnetism, which suppresses most of the excess degrees of freedom in the four-potential 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
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of the objects in a theory at the Lagrangian stage, before passing over to Hamiltonian mechanics
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
via the Legendre transform.
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 quantizationCanonical quantization
In physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...
. 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 connexion
Connexion
Connexion or Connexions may refer to:* Connexionalism, a system of ecclesiastical polity* Connexion by Boeing, an in-flight online connectivity service* PhyQuest Connexion, a risk-reduction reporting system...
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
Dyson series
In scattering theory, the Dyson series, formulated by British-born American physicist Freeman Dyson, is a perturbative series, and each term is represented by Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics at the second order the difference from...
.
For perturbative purposes, we gather the configuration of all the fields of our theory on an entire three-dimensional horizontal spacelike cross section of
into one object (a Fock stateFock state
A Fock state , in quantum mechanics, is any element of a Fock space with a well-defined number of particles . These states are named after the Soviet physicist, V. A. Fock.-Definition:...
), and then describe the "evolution" of this state over time using the interaction picture
Interaction picture
In quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...
. The Fock space
Fock space
The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...
is spanned by the multi-particle eigenstates of the "unperturbed" or "non-interaction" portion
of the HamiltonianHamiltonian system
In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
. 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 energyEnergy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
(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 constantCoupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
; 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
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
; 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
Gauge anomaly
In theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetry of a quantum field theory; i.e...
. 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 QEDQuantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
, 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 SU(2) of Yang–Mills and electroweak theory and the SU(3) of quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
. 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
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 quantizationCanonical quantization
In physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...
. 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
Functional determinant
In mathematics, if S is a linear operator mapping a function space V to itself, it is sometimes possible to define an infinite-dimensional generalization of the determinant. The corresponding quantity det is called the functional determinant of S.There are several formulas for the functional...
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 on a phase spacePhase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
. Let be the Lie algebra of and a regular value of the moment mapMoment map
In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum...
. Let . Assume the -action on is free and proper, and consider the space of -orbits on , which is also known as a Symplectic Reduction quotient .First, using the regular sequence of functions defining
inside , construct a Koszul complexKoszul complex
In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul...
. The differential, , on this complex is an odd -linear derivation of the graded -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 , where is the symmetric algebra of , and the module structure comes from a ring homomorphism induced by the hamiltonian action .This Koszul complex
Koszul complex
In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul...
is a resolution of the
-module , i.e.,, if and zero otherwise.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 , whose Lie algebra is .Let
be a complex such that with a differential . The cohomology groups of 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
:, if 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
on :, if and zero otherwise.The spectral sequence collapses at the second term, so that
, which is concentrated in degree zero.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 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"
with respect to a local gauge transformation 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
. As we shall see, this implies that there is a related operator on the state space for which —i. e., the BRST operator on Fock states is a conserved charge of the Hamiltonian systemHamiltonian system
In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....
. 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
Image
An image is an artifact, for example a two-dimensional picture, that has a similar appearance to some subject—usually a physical object or a person.-Characteristics:...
(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
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 , while exact states are those obtainable by applying to some arbitrary field configuration.)We can also suppress states that lie inside the image of
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
Fock space
The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...
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
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
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
Krein space
In mathematics, in the field of functional analysis, an indefinite inner product spaceis an infinite-dimensional complex vector space K equipped with both an indefinite inner product\langle \cdot,\,\cdot \rangle \,...
for the BRST-closed intermediate Fock states, with the time reversal operator playing the role of the "fundamental symmetry" relating the Lorentz-invariant 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
Interaction picture
In quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...
. 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 theoremAdiabatic theorem
The adiabatic theorem is an important concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock , can be stated as follows:...
by "turning on" the interaction Hamiltonian (the gauge coupling). The expansion of the Dyson series
Dyson series
In scattering theory, the Dyson series, formulated by British-born American physicist Freeman Dyson, is a perturbative series, and each term is represented by Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics at the second order the difference from...
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
or inside the imageImage
An image is an artifact, for example a two-dimensional picture, that has a similar appearance to some subject—usually a physical object or a person.-Characteristics:...
of
) 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.The Faddeev–Popov ghost field
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 , gauge bosons , and the ghost itself) which carry a non-trivial representation of the gauge group. The BRST transformation with respect to is therefore:Here we have omitted the details of the matter sector
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 . 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 are entirely dictated by the relationship . (The new fields are all Hermitian in Kugo–Ojima conventions, but the parameter is an anti-Hermitian "anti-commuting -numberC-number
The term c-number is an old nomenclature used by Paul Dirac which refers to real and complex numbers. It is used to distinguish from operators in quantum mechanics....
". 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
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 must carry the same representation of (up to a phase) as , as well as the representation of dual to its trivial representation on —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
. 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 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 and the scalar field , 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
Lorentz scalar
In physics, a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. A Lorentz scalar may be generated from multiplication of vectors or tensors...
for the sake of Poincaré invariance in
. However, its (anti-)commutation law relative to —i. e., its quantization prescription, which ignores the spin-statistics theoremSpin-statistics theorem
In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...
by giving Fermi–Dirac statistics to a spin-0 particle—will be given by the requirement that the inner product on our Fock space
Fock space
The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...
of asymptotic states be singular
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
along directions corresponding to the raising and lowering operators of some combination of non-BRST-closed and BRST-exact 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 bundleFiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
s, 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
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
as a passive transformation
Active and passive transformation
In the physical sciences, an active transformation is one which actually changes the physical position of a system, and makes sense even in the absence of a coordinate system whereas a passive transformation is a change in the coordinate description of the physical system . The distinction between...
) or as the pullback of the field configuration along a vertical diffeomorphism of the principal bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
. 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; this is important in several approaches to a Theory of Everything
Theory of everything
A theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle....
. (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
over a 4-manifold is locally isomorphic to , where the fiberFiber
Fiber is a class of materials that are continuous filaments or are in discrete elongated pieces, similar to lengths of thread.They are very important in the biology of both plants and animals, for holding tissues together....
is isomorphic to a Lie groupLie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
, the gauge group of the field theory. (This is an isomorphism of manifold structures, not of group structures; there is no special surface in corresponding to , so it is more proper to say that the fiber is a -torsor.) Its most basic property as a fiber bundleFiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
is the "projection to the base space"
, which defines the "vertical" directions on (those lying within the fiber over each point ). As a gauge bundle it has a left actionGroup action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of
on which respects the fiber structure, and as a principal bundlePrincipal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
it also has a right action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of
on which also respects the fiber structure and commutes with the left action.The left action of the structure group
on corresponds to a mere change of coordinate systemCoordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
on an individual fiber. The (global) right action
of a (fixed) corresponds to an actual automorphismAutomorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
of each fiber and hence to a map of
to itself. In order for to qualify as a principal -bundle, the global right action of each must be an automorphism with respect to the manifold structure of with a smooth dependence on —i. e., a diffeomorphism from to .The existence of the global right action of the structure group picks out a special class of right invariant geometric objects on
—those which do not change when they are pulled back along for all values of . The most important right invariant objects on a principal bundle are the right invariant vector fields, which form an idealIdeal (set theory)
In the mathematical field of set theory, an ideal is a collection of sets that are considered to be "small" or "negligible". Every subset of an element of the ideal must also be in the ideal , and the union of any two elements of the ideal must also be in the ideal.More formally, given a set X, an...
of the Lie algebraLie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
of infinitesimal diffeomorphisms on
. Those vector fields on which are both right invariant and vertical form an ideal of , which has a relationship to the entire bundle analogous to that of the Lie algebraLie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
of the gauge group to the individual -torsor fiber .We suppose that 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
. Different fields carry different representationsRepresentations
Representations is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journals was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It covers topics including literary, historical, and...
of the gauge group
, and perhaps of other symmetry groupSymmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
s of the manifold such as the Poincaré group
Poincaré group
In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
. One may define the space
of local polynomials in these fields and their derivatives. The fundamental Lagrangian density of one's theory is presumed to lie in the subspace 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
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 derivativeLie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
, 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
Inner derivative
In mathematics, the interior product is a degree −1 antiderivation on the exterior algebra of differential forms on a smooth manifold. It is defined to be the contraction of a differential form with a vector field...
and the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
. (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
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
is not. The proper generalization of Clairaut's theorem
Clairaut's theorem
Clairaut's theorem, published in 1743 by Alexis Claude Clairaut in his Théorie de la figure de la terre, tirée des principes de l'hydrostatique, synthesized physical and geodetic evidence that the Earth is an oblate rotational ellipsoid. It is a general mathematical law applying to spheroids of...
to the non-trivial manifold structure of
is given by the Lie bracket of vector fieldsLie bracket of vector fields
In the mathematical field of differential topology, the Lie bracket of vector fields, Jacobi–Lie bracket, or commutator of vector fields is a bilinear differential operator which assigns, to any two vector fields X and Y on a smooth manifold M, a third vector field denoted [X, Y]...
and the nilpotence of the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
. 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
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
techniques such as dimensional regularization
Dimensional regularization
In theoretical physics, dimensional regularization is a method introduced by Giambiagi and Bollini for regularizing integrals in the evaluation of Feynman diagrams; in other words, assigning values to them that are meromorphic functions of an auxiliary complex parameter d, called the...
as long as the surface term can be made gauge invariant.)
BRST formalism
In theoretical physicsTheoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, the BRST formalism is a method of implementing first class constraint
First class constraint
In a constrained Hamiltonian system, a dynamical quantity is called a first class constraint if its Poisson bracket with all the other constraints vanishes on the constraint surface .-Poisson brackets:In Hamiltonian mechanics, consider a symplectic manifold M with a smooth Hamiltonian over...
s. The letters BRST stand for Becchi
Carlo Becchi
Carlo Maria Becchi is an Italian theoretical physicist.Becchi studied at the University of Genoa, where he received his university degree in physics in 1962. He is since 1976 full professor for theoretical physics at the University of Genoa. Twice he was chairman of the physics faculty there...
, 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
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
.
Quantum version
The space of states is not a Hilbert space (see below). This vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
is both Z2-graded
Graded vector space
In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.-N-graded vector spaces:...
and R-graded
Graded vector space
In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.-N-graded vector spaces:...
. If you wish, you may think of it as a Z2×R-graded vector space
Graded vector space
In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.-N-graded vector spaces:...
. 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×R-graded
Graded algebra
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
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
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
.
The physical states are identified as elements of cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
of the operator
, i.e. as vectors in Ker Qn+1/Im Qn. The BRST theory is in fact linked to the standard resolution in Lie algebra cohomologyLie algebra cohomology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was defined by in order to give an algebraic construction of the cohomology of the underlying topological spaces of compact Lie groups...
.
Recall that the space of states is Z2-graded. If A is a pure graded operator, then the BRST transformation maps A to
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
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
associated with constraints that gauge-fix the symmetry.
The BRST is a supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
. It generates the Lie superalgebra
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...
with a zero-dimensional even part and a one dimensional odd part spanned by Q.
Lie superalgebra
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...
(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
Positive definiteness
Positive definiteness is a property of the following mathematical objects:* Positive-definite bilinear form* Positive-definite matrix* Positive-definite function* Positive-definite kernel* Positive-definite function on a group...
! This means it is not a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
.
For more general flows which can't be described by first class constraints, see Batalin–Vilkovisky formalism.
Example
For the special case of gauge theoriesGauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
(of the usual kind described by sections
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
of a principal G-bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
) with a quantum connection form
Connection form
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms....
A, a BRST charge (sometimes also a BRS charge) is an operator usually denoted
.Let the
-valued gauge fixingGauge fixing
In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field...
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 wheareas 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
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
.
where [,]L is the Lie bracket
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
, NOT the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
.
Q is an antiderivation.
The BRST Lagrangian density
While the Lagrangian density isn't BRST invariant, its integral over all of spacetime, the action is.
The operator
is defined aswhere
are the Faddeev–Popov ghosts and antighosts (fields with a negative ghost number), respectively, are the infinitesimal generators of the Lie groupLie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
, and
are its structure constants.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
Primary literature
Original BRST papers:- C. Becchi, A. Rouet and R. Stora, Phys. Lett. B52 (1974) 344.
- C. Becchi, A. Rouet and R. Stora, Commun. Math. Phys. 42 (1975) 127.
- C. Becchi, A. Rouet and R. Stora, "Renormalization of gauge theories", Ann. Phys. 98, 2 (1976) pp. 287–321.
- I.V. Tyutin, "Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism", Lebedev Physics Institute preprint 39 (1975), arXiv:0812.0580.
- The commonly cited Kugo–Ojima paper: T. Kugo and I. Ojima, "Local Covariant Operator Formalism of Non-Abelian Gauge Theories and Quark Confinement Problem", Suppl. Progr. Theor. Phys. 66 (1979) p. 14
- A more accessible version of Kugo–Ojima is available online in a series of papers, starting with: T. Kugo, I. Ojima, "Manifestly Covariant Canonical Formulation of the Yang–Mills Field Theories. I", Progr. Theor. Phys. 60, 6 (1978) pp. 1869–1889. 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", 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", 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.