In theoretical physics, the BRST formalism, or BRST quantization (where the BRST refers to the last names of Carlo Becchi, Alain Rouet, Raymond Stora and Igor Tyutin) denotes a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier quantum field theory (QFT) frameworks resembled "prescriptions" or "heuristics" more than proofs, especially in non-abelian QFT, where the use of "ghost fields" with superficially bizarre properties is almost unavoidable for technical reasons related to renormalization and anomaly cancellation.

The BRST global supersymmetry introduced in the mid-1970s was quickly understood to rationalize the introduction of these Faddeev–Popov ghosts and their exclusion from "physical" asymptotic states when performing QFT calculations. Crucially, this symmetry of the path integral is preserved in loop order, and thus prevents introduction of counterterms which might spoil renormalizability of gauge theories. Work by other authors a few years later related the BRST operator to the existence of a rigorous alternative to path integrals when quantizing a gauge theory.

Only in the late 1980s, when QFT was reformulated in fiber bundle language for application to problems in the topology of low-dimensional manifolds (topological quantum field theory), did it become apparent that the BRST "transformation" is fundamentally geometrical in character. In this light, "BRST quantization" becomes more than an alternate way to arrive at anomaly-cancelling ghosts. It is a different perspective on what the ghost fields represent, why the Faddeev–Popov method works, and how it is related to the use of Hamiltonian mechanics to construct a perturbative framework. The relationship between gauge invariance and "BRST invariance" forces the choice of a Hamiltonian system whose states are composed of "particles" according to the rules familiar from the canonical quantization formalism. This esoteric consistency condition therefore comes quite close to explaining how quanta and fermions arise in physics to begin with.

In certain cases, notably gravity and supergravity, BRST must be superseded by a more general formalism, the Batalin–Vilkovisky formalism.

Technical summary

BRST quantization is a differential geometric approach to performing consistent, anomaly-free perturbative calculations in a non-abelian gauge theory. The analytical form of the BRST "transformation" and its relevance to renormalization and anomaly cancellation were described by Carlo Maria Becchi, Alain Rouet, and Raymond Stora in a series of papers culminating in the 1976 "Renormalization of gauge theories". The equivalent transformation and many of its properties were independently discovered by Igor Viktorovich Tyutin. Its significance for rigorous canonical quantization of a Yang–Mills theory and its correct application to the Fock space of instantaneous field configurations were elucidated by Taichiro Kugo and Izumi Ojima. Later work by many authors, notably Thomas Schücker and Edward Witten, has clarified the geometric significance of the BRST operator and related fields and emphasized its importance to topological quantum field theory and string theory.

In the BRST approach, one selects a perturbation-friendly gauge fixing procedure for the action principle of a gauge theory using the differential geometry of the gauge bundle on which the field theory lives. One then quantizes the theory to obtain a Hamiltonian system in the interaction picture in such a way that the "unphysical" fields introduced by the gauge fixing procedure resolve gauge anomalies without appearing in the asymptotic states of the theory. The result is a set of Feynman rules for use in a Dyson series perturbative expansion of the S-matrix which guarantee that it is unitary and renormalizable at each loop order—in short, a coherent approximation technique for making physical predictions about the results of scattering experiments.

Gauge transformations in QFT

From a practical perspective, a quantum field theory consists of an action principle and a set of procedures for performing perturbative calculations. There are other kinds of "sanity checks" that can be performed on a quantum field theory to determine whether it fits qualitative phenomena such as quark confinement and asymptotic freedom. However, most of the predictive successes of quantum field theory, from quantum electrodynamics to the present day, have been quantified by matching S-matrix calculations against the results of scattering experiments.

In the early days of QFT, one would have had to say that the quantization and renormalization prescriptions were as much part of the model as the Lagrangian density, especially when they relied on the powerful but mathematically ill-defined path integral formalism. It quickly became clear that QED was almost "magical" in its relative tractability, and that most of the ways that one might imagine extending it would not produce rational calculations. However, one class of field theories remained promising: gauge theories, in which the objects in the theory represent equivalence classes of physically indistinguishable field configurations, any two of which are related by a gauge transformation. This generalizes the QED idea of a local change of phase to a more complicated Lie group.

QED itself is a gauge theory, as is general relativity, although the latter has proven resistant to quantization so far, for reasons related to renormalization. Another class of gauge theories with a non-Abelian gauge group, beginning with Yang–Mills theory, became amenable to quantization in the late 1960s and early 1970s, largely due to the work of Ludwig D. Faddeev, Victor Popov, Bryce DeWitt, and Gerardus 't Hooft. However, they remained very difficult to work with until the introduction of the BRST method. The BRST method provided the calculation techniques and renormalizability proofs needed to extract accurate results from both "unbroken" Yang–Mills theories and those in which the Higgs mechanism leads to spontaneous symmetry breaking. Representatives of these two types of Yang–Mills systems—quantum chromodynamics and electroweak theory—appear in the Standard Model of particle physics.

It has proven rather more difficult to prove the existence of non-Abelian quantum field theory in a rigorous sense than to obtain accurate predictions using semi-heuristic calculation schemes. This is because analyzing a quantum field theory requires two mathematically interlocked perspectives: a Lagrangian system based on the action functional, composed of fields with distinct values at each point in spacetime and local operators which act on them, and a Hamiltonian system in the Dirac picture, composed of states which characterize the entire system at a given time and field operators which act on them. What makes this so difficult in a gauge theory is that the objects of the theory are not really local fields on spacetime; they are right-invariant local fields on the principal gauge bundle, and different local sections through a portion of the gauge bundle, related by passive transformations, produce different Dirac pictures.

What is more, a description of the system as a whole in terms of a set of fields contains many redundant degrees of freedom; the distinct configurations of the theory are equivalence classes of field configurations, so that two descriptions which are related to one another by a gauge transformation are also really the same physical configuration. The "solutions" of a quantized gauge theory exist not in a straightforward space of fields with values at every point in spacetime but in a quotient space (or cohomology) whose elements are equivalence classes of field configurations. Hiding in the BRST formalism is a system for parameterizing the variations associated with all possible active gauge transformations and correctly accounting for their physical irrelevance during the conversion of a Lagrangian system to a Hamiltonian system.

Mathematical approach to BRST

This section only applies to classical gauge theories.

The Hamiltonian mechanics of a gauge theory is described by ${\displaystyle r}$ first class constraints ${\displaystyle \Phi _{i}}$ acting upon a symplectic space ${\displaystyle M}$. ${\displaystyle M_{0}}$ is the submanifold satisfying the first class constraints. The action of the gauge symmetry partitions ${\displaystyle M_{0}}$ into gauge orbits. The symplectic reduction is the quotient of ${\displaystyle M_{0}}$ by the gauge orbits.

According to algebraic geometry, the set of smooth functions over a space is a ring. The Koszul-Tate complex (the first class constraints aren't regular in general) describes the algebra associated with the symplectic reduction in terms of the algebra ${\displaystyle C^{\infty }(M)}$.

There are two antiderivations, ${\displaystyle \delta }$ and ${\displaystyle d}$ which anticommute with each other. The BRST antiderivation ${\displaystyle s}$ is given by ${\displaystyle \delta +d+\mathrm {more} }$. ${\displaystyle s}$ is also nilpotent. ${\displaystyle s^{2}=(\delta +d)^{2}=\delta ^{2}+d^{2}+(\delta d+d\delta )=0}$

Consider the supercommutative algebra generated by ${\displaystyle C^{\infty }(M)}$ and Grassman odd generators ${\displaystyle {\mathcal {P}}_{i}}$, i.e. the tensor product of a Grassman algebra and ${\displaystyle C^{\infty }(M)}$. There is a unique antiderivation ${\displaystyle \delta }$ satisfying ${\displaystyle \delta {\mathcal {P}}_{i}=-\Phi _{i}}$ and ${\displaystyle \delta f=0}$ for all ${\displaystyle f\in C^{\infty }(M)}$. The zeroth homology is given by ${\displaystyle C^{\infty }(M_{0})}$.

A longitudinal vector field on ${\displaystyle M_{0}}$ is a vector field over ${\displaystyle M_{0}}$ which is tangent everywhere to the gauge orbits. The Lie bracket of two longitudinal vector fields is itself another longitudinal vector field. Longitudinal ${\displaystyle p}$-forms are dual to the exterior algebra of ${\displaystyle p}$-vectors. ${\displaystyle d}$ is essentially the longitudinal exterior derivative defined by ${\displaystyle d\omega (V_{0},\ldots ,V_{k})=\sum _{i}(-1)^{i}d_{{}_{V_{i}}}(\omega (V_{0},\ldots ,{\widehat {V}}_{i},\ldots ,V_{k}))+\sum _{i<j}(-1)^{i+j}\omega ([V_{i},V_{j}],V_{0},\ldots ,{\widehat {V}}_{i},\ldots ,{\widehat {V}}_{j},\ldots ,V_{k})}$. The zeroth cohomology of the longitudinal exterior derivative is the algebra of gauge invariant functions.

The BRST construction applies when one has a Hamiltonian action of a compact, connected Lie group ${\displaystyle G}$ on a phase space ${\displaystyle M}$.^[1][2] Let ${\displaystyle {\mathfrak {g}}}$ be the Lie algebra of ${\displaystyle G}$ (via the Lie group–Lie algebra correspondence) and ${\displaystyle 0\in {\mathfrak {g}}^{*}}$ (the dual of ${\displaystyle {\mathfrak {g}})}$ a regular value of the momentum map ${\displaystyle \Phi :M\to {\mathfrak {g}}^{*}}$. Let ${\displaystyle M_{0}=\Phi ^{-1}(0)}$. Assume the ${\displaystyle G}$-action on ${\displaystyle M_{0}}$ is free and proper, and consider the space ${\displaystyle {\widetilde {M}}=M_{0}/G}$ of ${\displaystyle G}$-orbits on ${\displaystyle M_{0}}$, which is also known as a symplectic reduction quotient ${\displaystyle {\widetilde {M}}=M//G}$.

First, using the regular sequence of functions defining ${\displaystyle M_{0}}$ inside ${\displaystyle M}$, construct a Koszul complex

${\displaystyle \Lambda ^{\bullet }{\mathfrak {g}}\otimes C^{\infty }(M).}$

The differential, ${\displaystyle \delta }$, on this complex is an odd ${\displaystyle C^{\infty }(M)}$-linear derivation (differential algebra) of the graded ${\displaystyle C^{\infty }(M)}$-algebra ${\displaystyle \Lambda ^{\bullet }{\mathfrak {g}}\otimes C^{\infty }(M)}$. This odd derivation is defined by extending the Lie algebra homomorphism ${\displaystyle {\mathfrak {g}}\to C^{\infty }(M)}$ of the Hamiltonian action. The resulting Koszul complex is the Koszul complex of the ${\displaystyle S({\mathfrak {g}})}$-module ${\displaystyle C^{\infty }(M)}$, where ${\displaystyle S({\mathfrak {g}})}$ is the symmetric algebra of ${\displaystyle {\mathfrak {g}}}$, and the module structure comes from a ring homomorphism ${\displaystyle S({\mathfrak {g}})\to C^{\infty }(M)}$ induced by the Hamiltonian action ${\displaystyle {\mathfrak {g}}\to C^{\infty }(M)}$.

This Koszul complex is a resolution of the ${\displaystyle S({\mathfrak {g}})}$-module ${\displaystyle C^{\infty }(M_{0})}$, that is,

${\displaystyle H^{j}(\Lambda ^{\bullet }{\mathfrak {g}}\otimes C^{\infty }(M),\delta )={\begin{cases}C^{\infty }(M_{0})&j=0\\0&j\neq 0\end{cases}}}$

Then, consider the Chevalley–Eilenberg complex for the Koszul complex ${\displaystyle \Lambda ^{\bullet }{\mathfrak {g}}\otimes C^{\infty }(M)}$ considered as a differential graded module over the Lie algebra ${\displaystyle {\mathfrak {g}}}$:

${\displaystyle K^{\bullet ,\bullet }=C^{\bullet }\left({\mathfrak {g}},\Lambda ^{\bullet }{\mathfrak {g}}\otimes C^{\infty }(M)\right)=\Lambda ^{\bullet }{\mathfrak {g}}^{*}\otimes \Lambda ^{\bullet }{\mathfrak {g}}\otimes C^{\infty }(M).}$

The "horizontal" differential ${\displaystyle d:K^{i,\bullet }\to K^{i+1,\bullet }}$ is defined on the coefficients

${\displaystyle \Lambda ^{\bullet }{\mathfrak {g}}\otimes C^{\infty }(M)}$

by the action of ${\displaystyle {\mathfrak {g}}}$ and on ${\displaystyle \Lambda ^{\bullet }{\mathfrak {g}}^{*}}$ as the exterior derivative of right-invariant differential forms on the Lie group ${\displaystyle G}$, whose Lie algebra is ${\displaystyle {\mathfrak {g}}}$.

Let Tot(K) be a complex such that

${\displaystyle \operatorname {Tot} (K)^{n}=\bigoplus \nolimits _{i-j=n}K^{i,j}}$

with a differential D = d + δ. The cohomology groups of (Tot(K), D) are computed using a spectral sequence associated to the double complex ${\displaystyle (K^{\bullet ,\bullet },d,\delta )}$.

The first term of the spectral sequence computes the cohomology of the "vertical" differential ${\displaystyle \delta }$:

${\displaystyle E_{1}^{i,j}=H^{j}(K^{i,\bullet },\delta )=\Lambda ^{i}{\mathfrak {g}}^{*}\otimes C^{\infty }(M_{0})}$, if j = 0 and zero otherwise.

The first term of the spectral sequence may be interpreted as the complex of vertical differential forms

${\displaystyle (\Omega ^{\bullet }{\operatorname {vert} }(M_{0}),d_{\operatorname {vert} })}$

for the fiber bundle ${\displaystyle M_{0}\to {\widetilde {M}}}$.

The second term of the spectral sequence computes the cohomology of the "horizontal" differential ${\displaystyle d}$ on ${\displaystyle E_{1}^{\bullet ,\bullet }}$:

${\displaystyle E_{2}^{i,j}\cong H^{i}(E_{1}^{\bullet ,j},d)=C^{\infty }(M_{0})^{g}=C^{\infty }({\widetilde {M}})}$, if ${\displaystyle i=j=0}$ and zero otherwise.

The spectral sequence collapses at the second term, so that ${\displaystyle E_{\infty }^{i,j}=E_{2}^{i,j}}$, which is concentrated in degree zero.

Therefore,

${\displaystyle H^{p}(\operatorname {Tot} (K),D)=C^{\infty }(M_{0})^{g}=C^{\infty }({\widetilde {M}})}$, if p = 0 and 0 otherwise.

Introduction to / Summary of Mathematical Approach to BRST

BRST quantization is a sophisticated method used in theoretical physics, particularly in gauge theories which are a type of field theory. These theories are fundamental in understanding forces and particles in quantum mechanics. This section outlines the mathematical intricacies involved in the BRST quantization approach in classical gauge theories. By understanding these concepts, one can gain deeper insights into the fascinating world of quantum mechanics and gauge theories.

1.Hamiltonian Mechanics in Gauge Theories:

- Basics: In gauge theories, the Hamiltonian mechanics, a framework to describe the dynamics of a system, is constrained by certain conditions, termed “first-class constraints.”

- Symplectic Space: The system is described within a mathematical structure called symplectic space. A subset of this space, satisfying the constraints, forms what’s known as gauge orbits, which are essentially paths defined by the gauge symmetry (a kind of symmetry in physics).

2. Koszul-Tate Complex:

- Function Ring: The functions over this space form a ring (an algebraic structure), and the Koszul-Tate complex is used to describe the algebra associated with this space.

- Antiderivations: Two operations, δ and d, which have properties of reversing the process of derivation, are used. The BRST antiderivation, a crucial operation in this approach, is a combination of these two operations and some additional elements. It’s characterized by its nilpotence, meaning applying it twice results in zero.

3. Supercommutative Algebra:

- Construction: This involves creating an algebra that combines the smooth functions on the space and elements called Grassmann odd generators through a process called tensor product.

- Homology: The zeroth homology of this algebra, which is a measure of the ‘holes’ in the space, is given by the smooth functions on the constrained subset of the symplectic space.

4. Longitudinal Vector Fields:

- Definition: These are vector fields that are tangent to gauge orbits and are important in understanding the behavior of the gauge theories.

- Properties: The operation d, the longitudinal exterior derivative, describes how these fields change, and their zeroth cohomology is the algebra of gauge-invariant functions, which are functions unaffected by gauge transformations.

5. BRST Construction with Lie Groups:

- Application: This approach is used when a Hamiltonian action (a type of dynamic action in physics) of a certain kind of group (compact and connected Lie group) acts on a phase space.

- Symplectic Reduction: The quotient space formed by the gauge orbits in this setting is known as the symplectic reduction quotient.

6. Koszul and Chevalley-Eilenberg Complexes:

- Formation: Using these mathematical structures, the differential (an operator that generalizes the concept of differentiation) δ is applied to study the properties of the system.

- Differential Graded Module: The Chevalley-Eilenberg complex is considered over the Lie algebra (a type of algebraic structure) of the group, which helps in understanding the deeper geometric properties of the gauge theories.

7. Spectral Sequence and Cohomology:

- Analysis: Using a method known as spectral sequence associated with these complexes, the cohomology groups are computed. These groups provide crucial information about the structure and properties of the space in question, particularly in terms of gauge invariance.

The BRST operator and asymptotic Fock space

Two important remarks about the BRST operator are due. First, instead of working with the gauge group G one can use only the action of the gauge algebra ${\displaystyle {\mathfrak {g}}}$ on the fields (functions on the phase space).

Second, the variation of any "BRST exact form" s_BX with respect to a local gauge transformation dλ is

${\displaystyle \left[i_{\delta \lambda },s_{B}\right]s_{B}X=i_{\delta \lambda }(s_{B}s_{B}X)+s_{B}\left(i_{\delta \lambda }(s_{B}X)\right)=s_{B}\left(i_{\delta \lambda }(s_{B}X)\right),}$

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 s_BX = 0. As we shall see, this implies that there is a related operator Q_B on the state space for which ${\displaystyle [Q_{B},{\mathcal {H}}]=0}$—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 ${\displaystyle Q_{B}|\Psi _{i}\rangle =0}$ into a later configuration with ${\displaystyle Q_{B}|\Psi _{f}\rangle \neq 0}$ (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 Q_B 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 Q_B, while exact states are those obtainable by applying Q_B to some arbitrary field configuration.)

We can also suppress states that lie inside the image of Q_B 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 ${\displaystyle {\mathcal {H}}_{0}}$, 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 s_B or inside the image of s_B) and vertices that couple "unphysical" particles to one another.

Gauge bundles and the vertical ideal

In order to do the BRST method justice, we must switch from the "algebra-valued fields on Minkowski space" picture typical of quantum field theory texts (and of the above exposition) to the language of fiber bundles, in which there are two quite different ways to look at a gauge transformation: as a change of local section (also known in general relativity as a passive transformation) or as the pullback of the field configuration along a vertical diffeomorphism of the principal bundle. It is the latter sort of gauge transformation that enters into the BRST method. Unlike a passive transformation, it is well-defined globally on a principal bundle with any structure group over an arbitrary manifold. (However, for concreteness and relevance to conventional QFT, this article will stick to the case of a principal gauge bundle with compact fiber over 4-dimensional Minkowski space.)

A principal gauge bundle P over a 4-manifold M is locally isomorphic to U × F, where U ⊂ R⁴ and the fiber F is isomorphic to a Lie group G, the gauge group of the field theory (this is an isomorphism of manifold structures, not of group structures; there is no special surface in P corresponding to 1 in G, so it is more proper to say that the fiber F is a G-torsor). Thus, the (physical) principal gauge bundle is related to the (mathematical) principal G-bundle but has more structure. Its most basic property as a fiber bundle is the "projection to the base space" π : P → M, which defines the "vertical" directions on P (those lying within the fiber π⁻¹(p) over each point p in M). As a gauge bundle it has a left action of G on P which respects the fiber structure, and as a principal bundle it also has a right action of G on P which also respects the fiber structure and commutes with the left action.

The left action of the structure group G on P corresponds to a mere change of coordinate system on an individual fiber. The (global) right action R_g : P → P for a fixed g in G corresponds to an actual automorphism of each fiber and hence to a map of P to itself. In order for P to qualify as a principal G-bundle, the global right action of each g in G must be an automorphism with respect to the manifold structure of P with a smooth dependence on g—i. e., a diffeomorphism P × G → P.

The existence of the global right action of the structure group picks out a special class of right invariant geometric objects on P—those which do not change when they are pulled back along R_g for all values of g in G. The most important right invariant objects on a principal bundle are the right invariant vector fields, which form an ideal ${\displaystyle {\mathfrak {E}}}$ of the Lie algebra of infinitesimal diffeomorphisms on P. Those vector fields on P which are both right invariant and vertical form an ideal ${\displaystyle V{\mathfrak {E}}}$ of ${\displaystyle {\mathfrak {E}}}$, which has a relationship to the entire bundle P analogous to that of the Lie algebra ${\displaystyle {\mathfrak {g}}}$ of the gauge group G to the individual G-torsor fiber F.

The "field theory" of interest is defined in terms of a set of "fields" (smooth maps into various vector spaces) defined on a principal gauge bundle P. Different fields carry different representations of the gauge group G, and perhaps of other symmetry groups of the manifold such as the Poincaré group. One may define the space Pl of local polynomials in these fields and their derivatives. The fundamental Lagrangian density of one's theory is presumed to lie in the subspace Pl₀ of polynomials which are real-valued and invariant under any unbroken non-gauge symmetry groups. It is also presumed to be invariant not only under the left action (passive coordinate transformations) and the global right action of the gauge group but also under local gauge transformations—pullback along the infinitesimal diffeomorphism associated with an arbitrary choice of right-invariant vertical vector field ${\displaystyle \epsilon \in V{\mathfrak {E}}}$.

Identifying local gauge transformations with a particular subspace of vector fields on the manifold P equips us with a better framework for dealing with infinite-dimensional infinitesimals: differential geometry and the exterior calculus. The change in a scalar field under pullback along an infinitesimal automorphism is captured in the Lie derivative, and the notion of retaining only the term linear in the scale of the vector field is implemented by separating it into the inner derivative and the exterior derivative. (In this context, "forms" and the exterior calculus refer exclusively to degrees of freedom which are dual to vector fields on the gauge bundle, not to degrees of freedom expressed in (Greek) tensor indices on the base manifold or (Roman) matrix indices on the gauge algebra.)

The Lie derivative on a manifold is a globally well-defined operation in a way that the partial derivative is not. The proper generalization of Clairaut's theorem to the non-trivial manifold structure of P is given by the Lie bracket of vector fields and the nilpotence of the exterior derivative. And we obtain an essential tool for computation: the generalized Stokes theorem, which allows us to integrate by parts and drop the surface term as long as the integrand drops off rapidly enough in directions where there is an open boundary. (This is not a trivial assumption, but can be dealt with by renormalization techniques such as dimensional regularization as long as the surface term can be made gauge invariant.)

BRST formalism

In theoretical physics, the BRST formalism is a method of implementing first class constraints. The letters BRST stand for Becchi, Rouet, Stora, and (independently) Tyutin who discovered this formalism. It is a sophisticated method to deal with quantum physical theories with gauge invariance. For example, the BRST methods are often applied to gauge theory and quantized general relativity.

See also

• Batalin–Vilkovisky formalism
• Quantum chromodynamics

References

External links

• Brst cohomology on arxiv.org