- BRST quantization
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 agauge symmetry . Quantization rules in earlier QFT frameworks resembled "prescriptions" or "heuristics" more than proofs, especially innon-abelian QFT, where the use of "ghost fields " with superficially bizarre properties is almost unavoidable for technical reasons related torenormalization andanomaly cancellation . The BRSTsupersymmetry was introduced in the mid-1970s and was quickly understood to justify the introduction of theseFaddeev-Popov ghost s 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 ofHamiltonian mechanics to construct a perturbative framework. The relationship betweengauge invariance and "BRST invariance" forces the choice of a Hamiltonian system whose states are composed of "particles" according to the rules familiar from thecanonical quantization formalism. This esoteric consistency condition therefore comes quite close to explaining howquanta andfermions arise in physics to begin with.**Technical summary**BRST quantization (or the

BRST formalism ) is a differential geometric approach to performing consistent, anomaly-free perturbative calculations in anon-abelian gauge theory . The analytical form of the BRST "transformation" and its relevance torenormalization andanomaly cancellation were described byCarlo Maria Becchi ,Alain Rouet , andRaymond 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 byI. V. Tyutin . Its significance for rigorouscanonical quantization of aYang-Mills theory and its correct application to theFock space of instantaneous field configurations were elucidated byKUGO Taichiro andOJIMA Izumi . Later work by many authors, notablyThomas Schücker andEdward Witten , has clarified the geometric significance of the BRST operator and related fields and emphasized its importance totopological quantum field theory andstring theory .In the BRST approach, one selects a perturbation-friendly

gauge fixing procedure for theaction principle of a gauge theory using thedifferential geometry of the gauge bundle on which the field theory lives. One then quantizes the theory to obtain aHamiltonian system in theinteraction 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 ofFeynman rules for use in aDyson series perturbative expansion of theS-matrix which guarantee that it is unitary andrenormalizable 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 anaction principle and a set of procedures for performingperturbative 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 asquark confinement andasymptotic freedom . However, most of the predictive successes of quantum field theory, fromquantum electrodynamics to the present day, have been quantified by matchingS-matrix calculations against the results ofscattering 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 theLagrangian 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 representequivalence classes of physically indistinguishable field configurations, any two of which are related by agauge transformation . This generalizes the QED idea of alocal change of phase to a more complicatedLie 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 anon-Abelian gauge group, beginning withYang-Mills theory , became amenable to quantization in the late 1960s and early 1970s, largely due to the work ofLudwig D. Faddeev ,Victor N. Popov ,Bryce DeWitt , andGerardus '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 theHiggs mechanism leads tospontaneous symmetry breaking . Representatives of these two types of Yang-Mills systems—quantum chromodynamics andelectroweak theory —appear in theStandard Model ofparticle 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 theaction functional , composed of "fields" with distinct values at each point in spacetime andlocal operators which act on them, and aHamiltonian system in theDirac picture , composed of "states" which characterize the entire system at a given time andfield 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 areright-invariant local fields on theprincipal gauge bundle , and differentlocal 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 aquotient space (orcohomology ) 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 theLagrangian density of the action principle which "break the gauge symmetry" to suppress these "unphysical" degrees of freedom. The idea ofgauge fixing goes back to theLorenz gauge approach to electromagnetism, which suppresses most of the excess degrees of freedom in thefour-potential while retainingmanifest Lorentz invariance . The Lorenz gauge is a great simplification relative to Maxwell's field-strength approach toclassical 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 toHamiltonian mechanics via theLegendre 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 $i\; hbar$. 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, ahorizontal lift to the bundle space, and a spacelike surface "normal" (in theMinkowski metric ) to the unit time vector field at each point on the base manifold, it is dependent both on theconnexion and the choice ofLorentz 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 theDyson 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 $P$ into one object (a

Fock state ), and then describe the "evolution" of this state over time using theinteraction picture . TheFock space is spanned by themulti-particle eigenstates of the "unperturbed" or "non-interaction" portion $mathcal\{H\}\_0$ of the Hamiltonian $mathcal\{H\}$. Hence the instantaneous description of any Fock state is a complex-amplitude-weighted sum of eigenstates of $mathcal\{H\}\_0$. 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 itsenergy (the correspondingeigenvalue 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 inhigh-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 $mathcal\{H\}\_0$ and the true Hamiltonian $mathcal\{H\}$, in the form of a power series in thecoupling constant $g$; it is the principal tool for making quantitative predictions from a quantum field theory.To use the Dyson series to calculate anything, one needs more than a gauge-invariant Lagrangian density; one also needs the quantization and gauge fixing prescriptions that enter into the

Feynman rules of the theory. The Dyson series produces infinite integrals of various kinds when applied to the Hamiltonian of a particular QFT. This is partly because all usable quantum field theories to date must be consideredeffective field theories , describing only interactions on a certain range of energy scales that we can experimentally probe and therefore vulnerable toultraviolet divergences . These are tolerable as long as they can be handled via standard techniques ofrenormalization ; 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 uncancelledgauge anomaly . There is a deep relationship between renormalizability and gauge invariance, which is easily lost in the course of attempts to obtain tractable Feynman rules by fixing the gauge.**Pre-BRST approaches to gauge fixing**The traditional gauge fixing prescriptions of

continuum electrodynamics select a unique representative from each gauge-transformation-related equivalence class using aconstraint equation such as theLorenz gauge $partial^mu\; A\_mu\; =\; 0$. This sort of prescription can be applied to anAbelian gauge theory such as QED, although it results in some difficulty in explaining why theWard identities of the classical theory carry over to the quantum theory—in other words, whyFeynman diagrams containing internallongitudinally polarized virtual photons do not contribute toS-matrix calculations. This approach also does not generalize well tonon-Abelian gauge groups such as the SU(2) of Yang-Mills andelectroweak theory and the SU(3) ofquantum chromodynamics . It suffers fromGribov 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 theFeynman 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 $R\_xi$ gauge. It reduces in the case of an Abelian U(1) gauge to the same set ofFeynman rules that one obtains in the method ofcanonical quantization . But there is an important difference: the broken gauge freedom appears in thefunctional 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 unremovableanomalies .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 thefunctional determinant of theJacobian of this embedding, and the properties of the ghost field are dictated by the exponent desired on the determinant in order to correct thefunctional measure on the remaining "physical" perturbation axes.**The BRST operator and asymptotic Fock space**In the BRST formalism, one sets aside the gauge "group" action on the fields and focuses on an approximation linear in its gauge algebra, which is known in

representation theory as theWard operator $W(deltalambda)$). The Ward operator on each field may be identified (up to a sign convention) with theLie derivative along thevertical vector field associated with the parameter $deltalambda$ of the Ward operator. The**BRST operator**$s\_B$ on fields resembles theexterior derivative on the gauge bundle, or rather to its restriction to a reduced space ofalternating forms which are defined only on vertical vector fields. On 0-forms $X\; in\; \{Pl\}\_0$ the Ward and BRST operators are related (up to a phase convention introduced by Kugo and Ojima, whose notation we will follow in the treatment of state vectors below) by $W(deltalambda)\; X\; =\; deltalambda\; s\_B\; X$.Like the exterior derivative, the BRST operator is

nilpotent of degree 2, i. e., $(s\_B)^2\; =\; 0$. The variation of any "BRSTexact form " $s\_B\; X$ with respect to a local gauge transformation is $[i\_\{deltalambda\},,\; s\_B]\; s\_B\; X\; =\; i\_\{deltalambda\}\; (s\_B\; s\_B\; X)\; +\; s\_B\; (i\_\{deltalambda\}\; (s\_B\; X))\; =\; s\_B\; (i\_\{deltalambda\}\; (s\_B\; X))$, which is itself an exact form. This implies that its integral over each (compact) fiber of $P$ is zero.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 $s\_B\; L\; =\; 0$. As we shall see, this implies that there is a related operator $Q\_B$ on the state space for which $[Q\_B,,\; mathcal\{H\}]\; =\; 0$—i. e., the BRST operator on Fock states is a

conserved charge of theHamiltonian system . This implies that thetime evolution operator in a Dyson series calculation will not evolve a field configuration obeying $Q\_B\; |Psi\_i\; angle\; =\; 0$ into a later configuration with $Q\_B\; |Psi\_f\; angle\; eq\; 0$ (or vice versa).Another way of looking at the nilpotence of the BRST operator is to say that its

image (the space of BRSTexact forms ) lies entirely within its kernel (the space of BRSTclosed 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 withladder operators for the energy-momentum eigenconfigurations (particles) of each field, complete with appropriate (anti-)commutation rules, as well as apositive semi-definite inner product . We require that theinner 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 $mathcal\{H\}\_0$, gradually transformed into states of the full Hamiltonian in accordance with theadiabatic theorem by "turning on" theinteraction Hamiltonian (the gauge coupling). The expansion of theDyson series in terms ofFeynman 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 theimage of $s\_B$) and vertices that couple "unphysical" particles to one another.**The Kugo-Ojima answer to unitarity questions**T. Kugo and I. Ojima are commonly credited with the discovery of the principal QCD

color confinement criterion. Their role in obtaining a correct version of the BRST formalism in the Lagrangian framework seems to be less widely appreciated. It is enlightening to inspect their variant of the BRST transformation, which emphasizes the hermitian properties of the newly introduced fields, before proceeding from an entirely geometrical angle. The gauge fixed Lagrangian density is below; the two terms in parentheses form the coupling between the gauge and ghost sectors, and the final term becomes a Gaussian weighting for the functional measure on the auxiliary field $B$.:$mathcal\{L\}\; =\; mathcal\{L\}\_\; extrm\{matter\}(psi,,A\_mu^a)\; -\; frac\{1\}\{4\}\; F^a\_\{mu\; u\}\; F^\{a,,mu\; u\}\; -\; (i\; (partial^mu\; ar\{c\}^a)\; D\_mu^\{ab\}\; c^b\; +\; (partial^mu\; B^a)\; A\_mu^a)\; +\; frac\{alpha\_0\}\{2\}\; B^a\; B^a$

The

Faddeev-Popov ghost field $c$ is unique among the new fields of our gauge-fixed theory in having a geometrical meaning beyond the formal requirements of the BRST procedure. It is a version of theMaurer-Cartan form on $Vmathfrak\{E\}$, which relates each right-invariant vertical vector field $deltalambda\; in\; Vmathfrak\{E\}$ to its representation (up to a phase) as a $mathfrak\{g\}$-valued field. This field must enter into the formulas for infinitesimal gauge transformations on objects (such as fermions $psi$, gauge bosons $A\_mu$, and the ghost $c$ itself) which carry a non-trivial representation of the gauge group. The BRST transformation with respect to $deltalambda$ is therefore::$delta\; psi\; =\; W(deltalambda)\; psi$:$delta\; A\_mu\; =\; deltalambda\; D\_mu\; c$:$delta\; c\; =\; -\; deltalambda\; frac\{g\}\{2\}\; [c,,c]$:$delta\; ar\{c\}\; =\; i\; deltalambda\; B$:$delta\; B\; =\; 0$

Here we have omitted the details of the matter sector $psi$ 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 $delta\; A\_mu$. The properties of the other fields we have added are fundamentally analytical rather than geometric. The bias we have introduced towards connexions with $partial^mu\; A\_mu\; =\; 0$ is gauge-dependent and has no particular geometrical significance. The anti-ghost $ar\{c\}$ is nothing but a Lagrange multiplier for the gauge fixing term, and the properties of the scalar field $B$ are entirely dictated by the relationship $delta\; ar\{c\}\; =\; i\; deltalambda\; B$. (The new fields are all Hermitian in Kugo-Ojima conventions, but the parameter $deltalambda$ is an anti-Hermitian "anti-commuting $c$-number". This results in some unnecessary awkwardness with regard to phases and passing infinitesimal parameters through operators; this will be resolved with a change of conventions in the geometric treatment below.)

We already know, from the relation of the BRST operator to the exterior derivative and the Faddeev-Popov ghost to the Maurer-Cartan form, that the ghost $c$ corresponds (up to a phase) to a $mathfrak\{g\}$-valued 1-form on $Vmathfrak\{E\}$. In order for integration of a term like $-i\; (partial^mu\; ar\{c\})\; D\_mu\; c$ to be meaningful, the anti-ghost $ar\{c\}$ must carry representations of these two Lie algebras—the vertical ideal $Vmathfrak\{E\}$ and the gauge algebra $mathfrak\{g\}$—dual to those carried by the ghost. In geometric terms, $ar\{c\}$ must be fiberwise dual to $mathfrak\{g\}$ and one rank short of being a

top form on $Vmathfrak\{E\}$. Likewise, theauxiliary field $B$ must carry the same representation of $mathfrak\{g\}$ (up to a phase) as $ar\{c\}$, as well as the representation of $Vmathfrak\{E\}$ dual to its trivial representation on $A\_mu$—i. e., B is a fiberwise $mathfrak\{g\}$-dual top form on $Vmathfrak\{E\}$.Let us focus briefly on the one-particle states of the theory, in the adiabatically decoupled limit $g\; ightarrow\; 0$. There are two kinds of quanta in the Fock space of the gauge-fixed Hamiltonian that we expect to lie entirely outside the kernel of the BRST operator: those of the Faddeev-Popov anti-ghost $ar\{c\}$ and the forward polarized gauge boson. (This is because no combination of fields containing $ar\{c\}$ is annihilated by $s\_B$ and we have added to the Lagrangian a gauge breaking term that is equal up to a divergence to $s\_B\; (ar\{c\}\; (i\; partial^mu\; A\_mu\; -\; frac\{alpha\_0\}\{2\}\; s\_B\; ar\{c\}))$.) Likewise, there are two kinds of quanta that will lie entirely in the image of the BRST operator: those of the Faddeev-Popov ghost $c$ and the scalar field $B$, which is "eaten" by completing the square in the functional integral to become the backward polarized gauge boson. These are the four types of "unphysical" quanta which will not appear in the asymptotic states of a perturbative calculation—"if" we get our quantization rules right.

The anti-ghost is taken to be a

Lorentz scalar for the sake of Poincaré invariance in $-i\; (partial^mu\; ar\{c\})\; D\_mu\; c$. However, its (anti-)commutation law relative to $c$—i. e., its quantization prescription, which ignores thespin-statistics theorem by givingFermi-Dirac statistics to a spin-0 particle—will be given by the requirement that theinner product on ourFock 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".(Needs to be completed in the language of BRST cohomology, with reference to the Kugo-Ojima treatment of asymptotic Fock space.)

**Gauge bundles and the vertical ideal**In order to do the BRST method justice, we must switch from the "algebra-valued fields on Minkowski space" picture typical of quantum field theory texts (and of the above exposition) to the language of

fiber bundles , in which there are two quite different ways to look at a gauge transformation: as a change oflocal section (also known ingeneral relativity as a passive transformation) or as the pullback of the field configuration along avertical diffeomorphism of theprincipal 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 aTheory 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 $P$ over a 4-manifold $M$ is locally isomorphic to $(U\; subset\; mathbb\{R\}^4)\; imes\; F$, where thefiber $F$ is isomorphic to aLie group $G$, thegauge 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 .) Its most basic property as afiber bundle is the "projection to the base space" $pi:,\; P\; ightarrow\; M$, which defines the "vertical" directions on $P$ (those lying within the fiber $pi^\{-1\}(p)$ over each point $p\; in\; M$). As agauge bundle it has a left action of $G$ on $P$ which respects the fiber structure, and as aprincipal 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 ofcoordinate system on an individual fiber. The (global) right action $R\_g:,\; P\; ightarrow\; P$ of a (fixed) $g\; in\; G$ corresponds to an actualautomorphism 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 from $P\; imes\; G$ to $P$.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 arepulled back along $R\_g$ for all values of $g\; in\; G$. The most important right invariant objects on a principal bundle are the right invariantvector fields , which form an ideal $mathfrak\{E\}$ of theLie algebra ofinfinitesimal diffeomorphisms on $P$. Those vector fields on $P$ which are both right invariant and vertical form an ideal $Vmathfrak\{E\}$ of $mathfrak\{E\}$, which has a relationship to the entire bundle $P$ analogous to that of theLie algebra $mathfrak\{g\}$ of thegauge group $G$ to the individual $G$-torsor fiber $F$.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 $P$. Different fields carry different

representations of the gauge group $G$, and perhaps of othersymmetry groups of the manifold such as thePoincaré group . One may define the space $Pl$ oflocal polynomials in these fields and their derivatives. The fundamental Lagrangian density of one's theory is presumed to lie in the subspace $\{Pl\}\_0$ 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 underlocal gauge transformations —pullback along theinfinitesimal diffeomorphism associated with an arbitrary choice of right invariant vertical vector field $epsilon\; in\; Vmathfrak\{E\}$.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 theexterior calculus . The change in a scalar field under pullback along an infinitesimal automorphism is captured in theLie derivative , and the notion of retaining only the term linear in the scale of the vector field is implemented by separating it into theinner derivative and theexterior 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 ofClairaut's theorem to the non-trivial manifold structure of $P$ is given by theLie bracket of vector fields and thenilpotence of theexterior derivative . And we obtain an essential tool for computation: thegeneralized 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 byrenormalization techniques such asdimensional regularization as long as the surface term can be made gauge invariant.)**ee also***

Quantum chromodynamics **References****Textbook treatments:**Chapter 16 of Peskin & Schroeder (ISBN 0-201-50397-2 or ISBN 0-201-50934-2) applies the "BRST symmetry" to reason about anomaly cancellation in the Faddeev-Popov Lagrangian. This is a good start for QFT non-experts, although the connections to geometry are omitted and the treatment of asymptotic Fock space is only a sketch.

Chapter 12 of M. Göckeler and T. Schücker (ISBN 0-521-37821-4 or ISBN 0-521-32960-4) discusses the relationship between the BRST formalism and the geometry of gauge bundles. It is substantially similar to Schücker's 1987 paper:

^{ [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1104116716] }**Primary literature:**The commonly cited BRS paper: C. Becchi, A. Rouet, R. Stora, "Renormalization of gauge theories", Ann. Phys. 98, 2 (1976) pp. 287-321.

^{ [http://citeseer.ist.psu.edu/context/626475/0] }The commonly cited Kugo-Ojima paper: T. Kugo, I. Ojima, "Local Covariant Operator Formalism of Non-Abelian Gauge Theories and Quark Confinement Problem", Suppl. Progr. Theor. Phys. 66 (1979) p. 14

^{ [http://citeseer.ist.psu.edu/context/359616/0] }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.

^{ [http://ptp.ipap.jp/link?PTP/60/1869/] }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", Comm. Math. Phys. 117, 3 (1988), pp. 353–386

^{ [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1104161738] }**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

^{ [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1104178989] }A measure-theoretic perspective on the BRST method may be found in Carlo Becchi's 1996 lecture notes:

^{ [http://arxiv.org/abs/hep-th/9607181] }**External links**

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