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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 [[Quantization (physics)|quantizing]] a [[quantum field theory|field theory]] with a [[gauge symmetry]]. [[Quantization (physics)|Quantization]] rules in earlier [[quantum field theory]] (QFT) frameworks resembled "prescriptions" or "heuristics" more than proofs, especially in [[non-abelian group|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 ghost]]s 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 [[Renormalization#Renormalizability|renormalizability]] of gauge theories. Work by other authors {{By whom|section=
Only in the late 1980s, when QFT was reformulated in [[fiber bundle]] language for application to problems in the [[Donaldson theory|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 [[quantum|quanta]] and [[fermions]] arise in physics to begin with.
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