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| BRST formalism | |
|---|---|
| Name | BRST formalism |
| Field | Theoretical physics |
| Introduced | 1970s |
| Contributors | Becchi, Rouet, Stora, Tyutin |
BRST formalism is a method in theoretical physics for quantizing gauge-invariant systems that combines symmetry, cohomology, and ghost fields to enforce gauge conditions while preserving unitarity and locality. Developed in the 1970s by Becchi, Carlo, Rouet, Alain, Stora, Raymond and Tyutin, Igor, the formalism unifies ideas from canonical quantization, path integrals, and algebraic topology, drawing on tools used in studies by Dirac, Paul, Faddeev, Ludvig, Popov, Victor and later refined in work influenced by Witten, Edward, Weinberg, Steven, Schwinger, Julian and Feynman, Richard. It plays a central role in modern treatments of non-abelian gauge theories such as Yang–Mills theory, Quantum chromodynamics and in formulations related to String theory and Supersymmetry.
The BRST formalism arose amid developments in quantization by figures like Faddeev, Ludvig, Popov, Victor, Feynman, Richard, DeWitt, Bryce and Schwinger, Julian, providing an algebraic structure that encodes gauge symmetry via an odd nilpotent operator inspired by constructions in Cartan, Élie-style differential geometry and homological algebra used by Eilenberg, Samuel, Mac Lane, Saunders, Hochschild, Gerhard and Serre, Jean-Pierre. Influential researchers including Becchi, Carlo, Tyutin, Igor, Stora, Raymond, Rouet, Alain, Witten, Edward, Weinberg, Steven and Polchinski, Joseph extended the formalism to address issues in Quantum field theory, Renormalization group analysis, and perturbative treatments by teams working at institutions like CERN, Princeton University, Stanford University and Harvard University.
The mathematical foundations rely on graded vector spaces, differential graded algebras, and cohomology theories developed in the context of work by Cartan, Élie, de Rham, Georges, Hochschild, Gerhard, Chevalley, Claude and Eilenberg, Samuel. Structure constants and Lie algebraic inputs come from research on Lie groups and Lie algebras by Lie, Sophus and formalized by Cartan, Élie; the algebraic BRST operator parallels differentials used in studies by Noether, Emmy on conserved currents and constraints. Connections to algebraic topology and index theorems trace to Atiyah, Michael, Singer, Isadore, Bott, Raoul and Chern, Shiing-Shen, while categorical and homological techniques echo contributions by Grothendieck, Alexander, Verdier, Jean-Louis and Quillen, Daniel.
BRST transformations are generated by a nilpotent operator Q_B whose construction uses ghost fields introduced in the Faddeev–Popov approach by Faddeev, Ludvig and Popov, Victor. The BRST charge concept connects to canonical methods of Dirac, Paul and to the Hamiltonian treatments of constraints explored by Bergmann, Peter, Henneaux, Marc and Teitelboim, Claudio. In perturbative settings, practitioners like Becchi, Carlo, Rouet, Alain and Stora, Raymond implement Q_B in the operator formalism alongside procedures developed by Kugo, Taichiro and Ojima, Izumi for quartet mechanisms and ghost number grading, reflecting algebraic patterns studied by Lie, Sophus and Chevalley, Claude.
Physical states are identified with the cohomology of Q_B at ghost number zero, an idea leveraging cohomological principles advanced by Hochschild, Gerhard, Cartan, Élie and de Rham, Georges. This selection rule ensures equivalence classes under gauge transformations in contexts investigated by Weinberg, Steven and Peskin, Michael and underpins treatments of observables in analyses by Strominger, Andrew and Maldacena, Juan. The cohomological viewpoint interfaces with index-theory methods due to Atiyah, Michael and Singer, Isadore and with BRST extensions in topological field theories by Witten, Edward and Schwarz, Albert.
BRST methods are applied extensively in non-abelian gauge theories such as Yang–Mills theory, Quantum chromodynamics and electroweak models developed by Glashow, Sheldon, Weinberg, Steven and Salam, Abdus. The formalism informs perturbative quantization in the Standard Model contexts evolved at CERN, Fermilab and SLAC National Accelerator Laboratory, and it appears in constructive and axiomatic analyses pursued at Institute for Advanced Study and Mathematical Sciences Research Institute. BRST techniques also appear in studies of String theory by Polchinski, Joseph, in background field methods associated with DeWitt, Bryce and in supersymmetric gauge systems examined by Witten, Edward and Seiberg, Nathan.
In the path integral approach, BRST invariance replaces explicit gauge fixing by embedding gauge choices into BRST-exact terms, a practice rooted in the Faddeev–Popov procedure of Faddeev, Ludvig and Popov, Victor. Implementations of gauge-fixing functionals and ghost actions build on the work of DeWitt, Bryce, Feynman, Richard and Peskin, Michael and are central to computations at research centers like CERN and Brookhaven National Laboratory. Renormalizable gauge choices considered in treatments by ’t Hooft, Gerard and Veltman, Martinus preserve BRST invariance, facilitating loop calculations used in precision tests by collaborations including ATLAS Collaboration and CMS Collaboration.
Anomalies correspond to obstructions in BRST cohomology and were analyzed in influential work by Adler, Stephen, Bardeen, William, Bell, John and Jackiw, Roman. The cancellation and consistency conditions for anomalies connect to gauge and gravitational anomaly studies by Alvarez-Gaumé, Luis and Witten, Edward and to renormalization techniques pioneered by Bogoliubov, Nikolay, Parasyuk, Orest, Hepp, Klaus and Zimmermann, Wolfhart. BRST-preserving renormalization schemes are used in perturbative proofs elaborated by Weinberg, Steven, Collins, John and Zinn-Justin, Jean and implemented in lattice and continuum computations at institutions like CERN and Brookhaven National Laboratory.