"BRST quantization and cohomology"의 두 판 사이의 차이
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(피타고라스님이 이 페이지의 이름을 BRST quantization and cohomology로 바꾸었습니다.) |
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| 1번째 줄: | 1번째 줄: | ||
| − | <h5>introduction</h5> | + | <h5>introduction[http://empg.maths.ed.ac.uk/Activities/BRST/ ]</h5> |
| − | |||
* [[Gauge theory]] = principal G-bundle<br> | * [[Gauge theory]] = principal G-bundle<br> | ||
* We require a quantization of gauge theory<br> | * We require a quantization of gauge theory<br> | ||
* BRST quantization is one way to quantize the theory and is a part of path integral<br> | * BRST quantization is one way to quantize the theory and is a part of path integral<br> | ||
| − | * Gauge theory allows 'local symmetry' which should be ignored to be physical<br> | + | ** Gauge theory allows 'local symmetry' which should be ignored to be physical<br> |
| − | * this ignoring process leads to the cohomoloy theory.<br> | + | ** this ignoring process leads to the cohomoloy theory.<br> |
* BRST = quantization procedure of a classical system with constraints by introducing odd variables (“ghosts”)<br> | * BRST = quantization procedure of a classical system with constraints by introducing odd variables (“ghosts”)<br> | ||
| + | * re-packaging of Faddeev-Popov quantization<br> | ||
* the conditions D = 26 and α0 = 1 for the space-time dimension D and the zero-intercept α0 of leading trajectory are required by the nilpotency QB2 = 0 of the BRS charge<br> | * the conditions D = 26 and α0 = 1 for the space-time dimension D and the zero-intercept α0 of leading trajectory are required by the nilpotency QB2 = 0 of the BRS charge<br> | ||
| 14번째 줄: | 14번째 줄: | ||
| − | + | <h5 style="margin: 0px; line-height: 2em;">ghost variables</h5> | |
| − | * | + | |
| − | + | * [[Faddeev–Popov ghost fields|ghost fields]]<br> | |
| − | + | ||
| + | |||
| + | |||
| + | |||
| 25번째 줄: | 28번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;"> | + | <h5 style="margin: 0px; line-height: 2em;">nilpotency of BRST operator</h5> |
| − | * [[ | + | * [http://bolvan.ph.utexas.edu/%7Evadim/Classes/2008f.homeworks/brst.pdf http://bolvan.ph.utexas.edu/~vadim/Classes/2008f.homeworks/brst.pdf]<br> |
| + | * [http://www.nuclecu.unam.mx/%7Echryss/papers/brst_final.pdf http://www.nuclecu.unam.mx/~chryss/papers/brst_final.pdf]<br> | ||
| 35번째 줄: | 39번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;"> | + | <h5 style="margin: 0px; line-height: 2em;">BRST cohomology</h5> |
| + | |||
| + | * <math>\Lambda_{\infty}</math> semi-infinite form<br> | ||
| + | * <math>\mathfrak{g}</math> : <math>\mathbb{Z}</math>-graded Lie algebra<br> | ||
| + | * <math>\sigma</math> : anti-linear automorphism sending <math>\mathfrak{g}_{n}</math> to <math>\mathfrak{g}_{-n}</math><br> | ||
| + | * <math>H^2(\mathfrak{g})=0</math> (i.e. no non-trivial central extension)<br> | ||
| − | + | ||
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| 68번째 줄: | 76번째 줄: | ||
* <br> | * <br> | ||
| − | * | + | * Polchinski, vol. I. $3.1-3.4, 4.2-4.3 |
| + | * GSW, I. 3.1-3.2 | ||
2011년 10월 4일 (화) 07:48 판
introduction[1]
- Gauge theory = principal G-bundle
- We require a quantization of gauge theory
- BRST quantization is one way to quantize the theory and is a part of path integral
- Gauge theory allows 'local symmetry' which should be ignored to be physical
- this ignoring process leads to the cohomoloy theory.
- Gauge theory allows 'local symmetry' which should be ignored to be physical
- BRST = quantization procedure of a classical system with constraints by introducing odd variables (“ghosts”)
- re-packaging of Faddeev-Popov quantization
- the conditions D = 26 and α0 = 1 for the space-time dimension D and the zero-intercept α0 of leading trajectory are required by the nilpotency QB2 = 0 of the BRS charge
ghost variables
nilpotency of BRST operator
- http://bolvan.ph.utexas.edu/~vadim/Classes/2008f.homeworks/brst.pdf
- http://www.nuclecu.unam.mx/~chryss/papers/brst_final.pdf
BRST cohomology
- \(\Lambda_{\infty}\) semi-infinite form
- \(\mathfrak{g}\) \[\mathbb{Z}\]-graded Lie algebra
- \(\sigma\) : anti-linear automorphism sending \(\mathfrak{g}_{n}\) to \(\mathfrak{g}_{-n}\)
- \(H^2(\mathfrak{g})=0\) (i.e. no non-trivial central extension)
applications
- BRST approach to minimal models
- BRST approach to no-ghost theorem
- BRST approach to coset constructions
books
-
- Polchinski, vol. I. $3.1-3.4, 4.2-4.3
- GSW, I. 3.1-3.2
encyclopedia
- http://en.wikipedia.org/wiki/BRST_quantization
- http://www.scholarpedia.org/article/Becchi-Rouet-Stora-Tyutin_symmetry
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
expositions
- Introduction to Lie algebra cohomology with a view towards BRST cohomology ,Friedrich Wagemann, 2010-8
- PG minicourse: BRST cohomology (http://empg.maths.ed.ac.uk/Activities/BRST/Notes.pdf , very good introduction)
articles
- Igor B. Frenkel, Anton M. Zeitlin, Quantum Group as Semi-infinite Cohomology
- van Holten, J. W., The BRST complex and the cohomology of compact lie algebras
- D. Bernard and G. Felder, 1990, Fock representations and BRST cohomology inSL(2) current algebra
- BRST cohomology in classical mechanics
- Symplectic Reduction, BRS Cohomology, and Infinite-Dimensional Clifford algebras, B. Kostant, S. Sternberg, Ann. Physics 176 (1987) 49–113
- I. B. Frenkel,. H. Garland, and. G. J. Zuckerman, PNAS November 1, 1986 vol. 83 no. 22 8442-8446, Semi-infinite cohomology and string theory
- http://dx.doi.org/10.1007/BF02096498
blogs
- http://www.math.columbia.edu/~woit/notesonbrst.pdf
- http://www.math.columbia.edu/~woit/wordpress/?cat=12
- Notes on BRST I: Representation Theory and Quantum Mechanics
- Notes on BRST II: Lie Algebra Cohomology, Physicist’s Version
- Notes on BRST III: Lie Algebra Cohomology
- Notes on BRST IV: Lie Algebra Cohomology for Semi-simple Lie Algebras
- Notes on BRST V: Highest Weight Theory
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