"BRST quantization and cohomology"의 두 판 사이의 차이
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| − | + | ==gauge fixing== | |
| 22번째 줄: | 22번째 줄: | ||
| − | + | ==ghost variables== | |
* [[Faddeev–Popov ghost fields|ghost fields]]<br> | * [[Faddeev–Popov ghost fields|ghost fields]]<br> | ||
| 32번째 줄: | 32번째 줄: | ||
| − | + | ==Faddeev-Ghost determinant== | |
* [http://hitoshi.berkeley.edu/230A/FPghosts.pdf Faddeev-Popov ghosts], Hitoshi Murayama<br> | * [http://hitoshi.berkeley.edu/230A/FPghosts.pdf Faddeev-Popov ghosts], Hitoshi Murayama<br> | ||
| 42번째 줄: | 42번째 줄: | ||
| − | + | ==path integral and ghost sector== | |
* <math>Z = \int\!\mathcal{D}X\,\mathcal{D}c \mathcal{D}b \mathcal{D}\bar{c} \mathcal{D}\bar{b} \,e^{-\int\left(\partial X \partial \bar{X} -b_{zz}\partial_{\bar{z}}c^{z}+b_{\bar{z}\bar{z}}\partial_{z}c^{\bar{z}}\right)}</math> | * <math>Z = \int\!\mathcal{D}X\,\mathcal{D}c \mathcal{D}b \mathcal{D}\bar{c} \mathcal{D}\bar{b} \,e^{-\int\left(\partial X \partial \bar{X} -b_{zz}\partial_{\bar{z}}c^{z}+b_{\bar{z}\bar{z}}\partial_{z}c^{\bar{z}}\right)}</math> | ||
| 61번째 줄: | 61번째 줄: | ||
| − | + | ==nilpotency of BRST operator== | |
* new, global symmetry (BRST)<br> | * new, global symmetry (BRST)<br> | ||
| 73번째 줄: | 73번째 줄: | ||
| − | + | ==construction of Hilbert space of states== | |
* BRST charge acts on a huge space<br> | * BRST charge acts on a huge space<br> | ||
| 85번째 줄: | 85번째 줄: | ||
| − | + | ==BRST cohomology== | |
* <math>\Lambda_{\infty}</math> semi-infinite form<br> | * <math>\Lambda_{\infty}</math> semi-infinite form<br> | ||
| 96번째 줄: | 96번째 줄: | ||
| − | + | ==applications== | |
* BRST approach to minimal models BRST approach to minimal models [http://dx.doi.org/10.1016/0550-3213%2889%2990568-3 http://dx.doi.org/10.1016/0550-3213(89)90568-3]<br> | * BRST approach to minimal models BRST approach to minimal models [http://dx.doi.org/10.1016/0550-3213%2889%2990568-3 http://dx.doi.org/10.1016/0550-3213(89)90568-3]<br> | ||
| 108번째 줄: | 108번째 줄: | ||
| − | + | ==related items== | |
* [[물리학과 cohomology]]<br> | * [[물리학과 cohomology]]<br> | ||
| 118번째 줄: | 118번째 줄: | ||
| − | + | ==books== | |
* Polchinski, vol. I. $3.1-3.4, 4.2-4.3 | * Polchinski, vol. I. $3.1-3.4, 4.2-4.3 | ||
| 127번째 줄: | 127번째 줄: | ||
| − | + | ==encyclopedia== | |
* http://en.wikipedia.org/wiki/BRST_quantization | * http://en.wikipedia.org/wiki/BRST_quantization | ||
| 141번째 줄: | 141번째 줄: | ||
| − | + | ==expositions== | |
* [http://www.math.sciences.univ-nantes.fr/%7Ewagemann/LAlecture.pdf Introduction to Lie algebra cohomology with a view towards BRST cohomology] ,Friedrich Wagemann, 2010-8<br> | * [http://www.math.sciences.univ-nantes.fr/%7Ewagemann/LAlecture.pdf Introduction to Lie algebra cohomology with a view towards BRST cohomology] ,Friedrich Wagemann, 2010-8<br> | ||
| 151번째 줄: | 151번째 줄: | ||
| − | + | ==articles== | |
* Igor B. Frenkel, Anton M. Zeitlin, Quantum Group as Semi-infinite Cohomology | * Igor B. Frenkel, Anton M. Zeitlin, Quantum Group as Semi-infinite Cohomology | ||
| 163번째 줄: | 163번째 줄: | ||
| − | + | ==blogs== | |
* [http://www.math.columbia.edu/%7Ewoit/notesonbrst.pdf http://www.math.columbia.edu/~woit/notesonbrst.pdf] | * [http://www.math.columbia.edu/%7Ewoit/notesonbrst.pdf http://www.math.columbia.edu/~woit/notesonbrst.pdf] | ||
| 184번째 줄: | 184번째 줄: | ||
| − | + | ==TeX == | |
2012년 10월 28일 (일) 16:03 판
introduction
- 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
gauge fixing
ghost variables
Faddeev-Ghost determinant
- Faddeev-Popov ghosts, Hitoshi Murayama
path integral and ghost sector
- \(Z = \int\!\mathcal{D}X\,\mathcal{D}c \mathcal{D}b \mathcal{D}\bar{c} \mathcal{D}\bar{b} \,e^{-\int\left(\partial X \partial \bar{X} -b_{zz}\partial_{\bar{z}}c^{z}+b_{\bar{z}\bar{z}}\partial_{z}c^{\bar{z}}\right)}\)
- \(e^{S_1(X)+S_2(b,c,\bar{b},\bar{c},\cdots,X)\)
- DX : matter and Db : ghost Dc : antighost
- bc system of \epsilon=+1 (in Faddeev–Popov ghost fields)
- \lambda=2
- c_{b,c}=-26
- [c]=-1,[b]=2
- global issues
- discrepancies in conformal gauge
- moduli spaces
- CKV
- path integral and moduli space of Riemann surfaces
nilpotency of BRST operator
- new, global symmetry (BRST)
- Q is fermionic
- Q_{BRST}^2=0
- http://bolvan.ph.utexas.edu/~vadim/Classes/2008f.homeworks/brst.pdf
- [1]http://www.nuclecu.unam.mx/~chryss/papers/brst_final.pdf
construction of Hilbert space of states
- BRST charge acts on a huge space
- Q.v =0 <=> physical condition
- if the total central charge is not 0 but c, Q_{BRST}^2=c
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 minimal models http://dx.doi.org/10.1016/0550-3213(89)90568-3
- 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 José Figueroa-O’Farrill 2006
- D'Hooker, E., Phong, D.H.: The geometry of string perturbation theory. Rev. Mod. Phys. 60,. 917-1065 (1988)
articles
- Igor B. Frenkel, Anton M. Zeitlin, Quantum Group as Semi-infinite Cohomology
- J.W., van Holten. 1990. “The BRST complex and the cohomology of compact lie algebras”. Nuclear Physics B 339 (1) (7월 23): 158-176. doi:10.1016/0550-3213(90)90537-N
- 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/10.1103/RevModPhys.60.917
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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