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| − | ==introduction | + | ==introduction== |
| − | * [[Gauge theory|gauge theory]] = principal G-bundle | + | * [[Gauge theory|gauge theory]] = principal G-bundle |
| − | * we require a quantization of gauge theory | + | * we require a quantization of gauge theory |
| − | * BRST quantization is one way to quantize the theory and is a part of path integral | + | * 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 | + | ** gauge theory allows 'local symmetry' which should be ignored to be physical |
| − | ** this ignoring process leads to the cohomoloy theory. | + | ** this ignoring process leads to the cohomoloy theory. |
| − | * BRST = quantization procedure of a classical system with constraints by introducing odd variables (“ghosts”) | + | * BRST = quantization procedure of a classical system with constraints by introducing odd variables (“ghosts”) |
| − | * re-packaging of Faddeev-Popov quantization | + | * re-packaging of Faddeev-Popov quantization |
| − | * the conditions D = 26 and | + | * the conditions <math>D = 26</math> and <math>\alpha_0=1</math> for the space-time dimension <math>D</math> and the zero-intercept <math>\alpha_0</math> of leading trajectory are required by the nilpotency <math>Q_B^2 = 0</math> of the BRS charge |
| − | + | ==gauge fixing== | |
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| − | + | ==ghost variables== | |
| − | + | * [[Faddeev–Popov ghost fields|ghost fields]] | |
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| − | + | ==Faddeev-Ghost determinant== | |
| − | + | * [http://hitoshi.berkeley.edu/230A/FPghosts.pdf Faddeev-Popov ghosts], Hitoshi Murayama | |
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| − | + | ==path integral and ghost sector== | |
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* <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> | ||
| − | * <math>e^{S_1(X)+S_2(b,c,\bar{b},\bar{c},\cdots,X)</math> | + | * <math>e^{S_1(X)+S_2(b,c,\bar{b},\bar{c},\cdots,X)}</math> |
* DX : matter and Db : ghost Dc : antighost | * DX : matter and Db : ghost Dc : antighost | ||
* bc system of \epsilon=+1 (in [[Faddeev–Popov ghost fields]]) | * bc system of \epsilon=+1 (in [[Faddeev–Popov ghost fields]]) | ||
| 51번째 줄: | 47번째 줄: | ||
* c_{b,c}=-26 | * c_{b,c}=-26 | ||
* [c]=-1,[b]=2 | * [c]=-1,[b]=2 | ||
| − | * global issues | + | * global issues |
** discrepancies in conformal gauge | ** discrepancies in conformal gauge | ||
** moduli spaces | ** moduli spaces | ||
** CKV | ** CKV | ||
| − | * [[path integral and moduli space of Riemann surfaces]] | + | * [[path integral and moduli space of Riemann surfaces]] |
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| − | + | ==nilpotency of BRST operator== | |
| − | + | * new, global symmetry (BRST) | |
| + | * Q is fermionic | ||
| + | * Q_{BRST}^2=0 | ||
| + | * [http://bolvan.ph.utexas.edu/%7Evadim/Classes/2008f.homeworks/brst.pdf http://bolvan.ph.utexas.edu/~vadim/Classes/2008f.homeworks/brst.pdf] | ||
| + | * [http://www.nuclecu.unam.mx/%7Echryss/papers/brst_final.pdf ][http://www.nuclecu.unam.mx/%7Echryss/papers/brst_final.pdf http://www.nuclecu.unam.mx/~chryss/papers/brst_final.pdf] | ||
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| − | + | ==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 | ||
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| − | + | ==BRST cohomology== | |
| − | + | * <math>\Lambda_{\infty}</math> semi-infinite form | |
| + | * <math>\mathfrak{g}</math> : <math>\mathbb{Z}</math>-graded Lie algebra | ||
| + | * <math>\sigma</math> : anti-linear automorphism sending <math>\mathfrak{g}_{n}</math> to <math>\mathfrak{g}_{-n}</math> | ||
| + | * <math>H^2(\mathfrak{g})=0</math> (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%2889%2990568-3 http://dx.doi.org/10.1016/0550-3213(89)90568-3] | + | * 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] |
| − | * BRST approach to no-ghost theorem | + | * BRST approach to no-ghost theorem |
| − | * BRST approach to coset constructions | + | * BRST approach to coset constructions |
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| − | + | ==related items== | |
| − | * [[물리학과 cohomology]] | + | * [[물리학과 cohomology]] |
| − | * [[homological algebra|Homological algebra]] | + | * [[homological algebra|Homological algebra]] |
| − | * [[Lie algebra cohomology]] | + | * [[Lie algebra cohomology]] |
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| − | + | ==books== | |
| − | * Polchinski, vol. I. | + | * Polchinski, vol. I. <math>3.1-3.4, 4.2-4.3 |
* GSW, I. 3.1-3.2 | * GSW, I. 3.1-3.2 | ||
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| − | + | ==encyclopedia== | |
* http://en.wikipedia.org/wiki/BRST_quantization | * http://en.wikipedia.org/wiki/BRST_quantization | ||
* http://www.scholarpedia.org/article/Becchi-Rouet-Stora-Tyutin_symmetry | * http://www.scholarpedia.org/article/Becchi-Rouet-Stora-Tyutin_symmetry | ||
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| − | * [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 | + | ==expositions== |
| − | * [http://empg.maths.ed.ac.uk/Activities/BRST/ PG minicourse: BRST cohomology] (http://empg.maths.ed.ac.uk/Activities/BRST/Notes. | + | * http://www.mathcs.emory.edu/~rudolf/BRST.pdf |
| − | * D'Hooker, E., Phong, D.H.: [http://dx.doi.org/10.1103/RevModPhys.60.917 The geometry of string perturbation theory]. Rev. Mod. Phys. 60,. 917-1065 (1988) | + | * [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 |
| + | * [http://empg.maths.ed.ac.uk/Activities/BRST/ 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.: [http://dx.doi.org/10.1103/RevModPhys.60.917 The geometry of string perturbation theory]. Rev. Mod. Phys. 60,. 917-1065 (1988) | ||
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| − | + | ==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 | ||
| − | * J.W., van Holten. 1990. “The BRST complex and the cohomology of compact lie algebras”. <em>Nuclear Physics B</em> 339 (1) (7월 23): 158-176. doi:[http://dx.doi.org/10.1016/0550-3213%2890%2990537-N 10.1016/0550-3213(90)90537-N] | + | * J.W., van Holten. 1990. “The BRST complex and the cohomology of compact lie algebras”. <em>Nuclear Physics B</em> 339 (1) (7월 23): 158-176. doi:[http://dx.doi.org/10.1016/0550-3213%2890%2990537-N 10.1016/0550-3213(90)90537-N] |
* D. Bernard and G. Felder, 1990, [http://dx.doi.org/10.1007/BF02096498 Fock representations and BRST cohomology inSL(2) current algebra] | * D. Bernard and G. Felder, 1990, [http://dx.doi.org/10.1007/BF02096498 Fock representations and BRST cohomology inSL(2) current algebra] | ||
* [http://dx.doi.org/10.1007/BF01466770 BRST cohomology in classical mechanics] | * [http://dx.doi.org/10.1007/BF01466770 BRST cohomology in classical mechanics] | ||
| 161번째 줄: | 147번째 줄: | ||
* http://dx.doi.org/10.1103/RevModPhys.60.917 | * http://dx.doi.org/10.1103/RevModPhys.60.917 | ||
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| − | + | ==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] | ||
| 173번째 줄: | 159번째 줄: | ||
* [http://www.math.columbia.edu/%7Ewoit/wordpress/?p=1245 Notes on BRST V: Highest Weight Theory] | * [http://www.math.columbia.edu/%7Ewoit/wordpress/?p=1245 Notes on BRST V: Highest Weight Theory] | ||
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| + | [[분류:physics]] | ||
| + | [[분류:math and physics]] | ||
| + | [[분류:string theory]] | ||
| + | [[분류:migrate]] | ||
| − | + | ==메타데이터== | |
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q2752849 Q2752849] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'brst'}, {'LEMMA': 'quantization'}] | ||
| + | * [{'LOWER': 'becchi'}, {'OP': '*'}, {'LOWER': 'rouet'}, {'OP': '*'}, {'LOWER': 'stora'}, {'OP': '*'}, {'LOWER': 'tyutin'}, {'LEMMA': 'quantization'}] | ||
| + | * [{'LOWER': 'brst'}, {'LEMMA': 'quantisation'}] | ||
| + | * [{'LOWER': 'becchi'}, {'OP': '*'}, {'LOWER': 'rouet'}, {'OP': '*'}, {'LOWER': 'stora'}, {'OP': '*'}, {'LOWER': 'tyutin'}, {'LEMMA': 'quantisation'}] | ||
2021년 2월 17일 (수) 02:01 기준 최신판
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.
- 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 \(\alpha_0=1\) for the space-time dimension \(D\) and the zero-intercept \(\alpha_0\) of leading trajectory are required by the nilpotency \(Q_B^2 = 0\) 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. <math>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
expositions
- http://www.mathcs.emory.edu/~rudolf/BRST.pdf
- 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
메타데이터
위키데이터
- ID : Q2752849
Spacy 패턴 목록
- [{'LOWER': 'brst'}, {'LEMMA': 'quantization'}]
- [{'LOWER': 'becchi'}, {'OP': '*'}, {'LOWER': 'rouet'}, {'OP': '*'}, {'LOWER': 'stora'}, {'OP': '*'}, {'LOWER': 'tyutin'}, {'LEMMA': 'quantization'}]
- [{'LOWER': 'brst'}, {'LEMMA': 'quantisation'}]
- [{'LOWER': 'becchi'}, {'OP': '*'}, {'LOWER': 'rouet'}, {'OP': '*'}, {'LOWER': 'stora'}, {'OP': '*'}, {'LOWER': 'tyutin'}, {'LEMMA': 'quantisation'}]