"BRST quantization and cohomology"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
 
(사용자 3명의 중간 판 34개는 보이지 않습니다)
1번째 줄: 1번째 줄:
−
<h5>introduction</h5>
+
==introduction==
  
−
* [http://empg.maths.ed.ac.uk/Activities/BRST/ PG minicourse: BRST cohomology] (http://empg.maths.ed.ac.uk/Activities/BRST/Notes.pdf , very good introduction)<br>
+
* [[Gauge theory|gauge theory]] = principal G-bundle
−
* [[Gauge theory]] = principal G-bundle<br>
+
*  we require a quantization of gauge theory
−
*  We require a quantization of gauge theory<br>
+
*  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<br>
+
**  gauge theory allows 'local symmetry' which should be ignored to be physical
−
*  Gauge theory allows 'local symmetry' which should be ignored to be physical<br>
+
**  this ignoring process leads to the cohomoloy theory.
−
*  This ignoring process leads to the cohomoloy theory.<br>
+
*  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”)<br>
+
*  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<br>
+
*  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==
  
−
 
+
  
−
\Lambda_{\infty} semi-infinite form
+
  
−
\mathfrak{g} : \mathbb{Z}-graded Lie algebra
+
  
−
\sigma : anti-linear automorphism sending \mathfrak{g}_{n} to \mathfrak{g}_{-n}
+
==ghost variables==
  
−
H^2(\mathfrak{g})=0 (i.e. no non-trivial central extension)
+
* [[Faddeev–Popov ghost fields|ghost fields]]
  
−
 
+
  
−
 
+
  
−
 
+
  
−
 
+
==Faddeev-Ghost determinant==
  
−
 
+
* [http://hitoshi.berkeley.edu/230A/FPghosts.pdf Faddeev-Popov ghosts], Hitoshi Murayama
  
−
<h5 style="margin: 0px; line-height: 2em;">ghost variables</h5>
+
  
−
* [[Faddeev–Popov ghost fields|ghost fields]]<br>
+
  
−
 
+
  
−
 
+
==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>e^{S_1(X)+S_2(b,c,\bar{b},\bar{c},\cdots,X)}</math>
 +
* 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]]
  
−
<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>
 
  
−
 
+
==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]
  
−
<h5 style="margin: 0px; line-height: 2em;">applications</h5>
+
  
−
* BRST approach to minimal models<br>
+
   
−
*  BRST approach to no-ghost theorem<br>
 
−
*  BRST approach to coset constructions<br>
 
  
−
 
+
==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
  
−
 
+
  
−
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">related items</h5>
+
  
−
* [[물리학과 cohomology]]<br>
+
−
* [[homological algebra|Homological algebra]]<br>
 
  
−
 
+
==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)
  
−
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">books</h5>
+
==applications==
  
−
*   <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]
−
* [[2009년 books and articles|찾아볼 수학책]]
+
*  BRST approach to no-ghost theorem
 +
* BRST approach to coset constructions
  
−
 
+
  
−
 
+
  
−
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
+
 +
 
 +
==related items==
 +
 
 +
* [[물리학과 cohomology]]
 +
* [[homological algebra|Homological algebra]]
 +
* [[Lie algebra cohomology]]
 +
 
 +
 +
 
 +
 +
 
 +
==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://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
−
* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
−
 
−
 
 
−
 
−
[http://bomber0.byus.net/ ]
 
  
−
 
 
  
−
 
 
  
−
<h5 style="margin: 0px; line-height: 2em;">expositions</h5>
 
  
−
* [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>
+
==expositions==
−
* [http://empg.maths.ed.ac.uk/Activities/BRST/ PG minicourse: BRST cohomology] (http://empg.maths.ed.ac.uk/Activities/BRST/Notes.pdf , very good introduction)<br>
+
* http://www.mathcs.emory.edu/~rudolf/BRST.pdf
 +
* [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)
  
−
 
+
  
−
 
+
  
−
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
+
==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
−
* van Holten, J. W., The BRST complex and the cohomology of compact lie algebras
+
* 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]
 
* Symplectic Reduction, BRS Cohomology, and Infinite-Dimensional Clifford algebras, B. Kostant, S. Sternberg, Ann. Physics 176 (1987) 49–113
 
* 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, [http://www.pnas.org/content/83/22/8442.abstract Semi-infinite cohomology and string theory]
 
* I. B. Frenkel,. H. Garland, and. G. J. Zuckerman, PNAS November 1, 1986 vol. 83 no. 22 8442-8446, [http://www.pnas.org/content/83/22/8442.abstract Semi-infinite cohomology and string theory]
−
* http://dx.doi.org/10.1007/BF02096498
+
* http://dx.doi.org/10.1103/RevModPhys.60.917
  
−
 
+
  
−
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">blogs</h5>
+
==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]
126번째 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]
  
−
*  구글 블로그 검색<br>
 
−
** http://blogsearch.google.com/blogsearch?q=
 
−
** http://blogsearch.google.com/blogsearch?q=
 
−
** http://blogsearch.google.com/blogsearch?q=
 
−
 
−
 
 
−
 
−
 
 
  
−
 
+
 +
[[분류:physics]]
 +
[[분류:math and physics]]
 +
[[분류:string theory]]
 +
[[분류:migrate]]
  
−
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">TeX </h5>
+
==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) 03:01 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




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



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




related items



books

  • Polchinski, vol. I. <math>3.1-3.4, 4.2-4.3
  • GSW, I. 3.1-3.2



encyclopedia



expositions



articles


blogs

메타데이터

위키데이터

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'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=BRST_quantization_and_cohomology&oldid=51876"