"BRST quantization and cohomology"의 두 판 사이의 차이

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8번째 줄: 8번째 줄:
 
*  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 $\alpha_0=1$ 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
+
*  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==
 
==gauge fixing==
84번째 줄: 84번째 줄:
  
 
* <math>\Lambda_{\infty}</math> semi-infinite form
 
* <math>\Lambda_{\infty}</math> semi-infinite form
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* <math>\mathfrak{g}</math> : $\mathbb{Z}$-graded Lie algebra
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* <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>\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)
 
* <math>H^2(\mathfrak{g})=0</math> (i.e. no non-trivial central extension)
112번째 줄: 112번째 줄:
 
==books==
 
==books==
  
−
* Polchinski, vol. I. $3.1-3.4, 4.2-4.3
+
* Polchinski, vol. I. <math>3.1-3.4, 4.2-4.3
 
* GSW, I. 3.1-3.2
 
* GSW, I. 3.1-3.2
  

2020년 11월 13일 (금) 11:05 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.
  • 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


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