"BRST quantization and cohomology"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
1번째 줄: 1번째 줄:
 
<h5>introduction</h5>
 
<h5>introduction</h5>
  
−
* [[Gauge theory]] = principal G-bundle<br>
+
* [[Gauge theory|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>
55번째 줄: 55번째 줄:
 
** moduli spaces
 
** moduli spaces
 
** CKV
 
** CKV
−
*  
+
* [[path integral and moduli space of Riemann surfaces]]<br>
−
 
 
−
 
 
  
 
 
 
 

2011년 10월 22일 (토) 07:22 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 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

 

 

 

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. $3.1-3.4, 4.2-4.3
  • GSW, I. 3.1-3.2

 

 

encyclopedia

 

[2]

 

 

expositions

 

 

articles

 

blogs

 

 

 

TeX 
원본 주소 "https://wiki.mathnt.net/index.php?title=BRST_quantization_and_cohomology&oldid=36962"