"BRST quantization and cohomology"의 두 판 사이의 차이

2011년 10월 4일 (화) 07:29 판

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

Faddeev-Ghost determin

$Z = \int\!\mathcal{D}X\,\mathcal{D}b \mathcal{D}c \,e^{-\int\left(\partial X \partial \bar{X} -b_{zz}\partial_{\bar{z}}c^{z}+b_{\bar{z}\bar{z}}\partial_{z}c^{\bar{z}}\right)}$
DX : matter and DbDc : ghost sector
bc system of \epsilon=+1 (in Faddeev–Popov ghost fields)
\lambda=2
c_{b,c}=-26
[c]=-1,[b]=2

$\Lambda_{\infty}$ 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)

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

@@ 31번째 줄: / 31번째 줄: @@
+Faddeev-Ghost determin
-<h5 style="margin: 0px; line-height: 2em;">path integral</h5>
+<h5 style="margin: 0px; line-height: 2em;">path integral and ghost sector</h5>
+*
+* <math>Z = \int\!\mathcal{D}X\,\mathcal{D}b \mathcal{D}c \,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>
+* DX : matter and DbDc : ghost sector
+* bc system of \epsilon=+1 (in [[Faddeev–Popov ghost fields]])
+* \lambda=2
+* c_{b,c}=-26
+* [c]=-1,[b]=2