前回のSummary Path Integral Quantization path Integral

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Presentation transcript:

前回のSummary Path Integral Quantization path Integral generating functional

Gauge Theory for gauge field & spinor field Assume Lorentz inv., locality, superficial renormalizability & local gauge symmetry with Lie group G. Lie algebra g. gauge transformation b i :parameter, depends on xm Xi: generator of G f ijk: structure constants of G T i: representation of Xi on y Lagrangian field strength covariant derivative

SU(3)=group of complex 3×3 matrices U example 　　　 SU(3)=group of complex 3×3 matrices U withUU †=1 (unitary) & det U = 1 (special) generator li Gell-mann matrices commutators f ijkは完全反対称 irreducible representations are specified by two integers 1 2 3 6 10 3* 8 15 6* 15* 27 10* ......

Path intdegral quantization of gauge theories としてみる ∂mKmn =-∂n∂2+∂2∂n =0 generating functional ∂m is inappropriate ∵ does not exist. ∂m = = need gauge fixing ∂l 矛盾 we choose the gauge with (K-1)mn does not exist.

gauge fixing need gauge fixing we choose the gauge with

gauge fixing xi = = yj

gauge fixing = Gm i (gauge不変性より) = 無限大の定数物理はBi によらない無限大の定数 =

Grassman number Faddeev Popov ghost

fermionも加える Lagrangian

fermionも加える Lagrangian