Weinberg Salam Model Higgs field SU(2) gauge field U(1) gaugefield complex scalar, SU(2) doublet Y  =1 quark lepton SU(2) U(1)hypercharge 1/3 11 4/3 0  2/3 22 Lagrangian density SU(2)×U(1)gauge symmetry 2 SU(3) 3 13 Lorentz group quark lepton

SU(2)×U(1)gauge sym. is broken spontaneously v.e.v. redefinition mass of gauge fields Weinberg angle gauge field mixing mass of  W & Z get massive absorbing . The electromagnetic U(1) gauge symmetry is preserved., electromagnetic coupling constant

Yukawa interaction fermion mass term

diagonalization Cabibbo-Kobayashi-Maskawa matrix Maki-Nakagawa-Sakata matrix diagonal +h.c.

Path Integral Quantization fields eigenstate completeness probability amplitude xnxn xnxn xnxn tntn tntn x1x1 x1x1 x1x1 t1t1 t1t1 x1x1 xnxn cf. coordinate xixi xixi xixi titi titi xixi

Path Integral Quantization fields eigenstate completeness x1x1 x1x1 x1x1 t1t1 t1t1 x1x1 cf. coordinate xixi xixi xixi titi titi xixi provability amplitude xnxn xnxn xnxn tntn tntn xnxn H : Hamiltonian

: canonical conjugate of eigenstate completeness : canonical conjugate of eigenstate completeness  O((  t i ) 2 ) H titi titi  xixi ・

H : Hamiltonian  O((  t i ) 2 ) H L : Lagrangian  xi22Vxi22V  L    N'

Path Integral Quantization fields eigenstate completeness x1x1 x1x1 x1x1 t1t1 t1t1 x1x1 cf. coordinate xixi xixi xixi titi titi xixi provability amplitude xnxn xnxn xnxn tntn tntn xnxn  O((  t i ) 2 ) H H : Hamiltonian L : Lagrangian  N'

Path Integral Quantization fields eigenstate completeness x1x1 x1x1 x1x1 t1t1 t1t1 x1x1 cf. coordinate xixi xixi xixi titi titi xixi provability amplitude xnxn xnxn xnxn tntn tntn xnxn  N' : Lagrangian density

x jx j operator eigenvalue x jx j (x)(x)

x jx j x jx j (x)(x)

xx a xx b xx a xx b (xa)(xb)(xa)(xb)

xx a xx b xx a xx b (xa)(xb)(xa)(xb)

generating functional functional derivative JJ cf. partial derivative JJ JJ JJ  ( x ) JJ  ( y ) JJ h (xy)(xy)

JJ JJ JJ JJ

JJ JJ

commuting c- 数 anti-commuting c- 数 (Grassman 数 ) 微分 積分

cf

scalar  と fermion  の系 generating functional

need gauge fixing gauge theory is inappropriate because anddoes not have inverse. generating functional gauge boson と fermion  の系

gauge fixing

Faddeev Popov ghost =1