前回まとめ 自由scalar場の量子化 Lagrangian 密度 運動方程式 Klein Gordon方程式 正準共役運動量 量子条件 Hamiltonian 一般解 Fock space 真空状態 生成演算子 消滅演算子 Hamiltonian

Dirac spinor = Dirac行列 g 0 = = Lagrangian密度 正準共役運動量 Dirac equation quantization condition　 solution　 vacuum state Fock space particle creation operator annihilation operator antiparticle creation operator annihilation operator

Electromagnetic field electric field magnetic field Maxwell equation　　 4-dimensional description　　 4-current 4-field strength 4-current

Electromagnetic field electric field magnetic field Maxwell equation　　 4-dimensional description　　 * 4-current * - - ( -( ) - - 4-field strength dual of Fmn 4-current * * F01=e0123F 23 F23 =(×B)23 =-B1 F01 = (-E)1 =-E1 =-B1 * F23=e2301F 01 =-E1 = (×E)23 : totally anti-symmetric tensor

Electromagnetic field electric field magnetic field Maxwell equation　　 4-dimensional description　　 * scalar potential j, vector potential A,　　 4-dim. vector field Am　　 field strength field strength

Electromagnetic field electric field magnetic field Maxwell equation　　 4-dimensional description　　 * scalar potential j, vector potential A,　　 4-dim. vector field Am　　 field strength ∂0An -∂nA0 n = = n m = m = = ∂mA0 ∂mAn-∂nAm -∂0Am ( )23 ×(∇×A) = - (∇×A)1 =-(∂2A3-∂3A2) =∂2A3-∂3A2 : totally anti-symmetric tensor

Electromagnetic field electric field magnetic field Maxwell equation　　 4-dimensional description　　 * scalar potential j, vector potential A,　　 4-dim. vector field Am　　 field strength gauge transformation L: function of xm Fmn : invariant under gauge transformation , E, B

* gauge transformation L: function of xm Fmn : invariant under gauge transformation , E, B

require (i) vector field (dynamical variable) Maxwell eq.　　 (ii) Lorentzian invariance, locality Maxwell equation　　 (iii) gauge invariance * (iv) simple interaction with the current Lagrangian density gauge transformation L: function of xm Fmn : invariant under gauge transformation , E, B

require (i) vector field (dynamical variable) Maxwell eq.　　 (ii) Lorentzian invariance, locality (iii) gauge invariance * (iv) simple interaction with the current Lagrangian density * Euler equation symmetric symmetric = = = = = anti-symmetric anti-symmetric Maxwell equation　　 *

Quantization of free electromagnetic field Am free-field Lagrangian

positive frequency part Quantization of free electromagnetic field Am free-field Lagrangian canonical conjugate momentum quantization condition = ≠ ??? gauge fixing　 positive frequency part add 　 to　 and impose 　 the subsidiary condition physical states good! canonical conjugate momentum quantization condition

eq. of motion　 solution polarization vectors general solution Fock space vacuum state subsidiary condition　 creation operator annihilation operator

complex scalar 荷電場の記述　 Dirac spinor 自由場Lagrangian matter 場のgauge変換　 は不変 は不変でない covariant derivative gauge場の変換　 gauge invariant Lagrangian density　

Schrödinger Representation & Heisenberg Representation operators depend on time Heisenberg eq. H: Hamiltonian states : do not depend on time. Schrödinger representation states Schrödinger equation Operators do not depend on time. in state out state scattering matrix (S-matrix)　　 scattering matrix elements　　 scattering provability ∝　

perturbation Interaction representation　　 operators states solution T : time ordered product (T-product)

The solution of is proof T-product ordered! differentiation n! tn t3 t2 n! The n! terms with different orders become the same after T-ordering! differentiation

def. positsve frequency part (annihilation operator) negative frequency part (creation operator) def. N : normal product place to the left of Wick's theorem def. vacuum expatation value e. g.

Wick's theorem e. g.

の証明 -) = よって、このとき、与式は成立する　　

-) よって、このとき、与式は成立する　　 の証明 -) = よって、このとき、与式は成立する　　

spinor yi vector Am locality, Lorentz inv. gauge inv. perturbation interaction reprensatation probability S-matrix Wick's theorem def.

locality, Lorentz inv. gauge inv. spinor yi vector Am interaction reprensatation perturbation S-matrix probability Wick's theorem def.

scalar field If x0>y0 If x0<y0 2E(2p)3d (k-k' ) 2E(2p)3d (k-k' ) [ [ ] ] for = for にpole

Dirac field = for にpole for

electromagnetic field = for にpole for

locality, Lorentz inv. gauge inv. spinor yi vector Am interaction reprensatation perturbation S-matrix probability Wick's theorem def.