Quantization of free scalar fields scalar field  equation of motin Lagrangian density  (i) Lorentzian invariance (ii) invariance under  →  require (iii) at most quadratic in  Klein Gordon equation canonical conjugate momentum canonical commutator relation =quantization condition Lagrangian Hamiltonian density time-development of F Hamiltonian

Klein Gordon equation vacuum state solution :operator put general solution Fock space Hamiltonian negative energy! take  as an operator creation operator annihilation operator

Weyl spinor field representation rep. Lorentz invariant hermitian operators Lorentz invariant operators representation

Lagrangian density Dirac spinor Cliford algebra

equation of motion Dirac equation Lagrangian density

quantization vacuum state Fock space canonical conjugate momentum quantization condition solution creation operator annihilation operator particle antiparticle

discrete symmetry P, T, C

P T C CPT Lorentzian invariant Lagrangian density

Maxwell equation Lagrangian gauge transformation electromagnetic fieldelectric fieldmagnetic field vector potential

free-field Lagrangian quantization canonical conjugate momentum quantization condition eq. of motion solution gauge fixing Feynman gauge polarization vector

general solution 補助条件 vacuum state Fock space creation operator annihilation operator gauge invariant Lagrangian density gauge transformation for matter field covariant derivative complex scalar