Quantization of free scalar fields scalar field f

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Quantization of free scalar fields scalar field f require (i) Lorentzian invariance (ii) invariance

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

Klein Gordon equation solution put negative energy! general solution vacuum state Fock space Hamiltonian

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

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

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

Lagrangian density Dirac spinor Cliford algebra

Lagrangian density Dirac spinor Cliford algebra

Lagrangian density equation of motion Dirac equation

Lagrangian density equation of motion Dirac equation

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

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

discrete symmetry P, T, C

discrete symmetry P, T, C

P T C CPT Lorentzian invariant Lagrangian density

P T C CPT Lorentzian invariant Lagrangian density

electromagnetic field electric field vector potential Maxwell equation Lagrangian gauge transformation magnetic field

electromagnetic field electric field vector potential Maxwell equation Lagrangian gauge transformation magnetic field

gauge fixing Feynman gauge free-field Lagrangian quantization canonical conjugate momentum quantization condition eq. of

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

general solution vacuum state Fock space 補助条件 annihilation operator creation operator gauge invariant Lagrangian

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

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