Spin 0

Spin  

Spin 1

Free field equations

Lagrangian density, free

Minimal substitution

Not treated here  

 In free Lagrangian (density):   

Lagrangian, free plus interactions

Not treated here

Interacting fields equations

Not treated here

Conjugate momentum

Hamiltonian density

Free field solutions

Discrete eigenstates

(Plane waves, finite volume or periodic B.C.’s)

Continuous eigenstates

(Plane waves, infinite volume, no B.C.’s)

 spinor indices on u_r, v_r, and ψ  suppressed.  r = 1,2.

Second quantization

                  Ψ_r = any quantized field

All other commutators = 0.    Anti-commutator for spin  fields (use plus sign.)

  FREE FIELDS ONLY   

Using conjugate momenta expressions in the second quantization relation above yields

Equal time commutators

outer product, spinor indices α, β shown with α,β = 1,2,3,4.

Using free field solutions in the above and the 3D Dirac delta function (e.g., for discrete solutions,  ) yields the coefficient commutators (below).

Coefficient commutators

discrete

continuous

Using the free field solutions in the LHS below (covariant field commutators) and the relations above yields RHS of below.  (Note that different authors define the terms below slightly differently.)

Covariant commutators, continuous only

3-momentum space form

contour integral form

The coefficient commutation relations used with the Hamiltonian acting on states lead to creation and destruction operator interpretation of the coefficients and to the number operators below.

Operators:

creation

destruction

number

tot partic num

particle num:

lowering

raising

Normaliz factors

lowering

as with scalars

raising

as with scalars

Operations on states with creation, destruction, and number operators above yield the properties below.

Properties of states:

state symmetric under particle exchange, Bose-Einstein stats

state changes sign under particle exchange, Fermi-Dirac statistics

state symmetric under particle exchange, Bose-Einstein stats

Four currents (operators)

      = 0 for photons  

Emphasis in field theory is usually on the number of particles (N(k) operator), and particle probability densities are rarely used.  For completeness, however, and to make the connection with quantum mechanics, they are included below.  (Antiparticles would have negative values of those below!)

Single particle probability density (not operator)

 Note integration over , not x

For plane wave,  

As at left, but with Dirac j⁰ above.

= 0 for chargeless particles.  Led to conclusion that j⁰ is really proportional to charge probability density.

Creation and destruction of free particles (& antiparticles) and their propagation visualized below.

Feynman diagrams

Time ordered operator T

If t_y < t_x , , i.e., the  operates first, and should be placed on the right.

If t_x < t_y , , i.e, the  (x) operates first, and should be placed on the right.

Note that  (x) commutes with .   [Fermions would anti-commute.]

The operator fields in the T operator below will create and destroy kets on RHS.  In the wave mechanics formulation, bracket integration is over the dummy  variable in the bra and ket, not x,y of T operator. 

Transition amplitude density

The above represent both

1)      creation of a particle at y, destruction at x, and

2)      creation of an antiparticle at x, destruction at y

Fields such as  represent integration over all momenta.  The transition amplitude (density) above for fixed x and y equals the Feynman propagator  between x and y.  Note this is a number, not an operator.  Only the continuous field solutions are relevant as no boundary conditions exist in this case.

Feynman propagators

Evaluating the above in complex space and taking certain limits with contour integrals yields a form for Feynman propagators which works for any time ordering and will prove more convenient.

in physical space