Quantization of free fields -- massive spin 1 field (Moulders: chapter 7)
Additional condition to eliminate the spin-0 component from the
4-vector field φμ
∂μφμ = 0 -- (often called the "Lorentz condition" in
classical electrodynamics)
Lagrangian,second order in φ with first order derivatives
∂μφnu :
L = -∂μ φν*∂μ φν +
m2φν*φν
(remember that φνφν=φ0φ0-φφ)
Current:
jμ = -i(∂μ φν*φν +
φν*∂μ φν)
Massless spin 1 field
Gauge invariance
Aμ→Aμ+∂μφ
Lorentz condition
∂μAμ=0
Lagrangian
L = (1/8π)∂μ Aν∂μ Aν
L = -(1/16π)FμνFμν
Coulomb gauge
e = {0,e}, k⋅e=0
or A0=0, ∇⋅A=0
Plane-wave expansion
A=∑kλ√(2π/ω)
(akλeλe-ikx +
akλ+eλ*eikx)
Exercises
Rewrite the Dirac equation into the form
i∂ψ/∂t
=Hψ. Show that the spin operator
S=½[σ00σ] does not commute with H
while the "helicity" operator S⋅p does.