Discrete symmetries (for the scalar field). CPT theorem.(Moulders: chapter 8)
Charge conjugation:
ap → Uc ap Uc+ = bp ,
φ(x) → Uc φ(x) Uc+ = φ+(x)
Parity transformation:
ap→ Up ap Up+ = ± a-p ,
φ(t,x) → Up φ(t,x) Up+ = ± φ(t,-x)
Time reversal:
ap→ Ut apT Ut+ = a-p+ ,
φ(t,x) → Ut φ(t,x)T Ut+ = φ+(-t,x)
CPT theorem: under assumptions of locality and the spin-statistics theorem
the Lagrangian L(x) under the combined CPT transformation changes
sign of the argument L(x) → L(-x) and therefore the action
S = ∫ d4x L(x) is invariant under CPT transformation.
Interaction Quantum Fields (Moulders: chapter 10)
Interaction Lagrangian Lint and coupling constant g.
For example Lint = -gψψφ
Natural dimensions of fields
h =c=1 , x ∼ t ∼ m-1 , S ∼ 1 , L ∼ m4 ,
φ ∼ m , ψ ∼ m3/2
Natural dimensions of coupling constants
Lint = -gψψφ → g 1 :) - good for renormalization
Lint = -gψγ μ ψ∂μφ
→ g m-1 :( - bad for renormalization
Interaction representation for the time evolution of
the state vector Φ(t):
i∂/∂tΦ(t) = V(t)Φ(t)
S-matrix: Φ(+∞) = SΦ(-∞)
S = Texp(i∫ d4x Lint(x))
Exercises
Show that the classical interaction Lagrangian Lint =
-gAμjμ is invariant under the gauge transformation
Aμ-∂μf(x) if jμ is a conserved current,
that is ∂μjμ=0.