Quantum Field Theory. Note 9

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Cross sections and decay probabilities (Moulders: chapter 11)

Reaction matrix T and reaction amplitude M_fi: S = 1 - i T
S_fi = δ_fi - i (2π)⁴δ(p_f - p_i) M_fi
Decay probability per unit time: dλ_fi = |M_fi|² (2π)⁴δ(p_f - p_i) Π_n^d³p_n/_(2π)³
Cross section: dσ_fi = ¹/_u |M_fi|² (2π)⁴δ(p_f - p_i) Π_n^d³p_n/_(2π)³

Invariant perturbation theory

Time ordered product and Normal ordered product of field operators

T(A(t)B(t')) = A(t)B(t'), if t>t'
T(A(t)B(t')) = B(t')A(t), if t'>t (! sign change for fermions)
N(AB) = (all generators to the left)(all anihilators to the right)

Propagator

Δ(AB)=T(AB)-N(AB)

For a scalar field: Δ[φ(x)φ⁺(x')] = i∫ ^d⁴k/_(2π)⁴ (¹/_k²-m²+i0)e^-ik(x-x')
Propagator and the Green function G: G(x-x') = i Δ[φ(x)φ⁺(x')]

Wicks theorem

For two operators: T(AB) = N(AB) + Δ(AB)
For many operators: T(ABCDE...Z) = N(ABCDE...Z)
+ Δ(AB)N(CDE...Z) + Δ(AB)Δ(CD)N(E...Z)
+ (all other possible combinations of propagators times normal product of the rest)

Exercises

Prove that d³p/(2E) is Lorentz invariant. (Hint: consider δ(p²-m²)d⁴p).

Consider an unstable particle that decays into two other particles with a given amplitude M_fi. Calculate the differential decay probability dλ/dΩ in the rest frame of the decaying particle.
Answer: dλ(1→2+3)/dΩ= (^k/_4π²)|M_fi|²(^E₂E₃/_m₁)