Quantum Field Theory

Quantum Field Theory

Set 24

Exercise 1: Decay of a massive scalar particle

Consider the Lagrangian

L=1

2∂µφ∂^µφ−1

2m²φ²+1

2∂µΦ∂^µΦ−1

2M²Φ²−λ 2φ²Φ,

withM >2mandλa real, positive parameter. Using the Lagrangian above, compute the lifetimeτof the particle of massM.

To perform the computation, use the relation betweenM_Φ→2φ, entering the definition of the decay width, and the S-matrix element, and also the explicit expression of the latter in terms of the matrix element of the interaction Hamiltonian between initial and final states. In order to evaluate this, expand the scalar fields in terms of creation and annihilation operators.

Exercise 2: Microcausality

Consider a Klein-Gordon real scalar fieldφ.

• By using the decomposition in of the fieldφin terms of ladder operators, express the commutator [φ(x), φ(y)]

in terms of the functionD(x−y):

D(x)≡ Z

dΩpe^−ipx, (p0≡Ωp) (1)

• Check the Lorentz transformation properties ofD(x)

• Use Lorentz invariance to show that [φ(x), φ(y)] = 0 whenx−y is space-like.

This implies immediately that also [φ(x), π(y)] = 0 forx−y space-like.

Exercise 3: Scattering in λφ

Consider the Lagrangian

L=1

2∂_µφ∂^µφ− λ 4!φ⁴.

Using the above Lagrangian and Wick’s theorem, compute the matrix elementM2φ→2φand the total cross section for the scattering processφφ→φφ.