Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 301 / 4 / i

Table of contents
- Solution i

Solution

For bosonic fields, time ordering places operators with later time arguments to the left:

T {ϕ_{1} \dots ϕ_{n}} = ϕ_{π (1)} \dots ϕ_{π (n)}, t_{π (1)} \geq \dots \geq t_{π (n)} .

Normal ordering places all creation operators to the left of all annihilation operators and is denoted by colons. A Wick contraction is

contr (ϕ_{i}, ϕ_{j}) = ⟨ 0∣ T {ϕ_{i} ϕ_{j}} ∣0 ⟩ = D_{F} (x_{i} - x_{j}) .

Wick theorem states

T {ϕ_{1} \dots ϕ_{n}} =: ϕ_{1} \dots ϕ_{n} : + single contractions \sum : \dots : + double contractions \sum : \dots : + \dots,

where each sum runs over inequivalent disjoint pairings and contracted fields are replaced by their Feynman propagator.

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