Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 48 / 2 / a

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- Solution a

Solution

Use a real scalar field and Minkowski action

S [ϕ]

, with the vacuum selected by a Feynman propagator pole prescription or adiabatic vacuum boundary conditions. The normalized correlation functions are

G_{n} (x_{1}, \dots, x_{n}) = \frac{\int D ϕ ϕ ( x _{1} ) \dots ϕ ( x _{n} ) e ^{i S [ϕ]}}{\int D ϕ e ^{i S [ϕ]}} = ⟨ Ω∣ T {ϕ_{H} (x_{1}) \dots ϕ_{H} (x_{n})} ∣Ω ⟩ .

Here

∣Ω ⟩

is the interacting vacuum and

T

denotes time ordering. These are the canonical vacuum expectation values, rather than unordered products of noncommuting fields. The functional integral formula assumes a regulated measure and the same vacuum prescription on both sides.

Couple an external source

J

linearly to the scalar field and choose the normalized vacuum generating functional

Z [J] = \frac{\int D ϕ e x p ( i S [ ϕ ] + i \int d ^{d} x J ϕ )}{\int D ϕ e ^{i S [ϕ]}}, Z [0] = 1.

Then functional derivatives insert the fields:

G_{n} = i^{- n} \frac{δ ^{n} Z}{δ J ( x _{1} ) \dots δ J ( x _{n} )}_{J = 0}, Z [J] = \sum_{n = 0}^{\infty} \frac{i ^{n}}{n !} \int G_{n} (x_{1}, \dots, x_{n}) \prod_{j = 1}^{n} J (x_{j}) d^{d} x_{j} .

In the convention used below, the connected generating functional is

Z [J] = e^{iW [J]}, W [J] = - i lo g Z [J], G_{n}^{c} = i^{1 - n} \frac{δ ^{n} W}{δ J ( x _{1} ) \dots δ J ( x _{n} )}_{J = 0} .

The logarithm removes products of disconnected components. For example the connected correlation function

G_{2}^{c} = G_{2} - G_{1} G_{1}

; at three points one subtracts all three

G_{2} G_{1}

products and adds

2 G_{1} G_{1} G_{1}

. Normalization has already removed vacuum bubbles.

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