Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 1 / e / ii

Table of contents
- Solution ii

Solution

The eliminated field obeys

ϕ = - c p^{2} / (2 a)

, so its connected correlation function is

C_{ϕ} (r) = \frac{c ^{2}}{4 a ^{2}} [⟨ p^{2} (0) p^{2} (r)⟩ - ⟨ p^{2} ⟩^{2}] .

For the zero-mean Gaussian field of part (i), Wick theorem gives

C_{ϕ} (r) = \frac{c ^{2}}{2 a ^{2}} C_{p} (r)^{2} .

It is therefore nonnegative and has the square of the Ornstein--Zernike spatial form. In particular, if

C_{p} (r)

has exponential factor

e^{- r / ξ_{p}}

, then

C_{ϕ} (r)

has

e^{- 2 r / ξ_{p}}

and correlation length

ξ_{p} /2

, with the algebraic prefactor also squared.

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