The eliminated field obeys $\phi=-cp^2/(2a)$, so its connected <correlation function> is
$$
C_\phi(r)=\frac{c^2}{4a^2}\left[\langle p^2(0)p^2(r)\rangle-\langle p^2\rangle^2\right].
$$
For the zero-mean Gaussian field of part (i), <Wick theorem> gives
$$
\boxed{C_\phi(r)=\frac{c^2}{2a^2}C_p(r)^2}.
$$
It is therefore nonnegative and has the square of the Ornstein--Zernike spatial form. In particular, if $C_p(r)$ has exponential factor $e^{-r/\xi_p}$, then $C_\phi(r)$ has $e^{-2r/\xi_p}$ and correlation length $\xi_p/2$, with the algebraic prefactor also squared.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-344/1/e/ii/solution