Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 44 / 2 / c / Solution

It is useful to regulate the number of modes first, so the functional integral identities reduce to ordinary integration by parts. Let $C$ be the Gaussian covariance, $F=C^{-1}$, $W=e^{-S_1}$ and put a dot for $\Lambda {\partial }_{\Lambda }$. The matrix identity $\dot{F}=-F\dot{C}F$ givesThe second field derivative of the Gaussian isHence, after two integrations by parts,The imposed flow makes the bracket vanish. The remaining trace is independent of the fields and only changes the Gaussian normalization. Since the free Gaussian normalization is $\mathcal{N}=(\det (2\pi C){)}^{1/2}$, $\dot{\log \mathcal{N}}=\mathrm{Tr}(F\dot{C})/2$. ThereforeEquivalently, $Z$ is cutoff independent after discarding the stated overall rescaling. This is the Gaussian covariance differentiation identity behind the Polchinski equation.

With the Fourier convention above and functional derivatives satisfying $\delta \tilde{\varphi }(p)/\delta \tilde{\varphi }(q)={\delta }^{(4)}(p-q)$, contraction with $\dot{C}$ becomes $\int d^{4}p(2\pi {)}^4{\dot{C}}_{\Lambda }(p){\delta }^2/[\delta \tilde{\varphi }(p)\delta \tilde{\varphi }(-p)]$. Thus the numerator $(2\pi {)}^4$ in the printed flow is consistent with this derivative convention; it must not be changed independently of the convention.