The Analytic Renormalization Group

Abstract

Finite temperature Euclidean two-point functions in quantum mechanics or quantum field theory are characterized by a discrete set of Fourier coefficients $G_k$, $k\in \mathbb{Z}$, associated with the Matsubara frequencies ${\nu }_k=2\pi k/\beta$. We show that analyticity implies that the coefficients $G_k$ must satisfy an infinite number of model-independent linear equations that we write down explicitly. In particular, we construct "Analytic Renormalization Group" linear maps ${\mathsf{A}}_{\mu }$ which, for any choice of cut-off $\mu$, allow to express the low energy Fourier coefficients for $|{\nu }_k|<\mu$ (with the possible exception of the zero mode $G_0$), together with the real-time correlators and spectral functions, in terms of the high energy Fourier coefficients for $|{\nu }_k|\ge \mu$. Operating a simple numerical algorithm, we show that the exact universal linear constraints on $G_k$ can be used to systematically improve any random approximate data set obtained, for example, from Monte-Carlo simulations. Our results are illustrated on several explicit examples.

Comments: 52 pages, 25 figures; v2: a few comments and explanations added