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The Analytic Renormalization Group

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 $ν_{k}=2πk/β$. 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 $\mathsf A_μ$ which, for any choice of cut-off $μ$, allow to express the low energy Fourier coefficients for $|ν_{k}|<μ$ (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 $|ν_{k}|\geqμ$. 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.