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 GkGk, k∈Zk∈Z, associated with the Matsubara frequencies νk=2πk/βνk=2πk/β. We show that analyticity implies that the coefficients GkGk must satisfy an infinite number of model-independent linear equations that we write down explicitly. In particular, we construct “Analytic Renormalization Group” linear maps AμAμ which, for any choice of cut-off μ  , allow to express the low energy Fourier coefficients for |νk|<μ|νk|<μ (with the possible exception of the zero mode G0G0), together with the real-time correlators and spectral functions, in terms of the high energy Fourier coefficients for |νk|≥μ|νk|≥μ. Operating a simple numerical algorithm, we show that the exact universal linear constraints on GkGk 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.