The Analytic Renormalization Group

Understand

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 &#34;Analytic Renormalization Group&#34; 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 $|\nu_{k}|\geq\mu$.
• 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.