Fermi's Golden Rule and Exponential Decay as a RG Fixed Point

• We discuss the decay of unstable states into a quasicontinuum using Hamiltonian models. We show that exponential decay and the golden rule are exact in a suitable scaling limit, and that there is an associated renormalization group (RG) with these properties as a fixed point. The method is inspired by a limit theorem for infinitely divisible distributions in probability theory, where there is a RG with a Cauchy distribution, i.e. a Lorentz line shape, as a fixed point. Our method of solving for the spectrum is well known; it does not involve a perturbation expansion in the interaction, and needs no assumption of a weak interaction. Using random matrices for the interaction we show that the ensemble fluctuations vanish in the scaling limit. For non-random models we can use uniformity assumptions on the density of states and the interaction matrix elements to estimate the deviations from the decay rate defined by the golden rule.

Nyckelord

Fermi's golden rule

Renormalization group

Feshbach method

Random

matrix

Unstable states

Lorentz line shape

nuclear reactions

unified theory

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