On quantum tunnelling in the presence of Noether charges

We provide a complete first-principles based discussion of quantum tunnelling out of initial states carrying a conserved Noether charge. Our main result is a simple, unambiguous Euclidean-time prescription for the calculation of tunnelling rates out of such states. By relying on a combination of the direct approach and the steadyon framework for the evaluation of real-time path integrals, our derivation offers full transparency of its underlying assumptions, and is independent of any ad-hoc generalisations. This strategy also offers a simple explanation for the emergence of complex saddle points for such systems, justifying techniques postulated by earlier works. Our analysis furthermore offers the first results for initial states with both a conserved Noether charge and a non-trivial energy. We first illustrate the main conceptual points of our analysis for the simple example of a point particle in two spatial dimensions carrying a conserved angular momentum. Then, we generalise our results to the case of multiple dimensions and an arbitrary conserved Noether charge, providing an easy-to-implement prescription for the calculation of the tunnelling rate. We furthermore apply our results to the example of a complex scalar field subject to a global U(1)-symmetry with associated charge. These results provide a reliable foundation for the calculation of tunnelling rates in applications in finite-density and charge-asymmetric systems.