Instantonic Methods for Quantum Tunneling in Finite Size

TL;DR

This paper revisits instantonic methods for quantum tunneling in finite systems, deriving a consistent finite size instantonic theory using Jacobi elliptic functions, addressing singularities and formal contradictions present in infinite size assumptions.

Contribution

It develops a finite size instantonic framework for quantum tunneling, overcoming limitations of traditional infinite size instanton solutions through Jacobi elliptic functions.

Findings

Derived the classical background solving finite size Euler-Lagrange equations.

Formulated the general path integral in finite size systems.

Resolved the singularity issue in the path integral for finite systems.

Abstract

The instantonic approach for a $\phi^4$ model potential is reexamined in the asymptotic limit. The path integral of the system is derived in semiclassical approximation expanding the action around the classical background. It is shown that the singularity in the path integral, arising from the zero mode in the quantum fluctuation spectrum, can be tackled only assuming a {\it finite} (although large) system size. On the other hand the standard instantonic method assumes the (anti)kink as classical background. But the (anti)kink is the solution of the Euler-Lagrange equation for the {\it infinite} size system. This formal contradiction can be consistently overcome by the finite size instantonic theory based on the Jacobi elliptic functions formalism. In terms of the latter I derive in detail the classical background which solves the finite size Euler-Lagrange equation and obtain the general path integral in finite size. Both problem and solution offer an instructive example of fruitful interaction between mathematics and physics. {\bf Keywords: Path Integral Methods, Finite Size Systems, Instantons}