Tunneling in a quantum field theory on a compact one-dimensional space

TL;DR

This paper investigates quantum tunneling in a 1+1 dimensional quantum field theory on a compact space, revealing resonant tunneling phenomena and the effects of mode excitations on the tunneling process.

Contribution

It introduces a real-time simulation approach using the time-dependent Hartree-Fock method with a non-Gaussian zero mode wave function to study tunneling dynamics.

Findings

Tunneling occurs resonantly due to degeneracies in energy levels.

Weak excitation of nonzero modes leads to quantum-mechanical-like oscillations.

Strong excitation causes back-reaction, potentially eliminating the barrier and inducing wave function sliding.

Abstract

We compute tunneling in a quantum field theory in 1+1 dimensions for a field potential $U(\Phi)$ of the asymmetric double well type. The system is localized initially in the ``false vacuum''. We consider the case of a {\em compact space} ($S_1$) and study {\em global} tunneling. The process is studied in real-time simulations. The computation is based on the time-dependent Hartree-Fock variational principle with a product {\em ansatz} for the wave functions of the various normal modes. While the wave functions of the nonzero momentum modes are treated within the Gaussian approximation, the wave function of the zero mode that tunnels between the two wells is not restricted to be Gaussian, but evolves according to a standard Schr\"odinger equation. We find that in general tunneling occurs in a resonant way, the resonances being associated with degeneracies of the approximate levels in the two separated wells. If the nonzero momentum modes of the quantum field are excited only weakly, the phenomena resemble those of quantum mechanics with the wave function of the zero mode oscillating forth and back between the wells. If the nonzero momentum modes are excited efficiently, they react back onto the zero mode causing an effective dissipation. In some region of parameter space this back-reaction causes the potential barrier to disappear temporarily or definitely, the tunneling towards the ``true vacuum'' is then replaced by a sliding of the wave function.