Abstract
We investigate the nature of resonant tunneling in Quantum Field Theory. Following the pioneering work of Banks, Bender and Wu, we describe quantum field theory in terms of infinite dimensional quantum mechanics and utilize the ``Most probable escape path'' (MPEP) as the class of paths which dominate the path integral in the classically forbidden region. Considering a 1+1 dimensional field theory example we show that there are five conditions that any associated bound state in the classically allowed region must satisfy if resonant tunnelling is to occur, and we then proceed to show that it is impossible to satisfy all five conditions simultaneously.
Highlights
We investigate the nature of resonant tunneling in standard scalar Quantum Field Theory
The question is what should such a bound state look like? We shall argue that the natural choice for such a bound state, at least in standard scalar quantum field theory1, is the oscillon [8, 9, 10, 11, 12, 13, 14] a lump-like configuration of the scalar field whose amplitude varies in time
Given that quantum mechanics can be recovered from quantum field theory in the homogeneous limit, perhaps it is natural to expect resonant tunneling to occur in field theory
Summary
Let us begin with a review of tunneling in quantum mechanics (closely following [15, 17, 18]). The particle is described by its wavefunction, ψ(q), satisfying the time independent Schrodinger equation,. In the classically allowed region, E > V (q). We see that it is composed of a positive momentum piece (α+) and a negative momentum piece (α−). In the classically forbidden region, E < V (q), we have ψ(q) ∼=. By matching the general solution in the forbidden region, II, onto the adjacent regions, I and III, it can be shown using the WKB connection formulae (appendix A)
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