Fake Instability in the Euclidean Formalism of Quantum Tunneling

Abstract

We study the path-integral formalism in the imaginary-time to show its validity in a case with a metastable ground state. The well-known method based on the bounce solution leads to the imaginary part of the energy even for a state that is only metastable and has a simple oscillating behavior instead of decaying. Although this has been argued to be the failure of the Euclidean formalism, we show that proper account of the global structure of the path-space leads to a valid expression for the energy spectrum, without the imaginary part. For this purpose we use the proper valley method to find a new type of instanton-like configuration, the "valley instantons". Although valley instantons are not the solutions of equation of motion, they have dominant contribution to the functional integration. A dilute-gas approximation for the valley instantons is shown to lead to the energy formula. This method extends the well-known imaginary-time formalism so that it can take into account the global behavior of the theory.

Valley views: instantons, large order behaviors, and supersymmetry

“…For ǫ = 0, the valley configurations were found to behave in a qualitatively similar manner [9,11]. There are configurations that smoothly connect one minimum to the other minimum, which we call "valley-instanton" and "antivalley-instanton", and configurations that are similar to the pairs of these valley-instantons and anti-valley-instantons exist as solutions of the valley equation.…”

“…For ǫ = 0, the valley configurations were found to behave in a qualitatively similar manner [9,11]. There are configurations that smoothly connect one 1 In an earlier paper [9], we have used a (4g 3 q 3 − 3g 4 q 4 ) term in place of gq in the ǫ term of the potential.…”

“…There are configurations that smoothly connect one 1 In an earlier paper [9], we have used a (4g 3 q 3 − 3g 4 q 4 ) term in place of gq in the ǫ term of the potential. This was for the sake of the exact relations q + = 0 and q − = 1/g.…”

“…In Section 2, following Refs. [7,8,9,10], we review a method to define the valleys in the configuration space and present the construction of the valley configurations and their basic constituents, the valley-instantons. Section 3 is the core of this paper; we will give a detailed analysis of a valley that contains a pair consisting of an instanton and an anti-instanton, and show that the interplay between the perturbative effects and the tunneling effects can be summarized in a unified way in terms of the collective coordinate of the valley.…”

Theoretical studies on Josephson π states

“…We assume here that our potential in eqn (5) can be approximated by an array of asymmetric double-well potentials. Then, we can employ the valley instanton technique developed by Aoyama et al [15]. The action of the valley instanton is given by…”