Exact WKB Formulation of Quantization and Particle Production in Time-Dependent Backgrounds

Abstract

Divergence in perturbative expansions is where interesting physics takes place. Particle production on time-dependent backgrounds, as one such example, is interpreted as transition from one vacuum to another. Vacuum is typically defined as an asymptotic state in which the WKB approximation is valid. The use of the WKB method, however, poses several conceptual and computational issues, as the WKB series is divergent in general, quantization is insensitive to higher orders in the series, and the global behavior of solutions cannot be captured. Exact WKB analysis is a powerful resummation technology that provides an analytical tool for a global structure of exact solutions to overcome these problems. In this paper, we establish quantization by fully employing the exact WKB solutions as mode functions and by defining the vacua with respect to them. We provide a self-contained exact WKB formulation to obtain evolution matrices without resorting to the use of known special functions and without approximations. We find that the quantity called Voros coefficient plays an important role to re-normalize the exact WKB solutions compatible with asymptotic states. We show that the ambiguity that coexists with nontrivial Voros coefficients is eliminated by requiring physical quantization conditions. Our formalism provides a conceptual as well as practical framework to upgrade our treatment of quantization and particle production. Combined with other approximating techniques, it can form a basis to tackle a broad class of problems that are beyond technical ability of the existing formulations.