Equivalent Relations between Quantum Dynamics as Derived from a Gauge Transformation

Equivalent relations between quantum mechanical systems in the RobertsonWalker (RW) background metric and quantum dynamics with an induced quadratic background potential are derived in this work. Two elementary applications, which include an algebraic derivation of the evolution operator for a simple harmonic oscillator without using any special function or the path integral technique, and a moving soliton solution of a free particle in an oscillating universe, are presented to illustrate the use of these equivalent relations. Typeset using REVTEX email address: [email protected] email address: [email protected] 1 I. MOTIVATION AND INTRODUCTION Recent advance of duality in string theory has not only greatly improved our understanding of string theory as an unified framework of quantum gravity, but also drastically changed our view of quantum field theory. For instance, the Maldacena conjecture gives an unforeseen correspondence between super-gravity and super-Yang-Mills theory which allows us to study the strongly coupled dynamics of gauge theories via black hole physics [1]. Another example regarding strong-weak duality which exchanges fundamental with solitonic degrees of freedom in apparently different models and establishes their equivalence [2]. On the other hand, it has also become clear that, as a technical tool, the use of duality facilitates investigations of nonperturbative or strong coupling aspects in quantum dynamics, which are certainly beyond the domain of perturbative calculations. While there exists vast literature providing substantial evidences for supporting various duality relations as exact quantum symmetries, it remains to be a great theoretical challenge to find mathematical derivations of these conjectured connections. In this work, we shall focus on a special duality transformation which generates equivalent relations between quantum systems with different backgrounds. Due to relatively simple structures of these systems, it is possible to derive these equivalent relations from a series of change of variables. The precise correspondence between two classes of quantum systems can be formulated as transformations between wave functions and evolution operators, and we can explore physical consequences based on such an equivalent relation. This paper is organized as follows. We first derive the equivalent relations between quantum systems in the Robertson-Walker metric (denoted as Class A) and quantum systems with an induced quadratic background potential (denoted as Class B) in Sec.II. The use of these equivalent relations is illustrated by two elementary applications in Sec.III and Sec.IV. In particular, we find an algebraic derivation of evolution operator for a non-relativistic simple harmonic oscillator and a moving solitonic wave of a free particle in the space with oscillating metric. After making comparison with other approaches and commenting on