Mostafazadeh PLA Time-dependent Hilbert spaces, geometric phases, and general convariance in quantum mechanics.pdf

Physics Letters A 320 (2004) 375–382
ate/pla
Time-dependent Hilbert spaces, geometric phases,
and general covariance in quantum mechanics
Ali Mostafazadeh
Department of Mathematics, Koç University, Rumelifeneri Yolu, 34450 Sariyer, Istanbul, Turkey
Received 10 November 2003; accepted 1 December 2003
Communicated by . Holland
Abstract
We investigate consequences of allowing the Hilbert space of a quantum system to have a time-dependent metric. For a
given possibly nonstationary quantum system, we show that the requirement of having a unitary Schrödinger time-evolution
identifies the metric with a positive-definite (Ermakov–Lewis) dynamical invariant of the system. Therefore the geometric
phases are determined by the metric. We construct a unitary map relating a given time-independent Hilbert space to the time-
dependent Hilbert space defined by a positive-definite dynamical invariant. This map defines a transformation that changes the
metric of the Hilbert space but leaves the Hamiltonian of the system invariant. We propose to identify this phenomenon with a
quantum mechanical analogue of the principle of general covariance of general relativity. ment on the implications of
this principle for geometrically equivalent quantum systems and investigate the underlying symmetry group.
 2003 Elsevier . All rights reserved.
1. Introduction Unlike GR that is a