Time-dependent Hilbert Spaces Research Articles

Consistent Treatment of Quantum Systems with a Time-Dependent Hilbert Space.

We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a time-independent Hilbert space. We show that in general the Hamiltonian operator does not represent an observable of the system even if it is a self-adjoint operator. This is related to a hidden geometric aspect of quantum mechanics arising from the presence of an operator-valued gauge potential. We also offer a careful treatment of quantum systems whose Hilbert space is obtained by endowing a time-independent vector space with a time-dependent inner product.

Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics.

A non-Hermitian operator H defined in a Hilbert space with inner product may serve as the Hamiltonian for a unitary quantum system if it is -pseudo-Hermitian for a metric operator (positive-definite automorphism) . The latter defines the inner product of the physical Hilbert space of the system. For situations where some of the eigenstates of H depend on time, becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM.

Unitary time-evolution in stochastic time-dependent Hilbert spaces

In this work we study the unitary time-evolutions of quantum systems defined on infinite-dimensional separable time-dependent Hilbert spaces. Two possible cases are considered: a quantum system defined on a stochastic interval and another one defined on a Hilbert space with stochastic integration measure (stochastic time-dependent scalar product). The formulations of the two problems and a comparison with the general theory of open quantum systems are discussed. Possible physical applications of the situations considered are analyzed.

Gradient flow for the Boltzmann entropy and Cheeger’s energy on time-dependent metric measure spaces

We introduce notions of dynamic gradient flows on time-dependent metric spaces as well as on time-dependent Hilbert spaces. We prove existence of solutions for a class of time dependent energy functionals in both settings. In particular we are interested in the case when the underlying spaces are metric measure spaces and the energy functional is given by the time-dependent Boltzmann entropy or the time-dependent Cheeger's energy. Under the assumption that each static space satisfies a lower Ricci curvature bound, we prove existence and uniqueness of the entropy gradient flow and we identify it with the forward adjoint heat flow introduced in "Super-Ricci Flows for Metric Measure Spaces" by Kopfer/Sturm. We identify the gradient flow for the time-dependent Cheeger's energy with the heat flow introduced by Kopfer/Sturm.

Time-dependent Hilbert spaces, geometric phases, and general covariance in quantum mechanics

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. We comment on the implications of this principle for geometrically equivalent quantum systems and investigate the underlying symmetry group.

Quantum fields in nonstatic background: A histories perspective

For a quantum field living on a nonstatic space–time no instantaneous Hamiltonian is definable, for this generically necessitates a choice of inequivalent representation of the canonical commutation relations at each instant of time. This fact suggests a description in terms of time-dependent Hilbert spaces, a concept that fits naturally in a (consistent) histories framework. Our primary tool for the construction of the quantum theory in a continuous-time histories format is the recently developed formalism based on the notion of the history group. This we employ to study a model system involving a 1+1 scalar field in a cavity with moving boundaries. The instantaneous (smeared) Hamiltonian and a decoherence functional are then rigorously defined so that finite values for the time-averaged particle creation rate are obtainable through the study of energy histories. We also construct the Schwinger–Keldysh closed-time-path generating functional as a “Fourier transform” of the decoherence functional and evaluate the corresponding n-point functions.