Quantization ambiguity, ergodicity and semiclassics

Abstract

A simple argument shows that eigenstates of a classically ergodic system are individually ergodic on coarse-grained scales. This has implications for the quantization ambiguity in ergodic systems: the difference between alternative quantizations is suppressed compared with the O(h 2 ) ambiguity in the integrable case. For two-dimensional ergodic systems in the high-energy regime, individual eigenstates are independent of the choice of quantization procedure, in contrast with the regular case, where even the ordering of eigenlevels is ambiguous. Surprisingly, semiclassical methods are shown to be much more precise for chaotic than for integrable systems.For many years, it has been widely recognized that "quantizing" a given classical system is inherently an ambiguous procedure, as a large family of quantum Hamiltonians may have the same classical limit [1]. For example, given the classical dynamics of a particle constrained to move on a closed loop, different choices of boundary condition give rise to different phases relating classical paths of different winding number. Knowledge of these Aharonov-Bohm phases is of course necessary to construct a semiclassical dynamics (which includes interference between classical paths), and thus many semiclassical theories correspond to the same classical dynamics. Physically, this O(h) or gauge ambiguity may be associated with the possibility of varying the magnetic flux enclosed by the loop. This is not all, however: there are also many quantum theories, differing at O(h 2 ) or higher in the Hamiltonian, which all have the same semi-classical limit. There are many ways of seeing this O(h 2 ) ambiguity; one of the simplest is to imagine making a canonical transformation on the classical phase space, applying the canonical quantization prescription to the new coordinates, and then transforming back to the original coordinate system. Generically, one then obtains a new quantum Hamiltonian which differs from the original by O(h 2 ) plus higher order terms:Ĥwhere the operator has a well-defined classical limit [2]. As classical dynamics is of course independent of the choice of coordinate system, this implies that quantization is inherently ambiguous at second order inh. An even more striking case is that of a particle constrained to move on a two-dimensional surface embedded * kaplan@physics.harvard.edu in three-dimensional space. Here, the non-trivial metric contained in the kinetic term gives rise to obvious operator-ordering ambiguities in canonical quantization; this has led to much discussion in the literature over whether a term proportional to the local Gaussian curvature R of the surface should be added to the Hamiltonian, and if so, what the proportionality constant should be (different prescriptions suggestingh 2 R/8,h 2 R/6, and h 2 R/12 as the "correct" answer) [3][4][5]. In the path integral approach, ambiguities at the same order arise in choosing how to incorporate the metric into the kernel and in deciding at what point in the infinitesimal time interval to eval...

A random matrix formulation of fidelity decay

“…The transforms of simple functions such as Gaussians are readily done analytically, while we would like to exploit the quantum algorithm for an arbitrarily complicated case. Yet it is also interesting to consider eigenfunctions partially because of the applications in the context of effects of different types of quantisation [5] but also because these display most markedly the effects of spectral fluctuations.…”

Dephasing representation: Employing the shadowing theorem to calculate quantum correlation functions

“…In particular, we have rigorously shown that DR is accurate for systems and times where ST applies. The many-to-one relationship between classical and quantum dynamics on which DR is based has been exploited before [15,16]. While the replacement-manifold method [16] is very simple and gives excellent results in a variety of problems, it requires finding the replacement manifolds.…”

Mode-resolved travel-time statistics for elastic rays in three-dimensional billiards

“…In the field of quantum chaos it is well established that one can construct semiclassical approximations, on nothing else than these classical trajectories [18]. These approximations may remain valid for much longer times -in the case of chaotic two degreeof-freedom systems up to times of the order of the Heisenberg time [19]. Ultimately, the present work might help to pave the way towards a similar semiclassical theory for elastodynamic systems.…”