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Journal of Physics A Mathematical and General

Quasiclassical approximation for ultralocal scalar fields

G Francisco

Published Mar 11, 1986• 2 citations
Publisher

Abstract

The author shows how to obtain the quasiclassical evolution of a class of field theories called ultralocal fields. Coherent states that follow the 'classical' orbit as defined by Klauder's weak correspondence principle and restricted action principles are explicitly shown to approximate to the quantum evolution as h(cross) to O.

DOI
10.1088/0305-4470/19/4/012

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The author shows how to obtain the quasiclassical evolution of a class of field theories called ultralocal fields. Coherent states that follow the 'classical' orbit as defined by Klauder's weak correspondence principle and restricted action principles are explicitly shown to approximate to the quantum evolution as h(cross) to O.
In a wide range of quantum gravity theories, quasiclassical geometries, which are solutions to the Einstein field equations approximately, are described by "coherent states." Here we propose a Hamiltonian formalism for gravitational dynamics with respect to this coherent state basis, which generates time evolution of the spacetime with respect to a clock at infinity. Since the coherent states are not orthogonal, an initial quasiclassical geometry is dynamically driven into a superposition of different amplitudes. Our framework provides a dynamical mechanism for tunneling between geometries that is ubiquitous in a number of approaches to quantum gravity, from loop quantum gravity to the Euclidean path integral. We apply our framework to the problem of black hole evaporation, providing a hint at how unitarity may be preserved with the inclusion of quantum corrections to the semiclassical evolution of the black hole.
The Kustaanheino-Stiefel transformation is used to transform the Coulomb problem to a four-dimensional isotropic harmonic oscillator with a constraint. Ordinary coherent states are introduced over this oscillator and are shown to evolve in a fictitious time. When projected back into the physical space, the states follow the classical Kepler orbits in physical time. The resulting quasiclassical picture is obtained without the Bohr correspondence principle of taking only contributions from high principal quantum numbers.
The symmetries of the gradient free ultralocal model quantum field theories are studied. The internal parameter λ introduced by J. R. Klauder is replaced by an r-component vector and used to obtain an r-vector ultralocal field operator φ̃. Then ultralocal many-body models with translational and rotational symmetry are set up by partitioning λ⃗ into N, 3-d ’’relative internal coordinates.’’ Hartree-type 1/N limits are studied in this model and found to be accurate only if contributions from ‖χ⃗‖≳≳0 are negligible. A brief sketch is given of how to produce more general UN and pairing interaction ultralocal models.
Many field states in quantum optics may be regarded as quasiclassical in that they possess a fairly well-defined phase and amplitude in the limit in which the average number of photons is large. Examples include coherent states and squeezed states that have a coherent component substantially larger than the squeezed noise contribution. For such states, calculations involving sums over the coefficients in the Fock-state basis can be considerably simplified in the limit of large photon number. The Jaynes-Cummings model and its extensions (to, e.g., three-level, two-mode configurations) may be advantageously studied from this perspective. New results on the three-level, two-mode problems, on the influence of cavity losses, and on the various time scales of interest will be presented here. The method typically identifies a set of privileged, long-lived atomic states, which lead to a quasiclassical evolution that agrees with the prediction of semiclassical treatments over a characteristic time scale; linear combinations of the long-lived states, on the other hand, become superpositions of macroscopically distinct states in time and result in an evolution that exhibits many uniquely quantum effects.
We study the dispersion of the `temporally stable' coherent states for the hydrogen atom introduced by Klauder. These are states which under temporal evolution by the hydrogen atom Hamiltonian retain their coherence properties. We show that in the hydrogen atom such wavepackets do not move quasiclassically; i.e. they do not follow, with no or little dispersion, the Keplerian orbits of the classical electron. The poor quantum - classical correspondence does not improve in the semiclassical limit.