Quantum Evolution Equation Research Articles

Off-shell electromagnetism in manifestly covariant relativistic quantum mechanics

Gauge invariance of a manifestly covariant relativistic quantum theory with evolution according to an invariant time τ implies the existence of five gauge compensation fields, which we shall call pre-Maxwell fields. A Lagrangian which generates the equations of motion for the matter field (coinciding with the Schrodinger type quantum evolution equation) as well as equations, on a five-dimensional manifold, for the gauge fields, is written. It is shown that τ integration of the equations for the pre-Maxwell fields results in the usual Maxwell equations with conserved current source. The analog of the O (3, 1) symmetry of the usual Maxwell theory is found to be O (3, 2) or O (4, 1), depending on the space-time Fourier spectrum of the field. We argue that the structure that is relevant to the description of radiation in interaction with matter evolving in a timelike sense is that of O (3, 2). The noncovariant form of the field equations is given; there are two fields of electric type and one (divergenceless) magnetic type field. The Noether currents are studied, and some remarks are made on second quantization.

Two time localization

Although quantum evolution equations are first order in time it is possible to define a state through partial information at multiple times. We consider the specification of position at two times and for a given Hamiltonian seek the wavefunction that minimizes the sum of the spreads at those times. We also study the time dependence of that minimal spread for a variety of Hamiltonians and boundary conditions.

Quantum motion of particles in random dynamic fields and quantum dissipation: Schrodinger equation with Gaussian fluctuating potentials

The Schrodinger equation with Gaussian dynamic fluctuating potentials is considered. This model gives a convenient example of an open quantum system coupled to a given external environment. The evolution equations for averaged correlators are derived and it is shown that a system coupled to a thermoequilibrium environment tends asymptotically with time to a thermodynamic density matrix. At high temperatures the quantum evolution equation for the density matrix can be approximated by the quasiclassical Fokker-Planck equation. The generalisation for multiparticle systems is also given. The results are illustrated by the particular examples of coupling with phonons and of two-level systems. Finally, it is shown how the problem can be reformulated in terms of continuous stochastic evolution in the space of psi functions and a corresponding functional formalism based on the continuous Fokker-Planck equation in the space of psi functions.

Aspects of quantum physics in multidimensional spaces

Several authors have analysed the possibility that gravity, both at the semi-classical and at the full quantum level, may determine an apparent modification of the quantum evolution equation (QEE). Gross has also suggested that Kaluza-Klein (KK) theories could provide more suitable theoretical laboratories for studying the problems of quantum gravity, and particularly those concerning the QEE, than ordinary fourdimensional gravitation. We indicate here a simple reason why nonzero modes of the metric in KK theories could affect low-energy physics, not only in the sense discussed by Appelquist, and Chodos, but also as far as the QEE is concerned. We also discuss, in terms of a simple model, the circumstances under which the modified QEE may simplify to a Markovian form.

Quantum and classical Liouville dynamics of the anharmonic oscillator.

We consider the dynamics of a quantum joint phase-space probability density in an exactly solvable model. The density is defined to be a true (i.e., positive) probability distribution for approximate (and thus simultaneously measurable) position and momentum variables. The dynamics of the quantum density is governed by a second-order partial-differential equation with non-positive-definite second-order coefficients. The quantum dynamics is contrasted with the dynamics of a similar joint density in a classical description. The non-positive-definite second-order terms in the quantum evolution equation, not present in the classical case, are responsible for quantum recurrences and prevent the appearance of fine-scale-structure ``whorls'' predicted in a classical description. The generation of ``squeezing'' in the model is also discussed.

The classical field limit of scattering theory for non-relativistic many-boson systems. II

We study the classical field limit of non relativistic many-boson theories in space dimensionn≧3, extending the results of a previous paper to more singular interactions. We prove the expected results: when ħ tends to zero, the quantum theory tends in a suitable sense to the corresponding classical field theory, and the quantum fluctuations are governed by the equation obtained by linearizing the quantum evolution equation around the classical solution. These results hold uniformly in time and therefore apply to scattering theory. The interactions considered here are so singular as to require a change of domain in order to define the generator of the evolution of the fluctuations, but sufficiently regular so that no energy renormalization is needed.