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arXiv:chao-dyn/9309004 · Physical Review Letters

Semiquantum chaos

Fred Cooper, John F. Dawson, Dawn C. Meredith, Harvey K. Shepard

Published Feb 28, 1994• 48 citations
arXivHTMLPDF

Abstract

We consider a system in which a classical oscillator is interacting with a purely quantum mechanical oscillator, described by the Lagrangian L=1/2x${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}^{2}$+1/2A${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}^{2}$-1/2(${\mathit{m}}^{2}$+${\mathit{e}}^{2}$${\mathit{A}}^{2}$)${\mathit{x}}^{2}$, where A is a classical variable and x is a quantum operator. With 〈x(t)〉=0, the relevant variable for the quantum oscillator is 〈x(t)x(t)〉=G(t). The classical Hamiltonian dynamics governing the variables A(t),${\mathrm{\ensuremath{\Pi}}}_{\mathit{A}}$(t),G(t), and ${\mathrm{\ensuremath{\Pi}}}_{\mathit{G}}$(t) is chaotic so that the results of making measurements on the quantum system at later times are sensitive to initial conditions. This system arises as the zero momentum part of the problem of pair production of charged scalar particles by a strong external electric field.

DOI
10.1103/physrevlett.72.1337

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We consider a system in which a classical oscillator is interacting with a purely quantum mechanical oscillator, described by the Lagrangian L=1/2x\ifmmode˙\else˙\fi2{\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}^{2}\ifmmode˙\else˙\fi2+1/2A\ifmmode˙\else˙\fi2{\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}^{2}\ifmmode˙\else˙\fi2-1/2(m2{\mathit{m}}^{2}m2+{\mathit{e}}^{2}$${\mathit{A}}^{2})x2{\mathit{x}}^{2}x2, where A is a classical variable and x is a quantum operator. With 〈x(t)〉=0, the relevant variable for the quantum oscillator is 〈x(t)x(t)〉=G(t). The classical Hamiltonian dynamics governing the variables A(t),\ensuremathΠA{\mathrm{\ensuremath{\Pi}}}_{\mathit{A}}\ensuremathΠA​(t),G(t), and \ensuremathΠG{\mathrm{\ensuremath{\Pi}}}_{\mathit{G}}\ensuremathΠG​(t) is chaotic so that the results of making measurements on the quantum system at later times are sensitive to initial conditions. This system arises as the zero momentum part of the problem of pair production of charged scalar particles by a strong external electric field.
A semiquantum system, composed of a quantum part coupled to a classical part, can exhibit dynamical chaos in the motion of the quantum state vector. However, there has been disagreement as to whether this mathematical chaos is physically real or merely an artifact of the semiquantum approximation. It is shown, for a model of a quantum spin coupled to an approximately classical nonlinear oscillator, that the semiquantum chaos disappears rapidly as the mass of the oscillator is increased to make it more classical. The time interval during which the semiquantum approximation remains accurate increases with the mass, but is not closely related to the Lyapunov time.
The possibility of quantum chaos for vectorial interactions is here described in some detail, along with its apparent suppression when the radiative corrections of quantum field theory (quantum fluctuations of the classical, “external” electromagnetic field) are introduced. Based on these quantum-mechanical ideas, an application is made to classical-chaotic systems, of perhaps the simplest form – a forced Duffing model, without damping – where it is found that the chaos is first suppressed and then (apparently) removed by introducing couplings to random and/or chaotic sources. This may be characterized as “quantum mechanics with ħ ∼ 1”, and suggests a brute-force method by which the chaos of a classical system may be at least diminished. A similar effect is noted for a different classical system that displays chaos – a pair of coupled oscillators – independently of any external forcing. First-quantization chaos The reader is now asked to return to the G c [ A ] representations for vectorial interactions of the previous chapter; for simplicity, the arguments of this chapter are presented only for QED in a relativistic context, but have obvious generalizations to non-relativistic QED, and to relativistic QCD. The map (5.31) defines the quantity Ω μ ( s′ ), which is needed for the explicit solution of (5.33). It is the existence of such a map which carries with it the inescapable possibility of chaotic behavior, at least in the present context of vectorial interactions in potential theory. The analysis used here is given directly in terms of proper time τ, of which x μ (τ) is a function.
A relativistically invariant representation of the generalized momentum of a particle in an external field is proposed. In this representation, the dependence of the potentials of the interaction of the particle with the field on the particle velocity is taken into account. The exact correspondence of the expressions of energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions to the problems of mechanics with relativistic and nonrelativistic approaches. The invariance of the proposed representation of the generalized momentum makes it possible to equivalently describe a physical system in geometrically conjugate spaces of kinematic and dynamic variables. Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the constant interaction of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus is Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion, and the expression for the force acting on the charge are derived when moving in an external electromagnetic field. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), and also the problems of a hydrogen atom are given.
Relativistic, scalar particles are considered, contained in a box with periodic boundary conditions. Although interactions are not expected to be a fundamental problem, we concentrate on free particles. By considering them to be harmonic oscillators, it is found that their dynamical variables can be replaced by a completely ontological set, which means that, here, quantum mechanics does not deviate from a purely geometric, ontological particle system. The effects of the mass terms are included. Locality holds for the quantum theory, and seems to be fully obeyed also by the classical treatment, although further discussion will be needed. Quantised interactions are briefly speculated on, but mostly postponed to later. We do discuss extensively the distinction between the quantum treatment and the classical one, even though they produce exactly the same equations mathematically. We briefly explain how this result can be squared with the usual quantum no-go theorems. It is suggested to apply this theory for real time quantum model simulations.