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International Journal of Modern Physics B

TOWARDS A THEORY OF DECOHERENCE

Gianfausto Dell’Antonio

Published Feb 20, 2004• 8 citations
Publisher

Abstract

Consider a quantum particle of mass M in R3, described at time 0 by a wave function ϕ(x) with dispersion Δ, interacting independently with a collection of N particles of mass m. Using only Schroedinger's Quantum Mechanics we prove that when N becomes large and m/M becomes small, and if the information at time t>0 about the N particles of small mass in negleted, the system admits a "classical" description, i.e. a description in which the coherence of the wave function over distances of the order of mM-1N-1Δ have disappeared. We consider this a first step towards proving that most "sufficiently large" quantum systems interacting with an uncontrolled environment admit a classical description at least for position measurements.

DOI
10.1142/s0217979204024264

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Consider a quantum particle of mass M in R3, described at time 0 by a wave function ϕ(x) with dispersion Δ, interacting independently with a collection of N particles of mass m. Using only Schroedinger's Quantum Mechanics we prove that when N becomes large and m/M becomes small, and if the information at time t>0 about the N particles of small mass in negleted, the system admits a "classical" description, i.e. a description in which the coherence of the wave function over distances of the order of mM-1N-1Δ have disappeared. We consider this a first step towards proving that most "sufficiently large" quantum systems interacting with an uncontrolled environment admit a classical description at least for position measurements.
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A model for intrinsic decoherence in quantum mechanics is proposed, based on a simple modification of unitary Schr\"odinger evolution. On sufficiently small time scales the system evolves by a random sequence of unitary phase changes generated by the Hamiltonian. The Schr\"odinger equation is obtained to zeroth order in the expansion parameter. Higher-order corrections lead to a loss of coherence in the energy basis. The rate of coherence loss becomes very large as the energy scale of the system is increased. The expansion parameter determines an uncertainty in the time step on very short times scales. A number of testable consequences are derived including anomalous dispersion of a free particle, decay of oscillatory systems, destruction of interference-fringe visibility, and a phase shift of interference fringes.
We consider the claim that decoherence explains the emergence of classicality in quantum systems, and conclude that it cannot. We show that, given a universe composed of a variety of subsystems, some of which are macroscopic and subject to decoherence-inducing interactions, and some of which are microscopic, the macroscopic subsystems do not display any distinctively classical behavior. Therefore, a universe in which macroscopic and microscopic do display distinct behavior must be in a very special, highly nongeneric quantum state.
The mechanism of the transition of a dynamical system from quantum to classical mechanics is of continuing interest. Practically it is of importance for the interpretation of multi-particle coincidence measurements performed at macroscopic distances from a microscopic reaction zone. Here we prove the generalized iimaging theorem which shows that the spatial wave function of any multi-particle quantum system, propagating over distances and times large on an atomic scale but still microscopic, and subject to deterministic external fields and particle interactions, becomes proportional to the initial momentum wave function where the position and momentum coordinates define a classical trajectory. Currently, the quantum to classical transition is considered to occur via decoherence caused by stochastic interaction with an environment. The imaging theorem arises from unitary Schroedinger propagation and so is valid without any environmental interaction. It implies that a simultaneous measurement of both position and momentum will define a unique classical trajectory, whereas a less complete measurement of say position alone can lead to quantum interference effects.
The mechanism of the transition of a dynamical system from quantum to classical mechanics is one of the remaining challenges of quantum theory. Currently, it is considered to occur via decoherence caused by entanglement and/or stochastic interaction with a quantum environment. Here we prove that, in the absence of de-phasing environmental interaction, the asymptotic spatial wave function of any quantum system, propagating over distances and times large on an atomic scale but still microscopic, and subject to arbitrary deterministic fields, becomes proportional to the initial momentum wave function, \emph{where the particle positions and momenta are related by classical mechanics.} This implies that detection of particle positions and momenta at different times will yield results in accordance with classical mechanics, without the need to postulate decoherence of the wave function.
In recent years decoherence has been widely discussed and accepted as a mechanism responsible for the emergence of classicality in quantum measurements and the absence in the real world of Schrödinger-cat-like states [ 2 , 6 , 7 , 11 , 12 ]. Let me start with recalling the Schrödinger cat paradox. It was a hypothetical experiment proposed by Schrödinger in 1935 which can be briefly described as follows. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.