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How the Classical World Got Its Localization: an Elementary Account of How the Age of the Universe May be Implicated in the Quantum-Classical Transition

C. L. Herzenberg

Published Sep 1, 2009• 0 citations
OpenAlex

Abstract

An expanding universe of finite duration appears to impose limits on the temporal and spatial extent of quantum waves. These limitations seem to be able to bring about localization for sufficiently large quantum objects that can resemble classical behavior. A threshold for a transition from quantum to classical behavior of a physical object is derived in terms of the magnitude of its moment of inertia.

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An expanding universe of finite duration appears to impose limits on the temporal and spatial extent of quantum waves. These limitations seem to be able to bring about localization for sufficiently large quantum objects that can resemble classical behavior. A threshold for a transition from quantum to classical behavior of a physical object is derived in terms of the magnitude of its moment of inertia.
Independent studies by different authors have proposed that classicality may be induced in quantum objects by cosmological constraints presented by an expanding universe of finite extent in space-time. Cosmological effects on a quantum system can be explored in one approach by considering an object at rest in space with a universal Hubble expansion taking place away from it, and developing a Schroedinger type governing differential equation incorporating an intrinsic expansion speed. Wave function solutions to this governing equation exhibit pronounced central localization. The extent of concentration of probability depends on mass; objects with small masses tend to behave in a delocalized manner as ordinary quantum objects do in a static space, while quantum objects with large masses are concentrated into much smaller regions. To develop a criterion for classicality, we consider that if the size of the localized region of concentrated probability density is larger than the size of the corresponding extended object, then quantum behavior could be expected; whereas if the region of high probability density for the location of the center of mass is smaller than the size of the object, the object would behave in a more classical manner. The resultant size threshold for classicality accords with results of other studies examining these issues based on uncertainty relations and wave packets. This size threshold is informative for the case of compact extended objects and, as the constraint applies to the center of mass of the system, does not lead to inconsistencies for quantum correlations between distant entangled quantum objects. While local decoherence may lead to classicality under a variety of conditions, cosmologically induced classicality would appear to cause fundamental limitations on quantum behavior in our universe.
During the last few years, several studies have proposed the existence of a threshold separating classical from quantum behavior of objects that is dependent on the size and mass of an object as well as being dependent on certain properties usually associated with the universe as a whole. Here, we reexamine the results of these studies and recast the threshold criteria in terms of a critical threshold value for the moments of inertia of physical objects. Physical objects having moments of inertia above this critical threshold value would be expected to behave necessarily in a classical manner in terms of their center of mass motion as entire objects, while physical objects having moments of inertia lower than this threshold value could exhibit quantum behavior unless brought into classicality by other effects. A comparison with observed values of moments of inertia is presented, and the moment of inertia is suggested as a classifying parameter for examination of the quantum versus classical behavior of objects in the mesoscale domain.
Although cosmic expansion at very small distances is usually dismissed as entirely inconsequential, it appears that these extraordinarily small effects may in fact have a real and significant influence on our world. Calculations suggest that the minute recessional velocities associated with regions encompassed by extended bodies may have a role in creating the distinction between quantum and classical behavior. Using an uncertainty in position estimated from the spread in velocities associated with its size, the criterion that the uncertainty in position should be smaller than the extension of the object leads to a threshold size that could provide a fundamental limit distinguishing the realm of objects governed by classical laws from those governed by quantum mechanics.
We consider an object at rest in space with a universal Hubble expansion taking place away from it. We find that a governing differential equation developed from the Schroedinger equation leads to wave functions which turn out to exhibit pronounced central localization. The extent of concentration of probability depends on the mass; objects with small masses tend to behave in a delocalized manner as ordinary quantum objects do in a static space, while quantum objects with large masses have wave functions that are largely concentrated into much smaller regions. This in turn suggests the possibility that classical behavior is being induced in quantum objects by the presence of the Hubble expansion. If the size of the localized region of concentrated probability density is larger than the size of the corresponding extended object, quantum behavior might be expected; whereas classical behavior might be expected for cases in which the region of high probability density is smaller than the size of the object. The resulting quantum-classical boundary due to Hubble expansion may be expressed in terms of a relationship between the size and mass of an object, or may be expressed in terms of a threshold moment of inertia.
Recently, what appears to be a fundamental limit associated with the size of an object that separates the quantum behavior characterizing small objects from the classical behavior characterizing large objects has been derived from the Hubble velocity spread in an extended object. This threshold is now examined further and interpreted in terms of diffusion processes in stochastic quantum mechanics. This limiting size that separates quantum behavior from classical behavior is shown to correspond approximately to the diffusion distance of the object over the Hubble time.