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arXiv:1509.02107 · International Journal of Modern Physics A

Inconstant Planck’s constant

G. Mangano, Fedele Lizzi, A. Porzio

Published Dec 9, 2015• 30 citations
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Abstract

Motivated by the Dirac idea that fundamental constants are dynamical variables and by conjectures on quantum structure of space–time at small distances, we consider the possibility that Planck constant [Formula: see text] is a time depending quantity, undergoing random Gaussian fluctuations around its measured constant mean value, with variance [Formula: see text] and a typical correlation timescale [Formula: see text]. We consider the case of propagation of a free particle and a one-dimensional harmonic oscillator coherent state, and show that the time evolution in both cases is different from the standard behavior. Finally, we discuss how interferometric experiments or exploiting coherent electromagnetic fields in a cavity may put effective bounds on the value of [Formula: see text].

DOI
10.1142/s0217751x15502097

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Motivated by the Dirac idea that fundamental constants are dynamical variables and by conjectures on quantum structure of space–time at small distances, we consider the possibility that Planck constant [Formula: see text] is a time depending quantity, undergoing random Gaussian fluctuations around its measured constant mean value, with variance [Formula: see text] and a typical correlation timescale [Formula: see text]. We consider the case of propagation of a free particle and a one-dimensional harmonic oscillator coherent state, and show that the time evolution in both cases is different from the standard behavior. Finally, we discuss how interferometric experiments or exploiting coherent electromagnetic fields in a cavity may put effective bounds on the value of [Formula: see text].
By successive applications of time-dependent rotation, translation and dilatation, it is shown that a charged time-dependent harmonic oscillator in a uniform time-dependent electromagnetic field is equivalent to a free particle. This equivalence is employed to present a simple derivation of the Feynman kernel as well as to obtain a concrete picture of the time evolution of a generic wave packet. A compact formula for an evolving anisotropic Gaussian packet is also derived. A special case of this formula gives a squeezed coherent state for a harmonic oscillator whose frequency is an arbitrary function of time. Finally we discuss an Aharonov-Bohm-Berry-like phase associated with a wave packet whose center traces a closed circuit.
Relations between Hamiltonian mechanics and quantum mechanics are studied. It is stressed that classical mechanics possesses all the specific features of quantum theory: operators, complex variables, probabilities (in case of ergodic systems). The Planck constant and the Fock space appear after putting a dynamical system in a thermal bath. For harmonic oscillator in a thermal bath, the probability amplitudes can be identified with the complex valued phase functions f(q+ip)f(q+ip)f(q+ip) describing small deviations from the equilibrium state, when the time of relaxation is large. A chain of such oscillators models both the one-dimensional space (or string), and one-dimensional quantum field theory.
The quantum measurement problems are revisited from a new perspective. One of the main ideas of this work is that the basic entities of our world are various types of particles, elementary or composite. It follows that each elementary process, hence each measurement process at its core, is a space–time, pointlike, event. Another key idea is that, when a microsystem [Formula: see text] gets into contact with the experimental device, factorization of [Formula: see text] rapidly fails and entangled mixed states appear. The wave functions for the microsystem-apparatus coupled systems for different measurement outcomes then lack overlapping space–time support. It means that the aftermath of each measurement is a single term in the sum: a “wave function collapse.” Our discussion leading to a diagonal density matrix, [Formula: see text] shows how the information encoded in the wave function [Formula: see text] gets transcribed, via entanglement with the experimental device and environment, into the relative frequencies [Formula: see text] for various experimental results [Formula: see text]. These results represent the first, significant steps towards filling in the logical gaps in the standard interpretation based on Born’s rule, and replacing it with a more natural one. Accepting objective reality of quantum fluctuations, independent of any experiments, and independently of human presence, one renounces the idea that in a fundamental, complete theory of Nature the result of each single experiment must necessarily be predictable. A few well-known puzzles such as the Schrödinger cat conundrum and the Einstein–Podolsky–Rosen (EPR) paradox are briefly reviewed: they can all be naturally explained away.
By successive applications of time-dependent rotation, translation and dilatation, it is shown that a charged time-dependent harmonic oscillator in a uniform time-dependent electromagnetic field is equivalent to a free particle. This equivalence is employed to present a simple derivation of the Feynman kernel as well as to obtain a concrete picture of the time evolution of a generic wave packet. A compact formula for an evolving anisotropic Gaussian packet is also derived. A special case of this formula gives a squeezed coherent state for a harmonic oscillator whose frequency is an arbitrary function of time. Finally we discuss an Aharonov-Bohm-Berry-like phase associated with a wave packet whose center traces a closed circuit.
The propagation of an initially Gaussian wave packet of width [Formula: see text] in Gross–Pitaevskii equation is extensively studied both for attractive and repulsive interactions. It is predicted analytically and verified numerically that for a free particle with attractive interaction, the dynamics of the width is governed by an effective potential which is sensitive to initial conditions. If [Formula: see text] is equal to a corresponding critical width [Formula: see text], then the packet will propagate in time with very little change in shape. These are in essence like coherent states. Whereas, if [Formula: see text], depending on the nature of the effective potential for chosen [Formula: see text] and the interaction strength [Formula: see text], the width of the packet in course of time, either oscillates with bounded width or will spread like a free particle. For a simple harmonic oscillator (SHO) also, we find that for [Formula: see text] smaller than a critical value, there always exists a coupling strength for which the packet simply oscillates about the mean position without changing its shape, once again providing a resemblance to a coherent state. We also consider the Morse potential, which interpolates between the free particle and the oscillator. For large attractive interactions, the two limiting dynamics (free and simple harmonic) are indeed observed but in the intermediate form of the potential where the nonlinear terms dominate in the dynamics, an initial Gaussian wave packet does not retain its shape. For repulsive interaction, the Gaussian packet always changes shape no matter what the system parameters are.