What is it about?
Fisher information is a cornerstone of both statistical inference and physical theory, leading to debate about whether its latter role is active or passive. Motivated by connections between Fisher information, entropy, and the quantum potential in the de Broglie-Bohm causal interpretation of quantum mechanics, the purpose of this article is to derive the position probability density when there a is a close and ubiquitous bonding of Fisher information and quantum potential. This is done by exploring a case in which a particle moves in a straight-line and the integrands in Fisher information and expected quantum potential are proportional. It is found that in this case the probability density given by the Schrödinger wave equation has a Laplace distribution and that quantum potential is a negative constant at all points and times. It is noted that the rate of change of the entropy of the particle is bounded above by a limit that is proportional to the square roots of both Fisher information and the absolute value of quantum potential. Unlike Fisher information, quantum potential is a measure of a real physical potential, and it is proposed that it is quantum potential that puts an upper bound on the rate of change of the particle’s entropy and that, being negative in this case, may also act to contain the particle on its straight-line path. It is suggested that Fisher information does not have an active role in the physics, at least in this case, and only provides information about entropy.
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Why is it important?
The special case explored in the article is significant because the Laplace probability density function that applies to it describes a quantum mechanical process with the following unique set of properties. A particle can show up anywhere its straight-line path once its motion begins regardless of where it was emitted, giving the process a form of nonlocality. The parameter that determines the particle’s mean position can be set anywhere in the direction of travel and the scale parameter, which determines the spread and entropy of the particle’s wave packet, may be set at any positive number, and stay within the Rao-Cramer bound. The variance of the pdf is constant, so the wave packet containing the particle neither spreads nor contracts over time. There is a linear relationship between quantum potential and the contribution to Fisher information at every spatial point and time so that, the constant of proportionality permitting, the role of quantum potential in a process with this property can equally apply, or be attributed, to Fisher information, raising the question of which is active. The quantum potential is constant at every spatial point and time, and being constant, exerts no force on the particle, whose velocity therefore remains constant.
Perspectives
The close bonding of quantum potential and Fisher information and the probability density function derived from it, resonate with the concept presented by Bohm and Hiley in 1975 when, building on Bohm’s interpretation of quantum theory in 1952, they presented how the concept of a quantum potential leads to the notion of an “unbroken wholeness of the entire universe”, proposing that the fundamental new quality introduced by quantum physics is nonlocality. To quote from the synopsis of the book they co-authored in 1993, bringing their earlier proposals together “In The Undivided Universe, David Bohm, and Basil Hiley develop an interpretation of quantum mechanics which gives a clear, intuitive understanding of its meaning and in which there is a coherent notion of the reality of the universe without assuming a fundamental role for the human observer. With the aid of new concepts such as active information together with nonlocality, they provide a comprehensive account of all the basic features of quantum mechanics,” Perhaps then, the intrinsic property explored here and the probability density function it uncovers, and its ubiquitous bonding of Fisher information with quantum potential, might fit somewhere into the notion of the universe’s “unbroken wholeness” and its nonlocality.
Mr Tim C Jenkins
Independent Researcher
Read the Original
This page is a summary of: An exploration of a special case in the relationship between Fisher information and quantum potential in the causal interpretation of quantum mechanics, The European Physical Journal Plus, January 2021, Springer Science + Business Media,
DOI: 10.1140/epjp/s13360-020-01050-y.
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