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arXiv:quant-ph/0604178 · Physical Review A

Born rule in quantum and classical mechanics

Paul W. Brumer, Jiangbin Gong

Published May 19, 2006• 74 citations
arXivHTMLPDF

Abstract

Considerable effort has been devoted to deriving the Born rule [i.e., that ${\ensuremath{\mid}\ensuremath{\psi}(x)\ensuremath{\mid}}^{2}dx$ is the probability of finding a system, described by $\ensuremath{\psi}$, between $x$ and $x+dx$] in quantum mechanics. Here we show that the Born rule is not solely quantum mechanical; rather, it arises naturally in the Hilbert-space formulation of classical mechanics as well. These results provide insights into the nature of the Born rule, and impact on its understanding in the framework of quantum mechanics.

DOI
10.1103/physreva.73.052109

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Considerable effort has been devoted to deriving the Born rule [i.e., that \ensuremath∣\ensuremathψ(x)\ensuremath∣2dx{\ensuremath{\mid}\ensuremath{\psi}(x)\ensuremath{\mid}}^{2}dx\ensuremath∣\ensuremathψ(x)\ensuremath∣2dx is the probability of finding a system, described by \ensuremathψ\ensuremath{\psi}\ensuremathψ, between xxx and x+dxx+dxx+dx] in quantum mechanics. Here we show that the Born rule is not solely quantum mechanical; rather, it arises naturally in the Hilbert-space formulation of classical mechanics as well. These results provide insights into the nature of the Born rule, and impact on its understanding in the framework of quantum mechanics.
Abstract Logical inference leads to one of the major interpretations of probability theory called logical interpretation, in which the probability is seen as a measure of the plausibility of a logical statement under incomplete information. Assuming that our usual inference procedure is working rationally for every set of logical propositions represented in terms of commuting projectors on a given Hilbert space, we extend the logical interpretation to quantum mechanics and derive the Born rule. Our result implies that, from epistemological viewpoints, we can regard quantum mechanics as a natural extension of classical probability theory.
The predictions of quantum mechanics are probabilistic. Quantum probabilities are extracted using a postulate of the theory called the Born rule, the status of which is central to the problem of quantum mechanics. Efforts to justify the Born rule from other physical principles, and thus elucidate the measurement process, have involved lengthy statistical or information-theoretic arguments. Here we show that Bohm's deterministic formulation of quantum mechanics allows the Born rule for measurements on a single system to be derived, without any statistical assumptions. We solve a simple example where the creation of an ensemble of identical quantum states, together with position measurements on those states, are described by Bohm's quantum dynamics. The calculated measurement outcomes agree with the Born-rule probabilities, which are thus a consequence of deterministic evolution. Our results demonstrate that quantum probabilities can emerge from simple dynamical laws alone, and they support the view that there is no underlying indeterminism in quantum phenomena.
The emergence of intrinsic probability has long been one of the most important and puzzling problems in quantum mechanics, and the law most directly related to this problem is the Born rule. For a century, there have been many attempts to derive the Born rule as a theorem rather than postulating it. However, existing derivations of the Born rule are each based on different frameworks and have attracted different criticisms. The assumptions from which they start are also highly divergent, and the connections between them have not been sufficiently studied. These possible connections are very likely to be the key to answering questions about the origin of probability in quantum mechanics. This paper focuses on proving the necessity and indispensability of the additivity assumption in the derivation of the Born rule. This supports the view that the Born rule cannot be derived solely from other non-probabilistic quantum or additional postulates. We first prove that additivity cannot be derived from two other commonly used non-probabilistic additional assumptions, non-contextuality and normalization. Then we analyze the crucial role of the additivity assumption in five important existing derivations of the Born rule. These include Gleason's Theorem, Busch's extension of Gleason's Theorem, the Deutsch-Wallace Theorem, Zurek's envariance proof, and the Finkelstein-Hartle Theorem. We show that these derivations either depend heavily on the additivity assumption or lead to obvious loopholes due to the lack of additivity. We also point out some problems arised from the lack of a non-contextuality assumption. Our results provide a novel insight into the important role of additivity assumption in quantum measurement, as well as into the origin of probability in quantum mechanics.
The Born Rule plays a critical role in quantum mechanics (QM) since it supplies the link between the mathematical formalism and experimental results in terms of probabilities. The Born Rule does not occur in ordinary probability theory. Where then does it come from? This has been a topic of considerable controversy in the literature. We take the approach of asking what is the simplest extension of ordinary probability theory where the Born rule appears. This is answered by showing that the Born Rule appears by adding the notion of superposition events (in addition to the ordinary discrete events) to finite probability theory. Hence the rule does not need any physics-based derivation. It is simply a feature of the mathematics of superposition when only superposition events are added to ordinary probability theory.