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arXiv:1101.3521

Counting Steps: A New Approach to Objective Probability in Physics

Amit Hagar, Giuseppe Sergioli

Published Jan 18, 2011• 1 citation
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Abstract

We propose a new interpretation of objective deterministic chances in statistical physics based on physical computational complexity. This notion applies to a single physical system (be it an experimental set--up in the lab, or a subsystem of the universe), and quantifies (1) the difficulty to realize a physical state given another, (2) the `distance' (in terms of physical resources) of a physical state from another, and (3) the size of the set of time--complexity functions that are compatible with the physical resources required to reach a physical state from another. This view (a) exorcises "ignorance" from statistical physics, and (b) underlies a new interpretation to non--relativistic quantum mechanics.

DOI
10.48550/arxiv.1101.3521

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We propose a new interpretation of objective deterministic chances in statistical physics based on physical computational complexity. This notion applies to a single physical system (be it an experimental set--up in the lab, or a subsystem of the universe), and quantifies (1) the difficulty to realize a physical state given another, (2) the `distance' (in terms of physical resources) of a physical state from another, and (3) the size of the set of time--complexity functions that are compatible with the physical resources required to reach a physical state from another. This view (a) exorcises "ignorance" from statistical physics, and (b) underlies a new interpretation to non--relativistic quantum mechanics.
We propose a new interpretation of objective probability in statistical physics based on physical computational complexity. This notion applies to a single physical system (be it an experimental set-up in the lab, or a subsystem of the universe), and quantifies (1) the difficulty to realize a physical state given another, (2) the 'distance' (in terms of physical resources) between a physical state and another, and (3) the size of the set of time-complexity functions that are compatible with the physical resources required to reach a physical state from another. This view (a) exorcises 'ignorance' from statistical physics, and (b) underlies a new interpretation to non-relativistic quantum mechanics.
Abstract Many theories and models from physics are probabilistic. This observation raises several philosophical questions: What are probabilities in physics? Do they reflect objective chances which exist independently of the human mind? Or do they only express subjective credences and thus capture our own uncertainty about the world? Finally, which metaphysical lessons, if at all, can one draw from the largely probabilistic character of physics? The chapters collected in this volume address these questions and provide a detailed philosophical appraisal of the status of probabilities in all of physics. Particular emphasis is laid upon statistical physics and quantum mechanics. Many chapters reflect a desire to understand probabilities from physics as objective chances. These chances are characterized, e.g., as time-averages, as probabilities from a best system in the terms of David Lewis, or using the Boltzmannian typicality approach. Other chapters are sympathetic to a Bayesian view of probabilities in physics. The chapters about quantum mechanics elucidate the peculiar characteristics of quantum correlations and discuss strategies to justify the Born Rule. Finally, the chapters of this volume demonstrate how closely interpretive issues about probabilities are entangled with other foundational problems of physics such as the Reversibility Paradox, the ontology of the quantum world and the question whether the world is deterministic.
The chance of a physical event is the objective, single-case probability that it will occur. In probabilistic physical theories like quantum mechanics, the chances of physical events play the formal role that the values of physical quantities play in classical (deterministic) physics, and there is a temptation to regard them on the model of the latter as describing intrinsic properties of the systems to which they are assigned.I argue that this understanding of chances in quantum mechanics, despite being a part of the orthodox interpretation of the theory and the most prevalent view in the physical community, is incompatible with a very wide range of metaphysical views about the nature of chance. The options that remain are unlikely to be attractive to scientists and scientifically minded philosophers.
The chance of a physical event is the objective, single-case probability that it will occur. In probabilistic physical theories like quantum mechanics, the chances of physical events play the formal role that the values of physical quantities play in classical (deterministic) physics, and there is a temptation to regard them on the model of the latter as describing intrinsic properties of the systems to which they are assigned.I argue that this understanding of chances in quantum mechanics, despite being a part of the orthodox interpretation of the theory and the most prevalent view in the physical community, is incompatible with a very wide range of metaphysical views about the nature of chance. The options that remain are unlikely to be attractive to scientists and scientifically minded philosophers.
In classical physics, probabilistic or statistical knowledge has been always related to ignorance or inaccurate subjective knowledge about an actual state of affairs. This idea has been extended to quantum mechanics through a completely incoherent interpretation of the Fermi-Dirac and Bose-Einstein statistics in terms of "strange" quantum particles. This interpretation, naturalized through a widespread "way of speaking" in the physics community, contradicts Born's physical account of Ψ as a "probability wave" which provides statistical information about outcomes that, in fact, cannot be interpreted in terms of 'ignorance about an actual state of affairs'. In the present paper we discuss how the metaphysics of actuality has played an essential role in limiting the possibilities of understating things differently. We propose instead a metaphysical scheme in terms of powers with definite potentia which allows us to consider quantum probability in a new light, namely, as providing objective knowledge about a potential state of affairs.
This paper deals with the notion of objective randomness in classical deterministic theories. After the introduction, section 2 establishes an important distinction between a strictly metaphysical thesis of determinism (as characterized in the Montague-Earman definition, for example), and the doctrine of determinism, which can be roughly characterized as a methodological set of principles. The doctrine of determinism is associated with the idea that probability assignments can only reflect our ignorance of facts, and it also grounds the (ontological) thesis of separability: A system or process can be characterized completely in terms of the properties that a system has when in a given state, independently of the properties of other systems, The key notions of "completeness" and "independence" are only briefly discussed, as they are examined more in depth elsewhere. Section three examines attempts to characterize a notion of objective randomness in ergodic theory. The characterization can be seen to be equivalent to the formulation of a notion of a "physically impossible process". One way of expressing this idea is grounded on the thesis of the "coarse graining" of our measuring instruments. This leads to the problem of distinguishing "objective" from "apparent" randomness. The problem seems to be intrinsic to any attempt of characterizing the required notion of physical impossibility (and thus of objective randomness) in terms of an ideal observer. The alternative of trying to characterize physical impossibility in terms of a theory of algorithms suffers from the difficulty that it is not clear what would be the required notion of (non-epistemic) computability. The "coarse graining" approach, as well as the alternative in terms of a theory of algorithms seem to share the usual confusion between a strictly metaphysical thesis of determinism and the (methodological) doctrine of determinism. In section four an alternative approach is suggested. It is noticed that a denial of the thesis of separability is compatible with a strict deterministic theory, and thus that at least some classical systems (the "statistical" ones) can be described in terms of non-separable states. We do not have to think that the impossibility of preparing a classical state is too only possible explanation for physical (objective) randomness in deterministic theories. This proposal is an elaboration of an idea of Blatt (1959), although Blatt was still trying to understand objective randomness within an ideal-observer framework.[S.M.]