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The Principal Principle and Probability in the Many‐Worlds Interpretation

Rüdiger Schack

Published Jun 24, 2010• 2 citations
Publisher

Abstract

Abstract In the decision-theoretic approach to the many worlds interpretation due to Deutsch and Wallace, probabilities are taken to be Bayesian degrees of belief. Their connection to the quantum mechanical wavefunction, derived from a set of decision-theoretic axioms, can be regarded as a quantum version of Lewis's principal principle. This chapter shows that applying the principal principle in quantum mechanics is problematical because it is impossible to give precise criteria for what constitutes a repeated quantum-mechanical trial. This difficulty is resolved in a full quantum Bayesian theory where quantum states, as well as probabilities, are interpreted as expressing an agent's degrees of belief, rather than corresponding to objective properties of physical systems. A similar resolution exists in the many worlds interpretation; this resolution, however, raises a new question for the latter.

DOI
10.1093/acprof:oso/9780199560561.003.0018

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Abstract In the decision-theoretic approach to the many worlds interpretation due to Deutsch and Wallace, probabilities are taken to be Bayesian degrees of belief. Their connection to the quantum mechanical wavefunction, derived from a set of decision-theoretic axioms, can be regarded as a quantum version of Lewis's principal principle. This chapter shows that applying the principal principle in quantum mechanics is problematical because it is impossible to give precise criteria for what constitutes a repeated quantum-mechanical trial. This difficulty is resolved in a full quantum Bayesian theory where quantum states, as well as probabilities, are interpreted as expressing an agent's degrees of belief, rather than corresponding to objective properties of physical systems. A similar resolution exists in the many worlds interpretation; this resolution, however, raises a new question for the latter.
A group of philosophers and physicists at Oxford have recently crafted, in light of developments in the theory of decoherence, a new version of the Everett or Many Worlds interpretation of quantum mechanics. I will focus on the work of David Wallace, who identifies the quasi-classical domains making up the various “worlds” or “branches” of the interpretation with macroscopic patterns that are functionally realized (to an approximate degree good enough for practical purposes) by the quantum wavefunction of the universe. I argue that Wallace’s reliance on Savage’s pragmatic account of probability fails, due to the unreasonableness of Savage’s Sure Thing Principle in a many-worlds setting. I also argue that Wallace’s program is vulnerable to a version of Putnam’s model-theoretic indeterminacy argument, with the result that almost any conceivable macroscopic “world” is in fact realized by that function and so is (by Wallace’s light) as real as our own. I offer a hylomorphic revision of the many-worlds model, the “traveling forms” interpretation of Alexander Pruss, and I argue that this revision solves both the indeterminacy problem and the problem of interpreting quantum probabilities.
Recent work on probability in the Everett interpretation of quantum mechanics yields a decision-theoretic derivation of David Lewis’ Principal Principle, and hence a general metaphysical theory of probability; part 1 is a discussion of this remarkable result. I defend the claim that the ‘subjective uncertainty’ principle is required for the derivation to succeed, arguing that it amounts to a theoretical identification of chance. In part 2, I generalize this account, and suggest that the Everett interpretation, in combination with a plausible view of natural laws, has the potential to provide a reductive theory of metaphysical modality. I defend the resulting naturalistic modal realism, and outline some of its implications for other parts of metaphysics.
Probabilities may be subjective or objective; we are concerned with both kinds of probability, and the relationship between them. The fundamental theory of objective probability is quantum mechanics: it is argued that neither Bohr's Copenhagen interpretation, nor the pilot-wave theory, nor stochastic state-reduction theories, give a satisfactory answer to the question of what objective probabilities are in quantum mechanics, or why they should satisfy the Born rule; nor do they give any reason why subjective probabilities should track objective ones. But it is shown that if probability only arises with decoherence, then they must be given by the Born rule. That further, on the Everett interpretation, we have a clear statement of what probabilities are, in terms of purely categorical physical properties; and finally, along lines laid out by Deutsch and Wallace, that there is a clear basis in the axioms of decision theory as to why subjective probabilities should track these objective ones. These results hinge critically on the absence of hidden-variables or any other mechanism (such as state-reduction) from the physical interpretation of the theory. The account of probability has traditionally been considered the principal weakness of the Everett interpretation; on the contrary it emerges as one of its principal strengths.
Abstract In the 1950s and 1960s it was causing interest among physicists that in the formalism of quantum mechanics (complex-valued) ‘probability amplitudes’ obeyed laws similar to the laws obeyed by probabilities in the ordinary probability calculus. But this did not then lead decisively to the claim that probabilities should be represented by complex numbers. It became fashionable instead to regard probability amplitudes as an abstract concept from which actual probabilities could be derived by taking the squared moduli of the amplitudes. In this monograph, however, we make another attempt to show how probability amplitudes might after all be identified with actual probabilities. To do this, probability itself is interpreted in a rational Bayesian manner (i.e. as an extension of logic) and a new (complex-valued) probability theory is formulated that incorporates the uncertainty principle (i.e. that takes account of the fact that acquisition of knowledge of a quantum mechanical process generally interferes with it). Taking this probability theory as the new logic of science, and assuming certain physical laws and properties of matter, an interpretation of non-relativistic quantum mechanics is built up. It is consistent with the usual quantum mechanical formalism but allows a clear distinction to be made between the physical world and our knowledge of it.
Abstract This chapter examines probabilities in views that take a realist stance on quantum mechanics, i.e., the Ghirardi-Rimini-Weber (GRW) theory, the Bohm theory and the Everett interpretation. The GRW theory is found to claim a world that is chancy at a fundamental level at which ‘hits’ occur following an objective probability distribution. Although the Bohm theory gives up indeterminism, it is shown to allow for objective probabilities, this time ranging over initial conditions. However, the theory has difficulties in explaining the validity of the Born Rule. The Everett interpretation is found to be more satisfying in this respect. As Wallace and Deutsch have shown, rational agents in a many-world scenario should set their credences following the Born Rule if they choose among quantum games.
In the Bayesian approach to probability theory, probability quantifies a degree of belief for a single trial, without any a priori connection to limiting frequencies. In this paper, we show that, despite being prescribed by a fundamental law, probabilities for individual quantum systems can be understood within the Bayesian approach. We argue that the distinction between classical and quantum probabilities lies not in their definition, but in the nature of the information they encode. In the classical world, maximal information about a physical system is complete in the sense of providing definite answers for all possible questions that can be asked of the system. In the quantum world, maximal information is not complete and cannot be completed. Using this distinction, we show that any Bayesian probability assignment in quantum mechanics must have the form of the quantum probability rule, that maximal information about a quantum system leads to a unique quantum-state assignment, and that quantum theory provides a stronger connection between probability and measured frequency than can be justified classically. Finally, we give a Bayesian formulation of quantum-state tomography.