arXiv subjects

Matthew J. Donald

Publications and source records attributed to Matthew J. Donald.

9 recordsLinked to original sources

arXiv2004

Instability, Isolation, and the Tridecompositional Uniqueness Theorem

The tridecompositional uniqueness theorem of Elby and Bub (1994) shows that a wavefunction in a triple tensor product Hilbert space has at most one decomposition into a sum of product wavefunctions with each set of component wavefunctions linearly independent. I demonstrate that, in many circumstances, the unique component wavefunctions and the coefficients in the expansion are both hopelessly unstable, both under small changes in global wavefunction and under small changes in global tensor product structure. In my opinion, this means that the theorem cannot underlie law-like solutions to the problems of the interpretation of quantum theory. I also provide examples of circumstances in which there are open sets of wavefunctions containing no states with various decompositions.

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arXiv2003

On the Work of Henry P. Stapp

For many years, Henry Stapp and I have been working separately and independently on mind-centered interpretations of quantum theory. In this review, I discuss his work and contrast it with my own. There is much that we agree on, both in the broad problems we have addressed and in some of the specific details of our analyses of neural physics, but ultimately we disagree fundamently in our views on mind, matter, and quantum mechanics. In particular, I discuss our contrasting opinions about the nature and randomness of quantum events, about relativity theory, and about the many-minds idea. I also suggest that Stapp's theories are inadequately developed.

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arXiv2003

Finitary and Infinitary Mathematics, the Possibility of Possibilities and the Definition of Probabilities

Some relations between physics and finitary and infinitary mathematics are explored in the context of a many-minds interpretation of quantum theory. The analogy between mathematical ``existence'' and physical ``existence'' is considered from the point of view of philosophical idealism. Some of the ways in which infinitary mathematics arises in modern mathematical physics are discussed. Empirical science has led to the mathematics of quantum theory. This in turn can be taken to suggest a picture of reality involving possible minds and the physical laws which determine their probabilities. In this picture, finitary and infinitary mathematics play separate roles. It is argued that mind, language, and finitary mathematics have similar prerequisites, in that each depends on the possibility of possibilities. The infinite, on the other hand, can be described but never experienced, and yet it seems that sets of possibilities and the physical laws which define their probabilities can be described most simply in terms of infinitary mathematics.

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arXiv2002

Neural Unpredictability, the Interpretation of Quantum Theory, and the Mind-Body Problem

It has been suggested, on the one hand, that quantum states are just states of knowledge; and, on the other, that quantum theory is merely a theory of correlations. These suggestions are confronted with problems about the nature of psycho-physical parallelism and about how we could define probabilities for our individual future observations given our individual present and previous observations. The complexity of the problems is underlined by arguments that unpredictability in ordinary everyday neural functioning, ultimately stemming from small-scale uncertainties in molecular motions, may overwhelm, by many orders of magnitude, many conventionally recognized sources of observed ``quantum'' uncertainty. Some possible ways of avoiding the problems are considered but found wanting. It is proposed that a complete understanding of the relationship between subjective experience and its physical correlates requires the introduction of mathematical definitions and indeed of new physical laws.

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arXiv2002

The Uniqueness Theorem for Entanglement Measures

We explore and develop the mathematics of the theory of entanglement measures. After a careful review and analysis of definitions, of preliminary results, and of connections between conditions on entanglement measures, we prove a sharpened version of a uniqueness theorem which gives necessary and sufficient conditions for an entanglement measure to coincide with the reduced von Neumann entropy on pure states. We also prove several versions of a theorem on extreme entanglement measures in the case of mixed states. We analyse properties of the asymptotic regularization of entanglement measures proving, for example, convexity for the entanglement cost and for the regularized relative entropy of entanglement.

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arXiv2001

A review of Johnjoe McFadden's book ``Quantum Evolution''

In ``Quantum Evolution: Life in the Multiverse'' (HarperCollins, 2000), ISBN 0-00-255948-X, 0-00-655128-9, Johnjoe McFadden makes far-reaching claims for the importance of quantum physics in the solution of problems in biological science. In this review, I discuss the relevance of unitary wavefunction dynamics to biological systems, analyse the inverse quantum Zeno effect, and argue that McFadden's use of quantum theory is deeply flawed.

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arXiv1999

Continuity of Relative Entropy of Entanglement

We show that an entanglement measure called relative entropy of entanglement satisfies a strong continuity condition. If two states are close to each other then so are their entanglements per particle pair in this measure. It follows in particular, that the measure is appropriate for the description of entanglement manipulations in the limit of an infinite number of pairs of particles.

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arXiv1999

Progress in a Many-Minds Interpretation of Quantum Theory

In a series of papers, a many-minds interpretation of quantum theory has been developed. The aim in these papers is to present an explicit mathematical formalism which constitutes a complete theory compatible with relativistic quantum field theory. In this paper, which could also serve as an introduction to the earlier papers, three issues are discussed. First, a significant, but fairly straightforward, revision in some of the technical details is proposed. This is used as an opportunity to introduce the formalism. Then the probabilistic structure of the theory is revised, and it is proposed that the experience of an individual observer can be modelled as the experience of observing a particular, identified, discrete stochastic process. Finally, it is argued that the formalism can be modified to give a physics in which no constants are required. Instead, `constants' have to be determined by observation, and are fixed only to the extent to which they have been observed.

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arXiv1997

On Many-Minds Interpretations of Quantum Theory

This paper is a response to some recent discussions of many-minds interpretations in the philosophical literature. After an introduction to the many-minds idea, the complexity of quantum states for macroscopic objects is stressed. Then it is proposed that a characterization of the physical structure of observers is a proper goal for physical theory. It is argued that an observer cannot be defined merely by the instantaneous structure of a brain, but that the history of the brain's functioning must also be taken into account. Next the nature of probability in many-minds interpretations is discussed and it is suggested that only discrete probability models are needed. The paper concludes with brief comments on issues of actuality and identity over time.

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