In “Quantum Evolution”, Johnjoe McFadden makes far-reaching claims for the importance of quantum ... more In “Quantum Evolution”, Johnjoe McFadden makes far-reaching claims for the importance of quantum physics in the solution of problems in biological science. In this review, I shall 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. In the first half of his book, McFadden both discusses the biological problems he is interested in solving and gives an introduction to quantum theory. This part of the book is excellent popular science. It is well-written, competent, and fun. As far as the biology is concerned, McFadden, who is a molecular microbiologist, has very specific, and often controversial, opinions. Nevertheless, he does refer to a wide range of alternative points of view. He certainly managed to convince me that I had swallowed too easily the prevailing dogma (as found, for example, in chapter 1 of Albert et al. 1989) about the earliest (“prebioti...
At least three books struggle to emerge from this volume. One book, at the level of popular scien... more At least three books struggle to emerge from this volume. One book, at the level of popular science, leads us through the development of physics, from Newton's laws to Bell's inequalities, in order to argue for the relevance of consciousness to the understanding of quantum theory. This is followed by a sketch of an interpretation of quantum mechanics. Interwoven with both is a memoir of Walker's teenage girlfriend, who died of Hodgkin's disease nearly fifty years ago. The theme which holds the volume together is Walker's insistence on the importance of looking beyond materialism.
It has been suggested, on the one hand, that quantum states are just states of knowledge; and, on... more 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. Plausible Ideas Confronted. Recently, some plausible ideas about quantum theory have led to claims about the interpretation of the theory which, in my opinion, are simplistic. On the one hand, it has, from time to time, been suggested that quantum states are merely states of knowledge (or of belief) (Wolfe 1936, Wigner 1961, Peierls 1991, Fuchs 2002). This idea has led to the claim that quantum theory "needs no interpretation" (Fuchs and Peres 2000). On the other hand, various authors have argued, in various ways, that quantum theory is fundamentally just a theory of relations or of correlations (
The tridecompositional uniqueness theorem of Elby and Bub (1994) shows that a wavefunction in a t... more 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 lawlike 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. 1. Introduction. The central problem of the interpretation of quantum theory is to explain and characterize the existence, or apparent existence, of state “collapse”. State collapse is the process by which, for ...
In a series of papers, a many-minds interpretation of quantum theory has been developed. The aim ... more 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.
In his long 1957 paper, “The Theory of the Universal Wave Function”, Hugh Everett III made some s... more In his long 1957 paper, “The Theory of the Universal Wave Function”, Hugh Everett III made some significant preliminary steps towards the application and generalization of Shannon’s information theory to quantum mechanics. In the course of doing so, he conjectured that, for a given wavefunction on a compound space, the Schmidt decomposition maximises the correlation between subsystem bases. This is proved here. Let H1 and H2 be separable Hilbert spaces and H = H1 ⊗ H2 be their tensor product. Let Ψ ∈ H be a wavefunction – by which I mean simply that ||Ψ|| = 1. Suppose that H1 has dimension D1 ≤ ∞ and H2 has dimension D2 ≤ ∞. Without loss of generality, suppose that D1 ≤ D2. A Schmidt decomposition (von Neumann 1932, Everett 1957, and many modern textbooks) of Ψ is an expansion of the form Ψ = ∑D1 n=1 √ pnφnψn where (φn) D1 n=1 is an orthornormal basis of H1, (ψn) n=1 is an orthornormal basis of H2, 0 ≤ pn ≤ 1 and ∑D1 n=1 pn = 1. Schmidt decompositions always exist. They are unique, ...
The tridecompositional uniqueness theorem of Elby and Bub (1994) shows that a wavefunction in a t... more 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 lawlike 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.
This paper is a response to some recent discussions of many-minds interpretations in the philosop... more 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.
Some relations between physics and finitary and infinitary mathematics are explored in the contex... more 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 proba...
In “Quantum Evolution”, Johnjoe McFadden makes far-reaching claims for the importance of quantum ... more In “Quantum Evolution”, Johnjoe McFadden makes far-reaching claims for the importance of quantum physics in the solution of problems in biological science. In this review, I shall 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. In the first half of his book, McFadden both discusses the biological problems he is interested in solving and gives an introduction to quantum theory. This part of the book is excellent popular science. It is well-written, competent, and fun. As far as the biology is concerned, McFadden, who is a molecular microbiologist, has very specific, and often controversial, opinions. Nevertheless, he does refer to a wide range of alternative points of view. He certainly managed to convince me that I had swallowed too easily the prevailing dogma (as found, for example, in chapter 1 of Albert et al. 1989) about the earliest (“prebioti...
At least three books struggle to emerge from this volume. One book, at the level of popular scien... more At least three books struggle to emerge from this volume. One book, at the level of popular science, leads us through the development of physics, from Newton's laws to Bell's inequalities, in order to argue for the relevance of consciousness to the understanding of quantum theory. This is followed by a sketch of an interpretation of quantum mechanics. Interwoven with both is a memoir of Walker's teenage girlfriend, who died of Hodgkin's disease nearly fifty years ago. The theme which holds the volume together is Walker's insistence on the importance of looking beyond materialism.
It has been suggested, on the one hand, that quantum states are just states of knowledge; and, on... more 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. Plausible Ideas Confronted. Recently, some plausible ideas about quantum theory have led to claims about the interpretation of the theory which, in my opinion, are simplistic. On the one hand, it has, from time to time, been suggested that quantum states are merely states of knowledge (or of belief) (Wolfe 1936, Wigner 1961, Peierls 1991, Fuchs 2002). This idea has led to the claim that quantum theory "needs no interpretation" (Fuchs and Peres 2000). On the other hand, various authors have argued, in various ways, that quantum theory is fundamentally just a theory of relations or of correlations (
The tridecompositional uniqueness theorem of Elby and Bub (1994) shows that a wavefunction in a t... more 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 lawlike 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. 1. Introduction. The central problem of the interpretation of quantum theory is to explain and characterize the existence, or apparent existence, of state “collapse”. State collapse is the process by which, for ...
In a series of papers, a many-minds interpretation of quantum theory has been developed. The aim ... more 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.
In his long 1957 paper, “The Theory of the Universal Wave Function”, Hugh Everett III made some s... more In his long 1957 paper, “The Theory of the Universal Wave Function”, Hugh Everett III made some significant preliminary steps towards the application and generalization of Shannon’s information theory to quantum mechanics. In the course of doing so, he conjectured that, for a given wavefunction on a compound space, the Schmidt decomposition maximises the correlation between subsystem bases. This is proved here. Let H1 and H2 be separable Hilbert spaces and H = H1 ⊗ H2 be their tensor product. Let Ψ ∈ H be a wavefunction – by which I mean simply that ||Ψ|| = 1. Suppose that H1 has dimension D1 ≤ ∞ and H2 has dimension D2 ≤ ∞. Without loss of generality, suppose that D1 ≤ D2. A Schmidt decomposition (von Neumann 1932, Everett 1957, and many modern textbooks) of Ψ is an expansion of the form Ψ = ∑D1 n=1 √ pnφnψn where (φn) D1 n=1 is an orthornormal basis of H1, (ψn) n=1 is an orthornormal basis of H2, 0 ≤ pn ≤ 1 and ∑D1 n=1 pn = 1. Schmidt decompositions always exist. They are unique, ...
The tridecompositional uniqueness theorem of Elby and Bub (1994) shows that a wavefunction in a t... more 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 lawlike 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.
This paper is a response to some recent discussions of many-minds interpretations in the philosop... more 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.
Some relations between physics and finitary and infinitary mathematics are explored in the contex... more 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 proba...
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