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The Standard Formulation Of Quantum Mechanics

Jeffrey Alan Barrett

Published Aug 16, 2001• 2 citations
Publisher

Abstract

Abstract This chapter begins by discussing the foundations of Hugh Everett III's relative state theory, which he regards as a new, more general, and complete theory than the standard von Neumann–Dirac theory. It then talks about why Albert Einstein rejected Max Born's formulation of quantum mechanics where he described his new statistical interpretation of the wave function. It also provides a discussion on von Neumann's formulation of quantum mechanics and the standard theory, as well as a summary of the said theory. The chapter goes on to detail several arguments presented by Einstein regarding von Neumann's interpretation of quantum mechanics. It then presents von Neumann's psychophysical parallelism and Wigner's friend experiment.

DOI
10.1093/acprof:oso/9780199247431.003.0002

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Abstract This chapter begins by discussing the foundations of Hugh Everett III's relative state theory, which he regards as a new, more general, and complete theory than the standard von Neumann–Dirac theory. It then talks about why Albert Einstein rejected Max Born's formulation of quantum mechanics where he described his new statistical interpretation of the wave function. It also provides a discussion on von Neumann's formulation of quantum mechanics and the standard theory, as well as a summary of the said theory. The chapter goes on to detail several arguments presented by Einstein regarding von Neumann's interpretation of quantum mechanics. It then presents von Neumann's psychophysical parallelism and Wigner's friend experiment.
Abstract This chapter provides a detailed explanation of Everett's three works — his thesis “On the Foundations of Quantum Mechanics”, a short paper entitled “‘Relative State’ Formulation of Quantum Mechanics”, and a long paper “The Theory of the Universal Wave Function”. It also discusses what was wrong with von Neumann's theory. It evaluates five alternatives presented by Everett for developing a satisfactory formulation of quantum mechanics. It provides an explanation for Everett's interpretation of pure wave mechanics and his principle of the relativity of states. Lastly, it evaluates the classical appearance and behaviour of macroscopic systems.
Abstract The book starts with a description of classical mechanics then discusses the quantum phenomena that require us to give up our commonsense classical intuitions. We consider the physical and conceptual arguments that led to the standard von Neumann-Dirac formulation of quantum mechanics and how the standard theory explains quantum phenomena. This includes a discussion of how the theory’s two dynamical laws work with the standard interpretation of states to explain determinate measurement records, quantum statistics, interference effects, entanglement, decoherence, and quantum nonlocality. A careful understanding of how the standard theory works ultimately leads to the quantum measurement problem. We consider how the measurement problem threatens the logical consistency of the standard theory then turn to a discussion of the main proposals for resolving it. This includes collapse formulations of quantum mechanics like Wigner’s extension of the standard theory and the GRW approach and no-collapse formulations like pure wave mechanics, the various many-worlds theories, and Bohmian mechanics. In discussing alternative formulations of quantum mechanics we pay particular attention to the explanatory role played by each theory’s empirical ontology and associated metaphysical commitments and the conceptual trade-offs between theoretical options.
The chief aim of this chapter is to give brief accounts of topics in nonrelativistic quantum mechanics that are not always treated in elementary texts. We begin with the Hilbert space formulation of quantum mechanics as set down by von Neumann in his book Mathematical Foundations of Quantum Mechanics (von Neumann, 1955), which will henceforth be referred to as von Neumann's book. Much of our concern will be with continuous spectra, which cannot be discussed adequately in the Dirac formalism. The density matrix, which will play a key role in Chapters 8 and 10, will be treated in some detail. The section on formalism is followed by one on the probability interpretation; the latter is included because Sewell's theory of measurement suggests a subtle reformulation of a part of it. These are followed by sections on superselection rules and the Galilei group, which is the relativity group of quantum mechanics. They are based on the pioneering works of Wigner and Bargmann. The last section is devoted to the fundamental theorems of von Neumann and Stone, and to Reeh's observation on the physical significance of the failure of the Stone–von Neumann uniqueness theorem at the Lie algebra level. The formalism of quantum mechanics By quantum mechanics we shall mean the nonrelativistic quantum theory of a system with a finite number, N, of particles. The number N is assumed fixed. It will be convenient, for later recall, to divide the material into subsections.
Abstract The standard von Neumann-Dirac formulation of quantum mechanics is presented as a set of five basic rules. We discuss each rule is discussed in turn paying particular attention to the conceptual history of the theory. Of central importance is the standard interpretation of states (the eigenvalue-eigenstate link) and the dynamical laws of the theory (the random collapse dynamics and the deterministic linear dynamics) and how the interpretation and dynamics work together to predict and explain the results of basic quantum experiments. While the focus is on the behavior of electrons, we also briefly consider how the theory uses the same mathematical formalism to treat other phenomena like the behavior of neutral K mesons and qbits in a quantum computer.
Abstract This chapter covers the completion of the formalism of quantum mechanics and its statistical interpretation in the first two installments of a remarkable trilogy of work by von Neumann in 1927. In the first, von Neumann introduces the modern Hilbert space formalism and demonstrates the equivalence of matrix and wave mechanics by showing that they are just two different instantiations of Hilbert space, one being the space of square-summable sequences, the other being the space of square integrable functions. In the second, drawing on work by von Mises on the foundations of probability theory, he provides a careful formulation of the statistical interpretation of this formalism. In doing so, he introduces such staples of modern quantum mechanics as density operators characterizing quantum statistical ensembles and the distinction between pure and mixed states.
The aim of this volume is to develop and evaluate the suggestion that if one adopts the interpretation of quantum mechanics initially proposed by Hugh Everett, today frequently referred to as the “many-worlds interpretation,” then one can avoid the kind of nonlocality that is often thought to be a consequence of quantum entanglement according to other approaches, and thus permit the compatibility of these two pillars of modern physics: quantum mechanics and special relativity. Although this is an idea that seems to have been recognized by Everett and is sometimes touted as a reason to prefer his approach to interpreting quantum mechanics over others, the precise way in which Everett’s interpretation achieves this locality, indeed whether it even does achieve locality, has been debated, including among the interpretation’s proponents. The aim of this introductory chapter is to provide readers with a background of this debate concerning the locality of Everettian quantum mechanics. We begin with a brief historical introduction to Everett’s proposed interpretation of quantum mechanics and its later development into the many-worlds interpretation. The following section discusses how the by now well-established phenomenon of quantum entanglement seems to imply what Albert Einstein, in a 1947 letter to Max Born, called “spooky action at a distance,” and a tension with special relativity. We then discuss why one might think that the many-worlds interpretation allows one to avoid nonlocality in this sense. The next sections bring the reader up to date with how the locality of Everettian quantum mechanics has been debated both by critics and proponents of the interpretation. As we will see, the desire to understand Everettian quantum mechanics as an interpretation that avoids nonlocality has opened up some interesting debates concerning both what it means to say an interpretation of quantum mechanics is local and over the metaphysical and mathematical interpretation of Everettian quantum mechanics itself. This introductory chapter concludes with a summary of the contributions to the present volume in the context of this ongoing debate.