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Cambridge University Press eBooks

Macroscopic observables in quantum physics

Rathindra Nath Sen

Published Feb 11, 2010• 0 citations
Publisher

Abstract

In a certain sense, object and apparatus were essentially on the same footing in von Neumann's measurement theory. They were systems with k and l degrees of freedom respectively (page 146). There were no constraints on the numbers k and l , which could, for example, be of the same order of magnitude. Quite possibly, it was this lack of differentiation between object and apparatus that led Wigner to assert that ‘the state of the apparatus has no classical description’ (page 137), a state of affairs that produced the infinite von Neumann chain which only ended in the observer's consciousness. In this and the following chapter we shall break with von Neumann and assume that k is small, i.e., of the order of unity, and that l is large, i.e., within a few orders of magnitude of Avogadro's number. The room for manoeuvre that this provides will allow us to break with Wigner and explore systems with states that do have classical descriptions. It will not surprise the informed reader that the room for manoeuvre created by our assumption will be filled, very substantially, by von Neumann's own work. This chapter is divided into three sections. Section 9.1 is devoted to a theorem of von Neumann on observables that commute with each other. This prepares the way to our treatment of macroscopic observables, which is based on the commuting approximations to P and Q devised by von Neumann; these results, together with their antecedents, are presented in Section 9.2. In Section 9.3, an attempt is made to resolve some of Wigner's doubts.

DOI
10.1017/cbo9780511674761.014

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In a certain sense, object and apparatus were essentially on the same footing in von Neumann's measurement theory. They were systems with k and l degrees of freedom respectively (page 146). There were no constraints on the numbers k and l , which could, for example, be of the same order of magnitude. Quite possibly, it was this lack of differentiation between object and apparatus that led Wigner to assert that ‘the state of the apparatus has no classical description’ (page 137), a state of affairs that produced the infinite von Neumann chain which only ended in the observer's consciousness. In this and the following chapter we shall break with von Neumann and assume that k is small, i.e., of the order of unity, and that l is large, i.e., within a few orders of magnitude of Avogadro's number. The room for manoeuvre that this provides will allow us to break with Wigner and explore systems with states that do have classical descriptions. It will not surprise the informed reader that the room for manoeuvre created by our assumption will be filled, very substantially, by von Neumann's own work. This chapter is divided into three sections. Section 9.1 is devoted to a theorem of von Neumann on observables that commute with each other. This prepares the way to our treatment of macroscopic observables, which is based on the commuting approximations to P and Q devised by von Neumann; these results, together with their antecedents, are presented in Section 9.2. In Section 9.3, an attempt is made to resolve some of Wigner's doubts.
The aim of this final chapter of Part II is rather different from that of the preceding chapters. It is to provide a glimpse into the generalizations of the formalism of quantum mechanics on Hilbert space that are required to describe the symmetries and dynamics of systems with infinitely many degrees of freedom, owing to the qualitative differences that arise when the number of degrees of freedom tends to infinity. The burden of the preceding chapters was that the notion of a geometrical point is as meaningful in quantum physics as it is in classical physics. The argument involved a lengthy excursion into quantum-mechanical measurement theory. During this excursion, we found that the notion of design of the experiment played an essential role. An experiment, no matter how ingenious, does not create laws of nature; it substantiates, or refutes, an assumed law. According to the dichotomy between laws of nature and initial conditions that was posited by Wigner (page 206), the role of human intervention in an experiment is to realize, under controlled and repeatable conditions, a conjunction of initial conditions that may be improbable in nature. Seen in this light: (i) The failure of von Neumann's theory proper to explain wave function collapse is either a failure of the theory to establish control over complicated initial conditions or an indicator of as-yet-undiscovered laws of nature; the success of Sewell's theory suggests that it is the former, and not the latter. […]
We use the term ‘measurement’ to refer to the interaction between an object and an apparatus on the basis of which information concerning the initial state of the object may be obtained from information on the resulting state of the apparatus. The quantum theory of measurement is a quantum theoretic investigation of such interactions in order to analyse the correlations between object and apparatus that measurement must establish. Although there is a sizeable literature on quantum measurements there appear to be just two sorts of interactions that have been employed. There are the ‘disturbing’ interactions consistent with the analysis of Landau and Peierls (8) as developed by Pauli (11) and by Landau and Lifshitz (7), and there are the ‘non-disturbing’ interactions explicitly set out by von Neumann ((10), chs. 5, 6), and that dominate the literature. In this paper we shall investigate the most general types of interactions that could possibly constitute measurements and provide a precise mathematical characterization (section 2). We shall then examine an interesting subclass, corresponding to Landau's ideas, that contains both of the above sorts of measurements (section 3). Finally, we shall discuss von Neumann measurements explicitly and explore the purported limitations suggested by Wigner(12) and Araki and Yanase (2). We hope, in this way, to provide a comprehensive basis for discussions of quantum measurements.
Instruments are necessarily of macroscopic size because our sense organs—which can be considered as instruments occupying a certain rank in the von Neumann hierarchy—are of macroscopic size. It is only natural, therefore, that physicists should have attempted to make use of this property for constructing quantum measurement theories. Moreover, one of the remarkable properties possessed by most macroscopic systems, including all instruments, but lacking in microscopic systems is irreversibility. This simple observation suggests very strongly that the phenomenon of irreversibility and the phenomenon of "wave packet reduction," which is associated in some way or other with the measurement process, should somehow be correlated. The theories in the chapter show that it is in practice always possible to attribute definite macroscopic properties to the measuring instruments. The danger that the error we thus make will ever be detected has been proved to be vanishingly small.
The newly revised edition of this well-known book categorizes prevalent views on measurement as described by quantum mechanics, discusses basics agreed to by many who on other issues disagree, and describes some of the problems. Although many paragraphs rest unchanged from the first (1991) edition, results found since 1991 are used to reorganize and sharpen the focus of the book, so that even the old paragraphs are now understood differently and more clearly. One advance in the second edition is a new and more general no-go theorem; another is separation of formal derivations from interpretation. The dozen or so definitions essential to following the book even casually are stated clearly. By this means the authors succeed pretty well in achieving their stated desire to make the book comprehensible to those who do not share their philosophic predispositions. While I differ in philosophy from the authors, attending to their discussion uncovered weaknesses in my own thinking and improved it. The mathematical language of quantum mechanics expresses a system under measurement by states (as rays in a Hilbert space) and a hamiltonian operator, distinct from the apparatus that produces outcomes, which is expressed by a positive operator-valued (POV) measure. In this way quantum language separates state preparation from state measurement. Often the Hilbert space is viewed as a tensor product of subspaces, allowing further conceptual separations. The authors take the stance that quantum mechanics calls for splitting the empirical world into parts, the most essential being (1) objective systems S (to be observed) and (2) apparatus A (preparation and registration devices that produces outcomes (or, as the authors write, definite pointer values) of the observed system). In some interpretations, one or both of an observer and an environment appear as additional parts. What is to be included in the system S versus the apparatus A is not specified by the language, but is open to the theoretician; a probe, for example, can be analysed as either part of the apparatus or as part of the system, as was discussed by von Neumann in work cited by the authors. With what should a theory of measurement be concerned? A broad division of approaches is defined by how one answers the question: should one try to do away with the cut between measuring apparatus and system? The authors phrase this as a choice of REFERENT, a term which in my mind puts them on shaky semantic ground; anyway The Quantum Theory of Measurement is concerned with implications of the `yes'. The book unfolds with the power of a tragedy, in which the logical consequences lead, step-by-step, to the statement and proof of theorem 6.2.1, which, roughly speaking, asserts the non-existence of a solution to the problem of getting definite pointer values from measuring apparatus that exhibits the superposition demanded by its description as a quantum system. Possible responses to this theorem are discussed, among them the many-worlds interpretation and a modification of quantum mechanics that elevates decoherence to a principle. Citations of the literature are thorough, and many points of view are crisply summarized. The book is indispensable to both those who work in the broad direction chosen by the authors and to those who wish to set any other direction in context.
After some 80 years, the meaning of quantum theory remains as controversial as ever. The theory, as presented in textbooks, involves a human observer performing experiments with microscopic quantum systems using macroscopic classical apparatus. The quantum system is described by a wavefunction – a mathematical object that is used to calculate probabilities but which gives no clear description of the state of reality of a single system. In contrast, the observer and apparatus are described classically and are assumed to have definite states of reality. For example, a pointer on a measuring device will show a particular reading, or a particle detector will "fire" a definite number of times. Quantum systems seem to inhabit a fuzzy, indefinite realm, while our everyday macroscopic world does not, even though the latter is ultimately built from the former.
Classical physics is approached from quantum mechanics in the macroscopic limit. The technical device to achieve this goal is the quantum version of the central limit theorem, derived for an observable at a given time and for the time-dependent expectation value of the coordinate. The emergence of the classical trajectory can be followed for the average of an observable over a large set of independent microscopical systems, and the deterministic classical laws can be recovered in all practical purposes, owing to the largeness of Avogadro’s number. This result refers to the observed system without considering the measuring apparatus. The emergence of a classical trajectory is followed qualitatively in Wilson’s cloud chamber.