Ruetsche argues that a problem of unitarily inequivalent representations arises in quantum theories with infinitely many degrees of freedom. I provide an algebraic formulation of classical field theories and show that unitarily inequivalent representations arise there as well. I argue that the classical case helps us rule out one possible response to the problem of unitarily inequivalent representations called Hilbert Space Conservatism.

In the mid to late 1920s, the emerging theory of quantum mechanics had two main competing formalisms — the wave mechanics of E. Schrödinger [61] and the matrix mechanics of W. Heisenberg, M. Born and P. Jordan [27][2][3].1 Though a connection between the two was quickly pointed out by Schrödinger himself — see paper III in [61] — among others, the folk-theoretic “equivalence” between wave and matrix mechanics continued to generate more detailed study, even into our times.

Within the traditional Hilbert space formalism of quantum mechanics, it is not possible to describe a particle as possessing, simultaneously, a sharp position value and a sharp momentum value. Is it possible, though, to describe a particle as possessing just a sharp position value (or just a sharp momentum value)? Some, such as Teller, have thought that the answer to this question is No - that the status of individual continuous quantities is very different in quantum mechanics than in classical (...) mechanics. On the contrary, I shall show that the same subtle issues arise with respect to continuous quantities in classical and quantum mechanics; and that it is, after all, possible to describe a particle as possessing a sharp position value without altering the standard formalism of quantum mechanics. (shrink)

Ruetsche claims that an abstract C*-algebra of observables will not contain all of the physically significant observables for a quantum system with infinitely many degrees of freedom. This would signal that in addition to the abstract algebra, one must use Hilbert space representations for some purposes. I argue to the contrary that there is a way to recover all of the physically significant observables by purely algebraic methods. 1 Introduction2 Preliminaries3 Three Extremist Interpretations3.1 Algebraic imperialism3.2 Hilbert space conservatism3.3 Universalism4 Parochial (...) Observables4.1 Parochial observables for the imperialist4.2 Parochial observables for the universalist5 Conclusion. (shrink)

A normal state on a von Neumann algebra defines a countably additive probability measure over its projection lattice. The von Neumann algebras familiar from ordinary QM are algebras of all the bounded operators on a Hilbert space H, aka Type I factor von Neumann algebras. Their normal states are density operator states, and can be pure or mixed. In QFT and the thermodynamic limit of QSM, von Neumann algebras of more exotic types abound. Type III von Neumann algebras, for instance, (...) have no pure normal states; the pure states they do have fail to be countably additive. I will catalog a number of temptations to accord physical significance to non-normal states, and then give some reasons to resist these temptations: pure though they may be, non-normal states on non-Type I factor von Neumann algebras can't do the interpretive work we've come to expect from pure states on Type I factors; our best accounts of state preparation don't work for the preparation of non-normal states; there is a sense in which non-normal states fail to instantiate the laws of quantum mechanics. (shrink)

It is well known that the process of quantization—constructing a quantum theory out of a classical theory—is not in general a uniquely determined procedure. There are many inequivalent methods that lead to different choices for what to use as our quantum theory. In this paper, I show that by requiring a condition of continuity between classical and quantum physics, we constrain and inform the quantum theories that we end up with.

ABSTRACT Ruetsche claims that an abstract C*-algebra of observables will not contain all of the physically significant observables for a quantum system with infinitely many degrees of freedom. This would signal that in addition to the abstract algebra, one must use Hilbert space representations for some purposes. I argue to the contrary that there is a way to recover all of the physically significant observables by purely algebraic methods. 1Introduction 2Preliminaries 3Three Extremist Interpretations 3.1Algebraic imperialism 3.2Hilbert space conservatism 3.3Universalism 4Parochial (...) Observables 4.1Parochial observables for the imperialist 4.2Parochial observables for the universalist 5Conclusion. (shrink)

The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is (...) Heisenberg's `quantum-theoretical Umdeutung (reinterpretation) of classical observables', which lies at the basis of quantization theory. Similarly, Bohr's correspondence principle (in somewhat revised form) and Schroedinger's wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely in the limit h -> 0 of small Planck's constant (in a finite system), in the limit N goes to infinity of a large system with $N$ degrees of freedom (at fixed h), and through decoherence and consistent histories. The first limit is closely related to modern quantization theory and microlocal analysis, whereas the second involves methods of C*-algebras and the concepts of superselection sectors and macroscopic observables. In these limits, the classical world does not emerge as a sharply defined objective reality, but rather as an approximate appearance relative to certain ``classical" states and observables. Decoherence subsequently clarifies the role of such states, in that they are ``einselected", i.e. robust against coupling to the environment. Furthermore, the nature of classical observables is elucidated by the fact that they typically define (approximately) consistent sets of histories. This combination of ideas and techniques does not quite resolve the measurement problem, but it does make the point that classicality results from the elimination of certain states and observables from quantum theory. Thus the classical world is not created by observation (as Heisenberg once claimed), but rather by the lack of it. (shrink)

We show that Bohr's principle of complementarity between position and momentum descriptions can be formulated rigorously as a claim about the existence of representations of the canonical commutation relations. In particular, in any representation where the position operator has eigenstates, there is no momentum operator, and vice versa. Equivalently, if there are nonzero projections corresponding to sharp position values, all spectral projections of the momentum operator map onto the zero element.