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Oxford University Press eBooks

Representation Without Taxation

Laura Ruetsche

Published Jun 2, 2011• 1 citation
Publisher

Abstract

Abstract This chapter aspires to be an unpunishing introduction to mathematical notions useful for framing and pursuing foundational questions that arise from the unitary inequivalent representations available in QM∞. One such notion is that of an abstract C* algebra, which turns out to capture the structure common to all unitarily inequivalent representations of the CCRs/CARs quantizing a theory of QM∞.

DOI
10.1093/acprof:oso/9780199535408.003.0004

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•Jan 1, 2016•0 citations
Abstract This chapter aspires to be an unpunishing introduction to mathematical notions useful for framing and pursuing foundational questions that arise from the unitary inequivalent representations available in QM∞. One such notion is that of an abstract C* algebra, which turns out to capture the structure common to all unitarily inequivalent representations of the CCRs/CARs quantizing a theory of QM∞.
This chapter catalogs interpretative responses to the breakdown of unitary equivalence in QM∞. These range from ‘Hilbert Space Conservatism’ to ‘Algebraic Imperialism’. Maintaining the ordinary quantum mechanics account of physical equivalence in terms of unitary equivalence, the Hilbert Space Conservative takes unitarily inequivalent representations of the CCRs/CARs for a theory of QM∞ to be rivals, at most one of which could convey the content of that theory. Asserting the physics of a theory of QM∞ to reside in the abstract algebraic structure all its representations share, the Algebraic Imperialist announces criteria for the physical equivalence of quantum theories — including the criterion of ‘weak equivalence’ — at odds with the traditional criterion of unitary equivalence.
Quantum theory is the soul of contemporary physics. It was discovered in an adventurous way, under the urge to solve the puzzles posed by atomic spectra and blackbody radiation. But after its invention, it immediately became clear that it was not just a theory of specific physical systems: it was rather a new language of universal applicability. Already in 1928, the theory had received solid mathematical foundations by Hilbert, von Neumann, and Nordheim, and this work was brought to completion in the monumental work of von Neumann, in the form that we still study nowadays. The theory is extraordinarily successful, and its predictions have been confirmed to an astonishing level of precision in a large spectrum of experiments. However, almost 90 years after von Neumann's book, quantum theory remains mysterious. Its mathematical formulation – based on Hilbert spaces and self-adjoint operators – is far from having an intuitive interpretation. The association of physical systems to Hilbert spaces whose unit vectors represent pure states, the representation of transformations by unitary operators and of observables by self-adjoint operators – all such postulates look artificial and ad hoc . A slightly more operational approach is provided by the C * -algebraic formulation of quantum theory – still, this formulation relies on the assumption that observables form an algebra , where the physical meanings of the multiplication and the sum are far from clear. In short, the postulates of quantum theory impose mathematical structures without providing any simple reason for this choice: the mathematics of Hilbert spaces is adopted as a magic blackbox that “works well” at producing experimental predictions. However, in a satisfactory axiomatization of a physical theory the mathematical structures should emerge as a consequence of postulates that have a direct physical interpretation. By this we mean postulates referring, e.g., to primitive notions like physical system, measurement, or process, rather than notions like, e.g., Hilbert space, C * -algebra, unit vector, or selfadjoint operator. The crucial question thus remains unanswered: why quantum theory? Which are the principles at the basis of the theory? A case that is often invoked in contrast is that of Special Relativity theory, which directly follows from the simple understandable principle of relativity.
This chapter explicates the Hamiltonian scheme for quantizing classical mechanical theories by finding a Hilbert space representation of the appropriate canonical commutation relations. The chapter also reviews the canonical anticommutation relations, which encapsulate the quantum mechanics of spin systems. Having catalogued reasons to regard unitary equivalence as a robust criterion of physical equivalence for quantum theories obtained by finding representations of the CCRs/CARs, the chapter presents a pair of theorems—the Stone-von Neumann and Jordan-Wigner theorems—that establish that, provided only finitely many degrees of freedom are involved, all representations of the CCRs/CARs for a given quantum theory are unitarily (and so presumptively physically) equivalent.
The availability of unitarily inequivalent representations of the canonical commutation relations constituting a quantization of a classical field theory raises questions about how to formulate and pursue quantum field theory. In a minimally technical way, I explain how these questions arise and how advocates of the Hilbert space and of the algebraic approaches to quantum theory might answer them. Where these answers differ, I sketch considerations for and against each approach, as well as considerations which might temper their apparent rivalry.
The received view in the philosophy of physics is that a quantum theory is given its mathematical formulation in a so-called Hilbert space, with the states of a quantum system given by the elements of this space and observables—measurable quantities of the system—given by so-called operators on this space. But the status of this fundamental postulate becomes threatened when one moves to complex systems, like the quantum fields underlying fundamental particle physics or statistical systems composed of infinite collections of quantum particles. In these more complex cases, one finds many inequivalent Hilbert space representations that appear to be competing quantum theories. In looking for a mathematical structure of the theory to interpret as representing the physical world, one is faced with a choice: either arbitrarily pick one Hilbert space representation (out of the many competing representations) as a Hilbert Space Conservative, or else look for the abstract algebraic structure that all of these competing representations have in common as an Algebraic Imperialist. This dissertation provides an extended argument for the second route.