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Interpreting Quantum Theories$
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Laura Ruetsche

Print publication date: 2011

Print ISBN-13: 9780199535408

Published to Oxford Scholarship Online: September 2011

DOI: 10.1093/acprof:oso/9780199535408.001.0001

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Beyond the Stone–von Neumann Theorem

Beyond the Stone–von Neumann Theorem

Chapter:
(p.46) 3 Beyond the Stone–von Neumann Theorem
Source:
Interpreting Quantum Theories
Author(s):

Laura Ruetsche

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199535408.003.0003

This chapter catalogs circumstances under which the reassuring uniqueness results presented in Chapter 2 break down. The Stone-von Neumann theorem presupposes a continuity condition; with that presupposition suspended, it becomes possible to equip the quantum theory of a single mechanical system with exact position eigenstates. The Stone-von Neumann theorem presupposes the classical theory to be quantized to be set in Euclidean space; when systems such as a bead on a circle are quantized, the theorem breaks down and unitarily inequivalent representations of the physics ensue. Finally, the Stone-von Neumann and Jordan-Wigner theorems presuppose finitely many degrees of freedom. They therefore fail to apply to quantum field theories (because a field, assigning a value to each point of space[time] has infinitely many degrees of freedom), or theories at the thermodynamic (infinite volume) limit of quantum statistical mechanics. The chapter presents simple examples of infinite systems and their unitarily inequivalent quantizations: the free Klein-Gordon field and the infinite spin chain.

Keywords:   Stone-von Neumann theorem, Jordan-Wigner theorem, quantization, position representation, Heisenberg group, quantum field theory, unitarily inequivalence, thermodynamic limit, Fock space, particle

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