*Laura Ruetsche*

- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199535408
- eISBN:
- 9780191728525
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199535408.003.0003
- Subject:
- Philosophy, Philosophy of Science

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 ...
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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.Less

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.

*Robert Alicki and Mark Fannes*

- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198504009
- eISBN:
- 9780191708503
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504009.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces an abstract algebraic language unifying the description of classical systems, finite quantum systems, and infinite ones, the last appearing in particle and statistical ...
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This chapter introduces an abstract algebraic language unifying the description of classical systems, finite quantum systems, and infinite ones, the last appearing in particle and statistical physics. In the first part, the general theory of C*-algebras is presented and illustrated by the following examples: general finite dimensional algebras, Abelian algebras and Gelfand's theorem, UHF-algebras, and algebras generated by group representations such as the CCR-algebra arising from the Heisenberg group. Then the theory of states on C*-algebras leading to the GNS-representation in terms of operators on Hilbert spaces is outlined. The basic notion of algebraic dynamical system is given in terms of automorphisms on a C*-algebra of observables and the link to the Hilbert space formalism based on unitary operators is provided by the theory of von Neumann algebras. The examples of the Koopman formalism and the rotation algebra are worked out.Less

This chapter introduces an abstract algebraic language unifying the description of classical systems, finite quantum systems, and infinite ones, the last appearing in particle and statistical physics. In the first part, the general theory of C*-algebras is presented and illustrated by the following examples: general finite dimensional algebras, Abelian algebras and Gelfand's theorem, UHF-algebras, and algebras generated by group representations such as the CCR-algebra arising from the Heisenberg group. Then the theory of states on C*-algebras leading to the GNS-representation in terms of operators on Hilbert spaces is outlined. The basic notion of algebraic dynamical system is given in terms of automorphisms on a C*-algebra of observables and the link to the Hilbert space formalism based on unitary operators is provided by the theory of von Neumann algebras. The examples of the Koopman formalism and the rotation algebra are worked out.