*Laura Ruetsche and John Earman*

- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199577439
- eISBN:
- 9780191730603
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577439.003.0010
- Subject:
- Philosophy, Philosophy of Science, Metaphysics/Epistemology

Philosophical accounts of quantum theory commonly suppose that the observables of a quantum system form a Type-I factor von Neumann algebra. Such algebras always have atoms, which are minimal ...
More

Philosophical accounts of quantum theory commonly suppose that the observables of a quantum system form a Type-I factor von Neumann algebra. Such algebras always have atoms, which are minimal projection operators in the case of quantum mechanics. However, relativistic quantum field theory and the thermodynamic limit of quantum statistical mechanics make extensive use of von Neumann algebras of more general types. This chapter addresses the question whether interpretations of quantum probability devised in the usual manner continue to apply in the more general setting. Features of non-type I factor von Neumann algebras are cataloged. It is shown that these novel features do not cause the familiar formalism of quantum probability to falter, since Gleason's Theorem and the Lüders Rule can be generalized. However, the features render the problem of the interpretation of quantum probability more intricate.Less

Philosophical accounts of quantum theory commonly suppose that the observables of a quantum system form a Type-I factor von Neumann algebra. Such algebras always have atoms, which are minimal projection operators in the case of quantum mechanics. However, relativistic quantum field theory and the thermodynamic limit of quantum statistical mechanics make extensive use of von Neumann algebras of more general types. This chapter addresses the question whether interpretations of quantum probability devised in the usual manner continue to apply in the more general setting. Features of non-type I factor von Neumann algebras are cataloged. It is shown that these novel features do not cause the familiar formalism of quantum probability to falter, since Gleason's Theorem and the Lüders Rule can be generalized. However, the features render the problem of the interpretation of quantum probability more intricate.

*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/9780199535408.003.0004
- Subject:
- Philosophy, Philosophy of Science

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 ...
More

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∞.Less

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_{∞}.

*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/9780199535408.003.0007
- Subject:
- Philosophy, Philosophy of Science

This chapter highlights other surprising aspects of QM∞: unlike theories of ordinary QM, theories of QM∞ can traffic in algebras none of whose countably additive states are pure. The chapter sketches ...
More

This chapter highlights other surprising aspects of QM∞: unlike theories of ordinary QM, theories of QM∞ can traffic in algebras none of whose countably additive states are pure. The chapter sketches and provides simple illustrations of the mathematical basis of this circumstance: atomless von Neumann algebras. It also illustrates the uses to which atomless von Neumann algebras are put in QM∞.Less

This chapter highlights other surprising aspects of QM_{∞}: unlike theories of ordinary QM, theories of QM_{∞} can traffic in algebras none of whose countably additive states are pure. The chapter sketches and provides simple illustrations of the mathematical basis of this circumstance: atomless von Neumann algebras. It also illustrates the uses to which atomless von Neumann algebras are put in QM_{∞}.