Conditional Probability, Three-Slit Experiments, and the Jordan Algebra Structure of Quantum Mechanics
Most quantum logics do not allow for a reasonable calculus of conditional probability. However, those ones which do so provide a very general and rich mathematical structure, including classical probabilities, quantum mechanics, and Jordan algebras. This structure exhibits some similarities with Alf...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
|
| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2012/156573 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832555045150785536 |
|---|---|
| author | Gerd Niestegge |
| author_facet | Gerd Niestegge |
| author_sort | Gerd Niestegge |
| collection | DOAJ |
| description | Most quantum logics do not allow for a reasonable calculus of conditional probability. However, those ones which do so provide a very general and rich mathematical structure, including
classical probabilities, quantum mechanics, and Jordan algebras. This structure exhibits some similarities with Alfsen and Shultz's noncommutative spectral theory, but these two mathematical approaches are not identical. Barnum, Emerson, and Ududec adapted the concept of higher-order interference, introduced by Sorkin in 1994, into a general probabilistic framework. Their adaption is used here to reveal a close link between the existence of the Jordan product and the nonexistence of interference of third or higher order in those quantum logics which entail a reasonable calculus of conditional probability. The complete characterization of the Jordan algebraic structure requires the following three further postulates: a Hahn-Jordan decomposition property for the states, a polynomial functional calculus for the observables, and the positivity of the square of an observable. While classical probabilities are characterized by the absence of any kind of interference, the absence of interference of third (and higher) order thus characterizes a probability calculus which comes close to quantum mechanics but still includes the exceptional Jordan algebras. |
| format | Article |
| id | doaj-art-4d6701d042c64620a358cb6b5707432c |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-4d6701d042c64620a358cb6b5707432c2025-02-03T05:49:41ZengWileyAdvances in Mathematical Physics1687-91201687-91392012-01-01201210.1155/2012/156573156573Conditional Probability, Three-Slit Experiments, and the Jordan Algebra Structure of Quantum MechanicsGerd Niestegge0Zillertalstrasse 39, 81373 Muenchen, GermanyMost quantum logics do not allow for a reasonable calculus of conditional probability. However, those ones which do so provide a very general and rich mathematical structure, including classical probabilities, quantum mechanics, and Jordan algebras. This structure exhibits some similarities with Alfsen and Shultz's noncommutative spectral theory, but these two mathematical approaches are not identical. Barnum, Emerson, and Ududec adapted the concept of higher-order interference, introduced by Sorkin in 1994, into a general probabilistic framework. Their adaption is used here to reveal a close link between the existence of the Jordan product and the nonexistence of interference of third or higher order in those quantum logics which entail a reasonable calculus of conditional probability. The complete characterization of the Jordan algebraic structure requires the following three further postulates: a Hahn-Jordan decomposition property for the states, a polynomial functional calculus for the observables, and the positivity of the square of an observable. While classical probabilities are characterized by the absence of any kind of interference, the absence of interference of third (and higher) order thus characterizes a probability calculus which comes close to quantum mechanics but still includes the exceptional Jordan algebras.http://dx.doi.org/10.1155/2012/156573 |
| spellingShingle | Gerd Niestegge Conditional Probability, Three-Slit Experiments, and the Jordan Algebra Structure of Quantum Mechanics Advances in Mathematical Physics |
| title | Conditional Probability, Three-Slit Experiments, and the Jordan Algebra Structure of Quantum Mechanics |
| title_full | Conditional Probability, Three-Slit Experiments, and the Jordan Algebra Structure of Quantum Mechanics |
| title_fullStr | Conditional Probability, Three-Slit Experiments, and the Jordan Algebra Structure of Quantum Mechanics |
| title_full_unstemmed | Conditional Probability, Three-Slit Experiments, and the Jordan Algebra Structure of Quantum Mechanics |
| title_short | Conditional Probability, Three-Slit Experiments, and the Jordan Algebra Structure of Quantum Mechanics |
| title_sort | conditional probability three slit experiments and the jordan algebra structure of quantum mechanics |
| url | http://dx.doi.org/10.1155/2012/156573 |
| work_keys_str_mv | AT gerdniestegge conditionalprobabilitythreeslitexperimentsandthejordanalgebrastructureofquantummechanics |