NDLI: Intuitionistic Quantum Logic of an n-level System

Content Provider	Springer Nature Link
Author	Caspers, Martijn Heunen, Chris Landsman, Nicolaas P. Spitters, Bas
Copyright Year	2009
Abstract	A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364 ). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C-algebra M n (ℂ) of complex n×n matrices. This leads to an explicit expression for the pointfree quantum phase space Σ n and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen–Specker Theorem.In our approach, the nondistributive lattice ℘(M n (ℂ)) of projections in M n (ℂ) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice $\mathcal{O}(\Sigma_{n})$ of functions from the poset $\mathcal{C}(M_{n}(\mathbb{C}))$ of all unital commutative C*-subalgebras C of M n (ℂ) to ℘(M n (ℂ)). The lattice $\mathcal{O}(\Sigma_{n})$ is essentially the (pointfree) topology of the quantum phase space Σ n , and as such defines a Heyting algebra. Each element of $\mathcal{O}(\Sigma_{n})$ corresponds to a “Bohrified” proposition, in the sense that to each classical context $C\in\mathcal{C}(M_{n}(\mathbb{C}))$ it associates a yes-no question (i.e. an element of the Boolean lattice ℘(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann.
Starting Page	731
Ending Page	759
Page Count	29
File Format	PDF
ISSN	00159018
Journal	Foundations of Physics
Volume Number	39
Issue Number	7
e-ISSN	15729516
Language	English
Publisher	Springer US
Publisher Date	2009-04-21
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Quantum logic Topos theory Intuitionistic logic Philosophy of Science Mechanics Statistical Physics Relativity and Cosmology Quantum Physics Physics
Content Type	Text
Resource Type	Article
Subject	Physics and Astronomy

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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