Results 1  10
of
37
Intuitionistic quantum logic of an nlevel system
, 2009
"... A decade ago, Isham and Butterfield proposed a topostheoretic 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 combin ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
A decade ago, Isham and Butterfield proposed a topostheoretic 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 socalled internal language of topos theory (see arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*algebra Mn(C) of complex n × n matrices. This leads to an explicit expression for the pointfree quantum phase space Σn and the associated logical structure and Gelfand transform of an nlevel system. We also determine the pertinent nonprobabilisitic stateproposition pairing (or valuation) and give a very natural topostheoretic reformulation of the Kochen–Specker Theorem. In our approach, the nondistributive lattice P(Mn(C)) of projections in Mn(C) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice O(Σn) of functions from the poset C(Mn(C))
The Gelfand spectrum of a noncommutative C*algebra: a topostheoretic approach
, 2010
"... ..."
(Show Context)
When champions meet: Rethinking the Bohr–Einstein debate
, 2006
"... Einstein’s philosophy of physics (as clarified by Fine and Howard) was predicated on his Trennungsprinzip, a combination of separability and locality, without which he believed “physical thought ” and “physical laws ” to be impossible. Bohr’s philosophy (as elucidated by Hooker, Scheibe, Folse, Howa ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
(Show Context)
Einstein’s philosophy of physics (as clarified by Fine and Howard) was predicated on his Trennungsprinzip, a combination of separability and locality, without which he believed “physical thought ” and “physical laws ” to be impossible. Bohr’s philosophy (as elucidated by Hooker, Scheibe, Folse, Howard, and others), on the other hand, was grounded in a seemingly different doctrine about the possibility of objective knowledge, namely the necessity of classical concepts. In fact, it follows from Raggio’s Theorem in algebraic quantum theory that within a suitable class of physical theories Einstein’s doctrine is mathematically equivalent to Bohr’s, so that quantum mechanics accommodates Einstein’s Trennungsprinzip if and only if it is interpreted à la Bohr through classical physics. Unfortunately, the protagonists themselves failed to discuss their differences in a constructive way, since in its early phase their debate was blurred by an undue emphasis on the uncertainty relations, whereas in its second stage it was dominated by Einstein’s flawed attempts to establish the “incompleteness ” of quantum mechanics. These two aspects of their debate may still be understood and appreciated, however, as reflecting a much deeper and insurmountable disagreement between Bohr and Einstein on the knowability of Nature. Using the theological controversy on the knowability of God as a analogy, Einstein was a Spinozist, whereas Bohr could be said to be on the side of Maimonides. Thus Einstein’s offthecuff characterization of Bohr as a ‘Talmudic philosopher ’ was spoton.
On the Verge of Umdeutung in Minnesota: Van Vleck and the Correspondence Principle. 2 Pts. Archive for History of Exact Sciences
, 2007
"... time, published a remarkable twopart paper by John H. Van Vleck, working in virtual isolation at the University of Minnesota. Using Bohr’s correspondence principle and Einstein’s quantum theory of radiation along with advanced techniques from classical mechanics, Van Vleck showed that quantum formu ..."
Abstract

Cited by 8 (6 self)
 Add to MetaCart
time, published a remarkable twopart paper by John H. Van Vleck, working in virtual isolation at the University of Minnesota. Using Bohr’s correspondence principle and Einstein’s quantum theory of radiation along with advanced techniques from classical mechanics, Van Vleck showed that quantum formulae for emission, absorption, and dispersion of radiation merge with their classical counterparts in the limit of high quantum numbers. For modern readers Van Vleck’s paper is much easier to follow than the famous paper by Kramers and Heisenberg on dispersion theory, which covers similar terrain and is widely credited to have led directly to Heisenberg’s Umdeutung paper. This makes Van Vleck’s paper extremely valuable for the reconstruction of the genesis of matrix mechanics. It also makes it tempting to ask why Van Vleck did not take the next step and develop matrix mechanics himself. Communicated by J.D. Norton. This paper was written as part of a joint project in the history of quantum physics of the Max Planck
UNSHARP QUANTUM REALITY
"... ABSTRACT. The positive operator (valued) measures (POMs) allow one to generalize the notion of observable beyond the traditional one based on projection valued measures (PVMs). Here, we argue that this generalized conception of observable enables a consistent notion of unsharp reality and with it a ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
ABSTRACT. The positive operator (valued) measures (POMs) allow one to generalize the notion of observable beyond the traditional one based on projection valued measures (PVMs). Here, we argue that this generalized conception of observable enables a consistent notion of unsharp reality and with it a concept of joint properties. A property manifests itself as an element of unsharp reality by its power, or tendency, of becoming actual or actualizing a specific measurement outcome, where this tendency of actualization is quantified by the associated quantum probability. The resulting singlecase interpretation of probability as a degree of reality will be explained in detail and its role in addressing the tensions between quantum and classical accounts of the physical world will be elucidated. It will be shown that potentiality can be viewed as a causal agency that evolves in a welldefined way. 1.
Equivalence of the Symbol Grounding and Quantum System Identification Problems
 INFORMATION
, 2014
"... ..."