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Fundamental theories of physics

Topos theory and quantum logic

Klaas Landsman

Published Jan 1, 2017• 1 citation
Publisher

Abstract

The topos-theoretic approach to quantum mechanics (also known as quantum toposophy ) has the same origin as the quantum logic programme initiated by Birkhoff and von Neumann, namely the feeling that classical logic is inappropriate for quantum theory and needs to be replaced by something else. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-319-51777-3_12

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The topos-theoretic approach to quantum mechanics (also known as quantum toposophy ) has the same origin as the quantum logic programme initiated by Birkhoff and von Neumann, namely the feeling that classical logic is inappropriate for quantum theory and needs to be replaced by something else. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Topos theory has been suggested by Döring and Isham as an alternative mathematical structure with which to formulate, in general, physical theories. However, the motivation for using it to express quantum theory, lies in the desire to solve certain interpretational problems inherent in the standard formulation of the theory. In fact, the topos approach to quantum theory overcomes the instrumentalist–Copenhagen interpretation, rendering the theory more realist. The caveat is that one ends up with a multivalued–intuitionistic logic, rather than a Boolean logic.
The fundamental laws of Nature that govern the physics of atoms, elementary particles, light quanta, and everything else that is small and light, have been found to fit in one grandiose framework called Quantum Mechanics. The internal logic of this framework appears to be counter intuitive and mysterious, while extremely coherent and effective. In this general introduction the author explains why he feels the need for expressing his views on how to understand these facts, even if many known scientists have already given extensive reviews of numerous ideas and approaches. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The notion of mixed strategy, as originally introduced by Von Neumann and Morgenstern ( 1944 ) is a basic ingredient of classical game theory. Yet, as pointed out by von Neumann and Morgenstern themselves, the idea that a rational player may have to use a randomizing device, such as a coin flip, to decide on their actions poses some insuperable conceptual difficulties. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We put forward a new take on the logic of quantum mechanics, following Schroedinger's point of view that it is composition which makes quantum theory what it is, rather than its particular propositional structure due to the existence of superpositions, as proposed by Birkhoff and von Neumann. This gives rise to an intrinsically quantitative kind of logic, which truly deserves the name `logic' in that it also models meaning in natural language, the latter being the origin of logic, that it supports automation, the most prominent practical use of logic, and that it supports probabilistic inference.
The authors present briefly the basic notions of the quantum formalism: pure and mixed states; quantum observables; quantum probability; Born’s rule, superposition, and ‘state collapse’; the projection postulate (von Neumann–Lüders postulate); Dirac’s ket and bra-vector notations; elements of quantum information theory and quantum logic; Schrödinger’s and von Neumann’s equations; unitary dynamics; and positive operator valued measures. They also define the basic mathematical notions related to the quantum formalism: Hilbert space; scalar product; norm; Hermitian operator (matrix); projector; and unitary operator and adjoint operator. This chapter may be useful for newcomers to the field, but for those readers who have preliminary knowledge about quantum mechanics (QM) they can proceed directly to advanced chapters. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.