The field of real numbers in axiomatic quantum mechanics

1973 ◽  
Vol 14 (10) ◽  
pp. 1469-1471 ◽  
Author(s):  
M. J. Maczyński
Keyword(s):  
Quantum Mechanics ◽  
Real Numbers ◽  
Axiomatic Quantum Mechanics

A THEOREM OF SOLÉR, THE THEORY OF SYMMETRY AND QUANTUM MECHANICS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260005 ◽  
Author(s):  
GIANNI CASSINELLI ◽  
PEKKA LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Logic ◽  
Hilbert Spaces ◽  
Classical Problem ◽  
Partial Solution ◽  
Space Realization ◽  
Symmetry Transformations ◽  
Axiomatic Quantum Mechanics

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

Representations of Groups as Automorphisms on Orthomodular Lattices and Posets

1971 ◽  
Vol 23 (4) ◽  
pp. 659-673 ◽  
Author(s):  
Stanley P. Gudder
Keyword(s):  
Quantum Mechanics ◽  
General Setting ◽  
Symmetry Groups ◽  
Orthomodular Lattices ◽  
Induced Representations ◽  
Complete Study ◽  
Logic Approach ◽  
Physical Symmetry ◽  
Groups Of Automorphisms ◽  
Axiomatic Quantum Mechanics

In this paper we study the problem of representing groups as groups of automorphisms on an orthomodular lattice or poset. This problem not only has intrinsic mathematical interest but, as we shall see, also has applications to other fields of mathematics and also physics. For example, in the “quantum logic” approach to an axiomatic quantum mechanics, important parts of the theory can not be developed any further until a fairly complete study of the representations of physical symmetry groups on orthomodular lattices is accomplished [1].We will consider two main topics in this paper. The first is the analogue of Schur's lemma and its corollaries in this general setting and the second is a study of induced representations and systems of imprimitivity.

scholarly journals Transfer principle in quantum set theory

2007 ◽  
Vol 72 (2) ◽  
pp. 625-648 ◽  
Author(s):  
Masanao Ozawa
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Set Theory ◽  
Von Neumann Algebra ◽  
Transfer Principle ◽  
Real Numbers ◽  
Von Neumann ◽  
Equality Relation ◽  
Adjoint Operators ◽  
Transfer Theorems

AbstractIn 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.

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