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.

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.

Invariance Groups in Classical and Quantum Mechanics

1973 ◽  
Vol 28 (3-4) ◽  
pp. 538-540 ◽  
Author(s):  
D. J. Simms
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Energy Levels ◽  
Classical Mechanics ◽  
Symmetry Groups ◽  
Casimir Invariants ◽  
Classical Phase Space ◽  
Invariance Groups ◽  
Classical And Quantum Mechanics

AbstractThis is a report on some new relations and analogies between classical mechanics and quantum mechanics which arise out of the work of Kostant and Souriau. Topics treated are i) the role of symmetry groups; ii) the notion of elementary system and the role of Casimir invariants; iii) energy levels; iv) quantisation in terms of geometric data on the classical phase space. Some applications are described.

Relativistic extension of non-compact symmetry groups

1966 ◽  
Vol 289 (1417) ◽  
pp. 177-191 ◽  
Keyword(s):  
Symmetry Group ◽  
Conventional Method ◽  
Space Time ◽  
Unitary Representations ◽  
Symmetry Groups ◽  
Infinite Dimensional ◽  
Induced Representations ◽  
Relativistic Extension ◽  
A Chain

The problem of relativistieally boosting the unitary representations of a non-compact spin-containing rest-symmetry group is solved by starting with non-unitary infinite-dimensional representations of a relativistic extension of this group, by adjoining to this extension four space-time translations and by the napplying Bargmann-Wigner equations to guarantee aunitary norm. The procedure has similarities to the conventional method of induced representations. The boosting problem considered here is the first step towards the solution of the problem of coupling of such infinite-dimensional representations which is also briefly investigated. Startin g from a rest-symmetry like U (6,6) a chain of subgroups GL (6), U (3,3), etc., is exhibited for collinear and coplanar processes, etc.

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