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Journal of Physics A Mathematical and General

An extended characterisation theorem for quantum logics

C. S. Sharma, M K Mukherjee

Published Oct 1, 1977• 3 citations
Publisher

Abstract

Two theorems are proved. In the first properties of an important mapping from an orthocomplemented lattice to itself are studied. In the second the authors extend the characterisation theorem of Zierler (1961) to get a very useful theorem characterising orthomodular lattices Since quantum logics are merely sigma -complete orthomodular lattices, the principal result is, for application in quantum physics, a characterisation theorem for quantum logics.

DOI
10.1088/0305-4470/10/10/003

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Journal of Symbolic Logic•Sep 1, 1973•13 citations
Two theorems are proved. In the first properties of an important mapping from an orthocomplemented lattice to itself are studied. In the second the authors extend the characterisation theorem of Zierler (1961) to get a very useful theorem characterising orthomodular lattices Since quantum logics are merely sigma -complete orthomodular lattices, the principal result is, for application in quantum physics, a characterisation theorem for quantum logics.
A class of quantum logics (orthomodular atomic lattices) is defined and studied. The obtained results suggest that the covering property acts as a selection criterion for theories in the sense that, given a set of orthomodular atomic lattices, those which may be theories for a fixed set of systems may be selected by verifyng if they have the covering property or not. A relation between physical meanings of the covering property, lattice - algebraic operations and the commutativity relation is also suggested
The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset. In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa. Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following: (1) the greatest lower bound of a and Na exists and is 0, (2) a ≤ b implies Nb ≤ Na, (3) NNa = a. Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfying where a ≤ b means a ∧ b = a, as usual. Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following: An orthomodular lattice gives a Finch implication algebra by defining → by A Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ by The orthocomplementation is unchanged in both cases.