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arXiv:1112.4616 · International Journal of Theoretical Physics

On the Lattice Structure of Probability Spaces in Quantum Mechanics

Federico Holik, César Massri, Angel Ricardo Plastino, Leandro Zuberman

Published Aug 8, 2012• 20 citations
arXivHTMLPDF
DOI
10.1007/s10773-012-1277-5

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A generalized probability theory arises naturally from an attempt to formulate a statistical model for quantum mechanics.One soon finds that the probability theory of Kolmogorov is inadequate for the description of a quantum mechanical system mainly because the quantum mechanical events fail to form a er-algebra of subsets of a set.Instead, they have a much less richly endowed algebraic structure which we shall a "logic".We therefore look upon generalized probability as a stochastic theory in which the events form a logic.Besides being useful in the study of quantum mechanics, this theory has a purely mathematical interest since it constructs a stochastic formalism on an algebraic structure and at the same time gives an abstract generalization of Hubert space theory.The first formulation of this theory is due to G. Bodiou [1], [2].More recently, V. S. Varadarajan [8] has emphasized the stochastic properties of an axiomatic formulation of quantum mechanics due to G. W. Mackey [5] and has proved some important results in this theory.The author heartily recommends these references for the background and motivation of this study.In this paper, the main interest centers on the spectral theory of the subject.The spectrum is important physically since it represents the "allowable" values of a quantum mechanical observable.It is also mathematically useful since it introduces topological methods into the formalism.2. Definitions and notation.Let L be a complemented lattice.That is, L is a lattice with first and last elements, 0,1 respectively and a-*a' satisfying the conditions (i) ia'Y = a for all a eL;(ii) a z% b implies b' zi a'; (iii) a V a ' -1 for all a e L.
This acceptance talk is a curious mixture of personal history and developing ideas in the context of the growing field of IR covering several decades. I want to concentrate on models and theories, interpreted loosely, and try and give an insight into where I have got to in my thinking, where the ideas came from, and where I believe we are going.In the last few years I have been working on the development of what might be coined as a design language for IR. It takes its inspiration from Quantum Mechanics, but by analogy only. The mathematical objects represent documents; these objects might be vectors (or density operators) in an n-dimensional vector space (usually a Hilbert space). A request for information, or a query, is taken as an observable and is represented as a linear operator on the space. Linear operators can be expressed as matrices. Such an operator, Hermitian, has a set of eigenvectors forming a basis for the space; which we interpret as a point of view or perspective from which to understand the space. Thus any document-vector can be located with respect to the basis, and we can calculate an inner product between such a vector and any basis vector, which may be interpreted as a probability of relevance. The probability of observing any given eigenvector is now given by the square of that inner product assuming all vectors are normalised. Hence we connect the probability of observation to the geometry of the space. Furthermore, the subspaces of the space make up a lattice structure which is equivalent to a logic. This makes up the entire mathematical structure, and the language for handling this structure is linear algebra: vectors, matrices, projections, inner-products, neatly captured by the Dirac notation used in quantum mechanics. Our probability is slightly different from classical probability, the same for logic; we end up with quantum logic and quantum probability.A commitment to this kind of mathematical structure, with which to model objects and processes in IR, depends on two critical assumptions.