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Theory of Probability and Its Applications

An Equation of the Kolmogorov--Feller Type in a Factorized Probability Space

Виктор Павлович Маслов

Published Jan 1, 2000• 1 citation
Publisher

Abstract

Calculations of probabilities are considered in the products of separable probability spaces, factored with respect to the permutation group of factors, by means of the trace operation with respect to the symmetric and antisymmetric orthonormal bases. A nonlinear Kolmogorov--Feller type equation is deduced which is the image of the forward linear Kolmogorov--Feller equation under factorization of the initial probability space with respect to the permutation group with certain rules of superselection.

DOI
10.1137/s0040585x97977586

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Calculations of probabilities are considered in the products of separable probability spaces, factored with respect to the permutation group of factors, by means of the trace operation with respect to the symmetric and antisymmetric orthonormal bases. A nonlinear Kolmogorov--Feller type equation is deduced which is the image of the forward linear Kolmogorov--Feller equation under factorization of the initial probability space with respect to the permutation group with certain rules of superselection.
A function η(i1, …, if) of ƒ quantities i, varying over the finite range i = 1,2, …, n, is usually called an n-dimensional tensor of rank ƒ. Any permutation p: 1 → 1, … , ƒ → ƒ… changes this tensor into a tensor pη according to the equation pη(i1, ...,if) = η(i1', ...,if')Thus the permutation p appears as a linear operator p in the n-dimensional space Σ = Σn,ƒ of all n-dimensional tensors of rank ƒ, η is symmetric if pη = η for all permutations p, it is antisymmetric if pη δp.η where δp = + 1 for the even and — 1 for the odd permutations.
Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Aufféves and Grangier, which in turn are inspired by Gleason's theorem.
The dynamics of the system in the space of random joint events is considered. The symmetric difference of events is introduced in space based on the Kolmogorov axioms. To describe quantum effects in the dynamics of the system, an additional axiom is introduced for random joint events: “the symmetric sum of random events.” In the generated space of random joint events, an equation is constructed for the probability of a system transition between two events. It is shown that for pairwise joint events it is equivalent to the equation of quantum mechanics.
In this paper we analyze a recently proposed approach for the construction of antisymmetric functions for atomic and molecular systems. It is based on the assumption that the main problems with Hartree-Fock wavefunctions stem from their lack of proper permutation symmetry. This alternative building approach is based on products of a space times a spin function with opposite permutation symmetry. The main argument for devising such factors is that the eigenfunctions of the non-relativistic Hamiltonian are either symmetric or antisymmetric with respect to the transposition of the variables of a pair of electrons. However, since the eigenfunctions of the non-relativistic Hamiltonian are basis for the irreducible representations of the symmetric group they are not necessarily symmetric or antisymmetric, except in the trivial case of two electrons. We carry out a simple and straightforward general analysis of the symmetry of the eigenfunctions of the non-relativistic Hamiltonian and illustrate our conclusions by means of two exactly-solvable models of N=2N=2N=2 and N=3N=3N=3 identical interacting particles.
Abstract In quantum mechanics, mutually unbiased bases (MUBs) represent orthonormal bases (ONBs) that are as ‘far apart’ as possible, and their classification reveals rich underlying geometric structure. Given a complex inner product space, we construct the space of its ONBs as a discrete quotient of the complete flag manifold. We introduce a metric on this space, which corresponds to the ‘MUBness’ distance. This allows us to describe equivalence between sets of MUBs in terms of the geometry of this space. The subspace of bases that are unbiased with respect to the standard basis decomposes into orbits under a certain group action, and this decomposition corresponds to the classification of complex Hadamard matrices. More generally, we consider a list of k MUBs, that one wishes to extend. The candidates are points in the subspace comprising all bases which are unbiased with respect to the entire list. This space also decomposes into orbits under a group action, and we prove that points in distinct orbits yield inequivalent MUB lists. Thus, we generalize the relation between complex Hadamard matrices and MUBs. As an application, we identify new symmetries that reduce the parameter space of MUB triples in dimension 4 by a factor of 4.