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Advances in physics theories and applications

The Concept of Moments of Order Invariant Quantum Processes.

I. G. Bassi, Markus Samaila, C E Okechukwu

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

Important information about stationary flows (B I ) I Î I is contained in moments of the corresponding process. We consider the case where we have an order invariance of the moments of the increments, in the sense that such moments do not change if we shift the increments against each other, as long as we do not change the relative position of the intervals. We then show that for large classes of quantum processes one can make precise statements about the time behaviour of their moments. Keywords: Von Neumann algebras, Stationary flows and quantum processes, order invariant, Trace and moments.

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Important information about stationary flows (B I ) I Î I is contained in moments of the corresponding process. We consider the case where we have an order invariance of the moments of the increments, in the sense that such moments do not change if we shift the increments against each other, as long as we do not change the relative position of the intervals. We then show that for large classes of quantum processes one can make precise statements about the time behaviour of their moments. Keywords: Von Neumann algebras, Stationary flows and quantum processes, order invariant, Trace and moments.
We introduce “Markovian” cocycle perturbations of quantum stochastic processes with stationary increments and the Kolmogorov flows generated by them, which are characterized by a localization of the perturbation to the algebra of events of the past. The Markovian perturbations of the Kolmogorov flow generated by the quantum white noise result in the groups of automorphisms on the algebras of events (the von Neumann algebras in the quantum case) possessing the restrictions being isomorphic to the initial Kolmogorov flow. The possibility of obtaining this restriction can be interpreted as some analogue (in the quantum case) of the Wold decomposition, which allows us to exclude the “nondeterministic” part of the process.
Stationary quantum stochastic process j is introduced as a *-homomorphism embedding an involutive graded algebra K~=⊕i=1∞Ki\tilde K=\oplus_{i=1}^{\infty}K_iK~=⊕i=1∞​Ki​ into a ring of (abelian) cohomologies of the one-parameter group ααα consisting of *-automorphisms of certain operator algebra in a Hilbert space such that every x from KiK_iKi​ is translated into an additive i−αi-αi−α-cocycle j(x). It is shown that (noncommutative) multiplicative markovian cocycle defines a perturbation of the stationary quantum stochastic process in the sense of such definition. The E0E_0E0​-semigroup β~\tilde ββ~​ on the von Neumann algebra N\cal NN associated with the markovian perturbation of K-flow j posseses the restriction β~∣N0,N0⊂N\tilde β|_{{\cal N}_0}, {\cal N}_0\subset {\cal N}β~​∣N0​​,N0​⊂N, which is conjugate to the flow of Powers shifts βββ associated with j. It yields for β~\tilde ββ~​ an analogue of the Wold decomposition for classical stochastic process on completely nondeterministic and deterministic parts. The examples of quantum stationary stochastic processes on the algebras of canonical commutation, anticommutation and square of white noise relations are considered. In the model situation of the space L2(R)L^2(\mathbb R)L2(R) all markovian cocycles of the group of shifts are described up to unitary equivalence of perturbations.
Abstract Moments are expectation values over wave functions (or averages over a set of classical particles) of products of powers of position and momentum. For the harmonic oscillator, the evolution in the quantum case is very closely related to that of the classical case. Here we consider the non-relativistic evolution of moments of all orders for the oscillator in one dimension and investigate invariant combinations of the moments. In particular, we find an infinite set of invariants that enable us to express the evolution of any moment in terms of sinusoids. We also find explicit expressions for the inverse of these relations, thus enabling the expression of the evolution of any moment in terms of the initial set of moments. More detailed attention is given to moments of the third and fourth order in terms of the invariant combinations.
Abstract Moments are expectation values of products of powers of position and momentum, taken over quantum states (or averages over a set of classical particles). For free particles, the evolution in the quantum case is closely related to that of a set of classical particles. Here we consider the evolution of symmetrized moments for free particles in one dimension, first examining the geometric properties of the evolution for moments up to the fourth order, as determined by their extrema and inflections. These properties are specified by combinations of the moments that are invariant in that they remain constant under free evolution. An inequality constrains the fourth-order moments and shows that some geometric types of evolution are possible for a quantum particle but not possible classically, and some examples are examined. Explicit expressions are found for the moments of any order in terms of their initial values, for the invariant combinations, and for the moments in terms of these invariants.