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Infinite Dimensional Analysis Quantum Probability and Related Topics

A REPRESENTATION OF THE BELAVKIN EQUATION VIA PHASE SPACE FEYNMAN PATH INTEGRALS

Sergio A Albeverio, Giuseppina Guatteri, Sonia Mazzucchi

Published Nov 27, 2004• 13 citations
Publisher

Abstract

The Belavkin equation, describing the continuous measurement of the momentum of a quantum particle, is studied. The existence and uniqueness of its solution is proved via analytic tools. A stochastic characteristics method is applied. A rigorous representation of the solution by means of an infinite dimensional oscillatory integral (Feynman path integral) defined on the phase space is also given.

DOI
10.1142/s0219025704001748

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The Belavkin equation, describing the continuous measurement of the momentum of a quantum particle, is studied. The existence and uniqueness of its solution is proved via analytic tools. A stochastic characteristics method is applied. A rigorous representation of the solution by means of an infinite dimensional oscillatory integral (Feynman path integral) defined on the phase space is also given.
In 1979 B. Menski suggested a formula for the linear propagator of a quantum system with continuously observed position in terms of a heuristic Feynman path integral. In 1989 the aposterior linear stochastic Schrödinger equation was derived by V. P. Belavkin describing the evolution of a quantum system under continuous (nondemolition) measurement. In the present paper, these two approaches to the description of continuous quantum measurement are brought together from the point of view of physics as well as mathematics. A self-contained deductions of both Menski's formula and the Belavkin equation is given, and the new insights in the problem provided by the local (stochastic equation) approach to the problem are described. Furthermore, a mathematically well-defined representations of the solution of the aposterior Schrödinger equation in terms of the path integral is constructed and shown to be heuristically equivalent to the Menski propagator.
Path integration methods are of crucial importance to quantum mechanics and quantum field theory. There are multiple ways the path integral can be constructed, one method uses the link between the Fokker-Planck and the Langevin equations, as is covered in this thesis. This is related to the standard derivation of the path integral in physics, in which the time is discretised. We study the mathematical problems related to the Feynman path integral, in particular the impossibility of a Lebesgue-type measure on the space of paths. It is discussed how oscillatory integrals can be used to have a well-defined `integral' on an infinite dimensional space. This formalism is subsequently applied to both the harmonic and the anharmonic oscillator. We prove an infinite dimensional oscillatory integral exists and obtain a convergent expression for both these cases, under conditions. Examples of these conditions are the initial wavefunctions being Schwartz functions as well as conditions on the endtime, angular frequency and coupling constant. Finally, a possible approach to establish an oscillatory integral for the hydrogen atom is discussed. It is proven that the result is no longer independent of the sequence of projection operators, which is a key step towards a rigorous path integral for this system.
It has been stated in the physics literature that the probability amplitude for a quantum system after a continuous position measurement can be expressed by the restricted or the weighted Feynman path integral containing information about the result and the resolution of its measurement. In the present paper it is proved that the restricted Feynman path integrals can be defined in a rigorous sense. More specifically, we consider a quantum system consisting of some particles and simultaneously perform a continuous measurement of the positions of all particles in the time interval [0,T][0,T][0,T]. Then, it is shown rigorously that the restricted Feynman path integral after the measurement can be defined in the L2L^2L2 space and satisfies the corresponding non-self-adjoint Schroeodinger equation. In addition, our results are generalized to a quantum spin system where a continuous position measurement for all spin components of all particles is simultaneously performed, though the results obtained are inadequate for general use.