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Birkhäuser Basel eBooks

Stochastic PDE’s of Schrödinger Type and Stochastic Mehler Kernels — a Path Integral Approach

Aubrey Truman, Tomasz Zastawniak

Published Jan 1, 1999• 4 citations
Publisher

Abstract

We present a rigorous path integral derivation of stochastic Mehler kernel formulae with applications to stochastic partial differential equations used in the theory of quantum measurement and filtering. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-0348-8681-9_19

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Revista Brasileira de Ensino de Física•Jan 1, 2024•1 citation
We present a rigorous path integral derivation of stochastic Mehler kernel formulae with applications to stochastic partial differential equations used in the theory of quantum measurement and filtering. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In Section 21 we have already seen how the classical stochastic processes with independent increments can be realised as suitable linear combinations of the creation, conservation and annihilation operators in the boson Fock space Γ s (ℋ) over a Hilbert space ℋ. This includes, in particular, the Brownian motion and Poisson process. Since a well-developed theory of stochastic integration with respect to these classical processes exists, it is natural to wonder whether it would be possible to introduce quantum probabilistic analogues of stochastic integrals with respect to processes defined by the creation, conservation and annihilation operators. To this end one needs the notions of filtration and adapted processes. This, in turn, depends on the notion of time. We adopt the view that time is an ℝ+-valued observable in a Hilbert space ℋ which, as a spectral measure, has no jumps. The absence of jumps is assumed more as a matter of mathematical convenience than a philosophical or conceptual necessity. We shall now examine how such a notion of time leads to a filtration and the definition of adapted processes. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In the preceding chapter we have formulated the equation of motion of relativistic quantum particles as a pair of evolution equations with fractional power generators which are in formal duality, and shown that the evolution equation has an equivalent formulation in terms of stochastic differential equations of pure-jumps. In this chapter the existence and uniqueness of solutions of the stochastic differential equations of pure-jumps will be proved, and then pure-jump Markov processes will be constructed. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is the fourth paper of a series devoted to mathematically derive the Schrödinger equation using stochastic constructs. We then show that Quantum Mechanics has a stochastic support by following two paths: we first present a stochastic derivation already known in the literature and show that it is equivalent to the characteristic function derivation, made by us in the first paper of this series. However, this approach does not furnish the true stochastic dynamics of the quantum mechanical system, but only an averaged one. We then present a new derivation of the Schrödinger equation based on Langevin equations that present all features of a true stochastic system. These two approaches then improve our understanding of Quantum Mechanics to encompass stochastic behavior.