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数理解析研究所講究録

ON A TOY FOCK SPACE GENERATED BY PERMUTATIONS

Naofumi Muraki

Published Nov 1, 1996• 0 citations
OpenAlex

Abstract

An example of independence, conditional expectation and martingales in quantum probability theory is constructed on the permutational Fock space which is a kind of discrete "Fock space" generated by an the permutations from the natural numbers.Besides a discrete time analogue of quantum Ito's formula for the permutational Fock space is obtained.

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Cambridge University Press eBooks•Feb 9, 2015•0 citations
An example of independence, conditional expectation and martingales in quantum probability theory is constructed on the permutational Fock space which is a kind of discrete "Fock space" generated by an the permutations from the natural numbers.Besides a discrete time analogue of quantum Ito's formula for the permutational Fock space is obtained.
We review Fock space based quantum probability and in particular, the theory of stop times based on it. In the Fock space , a stop time may be defined as a positive self-adjoint operator whose spectral resolution is adapted to the natural filtration based on the splittings Then if K is one of the basic quantum martingales (creation, preservation or annihilation) the corresponding process beginning anew at time T can be defined unambiguosly using functional calculus for the self-adjoint operator T with an operator-valued integrand since the integrand commutes with the integrator. An isometric operator , the forward shift through T, is defined by a similar spectral integral of sure forward shifts and conjugates with . An optional stopping theorem is formulated and proved using similar operator valued spectral integrals.
This chapter begins with introductions of symmetric Fock space Γ(ℍ) and symmetric Guichardet space Φ(ℍ) of a generic complex Hilbert space ℍ. It is shown that these 2 spaces are actually isomorphic and can and will be used interchangeably throughout the chapter. In particular, the simple and yet useful integral-summation formula (see Theorem (3.2.1)) for the Guichardet space is often employed to establish the essential results in quantum stochastic calculus. It has been shown that the symmetric Fock space (and hence the symmetric Guichadet space) provides a plausible mathematical tool for modeling phenomena in quantum optics or quantum electrodynamics. Many results in the quantum physics and quantum probability literature are actually established based on this concrete model space. When the generic complex Hilbert space ℍ = L 2 (ℝ + ; K) (where K is another complex Hilbert space), the class of exponential vectors along with 3 different types of quantum noise processes, namely, the creation, annihilation, and neutral (or guage) processes, can be introduced. It is shown that the subspace generated by the class of exponential vectors is dense in the Fock space. Therefore, it is convenient to verify the properties that hold for Fock space by verifying the same hold for the class of exponential vectors. Parallel to those of Itô integrals with respect to classical Brownian motion and/or Poisson process, the concepts of a quantum stochastic integral of a operator-valued process (as a member of the Fock space) with respect to each of the above mentioned quantum noise processes are constructed. The quantum stochastic calculus, parallel to those of classical Itô calculus, is developed. The theory of quantum stochastic calculus on the symmetric Fock space due originally to Hudson and Parthasarathy [HP84a] has been the topic of investigation by many researchers in the last 20 years. Much of he material presented in this chapter can be found in the books by Meyer [Mey95], Parthasarathy [Par92] and Attal [Att03].