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Some Fock spaces with depth two action

Part of: General quantum mechanics and problems of quantization Selfadjoint operator algebras

Published online by Cambridge University Press:  05 November 2024

Michael Anshelevich  and
Jacob Mashburn Open the ORCID record for Jacob Mashburn [Opens in a new window]
Michael Anshelevich
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States e-mail: [email protected]
Jacob Mashburn*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, United States e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

The subject of this article is operators represented on a Fock space which act only on the two leading components of the tensor. We unify the constructions from [Ans07, BL09, BL11, LS08] and extend a number of results from these articles to our more general setting. The results include the quadratic relation satisfied by the kernel of the free cumulant generating function, the resolvent form of the generating function for the Wick polynomials, and classification results for the case when the vacuum state on the operator algebra is tracial. We handle the generating functions in infinitely many variables by considering their matrix-valued versions.

Keywords

Fock spacefree cumulantsBoolean cumulantsWick polynomials

MSC classification

Primary: 46L54: Free probability and free operator algebras
Secondary: 81S25: Quantum stochastic calculus
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 29
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work was supported in part by a Simons Foundation Collaboration Grant.

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