A generalization of the canonical commutation and anticommutation relations

Robbins, Steven

doi:10.1090/S0002-9939-1978-0482363-2

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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A generalization of the canonical commutation and anticommutation relations
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Proc. Amer. Math. Soc. 71 (1978), 85-88 Request permission

Abstract:

The relation ${C^ \ast }C{C^ \ast } - {C^ \ast } = C{C^{ \ast 2}}$ involving a closed densely defined operator C generalizes both the canonical commutation relation of bosons and the canonical anticommutation relation of fermions. It is shown that the operators satisfying this relation can be classified by an index $p,p = 0,1,2, \ldots ,\infty$.

References

C. R. Putnam, Commutation properties of Hilbert space operators and related topics, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 36, Springer-Verlag New York, Inc., New York, 1967. MR 0217618