×

Representation of Lie superalgebras and generalized boson-fermion equivalence in quantum stochastic calculus. (English) Zbl 0882.60097

Summary: The boson-fermion equivalence scheme of the second author and K. R. Parthasarathy [ibid. 104, 457-470 (1986; Zbl 0604.60063)] can be generalized to \(N\) dimensions with \(r\) boson and \(N-r\) fermion creation and annihilation fields. The same stochastic integral prescription replaces the \(N^2\) generalized number processes \(\Lambda^i_j\), which form representations of the Lie algebra \(gl(N)\), by processes forming representations of the Lie superalgebra \(gl(N,r)\).

MSC:

60K40 Other physical applications of random processes
60H99 Stochastic analysis
81S25 Quantum stochastic calculus
17A70 Superalgebras

Citations:

Zbl 0604.60063
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Driver, K., On the Kakutani-Itô-Segal-Gross and the Segal-Bargmann-Hall isomorphisms, J. Funct. Anal., 133, 69-128 (1995) · Zbl 0846.43001 · doi:10.1006/jfan.1995.1120
[2] Evans, M. P., Existence of quantum diffusions, PTRF, 81, 473-82 (1989) · Zbl 0667.60060 · doi:10.1007/BF00367298
[3] Eyre, T.M.W.: Chaos expansions in Lie superalgebras associated with a quantum stochastic calculus. Nottingham. Preprint. To appear in Commun. Math. Phys.
[4] Hall, B., The Segal-Bargmann “coherent state” transform for compact Lie groups, J. Funct. Anal., 122, 103-151 (1994) · Zbl 0838.22004 · doi:10.1006/jfan.1994.1064
[5] Hudson, R. L.; Parthasarathy, K. R., Unification of Boson and Fermion stochastic calculus, Commun. Math. Phys., 104, 457-470 (1986) · Zbl 0604.60063 · doi:10.1007/BF01210951
[6] Hudson, R. L.; Parthasarathy, K. R., Quantum Itô’s formula and stochastic evolutions, Commun. Math. Phys., 93, 301-323 (1984) · Zbl 0546.60058 · doi:10.1007/BF01258530
[7] Hudson, R. L.; Parthasarathy, K. R., Casimir chaos in a Boson Fock space, J. Funct. Anal., 119, 319-339 (1994) · Zbl 0804.60052 · doi:10.1006/jfan.1994.1013
[8] Hudson, R. L.; Parthasarathy, K. R., Chaos map for the universal enveloping algebra ofU(N), Math. Proc. Camb. Phil. Soc., 117, 21-30 (1995) · Zbl 0815.60057 · doi:10.1017/S030500410007290X
[9] Hudson, R.L., Pulmannová, S.: Chaotic expansions of elements of the universal enveloping algebra of a Lie algebra associated with a quantum stochastic calculus. Nottingham, Preprint. To appear in Jour. London Math. Soc. · Zbl 0904.60084
[10] Gross, L., Malliavin, P.: Hall’s transform and the Segall-Bargmann map. To appear in Taniguchi Proceedings, Warwick, 1995 · Zbl 0869.22006
[11] Parthasarathy, K. R., An introduction to quantum stochastic calculus (1992), Basel: Birkhäuser, Basel · Zbl 0751.60046
[12] Scheunen, M., The theory of Lie superalgebras, springer LNM716 (1979), Berlin-Heidelberg-New York: Springer-Verlag, Berlin-Heidelberg-New York · Zbl 0407.17001
[13] Segal, I. E., Tensor algebras over Hilbert spaces II, Ann. Math., 63, 160-175 (1956) · Zbl 0073.09403 · doi:10.2307/1969994
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.