Time-orthogonal unitary dilations and noncommutative Feynman-Kac formulae. II. (English) Zbl 0575.46053
In an earlier paper [Commun. Math. Phys. 83, 261-280 (1982; Zbl 0485.46038)] a dilation was constructed of an arbitrary strongly continuous self-adjoint contraction semigroup in a Hilbert space \({\mathfrak h}\), second quantisation of which in Boson Fock space over \({\mathfrak h}\) gives rise to a Feynman-Kac type formula. In this paper corresponding Feynman-Kac formulae are found firstly for when Boson is replaced by Fermion second quantisation, and secondly for when Fock is replaced by second quantisation corresponding to an extremal universally invariant state of the CCR algebra. In the latter case it is found to be necessary to restrict the infinitesimal generator of the semigroup to be a trace class operator.
Although the dilation constructed in these papers is a two parameter evolution rather than a oneparameter group dilation, its Euclidean covariance property ensures that there is an associated group dilation. Thus the time orthogonal construction may be compared with other universal constructions of unitary dilations of contraction semigroups, to which it is of course equivalent. See for example B. Kümmerer and W. Schröder, A new construction of unitary dilations: singular coupling to white noise, Lect. Notes. Math. 1136, 332-347 (1985).
Although the dilation constructed in these papers is a two parameter evolution rather than a oneparameter group dilation, its Euclidean covariance property ensures that there is an associated group dilation. Thus the time orthogonal construction may be compared with other universal constructions of unitary dilations of contraction semigroups, to which it is of course equivalent. See for example B. Kümmerer and W. Schröder, A new construction of unitary dilations: singular coupling to white noise, Lect. Notes. Math. 1136, 332-347 (1985).
MSC:
46L60 | Applications of selfadjoint operator algebras to physics |
47D03 | Groups and semigroups of linear operators |
81P20 | Stochastic mechanics (including stochastic electrodynamics) |
46L51 | Noncommutative measure and integration |
46L53 | Noncommutative probability and statistics |
46L54 | Free probability and free operator algebras |
81S40 | Path integrals in quantum mechanics |
Keywords:
dilation; strongly continuous self-adjoint contraction semigroup in a Hilbert space; second quantisation; Boson Fock space; Feynman-Kac type formula; Fermion second quantisation; extremal universally invariant state of the CCR algebra; trace class operator; two parameter evolution; Euclidean covariance propertyCitations:
Zbl 0485.46038References:
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