Journal article
Index theory and second quantization of boundary value problems
Journal of functional analysis, Vol.104(2), pp.243-290
1992
DOI: 10.1016/0022-1236(92)90001-Y
Abstract
The second quantization functor associates to each skew-symmetric operator in one-particle space a derivation δ of the algebra which is based on the given commutation relations. In this paper, we characterize the spatial theory of δ (in the Fock representation) by an index which generalizes the one studied earlier by Powers and Arveson in connection with the spatial cohomological obstruction for semigroups of endomorphisms of B H . It is well known that such semigroups corresponding to one-sided boundary conditions are generated by derivations; but derivations associated to two-sided boundary conditions do not generate semigroups. We show that the known index theory for semigroups generalizes to the quantization of arbitrary boundary conditions in one-particle space. Our quantized two-sided abstract boundary conditions dictate representations in a certain indefinite inner product space (a Krein space), and our index is an isomorphism invariant for representation theory in Krein spaces. The representations are not unitarizable (i.e., are not equivalent to Hermitian representations in Hilbert space).
Details
- Title: Subtitle
- Index theory and second quantization of boundary value problems
- Creators
- Palle E.T JorgensenGeoffrey L Price
- Resource Type
- Journal article
- Publication Details
- Journal of functional analysis, Vol.104(2), pp.243-290
- DOI
- 10.1016/0022-1236(92)90001-Y
- ISSN
- 1096-0783
- eISSN
- 1096-0783
- Language
- English
- Date published
- 1992
- Academic Unit
- Mathematics
- Record Identifier
- 9983985850602771
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