Reinhard Honegger

Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic space

A systematic approach to the C*-Weyl algebra W(E,σ) over a possibly degenerate pre-symplectic form σ on a real vector space E of possibly infinite dimension is elaborated in an almost self-contained manner. The construction is based on the theory of Kolmogorov decompositions for σ-positive-definite functions on involutive semigroups and their associated projective unitary group representations. The σ-positive-definite functions provide also the C*-norm of W(E,σ), the latter being shown to be *-isomorphic to the twisted group C*-algebra of the discrete vector group E. The connections to related constructions are indicated. The treatment of the fundamental symmetries is outlined for arbitrary pre-symplectic σ. The construction method is especially applied to the trivial symplectic form σ=0, leading to the commutative Weyl algebra over E, which is shown to be isomorphic to the C*-algebra of the almost periodic continuous function on the topological dual Eτ′ of E with respect to an arbitrary locally convex Ha...

General glauber coherent states on the Weyl algebra and their phase integrals

Abstract Glaubers original definition of coherence in quantum optics is extended to an operator algebraic version, where now the coherent states are given on the Weyl algebra and the factorization function appearing in the coherence condition is replaced by a linear form on the associated pre-Hilbert space. This allows to consider more general cases than the Fock representation. Different kinds of factorization of the correlation functionals are used to classify coherent states. Quasi-free coherent states, gauge-invariant coherent states and their relationship are analyzed.

Squeezing Bogoliubov transformations on the infinite mode CCR‐algebra

A detailed analysis of and a general decomposition theorem for in general unbounded symplectic transformations on an arbitrary complex pre‐Hilbert space (one–boson test function space) are given. The structure of strongly continuous symplectic groups on such spaces is determined. The connection between quadratic Hamiltonians, Bogoliubov transformations, and symplectic transformations is discussed in the Fock representation, and their relevance for squeezing operations in quantum optics is pointed out. The results for this rather general class of transformations are proved in a self‐contained fashion.

Decomposition of Positive Sesquilinear Forms and the Central Decomposition of Gauge-Invariant Quasi-Free States on the Weyl-Algebra

Abstract A decomposition theory for positive sesquilinear forms densely defined in Hilbert spaces is developed. On decomposing such a form into its closable and singular part and using Bochners theorem it is possible to derive the central decomposition of the associated gauge-invariant quasifree state on the boson C*-Weyl algebra. The appearance of a classical field part of the boson system is studied in detail in the GNS-representation and shown to correspond to the so-called singular subspace of a natural enlargement of the one-boson testfunction space. In the example of Bose-Einstein condensation a non-trivial central decomposition (or equivalently a non-trivial classical field part) is directly related to the occurrence of the condensation phenomenon.

Limiting Gibbs states and dynamics for thermal photons

For thermal photons, the limiting Gibbs state is investigated in the framework of algebraic quantum theory. For the one‐particle Hamiltonian, the square root of the local Laplacians with Dirichlet boundary conditions is used. The local regions are only demanded to have the segment property and the thermodynamic limit is performed along a nearly arbitrary family of such local regions. Also, the limiting dynamics as a noncontinuous group of Bogoliubov transformations on an extension of the quasilocal Weyl algebra and as a W* dynamics on the GNS–von Neumann algebra is derived. After having performed the limits, the photon algebra is restricted to the physical (transversal) photons. The crucial mathematical method in verifying these results are comparisons of the semigroups of the square root of the Laplacians in different regions of the ν‐dimensional Euclidean space by means of the ordering of positivity preserving operators.

The general form of the microscopic coherent boson states

Abstract Recently the classical coherent states on the photonic C ∗-Weyl algebra have been classified. Non-classical coherence occurs only for states which are normal with respect to the Fock representation of the CCR. Here, the Fock-normal (=microscopic) coherent states are characterized completely. It suffices to consider Fock coherence only on the one-mode Weyl algebra W ( C ). The smoothness and coherence properties of a regular state on W ( C ) are expressed by the diagonal elements of the associated density matrix. With a Kolmogorov decomposition the off-diagonal matrix elements are replaced by a unique sequence of normalized vectors in a Hilbert space, which leads to the GNS representation and to a construction procedure for the whole set of all (microscopic) coherent states of arbitrary order. The variety of non-classical fully coherent states on W ( C ) is shown to be much larger than the one for classical fully coherent states. Moreover, it is proved that there exist (non-classical) elements of the extreme boundary of the weak ∗ -compact, convex set of fully coherent states, which are not pure states.

On Higher-Order Coherent States of the Weyl Algebra

It is shown that for classical, analytical states of the Weyl algebra, the quantum optical coherence condition of second order implies those of nth order for all n≥2.

On the Continuous Extension of States on the CCR Algebra

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Global cocycle dynamics for infinite mean field quantum systems interacting with the boson gas

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