Alfred Rieckers

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...

Power series of the free field as operator-valued functional on spaces of typeS

Infinite series of Wick powers of the free, massive Bose field are analysed in terms of test function spaces of typeS for arbitrary space dimension. By direct estimates of the smeared phase space integrals sufficiency conditions for the existence of the vacuum expectation values are derived. These conditions are shown to be precise. The field-operators are defined on a dense invariant domain in Fock space, where they satisfy the Wightman axioms with the possible exception of locality. Localisable and nonlocalisable fields are dealt within the same frame. The behaviour of spectral functions and the strength of singularities are discussed.

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.

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.

Coherence and incompatability inWu* -algebraic quantum theory

In the framework of generalized quantum theory using aW*-algebraic formalism, we introduce a completely symmetric coherence relation for states which is also applicable to nonpure states. Making use of lattice theoretic results the properties of this relation, especially its connection with incompatibility, are investigated. By means of algebraic decomposition theory the investigation is reduced to the case of factors where a complete classification of the coherence classes is given.

Limiting Gibbs States and Phase Transitions of a Bipartite Mean-Field Hubbard Model

In the frame of operator-algebraic quantum statistical mechanics we calculate the grand canonical equilibrium states of a bipartite, microscopic mean-field model for bipolaronic superconductors (or anisotropic antiferromagnetic materials in the quasispin formulation). Depending on temperature and chemical potential, the sets of statistical equilibrium states exhibit four qualitatively different regions, describing the normal, superconducting (spin-flopped), charge ordered (antiferromagnetic), and coexistence phases. Besides phase transitions of the second kind, the model also shows phase transitions of the first kind between the superconducting and the charge ordered phases. A unique limiting Gibbs state is found in its central decomposition for all temperatures, even in the coexistence region, if the thermodynamic limit is performed at fixed particle density (magnetization).

Perturbation Dynamics of the Infinite Dicke Model

Abstract By means of operator algebraic methods the dynamics of the Dicke model is investigated in the limit where the number of the two-level atoms goes to infinity and the interaction strength remains on a finite level. The infinite atomic system is treated as a mean field quantum lattice system. It is shown that the limiting dynamics is essentially determined by the collective behaviour of the atoms. With Trotters product formula and perturbation theoretical methods we obtain explicit expressions for the unitary time evolution operators in the uncoupled representation.

Squeezing operations in Fock space and beyond

Combining the signal and the idler modes we show that each nondegenerate squeezing quadratic Hamiltonian may be transformed into the form of a degenerate squeezing Hamiltonian, if one uses the smeared field formalism. For the case of infinitely many photon modes we discuss the existence of squeezing quadratic Hamiltonians in Fock space. This gives a limitation on the squeezing parameters, which guarantees that all squeezed vacua are representable as vectors in Fock space. If this condition is not satisfied (the case of strong squeezing) then all squeezed vacua are outside the Fock space and mutually inequivalent.

Limiting dynamics of generalized Bardeen–Cooper–Schrieffer models beyond the pair algebra

Bardeen–Cooper–Schrieffer (BCS)‐like models for permutation invariant electronic interactions are investigated in terms of recent rigorous mean‐field methods. Their limiting dynamics is constructed in the Heisenberg picture on (extensions of) the full canonical commutation relation (CAR) algebra and also restricted to (extensions of) the pair algebra. The classical part of the proper BCS dynamics on the CAR algebra is shown to exhibit 15 macroscopic collective variables, the time dependence of which is explicitly integrated and discussed.

Partially Classical States of a Boson Field

Employing positive-definiteness arguments we analyse Boson field states, which combine classical and quantum mechanical features (signal and noise), in a constructive manner. Mathematically, they constitute Bauer simplexes within the convex, weak-*-compact state space of the C*-Weyl algebra, defined by a presymplectic test function space (smooth one-Boson wave functions) and are affinely homeomorphic to a state space of a classical field. The regular elements are expressed in terms of weak distributions (probability premeasures) on the dual test function space. The Bauer simplex arising from the bare vacuum is shown to generalize the quantum optical photon field states with positive P-functions.