Joachim Kupsch

Functional Representations for Fock Superalgebras

The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite-dimensional superspaces, and construct super-analogs of the classical function spaces with a reproducing kernel — including the Bargmann–Fock representation — and of the Wiener–Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein–Uhlenbeck semigroup on the Fock space.

States of quantum systems and their liftings

Let H1, H2 be complex Hilbert spaces, H be their Hilbert tensor product and let tr2 be the operator of taking partial trace, with respect to the space H2, of trace class operators in H. The operation tr2 maps states in H (=positive trace class operators in H with trace equal to 1) into states in H1. In this paper we give the full description of mappings that are linear right inverse to tr2. More precisely, we prove that any affine mapping F(W) of the convex set of states in H1 into the states in H that is right inverse to tr2 is given by W⟼W⊗D for some state D in H2. In addition we investigate a representation of the quantum mechanical state space by probability measures on the set of pure states and a representation—used in the theory of stochastic Schrodinger equations—by probability measures on the Hilbert space. We prove that there are no affine mappings from the state space of quantum mechanics into these spaces of probability measures.

Hilbert norms for graded algebras

Abstract: This paper presents a solution to a problem from superanalysis about the existence of Hilbert-Banach superalgebras. Two main results are derived: 1) There exist Hilbert norms on some graded algebras (infinite-dimensional superalgebras included) with respect to which the multiplication is continuous. 2) Such norms cannot be chosen to be submultiplicative and equal to one on the unit of the algebra.

ULTRACOHERENCE AND CANONICAL TRANSFORMATIONS

The (in)finite dimensional symplectic group of homogeneous canonical transformations is represented on the bosonic Fock space by the action of the group on the ultracoherent vectors, which are generalizations of the coherent states.

Applications of canonical transformations

Canonical transformations are defined and discussed along with the coherent and the ultracoherent vectors. It is shown that the single-mode and the n-mode squeezing operators are elements of the group of canonical transformations. An application of canonical transformations is made, in the context of open quantum systems, by studying the effect of squeezing of the bath on the decoherence properties of the system. Two cases are analysed. In the first case, the bath consists of a massless bosonic field with the bath reference states being the squeezed vacuum states and squeezed thermal states while in the second case a system consisting of a harmonic oscillator interacting with a bath of harmonic oscillators is analysed with the bath being initially in a squeezed thermal state.