W. Pusz

Functional calculus for sesquilinear forms and the purification map

The paper gives a proposition of a functional calculus for positive sesquilinear forms. A definition of any homogeneous function of two positive sesquilinear forms is given. The purification map for states on C∗-algebras is described in terms of the geometrical mean of two positive forms related to states in a natural way. Properties of the geometrical mean are investigated.

Passive states and KMS states for general quantum systems

We characterize equilibrium states of quantum systems by a condition of passivity suggested by the second principle of thermodynamics. Ground states and β-KMS states for all inverse temperatures β≧0 are completely passive. We prove that these states are the only completely passive ones. For the special case of states describing pure phases, assuming the passivity we reproduce the results of Haag et al.

Twisted canonical anticommutation relations

Twisted canonical anticommutation relations (TCAR) connected with a formalism of a second quantization procedure based upon the twisted SU(N) group are described. Irreducible representations of TCAR are investigated and a uniqueness theorem is proved.

Irreducible unitary representations of quantum Lorentz group

A complete classification of irreducible unitary representations of a one parameter deformationSqL(2,C) (0<q<1) ofSL(2,C) is given. It shows that in spite of a popular belief the representation theory forSqL(2,C) is not “a smooth deformation” of the one forSL(2,C).

Form convex functions and the wydl and other inequalities

We show that the functional calculus of sesquilinear forms derived in [3] leads to a general theory of the Wigner, Yanase, Dyson-Lieb-type inequalities. In particular, we obtain the joint convexity and the monotonicity of the relative entropy functional.

REPRESENTATIONS OF QUANTUM LORENTZ GROUP ON GELFAND SPACES

A large class of representations of the quantum Lorentz group QLG (the one admitting Iwasawa decomposition) is found and described in detail. In a sense the class contains all irreducible unitary representations of QLG. Parabolic subgroup P of the group QLG is introduced. It is a smooth deformation of the subgroup of SL (2, C) consisting of the upper-triangular matrices. A description of the set of all 1-dimensional representations (the characters) of P is given. It turns out that the topological structure of this set is not the same as for the parabolic subgroup of the classical Lorentz group. The class of (in general non-unitary) representations of QLG induced by characters of its parabolic subgroup P is investigated. Representations act on spaces of smooth sections of (quantum) line boundles over the homogeneous space P\QLG (Gelfand spaces) as in the classical case. For any pair of Gelfand spaces the set of all non-zero invariant bilinear forms is described. This set is not empty only for certain pairs. We give a complete list of such pairs. Using this list we solve the problems of equivalence and irreducibility of the representation. We distinguish a class of Gelfand spaces carrying unitary representations of QLG.

On the implementation of Sμ U(2) action in the irreducible representations of twisted canonical commutation relations

The natural Sμ U(2) action on the creation and annihilation operators satisfying twisted canonical commutation relations (TCCR) is investigated. It is shown that the Fock representation is the only covariant irreducible representation of TCCR.

Quantum GL(2, C) group as double group over ‘az + b’ quantum group☆

In this paper the quantum GL(2, C) group constructed in [4] is investigated. The deformation parameter q = e2πeN, where N is an even natural number. It is shown that this group can be obtained by the double group construction applied to the quantum ‘az + b’ group. The latter is the quantum deformation of the group of affine transformations of complex plane introduced in [9].

A quantum group at roots of unity

Abstract We construct a quantum deformation of corresponding to the deformation parameter , where N is an even natural number. Hopf *-algebra, Hilbert space and C *-algebra levels are considered. The C *-algebra A , which may be interpreted as the algebra of all continuous functions on the group vanishing at infinity, is generated by five elements α, β, γ, δ and det −1 satisfying certain commutation relations completed by hermiticity conditions. The group structure is encoded by a comultiplication Φ ϵ Mor ( A , A ⊗ A ) acting on generators in the standard way. On the Hopf algebra level our deformation corresponds to a one-parameter family of the standard two-parameter deformations .

Passive states for finite classical systems

Starting from a passivity condition based on the second law of thermodynamics [12], we show that ground states and Gibbs states (0<β<∞) are essentially the only passive states.