P. Truini

RELATIVISTIC WAVEFUNCTIONS ON SPINOR SPACES

In extending the work done by Wigner (1963) the authors introduce the Poincare group representations of functions defined on the space of complex spinors. They give a geometrical interpretation of the spinor space for both the massive and massless cases. In the massive case they get a six-dimensional manifold, with three compact dimensions. In the massless case they get a four-dimensional manifold, with one compact dimension.

Representation theory approach to the polynomial solutions of q‐difference equations: Uq(sl(3)) and beyond

A new approach to the theory of polynomial solutions of q‐difference equations is proposed. The approach is based on the representation theory of simple Lie algebras G and their q‐deformations and is presented here for Uq(sl(n)). First a q‐difference realization of Uq(sl(n)) in terms of n(n−1)/2 commuting variables and depending on n−1 complex representation parameters, ri, is constructed. From this realization lowest weight modules (LWM) are obtained which are studied in detail for the case n=3 (the well‐known n=2 case is also recovered). All reducible LWM are found and the polynomial bases of their invariant irreducible subrepresentations are explicitly given. This also gives a classification of the quasi‐exactly solvable operators in the present setting. The invariant subspaces are obtained as solutions of certain invariant q‐difference equations, i.e., these are kernels of invariant q‐difference operators, which are also explicitly given. Such operators were not used until now in the theory of polynom...

The Jordan Pair content of the magic square and the geometry of the scalars in N=2 supergravity

The close connection between Jordan and Lie algebras makes these Jordan structures of interest to physicists. The Freudenthal-Tits Magic Square, which exemplifies this connection, has recently entered into constructing supergravity. We show how Jordan pairs-which are, from several points of view, a most natural Jordan structure-are imbedded in the Magic Square. We compare our approach with that of Gürsey and show show the Hermitian symmetric spaces parametrized by the scalars of N=2, d=4 supergravity theories are related either to Jordan pairs or to geometries of projective dimension two, whose elements belong to a Jordan pair.

Exceptional Lie algebras, SU(3) and Jordan pairs: part 2. Zorn-type representations

A representation of the exceptional Lie algebras reflecting a simple unifying view, based on realizations in terms of Zorn-type matrices, is presented. The role of the underlying Jordan pair and Jordan algebra content is crucial in the development of the structure. Each algebra contains three Jordan pairs sharing the same Lie algebra of automorphisms and the same external su(3) symmetry. The applications in physics are outlined.

The concept of a quantum semisimple group

We explore the possibility of introducing the concept of a quantum semisimple group by exhibiting a class of deformations of classical groups whose Dynkin diagrams are disconnected at the classical level, but become connected at the quantum level. The possibility of applications to the quantization of Lorentz and Poincaré groups are mentioned.

On some nonunitary representations of the Poincaré group and their use for the construction of free quantum fields

In this paper a class of nonunitary infinite dimensional Hilbert space representations of a semidirect product is investigated. The equivalence of this category with the category of finite dimensional representations of the stability subgroups is shown. This theory is applied to the Poincare group and to the construction of free quantum fields. In an appendix a method is introduced for building an infinite family of finite dimensional indecomposable representations of the noncompact Euclidean group in two dimensions. Such representations are used for carrying out the analysis of the massless fields.

Conditional probabilities on orthomodular lattices

Abstract A definition of generalized probability on an orthomodular lattice which includes as particular cases the classical probability space and non-commutative probability theory on a von Neumann algebra is proposed. In this generalized structure the problem of conditioning with respect to Boolean σ-subalgebras is examined.

QUANTIZATION OF REDUCTIVE LIE ALGEBRAS: CONSTRUCTION AND UNIVERSALITY

The present paper addresses the question of universality of the quantization of reductive Lie algebras. Quantization is viewed as a torsion free deformation depending upon several parameters which are treated formally and not as complex numbers. The coalgebra and algebra structures are shown to restrict very sharply the possibilities for the infinite series in the generators of the Cartan subalgebra. Under an Ansatz which can be viewed as requiring that the two Borel subalgebras are deformed as Hopf algebras we construct a multi-parameter quantization which has the required property of universality. We also show that such a quantization can be defined so that the algebra structure is the same as that of the standard one-parameter quantization, the remaining parameters being relegated to the coalgebra structure. We discuss an example in which only the latter parameters appear in the deformation. We then complete the study of the universal deformations by developing some aspects of the representation theory of the deformed algebras. Using this theory, especially the freeness of the irreducible modules, we prove the analogue of the Poincare-Birkhoff-Witt theorem, and, as a consequence, the torsion freeness of the universal deformations.

A remark on possible violations of the Pauli principle

Recently Ignatiev and Kuzmin (1988) developed a theoretical model to implement small violations of the Pauli principle. The authors remark that the Pauli principle is unique among discrete symmetries, and that in consequence any apparent violation actually signals new physical degrees of freedom, with no violation of the Pauli principle. They construct an algebraic model which incorporates the algebra of Ignatiev and Kuzmin and leads to the same apparent violations, yet preserves the Pauli principle. Their algebra is that of a Jordan pair, an algebraic concept which provides a natural framework for structures that do not assume bilinear commutation relations.

Universal deformations of reductive Lie algebras

We construct multiparameter quantizations of reductive Lie algebras which have the property of universality within a certain class of deformations. The universal deformations can be defined so that the algebra structure on each simple component is the same as that of the standard one-parameter quantization, the remaining parameters being relegated to the coalgebra structure. We discuss an example in which only the latter parameters appear, as a special case of deformations of a semisimple algebra whose simple components remain classical. Deformations are defined as algebras over power series rings and it is essential to require them to be torsion free to secure the universality. The Poincaré-Birkhoff-Witt theorem and the torsion freeness are established for the universal deformation on the basis of results on the representation theory of the deformed algebras.