Representation of Groups

Jędrzej Śniatycki

Gong-ou Xu, Shun-Jin Wang, Yang Ya-tian

The generator coordinate method approach to the dynamic group representation is discussed in general. In various cases, either in group space or in coset space, representations of the dynamic group can readily be obtained with the generator coordinate method. Boson representation is just one form of the generator coordinate method approach to the dynamic group representation. Various representations of the dynamic group are described by the generator coordinate method approach to the dynamic group representation in a unified way. Not only is the algebraic structure of generators preserved, but also conditions imposed by (a) the Pauli principle, (b) symmetry properties, (c) dynamic properties, and (d) other concrete properties of nuclear systems are well incorporated in these representations, so that the original fermion representation is faithfully realized. The generator coordinate method approach to the dynamic group representation is strictly a transformation theory from the fermion representation to continuous variable or boson representations of the dynamic group. Examples are given for showing the essence and the prospects of applications of the generator coordinate method approach to the dynamic group representation.

Luigi Accardi, Paolo Gibilisco

We consider a representation of the path groupoid over the group of diffeomorphisms of the base space of a Hilbert bundle (equipped with a unitary parallel transport). The Schroedinger representation in (Weyl form) appears as a flat and measure-invariant particular case of this representation. We prove that our representation induces a representation of Mensky's path group and that generators of the representation of one-parameter groups of the path group are given by the covariant derivative plus a volume correction. We prove that, as in the case of locally compact groups, the imprimitivity systems of this representation is irreducible if and only if the diffeomorphisms group acts transitively on the manifold and the inducing representation is irreducible. In our case the inducing subgroup is the loop group and the inducing representation is the holonomy group of the parallel transport.

Nikolai Borisovich Melnikov, Boris Il'ich Reser

Michel Talagrand

The orthogonal group admits projective unitary representations which do not derive from true representations, and we describe a fundamental family of such representations. As a consequence there exist quantum systems that change state under a full turn rotation along a given axis (although a second full turn rotation brings them back to the original state). Amazingly, Nature has made essential use of this structure. In order to study the projective representations of the orthogonal and Lorentz groups, it is convenient to replace them by “better versions“; the groups SU(2) and SL(2,C), which are groups of 2 by 2 matrices, and for which projective representations are simply related to true representations. The orthogonal and Lorentz groups are then images of these groups under two-to-one group homomorphisms, and it is these isomorphisms that concentrate the behavior of their projective representations. Finally we describe how the introduction of parity in our theory leads to the discovery of the Dirac matrices.

Nikolai Borisovich Melnikov, Boris Il'ich Reser