Representations of orthogonal and unitary groups and nuclear models

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.

George W. Mackey

Masahito Hayashi

Peter Krämer, D. Schenzle

The relation between the Hilbert space H of quantum mechanics and the Hilbert space F of entire analytic functions introduced by Bargmann is used to derive unitary projective representations of the group SL(2,R) of linear canonical transformations. These representations are applied to an analysis of various nuclear models. (auth)

Gerald A. Goldin

It is possible that a complete physical theory can be written entirely in terms of operators such as current densities, rather than in terms of field operators. The current densities in these models correspond in general to distributions of unbounded operators. Such a theory is reviewed for the case of nonrelativistic quantum mechanics. It is found that one can exponentiate the current algebra to obtain a group, which may then be represented by unitary (hence bounded) operators in Hilbert space. This procedure is analogous to exponentiating the canonical commutation relations and obtaining the Weyl group. For nonrelativistic quantum mechanics without spin, the group is the semidirect product J∧K Schwartz space J(R3) with a group K of certain C∞ diffeomorphisms from R3 onto itself. For f ∈ J and φ ∈ K, the composition mapping (f,φ)→f∘φ defines the semidirect product law. The Gel'fand-Vilenkin formalism for ``nuclear Lie groups'' is suitable for the representation theory of such a group. Almost all of the physical information is contained in a cylindrical measure μ on J′, the dual of the nuclear space J. The Fourier transform of μ can be interpreted as an expectation functional with respect to the state of lowest energy. The nonrelativistic Fock representation (including the theory of n particles in a box) is examined in this formalism. Conditions on μ are systematically developed which suffice to recover all of the infinitesimal generators on a common, dense, invariant domain in the Hilbert space. For the Weyl group slightly weaker conditions than for the semidirect product case would suffice. Gaussian measures in J′, as well as the measures defining n-particle representations of J ∧K, satisfy these conditions.

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