9. Projective Representations

Nikolai Borisovich Melnikov, Boris Il'ich Reser

I.W. Richardson, A. H. Louie

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.

Stephen G Low

The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is Wigner's formulation of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group. However, Wigner's formulation makes no mention of the Weyl-Heisenberg group and the hermitian representation of its algebra that are the Heisenberg commutation relations fundamental to quantum physics. We put aside the relativistic symmetry and show that the maximal quantum symmetry that leaves the Heisenberg commutation relations invariant is the projective representations of the conformally scaled inhomogeneous symplectic group. The Weyl-Heisenberg group and noncommutative structure arises directly because the quantum phase requires projective representations. We then consider the relativistic implications of the quantum phase that lead to the Born line element and the projective representations of an inhomogeneous unitary group that defines a noninertial quantum theory. (Understanding noninertial quantum mechanics is a prelude to understanding quantum gravity.) The remarkable properties of this symmetry and its limits are studied.

Veeravalli S. Varadarajan