28 research outputs found
Projective Group Representations in Quaternionic Hilbert Space
We extend the discussion of projective group representations in quaternionic
Hilbert space which was given in our recent book. The associativity condition
for quaternionic projective representations is formulated in terms of unitary
operators and then analyzed in terms of their generator structure. The
multi--centrality and centrality assumptions are also analyzed in generator
terms, and implications of this analysis are discussed.Comment: 16 pages, no figures, plain Te
On the realization of Symmetries in Quantum Mechanics
The aim of this paper is to give a simple, geometric proof of Wigner's
theorem on the realization of symmetries in quantum mechanics that clarifies
its relation to projective geometry. Although several proofs exist already, it
seems that the relevance of Wigner's theorem is not fully appreciated in
general. It is Wigner's theorem which allows the use of linear realizations of
symmetries and therefore guarantees that, in the end, quantum theory stays a
linear theory. In the present paper, we take a strictly geometrical point of
view in order to prove this theorem. It becomes apparent that Wigner's theorem
is nothing else but a corollary of the fundamental theorem of projective
geometry. In this sense, the proof presented here is simple, transparent and
therefore accessible even to elementary treatments in quantum mechanics.Comment: 8 page
Noncommutative geometry and physics: a review of selected recent results
This review is based on two lectures given at the 2000 TMR school in Torino.
We discuss two main themes: i) Moyal-type deformations of gauge theories, as
emerging from M-theory and open string theories, and ii) the noncommutative
geometry of finite groups, with the explicit example of Z_2, and its
application to Kaluza-Klein gauge theories on discrete internal spaces.Comment: Based on lectures given at the TMR School on contemporary string
theory and brane physics, Jan 26- Feb 2, 2000, Torino, Italy. To be published
in Class. Quant. Grav. 17 (2000). 3 ref.s added, typos corrected, formula on
exterior product of n left-invariant one-forms corrected, small changes in
the Sect. on integratio