On the realization of Symmetries in Quantum Mechanics

Details On the realization of Symmetries in Quantum Mechanics On the realization of Symmetries in Quantum Mechanics

2007-12-06

arxiv.org/abs/0712.0997v1

Math Semesterber (2008) 55: 149-160

Physics

Quantum Physics

8 pages

Scientific paper

10.1007/s00591-008-0035-5

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.

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