REPRESENTATION OF SYMMETRY TRANSFORMATIONS IN QUANTUM MECHANICS

Abstract

A generalization of Wigner's representation theorem is presented that may be applied when superselection rules (of a specified character) must be taken into account. At the same time some gaps in existing proofs of Wigner's theorem are eliminated. The connection between Wigner,s theorem and the so called fundamental theorem of projective geometry is pointed out, and a new, short proof of the latter theorem (valid for an arbitrary field of numbers) is presented. The equivalent possibilities of characterizing symmetry transformations as mappings leaving invariant the logical structure of quantum mechanics, as isometric linear or conjugated linear transformations connecting two complex Hilbert spaces, or as mappings leaving invariant the algebraic structure of quantum mechanics (or equivalently the structure of the equations of motion) are discussed and lead to an invariant (coordinate free) characterization of reversal of the direction of motion (time inversion). An analogue of Wigner's theorem for symmetry transformations in the recently discussed quaternion quantum mechanics is proved in an appendix, and it is pointed out how quaternion quantum mechanics, as well as ordinary complex quantum mechanics, may be consistently represented in the framework of real quantum mechanics, which in this sense appears as a super theory. The symmetry transformationsmore » of classical mechanics are discussed in relation to their quantum mechanical analogues in another appendix. They are intimately connected with the canonical formalism. (auth)« less