Hermann Weyl and the symmetry problem in Quantum Mechanics

Giorgio Jules Mastrobisi

Michael Hehenberger

Otávio Bueno

In this paper, I shall discuss the heuristic role of symmetry in the mathematical formulation of quantum mechanics. I shall first set out the scene in terms of Bas van Fraassen's elegant presentation of how symmetry principles can be used as problem-solving devices (see van Fraassen [1989] and [1991]). I will then examine in what ways Hermann Weyl and John von Neumann have used symmetry principles in their work as a crucial problem-solving tool. Finally, I shall explore one consequence of this situation to recent debates about structural realism (SR) and empiricism in physics (Worrall [1989], Ladyman [1998], and French [1999]).

H. Weyl

Freeman J. Dyson

Using mathematical tools developed by Hermann Weyl, the Wigner classification of group-representations and co-representations is clarified and extended. The three types of representation, and the three types of co-representation, are shown to be directly related to the three types of division algebra with real coefficients, namely, the real numbers, complex numbers, and quaternions. The author's theory of matrix ensembles, in which again three possible types were found, is shown to be in exact correspondence with the Wigner classification of co-representations. In particular, it is proved that the most general kind of matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of which belongs to one of the three known types.

William Martin McClain