Beyond Wigner's theorems: The role of symmetry equivalences in quantum systems

A generalization of the concept of Wigner's symmetrical operator, i.e., of the operator invariant under symmetry transformations forming a group, namely G, is proposed here. Symmetrical operators in the present general sense do constitute a set whose elements are mutually commuting and equivalent to each other under transformations S-i is an element of G. A generalization of a well known theorem attributed to Wigner is established: for a given operator (A) over cap and its equivalent ones (A) over cap ([i]) (S) over cap (i)(-1) (A) over cap(S) over cap (i), symmetrical as previously defined, it is always possible to find a complete set of common eigenfunctions that form canonical basis sets for representations of G induced by the largest subgroup H subset of G whose elements keep (A) over cap invariant (the particular case H equivalent to G reduces to Wigner's theorem itself). The formal generalization presented here opens the possibility of describing and exploiting two complementary aspects of symmetry: invariance and equivalence, within a unified theoretical approach. It is also shown that the symmetry properties of the stationary solutions of Boys' localization functional obey a straightforward extension of this theorem. A few examples are provided to highlight the implications of this alternative way to look at symmetry in quantum systems. (C) 2012 Wiley Periodicals, Inc.