Transformations on the set of all n-dimensional subspaces of a Hilbert space preserving principal angles

Wigner's classical theorem on symmetry transformations plays a fundamental role in quantum mechanics. It can be formulated, for example, in the follow ing way: Every bijective transformation on the set L of all l-dimensional s ubspaces of a Hilbert space H which preserves the angle between the element s of L is induced by either a unitary or an antiunitary operator on H. The aim of this paper is to extend Wigner's result from the l-dimensional case to the case of n-dimensional subspaces of H with n epsilon N fixed.