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Wigner's classical theorem on symmetry transformations plays a fundamentalrole in quantum mechanics. It can be formulated, for example, in the followingway: Every bijective transformation on the set L of all 1-dimensional subspacesof a Hilbert space H which preserves the angle between the elements of L isinduced by either a unitary or an antiunitary operator on H. The aim of thispaper is to extend Wigner's result from the 1-dimensional case to the case ofn-dimensional subspaces of H with n fixed.Comment: 20 pages. To appear in Commun. Math. Phy

Transformations on the Set of All n-Dimensional Subspaces of a Hilbert Space Preserving Principal Angles