Symmetry Principles in Old and New Physics by Eugene P. Wigner

Introduction and summary. Symmetry and invariance considerations have long played important roles in physics. The 32 crystal classes—that is, groups of rotations in three-dimensional space all the elements of which are of the order 2, 3, 4 or 6—were determined 137 years ago, in the same year in which group theory was born. The determination of the 230 space groups, by Schönflies and by Fedorov (these are the discrete subgroups of the Euclidean group which contain three noncoplanar translations) was a masterpiece of analysis and so was the determination by Groth of the possible properties of crystals with the symmetries of these space-groups. The groups of prime importance in classical physics were subgroups of the Euclidean group and the enumeration of these subgroups and the derivation of the properties which are invariant under them were the principal problems. The invariance groups of the relativity theories were, from the mathematical point of view, much more esoteric but their use by physicists did not contribute greatly to the mathematical theory of groups nor did it point to new interesting mathematical problems. When, however, the invariance arguments were applied to the present century's other great innovation of physical theory, to quantum theory, a score of new problems and several interesting mathematical theorems were uncovered. The basic reason is the difference in the characterization of states in quantum and in pre-quantum theories. In the latter, a state was characterized by the positions and the velocities of particles. These could be specified by points in three-dimensional space. Quantum theory, on the other hand, specifies the states by vectors in an abstract Hilbert space. Symmetry transformations in pre-quantum theories were rather obvious transformations of three-dimensional space; in quantum theory they became unitary transformations of Hilbert space. These form subgroups of all unitary transformations which are essentially homomorphic to the symmetry group in question, essentially homomorphic only because a unitary transformation in quantum mechanics is equivalent to any of its multiples by a numerical factor (of modulus 1). However, this essential homomorphy could be reduced, particularly