The Formalism of Symmetry

Abstract

This chapter develops a general symmetry formalism needed for the application of symmetry considerations in science, especially quantitative applications. These very general concepts are introduced: system, which is whatever we investigate the properties of; subsystem, a system wholly subsumed within a system; state of a system, a possible condition of the system; and state space of a system, which is the set of all states of the same kind. The concept of transformation, a mapping of a state space of a system into itself, is presented. The set of all invertible transformations of a state space of a system forms a group, called a transformation group of the system. A compilation of a number of transformations in space, in time, and in space-time are presented. The spatial transformations are: displacement, rotation, plane reflection, line inversion, point inversion, glide, screw, dilation, plane projection, and line projection. The temporal transformations are displacement, inversion, and dilation. And the spatiotemporal transformationsare the Lorentz and Galilei transformations. The possibility of an equivalence relation for a state space of a system is considered. Such a relation decomposes a state space into equivalence subspaces. That leads to the idea of a symmetry transformation, which is any transformation that maps every state to an image state that is equivalent to the object state, i.e., any transformation that preserves equivalence subspaces. The set of all invertible symmetry transformations of a state space for an equivalence relation forms the symmetry group of the state space for the equivalence relation, and is a subgroup of the transformation group. Approximate symmetry is brought into the symmetry formalism. The notion of approximate symmetry transformation is made precise by means of a metric in the state space of a system, and properties of metrics are shown. Quantification of symmetry is discussed. It is found that the order of a (finite-order) symmetry group, or any monotonically increasing function thereof, can reasonably serve as the degree of symmetry of a system possessing that symmetry group. The discussion of state equivalence for quantum systems in leads to the result that for a given quantum system and a given set of operators in its Hilbert space of states, the symmetry group is the group of all unitary operators commuting with the given set. If the given set consists solely of Hermitian operators, the symmetry group is the group of all unitary and antiunitary operators commuting with the set.