Abstract
Symmetry under discrete operations—mainly reflections of space or time coordinates—complements the symmetry under coordinate rotations and other continuous transformations. This chapter deals with reversal of the frame of coordinate rotations from right- to left-handed or vice versa–that is, with the reversal of chirality. The symmetry of matrix U for an irreducible representation emerges from repeating the operation of frame reversal. This repeated operation amounts physically to an identity but its analytical representation should not coincide with the unit matrix. The symmetry of the U matrix coincides with the symmetry of boson or fermion wave functions under particle permutations. The time-reflection operator T- defined in field theories has properties similar to U because Dirac field operators change sign under double time reflection while scalar and vector fields do not,. The chapter focuses on contragredience and the construction of invariants.
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