Pparity

Pparity is a theoretical extension of the parity symmetry concept in physics, conceived as a combined operation that incorporates both spatial inversion and an internal symmetry transformation. It is used in certain models to classify states and to explore how internal degrees of freedom interact with spatial reflection. In systems where a Pparity operation commutes with the Hamiltonian, Pparity can be conserved and states may be organized into Pparity eigenstates.

Mathematically, Pparity acts on a state labeled by spatial coordinates x and internal quantum numbers i. If the internal sector is described by a representation D of an internal symmetry group G, the action can be written as Pparity |x, i> = |−x, D(g) i> times a possible phase. The properties of Pparity depend on the underlying group structure; in many realizations Pparity^2 equals the identity up to a phase, and eigenvalues are taken from a finite set determined by the internal representation. In the simple case where the internal symmetry is trivial, Pparity reduces to ordinary spatial parity.

Physically, Pparity provides a framework for classification and selection rules in theories with coupled spatial and internal symmetries. It has been discussed in contexts such as hadron spectroscopy and certain condensed matter models, where combined or generalized parity symmetries arise naturally. The concept is distinct from, but related to, ordinary parity and other discrete symmetries like charge conjugation and time reversal. Important caveats include the fact that ordinary parity is violated by the weak interaction, so Pparity need not be a universal symmetry across all sectors of a theory. See also parity, symmetry operations, selection rules, and CP violation.