Parityreversing

Parityreversing, or parity reversal, is a term used across mathematics and physics to describe an operation that changes the parity of an object, typically switching between even and odd components or flipping spatial parity.

In mathematics, parity reversal is most commonly formalized by the parity-change functor, often denoted Pi. For a Z2-graded vector space V = V0 ⊕ V1, Pi V is the same underlying space with the grading swapped: (Pi V)0 = V1 and (Pi V)1 = V0. This operation is involutive, since Pi(Pi V) ≅ V. It plays a central role in superalgebra and supergeometry, where odd (fermionic) and even (bosonic) components are treated with graded structures. The Koszul sign rule governs signs that arise when parity-reversed objects are permuted or composed, and Pi is used to define odd coordinates, exterior algebras, and graded tensor products.

In physics, parity typically refers to spatial inversion, implemented by the parity operator P with (Pψ)(x) = ψ(−x). States can be labeled as having definite parity (even or odd) under this operation. Parity-reversing language in physics most often pertains to flipping spatial parity or, in supersymmetric contexts, to the parity-change perspective on supermultiplets, where fermionic and bosonic components can be related through a parity-reversing structure in the underlying representation.

Related concepts include parity, parity-changing transformations, parity operators, and the parity-shift (Pi) functor in category-theoretic and algebraic settings.