antivector

An antivector is a term that appears in several mathematical and physical contexts, but it is not a single universally standardized object. In many mathematical treatments, antivector is used as a synonym for a covector (an element of the dual space V*, the linear functional that pairs with vectors to produce scalars). In this usage, a vector v and an antivector φ form the scalar φ(v), and the two transform in opposite ways under a change of basis.

In physics, antivector is sometimes used to mean a pseudovector (also called an axial vector). A pseudovector behaves like a regular vector under proper rotations but changes differently under improper rotations (such as reflections). Pseudovectors arise, for example, from cross products or from dualization of antisymmetric tensors via the Levi-Civita symbol. Classic examples in three dimensions are angular momentum and the magnetic field. Under parity, a pseudovector does not transform exactly like a polar vector, reflecting its distinct transformation properties.

In differential geometry and geometric algebra, dualization with respect to a metric or a Hodge star operation can pair vectors with bivectors or higher-grade multivectors. Some authors may loosely call one member of such a dual pair an antivector, but this usage is not universal and can lead to confusion.

Because antivector has multiple meanings, it is important to specify the context. In rigorous mathematical texts, the term is more reliably replaced by covector (for dual vectors) or pseudovector/axial vector (for certain transformation properties in physics). Related concepts include the dual space, covectors, contravariant and covariant transformation rules, and the Hodge dual.