Four-vector

In special relativity, a four-vector is an object with four in general complex components that transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a 4-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (½,½) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations. They include spatial rotations, boosts (a change by a constant velocity to another inertial reference frame), and temporal and spatial inversions.

Four-vectors describe, for instance, position x^μ in spacetime modeled as Minkowski space, a particles 4-momentum p^μ, the amplitude of the electromagnetic four-potential A^μ(x) at a point x in spacetime, and the elements of the subspace spanned by the gamma matrices inside the Dirac algebra.

The Lorentz group may be represented by 4×4 matrices Λ. The action of a Lorentz transformation on a general contravariant four-vector X (like the examples above), regarded as a column vector with Cartesian coordinates with respect to an inertial frame in the entries, is given by