In mathematics, given a vector space X with an associated quadratic form q, written, a null vector or isotropic vector is a non-zero element x of X for which .
In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.
A quadratic space which has a null vector is called a pseudo-Euclidean space. The term isotropic vector v when q(v) = 0 has been used in quadratic spaces,[1] and anisotropic space for a quadratic space without null vectors.
A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B,, where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres:The null cone is also the union of the isotropic lines through the origin.
A composition algebra with a null vector is a split algebra.[2]
In a composition algebra (A, +, ×, *), the quadratic form is q(x) = x x*. When x is a null vector then there is no multiplicative inverse for x, and since x ≠ 0, A is not a division algebra.
In the Cayley–Dickson construction, the split algebras arise in the series bicomplex numbers, biquaternions, and bioctonions, which uses the complex number field
\Complex
(hi)2=h2i2=(-1)(-1)=+1.
(1+hi)(1+hi)*=(1+hi)(1-hi)=1-(hi)2=0
The light-like vectors of Minkowski space are null vectors.
The four linearly independent biquaternions,,, and are null vectors and can serve as a basis for the subspace used to represent spacetime. Null vectors are also used in the Newman–Penrose formalism approach to spacetime manifolds.[3]
In the Verma module of a Lie algebra there are null vectors.