Isotropy

In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix a- or an-, hence anisotropy. Anisotropy is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. Within mathematics, isotropy has a few different meanings: Isotropic manifolds A manifold is isotropic if the geometry on the manifold is the same regardless of direction. A similar concept is homogeneity. Isotropic quadratic form A quadratic form q is said to be isotropic if there is a non-zero vector v such that q(v) = 0; such a v is an isotropic vector or null vector. In complex geometry, a line through the origin in the direction of an isotropic vector is an isotropic line. Isotropic coordinates Isotropic coordinates are coordinates on an isotropic chart for Lorentzian manifolds. Isotropy groupAn isotropy group is the group of isomorphisms from any to itself in a groupoid. An isotropy representation is a representation of an isotropy group. Isotropic position A probability distribution over a vector space is in isotropic position if its covariance matrix is the identity. Isotropic vector field The vector field generated by a point source is said to be isotropic if, for any spherical neighborhood centered at the point source, the magnitude of the vector determined by any point on the sphere is invariant under a change in direction. For an example, starlight appears to be isotropic. Quantum mechanics or particle physics When a spinless particle (or even an unpolarized particle with spin) decays, the resulting decay distribution must be isotropic in the rest frame of the decaying particle regardless of the detailed physics of the decay. This follows from rotational invariance of the Hamiltonian, which in turn is guaranteed for a spherically symmetric potential.