Helicity

Helicity is a measure that characterizes the handedness of a system involving angular momentum and motion. In physics, it refers to the projection of a system’s spin or polarization onto its direction of travel. For a single particle with spin operator S and momentum p, the helicity is h = (S · p)/|p|. The possible helicity values for a particle with spin s lie between -s and +s; for spin-1/2 and momentum along a given direction, the eigenvalues are ±1/2. In relativistic theory, helicity for massless particles is Lorentz invariant and coincides with chirality, while for massive particles it can change under boosts.

In electromagnetism and field theory, helicity also appears as a property of fields. Magnetic helicity, defined as H_m = ∫ A · B d^3x with B = ∇×A, measures the knottedness or linkage of magnetic field lines. H_m is gauge dependent in general, but becomes physically meaningful under suitable boundary conditions and is conserved in ideal magnetohydrodynamics when conductivity is perfect.

Fluid dynamics provides another notion of helicity: h_f = ∫ v · ω d^3x, where v is the fluid velocity and ω = ∇×v is the vorticity. This fluid helicity quantifies the knottedness of vortex lines and is conserved in ideal, inviscid flows, making it a useful diagnostic in turbulence and geophysical flows.

Beyond its physical definitions, helicity connects to topology through linking numbers and knotted field configurations. It serves as a diagnostic in plasma physics, solar and astrophysical contexts, and the study of topological aspects of fields, offering a measure of the global geometry of angular momentum and magnetic or velocity fields.