helicities

Helicity is a property used in several branches of physics to describe how angular momentum aligns with a system’s direction of motion. In quantum mechanics, the helicity operator is ĥ = (S · p)/|p|, where S is the spin operator and p is the momentum. Its eigenvalues specify the spin projection along the momentum direction. For a particle with spin s, the possible helicities range from -s to +s in integer steps. In relativistic theories, helicity is a frame-dependent quantity for massive particles but becomes Lorentz-invariant for massless particles.

In quantum contexts, common examples include spin-1/2 fermions with helicity ±1/2 and massless vector bosons such as photons with helicity ±1. Experiments in weak interactions reveal a preference for left-handed helicity among neutrinos; right-handed antineutrinos are not observed within the Standard Model framework. At sufficiently high energies, helicity tends to align with chirality, a related but distinct property tied to the particle’s Dirac structure and its couplings.

Beyond quantum mechanics, helicity appears in classical and field theories. Hydrodynamic helicity H = ∫ v · ω d^3x, with ω = ∇ × v, measures the knottedness of a fluid’s flow and is (approximately) conserved in ideal, inviscid flows. Magnetic helicity H_m = ∫ A · B d^3x, where B is the magnetic field and A its vector potential, gauges the linkage of magnetic field lines; in ideal magnetohydrodynamics it is conserved under appropriate gauge conditions.

Overall, helicity serves as a unifying descriptor of alignment and topology across quantum and classical systems, with distinct implications in particle physics, fluid dynamics, and magnetism.