Chirality and Helicity are both properties of particles that relate to their spin.
They distinguish between left-handed and right-handed particles.
Spin Overview
Spin: A property of a particle, similar to charge or mass.
Vector Nature: Spin has magnitude and direction, unlike mass or charge (which are scalars).
Spin-1/2 Particle: Refers to the magnitude of the spin.
Operators: Quantum mechanical operators for spin behave like those for angular momentum, described by commutator relations.
Practical Demonstration: The Einstein-de-Haas effect illustrates this behavior.
Understanding Helicity
Definition: Helicity is the projection of spin onto the particle's momentum direction.
Helicity Examples:
Particle traveling in positive x-direction with spin 3/2 in the same direction has helicity 3/2.
Particle traveling in minus z-direction with spin 1 in positive z-direction has helicity -1.
Spin-1/2 particle in positive x-direction traveling in negative y-direction has helicity 0.
All spin-0 particles have zero helicity.
Helicity Operator: Obtained by projecting the spin operator onto a unit momentum vector.
Quantized Values: The eigenvalues of helicity are discrete, coinciding with those of the spin operator.
Reference Frame Dependence: Helicity changes with reference frame boosts for massive particles; massless particles (traveling at light speed) do not experience this.
Convention:
Positive helicity: Spin and momentum are parallel (right-handed).
Negative helicity: Spin and momentum are anti-parallel (left-handed).
Understanding Chirality
Definition: Chirality is another fundamental property of particles, categorized as left-chiral or right-chiral.
Distinction: Left- and right-chiral particles are fundamentally different, comparable to particles of different mass.
Separation: Left- and right-chiral parts can be separated mathematically using the fifth gamma matrix via projection operators.
Superposition: Left- and right-chiral parts together form the original particle state.
Connection to Helicity: For massless particles, chirality corresponds to helicity; left-chiral is left-handed and right-chiral is right-handed.
Origin of Chirality
Representation Theory: Chirality arises from the representation theory of the Lorentz group (SO(3,1)).
Representations:
The group is equivalent to two copies of SL(2,C).
Representations of SL(2,C) are labeled by half-integer numbers (m, n).
Example representations include:
(0,0): Scalars (unit matrix representation)
(1/2,1/2): Transforms 4-vectors (used in special relativity)
(1/2,0) and (0,1/2): Connected to left- and right-chiral spinors respectively.
Practical Implications of Chirality
Weak Interaction:
Only affects left-chiral particles and right-chiral antiparticles, not the opposite.
Neutrinos are observed as left-chiral due to their weak interaction properties; gravity's effect is too weak to measure.
Summary
Chirality: A fundamental property like mass, spin, or charge.
Helicity: An observable that depends on spin and momentum.