Chirality distinguishes the two irreducible Weyl representations of the proper Lorentz group. For a massless particle it agrees with helicity up to the particle-antiparticle convention, while a mass term couples opposite chiralities.
With Minkowski metric signature and , define . Then . The symmetric-tensor contraction gives twice the sum of the two possible dot-product pairings; the Levi-Civita symbol contraction, including the two factors of , gives twice their difference. Their sum is the displayed identity, explaining the single pairing in left-chiral weak rates.
A vectorlike gauge spectrum pairs complex gauge group representations and among independent left-handed Weyl spinors. Such a pair permits a gauge-invariant fermion mass and occurs in a full charged hypermultiplet of four-dimensional extended supersymmetry.
A chiral gauge spectrum has independent left-handed Weyl spinors in complex gauge group representations without an obligatory conjugate pair of left-handed Weyl spinors. The right-handed antiparticles required by the CPT theorem do not themselves make a spectrum vectorlike.
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Chirality in physics refers to the property of an object or system that cannot be superimposed on its mirror image. It is a concept that has important implications in various fields, including physics, chemistry, and biology. In physics and materials science, chirality often relates to the spatial arrangement of particles and their interactions.