gamma5

Gamma5, denoted γ5, is a Dirac matrix used in relativistic quantum field theory to encode fermion chirality. In the common four-dimensional Minkowski space with the metric signature (+, −, −, −), it is defined as γ5 = i γ0 γ1 γ2 γ3. The exact sign can depend on the chosen metric convention, but γ5 always anticommutes with each γμ and squares to unity.

Algebraically, γ5 is Hermitian (γ5† = γ5) and satisfies {γ5, γμ} = 0 for all μ = 0,1,2,3, meaning it flips the chirality of spinor components. Its trace vanishes, Tr(γ5) = 0. In the Weyl (chiral) basis, γ5 is diagonal with eigenvalues ±1, corresponding to left- and right-handed chirality.

Chirality projectors are defined using γ5: PL = (1 − γ5)/2 and PR = (1 + γ5)/2. These operators project a Dirac spinor onto its left- and right-handed components, a distinction that is central to the electroweak interactions where only certain chiralities participate. For massless fermions, chirality is a good quantum number and is conserved; for massive fermions, the mass term couples left- and right-handed components.

Key trace identities involve γ5, such as Tr(γ5 γμ γν γρ γσ) = 4i εμνρσ, linking γ5 to axial currents and anomalies. The axial-vector current jμ5 = ψ̄ γμ γ5 ψ plays a crucial role in spin and parity properties of fermions. While γ5 is well-defined in four dimensions, its extension to dimensional regularization introduces subtleties, leading to different schemes for preserving its properties in loop calculations. Gamma5 remains a foundational tool for describing chirality and axial structure in quantum field theory.