Standard Model and Beyond · Problem 2.12

Lie Groups, Lie Algebras, and Symmetries

Physics — Particle Physics · pauli-dirac-chiral-unitary-transformation

Problem

Show by explicit construction that the Pauli-Dirac and chiral representations are related by a unitary transformation, i.e., that there exists a unitary matrix UU such that UγPDμU†=γchμ,UuPD(p⃗,s)=uch(p⃗,s),UvPD(p⃗,s)=−vch(p⃗,s).U\gamma^\mu_{PD}U^\dagger = \gamma^\mu_{ch}, \qquad Uu_{PD}(\vec{p},s) = u_{ch}(\vec{p},s), \qquad Uv_{PD}(\vec{p},s) = -v_{ch}(\vec{p},s). (The extra sign in the vv-spinor transformation is due to a sign convention.)

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