Standard Model and Beyond · Problem 2.13

Lie Groups, Lie Algebras, and Symmetries

Physics — Particle Physics · angular-momentum-dirac-field

Problem

The angular momentum operator for a Dirac field can be written as Ji=∫d3x⃗ ψ†(x)14ϵijkσjkψ(x)+orbital,J^i = \int d^3\vec{x}\,\psi^\dagger(x)\frac{1}{4}\epsilon^{ijk}\sigma_{jk}\psi(x) + \text{orbital}, where normal ordering is implied. Show, in the free field limit, that J3J^3 has the expected behavior J3∣ψ(p⃗,s1,2)⟩=±12∣ψ(p⃗,s1,2)⟩,J3∣ψc(p⃗,s1,2)⟩=±12∣ψc(p⃗,s1,2)⟩,J^3|\psi(\vec{p},s_{1,2})\rangle = \pm\frac{1}{2}|\psi(\vec{p},s_{1,2})\rangle, \qquad J^3|\psi^c(\vec{p},s_{1,2})\rangle = \pm\frac{1}{2}|\psi^c(\vec{p},s_{1,2})\rangle, where ψ\psi and ψc\psi^c represent particle and antiparticle states, p⃗\vec{p} is in the z^\hat{z} direction (so that the orbital terms do not enter), and s=s1s = s_1 or s2s_2 represent spins in the ±z^\pm\hat{z} direction.
Intuitive explanation

The spin operator 14ϵijkσjk\frac{1}{4}\epsilon^{ijk}\sigma_{jk} encodes intrinsic angular momentum through the commutation relations of the Pauli matrices—specifically, σjk\sigma_{jk} generates rotations in spin space. When you apply J3J^3 to a one-particle state with momentum along z^\hat{z} (killing the orbital piece), you're asking whether the spinor components of the Dirac field are eigenstates of the spin-zz operator with eigenvalue ±12\pm\frac{1}{2}. The key is to expand the field operators in terms of creation and annihilation operators, let them act on the single-particle states, and verify that the spinor structure itself carries the correct spin eigenvalue.

Intuitive explanation

The spin operator 14ϵijkσjk\frac{1}{4}\epsilon^{ijk}\sigma_{jk} encodes intrinsic angular momentum through the commutation relations of the Pauli matrices—specifically, σjk\sigma_{jk} generates rotations in spin space. When you apply J3J^3 to a one-particle state with momentum along z^\hat{z} (killing the orbital piece), you're asking whether the spinor components of the Dirac field are eigenstates of the spin-zz operator with eigenvalue ±12\pm\frac{1}{2}. The key is to expand the field operators in terms of creation and annihilation operators, let them act on the single-particle states, and verify that the spinor structure itself carries the correct spin eigenvalue.

Related theory

Browse the problem library