Pseudoclassical spinning particle ($\mathcal{L}=\dot{x}\cdot p+\frac{i}{2}\zeta \cdot \dot{\zeta }-\frac{e}{2}{p}^2-i\chi p\cdot \zeta$)

A pseudoclassical spinning particle supplements ordinary position and momentum by odd Grassmann variables. An even multiplier imposes $p^2=0$ and an odd multiplier imposes $p\cdot \zeta =0$. The odd symplectic term gives $\{{\zeta }^m,{\zeta }^n{\}}_D=-i{\eta }^{mn}$. Quantization represents these variables by a Clifford algebra, producing a spinor wavefunction.

Quantizing the odd Dirac brackets yields $\{{\hat{\zeta }}^m,{\hat{\zeta }}^n{\}}_{+}={\eta }^{mn}$. The constraint $p\cdot \zeta =0$ is then the massless Dirac equation on a spinor. Its square enforces $p^2=0$, matching the even constraint. The spin bilinear is represented by $-i[{\gamma }^m,{\gamma }^n]/4$ in this action convention.

The odd particle variables form an even antisymmetric tensor through their bilinear. Adding it to orbital angular momentum gives a conserved Lorentz Noether charge. The particle and odd-variable equations make the two time derivatives cancel. After spinning-particle Dirac quantization, the bilinear becomes the spinor Lorentz generator rather than an ordinary commuting classical spin vector.