# Classical toy models of particles with intrinsic spin

Related to my question here (spacetime torsion, the spin tensor, and intrinsic spin in einstein cartan theory), I'd like to be able to put test particles on a manifold with non-zero torsion and see how this affects the motion.

The action for a free particle is usually given as:

$$S_{free} = -m\int d\tau = - m\int \sqrt{\frac{\partial x^\mu(\lambda)}{\partial \lambda}\frac{\partial x^\nu(\lambda)}{\partial \lambda}g_{\mu\nu}(\lambda)}\ \ d\lambda$$ where $\tau$ is the world line length, and $\lambda$ is some parameter to describe the particle path $x^\mu(\lambda)$. I assume this is a scalar particle, since rotations will not affect its description.

1. Is there a term I am leaving out if we consider non-zero torsion?
2. What is the corresponding model for a free spinor particle? (I've seen classical spinor fields discussed, but never a particle)
3. What about for higher spin?
4. What about for arbitrary spin? (or even in classical models, are we limited to representations of the manifold tangent space?)

Note that in classical systems, spin is not quantized but just a parameter, so the question of ''higher spin'' is not really meaningful. There remains only the question of statistics.

Lagrangian principles for classical Fermions were first discussed in: J.L. Martin, Generalized classical dynamics, and the ‘classical analogue’ of a Fermioscillator, Proc. R. Soc. Lond. A 251 (1959), 536-542.