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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?)
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CuriousKev
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1 Answers1

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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.

Arnold Neumaier
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