# A Orbital Angular momentum couples to Christoffel connection

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1. Apr 7, 2017

### psi*psi

I am trying to understand Wen and Zee's article on topological quantum numbers of Hall fluid on curved space: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.69.953

They passingly mentiond the fact that a spinning particle with orbital angular momentum $s$ moving on a manifold with Christoffel connection $\omega$ will acquire a Aharanov-Bohm like phase of $s \oint \omega$.

I can sort of see why the Christoffel connection will give rise to such a phase, since by analogy with the magnetic vector potential, the Christoffel connection will enter into the covariant derivative the same way as a magnetic vector potential. However, I do not understand why it would couple to the orbital angular momentum. I would really appreciate if someone could show me a derivation or point me to some references.

Thanks.

2. Apr 9, 2017

### Ben Niehoff

The Christoffel connection tells you which vectors in two nearby tangent spaces are "the same". A spinning particle is a gyroscope and wants to keep its angular momentum vector pointing in the same direction as it moves from point P to a nearby point. So the Christoffel connection is needed in order to define what that means.

For particles moving along geodesics, the motion is inertial, so one can just use parallel transport (i.e., the Christoffel term). For accelerated motion, there are fictitious forces, and the appropriate thing to use is called Fermi-Walker transport. I'm sure if you look that up, you can find a derivation.