# Path integral for spin?

I'm searching for the path integral formulation for a spin particle and haven't found any precise description yet. Is there a systematic way to construct a non-relativistic path integral formulation for a spin particle?

Edit: Here are some my thoughts on this problem.

In Pauli theory of spin (non-raletivistic theory of spin), if we only consider the dimension of spin, then after we set a particular $$z$$ direction, all states of a spin 1/2 particle can be described by an element in $$SU(2)$$, so the time evolution of spin state will form a connected path in $$SU(2)$$, I think maybe we can construct path integral over all possible path between two ends that we set. Just a guess.

• Yes, it is basically the Lagrangian of a point particle plus a spinor (it can be described by a supersymmeteic point particle) Commented Apr 4, 2020 at 23:07
• I'm appreciate if you can provide some reference on that. @Slereah Commented Apr 4, 2020 at 23:09
• There is the spin-coherent-state path integral. This does not need supersymmetry. Commented Apr 4, 2020 at 23:23
• The path integral that describes the Dirac field (i.e. Dirac eqn in the classical limit) is the usual construct with the Dirac Lagrangian, which involves spinors. That's where spin is built in. However, you must ensure to do the functional integration over Grassmann variables, in order to ensure that you get the correct spin-statistics relation. If you don't already understand the spin-statistics theorem, make sure to learn it, because it's absolutely essential for doing spin-1/2 path integrals correctly. Commented Apr 4, 2020 at 23:25
• You can try here : www-th.bo.infn.it/people/bastianelli/1-ch3-FT2-2016.pdf Commented Apr 4, 2020 at 23:45

As you suggested the basic idea is to parametrize the spin with a representation of SU(2). This is sometimes called a spin-coherent-state (call it $$|g\rangle$$). On SU(2) you can define a (left) invariant integral and with that you can represent the identity as
$$$$\mathrm{id} = C \int_{SU(2)}dg |g\rangle\langle g|,$$$$
Now the derivation of a path integral is quite similar to the bosonic case: you do a Trotter decomposition of the time evolution operator and insert identities in the above form. How the diverse spin operators act on $$|g\rangle$$ are described in the Altland; look there for reference.