In Feynmans book, "Quantum mechanics and Path Integrals" he writes in the conclusions (chapter 12-10)
With regards to quantum mechanics, path integrals suffer most grievously from a serious defect. They do not permit a discussion of spin operators or other such operators in a simple and lucid way. ... It is a serious limitation that the half-integral spin of the electron does not find a simple and ready representation.
This has been written in 1965. Has there been any progress on this problem? For example, it is possible nowadays to not only derive the Schrödinger equation, but also the Pauli equation from the path integral formulation of QM?
Answer
Reposting comment as an answer and expanding.
The answer is yes. You can find an exposition in Condensed Matter Field Theory by Altland and Simons, starting on page 134 in the second edition. The troubles come from that spin can't be described with a Hamiltonian that is a function of q:s and their conjugate p:s. However the more general formulation of Hamiltonian mechanics in terms of symplectic manifolds permits a description of spin. Altland and Simons cite Arnold's Mathematical Methods of Classical Mechanics as a reference for this. It's an under-appreciated gem of a book.
So, when we construct the path integral we consider the paths to be paths in phase space, p:s and q:s. I think to understand this in geometrical terms we have to back to "basics". The Lagrange formulation we have coordinates q and velocities ˙q. The coordinates can be coordinates on any manifold -- this is why the Lagrangian formalism is so neat for constrained systems -- so the velocities are really tangent vectors. Lagrangian mechanics is thus naturally formulated on tangent bundles. But we can take the Legendre transform and pass to the Hamiltonian formalism with p:s and q:s. This takes us to the cotangent bundle, for pi=∂L∂˙qi
However you can do all of Hamiltonian mechanics on any manifold that comes with a symplectic structure. A symplectic structure is 2-form ω such that dω=0 and for every vector v, ω(v,⋅) is not the zero 1-form. (You can think of ω as a sort of anti-symmetric metric.) This is what Arnold does in his book. The cotangent bundle naturally comes with such an ω: ω=dp1∧dq1+…dpn∧dqn
Now you can couple orbital angular momentum to the vector potential just fine with p:s and q:s. But how to do it with intrinsic angular momentum, that is spin? We want a Hamiltonian like H=B⋅S.
Therefore the paths you use in your construction of the path integral for spin should be paths on the sphere, and you can use θ,φ as coordinates. There is a formal complication in the quantum case since our states also have phases. This means that we should really use paths in SU(2), since an arbitrary spin state can always be written like |g⟩:=g|↑⟩
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