Phase, Bloch sphere versus Feynman path

[nomadreid]

I am pretty sure that I would be comparing apples and oranges in this question, but as I usually learn something from the responses telling me in detail that my question is silly, here goes: Does the phase used as a weight in Feynman's path integral formulation (i.e., the quantum action S in the factor e^iS/ħ) have anything to do with either the (ignored) phase as expressed in the development of the Bloch sphere (i.e., the γ in |ψ> = e^iγ(cos (θ/2)|0> + e^iφsin(θ/2)|1>), or perhaps the relative phase differences? On the face of it, it seems not, but I would like to make sure.

 

[Demystifier]

Bloch sphere describes a "particle" with spin but without a position. Feynman path integral usually describes the opposite, a particle with position but without spin. In this sense, there is usually no relation between the two phases.

On the other hand, a particle with spin can also be treated by a path integral method (which should not be confused with Grassmann-number path "integral" method for fermion fields), but that is a separate topic treated in some condensed-matter field theory books. I like the treatment in X-G Wen, Quantum Field Theory of Many-Body Systems, Sec. 2.3 Quantum spin, the Berry phase, and the path integral. In Fig. 2.4 you will see a relation with the Bloch sphere.

 

[nomadreid]

Thank you very much, Demystifier. You not only answered my question, but gave me an interesting reference. Very helpful.
(I will look at the chapter more carefully anon. I presume this is the figure that you are referring to.)