I am aware that due to the properties of spinors, to return a spinor to its initial state by rotation, the spinor must be rotated by 4*pi radians, i.e. rotation by 720 degrees.

While I was taught the fact as a mathematical curiosity in my quantum classes, I still don't know what the physical ramifications of this are. What does this mean for spins in real space? Does this fact influence spin dynamics due to external fields and potentials? The cartoon for Larmor precession uses an arrow to represent spin that seems to act like any other normal 3D object.

What is the relevance of this 4*pi symmetry?

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