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?
The 4π, in my opinion, involves going around a circle twice (hence the h/2 spin) I you want a physical answer, think of a wave - there are no nodes so it takes a complete cycle to get to the zero you arbitrarily started with. But a wave must have a crest and a trough, therefore it takes two cycles to get back to where you started. It requires the motion to be represented by a wave.
A spinor describes an object of spin 1/2. That to return such an object to its initial state by a rotation requires a rotation of a particular angle-here it's 720 degrees-isn't any more or less surprising than the fact that to return any object, that isn't spherically symmetric, to its initial state by rotations requires a particular choice of angles
What matters is that it is possible to distinguish, using the rules that define rotations, an object of spin 0 from another object, that can be consistently defined as having soon 1/2. And it turns out that there objects to which can be assigned spin 1/2 can be found in Nature.
Ian Miller To be clear, is rotation in the sense of Larmor precession the same kind of rotation that is shown mathematically when describing the properties of spinors (4pi symmetry)? Is it true that it takes two rotations around the external magnetic field in real space to return the phase of the electron to its original value?
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