9813

Rotation of Spinors

The electron, and other fermions with spin , is described in relativistic quantum mechanics by a spinor. A distinguishing feature of spinors is their behavior under rotation. Whereas a vector boson, with spin 1, will return to its initial state after a rotation by , a spinor requires two full rotations, with the angle advancing by to recover its initial state. A spinor is described by a complex phasor in addition to a helicity. This is represented in the graphic by rotation in a circle normal to its spin direction, with the complex phase color coded. A rotation in space by an angle is accompanied by a phase change of . Thus after rotation by , the spin direction of the particle is recovered but the phase changes by a factor . This can be observed experimentally in interference phenomena, most notably those done in neutron diffraction. In the course of rotation of by , the phasor traces out a Möbius band. This accords with the fact that a point on the surface of a Möbius band must go around twice in order to return to its initial location.
In the terminology of group theory, the Lie group describing spinors provides a double covering for the 3-dimensionsal rotation group .

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Snapshot 1: starting point
Snapshot 2: rotation by produces a phase difference of
Snapshot 3: rotation by is required to recover the initial state
Reference
R. Penrose and W. Rindler, Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry, Cambridge: Cambridge University Press, 1988.

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