# Rotation of Spinors

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.

1

2

2π

4π

ϕ

ϕ/2

2π

-1

ϕ

4π

In the terminology of group theory, the Lie group describing spinors provides a double covering for the 3-dimensionsal rotation group .

SU(2)

SO(3)