Rotation of Spinors

helicity

+1/2

-1/2

rotation angle ϕ/π

4.

animate

The electron, and other fermions with spin

1

2

, 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

2π

, a spinor requires two full rotations, with the angle advancing by

4π

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

ϕ

is accompanied by a phase change of

ϕ/2

. Thus after rotation by

2π

, the spin direction of the particle is recovered but the phase changes by a factor

-1

. This can be observed experimentally in interference phenomena, most notably those done in neutron diffraction. In the course of rotation of

ϕ

by

4π

, 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

SU(2)

describing spinors provides a double covering for the 3-dimensionsal rotation group

SO(3)

.