nLab spin

Idea

The spin of a particle in quantum mechanics/quantum field theory is its intrinsic angular momentum, but subject to the possibility that the particle may be a fermion so that that its angular momentum is not determined by the representation theory of the special orthogonal group, but of the spin group.

Properties

Spin-Statistics theorem

By the spin-statistics theorem, the spin of any elementary particle must be (in magnitude) half of an integer multiple of Planck's constant ($\hbar = h/2\pi$), an integer if the particle is a boson but only a half-integer if the particle is a fermion.

Related concepts

• spin group,

• spin representation, real spin representation

• Fierz identity

• proton spin crisis

• spinor

• Dirac spinor, Weyl spinor, Majorana spinor

• Dirac field

• Ising model, Heisenberg model

• quantization of the 2-sphere

• supersymmetry

• division algebras and supersymmetry

• geometry of physics – supersymmetry

References

The concept of intrinsic spin of the electron (making it a “spinor”) originates with:

• Samuel A. Goudsmit, George E. Uhlenbeck: Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons, Naturwissenschaften 13 47 (1925) 953 [pdf, pdf]

Historical recollection:

• Samuel A. Goudsmit, The discovery of the electron spin, in: Foundations of Modern EPR, World Scientific (1998) [pdf, webpage, doi:10.1142/3624]

• Richard G. Milner, A Short History of Spin, Proceedings of PSTP2013, POS 182 (2014) [arXiv:1311.5016, doi:10.22323/1.182.0003]