Spin-Statistics Theorem

In quantum mechanics, the spin- statistics theorem relates to the spin of a particle to the particle statistics that it obeys. The spin of an article is its intrinsic angular momentum. All particles have either integer spin or half- integer spin. The theorem states that: the wave function of a system of identical integer- spin particles has the same value when the positions of any two particles are swapped. Particles with wavefunctions symmetric   under exchange are called bosons. The wavefunction of a system of identical half- integer spin particles changes sign when two particles are swapped. Particles with wavefunctions anti- symmetric under exchange are called fermions. In other words, the spin- statistics theorem states that integer spin particles are bosons, while half- integer spin particles are fermions.

The spin- statistics theorem states that fields with integer spin are bosonic fields; fields with half- integer spin are Femionic fields. A better name for the theorem would therefore be spin- commutation theorem, the name spin- statistics theorem stems from the fact that bosons are social, and multiple particles which exist in the same quantum state. The statement and proof of the theorem depends on the framework for quantum field theory that is used.

 History

The spin- statistics theorem was first formulated by Markus Fierz in 1939 and was later re- formulated in a more systematic way by Wolfgang Pauli. Fierz and Pauli argued by enumerating all free field theories, requiring that there should be quadratic forms for locally commuting observables including a positive definite energy density. A more conceptual argument was provided by Julian Schwinger in 1950. Richard Feynman gave a demonstration by demanding unitarity for scattering as an external potential is varied, which then translated to field language is a condition on the quadratic operator that couples to the potential.

Proof

The essential ingredient in providing the spin - statistics theorem is relativity that the physical laws do not change under Lorentz transformations. The field operators transform under Lorentz transformations according to the spin of the particle that they create.

The proof requires the following assumptions:

The theory Lorentz invariant Lagrangian, the vacuum is Lorentz invariant, the particle is a localized excitation, microscopically not attached to a string or a domain wall, the particle3 is propagating meaning that it has a finite and not infinite mass, and the particle is a real excitation, meaning that states containing this particle have a positive definite norm.

Questions:

• What is Spin- Statistics Theorem?
• What are the proofs for a spin statistics theorem?