Spin-statistics Theorem - Proof

Proof

The essential ingredient in proving the spin/statistics relation 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, by definition.

Additionally, the assumption (known as microcausality) that spacelike separated fields either commute or anticommute can be made only for relativistic theories with a time direction. Otherwise, the notion of being spacelike is meaningless. However, the proof involves looking at a Euclidean version of spacetime, in which the time direction is treated as a spatial one, as will be now explained.

Lorentz transformations include 3-dimensional rotations as well as boosts. A boost transfers to a frame of reference with a different velocity, and is mathematically like a rotation into time. By analytic continuation of the correlation functions of a quantum field theory, the time coordinate may become imaginary, and then boosts become rotations. The new "spacetime" has only spatial directions, and is termed Euclidean.

A π rotation in the Euclidean x–t plane can be used to rotate vacuum expectation values of the field product of the previous section. The time rotation turns the argument of the previous section into the spin/statistics theorem.

The proof requires the following assumptions:

1. The theory has a Lorentz invariant Lagrangian.
2. The vacuum is Lorentz invariant.
3. The particle is a localized excitation. Microscopically, it is not attached to a string or domain wall.
4. The particle is propagating, meaning that it has a finite, not infinite, mass.
5. The particle is a real excitation, meaning that states containing this particle have a positive definite norm.

These assumptions are for the most part necessary, as the following examples show:

1. The spinless anticommuting field shows that spinless fermions are nonrelativistically consistent. Likewise, the theory of a spinor commuting field shows that spinning bosons are too.
2. This assumption may be weakened.
3. In 2+1 dimensions, sources for the Chern–Simons theory can have exotic spins, despite the fact that the three dimensional rotation group has only integer and half-integer spin representations.
4. An ultralocal field can have either statistics independently of its spin. This is related to Lorentz invariance, since an infinitely massive particle is always nonrelativistic, and the spin decouples from the dynamics. Although colored quarks are attached to a QCD string and have infinite mass, the spin-statistics relation for quarks can be proved in the short distance limit.
5. Gauge ghosts are spinless fermions, but they include states of negative norm.

Assumptions 1 and 2 imply that the theory is described by a path integral, and assumption 3 implies that there is a local field which creates the particle.

The rotation plane includes time, and a rotation in a plane involving time in the Euclidean theory defines a CPT transformation in the Minkowski theory. If the theory is described by a path integral, a CPT transformation takes states to their conjugates, so that the correlation function

must be positive definite at x=0 by assumption 5, the particle states have positive norm. The assumption of finite mass implies that this correlation function is nonzero for x spacelike. Lorentz invariance now allows the fields to be rotated inside the correlation function in the manner of the argument of the previous section:

Where the sign depends on the spin, as before. The CPT invariance, or Euclidean rotational invariance, of the correlation function guarantees that this is equal to G(x). So

for integer spin fields and

for half-integer spin fields.

Since the operators are spacelike separated, a different order can only create states that differ by a phase. The argument fixes the phase to be −1 or 1 according to the spin. Since it is possible to rotate the space-like separated polarizations independently by local perturbations, the phase should not depend on the polarization in appropriately chosen field coordinates.

This argument is due to Julian Schwinger.