Spin-statistics theorem

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The spin-statistics theorem in quantum mechanics relates the spin of a particle to the statistics obeyed by that particle. The Spin of a particle is the angular momentum that the particle has when it is not moving, and in quantum mechanics, it gives the change in wavefunction phase when you do certain rotations. All particles have either integer spin or half-integer spin (in multiples of $\ \hbar$ (Dirac's constant)). Integer spin means that the phase change for a 360 degree rotation is 1, while half integer spin means that the phase change for a 360 degree rotation is -1.

The theorem states that:

The wave functions of a system of identical integer-spin particles, spin 0, 1, 2, 3, has the same value when the positions of any two particles are exchanged. Particles with wavefunctions symmetric under exchange are called bosons.

The wave functions of a system of identical half-integer-spin s = 1/2, 3/2, 5/2, are anti-symmetric under exchange, meaning that the wavefunction changes sign when the positions of any pair of particles are swapped. Particles whose wavefunction changes sign are called fermions.

So the spin statistics theorem states that integer spin particles are bosons while half-integer spin particles are fermions.

The Spin-Statistics relation was first formulated in 1939 by Markus Fierz^[1], and was rederived in a more systematic way by Wolfgang Pauli ^[2]. 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,^[3], which translated to field language is a condition on the quadratic operator that couples to the potential^[4].

1 General Discussion
2 Proof
3 Consequences
4 References
5 External links

[edit] General Discussion

Two identical particles, occupying two separate points, have only one state, not two. Bosons are particles whose wavefunction is symmetric under exchange, while Fermions are antisymmetric.

In a field theory, the quanta are created by a field operator. In order for the operators to project out the symmetric or antisymmetric component of the creating wavefunction, they must have the appropriate commutation law. The operator

$\int \psi(x,y) \phi(x)\phi(y) dx dy \,$

creates a two-particle state with wavefunction $ψ(x, y)$ , and depending on the commutation properties of the fields, either only the antisymmetric parts or the symmetric parts matter.

If the field has the property that at spacelike separation

$\phi(x)\phi(y)=\phi(y)\phi(x)\,$ ,

only the symmetric part of $ψ$ contributes, and the field will create bosonic particles. On the other hand if the field has the property that

$\phi(x)\phi(y)=-\phi(y)\phi(x)\,$

for two spacelike separated x and y, the particles will be fermionic.

Naively, neither has anything to do with the spin, which determines the rotation properties of the particles not the exchange.

[edit] Proof

Assuming that the theory is Lorentz invariant, we can rotate and boost the field operators, and they will transform according to the spin of the particle that they create. In the Euclidean theory, these boost generators become the generators of rotations. It is important to note that unlike the boost generators, a Euclidean rotation comes back to itself after $2\pi \$ .

Now consider the two-field product

$R(\pi)\phi(x) \phi(-x) \$

which creates two particles with polarizations which are rotated by $\pi \$ relative to each other. Now rotate this configuration by $\pi \$ around the origin. Under this rotation, the two points $x \$ and $-x \$ switch places, and the two field polarizations are additionally rotated by a $\pi \$ . So you get

$R(2\pi)\phi(-x) R(\pi)\phi(x) \$

which for integer spin is equal to

$\phi(-x) R(\pi)\phi(x) \$

and for half integer spin is equal to

$- \phi(-x) R(\pi)\phi(x) \$

So that exchanging the order of two appropriately polarized operator insertions into the vacuum can be done by a rotation, at the cost of a sign in the half integer case.

To turn this argument into a proof requires the following assumptions:

The theory has a Lorentz invariant lagrangian.
The vacuum is Lorentz invariant.
The particle is a localized excitation. Microscopically, it is not attached to a string or domain wall.
The particle is propagating, meaning that it has a finite, not infinite, mass.
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:

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.
This assumption may be weakened.
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.
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.
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 $\pi \$ 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

$G(x) = \langle 0| R\phi(-x) \phi(x) |0\rangle \,$

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 heuristic argument:

$G(Rx) = \langle 0| RR\phi(x) R\phi(-x)|0\rangle = \pm \langle 0|\phi(x) R\phi(-x)|0\rangle \,$

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

$\langle 0| [\phi(x),R\phi(-x)] |0\rangle = 0 \,$

for integer spin fields and

$\langle 0| \{\phi(x),R\phi(-x)\} |0\rangle = 0 \,$

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.

[edit] Consequences

Spin statistics theorem implies that fermions are subject to the Pauli exclusion principle, while bosons are not. This means that only one fermion can occupy a given quantum state, while the number of bosons that can occupy a quantum state is not restricted. The basic building blocks of matter such as protons, neutrons, and electrons are fermions. Particles such as photons, which mediate forces between matter particles, are bosons.

There are a couple of interesting phenomena arising from the two types of statistics. The Bose-Einstein distribution which describes bosons leads to Bose-Einstein condensation. Below a certain temperature, most of the particles in a bosonic system will occupy the ground state (the state of lowest energy). Unusual properties such as superfluidity can result. The Fermi-Dirac distribution describing fermions also leads to interesting properties. Since only one fermion can occupy a given quantum state, the lowest single-particle energy level can contain only two fermions, with the spins of the particles oppositely aligned. Thus, even at absolute zero, the system still has a significant amount of energy. As a result, a fermionic system exerts an outward pressure. Even at non-zero temperatures, such a pressure can exist. This degeneracy pressure is responsible for keeping certain massive stars from collapsing due to gravity. See white dwarf, neutron star, and black hole.

Ghost fields do not obey the spin-statistics relation. See Klein transformation on how to patch up a loophole in the theorem.

[edit] References

^ M. Fierz "Uber die relativistiche Theorie krafterfreier Teilchen mit Beliebigem Spin" Helvetica Physica Acta 12:3-37, 1939
^ W. Pauli "The Connection Between Spin and Statistics", Phys. Rev. 58, 716-722 (1940)
^ R.P. Feynman "Quantum Electrodynamics", Basic Books, 1961
^ W. Pauli "On the Connection Between Spin and Statistics" Progress of Theoretical Physics vol 5 no. 4, 1950