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. Spin is the intrinsic angular momentum of a particle such as an electron. All particles have either integer spin or half-integer spin (in multiples of Dirac's constant). These two classes of particles are known respectively as bosons and fermions.
The theorem states that:
- The wavefunctions of a systems of identical integer-spin particles, known as bosons, (i.e. s = 0, 1, 2,...) are symmetric under interchange of any pair of particle labels.
- The wavefunctions of a systems of identical half-integer-spin particles, known as fermions, (i.e. s = 1/2, 3/2, 5/2,...) is anti-symmetric under interchange of any pair of particle labels
This 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.
A novel variation on the Spin-Statistics Theorem appeared in 2003 in the Foundations of Physics. The article entitled "Rotational Invariance and the Spin-Statistics Theorem" directly relates the Pauli exclusion principle to Rotatational Invariance and Entanglement. It also establishes a link between spin-statistics and Bell's Inequality.
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.
See Klein transformation on how to patch up a loophole in the theorem.
Ghost fields do not obey the spin-statistics relation.
[edit] References
- W. Pauli, The Connection Between Spin and Statistics, Phys. Rev. 58, 716-722(1940). (abstract)
- Paul O'Hara, Rotational Invariance and the Spin-Statistics Theorem, Foun. Phys. 33, 1349-1368(2003).
[edit] External links
- A nice nearly-proof at John Baez home page
- parastatistics, anyonic statistics and braid statistics
