A Suggestive Bogus Argument
Consider the two-field operator product
where R is the matrix which rotates the spin polarization of the field by 180 degrees when one does a 180 degree rotation around some particular axis. The components of phi are not shown in this notation, has many components, and the matrix R mixes them up with one another.
In a non-relativistic theory, this product can be interpreted as annihilating two particles at positions x and −x with polarizations which are rotated by π (180°) relative to each other. Now rotate this configuration by π around the origin. Under this rotation, the two points and switch places, and the two field polarizations are additionally rotated by a . So you get
which for integer spin is equal to
and for half integer spin is equal to
(proved here). Both the operators still annihilate two particles at and . Hence we claim to have shown that, with respect to particle states: . So 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.
This argument by itself does not prove anything like the spin/statistics relation. To see why, consider a nonrelativistic spin 0 field described by a free Schrödinger equation. Such a field can be anticommuting or commuting. To see where it fails, consider that a nonrelativistic spin 0 field has no polarization, so that the product above is simply:
In the nonrelativistic theory, this product annihilates two particles at x and −x, and has zero expectation value in any state. In order to have a nonzero matrix element, this operator product must be between states with two more particles on the right than on the left:
Performing the rotation, all that you learn is that rotating the 2-particle state gives the same sign as changing the operator order. This is no information at all, so this argument does not prove anything.
Read more about this topic: Spin-statistics Theorem
Famous quotes containing the words suggestive and/or argument:
“Many of the phenomena of Winter are suggestive of an inexpressible tenderness and fragile delicacy. We are accustomed to hear this king described as a rude and boisterous tyrant; but with the gentleness of a lover he adorns the tresses of Summer.”
—Henry David Thoreau (1817–1862)
“There is no good in arguing with the inevitable. The only argument available with an east wind is to put on your overcoat.”
—James Russell Lowell (1819–1891)