标题: Spin and Statistics
作者: sage
As a result, the irreducible representations of Lorentz group could be classified according to their transformation properties under the rotational group, also known as spin. The well-known representations are: trivial j=0 (scalar), j=1/2 ((1/2,0) and (0,1/2),fermion), and j=1((0,1) and (1,0), vector). All of them are used by nature. It is also possible to Rarita-Schwinger j=3/2 (1,1/2) and (1/2,1) (such as gravitino) and j=2 such as the graviton. Higher spin representations exist, but somewhat less useful.
Spin is ``intrinsic'' in the sense that it is not possible change the spin of a particle by kinematical method. A particle of particular kind is born with its definite spin and states with different spin will not mix. In comparison, we could change spatial angular momentum by accelerate a particle. Beside this, I don't know a meaningful way to talk about ``intrinsic''.
Spin statistics can be divide into two questions:
1) how is the spin of the particle connect to the statistics?
This is the so-called spin statistics connection, first point out by Fierz.
It has everything to do with Lorentz invariance.
It is a long story with details. However, the basic story line is that Lorentz invariance of the S-matrix requires that the commutator of interaction Lagrangian to commute outside of the light-cone (a somewhat similar argument leads to the same conclusion is causality). Lorentz invariance again together with cluster decomposition principle require us to construct it with relativistic quantum field (Very crudely speaking, cluster decomposition requires we use creation and annihilation operators with specific momentum. So we are in general talking about many particles. Lorentz covariance requires we consider a linear combination of them, also known as fields.).
Putting these together leads to the conclusion that quantum fields will either have to commute or anti-commute. The form of the relativistic quantum fields are also constrained by transformation laws under Lorentz.
These two constraints fixes that spin half integer must anti-commute and spin integer must commute.
2) How many different statistics are possible? This has nothing to do with Lorentz invariance.
Let's ask how we distinguish particles. One way to do it is the following:
We start with a set of particles in a particular initial state, then we permute particles. If the particles are indistinguishable, this at most creates some complex phase in front of the state. We now test this result by asking the amplitude this state evolve into some particular final state. The physical amplitude is the sum of all indistinguishable permutations. In the path integral, we have integrate over all possible paths. These include all paths which can be continuously deform into each other, and sum over all topologically
different paths, different term in this sum in principle could contribute different phases.
In d>=3, all paths for the same permutation are equivalent. So there is no second part (the sum over topologies). Therefore, there is only permutation of initial state. The complex number must form a rep of permutation group. Moreover, two consecutive permutation (product of two complex numbers) should be the same as permute once with the product permutation. There are only two reps of the permutation group which has this property, trivial rep (1), and Z_2(-1, 1). Therefore, for d>=3, there are only two statistics possible, bose-einstein or fermi-dirac, independent of spin.
In d=2, there is a topology sum since there are inequivalent paths (which circles some particle any number of times). Therefore, the complex number does not have to for a rep of the permutation group (It has a name called braid group). In fact, it could be any thing. Therefore, we could have any kind of
statistics, the so called anyons.