Permutations

Two particles are called indistinguishable when they have the same `elementary' parameters such as mass, charge, total spin. As an example, it is believed that all electrons are the same in the sense that they all have the same mass, the same charge, and the same spin $1/2$. The evidence for this comes from experiments.

If some of the $N$ particles described by the wave function $\Psi$, Eq. (III.1.1), are indistinguishable, this restricts the form of $\Psi$. Let us assume that all $N$ particles are pairwise indistinguishable. We define the abbreviations $\xi_i\equiv ({\bf r}_i,\sigma_i)$. Since particle $j$ is indistinguishable from particle $k$, the $N$-particle wave functions with $\xi_j$ and $\xi_k$ swapped should describe the same physics: they may only differ by a phase factor,

$$\Psi(\xi_1,...,\xi_j,...,\xi_k,...,\xi_N)= e^{i\phi_{jk}} \Psi(\xi_1,...,\xi_k,...,\xi_j,...,\xi_N).$$ (1.2)

Swapping $j$ and $k$ a second time must yield the original wave function and therefore

$$e^{2i\phi_{jk}}=1\leadsto \phi_{jk} = 0,\pm \pi, \pm 2\pi, \pm 3\pi,$$ (1.3)

In fact, the phases $0, \pm 2\pi$ etc. are all equivalent: they lead to symmetrical wave functions. The phases $\pm \pi, \pm 3\pi$ etc. are also all equivalent: they lead to antisymmentrical wave functions.

It turns out that this argument (swapping the coordinates) depends on the dimension of the space in which the particles live, and that there is a connection to the spin of the particles. For $d\ge 3$, indistinguishable particles with half-integer spin are called Fermions which are described by antisymmentrical wave functions. For $d\ge 3$, indistinguishable particles with integer spin are called Bosons which are described by symmentrical wave functions. For $d=3$, this connection between spin and statistics can be proved in relativistic quantum field theory (Spin-Statistics-Theorem, W. Pauli 1940).

$$\Psi(\xi_1,..,\xi_j,...,\xi_k,..,\xi_N) = - \Psi(\xi_1,..,\xi_k,...,\xi_j,..,\xi_N),\quad {\rm Fermions}$$

$$\Psi(\xi_1,..,\xi_j,...,\xi_k,..,\xi_N) = \Psi(\xi_1,..,\xi_k,...,\xi_j,..,\xi_N),\quad {\rm Bosons}.$$ (1.4)

In two dimensions, things become more complicated. First of all, the connection with spin (integer, half integer in $d=3$) is different in $d=2$ because angular momentum in general is no longer quantized: rotations in the $x$- $y$ plane commuted with each other, i.e. the rotation group $SO(2)$ is abelian and has only one generator which can have arbitrary eigenvalues. Second, topology is different in two dimensions, in particular when discussing wave functions excluding two particles sitting on the same place ${\bf x}_k={\bf x}_j$ which leads to effective configuration spaces which are no longer simply connected.

In two dimensions, one obtains a plethora of possibilities with exciting new possibilities for `fractional spin and statistics'. These are important and have been discovered recently in, e.g., the fractional quantum Hall effect. For further literature on this topic, cf. S. Forte, `Quantum mechanics and field theory with fractional spin and statistics', Rev. Mod. Phys. 64, 193.