STATISTICAL MECHANICS OF BOSONS AND FERMIONS

The most general system of noninteracting particles will have a variety of single particle energy eigenstates, labeled by a parameter p, which can be discrete or continuous and range over a finite or infinite number of values. The space of p values might also have a geometrical structure corresponding to some number of spatial dimensions. As an example, p could label both the three vector momentum P and the J3 component of the spin σ of a nonrelativistic particle of any spin j. The momentum would be discrete if space were a torus, and would live on a torus, the first Brillouin zone, if space were a lattice. Denote the energy of the p-th single particle state by ∈_p.
The N-particle states are labeled by N copies of the quantum number p₁, . . . p_N, which are either symmetric or totally antisymmetric under permutations of the copies. The key to statistical mechanics is the realization that a nonredundant labeling of the states is just to give the number of particles n(p) occupying each single particle state p. For bosons, n(p) can be any nonnegative integer, while for fermions, it is either zero or one. For either type of statistics, we have

which means that the Hilbert space breaks up into a tensor product H = ⊗ pH_p, where n(p) acts as 1 ⊗ . . . n(p) ⊗ 1 . . . ⊗ 1. That is, n(p) acts in a nontrivial way only on the factor H_p.
Furthermore, because p is a full set of single particle quantum numbers, the states in H_p are completely characterized by the eigenvalue of n(p). For fermions, H_p is two dimensional, while for bosons, it is infinite dimensional.
In nonrelativistic physics, the numbers of each type of stable elementary particle (electrons, protons, neutrons inside nuclei which do not undergo beta decay) are conserved quantum numbers. From a relativistic perspective, this is because we never consider transitions between energy levels differing by more than about 10⁻⁴ of the electron mass. For the purposes of this chapter, we will consider only a single type of elementary particle, so that there is only one such conserved quantum number

Conservation means that [N,H] = 0, which is evident from