Astroparticle Physics

B.1 Statistical Mechanics Review 391The total wave function is thusψ(x 1 ,...,x N ) == √ 1 ∑P(i,j,...)ψA (x i )ψ B (x j ) ··· ,N!(B.3)where the sum includes all possible permutations of the coordinatesx i . For a system of identical bosons, the factor P isequal to one, whereas for identical fermions it is plus or minusone depending on whether the permutation is obtainedfrom an even or odd number of exchanges of particle coordinates.This results in a wave function that is symmetric forbosons and antisymmetric for fermions upon interchange ofany pair of coordinates. As a consequence, the total wavefunction for a system of fermions is zero if the same oneparticlewave function appears more than once in the productof terms; this is the Pauli exclusion principle.Although the most general solution to the N-particleSchrödinger equation does not factorize in the way ψ hasbeen written in (B.3), this form will be valid to good approximationfor systems of weakly interacting particles. Forhigh-temperature systems such as the early universe, (B.3)is assumed to hold.Further, one assumes that the one-particle wave functionsshould obey periodic boundary conditions in the volumeV = L 3 . The plane-wave form for the one-particlewave functions in (B.2) then implies that the momentumvectors p cannot take on arbitrary values but that they mustsatisfysymmetrization for bosonsantisymmetrizationfor fermionsPauli exclusion principleperiodic boundaryconditions: discretizingmomentump = 2π L (n x,n y ,n z ),(B.4)where n x , n y ,andn z are integers. Thus, the possible momentafor the one-particle states are given by a cubic latticeof points in momentum space with separation 2π/L.For a given N-particle wave function, where N will ingeneral be very large, the possible momentum vectors forthe one-particle states will follow some distribution in momentumspace. That is, one will find a certain number dNof one-particle states for each element d 3 p in momentumspace, andmomentum distributionf(p) = d3 Nd 3 p(B.5)will be called the momentum distribution.