This is a problem (Problem 3.16) from the book Statistical Mechanics 2nd Ed. by Pathria.
In the problem I have to calculate the partition function of an ultra-relativistic 1D gas ($E_i=cp_i$) consisting in $3N$ particles moving in one dimension. I know that the partition function is given by
$$Q_{3N}=\frac{1}{(3N)!h^{3N}} \int e^{-\beta H(q,p)}d\omega,$$
where $d\omega$ denotes a volume element of the phase space. In this case
$$d\omega=dq_1dp_1\cdots dq_{3N}dp_{3N},$$
and
$$H(q,p)=\sum_{i=1}^{3N}cp_i.$$
Then, making the substitution I find that
$$Q_{3N}=\frac{L^{3N}}{(3N)!h^{3N}} \left[\int_{-\infty}^{+\infty} e^{-\beta c p_j}dp_j\right]^{3N}.$$
$L$ being the "length" of the space available. But I'm pretty sure that this integral does not converge.
Where am I wrong?
Best Answer
Energy has to be bounded below. The relationship between energy & momentum in this case is $E_i = c |p_i|$, not $E_i =c p_i$. So the integral you should be trying is $\int_{-\infty}^\infty e^{-\beta c |p_i|} dp_i$, which converges just fine.