  We show that the spectral gap for the interchange process (and the symmetric
exclusion process) in a $d$-dimensional box of side length $L$ is asymptotic to
$\pi^2/L^2$. This gives more evidence in favor of Aldous's conjecture that in
any graph the spectral gap for the interchange process is the same as the
spectral gap for a corresponding continuous-time random walk. Our proof uses a
technique that is similar to that used by Handjani and Jungreis, who proved
that Aldous's conjecture holds when the graph is a tree.
