Consider the determinant of the euclidean form of the Dirac operator:
det(iD),iD=iγμ(∂μ+Aμ)
It is an elliptic operator, so has a discrete spectrum on compact manifolds. The euclidean manifold R4 is not a compact manifold. However, people typically writes
det(iD)=∞∏i=1λi,
where λi defines the ith eigenvalue:
iDψi=λiψi
Why can they do this?