Suppose the Dirac operator determinant in Euclidean space-time with manifold $\mathbb R^{4}$: $$ d = \text{det}(iD), \quad iD = i\gamma^\mu (\partial_\mu +A_{\mu}) $$ The Dirac operator is elliptical, so it has discrete spectrum $\{ \lambda_{i}\}$ on the compact manifold, say, in $S^{4}$. In non-compact spaces as $R^{4}$ the spectrum is continuous. However, people often write $$ \tag 1 d = \prod_{i = 1}^{\infty}\lambda_{i}, $$
which (seems to) mean that they make the stereographic projection of $R^{4} $ onto $S^{4}$. The projection is possible if some restrictions on a theory is imposed.

The thing doubting me is that they don't point out that they make stereographic projection (see, for example, Fujikawa, Suzuki, "Path Integral and anomalies"). So the question is: is it possible to write $(1)$ without any assumptions about the properties of the theory?


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