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Consider the determinant of the euclidean form of the Dirac operator: $$\text{det}(iD), \quad iD = i\gamma_{\mu}(\partial_{\mu}+A_{\mu})$$

It is an elliptic operator, so has a discrete spectrum on compact manifolds. The euclidean manifold $R^{4}$ is not a compact manifold. However, people typically writes $$ \text{det}(iD) = \prod_{i = 1}^{\infty}\lambda_{i}, $$

where $\lambda_{i}$ defines the $i$th eigenvalue: $$ iD\psi_{i} = \lambda_{i}\psi_{i} $$

Why can they do this?

If it is done in the framework of QFT, it is usually quite a formal expressions which then must be regularized in some way. The same story as for other determinants, traces, etc.

@NAME_XXX It may be a kind of philosophic question. How do we know that the Path Integral is correct? Basically, it is because it works, and in all the experimentally verified situations it works correctly with enormous precision.

One not only replaces the manifold by a compact one but afterwards takes the infrared limit (infinite radius) that restores the original manifold. Under this limit the spectrum becomes continuous. of course this is not fully rigorous, and for massless fields unexpected things may happen....

@AndreyFeldman : but is there some reason which allows us to do that? All of the theory quantities must be projected on $S^{4}$ coordinates, and corresponding action must be invariant.

@ArnoldNeumaier : but is there some projection rule? I.e., if I want to construct the theory on $S^{4}$, I need to construct corresponding quantities defined on $S^{4}$.

@NAME_XXX In general, compactification of $\mathbb{R}^4$ to get $S^4$ is subtle -- it may give masses to massless fields, etc. If you are looking for some universal recipe, I'm afraid that it doesn't exist.

But does this regularization imposethe discrete spectrum? Ii rather seems that when computing the Dirac determinant, we regularize alreary discrete spectrum from above. The fact that the spectrum is discrete is unexplained.

@NAME_XXX Usually one replaces a non-compact space by a compact one, which gives the former in the limit of infinite radius (say, $\mathbb{R}$ may be replaced by $S^1$), evaluates the index on this manifold, and then tends the radius to infinity.

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