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In (mathematical) physics in order to compute path integrals one often makes an infinite dimensional change of variables and uses infinite Jacobian as a purely formal expression. This step is done in formal analogy with the finite dimensional case (e.g. in the Faddeev-Popov construction).

Next one often needs to compute this Jacobian. In some situations it is a "determinant" of a self-adjoint elliptic differential operator on a compact manifold. A conventional procedure to compute it is to use the $\zeta$-function regularization. This procedure seems to me to be quite arbitrary. While I am not an expert, I have never seen in literature any motivation of it except that it gives the right answer for finite dimensional transformations. To compare with, in classical analysis there are many different methods to sum divergent series, but they may lead to different answers.

QUESTIONS. 1) Are there other conventional methods which are used in explicit computations of infinite Jacobians?

2) Are there any nice pleausible general properties of the $\zeta$-function regularized determinants which distingush them from other methods?

ADDED: This method leads to the identity mentioned in the answer below by Zurab Silagadze $$1+2+3+4+\dots=-\frac{1}{12}$$ which seems to me completely counter-intuitive.

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    $\begingroup$ $\zeta$-regularised determinants are used in the definition of analytic torsion (Ray-Singer torsion). By the Cheeger-Müller theorem, this invariant is closely related to Reidemeister torsion, which is defined using determinants of finite size. So at least in this case, the $\zeta$-regularisation is the "correct guess". $\endgroup$
    – Sebastian Goette
    Commented Mar 1, 2017 at 19:21
  • $\begingroup$ By far not enough to be a full answer: in situations where there are analytic relations that make sense and arguably hold at other values of the continuation parameter $s$, then by the Identity Principle from complex analysis the same relation holds for the continuation. I do not know whether most of these physics-y issues fall into this category, but I know of some that do. A classic is Hadamard's "finite part" device, which simply dropped an infinity, and was shown a little later by Riesz to be a meromorphic continuation. $\endgroup$
    – paul garrett
    Commented Apr 25, 2017 at 20:24

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Another chance to link to my favorite math blog post on the internet.

The answer is that zeta-function regularization is determining the constant part in a divergent series when you add in a smooth cutoff. So, you're secretly just subtracting off the infinities in a fancy way. This is shown in simple examples Terry Tao's blog post here. (Honestly, I haven't done the work to check that this all generalizes to more complicated cases, but it's hard to imagine it doesn't -- I do wonder if someone's written this up in full generality.)

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    $\begingroup$ But then again, why do you cut off the infinities in exactly that particular series? $\endgroup$
    – Sebastian Goette
    Commented Mar 1, 2017 at 19:22
  • $\begingroup$ The divergent series is "$\sum \log \lambda_i$", but I think the question you're asking is why compute what is effectively $\sum \frac{\log \lambda_i}{\lambda_i^s}$ rather than something like $\sum\frac{1}{\left(\log\lambda_i\right)^s}$, right? $\endgroup$
    – Aaron Bergman
    Commented Mar 1, 2017 at 21:54
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Of course that $1+2+3+4+\ldots =-1/12$ is a "correct" result is somewhat mysterious, as the following excerpt from Ramanujan's second letter to Hardy lively demonstrates:

"I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfall of divergent series. I have found a friend in you who views my labors sympathetically. This is already some encouragement to me to proceed with an onward course. I find in many a place in your letter rigourous proofs are required and so on and you ask me to communicate the method of proof. If I had given you my methods of proof I am sure you will follow the London Professor. But as a fact I did not give him any proof but made some assertions as the following under my new theory. I told him that the sum of an infinite number of terms in the series $1+2+3+4+\ldots =-1/12$ under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal."

In fact, as John Baez mentions in http://math.ucr.edu/home/baez/week126.html, Hardy later in his book Divergent Series gave almost a "physicist's proof" of this result.

A good account why physicists consider the zeta-function regularization meaningful and unique is given in https://arxiv.org/abs/hep-th/9308028 (Zeta-Function Regularization is Uniquely Defined and Well, by E. Elizalde). See also https://arxiv.org/abs/0909.0795 (On Twisted Virasoro Operators and Number Theory, by A. Huang).

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