I’ve been trying to understand (not entirely successfully yet…) Wick rotation using hyperfunctions, which are a sort of distributions not as well-known as they should be. Some notes about them are in a separate pdf.
What is well-known is that one often needs to generalize the notion of what a function is to include things like the Dirac delta function $\delta(x-a)$. The usual way to make this well-defined is the theory of distributions, but hyperfunctions provide an interesting and in many ways more useful alternative. See the notes for more detail, but an important example is that (up to a constant), the Dirac delta function is the function
$$\frac{1}{z-a}$$
of a complex variable $z=x+iy$, reinterpreted as a hyperfunction. This way of thinking about the delta function makes available the powerful methods of complex analysis.
There have been some previous attempts to do the sort of thing I’m thinking about. In particular see this paper. Such attempts to reformulate QFT, with Wightman functions taken to be hyperfunctions, have generally had as one motivation to resolve the problems with OS reconstruction discussed in an earlier blog posting. My own motivation is rather different (providing a formalism in which one can understand Wick rotation of spinor fields in a new way).
The idea of using hyperfunctions this way in QFT has attracted relatively little interest over the years. I’m guessing one reason for this is that hyperfunctions become much harder to work with when one is dealing with more than one complex variable. In the usual rigorous QFT framework one tries to understand Wick rotation by complexifying all spacetime variables, not just time, with just complexifying time violating usual ideas of the necessity of preserving Lorentz invariance.
In a later posting I’ll discuss how hyperfunctions show up in twistor theory, where the way complexification and hyperfunctions work is quite different.
Peter – thanks for the nice post! As a fellow fan of hyperfunctions, I wanted to point out what I thought was a gorgeous observation by Scholze https://mathoverflow.net/questions/470940/topology-on-space-of-hyperfunctions that in the world of condensed math one really can define hyperfunctions very cleanly just as the derived dual of compactly supported analytic functions. This extends the classical appearance of local cohomology as the derived dual of analytic functions with support along a subvariety.
Thanks David!
That’s great, had missed this. Very interesting to see an application of the condensed math motivation in this context.
I take the permission to mention my paper
cited below, which deals precisely with this subject
Wick rotation for D-modules
Math Physics Analysis Geometry
Vol 20 (3) 2017 pp 1-14.
Arxivmath 1702.00003
The following might be of interest in this context:
C.A. McCranie, Applications of Hyperfunctions in Quantum Field Theory and Mathematical Physics, PhD thesis, University of Colorado at Boulder, 2025.
Another relevant paper may be
A.U. Schmidt, Euclidean reconstruction in quantum field theory: between tempered distributions and Fourier hyperfunctions, arXiv:math-ph/9811002.
Pierre Schapira,
Yes, very much related. I’ll write soon something about the twistor story, hope to get a chance to ask you some questions about the D-module point of view.
Arnold Neumaier,
Yes, thanks, those are very definitely related. The McCranie thesis in particular has some things related to what I’ve been trying to do, your comment reminds me I should look at it more carefully.
Arnold Neumaier,
I took a look again at the McCranie thesis. There is an attempt to define field operators in terms of operator-valued hyperfunctions, and to write some Wightman functions as hyperfunctions. There is however nothing about using the hyperfunctions to do Wick rotation (i.e. going to Euclidean points and getting a Euclidean field theory).