Hyperfunctions

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.