### Quaternionic Analysis

#### Posted by David Corfield

Nobody else has mentioned it, but perhaps a few extracts from a paper by one of the founding fathers of categorification, Igor Frenkel, might be of interest, even if not on our topic:

Quaternionic Analysis, Representation Theory and Physics, Igor Frenkel and Matvei Libine.

Abstract

We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The requirement of unitarity of representations leads us to the extensions of these formulas in the Minkowski space, which can be viewed as another real form of quaternions. Representation theory also suggests a quaternionic version of the Cauchy formula for the second order pole. Remarkably, the derivative appearing in the complex case is replaced by the Maxwell equations in the quaternionic counterpart. We also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory, such as the spectrum of the hydrogen atom, polarization of vacuum, one-loop Feynman integrals. We also make some further conjectures. The main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis, representation theory and four-dimensional physics.

They say

But,Unfortunately…the quaternionic analogues of the ring structure of holomorphic functions, local conformal invariance, Riemann mapping theorem and many other classical results of complex analysis have never been found. Such a failure has even led R. Penrose to say, “[Quaternions] do have some very significant roles to play, and in a slightly indirect sense their influence has been enormous, through various types of generalizations. But the original ‘pure quaternions’ still have not lived up to what must undoubtedly have initially seemed to be an extraordinary promise… The reason appears to be that there is no satisfactory quaternionic analogue of the notion of a holomorphic function.”

The Minkowski space reformulation of quaternionic analysis brings us into a thorough study of Minkowski space realization of unitary representations of the conformal group SU(2, 2) by H. P. Jakobsen and M. Vergne who were motivated by the program of I. E. Segal on the foundational role of representation theory of SU(2, 2) in physics.

Irving Segal was John’s PhD. supervisor.

Furthermore,

Another unique feature of quaternionic analysis is its deep relation to physics, in particular, to the four-dimensional classical and quantum field theories for massless particles…We have already mentioned the implication of the Poisson formula…to the spectral decomposition of the Hamiltonian of the hydrogen atom. In general, the Minkowski formulation of various results of quaternionic analysis provides a link to the four-dimensional field theories. This is hardly surprising since the Minkowski space is the playground for these physical theories, but it is still quite remarkable that we encounter some of the most fundamental objects of these theories. It is certainly clear that the equations for the left- and right-regular functions (4) and (5) are nothing but the massless Dirac equation. But it comes as a surprise that the quaternionic analogue of the Cauchy formula for the second order pole (8) is precisely the Maxwell equations for the gauge potential. Moreover, the integral itself appears in the Feynman diagram for vacuum polarization and is responsible for the electric charge renormalization. Also, the quaternionic double pole formula in the separated form has a kernel (14) represented by the one-loop Feynman integral. There is no doubt for us that these relations are only a tip of the iceberg, and the other Feynman integrals also admit an interpretation via quaternionic analysis and representation theory of the conformal group. In fact, we make some explicit conjectures at the end of our paper. Thus we come to the conclusion that the quaternionic analysis is very much alive and well integrated with other areas of mathematics, since it might contain a great portion – if not the whole theory – of Feynman integrals. On the other hand, the latter theory – a vast and central subject of physics – might not seem so disconcerting and unmotivated anymore for mathematicians, and many of its beautiful results should be incorporated in an extended version of quaternionic analysis.

## Re: Quaternionic Analysis

Thanks a lot for mentioning this.

I don’t have the time to look into this article at the moment, but from the quotes you give I get the impression that it ought to be true that the excitement that the authors are expressing is essentially about the same kind of phenomena that drives the excitement that practitioners in what is called “Geometric Algebra” are notorious for.

Lectures on “geometric algebra” typically emphasize a lot the naturalness with which Maxwell’s equations and Dirac’s equation arise when making use of quaternionic structures.

As one can probably already see from these links, practitioners in “geometric algebra” tend to have a stronger tendency to invest energy into pedagogics and physics than into heavy-duty research math, which may be the reason why some of the very good ideas developed there are having a hard time propagating meme-wise in the pure-math community.

Maybe with Frenkel this is changing now. But I haven’t even looked at his article yet, so don’t trust me.