Split Quaternionic Analysis and Separation of the Series for SL(2,R) and SL(2,C)/SL(2,R)

TL;DR

This paper extends quaternionic analysis to split quaternions, relating it to SL(2,R) and hyperbolic space, and connects it to quantum field theory through series separation and representation theory.

Contribution

It introduces a split quaternionic analysis framework that relates to SL(2,R), hyperbolic space, and quantum field theory, revealing new formulas and representation connections.

Findings

Separation of discrete and continuous series on split quaternions and hyperbolic space.

Minimal representation of SL(4,R) from continuous series on H_R.

Connection between quantum field kernels and series projectors.

Abstract

We extend our previous study of quaternionic analysis based on representation theory to the case of split quaternions H_R. The special role of the unit sphere in the classical quaternions H identified with the group SU(2) is now played by the group SL(2,R) realized by the unit quaternions in H_R. As in the previous work, we use an analogue of the Cayley transform to relate the analysis on SL(2,R) to the analysis on the imaginary Lobachevski space SL(2,C)/SL(2,R) identified with the one-sheeted hyperboloid in the Minkowski space M. We study the counterparts of Cauchy-Fueter and Poisson formulas on H_R and M and show that they solve the problem of separation of the discrete and continuous series. The continuous series component on H_R gives rise to the minimal representation of the conformal group SL(4,R), while the discrete series on M provides its K-types realized in a natural polynomial basis. We also obtain a surprising formula for the Plancherel measure on SL(2,R) in terms of the Poisson integral on the split quaternions H_R. Finally, we show that the massless singular functions of four-dimensional quantum field theory are nothing but the kernels of projectors onto the discrete and continuous series on the imaginary Lobachevski space SL(2,C)/SL(2,R). Our results once again reveal the central role of the Minkowski space in quaternionic and split quaternionic analysis as well as a deep connection between split quaternionic analysis and the four-dimensional quantum field theory.