Hyper-complex fields in Supergravity?

[Professor_E]

Dear colleagues
I have recently come across a mathematical reference discussing all possible generalizations of complex numbers. A particularly interesting such generalization is known as split-complex numbers. These are defined:

z = x+jy
z* = x-jy

similarly to ordinary complex numbers except that j^2 = +1 not -1. This makes:

zz* = x^2 - y^2

Having learned about this for the first time, it immediately brought memories of a solution in supergravity that I have found but abandoned on physical grounds some time ago. It seems to me now that the solution I have found would be well-behaved, at least mathematically, if I had assumed it to be split-complex! But this begs the question: What does that mean physically? Do any of you know of any precedent in the literature.
I am just curious, but perhaps I will take a second look at that solution ...

 

[robphy]

The "Perplex numbers" can be used in special relativity.
http://dx.doi.org/10.1119/1.14605

(I am using them to develop a new way to teach relativity.)

 

[Professor_E]

The "Perplex numbers" can be used in special relativity.
http://dx.doi.org/10.1119/1.14605

(I am using them to develop a new way to teach relativity.)

Thanks - Yes I am aware to their application to special relativity. But what I am looking for is a precedent for a classical field Lagrangian that has split-complex solutions.