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Hyper-complex fields in Supergravity?

  1. Jan 11, 2009 #1
    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 ...
  2. jcsd
  3. Jan 11, 2009 #2


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  4. Jan 11, 2009 #3
    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.
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