Reference Request: Split-Complex Numbers

[sph3rical]

What's a good book on split-complex numbers?

If it also covers dual numbers or the relation between split-complex numbers and special relativity or Minkowski 4-space or some analysis of split-complex numbers then all the better, but that's just gravy. I really just want a good reference for the geometry of the plane as expressed via split-complex numbers.

 

[vanhees71]

What are "split-complex numbers"? I've never heard about this. For sure, you should not use complex numbers in special relativity (except for complex valued fields like the charged Klein-Gordon or the Dirac field). The idea to use an imaginary time coordinate to let the Minkowski product formally look like a Euclidean scalar product is not very good. It's very confusing, particularly when you want to learn general relativity, where this bad trick doesn't work anymore. Also in relativistic QFT, at one place you really might switch to Euclidean QFT by a Wick rotation, and then you really deal with an imaginary-time formalism. The same is possible for QFT at finite temperature (Matsubara formalism), but you should use the imaginary time coordinates only at these places where they make real physical sense!