Radial Complex Space

This is just the beginnings of a notion, taken from my small amount of work extending a vector field to a complex valued-vector field. What if, instead of writing complex numbers in the form x + iy, we write them in a radial form:

Here x would have the same meaning as the original x, but would be the dimensional “phase”. Since we are mostly interested in vectors, we could extend vectors to these new complex entities. Let’s look at some formula related to coordinate changes:

We can also look at some properties of the inner product:

We can guarantee this value is real (so as to maintain measurable lengths) by assuming the metric is hermitian. This then yields, after replacement:

where

and

© 2012, Jason Allen Blood. All rights reserved.