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18
Feb

## 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:

$z_{\mu}=x_{\mu}e^{i \theta_{\mu}}$

Here x would have the same meaning as the original x, but $\theta$ 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:

${z}'_{\mu}={x}'_{\mu}e^{i {\theta}'_{\mu}}$

${x}'_{\mu} = \frac{\partial x_{\mu}}{\partial {x}'_{\nu}}x_{\nu}$

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

$l^2=g_{\mu \nu^*} \zeta^{\mu} \zeta^{\nu^*}$

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

$l^2=Re(g_{\mu \nu}) A^{\mu} A^{\nu}cos(\theta^{\mu} - \theta^{\nu})$

where

$\zeta^{\mu}= A^{\mu} e^{\theta^{\mu}}$

and

$g_{\mu \nu^*} = g_{\nu \mu^*}$

© 2012, Jason Allen Blood. All rights reserved.