# Giving Vectors Phase

Let us dive quickly in and explore a basic idea: ascribing a “phase” to vector fields. Instead of writing a vector $\psi_{\mu}$ a simply 4 real components, we will allow the vector to take on an additional property which we call it’s phase:

$\psi_{\mu} \equiv B_{\mu}e^{i\theta}$(1.1)

The real-valued set B acts just like a normal vector, whereas the phase element $\theta$ is a scalar function. While we have added a new degree of freedom for vectors, it is important to note that we have NOT added another dimension to the space. This new degree of freedom will require us to adjust some definitions from differential geometry. For example, to ensure that distances are phase free (real valued), we will require the metric is a Hermitian matrix. The added degree of freedom also requires us to add an additional set of “coordinate changes” and expand on the affine connection to handle possible phase changes while a vector moves through a space.

$g_{\mu \nu}\rightarrow g_{\mu \nu^{*}} = g_{\nu \mu^{*}}$(1.2)

$l^{2} = g_{\mu \nu^{*}} \psi^{\mu} \psi^{\nu^{*}} = g_{\mu \nu}B^{\mu}B^{\nu}$(1.3)

$d \psi^{\mu} = \widehat{\Gamma}^{\mu}_{\nu \tau} \psi^{\nu} dx^{\tau}$(1.4)

Which, if we expand, gives:

$d B^{\mu} = \Gamma^{\mu}_{\nu \tau} B^{\nu} dx^{\tau}$(1.5)

$d \theta = A_{\tau} dx^{\tau} \theta$(1.6)

And hence:

$\widehat{\Gamma}^{\mu}_{\nu \tau} = \Gamma^{\mu}_{\nu \tau} + i \theta A_{\tau} \delta^{\mu}_{\nu}$(1.7)