People often write a complex scalar field via polar decomposition. What does this parametrization precisely mean?

To be more explicit consider the following Lagrangian of a complex scalar field with a $ U(1) $ symmetry,
\begin{equation}
{\cal L} = - m ^2 \left| \phi \right| ^2 - \frac{ \lambda }{ 4} \left| \phi \right| ^4 + \left| \partial _\mu \phi \right| ^2
\end{equation}

We can then make a field transformation,
\begin{equation}
\phi (x) = \rho (x) e ^{ i \theta (x) }
\end{equation}
We typically refer to $ \rho (x) $ and $ \theta (x) $ as real scalar fields, but this is a strange for a couple reasons. Firstly, the $ \rho (x) $ field is positive definite. This is a boundary condition that we don't normally see in QFT. Its especially weird if $ \rho $ is quantized around $0$ since it can't ``go in the negative direction''.

The second reason I think calling $ \rho (x) $ and $ \theta (x) $ quantum fields is strange is because after we write out the Lagrangian we get,
\begin{equation}
{\cal L} = - m ^2 \rho ^2 - \frac{ \lambda }{ 4} \rho ^4 + \partial _\mu \rho \partial ^\mu \rho + \rho ^2 \partial _\mu \theta \partial ^\mu \theta
\end{equation}
The $ \theta (x) $ field doesn't have a proper kinetic term and so can't be a propagating field!

If these objects aren't quantum fields then how should I think about them?

This post imported from StackExchange Physics at 2014-04-18 17:37 (UCT), posted by SE-user JeffDror