# Non-relativistic complex scalar field

This is spontaneous symmetry breaking problem.

1. Homework Statement

Temperature is ##T=0##.
For one component complex scalar field ##\phi##, non-relativistic Lagrangian can be written as
$$\mathcal{L}_{NR}=\varphi^* \Big( i\partial_t + \dfrac{\nabla^2}{2m} \Big)\varphi - g(|\varphi|^2-\bar{n})^2+const.$$
where ##\varphi## is non-relativistic complex scalar field, ##g## is composed of mass ##m## and coupling strength ##\lambda##, and ##\bar{n}## is ##\frac{\mu}{2g}## (##\mu## is chemical potential).
Vacuum expectation value can be calculated by
$$\dfrac{d}{d\varphi}\Big[ g(|\varphi|^2-\bar{n})^2 \Big]=0\\ \therefore \langle |\varphi| \rangle=\sqrt{\bar{n}}e^{i\theta}.$$
Let's think of fluctuation around the ground state:
$$\varphi(x)=\Big[ \sqrt{\bar{n}}+h(x) \Big] e^{i\theta(x)} . ~~~(h(x)\ll \sqrt{\bar{n}})$$
Use equation of motion of ##h(x)## and integrate with degrees of freedom of ##h(x)##, so that we can eliminate ##h##. Then, write the leading order for effective field theory of ##\theta## and derive dispersion relation of ##\theta##.

## Homework Equations

Lagrangian for (relativistic) complex scalar field is
$$\mathcal{L}=\partial_\mu \phi (\partial^\mu \phi)^* - m^2|\phi|^2-\lambda|\phi|^4.$$
By taking non-relativistic limit, we get
$$\phi(x)=\dfrac{1}{\sqrt{2m}}e^{-imt}\varphi(t,x).$$

## The Attempt at a Solution

First of all, I have no idea what "Use equation of motion of ##h(x)## and integrate with degrees of freedom of ##h(x)##, so that we can eliminate ##h## " part means.
I guess I can handle with "write the leading order for effective field theory of ##\theta## and derive dispersion relation of ##\theta##" part, but I don't know how to eliminate ##h##.
I thought 'equation of motion' part was about Euler-Lagrange equation. I calculated and got the result:
$$i\partial_t \varphi=-\dfrac{\nabla^2}{2m}\varphi+2g(|\varphi|^2-\bar{n})\varphi.$$