# Non-relativistic complex scalar field

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1. Jul 11, 2017

### Ken Gallock

This is spontaneous symmetry breaking problem.

1. The problem statement, all variables and given/known data

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 $\theta$.

2. Relevant 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
$$\phi(x)=\dfrac{1}{\sqrt{2m}}e^{-imt}\varphi(t,x).$$

3. 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:
$$i\partial_t \varphi=-\dfrac{\nabla^2}{2m}\varphi+2g(|\varphi|^2-\bar{n})\varphi.$$

2. Jul 16, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.