Anderson-Higgs mechanism for the (non-relativistic) $U(1)$ gauge theory under the unitarity gauge

Question

On Page 138, Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons by Xiaogang Wen, when he demonstrates the Anderson-Higgs mechanism for the $U(1)$ gauge theory, he starts with the general (real time) Lagrangian

$${\cal{L}} ~=~ \frac{i}{2}\left(\varphi^*(\partial_t + iA_0)\varphi -\varphi(\partial_t - iA_0)\varphi^*\right) - \frac{1}{2m} |(\partial_i +i A_i) \varphi|^2 $$ $$ + \mu |\varphi|^2 -\frac{V_0}{2}|\varphi|^4 + \frac{1}{8\pi e^2}(\mathbf{E}^2 -\mathbf{B}^2), \tag{3.7.5}$$ with $c=1$. (I wonder why this is the correct non-relativistic form because in my derivation I always have a term $A_0^2|\phi|^2/2m$.)

Then he chooses the gauge such that $\varphi$ is real (unitarity gauge according to Peskin and Schroeder) and obtains

$${\cal{L}} ~=~ -A_0 \phi^2 - \frac{1}{2m} (\partial_i \phi)^2 -\frac{\phi^2}{2m}A_i^2 + \mu \phi^2 -\frac{V_0}{2}\phi^4 $$ $$+ \frac{1}{8\pi e^2}(\mathbf{E}^2 -\mathbf{B}^2).\tag{3.7.16b}$$

He claims that if we have $\phi = \phi_0 +\delta \phi$ and integrate the small fluctuation $\delta \phi$, we can get

$${\cal{L}} ~=~ \frac{A_0^2}{2V_0} -\frac{\rho A_i^2}{2m} + \frac{1}{8\pi e^2}(\mathbf{E}^2 -\mathbf{B}^2).\tag{3.7.17}$$

I am curious what approximations he has done to get here.

Any help is appreciated.

This post imported from StackExchange Physics at 2015-04-15 10:43 (UTC), posted by SE-user L. Su