I was reading the theory of superfluidity in the language of field integrals from the book 'Condensed matter field theory' by A Altland and B Simons. The quantum partition function is given by $$Z=\int D(\rho,\phi)\exp(-S[\rho,\phi])$$ where $$S=\int d\tau \int dr[i\delta \rho \partial_\tau \phi + \frac{\rho_0}{2m}(\nabla \phi)^2+ \frac{g \rho^2}{2}].\tag{6.11}$$
It is mentioned that a gausssian integration over the density fluctuation $\delta \rho$ gives the effective action $$S=\frac{1}{2}\int d\tau \int dr[\frac{1}{g}(\partial_\tau \phi)^2 + \frac{\rho_0}{m}(\nabla \phi)^2 ]\tag{6.12}.$$
How does one perform the following integration? $$ \int_{-\infty}^{\infty} d\rho \exp(-\int d\tau \int dr [ i\delta \rho \partial_\tau \phi + \frac{\rho_0}{2m}(\nabla \phi)^2+ \frac{g \rho^2}{2}]). $$
Do we just complete the square as usual and ignore the integrals wrt time $\tau$ and $r$? Or rather perform this integration?,
$$\int_{-\infty}^{\infty} d\rho \exp(- [i\delta \rho \partial_\tau \phi + \frac{\rho_0}{2m}(\nabla \phi)^2+ \frac{g \rho^2}{2}])$$
I have attached a page for reference.
Best Answer
A&S are trying to say that one should perform a Gaussian path integral. Here the path integration variable $$\rho({\bf r},\tau)~=~\rho_0+\underbrace{\delta\rho({\bf r},\tau)}_{\text{fluctuations}}\tag{p.261}$$ is a field, i.e. a function of spacetime, so there are infinitely many Gaussian integrations: Formally speaking 1 integration for each spacetime point. (This should be regularized, e.g. via a discretization of spacetime.) A&S ignore$^1$ the functional determinant, and only sum up the action contributions (=the classical contributions) in eq. (6.12).
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$^1$ A&S also seem to have added a total derivative term $i\rho_0\partial_{\tau}\phi$ to the action (6.11).