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Starting with a classical Ginzburg-Landau theory of a superconductor, $\mathcal{L} = a |\nabla\psi|^2 + b |\psi|^2 + c |\psi|^4 +...$. suppose I want to include quantum (i.e temporal) fluctuations. What term should I add? It seems to me that there are many options.

  1. If the system is Lorentz invariant, then surely we just add a $|\partial_\tau \psi|^2$ term.
  2. If it's a Galilean invariant superfluid such as Helium, I guess we add a $\psi^\dagger \partial_\tau \psi$ term.

But what if the system is not Galilean or Lorentz invariant? Is there some generic answer?

Also, often, the normal state is a Fermi-liquid. If we take non-interacting fermions and calculate the superconducting correlations, we should be able to estimate the parameters of the GL theory. But aren't these correlations singular at long times (small frequencies)? Then shouldn't the dynamical term be long ranged in time?

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