In a BCS superconductor, by Hubbard-Stratonovich transformation, we can integrate out the electrons and get an effective theory for Cooper pairs like $$ S = \int \mathrm{d} \tau \mathrm{d}^3 x \left( (\nabla \Delta)^2 + \frac{1}{2} r \bar{\Delta} \Delta + u |{\Delta}|^4 + \cdots \right), $$ for example see Eq. (6.32) in Altland and Simons. We can then note that there is not a term like $\bar{\Delta} \partial_\tau \Delta$. That is strange, because it means that we cannot get an effective Hamiltonian of Cooper pairs by Legendre transformation, and that the classical motion of Cooper pairs is always no time evolution at all. Are these conclusions really correct for Cooper pairs?
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1$\begingroup$ Eq. (6.32) in Altland and Simons does have temporal derivative terms, they are just contained in the function they call $\mathcal{O}$. If your original electron lagrangian had a time-derivative term (as it must), after a Hubbard-Stratonovich transformation and integrating out the electrons should generate time-derivative terms for the HB field $\Delta$. $\endgroup$– Seth WhitsittCommented Feb 2, 2022 at 4:30
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