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On page 24 of Baym and Pethick's Landau Fermi-Liquid Theory book, they mention that the stress tensor is given by

$$ \Pi_{ij}=T_{ij}+\delta_{ij}\left(\sum_{\sigma}\int \frac{d^3 p}{(2\pi \hbar)^3}\epsilon_{p\sigma}n_{p\sigma}-E\right)$$

where

$$T_{ij}=\sum_{\sigma}\int \frac{d^3 p}{(2\pi \hbar)^3}p_i\frac{\partial \epsilon_{p\sigma}}{\partial p_j}n_{p\sigma}$$

Is it correct to say that this stress tensor is identical to the stress energy tensor that corresponds to the current that comes from imposing spacetime invariance of some quantum field? That is, given $x^\mu\rightarrow x^\mu+\epsilon^\mu(x)$, we have a conserved current given by

$$j_\mu=T_{\mu \nu}\epsilon^\mu$$

Is $\Pi_{ij}$ in the Fermi liquid identical to the $T_{\mu \nu}$ in the field theory? If they are different, then what would be the energy-momentum tensor $T_{\mu\nu}$ of a Fermi liquid?

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I think that this is indeed a very interesting question. We certainly believe that the answer is "yes", but I don't think that there are any formal proofs (other than simple, perturbative, examples). A few comments:

1) Deriving Landau Fermi Liquid Theory (FLT) from QFT is a non-trivial exercise. This is because in QFT we do not express thermodynamic quantities and $n$-point functions as functionals of non-equilibrium occupation numbers. This is problem is tackled in a number of text books and reviews, see for example, Abrikosov, Polchinski, Shankar, or this review.

2) The relation $j_\mu=T_{\mu\nu}\epsilon^\mu$ applies to the the Noether stress tensor. In general, we would expect that if the theory is invariant under diffeomorphisms then the stress tensor is given by $$ T^{\mu\nu} = \frac{1}{\sqrt{-g}}\frac{\delta S}{\delta g_{\mu\nu}} $$

3) Note that FLT is a non-relativistic, Galilean invariant, theory. Non-relativistic diffeo invariance, and the associated stress tensor have been studied by Son, Jensen, and others. Generalizations of FLT to relativistic theories have also been studied, but this is a complicated story because relativistic theories are typically gauge theories, and gauge interactions destroy FLT behavior.

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