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I'd like to understand the limits to which effective field theory works to sweep microscopic details under the rug in quantum many body problems.

In standard many body text, we start with the picture of second quantization. We start with a many body hamiltonian: $$ H = {\sum_i} {{p_i}^2 \over 2 m_i} + V(x_i) + \sum_{ij} V^{(2)}(x_i-x_j) + \text{higher order} $$ We then re-express it as a quantum field problem where all the physics is encoded in a path integral: $$ Z(J) = \int d \psi e^{-\int d^d x \mathcal{L}(\psi(x))}$$

We then use effective field theory, or the principle of selective inattention: since the distance from the UV to the IR is large, we hope all irrelevant terms go away and we can do physics by just guessing the terms consistent with the symmetries in the IR.

  1. Is the above argument correct? Are there any loopholes to the step between schrodinger's equation to field theory representation...

  2. What caveats are there to applying effective field theory in condensed matter systems that do not apply for usual high energy problems? (lattice issues? weakly irrelevant operators that affects IR physics?)

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