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I have a naive question about the meaning of effective theory in particle physics and condensed matter physics.

In particle physics, from what I know, the effective theory comes from the Wilsonian renormalization. We start from a theory with a fundamental cutoff $\Lambda$, e.g. Planck scale, and consider a smaller cutoff $\Lambda'$, e.g. LHC scale, $\Lambda'<\Lambda$. The Lagrangian then reshuffle into an infinity series including all possible powers of fields. The theory with $\Lambda'$ cutoff is called effective theory (possibly only a few leading terms)

In condensed matter theory, I heard the effective theory means neglect some details, like the BCS theory.

Roughly speaking, "the effective" in both fields has similar meaning, that we ignore something ($>\Lambda'$ in particle physics or uninteresting detail in condensend matter).

My question is, strictly speaking, does the two "effective" in particle and condensed physics have the same meaning? Can we start from QED, lower the cutoff, lead to BCS theory? (perhaps such crazy attempt will never work out...) or any deeper connection than heuristic similarities?

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Yes, the term "effective action" has the same meaning in particle physics and condensed matter physics. After all, the discovery of the concepts of the Renormalization Group by Ken Wilson and others was being done simultaneously in both disciplines and the exchange of ideas was fruitful in both directions.

On the other hand, what the effective theories actually are depends on the context i.e. on the disciplines, too. In particle physics, people study effective actions for excitations of the vacuum – most of the environment is the vacuum. That's why any effective action is still a relativistic theory. On the other hand, condensed matter physics studies dynamics in a different environment or medium – various solids – so the Lorentz symmetry is spontaneously broken and one obtains non-relativistic effective field theories.

But the Cooper pairs in the BCS theory are exactly analogous to hadrons in particle physics – two examples of degrees of freedom in an effective theory that is valid beneath an energy scale but ignores some detailed phenomena that are only seen with high energies (per excitation).

In particle physics, the quantity deciding about the scale is roughly speaking $|p_\mu|$ – in the $c=1$ units, it doesn't matter which component of the energy-momentum we consider. In condensed matter physics, one must distinguish them because they don't obey $E\sim pc$ in general due to the breaking of relativity by the environment. The usual quantity deciding about the scale is the energy, not the momentum.

One must also be ready for different conventions. In condensed matter physics, they use the opposite sign of the beta-function for running couplings, for example, as they imagine the flow to go in the opposite direction. This is a purely sociological difference, much like the difference between mostly-plus and mostly-minus metric tensors.

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Thank you very much for your explanations. >Lorentz symmetry is spontaneously broken, which Lorentz symmetry is broken in condensed matter? parity? – user26143 Sep 17 '13 at 17:22

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