# Effective Lagrangian

• A
emanaly
As I have read, the effective Lagrangian are non local sometimes, does that mean they break causlaity ? Are they non local because the heavy particles ( propagators) are integrated out?

Gold Member
2022 Award
Do you have a reference? All effective relativistic models I know of are local in the usual sense (the Hamilton density depends only on one spacetime argument).

ohwilleke
Gold Member
Do you have a reference? All effective relativistic models I know of are local in the usual sense (the Hamilton density depends only on one spacetime argument).
See e.g. Schwartz, QFT and the Standard Model, Eqs. (22.2), (22.3), (22.6), (33.5).

ohwilleke
Gold Member
2022 Award
And what this has to do with "non-locality"? (22.6) is a local Lagrangian as it must be. It doesn't matter that it consists of an infinite number of terms as it must for any "effective theory" that is not Dyson renormalizable. It's renormalizable in a more general sense, i.e., order by order in the expansion wrt. powers of energy-momentum scales (which must be small compared to some "cut-off scale", beyond which the theory is not valid anymore), with counterterms obeying the underlying symmetries (like chiral symmetry for effective models of hadrons).

ohwilleke
Gold Member
As I have read, the effective Lagrangian are non local sometimes, does that mean they break causlaity ?
They are derived from local causal Lagrangians, so their physical effect should not break causality. Typically the effective theory of this kind contains an infinite number of terms with higher and higher derivatives. If you truncate the series by retaining only a finite number of terms, then you may get a violation of causality at high energies, but this may be irrelevant when you apply the effective theory at low energies only. In any case, if you do a resummation of the whole series, the acausality problems should go away.

ohwilleke and vanhees71
Gold Member
2022 Award
That's indeed why effective theories always have an "energy cut-off", and are only valid for energies (very much) below that cut-off. For the same reason you also have to resum to restore unitarity of the S-matrix (leading to "unitarized effective theories", though a part of the physics community doesn't like this, because they only accept the order-by-order argument, because the resummation is not considered a controlled approximation).

One should also be aware that the series of perturbation theory (including this series of powers of energy-momentum scales of effective theories) are usually asymptotic series (with a convergence radius of 0!).

ohwilleke and Demystifier
Gold Member
ohwilleke, Astronuc and vanhees71