Effective Lagrangian

[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?

 

[vanhees71]

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).

 

[Demystifier]

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).

 

[vanhees71]

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).

 

[Demystifier]

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.

 

[vanhees71]

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!).

 

[Demystifier]

And what this has to do with "non-locality"?

Generally, an infinite series in higher and higher derivatives is an indication of possible non-locality. And even without infinite derivatives, an equation of motion with higher than second derivatives may lead to serious problems http://www.scholarpedia.org/article/Ostrogradsky's_theorem_on_Hamiltonian_instability

 

[vanhees71]

Well, yes, but you have to use effective theories in a correct way. You always do a calculation to a given finite order in energy-momentum powers. Usually you cannot solve the exact equations (which you can't even if you have a Dyson renormalizable theory with a finite number of terms in the Lagrangian) and even if you could, you cannot expect to get a well-defined physical theory.

 

[paralleltransport]

Because under RG flow you are coarse-graining the theory, you definitely lose information. My guess is that the effective lagrangian could violate causality if you tried to ask questions at energies higher than the cut-off ΛΛ above which the modes have been integrated. There's no reason why lorentz symmetry is respected at scales higher than the expected range of validity of the EFT.

For example, some non-renormalizable EFT's will violate unitary (froissard bound) at high energies, simply because the theory is being extrapolated beyond its range of validity.