How does interacting Lagrangian have form of product of fields?

ndung200790

Please teach me this problem:
It seem that following Haag's theorem there not exist quantized equation of motion for interacting fields.So I dont understand how to know the form of interacting Lagrangian has form of product of fields(example Lagrangian of Fermi field interacting with electromagnetic field).
Thank you very much for advance.

strangerep

It just comes from quantizing the classical theory.

That's not quite what Haag's theorem says. The free representation and the interacting
representation are both constructed to be Poincare representations, but Haag's theorem
basically means you can't (rigorously, nonperturbatively) express the latter in terms of
the former. Getting around the manifestations of this is one of the reasons for
renormalization.

ndung200790

So,how to derive phi-4 interacting Lagrangian and Yukawa interaction,because there are not coresponding classical theory.
Please give me a favour to explain again.
Thank you very much.

dextercioby

There are some rules you should follow in the absence of gauge invariance when it comes to building interaction terms. One of this is locality, namely the power of the fields be finite. And then you can use the concept of relevance judged by whether the interacting theory is normalizable in 4D or not. By this judgement, we rule out the phi-3,5,6,.. theories in 4D either because they're not normalizable, or, if they are, they have no physical application so far. That's why we-re left with the phi-4 case which is normally thoroughly analyzed in the serious books.