Is the Interaction in the Canonical Formalism Lorentz Invariant?

[Roumpedakis]

In the attached file I have outlined the renormalization procedure for a very simple QFT as described in Weinberg's book and it seems that there is a contradiction. I have red some QFT books but I still have this question. Still I believe that this book is by far the most clear book that I have red. Also in order to avoid some answers the origin of the problem starts in the proof of the LSZ formula and the fact that we cannot impose the asymtpotic condition in the strong sense (this is mentioned in some books: Itzykson-Zuber, Magiorre... but the correct proof is not given)

 

[vanhees71]

I don't know, what's the question. It's a bit strange to work in the canonical formalism and expect a manifestly Poincare covariant formulation. There's no simple way to achieve a manifestly covariant Hamiltonian formulation already in classical relativistic physics (mechanics and field theory). That's why it's easier to use the path-integral formalism, which very often leads to a formulation in terms of the Lagrange formalism, which is manifestly covariant, and this leads to manifestly covariant Feynman rules for the (connected) n-point Green's functions, which however are not observables.

Observables are S-matrix elements, describing the transition-probability rates for observing an asymptotic free final state, given the asymptotic free initial state (usually two colliding particles in HEP experiments). These are always manifestly covariant, provided you work with a local microcausal field theory. For the details, see the excellent treatment in Weinberg's vol. 1 (the first few chapters are sufficient).

It turns out that the S-matrix elements are indeed covariant objects, no matter whether you use the canonical (Hamiltonian) or the path-integral formalism, where already the connected n-point functions are manifestly covariant. The latter way is much more convenient to deal with in practice. For the path-integral formalism, see Bailin and Love, Gauge Theories.

My own try to explain QFT you can find here:

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

 

[Roumpedakis]

First of all thank you very much for your reply! I appreciate it !

I agree that in the path integral formalism all these questions don't arise... So I'm talking about the canonical formalism.

The question is that using the standard canonical formalism you write down the interaction (starting from a Lagrangian) and then using the Dyson series you begin to calculate. After renormalization as in the file that I uploaded the interaction is not Lorentz invariant (again I know that in the path integral everything is perfect, no problems). So the question is: Is this correct, the interactions in not Lorentz invariant? How can we show that at the end the symmetry is restored in the quantities that we measure (cross sections, decay rates)?

The confusing thing is that even in the canonical formalism we always take the interaction to be V=δZ (∂φ) ² + (other counterterms)+(interactions)...
So it seems to me that we take it to be Lorentz invariant but if you calculate it step by step (as in the pdf) is not...

BTW I have an objection regarding your notes. In equation 3.103 page 66 you impose the asymptotic condition in the strong sense (as an operator equation). Then if we take the time-derivative and then impose the standard commutation relations it turns out that Z=1. And that's because we cannot impose it as condition on operators (see Itzykson-Zuber). So this equation is problematic.