Quantum filed theory in 1+1 dimension

[paweld]

Is it true that in 1+1 dimensional Minkowski spacetime scalar quantum filed theory defined
by the lagrangian (in the interaction picture, so that the normal ordering makes sense):
$$\mathcal{L} = : \frac{1}{2} (\partial_\mu \phi) (\partial^\mu \phi) - \frac{1}{2} m^2 \phi^2 - \frac{1}{4!} \lambda \phi^2 :$$
is finite, i.e. all Feynman graphs which can be constructed in this theory give finite result?

What about the series one obtains summing corrections coming from all orders of loop expansion?
Is there any proof that in case of this theory the perturbation series is convergent?

It is said that the people who work in constructive quantum filed theory managed to show
the existence of the interacting filed in 1+1 and 2+1 dimension. What is the the idea of
their proof?

 

[azntoon]

The answer to the first question is yes, this theory is finite. All Feynman graphs constructed in this theory give finite results as a result of the normal ordering in the lagrangian.The second question is a bit more complicated. There is no general proof that the perturbation series is convergent for this theory, as it depends on the particular values of the mass and coupling constant. However, there is theoretical evidence to suggest that the perturbation series is convergent in certain cases.The idea behind the constructive quantum field theory is to find a set of self-consistent axioms which define a quantum field theory in a given dimension. This is done by carefully analyzing the analytic structure of the theory, such as the renormalization group flow and scattering amplitudes. By studying these properties, the constructive quantum field theorists are able to construct a theory which is both consistent and finite.