Wednesday, December 19, 2012

1212.4188 (Jorge Gueron et al.)

Non-perturbative λΦ^4 in D=1+1: an example of the constructive
quantum field theory approach in a schematic way

Jorge Gueron, Mauricio Leston
During the '70 there were constructed, in a non-perturbative way, several relativistic quantum field theory models in $D=1+1$ and also in $D=2+1$. That was done in the so-called constructive quantum field theory approach, whose main results have been obtained by a clever use of Euclidean functional methods. Although in the construction of a single model there are several technical steps, some of them involving long proofs, the constructive quantum field theory approach contains conceptual insights about relativistic quantum field theory that deserved to be known and which are accessible without entering in technical details. The purpose of this note is to illustrate such insights by providing an oversimplified schematic exposition of the simple case of $\lambda\Phi^4$ (with $m>0$) in $D=1+1$. Because of the absence of ultraviolet divergences in its perturbative version, this simple example -although does not capture all the difficulties in the constructive quantum field theory approach- allows to stress those difficulties inherent to the non-perturbative definition. We have make an effort in order to avoid several of the long technical intermediate steps without missing the main ideas and making contact with the usual language of the perturbative approach.
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