RENORMALIZED SQUARES OF BOSON FIELDS

Abstract

The standard renormalization procedure consists of introducing a cutoff and then trying to remove it by some limiting procedure. In Ref. 2 a new renormalization technique was introduced based on the idea of renormalizing a closed set of commutation relations and then finding a nontrivial representation for them. In Ref. 5 it was proved that, in the case of quadratic fields the new renormalization procedure leads to quadratic field operator which is gamma distributed in the quadratic vacuum (as one would intuitively expect from the "square" of a white noise) and to Meixner or Pascal distributed Poisson fields. It is natural to ask if the same result can be obtained with the usual cutoff and take-limit procedure. In this paper we prove that the answer to this question is negative. More precisely, we show that, independently of the choice of the cutoff (cf. Sec. 7), if a quadratic field admits a limit in the sense of mixed moments, then this limit will be Gaussian distributed in the vacuum and consequently the associated Poisson fields will have a Poisson distribution.