Exactly solvable quantum mechanical models with St\"uckelberg divergences

O. Yu. Shvedov

Teoretičeskaâ i matematičeskaâ fizika, Tome 125 (2000) no. 1, pp. 91-106

Résumé

We consider an exactly solvable quantum mechanical model with an infinite number of degrees of freedom that is an analogue of the model of $N$ scalar fields $(\lambda/N)(\varphi^a\varphi^a)^2$ in the leading order in $1/N$. The model involves vacuum and $S$-matrix divergences and also the Stückelberg divergences, which are absent in other known renormalizable quantum mechanical models with divergences (such as the particle in a $\delta$-shape potential or the Lee model). To eliminate divergences, we renormalize the vacuum energy and charge and transform the Hamiltonian by a unitary transformation with a singular dependence on the regularization parameter. We construct the Hilbert space with a positive-definite metric, a self-adjoint Hamiltonian operator, and a representation for the operators of physical quantities. Neglecting the terms that lead to the vacuum divergences fails to improve and, on the contrary, worsens the renormalizability properties of the model.