An expression for the asymptotic level density of interacting quantum fields is formulated in terms of the functional integral. The energy, the momentum, the angular momentum, and any conserved internal quantum numbers, may be given definitely prescribed values. Of particular interest is the asymptotic mass spectrum of the bag model. It is shown that interactions, treated perturbatively, do not change the form of the asymptotic mass spectrum. Specifying the values of internal quantum numbers, such as baryon number, leads to results which are consistent with the statistical bootstrap model.