We summarize the situation regarding non-polynomial Lagrangians: I should make the qualification that an enor­ mous amount of verification is needed before the problems of renormalizability are all sorted out, but one may ten­ tatively state: 1) All matrix elements are finite for theories where the Dyson index D is less than two. 2) For the cases when D=2 or 3, counter-terms have been explicitly written which absorb all infinities and the theories are renormalizable. 3) Mixed theories of polynomial and non-polynomial fields appear to be renormalizable provided the Dyson in­ dices separately and jointly fulfill renormalizability criteria. We believe that weak interactions, chiral La­ grangians and Yang-Mills theory fall into this class though detailed proofs have not yet been constructed. 4) It seems likely that to each order in the major coupling (and to all orders in the minor coupling}the S­ matrix elements, as computed by methods outlined, satisfy the necessary unitarity and analyticity requirements.

We summarize the situation regarding non-polynomial Lagrangians: I should make the qualification that an enor­ mous amount of verification is needed before the problems of renormalizability are all sorted out, but one may ten­ tatively state: 1) All matrix elements are finite for theories where the Dyson index D is less than two. 2) For the cases when D=2 or 3, counter-terms have been explicitly written which absorb all infinities and the theories are renormalizable. 3) Mixed theories of polynomial and non-polynomial fields appear to be renormalizable provided the Dyson in­ dices separately and jointly fulfill renormalizability criteria. We believe that weak interactions, chiral La­ grangians and Yang-Mills theory fall into this class though detailed proofs have not yet been constructed. 4) It seems likely that to each order in the major coupling (and to all orders in the minor coupling}the S­ matrix elements, as computed by methods outlined, satisfy the necessary unitarity and analyticity requirements.

We summarize the situation regarding non-polynomial Lagrangians: I should make the qualification that an enor­ mous amount of verification is needed before the problems of renormalizability are all sorted out, but one may ten­ tatively state: 1) All matrix elements are finite for theories where the Dyson index D is less than two. 2) For the cases when D=2 or 3, counter-terms have been explicitly written which absorb all infinities and the theories are renormalizable. 3) Mixed theories of polynomial and non-polynomial fields appear to be renormalizable provided the Dyson in­ dices separately and jointly fulfill renormalizability criteria. We believe that weak interactions, chiral La­ grangians and Yang-Mills theory fall into this class though detailed proofs have not yet been constructed. 4) It seems likely that to each order in the major coupling (and to all orders in the minor coupling}the S­ matrix elements, as computed by methods outlined, satisfy the necessary unitarity and analyticity requirements.