Compositeness, Triviality and Bounds on Critical Exponents for Fermions and Magnets

We argue that theories with fundamental fermions which undergo chiral symmetry breaking have several universal features which are qualitatively different than those of theories with fundamental scalars. Several bounds on the critical indices $\delta$ and $\eta$ follow. We observe that in four dimensions the logarithmic scaling violations enter into the Equation of State of scalar theories, such as $\lambda\phi^4$, and fermionic models, such as Nambu-Jona-Lasinio, in qualitatively different ways. These observations lead to useful approaches for analyzing lattice simulations of a wide class of model field theories. Our results imply that $\lambda\phi^4$ {\it cannot} be a good guide to understanding the possible triviality of spinor $QED$.