Quantum Field - Theory Models in Less Than 4 Dimensions

The scalar ${\lambda }_0{\phi }^4$ interaction and the Fermi interaction $G_0{\left(\overline{\psi }\psi \right)}^2$ are studied for space-time dimension $d$ between 2 and 4. An unconventional coupling-constant renormalization is used: ${\lambda }_0=u_0{\Lambda }^{\epsilon }$ ($\epsilon =4-d$) and $G_0=g_0{\Lambda }^{2-d}$, with $u_0$ and $g_0$ held fixed as the cutoff $\Lambda \to \infty$. The theories can be solved in two limits: (1) the limit $N\to \infty$ where $\phi$ and $\psi$ are fields with $N$ components, and (2) the limit of small $\epsilon$, as a power series in $\epsilon$. Both theories exhibit scale invariance with anomalous dimensions in the zero-mass limit. For small $\epsilon$, the fields $\phi$, ${\phi }^2$, and $\phi {\nabla }_{{\alpha }_1}·{\nabla }_{{\alpha }_n}\phi$ all have anomalous dimensions, except for the stress-energy tensor. These anomalous dimensions are calculated through order ${\epsilon }^2$; they are remarkably close to canonical except for ${\phi }^2$. The ${\left(\overline{\psi }\psi \right)}^2$ interaction is studied only for large $N$; for small $\epsilon$ it generates a weakly interacting composite boson. Both the ${\phi }^4$ and ${\left(\overline{\psi }\psi \right)}^2$ theories as solved here reduce to trivial free-field theories for $\epsilon \to 0$. This paper is motivated by previous work in classical statistical mechanics by Stanley (the $N\to \infty$ limit) and by Fisher and Wilson (the $\epsilon$ expansion).