Quantum Field - Theory Models in Less Than 4 Dimensions

The scalar λ₀φ⁴ interaction and the Fermi interaction G₀(ψ¯ψ)² are studied for space-time dimension d between 2 and 4. An unconventional coupling-constant renormalization is used: λ₀=u₀Λ^ɛ (ɛ=4-d) and G₀=g₀Λ^2-d, with u₀ and g₀ held fixed as the cutoff Λ-->∞. The theories can be solved in two limits: (1) the limit N-->∞ where φ and ψ are fields with N components, and (2) the limit of small ɛ, as a power series in ɛ. Both theories exhibit scale invariance with anomalous dimensions in the zero-mass limit. For small ɛ, the fields φ, φ², and φ∇_α1.∇_αnφ all have anomalous dimensions, except for the stress-energy tensor. These anomalous dimensions are calculated through order ɛ² they are remarkably close to canonical except for φ². The (ψ¯ψ)² interaction is studied only for large N; for small ɛ it generates a weakly interacting composite boson. Both the φ⁴ and (ψ¯ψ)² theories as solved here reduce to trivial free-field theories for ɛ-->0. This paper is motivated by previous work in classical statistical mechanics by Stanley (the N-->∞ limit) and by Fisher and Wilson (the ɛ expansion).