Algebraic Approach to the 1/N Expansion in Quantum Field Theory

Abstract

The 1/N expansion in quantum field theory is formulated within an algebraic framework. For a scalar field taking values in the $N$ by $N$ hermitian matrices, we rigorously construct the gauge invariant interacting quantum field operators in the sense of power series in 1/N and the `t Hooft coupling parameter as members of an abstract *-algebra. The key advantages of our algebraic formulation over the usual formulation of the 1/N expansion in terms of Green's functions are (i) that it is completely local so that infra-red divergencies in massless theories are avoided on the algebraic level and (ii) that it admits a generalization to quantum field theories on general Lorentzian curved spacetimes. We expect that our constructions are also applicable in models possessing local gauge invariance such as Yang-Mills theories. The 1/N expansion of the renormalization group flow is constructed on the algebraic level via a family of *-isomorphisms between the algebras of interacting field observables corresponding to different scales. We also consider $k$-parameter deformations of the interacting field algebras that arise from reducing the symmetry group of the model to a diagonal subgroup with $k$ factors. These parameters smoothly interpolate between situations of different symmetry.

52 pages, Latex, 4 figures, includes self-contained review of the operator algebraic approach to perturbatively defined quantum field theory in Minkowski spacetime, updated version