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 infrared divergencies in massless theories are avoided on the algebraic level and (ii) that it admits a generalization to quantum field theories on globally hypberbolic 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.
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