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

TL;DR

This paper develops an algebraic framework for the 1/N expansion in quantum field theory, offering a local, curvature-compatible approach that could extend to gauge theories like Yang-Mills.

Contribution

It introduces a rigorous, algebraic formulation of the 1/N expansion that avoids infra-red divergences and generalizes to curved spacetimes, applicable to gauge-invariant models.

Findings

Constructed gauge-invariant quantum field operators as power series in 1/N

Provided a local algebraic formulation avoiding infra-red issues

Extended the 1/N expansion to curved spacetime and renormalization group flow

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.