{
  "id": "math-ph/0309042",
  "version": "v2",
  "published": "2003-09-17T19:49:52.000Z",
  "updated": "2004-04-19T20:40:12.000Z",
  "title": "Algebraic Approach to the 1/N Expansion in Quantum Field Theory",
  "authors": [
    "Stefan Hollands"
  ],
  "comment": "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",
  "journal": "Rev.Math.Phys. 16 (2004) 509",
  "doi": "10.1142/S0129055X04002072",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-04-19T20:40:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81T15",
      "81T05"
    ],
    "keywords": [
      "quantum field theory",
      "algebraic approach",
      "gauge invariant interacting quantum field",
      "invariant interacting quantum field operators",
      "models possessing local gauge invariance"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Reviews in Mathematical Physics",
      "year": 2004,
      "volume": 16,
      "number": 4,
      "pages": 509
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 628534,
      "adsabs": "2004RvMaP..16..509H"
    }
  }
}