Algebraic Approach to the 1/N Expansion in Quantum Field Theory
Stefan Hollands
备注：
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
量子场论中的1/N展开是在一个代数框架内建立的。对于取值于$N$ by $N$厄米矩阵的标量场，我们严格地构造了1/N幂级数意义下的规范不变相互作用量子场算符和耦合参数t Hooft作为抽象 *-代数的成员.我们的代数公式的关键优势，通常制定的1/N展开的格林函数是（i），它是完全本地的，使红外发散的无质量理论是在代数水平上避免和（ii），它承认一般的洛伦兹弯曲时空的量子场论的推广。我们期望我们的构造也适用于具有局部规范不变性的模型，如杨-米尔斯理论。通过对应于不同尺度的相互作用场观测量代数之间的 *-同构族，在代数水平上构造了重整化群流的1/N展开。我们还考虑了$k$参数变形的相互作用场代数所产生的对称群的模型减少到一个对角子群的$k$因素。这些参数在不同对称性的情况之间平滑地插值。
英文摘要：
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.
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