四维共形量子场论中的无限维李代数
Infinite dimensional Lie algebras in 4D conformal quantum field theory
摘要：
全局共形不变性（GCI）的概念开辟了将二维手征共形场论中发展起来的代数方法应用于更高维时空的途径。特别地，共形维数为2的GCI标量场系统产生调和双局部场的李代数V_m（x，y），其中m跨越在转置下闭的有限维实矩阵代数M。结合代数M是不可约的当且仅当它的交换子M'与三个实除环之一重合。双局部域（的模）的李代数在每种情况下都是无限维李代数：sp（infty，R）的中心扩张对应于实数域R，u（infty，infty）与复数域C相关，so*（4 infty）与四元数代数H相关。它们产生了量子场论模型，其超选择扇区分别由（全局）规范群O（N）、U（N）和U（N，H）=Sp（2N）支配。
英文摘要：
The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of 2-dimensional chiral conformal field theory, to a higher (even) dimensional space-time. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, V_m(x,y), where the m span a finite dimensional real matrix algebra M closed under transposition. The associative algebra M is irreducible iff its commutant M' coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite dimensional Lie algebra: a central extension of sp(infty,R) corresponding to the field R of reals, of u(infty,infty) associated to the field C of complex numbers, and of so*(4 infty) related to the algebra H of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N), and U(N,H)=Sp(2N), respectively.
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