MODULAR STRUCTURE OF THE LOCAL ALGEBRAS ASSOCIATED WITH THE FREE MASSLESS SCALAR FIELD-THEORY
MODULAR STRUCTURE OF THE LOCAL ALGEBRAS ASSOCIATED WITH THE FREE MASSLESS SCALAR FIELD-THEORY
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DOI：
10.1007/bf01208372
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发表时间：
1982-01-01
影响因子：
2.4
通讯作者：
LONGO, R
中科院分区：
文献类型：
--
作者：
HISLOP, PD;LONGO, R
本文描述了任意维数自由无质量标量场理论真空表象中与双锥相关的局部可观量的冯诺依曼代数的模结构。模自同构群是由Minkowski空间上的一族广义分式线性变换的酉实现导出的，是共形群的子群。模共轭算子是时间反演和相对论射线反演的产物的反幺正实现。双锥域族所对应的局部代数的模共轭算子所生成的群是真共形变换群。给出了一个定理，证明了与光锥、双锥和楔形区域相关的局部代数的酉等价性。对于双锥代数，这提供了类空对偶的明确实现，并建立了已知的III1型因子性质。证明了光锥代数的类时对偶性质对双锥代数不成立。引入了与区域相关联的von Neumann代数的一个不同的定义，该定义与光锥或双锥区域的标准定义一致，但允许双锥代数具有类时对偶性质.在一维情形下，与双锥区域相关联的标准局部代数同时满足类空和类时对偶。
The modular structure of the von Neumann algebra of local observables associated with a double cone in the vacuum representation of the free massless scalar field theory of any number of dimensions is described. The modular automorphism group is induced by the unitary implementation of a family of generalized fractional linear transformations on Minkowski space and is a subgroup of the conformal group. The modular conjugation operator is the anti-unitary implementation of a product of time reversal and relativistic ray inversion. The group generated by the modular conjugation operators for the local algebras associated with the family of double cone regions is the group of proper conformal transformations. A theorem is presented asserting the unitary equivalence of local algebras associated with lightcones, double cones, and wedge regions. For the double cone algebras, this provides an explicit realization of spacelike duality and establishes the known typeIII1factor property. It is shown that the timelike duality property of the lightcone algebras does not hold for the double cone algebras. A different definition of the von Neumann algebras associated with a region is introduced which agrees with the standard one for a lightcone or a double cone region but which allows the timelike duality property for the double cone algebras. In the case of one spatial dimension, the standard local algebras associated with the double cone regions satisfy both spacelike and timelike duality.