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Borchers' Commutation Relations for Sectors with Braid Group Statistics in Low Dimensions

机译：具有低维编织群统计量的部门的Borchers换向关系

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摘要

Borchers has shown that in a translation covariant vacuum representation of a theory of local observables with positive energy the following holds: The (Tomita) modular objects associated with the observable algebra of a fixed wedge region give rise to a representation of the subgroup of the Poincare group generated by the boosts and the reflection associated to the wedge, and the translations. We prove here that Borchers' theorem also holds in charged sectors with (possibly non-Abelian) braid group statistics in low space-time dimensions. Our result is a crucial step towards the Bisognano-Wichmann theorem for Plektons in d = 3, namely that the mentioned modular objects generate a representation of the proper Poincare group, including a CPT operator. Our main assumptions are Haag duality of the observable algebra, and translation covariance with positive energy as well as finite statistics of the sector under consideration.
机译：Borchers已经证明，在具有正能量的局部可观察物理论的平移协变真空表示中，以下条件成立：与固定楔形区域的可观察代数关联的（Tomita）模块化对象产生了Poincare子组的表示由与楔形相关的增强和反射以及平移生成的组。我们在这里证明，Borchers定理在具有低时空维度的（可能是非阿贝尔）编织群统计信息的带电部门中也成立。我们的结果是在d = 3的情况下朝Plektons的Bisognano-Wichmann定理迈出的关键一步，即所提到的模块化对象生成了包括CPT算子在内的适当Poincare组的表示。我们的主要假设是可观察代数的Haag对偶性，具有正能量的平移协方差以及所考虑领域的有限统计量。

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