Sufficient conditions for quasifree states and an improved uniqueness theorem for quantum fields on space–times with horizons
Sufficient conditions for quasifree states and an improved uniqueness theorem for quantum fields on space–times with horizons
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准自由态的充分条件和视界时空量子场改进的唯一性定理
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发表时间：
1993
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通讯作者：
B. S. Kay
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作者：
B. S. Kay
令 ω 为辛空间上外尔代数的状态。我们证明，如果 (i) ω 的“解放”是纯的，或者 (ii) ω 对两个生成 Weyl 子代数中的每一个的限制是拟自由和纯的，则 ω 是拟自由和纯的[并且，如果 (i) 等于其解放，则 (ii) 由其限制唯一确定]。 [在这里，我们将（足够规则的）态的解放定义为具有相同两点函数的准自由态。]结果（i）和（ii）允许人们放弃由于 Wald 和作者关于时空上具有分岔杀伤视界的线性标量量子场的结果中的准自由假设，从而得出结论，在这样一个系统的场代数的大子代数上，存在一个唯一的稳态，其两个点函数具有 Hadamard 形式。该论文包含许多进一步的相关进展，包括：（a）（i）暗示了唯一性结果，例如，对于闵可夫斯基空间中的通常自由场。我们将此与该系统的其他已知独特性结果进行比较和对比。 （b）对于“准自由”状态和“自由”，证明了与（i）和（ii）类似的一对结果，其中准自由的定义与我们这里所说的准自由不同，因为允许非零的单点函数，并且状态的自由被定义为具有相同的一和两个点函数的准自由状态。 (c) 我们对规范反交换关系得出了类似的结果。
Let ω be a state on the Weyl algebra over a symplectic space. We prove that if either (i) the ‘‘liberation’’ of ω is pure or (ii) the restriction of ω to each of two generating Weyl subalgebras is quasifree and pure, then ω is quasifree and pure [and, in case (i) is equal to its liberation, in case (ii) is uniquely determined by its restrictions]. [Here, we define the liberation of a (sufficiently regular) state to be the quasifree state with the same two point function.] Results (i) and (ii) permit one to drop the quasifree assumption in a result due to Wald and the author concerning linear scalar quantum fields on space–times with bifurcate Killing horizons and thus to conclude that, on a large subalgebra of the field algebra for such a system, there is a unique stationary state whose two point function has the Hadamard form. The paper contains a number of further related developments including: (a) (i) implies a uniqueness result, e.g., for the usual free field in Minkowski space. We compare and contrast this with other known uniqueness results for this system. (b) A similar pair of results to (i) and (ii) is proven for ‘‘quasiFree’’ states and ‘‘libeRations’’ where the definition of quasiFree differs from what we call here quasifree in that nonvanishing one point functions are permitted, and the libeRation of a state is defined to be the quasiFree state with the same one and two point functions. (c) We derive similar results for the canonical anticommutation relations.