论相对论量子场论中的可分态
On separable states in relativistic quantum field theory
AI总结：
本文研究了相对论量子场论中类空分离区域间可分态的存在性及其所需的最小能量密度，证明了可分准自由 Hadamard 态的存在性，并在 Minkowski 时空中为有质量标量场构造了具有特定能量密度上界的平稳可分态。
摘要：
我们开启了对相对论量子场论中可分但物理上合理的态的研究。特别地，我们将考虑为确保给定类空分离区域之间存在可分态所需的最小能量密度。这是增进我们对纠缠熵与能量（密度）之间平衡理解的第一步，该平衡本身具有极大的物理意义，在黑洞热力学背景下也是如此。我们将具体关注拓扑平凡的四维全局双曲时空中的线性标量量子场。对于相当一般的类空分离区域 $A$ 和 $B$，我们证明了可分准自由 Hadamard 态的存在性。在 Minkowski 时空中，我们为有质量自由标量场提供了更严格的构造：给定任意 $R>0$，我们构造一个准自由 Hadamard 态，它是平稳、均匀、空间各向同性的，并且在惯性时间切片 $t=\mathrm{const.}$ 上任意两点距离 $>R$ 的两个区域之间是可分的。我们还证明，这些态的正规序能量密度可以做到 $\le 10^{31}\frac{m^4}{(mR)^8}e^{-\frac14mR}$（普朗克单位制）。为获得这些结果，我们使用了正型测试函数 $f$ 的一种相当显式的构造，借此我们能对 $\hat{f}$ 的下界获得充分控制。
英文摘要：
We initiate an investigation into separable, but physically reasonable, states in relativistic quantum field theory. In particular we will consider the minimum amount of energy density needed to ensure the existence of separable states between given spacelike separated regions. This is a first step towards improving our understanding of the balance between entanglement entropy and energy (density), which is of great physical interest in its own right and also in the context of black hole thermodynamics. We will focus concretely on a linear scalar quantum field in a topologically trivial, four-dimensional globally hyperbolic spacetime. For rather general spacelike separated regions $A$ and $B$ we prove the existence of a separable quasi-free Hadamard state. In Minkowski spacetime we provide a tighter construction for massive free scalar fields: given any $R>0$ we construct a quasi-free Hadamard state which is stationary, homogeneous, spatially isotropic and separable between any two regions in an inertial time slice $t=\mathrm{const.}$ all of whose points have a distance $>R$. We also show that the normal ordered energy density of these states can be made $\le 10^{31}\frac{m^4}{(mR)^8}e^{-\frac14mR}$ (in Planck units). To achieve these results we use a rather explicit construction of test-functions $f$ of positive type for which we can get sufficient control on lower bounds on $\hat{f}$.
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