一维盒子中标量辐射局部热行为的不确定性
Ambiguities in the local thermal behavior of the scalar radiation in one-dimensional boxes
摘要：
本文报道了在计算平坦时空中一维盒子中任意耦合的无质量标量场的应力-能量-动量张量的系综平均$\left<T_μ_ν\right>$时的某些不确定性。该研究解决了一个盒子与周期性的边界条件（圆）和盒子反射边缘（与狄利克雷或诺依曼的边界条件在端点）。由有限温度格林函数得到了$\left<T^μ^ν\right>$的表达式。在附录中，为了控制发散典型的二维，这些格林函数计算的相关背景与任意数量的维度和任意质量的标量场，并专门在文本中的二维和无质量的领域。由于在$\left<T^μ^ν\right>$中存在不绝对收敛的二重级数，因此产生了歧义。两个相关求和的计算顺序很重要，导致每种盒子都有两种不同的热力学。在圆的情况下，它示出的模糊性对应于经典的争论在文献中是否零模式的贡献应考虑在计算的配分函数。在具有反射边的盒子的情况下，结果是，其中一个热力学对应于总能量（通过对空间上的非均匀能量密度积分得到），该总能量不像预期的那样依赖于曲率耦合参数$ξ$;而另一个热力学奇怪地对应于依赖于$ξ$的总能量。热力学要求（如局部和全局稳定性）和它们的限制值的$ξ$被认为是。
英文摘要：
This paper reports certain ambiguities in the calculation of the ensemble average $\left<T_μ_ν\right>$ of the stress-energy-momentum tensor of an arbitrarily coupled massless scalar field in one-dimensional boxes in flat spacetime. The study addresses a box with periodic boundary condition (a circle) and boxes with reflecting edges (with Dirichlet's or Neumann's boundary conditions at the endpoints). The expressions for $\left<T^μ^ν\right>$ are obtained from finite-temperature Green functions. In an appendix, in order to control divergences typical of two dimensions, these Green functions are calculated for related backgrounds with arbitrary number of dimensions and for scalar fields of arbitrary mass, and specialized in the text to two dimensions and for massless fields. The ambiguities arise due to the presence in $\left<T^μ^ν\right>$ of double series that are not absolutely convergent. The order in which the two associated summations are evaluated matters, leading to two different thermodynamics for each type of box. In the case of a circle, it is shown that the ambiguity corresponds to the classic controversy in the literature whether or not zero mode contributions should be taken into account in computations of partition functions. In the case of boxes with reflecting edges, it results that one of the thermodynamics corresponds to a total energy (obtained by integrating the non homogeneous energy density over space) that does not depend on the curvature coupling parameter $ξ$ as expected; whereas the other thermodynamics curiously corresponds to a total energy that does depend on $ξ$. Thermodynamic requirements (such as local and global stability) and their restrictions to the values of $ξ$ are considered.
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