再论相对论复合弦的Casimir理论及标量QFT中的一个形式相关问题
Casimir Theory of the Relativistic Composite String Revisited, and a Formally Related Problem in Scalar QFT
摘要：
本文的主要内容是对零温和有限温下分段均匀闭弦横向振动的Casimir能量进行了新的评述。我们使用三种不同的正则化：截断法，复轮廓积分法和zeta函数法。弦模型是相对论性的，在这个意义上，每个弦段的声速都等于光速。在这个意义上，该理论类似于介电介质中的电磁理论，其中介电常数和磁导率的乘积等于1（等折射介质）。我们演示了如何工作的形式主义的两件字符串，并为2N件字符串，并显示如何在后一种情况下，一个紧凑的递归关系，以促进形式主义相当。卡西米尔能量是负的，而且越是负的，弦中的碎片就越多。两段弦在D维时空中是量子化的，在两种张力之比非常小的极限。我们计算的自由能和其他热力学量，证明标度特性，并评论的意义上的Hagedorn临界温度的两件弦。此后，作为一个新的发展，我们提出了一个标量场理论的三维空间中的一个真正的领域，在一个潜在的线性上升与纵向坐标z在0<z<1的间隔，并在此后保持恒定的水平高原。在简化条件下，将两段弦势的长度比形式上用长度参数z代替，得到了两段弦势的一个粗略模型。
英文摘要：
The main part of this paper is to present an updated review of the Casimir energy at zero and finite temperature for the transverse oscillations of a piecewise uniform closed string. We make use of three different regularizations: the cutoff method, the complex contour integration method, and the zeta-function method. The string model is relativistic, in the sense that the velocity of sound is for each string piece set equal to the velocity of light. In this sense the theory is analogous to the electromagnetic theory in a dielectric medium in which the product of permittivity and permeability is equal to unity (an isorefractive medium). We demonstrate how the formalism works for a two-piece string, and for a 2N-piece string, and show how in the latter case a compact recursion relation serves to facilitate the formalism considerably. The Casimir energy turns out to be negative, and the more so the larger the number of pieces in the string. The two-piece string is quantized in D-dimensional spacetime, in the limit when the ratio between the two tensions is very small. We calculate the free energy and other thermodynamic quantities, demonstrate scaling properties, and comment on the meaning of the Hagedorn critical temperature for the two-piece string. Thereafter, as a novel development we present a scalar field theory for a real field in three-dimensional space in a potential rising linearly with a longitudinal coordinate z in the interval 0<z<1, and which is thereafter held constant on a horizontal plateau. The potential is taken as a rough model of the two-piece string potential under simplifying conditions, when the length ratio between the pieces is replaced formally with the mentioned length parameter z.
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