具有Dirichlet边界的正多边形的Casimir能量
Casimir energy for a Regular Polygon with Dirichlet Boundaries
摘要：
研究了N边正多边形标量场的Casimir能量。标量场在多边形的周边处服从狄利克雷边界条件。多边形本征值$λ_N$根据狄利克雷圆本征值$λ_C$表示为形式为$λ_N = λ_C (1 + \frac{4ζ(2)}{N^2} + \frac{4ζ(3)}{N^3} + \frac{28ζ(4)}{N^4}+...)$的$\frac{1}{N}$的展开。比较了用我们的方法得到的N=4的多边形上的Casimir能量和正方形上的标量场的Casimir能量。我们将结果推广到形式为$R^d\times P_N$的空间，其中$P_N$是N-多边形。同样，我们求出了一个无限长多边形截面的“圆柱体”的电场能量。利用我们所使用的方法和结果，我们有理由假定$D$ -球的Casimir能量与同形于$D$ -球的规则形状的Casimir能量具有相同的符号.在文章的最后，我们对结果进行了总结和讨论。
英文摘要：
We study the Casimir energy of a scalar field for a regular polygon with N sides. The scalar field obeys Dirichlet boundary conditions at the perimeter of the polygon. The polygon eigenvalues $λ_N$ are expressed in terms of the Dirichlet circle eigenvalues $λ_C$ as an expansion in $\frac{1}{N}$ of the form, $λ_N = λ_C (1 + \frac{4ζ(2)}{N^2} + \frac{4ζ(3)}{N^3} + \frac{28ζ(4)}{N^4}+...)$. A comparison follows between the Casimir energy on the polygon with N=4 found with our method and the Casimir energy of the scalar field on a square. We generalize the result to spaces of the form $R^d\times P_N$, with $P_N$ a N-polygon. By the same token, we find the electric field energy for a "cylinder" of infinite length with polygonal section. With the method we use and in view of the results, it stands to reason to assume that the Casimir energy of $D$-balls has the same sign with the Casimir energy of regular shapes homeomorphic to the $D$-ball. We sum up and discuss our results at the end of the article.
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