半球帽管中的Casimir效应
Casimir effect in hemisphere capped tubes
摘要：
本文研究了具有一般曲率耦合的有质量标量场在（2+1）维时空背景中的真空密度，该背景对应于一个带半球帽的圆柱管.一套完整的模式功能的构造和正频率Wightman函数的圆柱形和半球形子空间进行评估。在此基础上，讨论了真空场平方和能量动量张量的期望值。在分隔两个子空间的边界上，平均场平方和法向应力是有限的，而能量密度和平行应力则随着到边界的距离的倒数幂而发散。对于共形耦合场，真空能量密度在空间的圆柱部分为负。在半球上，它在顶部附近为负，在边界附近为正。在最小耦合的情况下，杯上的能量密度为负。在管上，它在边界附近为正，在很远的地方为负。虽然子空间的几何形状不同，但边界两侧的卡西米尔压力相等，净卡西米尔力为零。所得结果可应用于长波长近似下的有效场论所描述的加盖碳纳米管。
英文摘要：
In this paper we investigate the vacuum densities for a massive scalar field with general curvature coupling in background of a (2+1)-dimensional spacetime corresponding to a cylindrical tube with a hemispherical cap. A complete set of mode functions is constructed and the positive-frequency Wightman function is evaluated for both the cylindrical and hemispherical subspaces. On the base of this, the vacuum expectation values of the field squared and energy-momentum tensor are investigated. The mean field squared and the normal stress are finite on the boundary separating two subspaces, whereas the energy density and the parallel stress diverge as the inverse power of the distance from the boundary. For a conformally coupled field, the vacuum energy density is negative on the cylindrical part of the space. On the hemisphere, it is negative near the top and positive close to the boundary. In the case of minimal coupling the energy density on the cup is negative. On the tube it is positive near the boundary and negative at large distances. Though the geometries of the subspaces are different, the Casimir pressures on the separate sides of the boundary are equal and the net Casimir force vanishes. The results obtained may be applied to capped carbon nanotubes described by an effective field theory in the long-wavelength approximation.
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