三维标量模型作为四维理想导体极限：降维和变分边界条件
3D scalar model as a 4D perfect conductor limit: dimensional reduction and variational boundary conditions
摘要：
在降维下，一个D时空维的系统不一定会产生它的D-1维模拟版本。除此之外，这个结果将取决于边界条件和系统的维数D。我们研究这个问题的标量和阿贝尔规范场的边界条件下，遵守对称性的行动。我们将我们的研究结果应用于卡西米尔活塞，一个理想的系统，用于检测边界效应。我们的调查是不限于额外的维度，我们表明，在2004年提出的原始活塞的情况下，玩具模型，涉及一个标量场在3D（2+1）维，可以通过降维从一个更现实的4D电磁（EM）系统。我们表明，理想导体条件下，D维EM场减少到D-1标量场，而不是其低维版本。对于Dirichlet边界条件，在降维下没有恢复理论，Casimir压力在任何维度上都趋于零。这个“零狄利克雷”结果对于理解EM情形是有用的。然后，我们确定了两个特殊的系统，其中低维版本恢复在任何维度：系统与完美的磁导体（PMC）和诺依曼边界条件。我们表明，这两个边界条件可以得到从一个变分过程中的行动消失外的有界区域。场在表面上自由变化，并且在降维后具有零模式。
英文摘要：
Under dimensional reduction, a system in D spacetime dimensions will not necessarily yield its D-1-dimensional analog version. Among other things, this result will depend on the boundary conditions and the dimension D of the system. We investigate this question for scalar and abelian gauge fields under boundary conditions that obey the symmetries of the action. We apply our findings to the Casimir piston, an ideal system for detecting boundary effects. Our investigation is not limited to extra dimensions and we show that the original piston scenario proposed in 2004, a toy model involving a scalar field in 3D (2+1)dimensions, can be obtained via dimensional reduction from a more realistic 4D electromagnetic (EM) system. We show that for perfect conductor conditions, a D-dimensional EM field reduces to a D-1 scalar field and not its lower-dimensional version. For Dirichlet boundary conditions, no theory is recovered under dimensional reduction and the Casimir pressure goes to zero in any dimension. This "zero Dirichlet" result is useful for understanding the EM case. We then identify two special systems where the lower-dimensional version is recovered in any dimension: systems with perfect magnetic conductor (PMC) and Neumann boundary conditions. We show that these two boundary conditions can be obtained from a variational procedure in which the action vanishes outside the bounded region. The fields are free to vary on the surface and have zero modes which survive after dimensional reduction.
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