For a massive scalar field with general curvature coupling we evaluate the Wightman function in the geometry of two parallel branes perpendicular to the AdS boundary. On the separate branes, the field operator is constrained by Robin boundary conditions, in general, with different coefficients. In the region between the branes their contribution to the Wightman function is explicitly separated. By using this decomposition, the brane-induced effects on the vacuum expectation values for the field squared and energy-momentum tensor are investigated. The behavior of those expectation values is studied in various asymptotic regions of the parameters. The vacuum energy-momentum tensor in addition to the diagonal components has a nonzero off-diagonal stress. Depending on the boundary conditions and also on the distance from the branes, the vacuum energy density can be either positive or negative. The Casimir forces acting on the branes have two components. The first one corresponds to the standard normal force and the second one is parallel to the branes and presents the vacuum shear force. Unlike to the problem of parallel plates in the Minkowski bulk, the normal Casimir forces acting on separate branes differ if the boundary conditions on the branes are different. They can be either repulsive or attractive. In a similar way, depending on the coefficients in the boundary conditions, the shear force is directed toward or from the AdS boundary. The separate components may also change their signs as functions of the interbrane separation. At large proper separations between the branes, compared to the AdS curvature radius, both of the components of the Casimir forces exhibit a power-law decay. For a massive scalar field this behavior is in contrast to that for the Minkowski bulk, where the decrease is exponential.

中文翻译：

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垂直于 AdS 边界的膜的真空密度和卡西米尔力

对于具有一般曲率耦合的大规模标量场，我们在垂直于 AdS 边界的两个平行膜的几何中评估 Wightman 函数。在单独的膜上，场算子通常受 Robin 边界条件的约束，具有不同的系数。在膜之间的区域中，它们对 Wightman 函数的贡献是明确分离的。通过使用这种分解，研究了膜诱导对场平方和能量-动量张量的真空期望值的影响。在参数的各种渐近区域中研究了这些期望值的行为。除了对角分量外，真空能量-动量张量还具有非零非对角应力。根据边界条件以及与膜的距离，真空能量密度可以是正的也可以是负的。作用在膜上的卡西米尔力有两个组成部分。第一个对应于标准法向力，第二个平行于膜并呈现真空剪切力。与 Minkowski 块体中的平行板问题不同，如果膜上的边界条件不同，则作用在单独膜上的法向卡西米尔力也会不同。它们可以是令人厌恶的，也可以是有吸引力的。以类似的方式，根据边界条件中的系数，剪切力指向或来自 AdS 边界。作为膜间分离的函数，分离的组分也可以改变它们的符号。与 AdS 曲率半径相比，在膜之间有较大的适当间距时，卡西米尔力的两个组成部分都表现出幂律衰减。对于大规模标量场，这种行为与 Minkowski 体积的行为相反，后者的减少是指数级的。