Following a field-theoretical approach, we study the scalar Casimir effect upon a perfectly conducting cylindrical shell in the presence of spontaneous Lorentz symmetry breaking. The scalar field is modeled by a Lorentz-breaking extension of the theory for a real scalar quantum field in the bulk regions. The corresponding Green’s functions satisfying Dirichlet boundary conditions on the cylindrical shell are derived explicitly. We express the Casimir pressure (i.e. the vacuum expectation value of the normal–normal component of the stress–energy tensor) as a suitable second-order differential operator acting on the corresponding Green’s functions at coincident arguments. The divergences are regulated by making use of zeta function techniques, and our results are successfully compared with the Lorentz invariant case. Numerical calculations are carried out for the Casimir pressure as a function of the Lorentz-violating coefficient, and an approximate analytical expression for the force is presented as well. It turns out that the Casimir pressure strongly depends on the Lorentz-violating coefficient and it tends to diminish the force.

中文翻译：

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违反洛伦兹背景下导电圆柱体的标量卡西米尔效应

按照场论方法，我们研究了在自发洛伦兹对称性破缺的情况下，完美导电圆柱壳上的标量卡西米尔效应。标量场是通过洛伦兹破断扩展理论对体区域中的实标量量子场进行建模的。明确导出了满足圆柱壳上狄利克雷边界条件的相应格林函数。我们将 Casimir 压力（即应力-能量张量的法向-法向分量的真空期望值）表示为合适的二阶微分算子，该算子在重合参数处作用于相应的格林函数。通过使用 zeta 函数技术来调节分歧，并且我们的结果成功地与洛伦兹不变量情况进行了比较。对作为洛伦兹破坏系数的函数的卡西米尔压力进行了数值计算，并给出了力的近似解析表达式。事实证明，卡西米尔压力强烈依赖于洛伦兹破坏系数，并且趋向于减小力。