The Heisenberg uncertainty principle is modified by the introduction of an observer-independent minimal length. In this work, we have considered the resonant gravitational wave detector in the modified uncertainty principle framework, where we have used the position momentum uncertainty relation with a quadratic order correction only. We have then used the path integral approach to calculate an action for the bar detector in the presence of a gravitational wave and then derived the Lagrangian of the system, leading to the equation of motion for the configuration-space position coordinate in one dimension. We then find a perturbative solution for the coordinate of the detector for a circularly polarized gravitational wave, leading to a classical solution of the same for the given initial conditions. Using this classical form of the coordinate of the detector, we finally obtain the classical form of the on-shell action describing the harmonic oscillator–gravitational wave system. Finally, we have obtained the free particle propagator containing the quantum fluctuation term considering gravitational wave interaction.

中文翻译：

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广义不确定性原理框架中引力波谐振探测器的路径积分作用

通过引入独立于观察者的最小长度来修改海森堡不确定性原理。在这项工作中，我们在改进的不确定性原理框架中考虑了谐振引力波探测器，其中我们仅使用了具有二次阶校正的位置动量不确定性关系。然后，我们使用路径积分方法计算了存在引力波时条形探测器的动作，然后导出了系统的拉格朗日，从而得出了一维配置空间位置坐标的运动方程。然后，我们为圆​​偏振引力波的探测器坐标找到了一个微扰解，从而在给定的初始条件下得到了相同的经典解。使用探测器坐标的这种经典形式，我们最终获得了描述谐振子 - 引力波系统的壳上作用的经典形式。最后，我们得到了考虑引力波相互作用的包含量子涨落项的自由粒子传播子。