We derive the functional Schrödinger equation for quantum fields in curved spacetime in the semiclassical limit of quantum geometrodynamics with a Gaussian incoherent dust acting as a clock field. We perform the semiclassical limit using a WKB-type expansion of the wave functional in powers of the squared Planck mass. The functional Schrödinger equation that we obtain exhibits a functional time derivative that completes the usual definition of WKB time for curved spacetime, and the usual Schrödinger-type evolution is recovered in Minkowski spacetime.

中文翻译：

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具有高斯时钟场的量子引力半经典极限中的功能Schrödinger方程

我们在量子地球动力学的半经典极限中，以高斯非相干尘埃作为时钟场，导出了弯曲时空中量子场的函数Schrödinger方程。我们使用以平方的普朗克质量的幂为单位的波函数的WKB型展开来执行半经典极限。我们获得的泛函Schrödinger方程展示了函数时间导数，该函数时间导数完成了弯曲时空WKB时间的常规定义，并且在Minkowski时空中恢复了常规Schrödinger型演化。