Indeterminacy of holographic quantum geometry

Abstract

An effective theory based on wave optics is used to describe indeterminacy of position in holographic spacetime with a UV cutoff at the Planck scale. Wave functions describing spacetime positions are modeled as complex disturbances of quasimonochromatic radiation. It is shown that the product of standard deviations of two position wave functions in the plane of a holographic light sheet is equal to the product of their normal separation and the Planck length. For macroscopically separated positions the transverse uncertainty is much larger than the Planck length, and is predicted to be observable as a ``holographic noise'' in relative position with a distinctive shear spatial character, and an absolutely normalized frequency spectrum with no parameters once the fundamental wavelength is fixed from the theory of gravitational thermodynamics. The spectrum of holographic noise is estimated for the GEO600 interferometric gravitational-wave detector and is shown to approximately account for currently unexplained noise between about 300 and 1400 Hz. In a holographic world, this result directly and precisely measures the fundamental minimum interval of time.