Measurement of quantum fluctuations in geometry

A particular form for the quantum indeterminacy of relative spacetime position of events is derived from the context of a holographic geometry with a minimum length at the Planck scale. The indeterminacy predicts fluctuations from a classically defined geometry in the form of “holographic noise” whose spatial character, absolute normalization, and spectrum are predicted with no parameters. The noise has a distinctive transverse spatial shear signature and a flat power spectral density given by the Planck time. An interferometer signal displays noise due to the uncertainty of relative positions of reflection events. The noise corresponds to an accumulation of phase offset with time that mimics a random walk of those optical elements that change the orientation of a wavefront. It only appears in measurements that compare transverse positions and does not appear at all in purely radial position measurements. A lower bound on holographic noise follows from a covariant upper bound on gravitational entropy. The predicted holographic noise spectrum is estimated to be comparable to measured noise in the currently operating interferometric gravitational-wave detector GEO600. Because of its transverse character, holographic noise is reduced relative to gravitational wave effects in other interferometer designs, such as the LIGO observatories, where beam power is much less in the beam splitter than in the arms.

Figure 1

Inteferometer showing two branches of a self-interfering particle path with dotted lines. Reflection events on the beam splitter, and detection events on the detector $S$, are shown by dots. The signal phase measures the path difference, which includes noise due to the holographic uncertainty in the relative positions of the beamsplitter at the times of the two reflection events.

Figure 2

Holographic encoding of two reflection events on the beam splitter $B$. A null surface $N$ is shown at slice $N1$ corresponding to reflection event 1 and slice $N2$ corresponding to event 2. Once the position of event 1 is fixed, the relative position of a reflection event 2 on a different slice of the same null surface (the same wavefront in 3D) is encoded by a patch of fields with a finite transverse size given by holographic uncertainty, indicated by an ellipse. The reflection rotates the wavefront to land on a detector surface, $S$, where measurement collapses the uncertainty of event location in that direction to a definite value. The arrival time acquires the uncertainty in transverse position from event 2.