Interferometers as probes of Planckian quantum geometry

A theory of position of massive bodies is proposed that results in an observable quantum behavior of geometry at the Planck scale, $t_P$. Departures from classical world lines in flat spacetime are described by Planckian noncommuting operators for position in different directions, as defined by interactions with null waves. The resulting evolution of position wave functions in two dimensions displays a new kind of directionally coherent quantum noise of transverse position. The amplitude of the effect in physical units is predicted with no parameters, by equating the number of degrees of freedom of position wave functions on a 2D space-like surface with the entropy density of a black hole event horizon of the same area. In a region of size $L$, the effect resembles spatially and directionally coherent random transverse shear deformations on time scale $\approx L/c$ with typical amplitude $\approx \sqrt{c{t}_{P}L}$. This quantum-geometrical “holographic noise” in position is not describable as fluctuations of a quantized metric, or as any kind of fluctuation, dispersion or propagation effect in quantum fields. In a Michelson interferometer the effect appears as noise that resembles a random Planckian walk of the beam splitter for durations up to the light-crossing time. Signal spectra and correlation functions in interferometers are derived, and predicted to be comparable with the sensitivities of current and planned experiments. It is proposed that nearly colocated Michelson interferometers of laboratory scale, cross-correlated at high frequency, can test the Planckian noise prediction with current technology.

Figure 1

Spacetime diagram of the nested-causal-diamond construction associated with collective quantum states of matter position and rest frame in flat classical spacetime. One spatial dimension is suppressed. The two vertical wavy lines represent timelike trajectories or world lines of bodies. A causal diamond is the intersection of the future light cone of one point on a world line with the past light cone of some future point. Two nested causal diamonds associated with the left-hand trajectory are shown. It is conjectured that relative positions of two world lines in spacetime are encoded by Planck-limited wave functions on 2D boundaries of the diamonds tangent to the world lines; one of these 2D surfaces is shown as a space-like circle intersecting the right-hand world line at the position of the solid dot. Their rest-frame separation $L$ determines the amplitude of coherent random transverse fluctuations in measured position of amplitude $\Delta x\approx \sqrt{c{t}_{P}L}$ on a timescale $\approx L/c$.

Figure 2

Spacetime diagram showing a causal diamond associated with a Michelson interferometer. The spatial dimension out of the interferometer plane is suppressed. The central worldline represents the beam splitter, the other two represent end mirrors. The two arms are shown on one space-like surface, a particular time in the lab frame. The measured signal compares light reflected from the two end mirrors, in different directions, as a function of time. The interactions with the beam splitter in the two directions occur at two times separated by $2L/c$, where $L$ denotes the arm length. The solid dots represent reflection events that contribute to the signal at the time represented by the uppermost dot. The measurement compares positions nonlocally, and in different directions. The wave packet description of uncertainty in an interferometer refers to wave functions in the positions of a single beam splitter mirror trajectory at two different times, relative to end mirrors in two different directions.

Figure 3

(a) Projection of the causal diamond in Fig. 2 onto the plane of an interferometer. The time axis is not shown. The causal spheres around the two end mirrors are also shown, as solid circles. Beam splitter position (center of dashed circle) “jitters” in the directions transverse to laser wave fronts coming from the two end mirrors, along the tangent directions of each of their causal spheres. Holographic noise appears in the dark-port signal that measures the position difference of the beam splitter in the two directions. (b) Projection showing two interferometers slightly displaced from each other. Most of the spacetime volume overlaps. The Hilbert spaces of the their diamonds are highly entangled, leading to highly correlated holographic noise signals, as in Eq. (35). (c) Projection showing two interferometers with one nearly-collocated arm, with other arms extending in opposite directions. The causal diamonds from the left and right end mirrors do not overlap at all, so the holographic-noise part of their signals display zero correlation. This null configuration is a useful control for experiments.

Figure 4

Sketch showing the scaling of Planckian uncertainty and noise. The relative amplitude of position fluctuations is shown from typical wave components on two scales. Dots represent positions of bodies. Vertical displacement shows the (greatly exaggerated) difference from classical position; horizontal scale is time, or distance in the rest frame. The typical excursion grows with duration like $\sqrt{\tau }$, therefore, the total displacement is dominated by the longest scale measured, even though the angular fluctuations, here shown by the slopes of the waves, are largest on small scales (ultimately reaching unity on the holographic Planck scale). The longest wavelength measured corresponds to the scale of a measured causal diamond, which determines the overall excursion of measured amplitude fluctuations. Spatially overlapping diamonds collapse into the same modes on this scale, so nearby bodies share correlated, directionally coherent motion on scales much larger than their transverse separation, as indicated by dashed lines. These spatially and directionally coherent fluctuations from a classical geometry are shared by collections of particles and bodies in the same region of spacetime. Over timescales long compared with the size of a region, the fluctuations average away to become negligible.

Figure 5

Differential-position fluctuation as a function of time interval or wave period. Both scales are in meters. For interferometers, the radius of the causal diamond is the arm length, $L$. The holographic-noise line refers to the transverse displacement amplitude estimated in Eq. (40), for time interval $c\tau =2L$. For averaging time $c\tau \gg 2L$, corresponding to the flat low-frequency limit of the predicted spectrum, Eq. (41), the mean fluctuation amplitude falls off as $\sqrt{\tilde{\Xi }(f)\Delta f}\approx \sqrt{2L/c\tau }$, as shown by the dashed lines for the 600 m arms of GEO600 and 4 km arms of LIGO. Current GEO600 and LIGO sensitivities show the standard deviation of displacement at the minimum of their noise curves, about 800 Hz and 150 Hz, respectively, for gravitational wave induced displacements. This plot does not show the additional factor to correct for the reduced response of these particular layouts to holographic noise. Folded arms in GEO600, and Fabry-Perot arm cavities in LIGO, reduce sensitivities to holographic noise by about a factor of 2 and 100, respectively, at low frequencies, so current measurements remain above the holographic-noise predictions. The point labeled Holometer shows the estimated photon shot noise limit for two 40 m, colocated interferometers, with the same cavity power as GEO600, cross-correlated for 1 h up to frequency $c/2L=3.74\mathrm{MHz}$. An instrument of this kind should be able to convincingly rule out or detect holographic noise.

Figure 6

Comparison of measurement precision of a larger variety of experiments to position fluctuations. Differential position change in length or time is plotted as a function of system size or duration, both with decimal log scales in meters, extending from the Planck scale to the Hubble scale. The holographic-noise prediction and interferometer sensitivities are shown as in Fig. 5, with the addition of LISA. Rough estimates of precision with current technology for other experimental techniques are labeled. The upper dashed line shows rough scales of natural systems; the lower dashed line shows a fractional transverse position fluctuation, or angular indeterminacy, $\Delta x/x=\Delta \tau /\tau =\Delta \theta ={10}^{-20}$. Laser-based interferometry is the most sensitive technique by this measure, and the only one currently capable of detecting holographic noise.