FERMILAB-PUB-15-006-A

Interferometric Tests of Planckian Quantum Geometry Models

Ohkyung KwonUniversity of Chicago

Craig J. HoganUniversity of Chicago and Fermilab Center for Particle Astrophysics

The effect of Planck scale quantum geometrical effects on measurements with interferometers isestimated with standard physics, and with a variety of proposed extensions. It is shown that effectsare negligible in standard field theory with canonically quantized gravity. Statistical noise levels areestimated in a variety of proposals for non-standard metric fluctuations, and these alternatives areconstrained using upper bounds on stochastic metric fluctuations from LIGO. Idealized models ofseveral interferometer system architectures are used to predict signal noise spectra in a quantumgeometry that cannot be described by a fluctuating metric, in which position noise arises fromholographic bounds on directional information. Predictions in this case are shown to be close tocurrent and projected experimental bounds.

I. INTRODUCTION

It is often remarked that quantum gravity should leadto the creation and annihilation of high energy vir-tual particles that gravitationally alter spacetime to be“foamy” or “fuzzy” at the Planck scale. While variousmodels mathematically suggest such nonclassicality[1–5],it is unclear exactly how much the actual physical metricdeparts from the classical structure, particularly in sys-tems much larger than the Planck length. In particular,it is not known how the effects of Planck scale fluctua-tions perturb the geodesic of a macroscopic experimentalobject.

Laser interferometry is a particularly good experimen-tal tool to test theories of position, because it is the mostprecise measure of relative space-time position of massivebodies. The LIGO collaboration has recently publishedlimits on the stochastic gravitational-wave backgroundthat correspond to a spectral density of noise below thePlanck spectral density– that is, the formal experimentalbound on variance in dimensionless strain per frequencyinterval is now less than the Planck time. Having crossedthe Planck threshold in experimental technique, there isa need for better controlled theoretical predictions forthe characteristics of Planckian spacetime noise as theyactually appear in interterferometer data, and a moresystematic application of the experimental constraintsto model parameters. This paper presents a survey ofcandidate models of Planckian quantum geometry, andcomputes their effects using schematic theoretical mod-els of interferometers that are sufficiently detailed andwell characterized to provide useful constraints on newphysics.

A straightforward Planck-scale change in position isimpossible to measure experimentally. Moreover, weshow below that metric perturbations that fluctuate onlyat a microscopic scale average out in macroscopic mea-surements and thus become practically unobservable inreal experiments. However, a beam splitter suspended inan interferometer such as LIGO is in free fall (i.e. fol-

lowing a geodesic) in horizontal directions at frequenciesmuch higher than 100Hz [6–8]. Thus, the effects of cer-tain types of spacetime noise can accumulate over time,enough for some types of deviations from geodesics to beobservable in extensions of known physics.

We start by calculating, in standard field theory, mea-surable fluctuations in a macroscopic spacetime distanceexpected from vacuum fluctuations in graviton fields.This calculation confirms the conventional wisdom thatin standard field theory, Planck scale effects stay at thePlanck scale: they average out to create negligible depar-tures from classical behavior on macroscopic scales.

We then proceed to survey a variety of phenomenolog-ical proposals, based on conjectures about macroscopicquantum properties of the space-time metric. These ex-tensions of standard physics are still based on metricperturbations, so they can be compared directly withpublished bounds on metric fluctuations from gravita-tional wave backgrounds. We classify the scaling behav-ior of metric perturbation noise with respect to the lengthscales involved in the measurement[9–13], and comparethese with the most recent experimental data from LIGO.Among the alternative models surveyed, we find thatthey generally are either already conclusively ruled outby current data, or do not produce detectable effects.

An important exception lies in a class of models wherePlanckian geometrical degrees of freedom cannot be ex-pressed as fluctuations of a metric, treated in the usualway as quantized amplitudes of modes that have a deter-minate classical spatial structure. It has been knownthat field modes within a classical background space-time result in infrared paradoxes of states denser thanblack holes[14]. Macroscopic geometrical uncertainty ina different kind of model can be estimated from theholographic information content of gravitational systems,and from general symmetry principles [15–18]. Spatialinformation can be regarded as being carried by nullwaves subject to a Planck frequency cut-off or bandwidthlimit[19–22]. This hypothesis predicts directional space-time uncertainty that does not average away in the same

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way as fluctuations in field theory[23–25]; indeed by somemeasures, it grows with scale. It also allows nonlocalentanglement of position states of a kind not availablewithin the local framework of quantum field theory, whilepreserving causal relationships.

In these information-bounded models, the spatial co-herence of flucuations depends differently on the causalstructure of the space-time than in the case of gravita-tional waves, or in metric-based extension models. Inter-ferometers far apart from each other display a coherent,correlated response to a low frequency stochastic gravita-tional wave background. This is true even for small inter-ferometers, as long as their separation is not much largerthan the measured wavelength (on the order of 106m forgravitational-wave detectors). By contrast, fluctuationsfrom information bounds are only correlated if the twointerferometers measure causally overlapping space-timeregions. The response of interferometer systems dependsdifferently on their architectures – the layout of the op-tical paths in space. The apparatus must be modeled totake these differences into account in making predictionsfor signal correlations.

In the later sections of this paper we compare predic-tions in this kind of model with current experimentalbounds from LIGO and GEO-600, as well as future testswith the Fermilab Holometer, a pair of co-located, cross-correlated interferometers specifically designed to searchfor such effects. Since these perturbations are not equiv-alent to metric fluctuations, we develop models of theinterferometers to estimate their response to this kind ofgeometrical uncertainty. We find that this class of modelis close to being either ruled out or experimentally de-tectable.

Throughout this paper, where numerical values areneeded, we use the Particle Data Group values: c =299792458 m/s, ~ = 6.58211928(15)× 10−22 MeV s, andG = 6.70837(80)× 10−39 ~c (GeV/c2)−2 [26].

II. CONSTRAINTS ON PLANCKIAN NOISEFROM METRIC FLUCTUATIONS

A. Standard Field Theory

It is well-known that standard methods of quantizinggeneral relativity as graviton fields lead to results thatare not renormalizable. However, on macroscopic scalesgraviton self-interactions can be neglected, and it is con-sistent to use the zero-point amplitude of metric fluc-tuations to estimate the order of magnitude of metricfluctuations predicted by a model based on field theory.

Assume plane wave solutions in linearized gravity, andconsider a wave incident perpendicular to a length beingmeasured (e.g. an interferometer arm). The mean am-plitude of a perturbation vanishes, but a graviton modehas a typical fluctuation energy on the order of 1

2~ωg,where ωg is the frequency of the mode. Divided by thetypical scale of volume that such a field would occupy,

this corresponds to an energy density of:

u =12~ωg

(c/ωg)3(1)

A gravitational wave of strain amplitude h has an energydensity of:

u =c2

32πGω2gh

2 (2)

Equating the two equations gives us a typical strain am-plitude of:

h ≈ 4√πlpωgc

(3)

If a gravitational plane wave of this strain amplitudetravels in the z direction, in a length L orthogonal tothe propagation we expect a position fluctuation of thefollowing time-dependent magnitude:

∆L = F (z − ct) ≡ 1

2Lh cos(kz − ωgt) (4)

= 2√πL lpωg

ccos(

ωgcz − ωgt) (5)

However, this is not the actual length fluctuation mea-surable between two macroscopic objects (e.g. mirrors).To determine the position of a macroscopic object, weneed the object to interact with a measurable field, whichoccurs over a nonzero area. If we consider a light beamof wavelength λ extended over a length L, that would re-quire the beam to have a minimum cross section width onthe order of the diffraction scale,

√Lλ/2π. The function

F (z−ct) varies with z at a distance scale of 2πcωg

, which is

much smaller than√Lλ/2π for the higher-energy modes

that lead to larger metric strain. Therefore the inter-action between the surface of the object and the lightbeam would cause a significant amount of suppression inthe measurable length fluctuation as we average across amacroscopic boundary of this width.

Now model the cross section of the beam as a gaussian.A one-dimensional averaging in the direction of an opti-mally oriented gravitational plane wave mode will sufficeas a demonstrative example.

G(z) ≡ 1√Lλ

e−πz2

Lλ (6)

To find the measurable time-dependent fluctuation in L,we average the fluctuation F (z−ct) over the cross sectionof the beam by taking the following convolution withG(z):

x(t) =

∞

−∞

F (z − ct)G(z)dz (7)

= 2√πLlpωgc

cos(ωgt)e−Lλω2

g

4πc2 (8)

3

Figure 1. RMS length deviation in a 1µm laser beam extended over a 4km macroscopic distance, measured with two macroscopicboundary surfaces and generated by an optimally oriented graviton vacuum mode of zero-point energy 1

2~ωg. The result is also

shown in equivalent strain, calculated by assuming a perfect detector response. This is not the same as strain noise observed ininterferometers, as discussed below. Averaging the metric fluctuation over the minimum laser beam width causes exponentialsuppression at frequencies higher than ωg = c

√2π/Lλ, the inverse diffraction scale. The value of the peak, marked above at

2GHz, is 4× 10−30m or 5× 10−35Hz−12 .

Take the RMS value over time:

σ(ωg) =√

2πLlpωgc

e−Lλω2

g

4πc2 (9)

Figure 1 shows a plot of σ(ωg), assuming λ = 1µm andL = 4km to match the physical parameters of LIGO, cur-rently the most sensitive experiment for detecting suchmetric strains. As intuitively expected, higher frequencymodes lead to higher zero-point energy and larger met-ric strain, but averaging the fluctuation over the beamwidth exponentially suppresses the observable macro-scopic length deviation for those short-wavelength gravi-ton modes. The resultant peak occurs at:

ωg,max = c

√2π

Lλσ(ωg,max) = 2πlp

√Leλ

(10)

The plot has a peak value of 4× 10−30m at 2GHz. Thisis clearly too small to be observed, and occurs at a fre-quency many orders of magnitude higher than the opti-mal measurement band for LIGO, which causes the de-tectable effect to be even further suppressed.

We sum over all frequency modes to obtain the totallength fluctuation. Since the relevant length scale here is√Lλ/2π, we perform a dimensionless mode summation

assuming a 1-dimensional box of that size. We apply thesubstitution:

ωg = 2πfg = πc

√2π

Lλn (11)

and sum over values of n running from 0 to ∞. Sincethese are independent modes, to obtain the overall un-certainty in L, we sum their contributions in quadrature:

σtot =

ˆ ∞0

(√

2πLlpcπc

√2π

Lλne−

π2

2 n2

)2

dn

12

(12)

=

√Lλπ

34 lp (13)

For the physical parameters of LIGO, this gives a totaluncertainty of 2× 10−30m, dominated by the peak valuein Figure 1. Observing this would require either a muchlarger apparatus or a light beam of a much higher fre-quency, both without compromising sensitivity to othersources of noise.

In an ideal detector, we could make another substitu-tion to rewrite this integral in terms of frequency:

2πf ≡ πc√

2π

Lλn (14)

σ2tot =

ˆ ∞0

(4

√π3

c3Llp

(Lλ2π

) 14

fe−πLλc2

f2

)2

df (15)

=

ˆ ∞0

S(f)df (16)

√S(fmax) =

lp√ec

(25L3π3

λ

) 14

(17)

4

To be clear, this the positional uncertainty involvedin a light beam statically extended over a single lengthL, arising from optimally oriented vacuum plane wavemodes of the graviton field. An actual measurementof a length L involves a light round trip, in which casewe should consider the frequency response to this time-varying gravitational wave mode as the light makes for-ward and return passes. The result is a further suppres-sion of the measurable noise at higher frequencies relativeto inverse light travel time[27]. The angular response iseven more involved because the polarization of the grav-itational wave mode is flipped midway relative to thedirection of light propagation.

We should also note here that the frequency depen-dence of metric strain noise detected in a linear measure-ment of length is different from the frequency dependenceobserved in interferometric experiments, as we will dis-cuss in a future section.

B. Non-Standard Metric-Based Fluctuations

While field theory gives a spacetime position fluctua-tion that is clearly too small to be measurable, there havebeen proposed extensions of standard physics that givemeasurable phenomenological predictions. All of thesemodels assume that light propagates in a metric in theusual way, but apply different non-standard assumptionsabout quantum fluctuations in the metric, generally withsome degree of non-standard macroscopic coherence, todiscuss how metric perturbations affect macroscopic dis-tance measurements. It is assumed in all of these modelsthat the effect of metric fluctuations on a macroscopic ob-ject can be calculated by treating it as a coherent rigidbody and calculating the effects on the center-of-mass.

With these caveats, we categorize the proposed phe-nomenologies by how the magnitude of the uncertaintyscales with the macroscopic lengths and times being mea-sured. We characterize each suggested hypothesis by theroot-mean-square deviation σ and the power spectrumS(f) of the strain noise h, defined as follows[28, 29]:

⟨h2⟩≡⟨x2⟩

L2=σ2

L2=

ˆ fmax

1/t

S(f)df (18)

Here L and fmax are respectively a length scale and acutoff frequency associated with an experimental appa-ratus. In most cases, the integral will be dominated bythe region around f ∼ 1/t, where is t is the time scaleof the measurement, usually following t ∼ L/c; this iswhere the non-standard macroscopic coherence enters.

1. White Spacetime Noise

One of the simplest conjectures is that S(f) is not depen-dent on the physical properties of the space-time probe.

Under such an assumption, dimensional analysis leads toa low-frequency expansion of the type[10]:

S(f) = a0lpc

+a1

(lpc

)2

f+a2

(lpc

)3

f2+· · · ∼ lpc

(19)

Terms involving f−|n| are not included because they alsoinvolve factors of l−|n|+1

p and thus do not disappear inthe classical limit lp → 0. For cases in which f c/lp,the expansion reduces to a “white spacetime noise” thatis frequency independent. This prediction covers a classof theories in which the strain spectrum of the spatial un-certainty is not apparatus dependent, although the mea-sured noise spectrum in specific experiments (e.g. inter-ferometers) might not be flat, as will be discussed fur-ther below. In particular, some analogies interpretingthe spacetime foam as a quantum thermal bath suggestsuch results[36].

2. Minimum Uncertainty

Another simple hypothesis is that the RMS deviation inany length follows a “minimum uncertainty” close to thePlanck scale[9, 41, 42]:

σ ∼ lp S(f) ∼l2pfL2

(20)

The spectrum in (20) is only approximate, as small log-arithmic t-dependent corrections are ignored. This pre-diction might be consistent with some theories of criticalstrings [43] and the quantum-group structure describedin [44]; it is also implied by certain models of minimumdistance fluctuations from graviton effects[45, 46].

3. Random Walk Noise

One of the most frequently suggested is the “randomwalk” model. The rationale for this model is a gedankenexperiment suggested by Salecker and Wigner in whicha distance is measured by a clock that records the timetaken for a light signal to travel the distance twice, withthe light reflected by a mirror at the end[48, 49]. Ifwe apply Heisenberg’s uncertainty principle to the posi-tions and momenta of the mirrors and require that mea-surement devices be less massive than black holes whoseSchwarzschild radii are equal to the size of the devices,we obtain[9, 30, 31] (Also see [32]):

σ ∼√ctlp S(f) ∼ clp

f2L2(21)

Since this model predicts a noise level near the cur-rent limits of experimental accuracy, we will go througha slightly more involved description of the noise spec-trum. Note that the effect of this model is as if an object

5

RMSDeviation

σ

Strain Amplitude Spectrum√S(f)of Metric Noise(Scaling Behavior)

Strain Amplitude Spectrum√S∆(f) of InterferometerNoise (Low Frequency)

√S∆(f)

(LIGO)(Hz−1/2)

References

RandomWalk Noise

√ctlp f−1

√clpL−1 4π

√lp/c 3× 10−21 [5, 9, 30–35]

White Space-Time Noise

√lp/c 4πf

√lp/c3L 4× 10−24 [10, 36]

One-ThirdPower Noise

3√ctl2p f−

56 3√cl2pL−1 4πf

16 3√l2p/c2 4× 10−28 [9, 11, 30, 37–40]

MinimimUncertainty lp f−

12 lpL−1 4πf

12 lp/c 7× 10−42 [9, 41–46]

Field Theory(Graviton 0-pt)

√ctλπ

34 lp 4

√π3

c3Llp

(Lλ2π

) 14 fe−

πLλc2

f216√

π5

c5L2lp

(Lλ2π

) 14 f2 7× 10−44

Table I. A list of metric-based measurable spacetime uncertainty predictions for a length of scale L, corresponding to a traveltime of scale t. We give the predicted overall RMS length uncertainty from metric fluctuations, the corresponding metric strainamplitude spectra, and the strain spectra of the noise actually expected in gravitational-wave interferometers. Numbers forLIGO are calculated assuming L = 4km, λ = 1µm, and f = 100Hz. The scaling dependence on f listed in the

√S(f) column

are order of magnitude estimates only and likely omits a scalar coefficient from a more robust predictive theory. The√S∆(f)

column gives approximate behavior for frequencies lower than the inverse travel time for a single interferometer arm. LIGO’s95% upper bound at 100Hz on the

√S∆(f) values is 9× 10−25Hz−1/2[47]. This rules out the “random walk” model and places

a limit a0 < 0.06 on the scalar coefficient that included in (19) for the “white spacetime noise” model.

deviates from its classical gedesic in a traditional randomwalk; i.e. for every Planck time elapsed the object takes arandom step of size lp. This means that its RMS velocityis always the speed of light, similar to the “Zitterbewe-gung” of an electron studied by Schrodinger early in thedevelopment of quantum mechanics. Thus, representingthe deviation x as a Fourier integral:

x(t) =

ˆdfx(f)e2πift (22)

it is straightforward from Parseval’s theorem that theone-sided power spectrum of the velocity v = dx/dt iswhite:

Ξv(f) ∼

c2/fN = 2clp if f < fN = 1/2tp0 if f > fN = 1/2tp

(23)

where fN is the Nyquist frequency. This in turn impliesthat the one-sided power spectrum of the strain is givenby:

S(f) ∼ Ξv(f)

4π2f2L2∼

clp2π2f2L2

(24)

This prediction is associated with dimensionally de-formed Poincare symmetries [33, 34] and could also holdwithin Liouville (non-critical) string theory[5, 35].

4. One-Third Power Noise

Lastly, there is an intriguing prediction, called the “one-third power model,” also based on the same Salecker-Wigner gedanken experiment but with the added as-sumption that the uncertainty in a length measurement

is bounded by the size of the measurement device, forexample a light clock that measures travel time with itsticks[9, 30, 37] (Also see [38, 39]). This bound is known tobe in rough agreement with what we get if we divide up acube of side L into small cubes of side σ and crudely ap-ply the Holographic Principle in a directionally isotropicmanner, demanding that the number of degrees of free-dom (the number of small cubes) match the holographiclimit[11, 40]. This gives:

σ ∼ 3

√ctl2p S(f) ∼ c2/3l

4/3p

f5/3L2(25)

C. Predictions for Measured Noise inInterferometers

Most, if not all, previously published works assumethat the scaling behavior for the spatial length uncer-tainties translates into a similar behavior in the noisemeasured in interferometers (in the portion of the noisecaused by quantum gravitational effects). However, thisnaive conversion from the raw noise source to the mea-sured phenomenon is inaccurate because an interferome-ter measures the change in the position of the beam split-ter (relative to the two mirrors at the end of the arms)at two different times as light travels along the two arms.For most interferometers, this incorrect assumption over-estimates the order of magnitude of the measurable noise.

Let us take a concrete example in which the beam split-ter position variable x is measured along one of the twoarms. We will take the relevant length scale L of theapparatus to be the length of an interferometer arm (the

6

Figure 2. Spectra for strain amplitudes in interferometers, assuming metric fluctuations on classical determinate spacetime.The most recently published noise spectra are compared against the predictions from models. Absolute normalizations fornoise predictions are rough estimates only. The dashed line for GEO-600 is the sensitivity before light-squeezing techniqueswere applied. Since all predictions are metric-based, they should be compared against the “correlated” data point for LIGO’sstochastic gravitational-wave background limit, obtained by cross-correlating two interferometers far apart. [50, 51]

most likely candidate). The light traversing along thetwo orthogonal arms of an interferometer reflect off ofthe beam splitter at two times separated by an intervalτ = 2L/c. An interferometer measures the deviation inx over this time interval, given by:

∆x(t) = x(t+ τ/2)− x(t− τ/2) (26)

This one-dimensional model suffices to derive a scalingbehavior for the measured noise because we are assumingmetric strains that are coherently applied to both theoptics and the spatial paths traveled by the light beams.

In Fourier space, the power spectrum S∆(f) of thismeasured strain ∆h = ∆x/L is related to the power spec-trum S(f) of the raw spacetime strain h = x/L by:

S∆(f) = 4 sin2(πfτ)S(f) ≈16π2f2L2

c2S(f) (27)

where we have taken a low frequency (relative to c/2L)approximation as appropriate for graviational-wave in-terferometers.

Note that this implies that the “random walk” modelabove, expressed in terms of a dimensionless gravita-tional wave strain measured inside an interferometer, cor-responds to a flat spectrum of noise.

D. Comparison with Experimental Data

Currently the experiments closest to the Planckian orsub-Planckian sensitivity levels required to test the pre-

dictions listed in the previous sections are gravitationalinterferometers such as LIGO. Noise levels published bythe LIGO collaboration now rule out select classes ofmetric-based noise predictions. The LIGO collabora-tion reports that it has achieved a strain noise of 3 ×10−23Hz−1/2 around 100Hz in a single interferometer[52].However, by cross-correlating signals from pairs of in-terferometers separated by a large distance (the two 4-km LIGO detectors), the LIGO and Virgo collaborationshave obtained a more stringent bound on the stochasticgravitational-wave background, which may be used to setan upper limit on metric fluctuations as a source of noise.As long as the effects of Planckian fluctuations can be de-scribed in terms of a fluctuating metric, these limits canbe carried over into limits on Planckian physics.

The cross-correlations here are of a different naturethan the cross-correlations used in the Holometer, whichare explained in a later section. In the case of a stochas-tic gravitational-wave background, as long as two detec-tors are within a distance smaller than the wavelengthof the gravitational wave (on the order of 106m for a100Hz wave), they show a coherent response to strains inthe metric. LIGO and Virgo take the measured signalsfrom two interferometers s1(t) and s2(t) and generate atime-integrated product signal S ≡

´ T/2−T/2 dt s1(t)s2(t)

[53]. The two measured signals are each a mixture ofmetric strains and instrument noise, but 〈S〉 measuresthe correlated strain, while

√〈S2〉 − 〈S〉 2 measures the

uncorrelated noise.

7

The gravitational wave energy density is defined as:

ΩGW (f) =f

ρc

dρGWdf

(28)

ΩGW (f) is characterized by a power law dependenceon f in most models of interest. By assuming afrequency-independent spectrum over the frequency band41.5 − 169.25Hz, LIGO and Virgo obtained a result ofΩGW < 5.6 × 10−6 at 95% confidence[47]. Using therelationship[8],

√S∆(f) = 4× 10−22

√ΩGW

(100Hzf

) 32

Hz−12 (29)

this corresponds to a strain noise of 9.5× 10−25Hz−1/2.

Referring to Figure 2 and the√S∆(f) column of Ta-

ble I, it is clear that the “random walk” model is nowsafely ruled out. There were previous assertions that thismodel was ruled out by the Caltech 40-meter interferom-eter data[9, 11], but as discussed in the previous section,such claims were results of incorrect naive comparisonsbetween the predicted raw spacetime noise

√S(f) and

the noise levels measured specifically in interferometers,which should correspond to

√S∆(f). Also notable is

that the “white spacetime noise” model, approximatinga class of possibilities in which the strain spectrum ofthe raw spatial noise is not dependent on the character-istics of the apparatus, is also probably ruled out: the95% upper bound established by LIGO and Virgo is ap-parently lower than the predicted level of noise. How-ever the quoted expression for S(f) was only an estimateand equation (19) contains an uncalibrated scalar nor-malization factor that could be numerically much lessthan unity. We calculate the limit on this coefficient asa0 < 0.06, with the same 95% confidence level. The “one-third power” model is still out of reach, but could bewithin reach of the advanced phase of LIGO or proposedspace interferometers such as eLISA. The simple “mini-mum uncertainty” model is quite far out of reach. Ourprediction based on a zero-point field theory calculationalso gives a result that is too small for any foreseeableexperimental project. Again, all of the alternative met-ric fluctuation-based models are able to generate largermacroscopic effects only by suggesting non-standard co-herence that is not present in standard field theory ofPlanckian metric fluctuations.

Other phenomenological approaches have considereddifferent fundamental length scales for quantum gravityeffects, for example by using the string length instead ofthe Planck length[10]. This would increase the predictednoise level slightly, although not to an extent that wouldsignificantly change our conclusions about exclusion ofmodels.

E. Other Constraints Without MacroscopicCoherence

We have previously stated that these meta-models ofmetric based fluctuations make non-standard assump-tions of coherence in order to generate macroscopicallymeasurable effects. Without such assumptions, any noisethat scales with distance would eventually result in lo-cally measurable effects, making it difficult to preservelocality. There is now an experimental bound on sucheffects, made by observing TeV γ-rays from Cherenkovtelescopes[54]. If photons from a faraway source weresubject to uncertainties that scale with the distance oftravel, without an additional assumption of macroscopiccoherence, the local images we take (at the telescope) ofthe source would suffer a loss of resolution. The clar-ity of these images can be used to place a general upperbound on any strain noise that scales as σ ≈ lαPL1−α,at α ? 0.72, which rules out both the “random walk”model and the “one-third power” model if we considerthem without macroscopic coherence.

This constraint demonstrates the difficulty of satis-fying the holographic information bound while consid-ering spacetime degrees of freedom. Unlike in localfield theories that attain holographic scaling of informa-tion through dualities (ignoring infrared paradoxes[14]),quantum geometric states dominate the field ones innumber once they are counted (e.g. statistical mod-els of gravity [55, 56]). Examples like the “one-thirdpower” model suggest that quantum geometric uncertain-ties must scale with system size in order to sufficientlylimit the total degree of freedom. But an uncertaintythat scales with distance makes it difficult to create amodel that preserves locality, not to mention that a di-rectionally isotropic strain noise such as the “one-thirdpower” model violates the fundamental requirement ofdiffeomorphism covariance usually present in theories ofspacetime.

III. HOLOGRAPHIC FLUCTUATIONS

A. Theory of Planckian Directional Entanglement

1. Theoretical Motivation

All effective theories treated thus far have assumed aclassical determinate structure of spacetime and consid-ered metric fluctuations within that structure. However,there are motivations to relax this assumption.

The theory of black holes suggests that the fundamen-tal degrees of freedom present within a region of space-time is not an extensive quantity but instead limited bythe area of a surface bounding a given volume, measuredin Planck units[57–62]. Since it is well-known from con-siderations of quantum mechanical unitarity[63, 64] thatblack holes are objects of maximal entropy, it makes senseto consider the two-dimensional entropy of black holes as

8

an upper limit on the entropy contained within any givenvolume. Such arguments have led to a Lorentz covariantentropy bound that generalizes the same principles intoa more precise upper limit, known as the HolographicPrinciple or Covariant Entropy Bound: the area of an ar-bitrary bounding surface in Planck units must be largerthan the entropy contained throughout light sheets en-closed by that surface[15–18]. This projection of internaldegrees of freedom onto bounding surfaces inspires a non-local formulation of fundamental physics, as exemplifiedby the entanglement across spacelike separations in ba-sic quantum mechanics, that imposes a Planckian boundon quantum degrees of freedom stored within spacetime(which dominate the entropic budget).

Hogan has previously suggested a phenomenologicalmodel of how such holographic bounds might be mani-fested in the real universe[19–25, 65], leading to an exotickind of position fluctuation called “holographic noise.” Inthis model, the nonlocality discussed above arises in aspacetime that is not a fundamental entity but rather anemergent phenomenon in systems much larger than thePlanck scale. A classical concept of spacetime involvespointlike events on a determinate manifold, continuouslymapped onto real coordinates. But in quantum mechan-ics, “position” is a property represented by an operatoracting on a Hilbert space. An added assumption of adefinite background usually leads to contradictions withgravity[14, 66, 67], and there are arguments that the met-ric should also be considered an emergent entity[68, 69].Here, a Hilbert space describes the background space-time degrees of freedom, and operators that act on sucha Hilbert space generate positions of massive bodies.

In a wave picture, spatial information is carried bynull waves, and the Hilbert space describing spacetimedegrees of freedom “collapses” when null surfaces interactwith matter. The waves are Planck bandwidth-limited,so they are fundamentally indeterminate in directionalresolution and transverse localization[65].

A simple, well controlled model of such a geometricalstate can also be expressed by a commutator of geomet-rical operators xµ that approximate classical 4-positioncoordinates of a body, including time, in the macroscopiclimit[23–25]. We write a manifestly Lorentz-covariant 4-dimensional formulation of the commutator[25]:

[xµ, xν ] =i1

2√πxκUλεµνκλlp (30)

where Uλ ≡˙xλ√˙xα ˙xα

, ˙x ≡ ∂x

∂τ(31)

Here Uλ represents an operator in the same form as thedimensionless 4-velocity of a body, and τ denotes propertime.

The xµ are not coordinate variables, but rather op-erators acting on a Hilbert space from which spacetimeemerges. This is not conventional quantum mechanics, inwhich operators describe the position of microscopic bod-ies within a classical spacetime. Instead, these operators

describe the spacetime positions of macroscopic objects,ignoring in this approximation standard quantum me-chanics as well as gravity. Thus it is not a fundamentaltheory, but gives an effective model of quantum geome-try, that agrees with gravitational bounds on directionalinformation. In this model, spacetime itself can undergoquantum entanglement.

Strangely, x0 is an operator that represents propertime, but is not exactly the classical time variable. Thisproper time operator does not commute with the spaceoperators, and therefore implies slightly different clocksin different spatial directions. Since an interferometer canbe thought of as light clocks in two orthogonal directions,this commutator model lends itself well to measurementwith an interferometer. We have seen that the two or-thogonal macroscopic arms of a Michelson interferome-ter contain coherent states of photons that are spatiallyextended[50].

While this commutator describes a quantum relation-ship between macroscopically separated world lines basedon relative position and velocity, we still require thatcausality remains consistent. Causal diamonds surroundevery timelike trajectory, and an approximately classicalspacetime emerges that is consistent wherever the causaldiamonds overlap.

2. Normalization of Holographic Position Fluctuations

To estimate a precise normalization of this model, takea rest frame limit

[xi, xj ] = i1

2√πlpεijkxk (32)

Equation (32) is a spin algebra-like representation ofholographic information that can be used to count pre-cisely the degrees of freedom present in a 2-sphere[25].Consider |l〉, a radial spatial separation eigenstate of twobodies separated by a distance L, labeled by its quantumnumber.

|x|2 |l〉 =1

4πl(l + 1)l2p |l〉 = L2 |l〉 (33)

Within a 2-sphere of radius R = 12√π

√lR(lR + 1)lp exist

lR discrete radial position eigenstates (1 ≤ l ≤ lR). Eachof them have 2l + 1 eigenstates of direction:

N2S =

lR∑l=1

(2l + 1) = lR(lR + 2) ≈ 4π

(R

lp

)2

(34)

The numerical coefficient is chosen to agree withemergent spacetime theories that describe gravityentropically[56]. For a length measurement of scaleL, this gives a precisely characterized transverse meansquare position uncertainty of

〈∆x2⊥〉 =

1

2√πlpL, (35)

which increases with scale.

9

3. Properties of Holographic Position Noise

The sections that follow estimate the signal responsein various interferometer architectures. Because a met-ric with classical position coordinates is not assumed, aholistic analysis is required of the system that includesthe apparatus and the emergent geometry it resides in:a quantum-geometrical system of matter and light. In-stead of a fluctuating metric as in the previous sections,we adopt the following phenomenological model of mat-ter position and light propagation in an emergent space-time:

1. Light propagates in the vacuum of emergent space-time as if it is classical and conformally flat. Thus,longitudinal propagation and photon quantum shotnoise are standard.

2. In the nonrelativistic limit, the position of matteris described by a wavefunction that represents spa-tial information in the geometry, with a transversewidth estimated in in (35). This is not standardquantum mechanics: it is a geometrical wavefunc-tion shared by bodies close together in space, sep-arated by much less than L.

3. Position becomes definite relative to an observer—any timelike world-line— when the Hilbert spaceof the quantum geometry “collapses”, as null wavefronts, propagating in causal diamonds around theobserver, interact with other matter on null sur-faces. The finite wave function width thus givesrise to coherent fluctuations in transverse position,as well as emergent locality: nearby bodies fluc-tuate together with each other, relative to distantones. (Here, “null” is used to describe a set of eventsthat have lightlight separation from an observer’sworldline.)

4. Relationship Between Planckian DirectionalEntanglement and Other Violations of Relativity

In this model of quantum geometry, localized spatial po-sition coordinates do not have an exact physical mean-ing, but emerge from the quantum system. While thecommutation relation is covariant and has no preferreddirection, timelike surfaces are now frame-dependent inways that describe the observer-dependence of a space-time. The system violates Lorentz invariance, but asmentioned in the introduction, is within established ex-perimental bounds. It leads to a directional uncertaintythat decreases with separation. Thus the model reducesto classical geometry in the macroscopic limit.

Thus, the conjectured model is Lorentz covarant, butit is not Lorentz invariant. The violation of Lorentz in-variance here is qualitatively different from previouslyinvestigated possible effects, such as those predicted bysome effective field theories[70]. This type of Lorentz

violation does not result in any dispersive change inphoton propagation. Null particles of any energy inany one direction propagate exactly according to nor-mal special relativity, consistent with current experimen-tal limits from cosmic observations of gamma ray burstswith the Fermi/GLAST satellite[71]. The energy non-dependence of polarization position angle agrees withINTEGRAL/IBIS satellite bounds[72], and we do notpropose any kind of dispersive effect on the propagationof massive particles that could be tested by cold-atominterferometers[73].

Several recent proposals have suggested experimen-tally testable mascroscopic phenomena involving gravitywithin the framework of traditional quantum mechan-ics. One idea is to calculate and observe the quantumevolution of the center of mass of a many-body systemwithin a classical spacetime[74], which is clearly distinctfrom the type of quantum spacetime we are proposing.Another is to create and study quantum superpositionstates of many-particle systems extended across macro-scopic spatial distances[75, 76], which is different fromthe entanglement of spatial degrees of freedom we arediscussing. Lastly, there is a proposal to probe the canon-ical commutation relation of the center-of-mass mode ofa massive object using quantum optics, assuming certainmodifications to the Heisenberg uncertainty relation aris-ing from Planck-scale spacetime uncertainties[77]. Thistype of experiment is suited to test the kind of meta-models surveyed in section II B, especially the “minimumuncertainty” one, but not the type of transverse effect weare hypothesizing.

The phenomena discussed here are of course distinctfrom potential effects of primordial gravitational-wavebackgrounds[78]. The new physics proposed also dif-fers from microscopic non-commutative directional un-certainties in models that consider gravity as the gaugetheory of the deSitter group, in which Planck’s constantis kinematically introduced into gravity through non-commutative generators[79, 80]. These ideas have notled to an effective theory that describes a macroscopi-cally manifested phenomenon.

B. Predictions for Noise in Interferometers

1. Holometer: Simple Michelson

As previously mentioned, Fermilab is commissioningan experimental apparatus designed to be particularlysensitive to this type of spacetime uncertainty, namedthe Holometer. This experiment uses a simple Michel-son interferometer configuration, but is sensitive to rapidPlanckian transverse position fluctuations on a light-crossing time, whose spatial coherence is shaped bycausal structure, as expected for holographic noise.

To predict the noise profile measured in a Michelsoninterferometer, we again need to consider the reflectionsoff of the beam splitter at two different times. Because we

10

are positing a noncommutative spacetime with an uncer-tainty of transverse nature, the one-dimensional formu-lation in (26) is now insufficient. An idealized Michelsoninterferometer in a classical geometry, with arms orientedin the 1 and 2 directions, actually measures the followingmacroscopic quantity:

X(t) = x2(t)− x1(t− 2L/c) (36)

We will approximate the continuous interaction of matterwith null waves as a series of discrete measurements sepa-rated by Planck times, over which a transverse Planckianrandom walk accumulates[23]. We will give two predic-tions, a baseline prediction based on the most likely in-terpretation of the theory and a minimal prediction thatassumes the most conservative limit on the accumulationof spacetime uncertainty between successive collapses ofthe wavefunction.

Baseline Prediction

We write down the autocorrelation function for thearm-length difference X (t) at time lag τ in the followingform:

Ξ(τ) = 〈X(t)X(t+ τ)〉 (37)

=

ctp

2√π

(2L− c |τ |) 0 < c |τ | < 2L

0 c |τ | > 2L(38)

Detailed explanations of these calculations are laid outin previous papers and need not be repeated here, butthe justification for (38) is fairly straightforward. Theaforementioned considerations of an emergent spacetimeobeying causal structures leads us to conclude that thisholographic random walk is bounded by the light roundtrip time 2L/c, as causal boundaries dictate this to be thelongest time interval during which the relative phases intransverse directions deviate before the “memory” is “re-set” (see Figure 3). In concluding this, we are assum-ing that the spacetime degrees of freedom in the trans-verse direction do not collapse upon the interaction ofthe information-carrying null waves with the end mir-rors, and that these null waves complete the two-waytrips between the beam splitter and the end mirrors whileretaining this information. Once we establish that theautocorrelation Ξ(τ) must decrease to zero at 2L/c, it isstraightforward to conclude that the function must de-crease linearly from its peak value at zero lag in order tocontain causal diamonds and reflect causal structure in ascale-invariant way.

The zero-lag value in (38) naturally follows from (32)and (35), but it is important to clarify that the valuedoes not represent the propagation of one null wave overa distance 2L despite its algebraic appearance. SinceX (t) represents the arm-length difference, each arm con-tributes its own uncorrelated portion of the total vari-ance. The uncertainty posited is fundamental to thespacetime itself—that is, the positions of massive bodies,

Figure 3. Causal structures of the baseline and minimal mod-els compared, along with the corresponding behavior in thesignal power of the autocorrelation in differential arm length.Barely overlapping causal diamonds of two interferometersseparated in time by 2L/c leads to a correlation in the base-line model, but in the minimal model, the causal diamondof a single interferometer acts as the causal boundary for thecorrelation signal, a minimal departure from classicality.

such as mirrors—and not the applied to the spatial prop-agation of light, which is not susceptible to the transversefluctuations. The information-carrying null waves onlyaccummulate transverse uncertainties over the physicaldistance of a single arm length L even though the lightbeam travels a distance of 2L. The two reflections offof the beam splitter respectively measure its location intwo orthogonal directions at two different times, and witheach transverse reflection manifests a spatial uncertaintyin the direction transverse to the one being measured.

The corresponding spectrum in the frequency domainis calculated by the cosine transform (see Figure 4) [23]:

Ξ(f) = 2

ˆ ∞0

dτΞ(τ) cos(2πfτ) (39)

=c2tp√π(2πf)2

[1− cos(f/fc)] , fc ≡c

4πL(40)

Ξ(f fc) =2√πtpL

2

= [(2.46641± 0.00007)× 10−22Hz−12 ]2 · L2 (41)

As discussed below in more detail, the actual Holome-ter experiment design includes another feature that isnot described by this simple Michelson model: its sig-nal is obtained by cross-correlating the outputs of dualinterferometers in a nested configuration. Due to entan-glement, if the separation is much smaller than the size

11

Figure 4. Predicted noise spectra for the Holometer fromPlanckian directional entanglement using two different mod-els, calculated in eq. (40) and eq (43). For 40m arms,the low-frequency limits are 9.86 × 10−21m/

√Hz and 6.98 ×

10−21m/√Hz. The minimal model has a low-frequency value

that is smaller by 1/√

2, but the first zero occurs at a fre-quency twice as large compared to the baseline model.

of the apparatus, the cross signal approximates the au-tospectrum estimated here: the beam splitters of the twointerferometers “move together” because the light pathsencompass the same region of space-time and their emer-gent geometries collapse into the same position state.

Recall the fact that a random walk metric fluctuationresulted in a white noise spectrum in section IIC. Equa-tion (40) also gives the same flat spectrum (up to a shoul-der around inverse light travel time). However, the dif-ferent physical assumptions about the origin of the noiselead to different predictions for other interferometer ar-chitectures. The coherent quantum and transverse na-ture of holographic fluctuations lead to counterintuitivebehavior that cannot be reproduced in a model based ona fluctuating metric.

Minimal Prediction

In generating the baseline prediction, we made one as-sumption that is not obvious from the principles laid outin section IIIA 3. We now consider the alternative pos-sibility that the spacetime degrees of freedom collapsesin the transverse directions upon the interaction of theinformation-carrying null waves with the end mirrors.The end mirrors have the effect of isolating the quantumuncertainty of the spacetime inside the apparatus fromits uncertainty relative to the outside (see Figure 3). Ear-lier we considered these null waves making two-way tripsbetween the beam splitter and the end mirrors, but inactuality, we are talking about a conception of emergentspacetime that is covered by these null waves carryingwavefunctions of transverse quantum uncertainty, at allpoints in spacetime and in all directions. Simply con-sider only the incoming waves from the end mirrors tothe beam splitter, and the time-domain autocorrelation

for X (t) changes to:

Ξ(τ) =

ctp√π

(L− c |τ |) 0 < c |τ | < L

0 c |τ | > L(42)

The only difference here is that the “memory” has beenreduced to a single arm length. The zero-lag value andthe linear triangular shape of the function remains un-changed. The corresponding spectrum in the frequencydomain is given by (see Figure 4):

Ξ(f) =2c2tp√π(2πf)2

[1− cos(f/f0

c )], f0

c ≡c

2πL(43)

Ξ(f f0c ) =

1√πtpL

2

= [(1.74401± 0.00005)× 10−22Hz−12 ]2 · L2 (44)

2. GEO-600: Interferometer with Folded Arms

As there are several interferometric experiments at orclose to Planck sensitivity, it is important to generateconsistent predictions for those experiments in order tomake sure that this hypothesis is not already ruled out.Here we will focus on two of the most sensitive exper-iments, first GEO-600 and then LIGO in the followingsection. We will be mostly following the assumptionsmade for the baseline prediction in the previous section.

The GEO-600 detector uses a Michelson interferometerwith each arm folded once to double the distance traveledby the light (see Figure 5(a)). In such a case, we expectthe “memory” to last twice as long. But the actual phys-ical distance being measured is still just a single L, andthe holographic noise only accumulates over a single armlength, starting or ending at the beam splitter. Again,the uncertainty is fundamental to the spacetime itself andnot subject on the light propagating through space. Atransverse uncertainty is manifest in each measurementof the relative distance between two objects, the beamsplitter (BS) and either one of the two end mirrors (Band D). This directional entanglement only results in ob-servable noise at instances of orthogonal reflection of thelight, which happens twice at the beam splitter.

One might raise the objection that if these hypothe-sized uncertainties are inherent to the fabric of space-time itself, the mirrors near the beam splitter (A and C)should appear to move coherently with the beam splitter,allowing the device to accumulate spatial noise over thefull length of the folded arm. In fact, such posited space-time coherency is integral to the design of the Holometer.But for the GEO-600 setup, this coherence would not af-fect the contribution of folded arms to the signal.

As mentioned in the previous section, we may see theinterferometer as a pair of independent orthogonal lightclocks and consider the delocalized quantum modes ex-tended within the arms. Consider the fact that the

12

Figure 5. (a) Schematic diagram of GEO-600. (b) Schematicdiagram of LIGO.

proper time operator does not commute with either ofthe two non-commuting orthogonal space operators, andit becomes clear that a transverse uncertainty is manifestonly upon the measurement of a single non-folded armlength. For example, when the distance between the BSand B is being measured, the horizontal arm comprisedof C, D, and the BS acts as a light clock. Its horizontalposition (and the associated uncertainty), transverse tothe vertical distance being measured, is coherent withincausal bounds.

For a semi-classical explanation, consider localizedmodes of the information-carrying null waves. In thisformulation, the quantum wavefunction of the geometrycollapses every time the null waves interact with matter(e.g. a mirror) even though the actual measurement ismade only when the photon is observed. Consider look-ing at the interferometer from the perspective of B andobserving the location of the BS (and A) via a null wavealong the vertical arm. This reference null wave accumu-lates transverse phase uncertainty over a single verticalarm length. From this perspective we can consider thehorizontal arm as a light clock, and indeed the transverse(horizontal) uncertainty at the BS is coherently applica-ble to C, as the distance between the two is well withinthe causal bounds established by the travel time of said

vertical null wave.However, consider a light beam that has just reflected

off of C and traveling towards D. As the light beam makesthe trip over the length of the horizontal arm, we canimagine another null wave that carries the informationabout the horizontal deviation at C (transverse to the ref-erence null wave) also making the same trip and reachingD at the same time as the light beam. Thus, the uncer-tainties in the horizontal positions of the BS or C do notaffect the travel time of the horizontal light beam dur-ing 3/4 of the beam’s propagation within the arm. Thetransverse (horizontal) uncertainty really only comes intoplay on 1/4 of the beam path, as the reference null wavemeasures the vertical distance from B to the BS and therelative transverse uncertainty created affects the lightbeam’s travel time between D and the BS.

We can now conclude that the low-frequency limit ofthe displacement amplitude spectrum Ξ(f) calculated ineq. (41) must remain the same despite the folded arms,because at frequencies much below inverse light stor-age time the phenomenon described must behave almost“classically,” and the extended “memory” should not af-fect the measured uncertainty at all. If we imagine asimple Michelson interferometer co-occupying the spaceof the GEO-600 detector in parallel configuration, thetwo devices should demonstrate equal phase noise at suchlow frequencies. These requirements lead us to write (seeFigure 6):

X(t) =x2(t)− x1(t− 4L/c) (45)

Ξ(τ) = 〈X(t)X(t+ τ)〉

=

122

ctp2√π

(2 · 2L− c |τ |) 0 < c |τ | < 4L

0 c |τ | > 4L(46)

Ξ(f) =1

22

c2tp√π(2πf)2

[1− cos(f/f ′c)] (47)

where f ′c ≡1

2

c

4πL

This gives the same low-frequency limit as equation (41).We still cannot simply compare this prediction to the

noise in differential arm length measured at GEO-600,because noise from metric fluctuations behaves differ-ently from this posited holographic noise. Experimentsdesigned for gravitational waves look for strains in themetric applied to the entire arm length and the entirelight path. They assume that the perturbation in thelocation of the beam splitter is coherent with those ofthe inboard mirrors (e.g. A and C in Figure 5(a)) inall spatial directions, and hence can claim an increase insensitivity by reflecting light back and forth within thearms. GEO-600 attains a twofold enhancement in sensi-tivity at low frequencies (relative to inverse light storagetime) by making the laser beam do two round trips withineach arm, during which the strain in the metric will af-fect a light path that is twice as long. But for the kindof directional entanglement based on the assumptions in

13

Figure 6. Plots of time-domain autocorrelation functions for X (t) at time lag τ, using the baseline model: eq. (38) for theHolometer, eq. (46) for GEO-600, and eq. (49) for LIGO. All axes have been normalized to the same appropriate unitless scaleto provide a clear comparison between different interferometer configurations independent of experimental parameters. Thesimple Michelson configuration has a Ξ(τ = 0) value that shows the random walk variance in (35) applied over two independentarm lengths, along with a time memory of one light round trip time over one arm length. The folded Michelson configurationhas a time memory that lasts over two light round trip times but preserves the same area underneath the function. The modelfor an interferometer with Fabry-Perot cavities uses a probabilistic summation of the autocorrelation functions for Michelsoninterferometers with arms that are folded multiple times.

section IIIA 3, the inboard mirrors no longer contributethe same noise as the beam splitters, for reasons laidout in detail above. This means that for the purposes ofmeasuring the effects of this noncommutative emergentspacetime, for frequencies lower than inverse light storagetime f = c/4L (to account for the extended “memory”)we must correct the sensitivity of GEO-600 data by a fac-tor of 1

2 (i.e. multiply the position noise level by a factorof 2). Since the light storage time in the arms is quiteshort, this correction applies to the spectrum of GEO-600 over the entire relevant frequency range. (For thisdesign, we do not count the “storage time” representedby the entire power-recycled cavity, only that betweenencounters with the beamsplitter.)

If we convert the holographic noise prediction into agravitational wave strain equivalent expected to be ob-served in GEO-600 (instead of correcting the sensitivityof GEO-600 data for holographic noise), we get:

heq(f f ′c) =1

2·

√Ξ(f f ′c)

L=

1

2

√2tp√π

(48)

= (1.23320± 0.00004)× 10−22Hz−12

In the case of the minimal model, the folded arms ofGEO-600 does nothing to change the causal structure ofthe system (including the length of the “memory”) or thelow-frequency amplitude of the displacement spectrum.So the time and frequency-domain behaviors take theexact same functional forms as the minimal predictionfor the simple Michelson configuration of the Holome-ter, equations (42)∼(44), simply numerically adjustedfor the longer 600m arm length. Converted to a grav-itational wave strain equivalent, we get heq(f f0

c ) =12

√tp/√π = (8.72006± 0.00026)× 10−23Hz−

12 .

3. LIGO: Interferometer with Fabry-Perot Cavities

The prediction for LIGO is subject to more compli-cations than the GEO-600 example. The detector usesFabry-Perot (F-P) cavities within each arm with averagelight storage times (for the device as used in publishedbounds) of r = 35.6 light round trips[8] (see Figure 5(b)).We need to devise a simple model of the response such asystem has to holographic noise.

We will work within the assumptions of the baselinemodel. Think of each incoming light wavefront as hav-ing an exponentially decreasing probability e−(n−1)/r ofmaking at least n round trips within the cavity, and sumthe contribution of each possibility to the total noise. Anatural extension of the arguments made for the 2 roundtrips within a GEO-600 arm gives (see Figure 6):

Ξ(τ) =

∞∑n=1

(e−(n−1)/r − e−n/r)

×

1n2

ctp2√π

(2nL− c |τ |) 0 < c |τ | < 2nL

0 c |τ | > 2nL(49)

Ξ(f) =

∞∑n=1

(e−(n−1)/r − e−n/r)

× 1

n2

c2tp√π(2πf)2

[1− cos(f/f (n)

c )]

(50)

where f (n)c ≡ 1

n

c

4πL

The low-frequency limit is still equal to equation (41).However, in constructing the model above, we in-

troduced an additional assumption that these nonlocalmodes of light respond coherently to holographic noise.While it is tempting to think of localized light being “re-

14

flected” or “transmitted” by the input test mass (ITM),the actual quantum modes in the system are of coursedelocalized. A model of localized light modes can accu-rately be used to describe a light cavity responding totime-fluctuations in the metric, as long as we build intoour calculations the time delays after each successive re-flection. But this implcitly assumes that there is only one“clock,” and for this type of noise generated by a quantumspacetime, there are two independent clocks in noncom-muting orthogonal directions. Therefore the fact thatthe quantum wavefunction of the geometry collapses af-ter each interaction of null waves with matter invalidatessuch a classical calculation. So without a rigorously for-mulated structure for this emergent spacetime, let alonea precise quantum mechanical description of how thesenull wave modes interact with matter, it is unclear thatthis model gives an accurate description of the actualphysical phenomenon.

We could instead think of delocalized light modes beingstored within the cavity for an extended period of timebefore collapsing outside of the cavity, hence averagingthe position variance present at the beam splitter acrossa longer period of time. Such considerations lead us tosuggest eq. (50) only as a best guess, not a rigorousprediction of directional entanglement.

We also need to remove LIGO’s enhancement factorfor gravitational waves that do not apply to the conjec-tured effects of directional entanglement. The correctionhere necessary to generate the applicable noise spectrumis much more involved than the GEO-600 case. LIGO’slight storage time is long enough that the correction fac-tor is no longer constant across its entire measurementspectrum. Also, we might naively assume that an r -foldincrease in light storage time would result in an r -fold en-hancement in sensitivity to gravitational waves, but thisis not actually the case. The sensitivity to gravitationalwave strain is enhanced by a factor somewhat greaterthan r due to optical resonance. The transfer functionfrom an optimally oriented and polarized gravitationalwave strain to the optical phase shift measured at thelaser output is given by (at low frequencies f < c/4πL)[29, 81]:

φ

h(f) =

(16πLr

λ

)1√

1 + (8πfLr/c)2(51)

We have argued that at low frequencies, the predicteddirectional entanglement will manifest in a folded inter-ferometer such as GEO-600 in a manner that seems al-most “classical.” Since we used for LIGO a rough modelof probabilistically summing up light packets that gothrough n round trips before exiting the Fabry-Perotcavities, we argue that at low frequencies LIGO also re-sponds to holographic noise as if it was a simple Michel-son interferometer. The transfer function for a simpleMichelson interferometer operating at the low-frequencylimit, given the same kind of optimally configured gravi-

tational wave strain as above, is given by:

φ0

h(f → 0) =

(2π

λ

)(2L) · 2 · 1

2(52)

where the factor of 2 comes from the two arms contribut-ing equal parts, and the 1

2 comes from translating metricstrain into length strain. This means that the sensitivityof LIGO is reduced by the following factor when mea-suring the uncertainty in beam splitter position due toholographic noise (instead of gravitational waves):

φ

φ0(f) =

4r√1 + (8πfLr/c)2

(53)

The gravitational wave equivalent we expect to detect inLIGO can be calculated by writing

heq(f) =

√Ξ(f)

2 · 12 · L

(φ

φ0(f)

)−1

(54)

and substituting equations (50) and (53) into the expres-sion.

Assuming the minimal model has rather drastic conse-quences for the LIGO case. The ITM creates a boundarycondition that is defined very close to the beam splitter,where the quantum wavefunction containing the space-time degrees of freedom would collapse. One could ar-gue that the beam splitter never “sees” the entire armand the end mirror, leading to a tiny undetectable holo-graphic noise since the ITM is very close to the beamsplitter. We discounted this possibility in the baselinemodel, citing the view that the arms can be regarded as(non-commuting) directional light clocks whose phasesare compared at the beam splitter, but it cannot be con-clusively ruled out.

C. Comparison with Experimental Data

1. Expected Sensitivity for the Holometer

The Fermilab Holometer implements a design that isoptimized to look for this noise from Planckian direc-tional entanglement. By operating at high frequenciesabove 50kHz, it minimizes noise sources that are difficultto control such as seismic noise, thermal noise, or acous-tic noise[6–8]. A flat Poisson shot noise dominates thespectrum at a phase spectral density of[82, 83]:

Φshot =1√nBS

=

√EγPBS

= 9.6× 10−12rad/√Hz (55)

where nBS is the number of photons incident on the beamsplitter per unit time, Eγ = 1.2eV is the energy of theinfrared photons used, and PBS = 2kW is the intracavitypower of the laser after recycling. This noise is about afactor of 330 above the predicted signal from holographicfluctuations, calculated in (41) using the baseline model.

15

Figure 7. Spectra for beam splitter position fluctuations, assuming the noise source to be Planckian directional entanglement(baseline model). To give the appropriate reduced sensitivity in measuring such noise, we increase LIGO’s published noise levelsfor differential arm length by the sensitivity enhancement factor for gravitational wave strain calculated in (53). For GEO-600data, the amplification factor is simply 2. The dashed lines represent uncorrected raw published noise curves. Predicted noisespectra are from eq. (47) for GEO-600 (folded Michelson), eq. (50) for LIGO (Fabry-Perot cavity), and eq. (40) for theHolometer (simple Michelson). Expected levels of flat shot noise are plotted for comparison, as the Holometer operates at ahigh enough frequency that we expect this noise source to dominate. By using an integration time that is N times longer thansampling time, with data from two overlapping interferometers, we can significantly reduce the uncorrelated shot noise by afactor of 1/

√N while keeping the signal from correlated holographic noise constant. A frequency bin ranging from f = c/8L

to c/4L is shown as an example. [50, 51]

However, the Holometer is a set of two overlappingMichelson interferometers co-occupying almost the samespace. If the holographic noise is inherent to the space-time itself, we can reasonably expect the signal to be cor-related across the two devices since the two beam split-ters are close together, well within the causal boundsestablished by null wave travel times. In contrast, shotnoise will obviously be uncorrelated. So we may writethe phase fluctuations observed in the two detectorsas dφtot1i = dφcorri + dφuncorr1i and dφtot2i = dφcorri +dφuncorr2i , where i indexes the data taken each samplingtime. Therefore if we cross-correlate the measured fluc-tuations from the two interferometers and integrate overan extended period of time, the signal from the corre-lated noise from emergent quantum geometries will re-main at a constant level, while the cross-correlation ofthe two uncorrelated shot noises will decrease by a factorof 1/

√N , where N is the ratio of the integration time

over the sampling time[82, 83]:

dφtot1i × dφtot2i

≈ 1

N

N∑i

(dφcorri dφcorri + dφuncorr1i dφuncorr2i ) (56)

=⟨(dφcorr)2

⟩+

2

π

1√N

⟨(dφuncorr)2

⟩(57)

where 2/π is the average absolute value of the phasorinner product between two uncorrelated noise sources ofunit magnitude. This gives a signal-to-noise ratio of:

SNR2 =√Nπ

2

⟨(dφcorr)2

⟩〈(dφuncorr)2〉

(58)

Various choices are possible for frequency binning inthe 1 ∼ 10 MHz range for the actual experimental analy-sis, but here we will just provide one numerical example.

16

Figure 8. Spectra for holographic noise (baseline model),expressed as effective strain noise observed in gravitationalwave interferometers. Published noise curves from LIGO andGEO-600 have been left in their original form to reflect sen-sitivity to gravitational wave strain. Predicted spectra forholographic noise have been converted into gravitational wavestrain equivalents to demonstrate the levels observable in datapublished for gravitational wave strain. The predicted spec-trum for GEO-600 is given by dividing eq. (47) by 2L andobtaining eq. (48), as the folded arm enhances gravitationalwave sensitivity by an amplification factor of 2 that is notapplicable to holographic noise. The spectrum for LIGO isgiven in eq. (54), where the spectrum in eq. (50) is di-vided by the amplification factor from Fabry-Perot cavitiesgiven in eq. (53). The flat strain amplitude for GEO-600,at (1.23320± 0.00004)× 10−22Hz−1/2, is consistent with theunidentified noise detected. [50, 51, 84]

If we set the sampling rate at 2L/c = 2.7×10−7s, we canuse a frequency band ranging up to the Nyquist frequencyf = c/4L = 1.9 MHz. If we choose a bin ranging fromhalf that frequency, f = c/8L = 0.94 MHz, up to thatlimit, we get unity signal-to-noise after an integrationtime of 42 minutes, and highly significant detection afterlonger integration times, as shown in Figure 7. We shouldagain note here that the correlated behavior is only exhib-ited within causally overlapping spacetime regions, andthe kind of cross-correlation over large distances donein the LIGO analysis for stochastic gravitational-wavebackgrounds would not be able to detect this holographicnoise at all.

2. Comparison with Published Data from GEO-600 andLIGO

We present the predicted and measured noise spectrafrom the baseline models in two different ways. Figure 7shows the spectra plotted in terms of beam splitter po-sition noise, relevant to measurements of Planckian di-

rectional entanglement. This figure displays all of thepredicted spectra for holographic noise calculated in theprevious section, and applies the previously calculatedconversions to the noise spectra published by LIGO andGEO-600 in order to obtain the correct levels of sensitiv-ity to this type of effect from noncommutative emergentspacetime.

Figure 8, on the other hand, presents everything interms of equivalent levels of gravitational wave strain.Unlike the preceding plot, published noise curves fromLIGO and GEO-600 are left unaltered. Instead of con-verting the published noise curves to reflect sensitivity toholographic noise, for this plot we have converted the pre-dicted levels of holographic noise into gravitational wavestrain equivalents to show what we expect to be mea-surable in experiments designed to detect gravitationalwave strains coherently applied to folded light paths orFabry-Perot cavities.

It should again be noted that the predicted holographicnoise curves for LIGO are based on an effective modelwithout a rigorous theory of how this Planckian direc-tional entanglement would manifest in delocalized lightmodes within Fabry-Perot cavities. Hence the LIGOcurves should be subject to some uncertainty in regionswhere the prediction based on probabilistic summationdeviates from the reference curve for a simple Michelsonconfiguration (see Figure 7). The prediction for GEO-600is more quantitatively certain. But both predictions ac-tually depend on our lengthy discussion in section III B 2being entirely correct about the nature of orthogonal in-terferometer arms as independent light clocks. It is pos-sible that there are inaccuracies in our arguments aboutHilbert space collapse. We should allow for the possibil-ity that these estimates might be off by factors of 2 whendetermining the amplitude of transverse noise accumu-lated through the distances measured within the arms,or when adding the effects from the two arms in orthog-onal directions. Still, these simple models allow a directcomparison of the various different configurations.

We have also omitted from our analysis any possible ef-fect from the cavities created by signal recycling mirrorsin GEO-600 or LIGO. While we expect the correctionsnecessary to be small, as our posited noise is fundamentalto the spacetime itself, a more careful analysis is certainlydue if any meaningful signal is detected in future exper-imental data.

The data from GEO-600 is of particular note, as it iscurrently the experiment closest to the sensitivity lev-els required to detect the effects of this hypotheticalPlanckian directional entanglement. Current sensitivitylevels[84] are slightly better than the latest published lev-els [50] used in the figures, by approximately 10-15% instrain amplitude at the lowest point. Sources of non-holographic environmental and technical noise are dif-ficult to comprehensively identify[85], but the collab-oration has developed a reliable model for shot noiseand conducted a rough analysis of the frequency regionwhere this is the only significant source of noise. At

17

present, the data and model are consistent with unidenti-fied flat-spectrum noise close to our baseline prediction of(1.23320±0.00004)×10−22Hz−1/2 in eq. (48) and Figure8 [84]. We conclude that holographic noise is not ruledout, but if it exists, should be measurable with a reason-able improvement in sensitivity or with the experimentaldesign of the Holometer.

IV. CLOSING REMARKS

As noted previously, amongst the metric-based meta-models, recent LIGO data has made it clear that the “ran-dom walk” model with the flat measured noise spectrumcannot be valid, and indications are that the strain powerspectrum of the spacetime noise cannot be independentof the physical scales of the measurement appartatus (the“white spacetime noise” model). The only simple theoret-ically motivated prediction that remains untested (albeitlacking in covariance) is the “one-third power” model.

These alternative models assume non-standard coher-ences that do not exist in field theory. Without thosecoherences, a simple field theoretic consideration showsthat the interaction of a finite-width beam with the sur-face boundary conditions of a macroscopic object wouldaverage out the metric fluctuations in a way that wouldexponentially suppress the actually observable deviationin macroscopic distance. Experimental constraints alsorule out the “one-third power” model in this case withoutmacroscopic coherence[54].

The effective theory of position noise suggested byHogan[23] posits a noncommutative spacetime and de-

rives a Planckian random walk noise only in directionstransverse to separation between bodies whose positionis measured, over a time corresponding to that separa-tion. The fluctuations in this case are not describableas perturbations of any metric, but are instead consid-ered to be a result of a directionally entangled spacetimethat emerges from overlapping causal diamonds amongstevents and observers. Therefore more careful attentionmust be paid to the interferometer configuration to de-rive limits, by considering the phases of null waves whoseinteractions with matter define position operators. Ourestimates show that the predicted level of noise is com-parable to the unidentified noise observed in GEO-600,the detector currently most sensitive to this type of noise.The Holometer design should be able to reach a highlysignificant detection, or a constraining upper limit, bysampling data at high frequencies, and using two cross-correlated interferometers co-occupying the same spacein order to average out uncorrelated high frequency shotnoise over long integration times.

ACKNOWLEDGMENTS

We are grateful to Rainer Weiss, Bruce Allen, Hart-mut Grote, Chris Stoughton, and the Holometer team formany helpful discussions, especially their explanations ofexperimental design and data interpretation and analy-ses.

This work was supported by the Department of Energyat Fermilab under Contract No. DE-AC02-07CH11359.

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