Quantum Twists of Space
Exotic Rotational Correlations from Quantum Geometry, Their Effects on Interferometer Signals, and Their Possible Connection with Cosmic Acceleration
Craig Hogan University of Chicago and Fermilab
http://arxiv.org/abs/1509.07997 The Problem with Quantum Gravity
General Relativity agrees perfectly with experiments, but only if you ignore the quantum character of matter
The standard quantum theory of matter agrees perfectly with experiments, but only if you ignore dynamical geometry
So every experiment is OK with one theory or the other
But a blend of the two theories makes crazy predictions
How does dynamical geometry reconcile with the quantum?
Can experiments help guide theory?
2 Rotation in Standard Theory
Newton’s thought experiment: a rotating vessel of water proves the physical reality of absolute space
Mach asked why this local rotation agrees with distant stars
Einstein’s general theory of relativity explains why: local and global frames are connected
3 Absolute Space and Rotation in General Relativity
There is an absolute local inertial frame at every event
GR answers Mach’s question GR gives a complete and exact account of the relationship between space and classical matter, including rotation “Frame dragging” is measured and agrees with GR (Sometimes space rotates a lot, e.g., Kerr black holes) The issue is settled, except for quantum mechanics!
4 Apache Point Observatory lunar laser ranging Gravity Probe- B 5 Quantum systems, entanglement, and correlations
Matter is a quantum system Whole system = superposition of states Every subsystem of a whole is entangled with every other A measurement projects onto a subspace A measurement of one subsystem projects all the others Theory only predicts correlations among observables Nothing “happens” at a definite place or time: no “locality” (although correlations obey causality) entangled subsystems: cat and poison state of the cat subsystem
6 Absolute Space and Quantum Matter
In GR, the local inertial frame is absolute and deterministic, even for infinitesimal volumes
Quantum particles “know about” classical absolute space —- the local inertial frame—- which itself is not a quantum system But matter states are quantum superpositions:
+
In GR, space interacts with a classical model of matter, even though real matter is quantum
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2 C C ⇡ ↵ (in Planck units) (in Planck units) ⇡ /m / ⇡ ~ cl ihwavelength with scale ( ` GM ) m P nm 3 P / si h hnrska ytm hs navlm iha with volume a in Thus, system. Chandrasekhar the in as , e ⌘ ` /m ⌘ c/ P Chandra b 2 ( ↵ R ⇡ ( P ~ m/m m/m rnefo h ytmsuidb hnrska sthat is Chandrasekhar by studied system the from erence ) 3 ⇡ 3 m M . , 8 P 2 m Chandra P /m P P 2 ) /R ) /m R e 2 2 Chandra m , S . b 2 b , =2 nydpnson depends only , M n m M Chandra . e qasta feetos eoetheir Denote electrons. of that equals = ihe atce il agrcritical larger a yield particles lighter ; nPac units. Planck in m ~ e /mc . estesaefralstellar-mass all for scale the sets navolume a In . m b hc ae tcerwhy clear it makes which , ↵ c u discussion our ect R 3 h edhas field the , R C = M ↵ ect (9) (8) (7) (6) (5) (4) (3) C 4 , 3 No Absolute Space at the Planck Length Scale
Imagine a quantum Newton bucket a Planck length across It is a superposition of spin states, like an atom + Components of spin do not commute Rotational indeterminacy ~ one radian per Planck time Frame dragging by a quantum bucket makes it impossible to define determinate, classical local rotation at the Planck scale The local inertial frame does not exist at small scales
This is radically different from standard absolute space
Frames become nearly classical in larger, longer-lived systems
But small, exotic quantum-gravitational rotational correlations must exist that do not occur in standard theory
9 Conflict of Quantum Fields with Gravity on Large Scales
Consider a standard quantum field in a volume of size R Its general state is a superposition of excitations Some of these have ~1 particle per independent mode of energy m Mass of this state exceeds black hole mass for R>m-2 in Planck units This field state is impossible in General Relativity
The paradox occurs for large R, even at well studied values of m
Very large R ==> cosmological constant catastrophe (more later)
Conflict could simply be an artifact of disregard for exotic correlations with geometry on large scales Entanglement reduces independent degrees of freedom
10 Exotic Correlations of Quantum Geometry
Suppose space is a quantum subsystem, as well as matter There is no background absolute space or time Geometry is indeterminate— a superposition of states Geometry is entangled with the matter subsystem
Finite information ==> finite number of degrees of freedom ==> new, exotic correlations not in standard theory
What is the character of exotic correlations on large scales?
How can we study them?
11 Magnitude of Exotic Correlations in Large Systems
There is no standard theory of quantum gravity
But estimates of exotic correlation follow from simple thought experiments using standard quantum mechanics and gravity
Estimate from standard quantum theory: the Planck diffraction scale This is much larger than the Planck length (following slides) Agrees with current experiments Also about the right amount of correlation needed to resolve conflicts of dynamical space-time and quantum matter
12 3
FIG. 1. Causal diamonds. FIG. 2. A comparion of causal support in the two classes of correlation functions. and the positions of world-lines in some measurements can decohere gradually with time. The 2D density of position eigenstates— that is, the number of independent world-line eigenstates per 2-volume in a particularx ˆ projection— 1 has a finite value given by ⌅ . Its value depends on the projection defined by the measurement, and the invariant classical positional relationships between the world lines, particularly the causal diamonds traced by light propagation in the apparatus. The scaling with ⌧ can be estimated from dimensional considerations, or from analogy with other, standard quantum systems. It is plausible that world lines decohere slightly at long durations, by an amount that would not yet have been detected. As one example of a quantum system with this behavior, consider the standard position wave function for the state of a particle of mass m at rest, that lasts for duration ⌧. This form of standard Heisenberg uncertainty3 can be derived in several ways[23–26], for example from a nonrelativistic Schr¨odinger equation for a mass m,ina path integralHere, F (⌧ approach) denotes a by projection extremizing determined the action by the of configuration a particle of or a body measurement whose motion apparatus. is described In general, correlationsby a wave equation with decan Broglie be measured wavelength for two world/mc lines,, or from sayx ˆ aA Wheeler-Deandx ˆB; most Witt of this equation paper focuses for a on pendulum the configurations of mass wherem (ref.A and [1],B Eq. 5.75) causal diamonds are almost~ the same. in the lowThe frequency, positional correlation nearly free encoded particle in ⌅ limit.measures a lack of independence of values ofx ˆ. In a classical space-time, ⌅ Forvanishes; all of these this systems, is possible with because the the standard information assumption density in that a continuum directions is infinite. in space The are same independent, is true in field the theory, wave function in classicalwhich positionassumes a and classical time continuous obeys space-time. In an emergent quantum space-time, ⌅ in general does not vanish, and the positions of world-lines in some measurements can decohere gradually with time. The 2D density of position i eigenstates— that is, the number of independent(@i@ world-line2i(m/ eigenstates~)@t) =0 per. 2-volume in a particularx ˆ projection— (2) 1 has a finite value given by ⌅ . Its value depends on the projection defined by the measurement, and the invariant This equationclassical positional can be solvedrelationships with between pure (infinite) the world plane lines, particularly wave eigenmodes. the causal However, diamonds traced the state by light of propagation a body localized in space,in whose the apparatus. position is prepared and measured at two times separated by an interval much longer than the inverse de BroglieThe frequency, scaling with is better⌧ can described be estimated by from a symmetric dimensional gaussian considerations, solution or from for analogy(r, t), wherewith other,r2 = standardx xi, with quantum a probability systems. It is plausible that world lines decohere slightly at long durations, by an amount that wouldi not yet have densitybeen on detected. constant-time As one surfacesexample of given a quantum by system with this behavior, consider the standard position wave function for the state of a particle of mass m at rest, that lasts for duration ⌧. This form of standard Heisenberg uncertainty 2 r2/ (t)2 can be derived in several ways[23–26], for example ⇤ from a nonrelativistice . Schr¨odinger equation for a mass m,ina (3) path integral approach by extremizing the action|h of a| particlei| / or body whose motion is described by a wave equation Thesewith “paraxial” de Broglie solutions wavelength for~/mc matter, or from de Broglie a Wheeler-De waves Witt are mathematically equation for a pendulum the same of mass as them (ref. standard [1], Eq. normal 5.75) modes of light[35]in the low in a frequency, laser cavity. nearly InStandard free the particle matter Quantum limit. system, Correlationst takes the in place Extended of the States laser beam axis coordinate z (that is, constant-phaseFor all of de these Broglie systems, wavefronts with the standard are nearly assumption surfaces that of constant directions time), in space and are independent,position measurements the wave function at particular times,in over classical an interval position much and time smallerSchrödinger obeys than wave the equation interval for betweenposition of a measurements, free body of mass takesm: the place of thin mirror surface boundary conditions[23]. In this family of solutions,i the rate of spatial spreading depends on the preparation of a (@i@ 2i(m/~)@t) =0. (2) state: d /dt is smaller for larger . In the same way that a larger beam waist leads to a smaller dispersion angle, a largerThis position equation uncertainty can be solved leadsExtended with to pure slower waves (infinite) spreading. obey a plane form of wave Spreading the Heisenberg eigenmodes. of theuncertainty However, wave principle function the state with of a time body is localized unavoidable; in any space, whose position is prepared and measured at two times separated by an interval much longer than the inverse de solution of duration ⌧ has a meanExpress variance as a bound of position on position at correlation least as in large the time as domain: 2 i Broglie frequency, is better described by a symmetric gaussian solution for (r, t), where r = xix , with a probability density on constant-time surfaces given by 2 2 (r(t + ⌧) r(t)) = (⌧) > ~⌧/2m. (4) h i 0 2 r2/ (t)2 ⇤ e . (3) Minimum departure from|h classical| i| / body at rest grows linearly with time These “paraxial” solutions for matter de Broglie waves are mathematically the samewave as function the standard normal modes of light[35] in a laser cavity. In the matter system, t takes the place of the laser beamenvelope axis coordinate z (that is, constant-phase de Broglie wavefronts width areof nearly surfaces of constant time), and position measurements at particular times, over an interval much smallerposition thanwave the interval between measurements, takes the place of thin mirror surface boundary conditions[23]. In thisfunction family of solutions, the rate of spatial spreading depends on the preparation of a state: d /dt is smaller for larger . In the same way that a larger beam waist leads to a smaller dispersion angle, a larger position uncertainty leads to slower spreading.time Spreading duration between of the wave function with time is unavoidable; any h/mc2, de Broglie period solution of duration ⌧ has a mean variance of positionmeasurements at least as large as
2 13 2 (r(t + ⌧) r(t)) = (⌧) > ~⌧/2m. (4) h i 0 Thus, d 0/d⌧ = ~/2m gives the minimum rate of decoherence in the position state of a body at rest over time. Of course, these states also have an associated indeterminacy of momentum. Although the particle is classically “at rest”, and there is a well defined expectation value for its rest frame, the actual velocity is indeterminate. In this situation, the position variance between two times cannot be “squeezed away” into momentum uncertainty, or vice versa; Eq. (4) gives the minimum spread of position between two times for any state[32, 36]. The momentum uncertainty in the minimum-position-uncertainty state decreases with ⌧ like p (2m~/⌧)1/2,whichiswhyde Broglie wave states behave almost like classical world lines on large scales. ⇡ Similarly, if paths in a background quantum space time are described by quantum wave propagation similar to Eq. 2 (2), with a fundamental timescale set by the Planck time (that is, setting m = mP = ~/c tP ), then its emergent world lines decohere over duration ⌧ by about
⌅(⌧) (x(t + ⌧) x(t))2 c2⌧t /2. (5) ⇡h i⇡ P To be sure, this background decoherence cannot be detected by a local measurement, in the same way that quantum decoherence between independent paths of mass mP bodies can be measured by comparing their positions. It may also include exotic correlations between di↵erent directions and components of position, which we have not included. 2 However, the scaling of 0(⌧) and ⌅(⌧)with⌧ are the same: world lines should decohere at a rate given approximately by d⌅/d⌧ c2t /2. ⇡ P Mathematical equivalents: wave function and laser beam
wave equation for a massive body <=> paraxial wave equation for light quantum wave function of a body <=> field amplitude in a cavity body at rest <=> gaussian mode of cavity spreading of wave function in time <=> transverse spreading of beam preparation and measurement of state <=> curvature of mirrors
wave function of a body of mass m around world line
time duration h/mc2
2D width of laser light around beam axis
spatial cavity size wavelength 14 6
Exotic Correlations of Transverse Position and Direction
The arguments just given provide estimates of the amount of information, and the scale of entanglement of geometry and field states. However, they do not reveal the origin or nature of the entanglement. Based on these estimates alone it is possible to imaginePlanck thatDiffraction the entanglement Scale: variance may of beextended of a very state subtle nature. The following arguments more specifically suggest that new Planckian quantum behavior in large systems may appear as long-range, nonlocal geometrical correlations ofModel the the world geometrical lines of quantum massive state bodies. as a Planck They variance take di wave↵erent complementary approximations to the emergent space but all approximatelyclassical ray agree <=> onexpectation the magnitude value of wave and direction character of exotic Planck correlation, related to a transverse width for the geometrical position wave function of a geometrical position operatorx ˆ that depends on separation R approximatelyactual as state is a superposition:
wave function variance <=>2 exotic correlation ⌅(c⌧ = R) xˆ R = R`P , (10) ⇡h ?i Posit that exotic correlation at separation ~R is set by the directional or equivalently, a wave functionresolution of of emergent a Planck wave: direction with a separation-dependent width,
2 2 2 ✓P R xˆ R/R = `P /R (11) h i ⇡h ?i A consistent toy model of a quantum algebra to describe geometrical states with these properties is reviewed in the Appendix. Minimal angular spread of Planckian wave Quantum Geometrical Information Planck 15 radial separation R wavelength Quantum mechanics imposes a number of fundamental limits on the precision of position measurements[22]. One of these[39] enforces a minimum amount of energy associated with measurement of any time interval between events: the time it takes a quantum system to go from one state to an orthogonal state is greater than or equal to ⇡~/2E, where E is the expectation value of the energy of the system above the ground state energy. This result has been used to show[40] that in any volume there is a maximum number of distinguishable events, such that the density of clocks and signals used to measure their separation, GPS-style, not exceed the density of a black hole. In a spacetime volume of radius R and duration ⌧ in Planck units, the bound on the number of events is about R⌧.Sincec⌧ R for a covariantly defined volume (such as a causal diamond), this is the same amount of information as the holographic⇡ estimates from statistical gravity just discussed. Indeed, this argument has been used as a statistical basis for general relativity[40]. Note that the area here combines spatial and temporal dimensions, R and ⌧. This argument is not airtight; for example, the measurement bounds might not apply to unmeasured degrees of freedom if position information is subject to subtle squeezing. However, it does use an exact formulation of quantum information that applies specifically to event positions. It suggests that large scale exotic Planckian correlations indeed appear in measurements of relative positions of events and world lines, that could be measured by light reflecting o↵ of macroscopic objects.
Planck Di↵raction Limit on Directional Resolution
Holographic entanglement of fields in emergent geometry may be governed by a Planckian bound on directional µ information. An estimate of the directional spread follows directly from the relativistic wave equation, @µ@ = 0. For i!t waves of frequency ! propagating in a beam— a stationary, monochromatic wave state proportional to e (x, y, z), transversely localized in x and y and extended in z— the wave equation becomes
[@2 + @2 2i!@ ] =0, (12) x y z the same equation used to model laser beams. The minimum transverse angular spread is thus the di↵raction limit 2 for a Planck frequency beam[18], setting ! mP c /~: ⇡ ✓2 ` /R . (13) h P i⇡ P Physically, the role played by sharp longitudinal boundary conditions (mirrors) in a laser cavity is played in emergent geometry by causal structure: the requirement that radial position and causal structure have Planck length precision at all separations. The degrees of freedom can be divided into radial information, which describes causal structure, and directional information, which describes two-dimensional angular orientation and transverse position. One factor R/`P of holographic information content describes causal, longitudinal degrees of freedom; the other factor of R/`P Scale of Exotic Correlation in Planck Units
60
) 40 Planck / / 20 mass
Exotic correlation length ~ Planck diffraction scale 0 ~ R1/2 particle (quantum correlation -20 length is plotted as inverse particle mass) equivalent ( -40 log R ~ m-1 -60 0 10 20 30 40 50 60 log(system size /Planck length) 16 Character of Exotic Correlations: shared displacement
Base a model on a radical assumption:
quantum geometrical phase entangles with quantum field phase
==> quantum superpositions of shared spatial displacement
Displacement on Planck diffraction scale is shared by all fields and massive bodies
3+1D structure is not known but is constrained by general principles Covariance between directions ==> strain, rotation and/or shear Time projection: “Collapse of wavefunction” on covariant causal boundaries
Unlike all classical geometry since Euclid! 17 10 that can be combined into a quantity that scales like ✏ijkxk`P , in the same way as the exotic correlation on each hypersurface. In a Sagnac interferometer, the path of light is a closed circuit. The light follows the same path in two directions and + the signal records their phase di↵erence at the beamsplitter. Let ⌧ and ⌧ denote a ne parameters along the path in the two directions. Denote the classical path of the interferometer in space, in the rest frame of the beamsplitter, + by xi (⌧) and xi (⌧), in the two directions around the circuit. Here i =1, 2, 3 again denotes the 3D spatial indices, + although below we will assume a planar apparatus for simplicity. The functions xi (⌧) and xi (⌧) can be visualized as the trajectories of “tracer photons” in each direction around the circuit; they map positions on the circuit in 3-space to points on an interval on the real line, ( C0, +C0), where C0 denotes the circumference (or perimeter) of the circuit, the origin maps to the beamsplitter, and ⌧ represents a time interval in the proper time of the beamsplitter. The + e↵ects of quantum geometry on the measured correlation ⌅(⌧) depends only on the classical path, defined by xi (⌧) and xi (⌧), the positions of a pair of tracer photons that begin and end their circuit at the same time. The tangent vector to the path in each direction is @xj±/@⌧. It will be convenient to express a general form for the functional dependence of ⌅(⌧) on the classical path in terms of the “swept-out area”, A(⌧), similar to the concept used in Kepler’s second law of planetary motion. Define Ai(⌧) as an oriented area in the inertial rest frame, swept out by lines between the beamsplitter and the tracers in the two directions (see Fig. 2). The areas from the beamsplitter to the two tracers change as:
+ + + dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧] (27)
19 dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧]. (28) This operator takes discrete eigenvalues:
These quantities are completely2 determined2 by the geometrical configuration of the apparatus. xˆ l = l(l + 1)`P l . (A5) In general a transverse| swept| | i area can| i also be defined from from the swept area between the two tracers, adding The discreteadditional eigenvalues correspond cross terms: to classical values of separation R according to: R2 l(l + 1)`2 l2`2 (A6) ⌘ P ⇡ P 1 + + where l are positive integers, quantum numbers of eigenstatesdAi⇥ of/dxˆ⌧2.= ✏ijk [dxj /d⌧ + dxj /d⌧][xk (⌧) xk (⌧)]. (29) Let l denote eigenvalues of position components projected in direction| | i. In2 a state l of separation number l,the i | i projected positionThis component operatorx ˆi can corresponds have eigenvalues in to units shear, of `P , rather than rotation about the beamsplitter. We therefore assume here that l = l, l 1,..., l, (A7) the emergent rotational componenti just depends on a combination of the two terms directly related to rotation about giving 2l +the 1 possible observer/beamsplitter values. In an eigenstate with world a definite line, value Eqs. of position (27) in and direction (28).i, This assumption is also testable: if present, the shear terms would be detectable in a Michelson interferometer[15, 16], which is not sensitive to the rotational terms measured in xˆi l, li = li`P l, li . (A8) a Sagnac device. Experiments| i in both| i configurations would distinguish the modes, and10 explore the full character of The total number of position eigenstates in a 3-sphere is transverse exotic position fluctuations. 6 These quantitiesThe areexotic completely correlation determined amplitudelR by the geometrical in the signal configuration is given of by the the apparatus. projection of the exotic transverse position correlation 2 ThusIn any general volume a theretransverse is a maximum sweptQ3 areaS number(R)= can of also(2 distinguishablel + be 1) = definedlR(lR + events, from 2) = ( from suchR/`P that) the, the swept density area of between clocks and the signals(A9) two tracers, adding represented by theN apparatus. The swept area provides this: relations between dAi±(⌧)/d⌧, xk±(⌧) and dxj±/d⌧ combine usedadditional to measure cross their terms: separation, GPS-style, notXl=1 exceed the density of a black hole[37]. In a spacetime volume of radius R andinto duration a function⌧ in Planck of ⌧ units,that the depends bound on on the the number exotic of events correlation is about R✏⌧ijk.Sincexk`Pc⌧ onR eachfor a hypersurface, but where the spatial indices where the last equality applies in the large l limit. Thus, the number of quantum-geometrical position eigenstates covariantly defined volume (such as a causal diamond),1 this is the same amount of information as the holographic⇡ in a volumehave scales all holographically, contracted. as the surface area in Planck+ units. This simple+ Planckian quantization closely estimates from statistical gravity just discussed.dAi⇥/d⌧ = Indeed,✏ijk [ thisdxj argument/d⌧ + dx hasj /d been⌧][x usedk (⌧) as ax statisticalk (⌧)]. basis for general (29) resembles (but is not exactly the same as) the spectrum2 of area states more rigorously derived within the framework relativity[37].The Note that projection the area here can combines be visualized spatial and temporal geometrically dimensions, (Fig.R and 2).⌧. The essential assumptions are just the holographic scaling of loop quantum gravity (Eq. 17). This argument is not airtight; for example, the measurement bounds might not apply to unmeasured degrees of ThisDirect component calculationof transverse corresponds (e.g., ref.[63]) position toalso shear, leads correlations, to rather the following than rotation and product the of about amplitudes standard the beamsplitter. for projective measurements We properties of either therefore of the assume of emergent here that space-time that relate R, t and freedomthe emergent if position rotational information component is subject just to subtle depends squeezing. on a combinationHowever, it does of use the an two exact terms formulation directly of related quantum to rotation about transverse⌧ components. Assumex ˆj,with for simplicityj = i: a planar apparatus, and thereby suppress directional indices for Ai. The swept area rates informationthe observer/beamsplitter that applies specifically world6 to line,event Eqs. positions. (27) It and suggests (28). that This large assumption scale exotic is Planckian also testable: correlations if present, indeed the shear terms appear in measurementsdA±/d⌧ are of relative proportional positions of toevents separation and world lines, from that the could2 origin be measured at any by light point reflecting on o the↵ path, and to the projection of the path would be detectable in a Michelsonli xˆj li interferometer[15,1 li 1 xˆj li =(l + 16],li)(l whichli + 1) is`P not/2, sensitive to the rotational(A10) terms measured in of macroscopic objects. h | Exotic| ih rotational | | i correlations project twists onto causal boundaries a Sagnacon device. the Experimentstransverse in direction both configurations at that point. would distinguish Denote the the modes, angle andbetween explore the the light full character path tangent of and position vectors by again for any i. The left side of equation (A10) can be interpreted as the expected value for the operator transverse✓0 exotic(⌧). This position angle fluctuations. determines the a ne mapping between t and ⌧ via dt/d⌧ =sin✓0, and the swept area via The exotic correlation amplitudePlanckOrientation in Di↵ theraction signalofxˆ everythingl Limit is1 givenonl Directional1 insidexˆ by the boundary Resolution projection is of the exotic transverse(A11) position correlation a superpositionj| i of twists,ih i | relativej to outside represented by the apparatus. The swept area provides this:dA relations±/d⌧ =sin between✓0dAdAi±±(⌧/dt)/d⌧=, xk±R((⌧⌧))sin and dx✓0j±,/d⌧ combine (30) intoforHolographic components a function entanglement with of ⌧jthat= i, depends in of afields state in onl emergenti theof exoticdefinite geometry correlationx ˆi. For mayl>> be✏ 1 governedx and` li on byl each, athis Planckian hypersurface, corresponds bound to but theon directional expectedwhere the spatial indices 6 | i ijk k P ⇡ µ information.variance in components An estimate of of position the directional transverseProjection spread to separation: follows is determined directly from by theinterval relativistic on a wave equation, @µ@ covariant= 0. For have all contracted.where R(⌧)= xi(⌧) . The contribution to the exotic signal correlationi!t at each ⌧, apart from a constant o↵set fixed waves of frequency ! propagating in a beam— a stationary, monochromaticworld wave stateline proportional to e“causal (x, diamond” y, z), The projection can be visualized| | geometrically. The essential2 assumptions are just the holographic scaling of transverselyby localized the boundary in x and y and condition extendedxˆj inli z determined—1 theli wave1 xˆj equation byxˆj . closure becomes of the circuit, is also(A12) fixed by a projection onto the path, in this transverse position correlations, and theh | standard ih projective| i!h i properties of emergent space-time that relate R, t and We can rewritecase this of as transverse a formula for position the variance2 variance of the2 wave (Eqs. function 26,x ˆ in A13) any direction : transverse to separation, ⌧. Assume for simplicity a planar apparatus,[@x + @y and2i thereby!@z] =0 suppress, directional indices for Ai. The(12) swept area rates given by the right hand side of Eq. (A10) in the limit of l>> 1, for l l =? 1 (or indeed for any value of l l << l1/2): dA /d⌧ are proportional to separation from the origin at± anyi point on the path, and| to i the| projection of the path the same± equation used to model laser beams. The minimum transverse angular spread is2 thus1/ the2 di↵raction limit Exotic 2 (⌅(⌧) + constant) = x sin ✓0 = ` R(⌧)sin✓0. (31) foron a the Planck transverse frequency direction beam[14], at setting that! point.m c Denote2/ xˆ: = theR`P . angle between the light pathR( tangent⌧) and(A13) positionP vectors by correlationP h~ ?i h ?i ✓0(⌧). This angle determines the a ne⇡ mapping between t and ⌧ via dt/d⌧ =sin✓0, and the swept area via This simple result is used here to set theamplitude scale of transverse ✓2 = position` /R . variance, for example in Eqs. (32) and (41). The(13) Since this also scales linearlyh withP i RP (⌧), the swept area rate is simply related to the signal correlation by same algebra, with j and ~ in place of x and dA`P ,± yields/d⌧ =sin a formula✓0dA for±/dt the= widthR(⌧ of)sin the✓ wave0, function for transverse (30) componentsPhysically, of the standard role played angular by sharp momentum, longitudinal in the boundary limit of conditions large j : (mirrors) in a laser cavity is played in emergent | | geometry by causal structure: the requirement that radial position and causal(⌅(⌧) structure + constant) have Planck = length` dA/cd precision⌧. (32) where R(⌧)= xi(⌧) . The contribution to the2 exotic signal correlation at each ⌧, apartP from a constantBoundary o↵set fixed at all separations.| The degrees| of freedom can be dividedˆj = intoj ~. radial information, which describes causal structure,(A14) by the boundary condition determined by closureh ?i of| | the circuit,Planck iswave also fixed by a projection onto~2-sphere the path, in this and directional information, which describes two-dimensional angular orientation and transverse position. One factor caseHowever, of transverse thisDefine simple the position formula total does variance swept not appear (Eqs. area in 26,standardA( A13)⌧) as treatments : a sum offunction of angular the momentum. rotational components in the two directions, so that R/`P of holographic information content describes causal, longitudinal degrees of freedom; the other factor of R/`P envelope describes directional degrees of freedom. By contrast, in a full resolution1/2 3D space the two directional dimensions on 2 2 + their own would comprise (R/`P ) degrees(⌅(⌧) of + freedom. constant) = x R(⌧)dA/dsin ✓0 ⌧= `=P R(dA⌧)sin/d✓0.⌧ + dA /d⌧ , (31) (33) h ?i DirectionsAppendix in B: space Experiment emerge from Concept: the Planck Superluminally scale in such Cross-Correlated, a way that directional| Co-located,| resolution| Power-Recycled of the emergent Sagnac space on| Interferometers anySince scale this is given also by scales Eq. (13). linearly Field with statesR( cannot⌧), the exceed swept the area angular rate resolution is simply18 of related geometry, to so the they signal become correlation strongly by entangled with geometry at a separation where Planck di↵raction matches the resolution of matter fields, (m/R)2; The exotic correlation (Eq. 35) may be detectable using an apparatus similar to the Fermilab Holometer[64],⇡ that separation is given by the idealized Chandrasekhar(⌅(⌧) radius, + constant) Eq. (8). = Note`P dA/cd that a⌧. black hole of this size still has a (32) muchreconfigured larger entropy to measure than pure field rotationalstates of mass modes.m Planckian Wave Packet Spreading+ dA/d⌧ = dA /d⌧ + dA /d⌧ , (33) | | | | andThe the directional total enclosed blurring scale area can is also be derived from standard wave equations applied to quantum states of wave packets with Planck frequency[14, 16, 17]. In this estimation, wave functions of world lines of bodies of any mass spread at the same rate as those of particles of Planck mass. The1 scaleC0 can be estimated starting from the standard A = d⌧dA/d⌧ (34) wave function (xi,t) of a massive body in classical position0 2 andˆ time. In its classical rest frame, a massive body obeys a 3+1-dimensional version of the paraxial wave equation (Eq.⌧=0 12), with propagation in time in place of radial distance, and rest mass in place of frequency: when integrated over the total circumference C0. Adding the contributions from the two directions in Eq. (32) then leads to the following formula for exotic2 correlation2 2 in a planar Sagnac interferometer of any shape: [@ + @ + @ 2i(m/~)@t] =0. (14) x y z Again like a laser beam, spreading of this wave function⌅(⌧)/`P with= ⌅ time0/`P is unavoidable;dA(⌧)/cd⌧ any, solution of duration ⌧ has a (35) mean variance of position at least as large as | | where the correlation at zero lag, 2 2 (r(t + ⌧) r(t)) = (⌧) > ~⌧/2m. (15) h i 0 ⌅0 ⌅(⌧ = 0) = 2A0`P /C0, (36) the standard Heisenberg position uncertainty[38–41] for⌘ a body measured at times separated by ⌧.Inthiswave picture,gives the it is total derived mean in exact square analogy phase to di variation↵ractive spreading from the of exotic a light correlation beam. Here, in the length sharp preparatoryunits. This boundary constant is determined by the requirement that the correlation integrates to zero for c⌧ >C0, so that the average rotation vanishes in the classical limit of zero frequency. A specific example of the predicted| | correlation function for a square configuration is shown in Fig. (3). It is interesting to compare the exotic correlations with the classical Sagnac e↵ect. For an apparatus of any shape, the phase shift in length units between light going in two directions around a circuit rotating with angular velocity ⌦0, relative to the local nonrotating inertial frame, is c t =4A0⌦0/c. (37) 11 is determined by a choice of world line or timelike observer. In the emergent space, spatial structure is defined by intersecting light cones from the ends of any interval on the world line, called causal diamonds. World lines, intervals of proper time, and causal diamonds represent invariant emergent geometrical objects. In the emergent system, there is no fixed background space to define an inertial frame. From one moment to the next, the situation depends only on previous relationships. The causal dependences of rotation around a world line are mapped out by the light cones emanating from each event. The total geometrical state of a system includes entanglement of all of the nested causal diamonds, and the total variance in position is the sum of variances from the nested diamonds (see Figs. 2, ??). The e↵ect can be thought of as a quantum di↵erential rotation of space: everything closer than R rotates relative to everything more distant than R, by a very small amount. It is helpful to visualize the fluctuations as a sum of coherent discrete transverse spatial displacements, or quantum-gravitational twists of space, associated with discrete displacements in time. The amplitude corresponds to a Planck scale di↵raction bound on directional resolution in space, or to a discrete Planck random walk in transverse position. On a null trajectory from a distant source to an observer, for each Planck step, space experiences a random transverse displacement from the classical trajectory, in which it shares common with other radial trajectories at the same R. The exotic correlation arises from the sum of coherent transverse displacement at the 2D boundaries of the nested causal diamonds. In the quantum states of an interferometer, fields are entangled with geometry. The measured radiation depends on the phases of light incident on the beamsplitter, which combines geometrical information from macroscopically di↵erent paths through the space[47–49]. The projection of geometrical quantum states is determined by the true causal structure of the space, the therefore by the transverse components to the observer’s radial direction. The relationship between the two is analyzed in next section to compute the e↵ect on an interferometer signal. Behavior of Exotic Rotational Fluctuations in Large Systems Like zero point fluctuations of a field excitation in a vacuum, this quantum “motion” represents a quantum system with a nonzero width of a wave function that has a zero mean. In this picture, the zero point refers to a stationary direction in the classical nonrotating frame. A quantum relationship with matter out to the cosmic horizon prepares the global state and defines zero rotation. A zero net rotation state is defined by the classical geometry, but nearby there is a nonzero quantum variance that depends on distance from an observer. The wave function width is identified with the exotic correlations and fluctuations. Directional degrees of freedom are subject to the usual holographic bounds on information, so the above estimates of directional variance lead to an estimate of the exotic directional correlations and their associated fluctuations, as shown in Figure (2). The exotic correlation and rotational displacement variance at separation R from an observer are about 2 6 d⌅R(⌧)/d⌧ (d x /dR)(dR/d⌧) c`P . (22) ⇡ h ?i ⇡ The rotational displacement is coherentExotic Correlations in all directions of Transverse from the Position observer and at Direction a constant radial separation defined by its causal structure— that is, on a 2+1-D tube of spacelike 2-surfaces swept out in time by causal diamonds with fixed sizesTheR = argumentsc⌧/2. For just calculationalgiven provide estimates simplicity, of the in amount this paper of information, we focus and on the results scale of in entanglement nearly-flat of space geometry relevant for experiments,and field but states. the formulation However, they based do not on reveal causal the diamonds origin or nature should of thegeneralize entanglement. covariantly Based to on curved these estimates spaces. A toy alone it is possible to imagineVisualize that the exotic entanglement rotational may correlations be of a veryas fluctuations subtle nature. The following arguments model ofmore 3D specifically holographic suggest quantum that new positional Planckian states, quantum summarized behavior in large in Appendix systems may (A), appear based as long-range, on the 3D nonlocal spin algebra, provides an example of a consistent holographic model of quantum position states in 3D. We use classical space-time geometrical correlations of theMismatch world lines of “local” of massive and “global” bodies. rotation They around take any di observer↵erent complementary approximations to for the timethe emergent projection, space as but in all Eq. approximately“Twists” (22). Sinceof local agree frame the on states Planck Di↵raction Limit on Directional Resolution Holographic entanglement of fields in emergent geometry may be governed by a Planckian bound on directional µ information. An estimate of the directional spread follows directly from the relativistic wave equation, @µ@ = 0. For i!t waves of frequency ! propagating in a beam— a stationary, monochromatic wave state proportional to e (x, y, z), transversely localized in x and y and extended in z— the wave equation becomes [@2 + @2 2i!@ ] =0, (12) x y z the same equation used to model laser beams. The minimum transverse angular spread is thus the di↵raction limit 2 for a Planck frequency beam[18], setting ! mP c /~: ⇡ ✓2 ` /R . (13) h P i⇡ P Physically, the role played by sharp longitudinal boundary conditions (mirrors) in a laser cavity is played in emergent geometry by causal structure: the requirement that radial position and causal structure have Planck length precision at all separations. The degrees of freedom can be divided into radial information, which describes causal structure, and directional information, which describes two-dimensional angular orientation and transverse position. One factor R/`P of holographic information content describes causal, longitudinal degrees of freedom; the other factor of R/`P Quantum geometry entangles field states on large scales In this model of quantum geometry, Geometrical correlation ~ Planck diffraction scale Field phase is affected by geometric phase Propagating field states become less distinct from each other at large R than in standard theory This effect is not in standard field theory Planck correlation over distance R Planck length field wavefronts in one mode R1/2 field wavefronts in another mode 20 Exotic entanglement resolves conflict of field states with gravity on large scales In-common exotic displacement ~ particle wavelength at -2 R ~ RC (m) = m in Planck units, the Chandrasekhar radius for field states of mass m Fewer independent field states: no gravitational catastrophes Implements previously ad hoc “IR cutoff” to field theory at R>m-2 Consistent with particle experiments (Cohen et al. 1999) 21 log( mass m or inverse displacement/ Planck) - - - 60 40 20 20 40 60 entangles field states with field entangles Exotic quantum geometry geometry Exotic quantum 0 gravity on large scales large on gravity 0 (neutron mass) 10 log ( system 20 size 30 R / Planck standard field theory R 40 (neutron star mass) = m length R = 2M of freedom correlations reduce field degrees In this region: (tiny) exotic -1 50 ) background space (as usual) nearly independent of In this region: field states are Chandrasekhar radius wavelength~ particle Exotic correlation 60 R ~ m -2 at 22 A radical departure from foundations of field theory Planck scale correlations displace field states on much larger length scales The effect is far too small to detect in local particle experiments (about one wavelength at the Chandrasekhar radius) But may affect fields detectably in interferometers 23 log(equivalent particle energy/ Planck mass) - - - 60 40 20 20 40 60 Effect coherently fields quantum of correlations: exotic 0 0 LHC LHC events displaced by distance ~ by distance displaced 10 log ( system 20 size 30 / Planck ~ a macroscopic spacetime volume changes quantum geometry R , e.g. 40 length ~ R by rms displacement probe states with experiments particle ~100 meters ) 50 1/2 R 1/2 , e.g. ~ 30 attometers on scale scale on 60 Exotic correlation R~ m phase over R -1 24 log(equivalent particle energy/ Planck mass) - - - scale ~ scale on correlations phase that measure interferometers to correlations: study exotic program An experimental 60 40 20 20 40 60 0 0 10 R log in space and time, with precision ~ time, precision with and space in ( system 20 size 30 / Planck interferometer ~100 meter size of photon states in 40 length ~10 attometer precision ) 50 R Exotic correlation 60 ~ m -2 R 1/2 25 Interferometers measure large coherent quantum states States inside interferometer cavity Direction and location of are superpositions of two directions a photon inside the cavity is indeterminate A single photon follows two separated trajectories Exotic entanglement with geometry means that quantum fields share a geometrical component of path- dependent phase Sagnac Interferometer 26 15 + e↵ects of quantum geometry on the measured correlation ⌅(⌧) depends only on the classical path, defined by xi (⌧) and xi (⌧), the positions of a pair of tracer photons that begin and end their circuit at the same time. The tangent vector to the path in each direction is @xj±/@⌧. It will be convenient to express a general form for the functional dependence of ⌅(⌧) on the classical path in terms of the “swept-out area”, A(⌧), similar to the concept used in Kepler’s second law of planetary motion. Define Ai(⌧) as an oriented area in the inertial rest frame, swept out by lines between the beamsplitter and the tracers in the two directions (see Fig. 4). The areas from the beamsplitter to the two tracers change as: + + + dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧] (31) dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧]. (32) These quantities are completely determined by the geometrical configuration of the apparatus. In general a transverse swept area can also be defined from from the swept area between the two tracers, adding additional cross terms: 1 + + dA⇥/d⌧ = ✏ [dx /d⌧ + dx /d⌧][x (⌧) x (⌧)]. (33) i ijk 2 j j k k This component corresponds to shear, rather than rotation about the beamsplitter. We therefore assume here that Sagnac Interferometerthe emergent rotational component just depends on a combination of the two terms directly related to rotation about the observer/beamsplitter world line, Eqs. (31) and (32). This assumption is also testable: if present, the shear terms would be detectable in a Michelson interferometer[19, 23], which is not sensitive to theFERMILAB-PUB- rotational terms measured in Light path is a closeda Sagnac circuit device. Experiments in both configurations would distinguish the modes, and explore the full character of transverse exotic position fluctuations. Signal measures classical absolute Sagnacrotation relative Holometer to inertial frame General Formula for a Planar Interferometer The exotic correlation amplitude in the signal is given by the projection of the exotic transverse position correlation Rotating apparatus has different light sagnac holometer represented by the apparatus. The swept area provides this: relations between dAi±(⌧)/d⌧, xk±(⌧) and dxj±/d⌧ combine path length in oppositeinto a directions function of ⌧ that depends on the exotic correlation ✏ijkxk`P on each hypersurface, but where the spatial indices have all contracted. The projection can be visualized geometrically (Figs. 4, 5). Assume for simplicity a planar apparatus, and thereby Effect on phase in eachsuppress segment directional is indices for Ai. The essential assumptions are just the holographic scaling of transverse position given by the projectedcorrelations, component and the of standard projective propertiesmeasurement of emergent space-time that relate R and ⌧. The swept area rate rotationI. SOME on light pathdA/d NOTES ⌧ is proportional to separation from the origin at any point on the path, and to the projection of the path on the transverse direction at that point. Denote the angle between the light path tangent and position vectors by ✓0(⌧). This angle determines the a ne mapping and the swept area via Integral around circuit depends only on total area and rotation rate: dA/d⌧ = R(⌧)sin✓0/2, (34) where R(⌧)= xi(⌧) . The exotic signal correlation at each ⌧, apart from a constant o↵set fixed by the boundary condition determined| | by global layout of an interferometer, is also fixed by a projection onto the path, in this case of t =4Atransverse!/c2 position variance (Eqs. 30, A13) : (1) 2 (⌅(⌧) + constant) = x R(⌧) sin ✓0 = `P R(⌧)sin✓0. (35) h ?i Since this also scales linearly with R(⌧), the swept area rate is simply related to the signal correlation by 27 (⌅(⌧) + constant) = 2`P dA/cd⌧. (36) For the specific case of a Sagnac interferometer, define the total swept area A (⌧) as a sum of the rotational components in the two directions, so that ± + dA /d⌧ = dA /d⌧ + dA /d⌧ , (37) | ± | | | and the total physical enclosed area is 1 P0 A0 = d⌧dA /d⌧ (38) 2 ˆ⌧=0 ± 1925ApJ....61..140M Michelson-Gale experiment (1925): Measured rotation of the Earth Michelson’s team in suburban Chicago, winter 1924, with partial-vacuum pipes of 1000 by 2000 foot interferometer, measuring the rotation of the earth with light traveling in two directions around a loop 28 Exotic Rotational Correlation in Interferometer Signals If exotic rotational correlations exist, they create “holographic noise” in the signal of an interferometer whose light path encloses an area Signal responds to twists in the contained inertial frame, with response determined by world lines of mirrors Predicted signal correlations depends on the shape of interferometer, and spatial structure of exotic correlations 29 Signal in an interferometer is a time series on one world line In a (square) Sagnac it depends on events at four world lines connected by null trajectories The quantum state is delocalized over the whole 4-volume (e.g., Caves 1980) Photon field state is entangled with geometry on this scale World line of beamsplitter 30 3 produce quantum correlations among observable quantities, beyond those of quantum matter. Quantum correlations have an inverse relation to the information in a system: they vanish in the limit of a continuous classical system, which has an infinite amount of information. Theoretical extrapolations from standard theory suggest that quantum gravity limits the amount of information available to a system of matter and geometry, and therefore, creates new forms of correlation not predicted in standard theory. However, it is not known what form those correlations take. Here we consider the hypothesis that quantum-geometrical correlations can be expressed as exotic correlations in the spatial positions of world lines, as traced by trajectories of massive bodies. These correlations can be distinguished from standard quantum correlations by precise measurements of positions of bodies in space. They carry significant, specific and quantitative information about the structure of new physics at the Planck scale. We have previously considered one class of such measurements, those possible with Michelson interferometers[15– 21]. Here, we consider a related but di↵erent class of possible correlations, associated with rotational modes, or pure changes of direction in space.Interferometer In particular, Signal we argue Exotic that Correlation measurements Function sensitive to the phase of radiation whose path encloses a large area may detect correlations due to the imperfect emergence, in a laboratory-scale system, of a local nonrotating frame fromExotic quantum 3+1D relationshipsrotational correlations of Planck project scale onto elements. 1D signal correlation In an interferometer, a quantum-geometrical correlation is ultimately measured as the time domain correlation (or Statistical observable: expressed as a correlation function of equivalently, its frequency-domaindifferential transform) mirror position, of an observable x(t), with dimension of length: ⌅(⌧) x(t)x(t + ⌧) . (1) ⌘h it The quantity x is measuredSignal from correlation a light intensity function thatmust depend depends only on on the the phase path shape di↵erence between beams that travel through an arrangement of mirrors in space. It characterizes the departure from a perfectly static, classical, continuous geometry. A variety of estimates reviewed below suggest that quantum geometry in a system of size c⌧ produces exotic correlations in position with magnitude roughly ⌅(⌧)/c⌧ ` , (2) ⇡ P 3 35 where `P = ~G/c =1.616 10 m denotes the Planck length. They represent a measure of the (inverse of the) information density in the⇥ geometry. We refer to correlations of this magnitude as “exotic” Planck correlations p because they are not produced in standard classical geometry.31 Experiments with interferometers— in particular, the Fermilab Holometer[26, 27]— have shown that it is possible to measure position correlations with instrumental metrology at better than the Planck sensitivity in Eq. (2). Such experiments can in principle probe Planck scale quantum degrees of freedom [15–22]. A signal is produced by light interference from a system of mirrors in a specific spatial arrangment. The time correlation of the signal, as in Eq. (1), depends on the detailed spatial layout of the apparatus, and the detailed character of the exotic 3+1 D position correlations. An experimental program with a variety of configurations can map out the structure of Planck scale quantum correlations and their relationship with properties of matter fields. The projection operators for a given apparatus are not calculable in standard theory, since they depend on a phenomenology for a still-unknown theory of quantum geometry. On the other hand it is often still possible to compute the forms of correlation from the symmetries of the apparatus in the emergent system, that is, the arrangment of mirrors in space, and to make quantitative tests of specific hypotheses about the geometrical correlations. These hypotheses necessarily lay beyond the framework of standard theory, but provide theoretical estimates of the density and character of information in a space-time system that can be compared with the magnitude and character of the correlations in data. In the following sections of this paper, we compute new, specific manifestations of exotic correlation associated with rotation and emergent direction. First however, we review estimates of the magnitude of exotic correlations from extrapolations of well tested physics. Gravitational Bounds on Matter Quantum Degrees of Freedom A classical geometry is a continuous system, with an infinite density of position information. Significant deviations from this approximation are predicted at the Planck scale. A variety of theoretical estimates suggest that exotic correlations, originating from Planck scale quantum geometry, may have a magnitude in large systems that corresponds to position displacements much larger than the Planck length. Those estimates are reviewed here: first, general bounds on the density of information, and then, more specific indications that these should manifest in observable exotic position and direction correlations. 10 that can be combined into a quantity that scales like ✏ijkxk`P , in the same way as the exotic correlation on each hypersurface. In a Sagnac interferometer, the path of light is a closed circuit. The light follows the same path in two directions and + the signal records their phase di↵erence at the beamsplitter. Let ⌧ and ⌧ denote a ne parameters along the path in the two directions. Denote the classical path of the interferometer in space, in the rest frame of the beamsplitter, + by xi (⌧) and xi (⌧), in the two directions around the circuit. Here i =1, 2, 3 again denotes the 3D spatial indices, + although below we will assume a planar apparatus for simplicity. The functions xi (⌧) and x i (⌧) can be visualized as the trajectories of “tracer photons” in each direction around the circuit; they map positions on the circuit in 3-space to points on an interval on the real line, ( C0, +C0), where C0 denotes the circumference (or perimeter) of the circuit, the origin maps to the beamsplitter, and ⌧ represents a time interval in the proper time of the beamsplitter. The + e↵ects of quantum geometry on the measured correlation ⌅(⌧) depends only on the classical path, defined by xi (⌧) and xi (⌧), the positions of a pair of tracer photons that begin and end their circuit at the same time. The tangent vector to the path in each direction is @xj±/@⌧. It will be convenient to express a general form for the functional dependence of ⌅(⌧) on the classical path in terms of the “swept-out area”, A(⌧), similar to the concept used in Kepler’s second law of planetary motion. Define Ai(⌧) as an oriented area in the inertial rest frame, swept out by lines between the beamsplitter and the tracers in the two directions (see Fig. 2). The areas from the beamsplitter to the two tracers change as: + + + dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧] (27) 19 dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧]. (28) This operatorThese takes quantities discrete eigenvalues: are completely determined by the geometrical configuration of the apparatus. xˆ 2 l = l(l + 1)`2 l . (A5) In general a transverse swept| | | i area canP | alsoi be defined from from the swept area between the two tracers, adding The discreteadditional eigenvalues cross correspond terms: to classical values of separation R according to: R2 l(l + 1)`2 l2`2 (A6) ⌘ P ⇡ P 1 + + dAi⇥/d⌧ = ✏ijk [dxj /d⌧ + dxj /d⌧][xk (⌧) xk (⌧)]. (29) where l are positive integers, quantum numbers of eigenstates of xˆ 2. 2 Let l denote eigenvalues of position components projected in direction| | i. In a state l of separation number l,the i | i projectedThis position component operatorx ˆi can corresponds have eigenvalues to in shear, units of ` ratherP , than rotation about the beamsplitter. We therefore assume here that the emergent rotational componentli = l, l just1,..., dependsl, on a combination of the(A7) two terms directly related to rotation about the observer/beamsplitter world line, Eqs. (27) and (28). This assumption is also testable: if present, the shear terms giving 2l + 1 possible values. In an eigenstate with a definite value of position in direction i, would be detectable in a Michelson interferometer[15, 16], which is not sensitive to the rotational terms measured in a Sagnac device. Experimentsxˆi l, in li both= li`P l, configurations li . would distinguish the(A8) modes, and explore the full character of | i | i 10 Thetransverse total number of exotic position eigenstatesposition in fluctuations. a 3-sphere is 6 These quantitiesThe exotic are completely correlation determined amplitudelR by the in geometrical the signal configuration is given by of the the projection apparatus. of the exotic transverse position correlation (R)= (2l + 1) = l (l + 2) = (R/` )2, (A9) ThusIn generalanyrepresented volume a transverse there is by a maximumthe swept apparatus.NQ area3S number can of also Thedistinguishable be swept definedR R area events, from provides from suchP that the the swept this: density area relations of between clocks between and the signals twodA tracers,i±(⌧) adding/d⌧, xk±(⌧) and dxj±/d⌧ combine additionalused tointo measure cross a function their terms: separation, of ⌧ GPS-style,that dependsX notl=1 exceed on the the density exotic of correlationa black hole[37].✏ In ax spacetime` on each volume hypersurface, of but where the spatial indices radius R and duration ⌧ in Planck units, the bound on the number of events is about ijkR⌧.Sincek P c⌧ R for a where the last equality applies in the large l limit. Thus, the number of quantum-geometrical position eigenstates covariantly defined volume (such as a causal diamond),1 this is the same amount of information as the holographic⇡ in a volumehave scales all contracted. holographically, as the surface area in Planck+ units. This simple+ Planckian quantization closely estimates from statistical gravity justdA discussed.i⇥/d⌧ = Indeed,✏ijk [ thisdxj argument/d⌧ + dx hasj /d been⌧][x usedk (⌧) as ax statisticalk (⌧)]. basis for general (29) resemblesThe (but is projection not exactly the can same be as) visualizedthe spectrum2 of geometrically area states more rigorously (Fig. 2). derived The within essential the framework assumptions are just the holographic scaling relativity[37]. Note that the area here combines spatial and temporal dimensions, R and ⌧. of loop quantum gravity (Eq. 17). This argument is not airtight; for example, the measurement bounds might not apply to unmeasured degrees of ThisDirect componentof calculation transverse corresponds (e.g., position ref.[63]) to also shear, correlations, leads to rather the following than and rotation product the standard of about amplitudes the projective beamsplitter. for measurements properties We of either therefore of of the assume emergent here space-time that that relate R, t and thefreedom emergent if position rotational information component is subject just to subtle depends squeezing. on a combination However, it does of use the an two exact terms formulation directly of related quantum to rotation about transverse⌧. components Assumex ˆ forj,with simplicityj = i: a planar apparatus, and thereby suppress directional indices for Ai. The swept area rates theinformation observer/beamsplitter that applies specifically world6 to line, event Eqs. positions. (27) It and suggests (28). that This large assumption scale exotic is Planckian also testable: correlations if present, indeed the shear terms appeardA in measurements±/d⌧ are of proportional relative positions to of events separation and world from lines, that the could origin2 be measured at any by point light reflecting on the o↵ path, and to the projection of the path would be detectable in a Michelsonli xˆj li interferometer[15,1 li 1 xˆj li =(l + 16],li)(l whichli + 1)is` notP /2, sensitive to the rotational(A10) terms measured in of macroscopic objects. h | | ih | | i a Sagnacon device. the transverse Experiments direction in both configurations at that point. would Denote distinguish the the angle modes, between and explore the the light full path character tangent of and position vectors by again for any i. The left side of equation (A10) can be interpreted as the expected value for the operator transverse✓0(⌧ exotic). This position angle fluctuations. determinesSignal correlation the a ne from mapping projection between of t and ⌧ via dt/d⌧ =sin✓0, and the swept area via The exotic correlation amplitudePlanckexotic in Di the↵ ractionrotational signalxˆ l Limit is1 given onlcorrelations Directional1 x byˆ the Resolution projection onto of the exotic transverse(A11) position correlation j| i ih i | j represented by the apparatus. Theinterferometer swept area provides light this:pathdA± relations/d⌧ =sin between✓0dAdA±/dti±(⌧)=/d⌧R, x(⌧k±)sin(⌧) and✓0,dxj±/d⌧ combine (30) intoforHolographic components a function entanglement with of ⌧ jthat= i, depends in of a fields state on inli emergent theof exotic definite geometry correlationx ˆi. For mayl>> be✏ 1 governedx and` lion by eachl, a this Planckian hypersurface, corresponds bound to but on the directional expected where the spatial indices variance in components6 of position transverse| i to separation: ijk k P ⇡ “causalµ diamond” haveinformation. all contracted. An estimate of the directional spread follows directly from the relativistic wave equation, @µ@ = 0. For waves ofwhere frequencyR!(⌧propagating)= xi(⌧ in) a. beam— The a contribution stationary, monochromatic to the exotic wave state signal proportional correlation to e i!t (x, at y, z each), ⌧, apart from a constant o↵set fixed Path of2 light in transverselyThe projection localized can in x beand| visualizedy and| extended geometrically.xˆj inli z—1 theli wave1 xˆ Thej equation essentialxˆj . becomes assumptions are just the holographic(A12) scaling of transverseby the position boundary correlations, condition and theh determined| standard ih projective| i!hinterferometer by closurei properties of the of emergent circuit, space-time is also fixed that relate by aR projection, t and onto the path, in this We can rewrite this as a formula for the variance2 of the2 wave functionx ˆ in any direction transverse to separation, ⌧. Assumecase for of simplicity transverse a planar position apparatus, variance[@x + @y and2 (Eqs.i thereby!@z] =0 26, suppress, A13) directional : indices for Ai. The(12) swept area rates given by the right hand side of Eq. (A10) in the limit of l>> 1, for l l =? 1 (or indeed for any value of l l << l1/2): dA /d⌧ are proportional to separation from the origin at± anyi point on the path, and| to i the| projection of the path the± same equation used to model laser beams. The minimum transverse angular spread is thus the di↵raction limit 2 2 1/2 on the transverse direction at thatExotic point. Denote2 xˆ = theR` angle. between the light path tangent and(A13) position vectors by for a Planck frequency beam[14], setting ! mP c / (:⌅(⌧) +P constant) = x sin ✓0 = `P R(⌧)sin✓0. (31) rotational h~ ?i R(⌧) ✓0(⌧). This angle determines the a ne⇡ mapping between t and ⌧ via dt/dh ⌧ =sin?i ✓0, and the swept area via This simple result is used here to set thecorrelations scale of transverse ✓2 = position` /R . variance, for example in Eqs. (32) and (41). The(13) h P i P same algebra,Since with thisj and also~ in scales place of linearlyx anddA`P ,± with yields/d⌧ =sin aR formula(⌧✓),0dA the for±/dt the swept= widthR(⌧ area of)sin the✓ rate wave0, function is simply for transverse related to the signal(30) correlation by componentsPhysically, of the standard role played angular by sharp momentum, longitudinal in the boundary limit of conditions large j : (mirrors) in a laser cavity is played in emergent | | geometry by causal structure: the requirement that radial position and causal structure have Planck length precisionBoundary where R(⌧)= xi(⌧) . The contribution to the2 exotic signal correlation at each ⌧, apart from a constant o↵set fixed at all separations.| The| degrees of freedom can be dividedˆj = intoj ~. radial(⌅ information,(⌧) + constant) which describes = `P causaldA/cd structure,⌧(A14).2-sphere (32) by the boundary condition determined by closureh ?i of| | the circuit, is also fixed by a projection onto the path, in this and directional information, which describes two-dimensionalworld line of beamsplitter angular orientation and transverse position. One factor caseHowever, of transverse this simple position formula does variance not appear (Eqs. in 26, standard A13) treatments : of angular momentum. R/`P ofDefine holographic the information total swept content area describesA(⌧ causal,) as a longitudinal sum of degrees the rotational of freedom; the components other factor of inR/` theP two directions, so that describes directional degrees of freedom. By contrast, in a full resolution1/2 3D space the two directional dimensions on 2 2 their own would comprise (R/`P ) degrees(⌅(⌧) of + freedom. constant) = x R(⌧) sin ✓0 = `P R(+⌧)sin✓0. (31) h ?i DirectionsAppendix in B: space Experiment emerge from Concept: the Planck Superluminally scale in such Cross-Correlated, a way thatdA/d directional⌧ Co-located,= resolutiondA Power-Recycled/d of⌧ the+ emergentdA /d Sagnac space⌧ , on (33) Interferometers Sinceany scale this is given also scales by Eq. linearly (13). Field with statesR( cannot⌧), the exceed swept the area angular rate| resolution is simply| 32 of related geometry,| to so the they signal become correlation strongly| by entangled with geometry at a separation where Planck di↵raction matches the resolution of matter fields, (m/R)2; The exotic correlation (Eq. 35) may be detectable using an apparatus similar to the Fermilab Holometer[64],⇡ that separation is given by the idealized Chandrasekhar(⌅(⌧) radius,+ constant) Eq. (8). = Note`P dA/cd that a⌧. black hole of this size still has a (32) muchreconfigured larger entropy to measure than pure field rotational states of mass modes.m Planckian Wave Packet Spreading+ dA/d⌧ = dA /d⌧ + dA /d⌧ , (33) | | | | andThe the directional total enclosed blurring area scale is can also be derived from standard wave equations applied to quantum states of wave packets with Planck frequency[14, 16, 17]. In this estimation, wave functions of world lines of bodies of any mass spread at the same rate as those of particles of Planck mass.1 The scaleC0 can be estimated starting from the standard A = d⌧dA/d⌧ (34) wave function (xi,t) of a massive body in classical position0 2 andˆ time. In its classical rest frame, a massive body obeys a 3+1-dimensional version of the paraxial wave equation (Eq.⌧=0 12), with propagation in time in place of radial distance, and rest mass in place of frequency: when integrated over the total circumference C0. Adding the contributions from the two directions in Eq. (32) then leads to the following formula for exotic2 correlation2 2 in a planar Sagnac interferometer of any shape: [@ + @ + @ 2i(m/~)@t] =0. (14) x y z Again like a laser beam, spreading of this wave⌅ function(⌧)/`P with= ⌅ time0/`P is unavoidable;dA(⌧)/cd⌧ any, solution of duration ⌧ has a (35) mean variance of position at least as large as | | where the correlation at zero lag, 2 2 (r(t + ⌧) r(t)) = (⌧) > ~⌧/2m. (15) h i 0 ⌅0 ⌅(⌧ = 0) = 2A0`P /C0, (36) the standard Heisenberg position uncertainty[38–41] for⌘ a body measured at times separated by ⌧.Inthiswave givespicture, the it is total derived mean in exact square analogy phase to variation di↵ractive spreading from the of exotic a light correlation beam. Here, in the length sharp preparatory units. This boundary constant is determined by the requirement that the correlation integrates to zero for c⌧ >C0, so that the average rotation vanishes in the classical limit of zero frequency. A specific example of the predicted| | correlation function for a square configuration is shown in Fig. (3). It is interesting to compare the exotic correlations with the classical Sagnac e↵ect. For an apparatus of any shape, the phase shift in length units between light going in two directions around a circuit rotating with angular velocity ⌦0, relative to the local nonrotating inertial frame, is c t =4A0⌦0/c. (37) 13 We do not have a complete theory of the apparatus projection Pijk for the emergent 3+1 D system, but we can guess at relations between the 3D correlations and the 1D correlation in a Sagnac signal, based on the antisymmetric 15 13 character of rotational motion. The approach is to define geometric quantities in the rest frame of the apparatus + that can be combined into ae↵ quantityects of quantum that scales geometry like ✏ijk onx thek`P measured, in the same correlation way as⌅( the⌧) depends exotic correlation only on the on classical each path, defined by xi (⌧) hypersurface.We do not have a completeand theoryxi (⌧), of the the positions apparatus of a projection pair of tracerPijk photonsfor the that emergent begin 3+1 and D end system, their circuit but we at can the same time. The tangent guessIn a Sagnac at relations interferometer, between thevector the 3D path to correlations the of path light in and is each a the closed direction 1D correlation circuit. is @x Thej±/ in@⌧ light a. Sagnac follows signal, the based same path on the in antisymmetric two directions character of rotational motion. The approach is to define geometric quantities+ in the rest frame of the apparatus and the signal records their phaseIt will di be↵erence convenient at the to express beamsplitter. a general Let form⌧ forand the⌧ functional denote dependencea ne parameters of ⌅(⌧) along on the classical path in terms thethat path can in be the combined two directions. intoof a the quantity Denote “swept-out that the scales classical area”, likeA( path⌧),✏ijk similarx ofk`P the, to in interferometer the the concept same way used in as inspace, the Kepler’s exotic in the second correlation rest law frame of on planetary of each the motion. Define Ai(⌧) hypersurface. + as an oriented area in the inertial rest frame, swept out by lines between the beamsplitter and the tracers in the two beamsplitter, by xi (⌧) and xi (⌧), in the two directions around the circuit. Here i =1, 2, 3 again denotes the 3D In a Sagnac interferometer,directions the path (see of Fig. light 4). is aThe closed areas circuit. from the The beamsplitter light follows to the the same two tracers path+ in change two directions as: spatial indices, although below we will assume a planar apparatus for simplicity.+ The functions xi (⌧) and xi (⌧) can and the signal records their phase di↵erence at the beamsplitter. Let ⌧ and ⌧ denote a ne parameters along be visualized as the trajectories of “tracer photons” in each direction around+ the circuit;+ they+ map positions on the the path in the two directions. Denote the classical path of the interferometerdAi /d⌧ = ✏ijk in[ space,xk (⌧)dx inj /d the⌧] rest frame of the (31) circuit in 3-space to+ points on an interval on the real line, ( P0, +P0), where P0 denotes the perimeter of the circuit, 10 beamsplitter, by xi (⌧) and xi (⌧), in the two directions around the circuit. Here i =1, 2, 3 again denotes the 3D the origin maps to the beamsplitter, and ⌧ represents a time interval in the proper time of the+ beamsplitter. The spatial indices, although below we will assume a planar apparatus for simplicity. The functions xi (⌧) and xi (⌧) can+ e↵ects of quantum geometry on the measured correlation ⌅(⌧) dependsdA only /d⌧ on= ✏ the[x classical (⌧)dx path,/d⌧]. defined by x (⌧) (32) Thesebe visualized quantities as are the completely trajectories determined of “tracer photons” by the geometrical in each direction configuration aroundi the ofijk circuit; thek apparatus. theyj map positions oni the and xi (⌧), the positions of a pair of tracer photons that begin and end their circuit at the same time. The tangent Incircuit general in 3-space a transverse to points swept on an area interval can on also the be real defined line, ( fromP0, + fromP0), wherethe sweptP0 denotes area between the perimeter the two of the tracers, circuit, adding vector to the path in each directionThese quantities is @x±/@⌧ are. completely determined by the geometrical configuration of the apparatus. additionalthe origin cross maps terms: to the beamsplitter, andj ⌧ represents a time interval in the proper time of the beamsplitter. The It will be convenient to expressIn general a general a transverse form for the swept functional area can dependence also be defined of ⌅( from⌧) on from the theclassical swept path area in between+ terms the two tracers, adding e↵ects of quantum geometry on the measured correlation ⌅(⌧) depends only on the classical path, defined by xi 15(⌧) of the “swept-out area”, A(⌧additional), similar crossto the terms: concept1 used in Kepler’s second law of planetary motion. Define A (⌧) 16 and x (⌧), the positions of a pair of tracer photons+ that begin and end+ their circuit at the same time. The tangenti i dA⇥/de⌧↵ects= of✏ quantum[dx geometry/d⌧ on+ thedx measured /d⌧ correlation][x (⌧)⌅(⌧)x depends (⌧)] only. on the classical path, defined by x+(⌧) (29) as an oriented area in the inertial resti frame,ijk swept outj by linesj betweenk the1 beamsplitterk and the tracers in thei two vector to the path in each direction is @andx±x /(@⌧⌧),. the2 positions of a pair of tracer photons that begin + and end their circuit at+ the same time. The tangent j i dA⇥/d⌧ = ✏ijk [dx /d⌧ + dx /d⌧][x (⌧) x (⌧)]. (33) directions (see Fig. 4). The areas from thevector beamsplitter to the path in each directionto thewheni is two@x±/ integrated@⌧ tracers. change overj the as: totalj perimeterk P0. k It will be convenient to express a general form for the functionalj dependence2 of ⌅(⌧) on the classical path in terms This component corresponds to shear, ratherIt will be thanconvenient rotation to express a aboutgeneralAdding form the the for the contributionsbeamsplitter. functional dependence from We the of ⌅ two( therefore⌧) on directions the classical assume in path10 Eq. in terms(36) here then that leads to the following formula for exotic correlation of the “swept-out area”, A(This⌧), similar component toof the the “swept-out corresponds concept+ area”, usedA( to⌧), similarshear,in+ in Kepler’s ato planar ratherthe+ concept second Sagnac than used in rotation law interferometer Kepler’s of second planetary about law of of theany planetary motion. shape: beamsplitter. motion. Define Define AA Wei(i⌧()⌧ therefore) assume here that the emergent rotational component justas depends andA orientedi /d area on⌧ = in a the✏ combinationijk inertial[xk rest(⌧) frame,dxj swept/d of⌧ the] out by two lines terms between the directly beamsplitter related and the tracers to rotation in the two (31) about asThese an oriented quantities area are completelyin the inertialthe determined emergent rest frame,directions rotational by the (seeswept geometrical Fig. component 4). out The by areas lines configuration from just betweenthe depends beamsplitter of the theon to beamsplitter the a apparatus. twocombination tracers change and as: of the the tracers two terms in the directly two related to rotation about the observer/beamsplitter world line, Eqs. (27) and (28). This assumption is also testable: if present,⌅(⌧)/`P = the⌅0/` shearP dA terms(⌧)/cd⌧ , (39) directions (see Fig. 4). The areas from the beamsplitter to the two tracers+ change+ + as: | ± | In general a transverse sweptthe area observer/beamsplitter can also be defined world from line,from Eqs. thedA swept (31)/d⌧ = and area✏ijk[x (32).( between⌧)dx /d This⌧] the assumption two tracers, is alsoadding testable: (31) if present, the shear terms would be detectable in a Michelson interferometer[15, 16], whichwhere is thei not correlation sensitivek atj zero to the lag, rotational terms measured in additional cross terms: would be detectabledA /d in⌧ a= Michelson✏ijk[x (⌧ interferometer[19,)dx /d⌧]. 23], which is not sensitive to the rotational(32) terms measured in dAi +/d⌧ = ✏ [xk+(⌧)dxj+/d⌧] (31) a Sagnac device. Experiments in both configurationsi wouldijk k distinguishdAj i /d⌧ = ✏ijk the[xk ( modes,⌧)dxj /d⌧]. and explore⌅0 the⌅ full(⌧ = character 0)(32) = 2A0`P /P of0, (40) a Sagnac device.1 Experiments+ in both+ configurations would distinguish the⌘ modes, and explore the full character of transverseThese quantities exotic are position completely fluctuations.transversedA determinedi⇥/d⌧ exotic=These✏ijk quantities by position[dx thej /d are geometrical⌧ completely fluctuations.+ dxj /d determinedgives⌧][ configurationxk the( by⌧) the total geometricalxk mean(⌧)] of. square configuration the apparatus. phase of the variation apparatus. from the(29) exotic correlation in length units. This constant is determined In general2 a transverse swept area can also be defined from from the swept area between the two tracers, adding TheIn general exotic correlationa transverse amplitude swept area in can the also signal be is defined given from by theby from the projection requirement the swept of that the area theexotic between correlation transverse the integrates two position tracers, to zero addingcorrelation for c⌧ >P , so that the average rotation vanishes in the additionaldA cross /d terms:⌧ = ✏ [x (⌧)dx /d⌧]. (32) 0 additionalThis component cross terms: corresponds to shear, rather thani rotationijk aboutk classical the beamsplitter.j limit of zero frequency. We therefore A specific assume example here that of the predicted| | correlation function for a square configuration is represented by the apparatus. The swept area provides this: relations1 between+ dA±+(⌧)/d⌧, x±(⌧) and dx±/d⌧ combine dA⇥/d⌧ = ✏ [dx /d⌧ + dx /d⌧][ix (⌧) x (⌧)]. k j (33) the emergent rotational component just depends on a combinationshowni ofGeneral theijk in two2 Fig. Formulaj terms (6). directlyj for ak Planar related k Interferometer to rotation about intoThese a function quantities of ⌧ arethat completely depends on determined the exotic by1 correlation the geometrical✏ijkx configurationk`P on each hypersurface, of the apparatus. but where the spatial indices the observer/beamsplitter world line, Eqs.This (27) component and (28). corresponds+ This to assumption shear, rather than+ is also rotation testable: about the beamsplitter.if present, We the therefore shear assume terms here that In general a transverse sweptdA areai⇥/d can⌧ = also✏ijk be[dx definedj /d⌧ + fromdxj from/d⌧][x the(⌧ swept) x area(⌧)]. between the two tracers, adding(33) havewould all contracted. be detectable in a Michelson interferometer[15,the emergent2 rotational 16], component which is just not depends sensitivek on a combination to thek rotationalof the two terms terms directly measured related to rotation in about the observer/beamsplitter world line, Eqs. (31) and (32). This assumption is also testable: if present, the shear terms Theadditionala Sagnac projection device. cross terms: Experiments can be visualizedThe in both exotic configurations geometrically. correlation would amplitude The distinguish essential in the the signal assumptions modes, is given and by explore are the just projection the the full character holographic of theClassical exotic of Sagnac scaling transverse E↵ect of position correlation This component corresponds to shear, ratherwould be detectable than rotation in a Michelson about interferometer[19, the beamsplitter. 23], which is not sensitive We therefore to the rotational assume terms measured here in that transverse exotic position fluctuations.represented bya Sagnac the device. apparatus. Experiments The in both swept configurations area provides would distinguish this: therelations modes, and between explore thedA full±i character(⌧)/d⌧ of, xk±(⌧) and dxj±/d⌧ combine transverse position correlations, and thetransverse standard exotic1 position+ projective fluctuations. properties+ of emergent space-time that relate R, t and the emergentThe exotic rotational correlation component amplitudeinto adA function in justi⇥ the/d⌧ dependssignal= of✏⌧ijk isthat given on[dx depends aj by combination/d the⌧ + projection ondxj theIt/d is⌧ exotic of][ interestingx theofk ( the⌧ correlation two) exoticx to termsk compare(⌧ transverse)].✏ directlyijkx thek`P exotic position relatedon each correlations correlation to hypersurface, rotation with the about(33) but classical where Sagnac the spatial e↵ect. For indices an apparatus of any shape, ⌧. Assume for simplicity a planar apparatus, and2 thereby suppressthe phase shiftdirectional in length indices units between for A lighti. The going swept in two area directions rates around a circuit rotating with angular velocity therepresented observer/beamsplitter by the apparatus. worldhave The line, all swept contracted. Eqs. area (31) provides and this: (32). relations This assumption between dA is±( also⌧)/d testable:⌧, x±(⌧) and if present,dx±/d⌧ combine the shear terms General Formula fori a Planar Interferometerk j dA±/d⌧ are proportional to separation from the origin at any!0, relative point on to the the local path, nonrotating and to inertial the projection frame, is of the path wouldThisinto a be component function detectable of ⌧ correspondsthat in a depends MichelsonThe to on shear, the projection interferometer[7, exotic rather correlation can than be visualized rotation 19],✏ijkx whichk`P about geometricallyon is each not the hypersurface, sensitivebeamsplitter. (Figs. to 4, but the 5). We where rotational Assume therefore the spatial for terms assume simplicity indices measured here a planarthat in apparatus, and thereby on the transverse direction at that point.The Denote exotic correlation the amplitude angle in between the signal is given the by light the projection path of tangent the exotic transverse and position correlation vectors by a Sagnacthehave emergent all device. contracted. rotational Experiments componentsuppress in both just directional configurations depends indices on a would combination for Ai distinguish. The of essential the the two assumptions modes, terms directly and areexplore related just the to holographic full rotationc character t =4 aboutA0 scaling!0 of/c. of transverse position (41) represented by the apparatus. The swept area provides this: relations between dA±(⌧)/d⌧, x±(⌧) and dx±/d⌧ combine ✓0(⌧).The This projection angle determines can be visualized the a geometrically.ne mapping between The essentialt and assumptions⌧ via dt/d are⌧ =sin just✓ the0, and holographici the sweptk scaling areaj of via transversethe observer/beamsplitter exotic position fluctuations. worldcorrelations, line, Eqs.into and a(31) function the and standard of ⌧ that (32). depends projective This on the assumptionThis exotic properties formula correlation is can✏ijk also ofx bek`P emergent testable: derivedon each hypersurface, by space-time if summing present, but where the the that the extra shear spatial relate contributions indices termsR and ⌧ from. The each swept straight area photon rate path segment around Projection onto single time transverse position correlations, and the standardhave all contracted. projective propertiesa circuit. of Let emergent t denote space-time the “extra” that propagation relate R, timet and for a photon— the extra time to the next reflection, say— in a wouldThe exotic be detectable correlation in amplitude a MichelsondA/d⌧ is in interferometer[7, proportional theThe signal projection is givento can separation be 19], visualized by which the geometrically projection from is not the (Figs. sensitive origin of 4, the 5). Assume at exotic to any the for point simplicitytransverse rotational on a planar the termsposition apparatus, path, measured and and correlation thereby to the in projection of the path on ⌧. Assume for simplicity a planar apparatus,suppressdA± and/d directional⌧ thereby=sin indices✓ suppress0dA for A±./dtsmall The directional essential= intervalR(⌧ assumptions)sin of indices a✓ straight0, are for just path, theAi holographic. The due to swept scalingthe fact area of transverse that rates the position apparatus(30) is rotating. It is related to the transverse rotation representeda Sagnac device. by the apparatus. ExperimentsProjectionthe The transverse in swept both of 3+1D configurations area direction provides at that this: would point. relationsi distinguish Denote between the the modes, angledAi±( between⌧ and)/d⌧ explore, x thek±(⌧ light) theand path fulldxj± character tangent/d⌧ combine and of position vectors by ✓0(⌧). dA±/d⌧ are proportional to separation fromcorrelations, the origin and the standardat any projective pointd x on properties= theR!0 path,d of⌧ emergentin the and space-timesame to the interval that projection relate by R and of⌧. the The swept path area rate intotransverse a function exotic of ⌧ positionthat dependscorrelation fluctuations.This on angle the onto determines exoticdA/d 1D⌧ is proportional correlation signal the a to separationne✏ijk mappingxk from`P? on the and origin each the at hypersurface, any swept point on area the path, via but and where to the projection the spatial of the path indices on whereon theR(⌧ transverse)= xi(⌧ direction) . The at contribution that point. Denote to the the exotic angle signal between correlation the light path at each tangent⌧, apart and position from a vectors constant by o↵set fixed haveThe all contracted. exotic correlation amplitude in thethe transverse signal is direction given at by that the point. projection Denote the angle of between the exotic the light transverse path tangent and position positioncd vectors correlationt/d by x✓0(⌧=sin). ✓0, (42) | | This angle determines the a ne mapping and the swept area via ? by✓ the0(⌧). boundary This angle condition determines determined the a ne mapping by closure between oft theand circuit,⌧ via dt/d is⌧ also=sindA/d fixed✓0,⌧ and= byR the( a⌧)sin swept projection✓0/ area2, via onto the path, in this (34) representedThe projection by the can apparatus. be visualized The geometrically swept area provides (Fig. this: 4). The relationsusing essential the between same assumptions notationdAi±(⌧ as)/d are before⌧, x justk± (Fig(⌧ the) and 5). holographic Combiningdxj±/d⌧ combine with scaling the swept area (Eq. 34), the total time lab around a case of transverse positionDetermined variance (Eqs. by the 26, path A13) of : dA/d⌧ = R(⌧)sin✓0/2, (34) ofinto transverse a function position of ⌧ that correlations, depends on and thedA the exotic±/d standard⌧ =sin correlation✓0 projectivedA±/dt✏ijk=xcircuitRk properties`(P⌧)sinon is each✓0, of hypersurface, emergent space-time but where that the spatial relate(30)R indices, t and where R(⌧)=wherexRi((⌧⌧)=) .x The(⌧) . The exotic exotic signal correlation correlation at each ⌧, at apart each from⌧ a, constant apart o↵ fromset fixed a by constant the boundary o↵set fixed by the boundary ⌧.have Assume all contracted. for simplicitylight a planar at the apparatus, corresponding| and| | i thereby| suppress directional indices for A . The swept area rates condition determinedcondition determined by global by global2 layout layout1/2 of of an interferometer,an interferometer, is also fixed by is a also projection fixedi onto by the a path, projectionc ind this t =2 case! of ontoA /c. the path, in this case of (43) where R(⌧)= xi(⌧) . Theaffine contribution time(⌅(⌧ lag) + totransverse constant)from the exotic positionthe signal= variance x correlation (Eqs. 30, A13)sin : at✓ each= ` ⌧,R apart(⌧)sin from✓ . a constant o↵set fixed 0 0 (31) dA±The/d⌧ are projection proportional| can| be to visualized separation geometrically from the origin (Fig. 4). at any TheR(⌧ point essential) on0 assumptions theP path, and are0 to just the the projection holographicˆ of the scaling path by the boundary condition determinedtransverse by position closure variance of theh circuit, (Eqs.?i 30, is also A13) fixed : by2 a projection onto the path, in this onof the transverse transverse position direction correlations,beamsplitter, at that point. and in the the Denote standard rest the projective angle(⌅ between(The⌧) properties + constant) total the Sagnac = path of lightx emergent eR ↵of(⌧ect) path sinlight✓ is0 = twice tangent`P space-timeR(⌧ this)sin✓0 amount. and that position because relate vectors itR is, t the(35)and total by di↵erence between opposite directions. case of transverse position variance (Eqs. 26, A13) : h ?i ⌧. Assume for simplicity a planar apparatus,Since this also and scales thereby linearly with suppressR(⌧The), the exotic swept directional area e↵ect rate produces is indicessimply2 related the for to tinyA the. fluctuations signal The correlation swept around by area this rates value, due to the imperfect definition of the frame. The Since✓0(⌧). this This also angle scales determines linearlyframe the with a Rne(⌧ mapping), the swept between areat rate(and⌅(⌧ is)⌧ + simplyvia constant)dt/d related⌧ =sin = to✓x0, the andR(⌧) signal thesin i✓ swept0 = correlation`P areaR(⌧)sin via by✓0. (35) dA±/d⌧ are proportional to separation from the origin2 1 at/2 anyexotic point(⌅ correlation(⌧) on + constant) the path, amplitude =h 2` dA/cd and?i ⌧ for. to a the Sagnac projection device on of any the scale(36) path (Eq. 40), which is a mean square displacement, is equal (⌅(⌧) + constant) = x beamsplittersin ✓0 = ` R(⌧)sin✓0. P (31) R(⌧) to the classicalP displacement for an apparatus rotating at angular velocity !0 = c/2P0, times the Planck length. Once on the transverse directionSince at that this point. alsodAFor the± scales Denote(/d specific⌅(⌧⌧=sin) linearly caseh + the ofconstant)?✓ a Sagnac0i angledA with± interferometer,/dt betweenR =(=⌧`),RP thedA/cd(⌧ define the)sin swept the light⌧✓ total.0, area swept path rate area tangentA is(⌧ simply) as a and sum of related position the rotational to vectors components the signal(34) by correlation(32) by again, the exotic phase displacement± resembles approximately the value of displacement a Planckian random walk, ✓ (⌧). This angle determines the a nein mapping the two directions, between so thatt and ⌧ via dt/d⌧ =sin✓ , and the swept area via Since0 this also scales linearly with R(⌧), the swept area rate is simplybut with related symmtries to the and signal0 projection correlation factors by appropriate to rotation. + Definewhere theR(⌧)= totalx swepti(⌧) . The area contributionA(⌧) as a sum to the of the exotic rotational signal correlation componentsdA (/d⌅⌧(⌧ at=)dA + eachin constant)/d the⌧ +⌧dA, two apart /d⌧ = directions,, 2 from`P dA/cd a constant⌧ so. that o↵set(37) fixed (36) | | | ± | | | by the boundary condition determined bydA(⌅ closure(±⌧/d) +⌧ constant)=sin of the✓0dA =circuit,±`/dtP dA/cd= isR⌧ also(.⌧)sin fixed✓0, by a projection onto the path,(32) in(34) this and the total physical enclosed+ area is case of transverse position varianceFor the specific (Eqs. 30, casedA/d A13) of a⌧ Sagnac := dA interferometer,/d⌧ + dA /d define⌧ , the totalGeneralization, swept area ExperimentA (⌧) as aDesign, sum of Exotic the(33) rotational Nonlocal Correlations components 1 P0 ± whereDefineR the(⌧ total)= x swepti(⌧) . area TheAin contribution(⌧ the) as two a sum directions, of to| the the rotational exotic so| that signal| components correlation inA0 the= at two each| d⌧ directions,dA⌧,/d apart⌧ so from that a constant o↵set fixed(38) ˆ ± | | 2 ⌧=0 by the boundary condition determined by closure of the2 circuit,1/2 is also fixed by a projection onto the path, in this and the total enclosed area is (⌅(⌧) + constant) = +x R(⌧) sinThe✓0 general= `P R formula(⌧)sin (Eq.✓0+. 39) respects the symmetries expected(35) for an emergent geometry, and for a Sagnac dA/d⌧ = dAh /d?⌧i+ dA /d⌧ dA, /d⌧ = dA /d⌧ + dA /d⌧ , (33) (37) case of transverse position variance (Eqs.| 30, A13)| | : apparatus.| ± It appears to have the correct spatial scaling and transformation properties to describe the 1D time C0 | | | | 1 33 domain correlation for a planar interferometer of any shape. It also makes physical sense; for example, in sections Sinceand this the total also enclosed scales linearly area isand with theR total(⌧), the physical swept enclosed area rate2 area1/ is2 simply is related to the signal correlation by (⌅(⌧) + constant)A0 == x ofd⌧sin thedA/d✓ circuit0 =⌧` whereR(⌧)sin light✓0. is not interacting with matter, the(35) path(34) is straight, so that dA (⌧)/cd⌧ is constant for 2 ˆ R(⌧) P | | 1 h C0 ?⌧i=0 corresponding intervals of ⌧. Thus as expected, the correlation is constant over those intervals; the phase of radiation (⌅(A⌧)= + constant)d⌧dA/d = `⌧P dA/cd⌧. 1 P0 (34) (36) Since this also scales linearly with R(⌧), the swept0 2 areaˆ rate isis simply not a↵ected related by trajectories to the signal of emergent correlation world by lines it does not interact with. At reflections, the light path changes when integrated over the total circumference C0. Adding⌧=0 the contributionsA0 = from thed⌧dA two/d directions⌧ in Eq. (32) then (38) directions and the correlation2 ˆ⌧=0 changes± value discontinuously, as expected if the matter fields associated with the leads to the following formula for exotic correlation in a planarpositions Sagnac of the interferometer mirrors are directionally of any entangled shape: with emergent geometry. when integrated over the total circumference (C⌅0(.⌧ Adding) + constant) the contributions = `P dA/cd from⌧. the two directions in Eq. (32) then (36) leads to the following formula for exotic correlation in a planar SagnacThe interferometer Sagnac layout isof not any unique shape: in displaying an e↵ect; an enclosed area and nonzero ⌅0 can be achieved for Mach-Zender, or bent Michelson layouts. In all of these cases, similar formulas can be derived for the correlation, ⌅(⌧)/`P = ⌅0/`P dA(⌧)/cd⌧ , (35) ⌅(⌧)/` = ⌅ /` dA (based⌧)|/cd on⌧ , swept out| area from the beamsplitter. (35) P 0 P | | where the correlation at zero lag, The nonlocal character of the exotic correlations provides opportunities for experiments to distinguish them from where the correlation at zero lag, ordinary sources of environmental position noise. For example, two interferometers very close to each other should be almost entirely entangled, and show a cross correlation almost the same as their individual ⌅(⌧). However, if ⌅0 ⌅0⌅(⌧ ⌅=( 0)⌧ = = 0)2A0 =`oneP 2/CA of0,` themP /C has0, a layout with zero area, say from(36) folding the arms(36) back to the beamsplitter, the exotic correlation ⌘ ⌘ disappears. The formalism here should generalize to computations of cross correlations between interferometers due givesgives the the total total mean mean square square phase phase variation variation from from the exoticthe exotic correlationto correlation entanglement, in length in basedunits. length on This overlap units. constant between This is constant determined their swept is areas. determined by the requirement that the correlation integrates to zero for c⌧ >C , so that the average rotation vanishes in the by the requirement that the correlation integrates to zero| for| c⌧0 >C0, so that the average rotation vanishes in the classicalclassical limit limit of of zero zero frequency. frequency. A A specific specific example example of the of predicted the predicted| correlation| correlation function function for a square for configuration a square configuration is is shown in Fig. (3). shownIt in is interesting Fig. (3). to compare the exotic correlations with the classical Sagnac e↵ect. For an apparatus of any shape, Itthe is phase interesting shift in to length compare units the between exotic light correlations going in two with directions the classical around a Sagnac circuit rotating e↵ect. For with an angular apparatus velocity of any shape, the⌦ phase0, relative shift to in the length local nonrotating units between inertial light frame, going is in two directions around a circuit rotating with angular velocity ⌦0, relative to the local nonrotating inertial frame, is c t =4A0⌦0/c. (37) c t =4A0⌦0/c. (37) 15 + e↵ects of quantum geometry on the measured correlation ⌅(⌧) depends only on the classical path, defined by xi (⌧) and xi (⌧), the positions of a pair of tracer photons that begin and end their circuit at the same time. The tangent vector to the path in each direction is @xj±/@⌧. It will be convenient to express a general form for the functional dependence of ⌅(⌧) on the classical path in terms of the “swept-out area”, A(⌧), similar to the concept used in Kepler’s second law of planetary motion. Define Ai(⌧) as an oriented area in the inertial rest frame, swept out by lines between the beamsplitter and the tracers in the two directions (see Fig. 4). The areas from the beamsplitter to the two tracers change as: + + + dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧] (31) dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧]. (32) These quantities are completely determined by the geometrical configuration of the apparatus. In general a transverse swept area can also be defined from from the swept area between the two tracers, adding additional cross terms: 1 + + dA⇥/d⌧ = ✏ [dx /d⌧ + dx /d⌧][x (⌧) x (⌧)]. (33) i ijk 2 j j k k This component corresponds to shear, rather than rotation about the beamsplitter. We therefore assume here that Exotic signal correlationthe emergent rotational function component expressed just depends as on a swept combination area of the two terms directly related to rotation about the observer/beamsplitter world line, Eqs. (31) and (32). This assumption is also testable: if present, the shear terms would be detectable in a Michelson interferometer[19, 23], which is not sensitive to the rotational terms measured in For any shape ofa Sagnaccircuit, device. correlation Experiments depends in both configurations only on would one distinguish function the modes, and explore the full character of of time lag: transverse exotic position fluctuations. General Formula for a Planar Interferometer A = area swept out by the light path, as a function of time The exotic correlation amplitude in the signal is given by the projection of the exotic transverse position correlation represented by the apparatus. The swept area provides this: relations between dAi±(⌧)/d⌧, xk±(⌧) and dxj±/d⌧ combine into a function of ⌧ that depends on the exotic correlation ✏ijkxk`P on each hypersurface, but where the spatial indices have all contracted. Projects exotic 4DThe rotational projection can fluctuations be visualized geometrically onto time (Figs. domain 4, 5). Assume for simplicity a planar apparatus, and thereby measured in an apparatussuppress directional indices for Ai. The essential assumptions are just the holographic scaling of transverse position correlations, and the standard projective properties of emergent space-time that relate R and ⌧. The swept area rate dA/d⌧ is proportional to separation from the origin at any point on the path, and to the projection of the path on the transverse direction at that point. Denote the angle between the light path tangent and position vectors by ✓0(⌧). This angle determines the a ne mapping and the swept area via dA/d⌧ = R(⌧)sin✓0/2, (34) where R(⌧)= xi(⌧) . The exotic signal correlation at each ⌧, apart from a constant o↵set fixed by the boundary condition determined| | by global layout of an interferometer, is also fixed by a projection onto the path, in this case of transverse position variance (Eqs. 30, A13) : 2 (⌅(⌧) + constant) = x R(⌧) sin ✓0 = `P R(⌧)sin✓0. (35) h ?i Since this also scales linearly with R(⌧), the swept area rate is simply related to the signal correlation by 34 (⌅(⌧) + constant) = 2`P dA/cd⌧. (36) For the specific case of a Sagnac interferometer, define the total swept area A (⌧) as a sum of the rotational components in the two directions, so that ± + dA /d⌧ = dA /d⌧ + dA /d⌧ , (37) | ± | | | and the total physical enclosed area is 1 P0 A0 = d⌧dA /d⌧ (38) 2 ˆ⌧=0 ± 13 The projection can be visualized geometrically (Figs. 3, 4). Assume for simplicity a planar apparatus, and thereby suppress directional indices for Ai. The essential assumptions are just the holographic scaling of transverse position correlations, and the standard geometrical relations between R and ⌧. The swept area rate dA/d⌧ is proportional to separation from the origin at any point on the path, and to the projection of the path on the transverse direction at that point. Denote the angle between the light path tangent and position vectors by ✓0(⌧). This angle determines the a ne mapping and the swept area via dA/d⌧ = R(⌧)sin✓0/2, (30)15 + e↵ects of quantum geometry on the measured correlation ⌅(⌧) depends only on the classical path, defined by xi (⌧) where R(⌧)= xi(⌧) . The exotic signal correlation at eachand xi ⌧(⌧,), the apart positions from of a pair of a tracer constant photons that begin o↵ andset end fixed their circuit by at the the same time. boundary The tangent | | vector to the path in each direction is @xj±/@⌧. condition determined by global layout of an interferometer,It will is be convenientthe projection to express a general of form forthe the functional exotic dependence 3D of transverse⌅(⌧) on the classical position path in terms of the “swept-out area”, A(⌧), similar to the concept used in Kepler’s second law of planetary motion. Define Ai(⌧) variance (Eqs. 26, A13) onto the light path: as an oriented area in the inertial rest frame, swept out by lines between the beamsplitter and the tracers in the two directions (see Fig. 4). The areas from the beamsplitter to the two tracers change as: 2 dA+/d⌧ = ✏ [x+(⌧)dx+/d⌧] (31) (⌅(⌧) + constant) = x R(⌧) sin ✓0 = `P R(⌧)sini ✓0. ijk k j (31)14 h ?i dA /d⌧ = ✏ [x (⌧)dx /d⌧]. (32) The swept area rate is then simply related to the signal correlation by i ijk k j Define the total swept area A(⌧) as a sum of the rotationalThese components quantities are completely in determined the two by the geometrical directions, configuration so of the that apparatus. In general a transverse swept area can also be defined from from the swept area between the two tracers, adding (⌅(⌧) + constant)additional = 2 cross`P dA/cd terms: ⌧. (32) + 1 + + dA⇥/d⌧ = ✏ [dx /d⌧ + dx /d⌧][x (⌧) x (⌧)]. (33) Equation (32) applies separately to eachdA/d arm⌧ or= legdA of an/d interferometer⌧ + dA /d⌧ , ofi anyijk shape.2 j Thej k fullk correlation function(37) | | | | depends on the overall layout. For the specific case of a SagnacThis component interferometer, corresponds to shear, rather define than rotation the about total the beamsplitter. swept We thereforearea assumeA ( here⌧) that as the emergent rotational component just depends on a combination of the two terms directly related to rotation± about and the total enclosed area is the observer/beamsplitter world line, Eqs. (31) and (32). This assumption is also testable: if present, the shear terms a sum of the contributions from both directions, so that would be detectable in a Michelson interferometer[19, 23], which is not sensitive to the rotational terms measured in + a Sagnac device. Experiments in both configurations would distinguish the modes, and explore the full character of dA/d⌧ =1dA P0/dtransverse⌧ + exoticdA position/d⌧ fluctuations., (33) A0 = d⌧dA/d⌧ (38) General Formula for a Planar Interferometer and the total physical enclosed area is 2 ˆ⌧=0 The exotic correlation amplitude in the signal is given by the projection of the exotic transverse position correlation P0 represented by the apparatus. The swept area provides this: relations between dA±(⌧)/d⌧, x±(⌧) and dx±/d⌧ combine 1 i k j when integrated over the total perimeter P0. Adding the contributionsinto a function of ⌧ that depends from on the the exotic two correlation directions✏ijkxk`P on each hypersurface, in Eq. but (36) where the then spatial indices leads A0 = haved all⌧ contracted.dA/d⌧ (34) to the following formula for exotic correlation in a planar2 ˆ⌧ Sagnac=0The projection interferometer can be visualized geometrically of (Figs. any 4, 5). shape: Assume for simplicity a planar apparatus, and thereby suppress directional indices for Ai. The essential assumptions are just the holographic scaling of transverse position correlations, and the standard projective properties of emergent space-time that relate R and ⌧. The swept area rate when integrated over the total perimeter P0.Predicted Adding the exotic contributionsdA/d signal⌧ is proportional correlation to from separation function the from the two origin directions at any point on the path, in Eq. and to the (32) projection then of the leads path on to the following formula for exotic correlation⌅(⌧)/` inP a= planar⌅0/`P interferometerthe transversedA( direction⌧)/cd at that⌧ of, point. any Denote shape: the angle between the light path tangent and position vectors by ✓0(⌧(39)). This angle| determines the a ne| mapping and the swept area via dA/d⌧ = R(⌧)sin✓0/2, (34) where the correlation at zero lag, ⌅(⌧)/`P = ⌅0/`P dA(⌧)/cd⌧, (35) where R(⌧)= xi(⌧) . The exotic signal correlation at each ⌧, apart from a constant o↵set fixed by the boundary condition determined| | by global layout of an interferometer, is also fixed by a projection onto the path, in this case of where the correlation at zero lag, transverse position variance (Eqs. 30, A13) : 2 ⌅0 ⌅(⌧ = 0) = 2A0`P /P0, (⌅(⌧) + constant) = x R(⌧) sin ✓0 = `P R(⌧)sin✓0. (35)(40) h ?i ⌅0⌘ ⌅(⌧ = 0) = 2A0`P /P0, (36) ⌘ Since this also scales linearly with R(⌧), the swept area rate is simply related to the signal correlation by givesgives the the total total mean mean square square phase phase variation variation from from the the exotic exotic correlation correlation in in length(⌅(⌧) units. + units. constant) = This 2 This`P dA/cd constant⌧. constant is is determined determined(36) Correlation expressed inFor terms the specific of case swept of a Sagnac area interferometer, A define the total swept area A (⌧) as a sum of the rotational components by the requirement that the correlation integrates to zero for c⌧ >P , so that the average rotation± vanishes in the by the requirement that the correlation integrates to zeroin for the twoc⌧ directions,>P so00 that, so that the average rotation vanishes in the A0= total enclosed area| | | | + classical limit of zero frequency. A specific example of the predicted correlationdA function/d⌧ = dA /d⌧ + fordA /d a⌧ , square configuration(37) is classical limit of zero frequency. A specific exampleP0= perimeter of the (inverse predicted of free spectral correlation range)| function± | | for a square| configuration is shownshown in Fig. in Fig. (5). (5). lP= Planck length and the total physical enclosed area is 1 P0 It isThe interesting general formula to compare (Eq. the 35) exotic respects correlations the symmetries with expected the classical for an Sagnac emergent e↵Aect. geometry.0 = Ford⌧dA an/d⌧ It apparatus has the correct of any spatial shape,(38) Formula is valid for any shape of light path 2 ˆ⌧=0 ± the phasescaling shift and transformation in length units properties between light toIntegral describe going over the inany two complete 1D time directions circuit domain vanishes: around correlation zero a DC circuit fluctuation for a rotating planar interferometer with angular of velocity any ⌦0, relativeshape. It to also the makes local physical nonrotating sense; inertial for example,Variance frame, scales in is sections like Sagnac of effect the circuit==> covariant where light is not interacting with matter, the path is straight, so that dA(⌧)/cd⌧ is constant for corresponding intervals of ⌧. Thus as expected, the correlation arxiv.org/abs/1509.07997 is constant over those intervals; the phase of radiationc t =4 isA not0⌦ a0/c.↵ected35 by trajectories of emergent world lines it does(41) not interact with. At reflections where the rotational position of the mirror a↵ects the phase of the light (that is, Thewhere exotic the e↵ect mirror produces surface the normal tiny is fluctuations not in the radial around direction), this value, the due signal to correlation the imperfect changes definition value discontinuously, of the frame. At theas Planck expected scale, from the exotic two formulas rotational agree. correlations The exotic in the positions correlation of the amplitude mirrors. for a Sagnac device on any scale (Eq. 40), which is a mean square displacement, is proportional to the classical displacement for an apparatus rotating at angular velocity ⌦0 = c/2P0, times the Planck length; that is, the exotic phase displacement resembles a Planckian random walk in transverseOther spatial Experiment position. Layouts The correlation and Cross-Correlation can also be Between visualized Interferometers as due to rotational fluctuations in the inertial frame, as discussed below. TheThe general Sagnac formula layout (Eq. illustrated 39) is respects not unique: the symmetries the same derivation expected applies for an for emergent any layout geometry, with an enclosed and for area, a Sagnac such apparatus.as a Michelson It appears interferometer to have the with correct a bent spatial arm. The scaling derivation and transformation above can be boiled properties down to to describe a simple conjecture:the 1D time domainthe swept-area correlation formulas, for a planar Eqs. interferometer (35) and (36), gives of any the shape. full form It also of the makes function, physical⌅(⌧), sense; for an for individual example, arm in of sections any of theinterferometer. circuit where In light the general is not case, interacting to satisfy with the matter, integral the constraint, path is the straight, perimeter so thatP0 correspondsdA (⌧)/cd to⌧ theis total constant a ne for correspondingpath length, intervals that is, off0 ⌧=.c/P Thus0,where as expected,f0 denotes the the correlation free spectral is constant range. over those intervals;| the phase| of radiation is notThe a↵ected nonlocal by trajectories character of of the emergent exotic correlations world lines provides it does not opportunities interact with. for experiments At reflections, to distinguish the light path them changes from ordinary sources of environmental position noise. For example, two interferometers close to each other, with overlap- directions and the correlation changes value discontinuously, as expected if the matter fields associated with the ping areas, should be almost entirely entangled, and show a cross correlation almost the same as their individual ⌅(⌧). positions of the mirrors are directionally entangled with emergent geometry. However, if one of them has a layout with zero area, say from folding the arms back to the beamsplitter, the exotic Thecorrelation area quantization disappears. from The Loopformalism Quantum here generalizes Gravity described to computations above (Eq. of cross 19) correlations has an exact between normalization interferometers derived fromdue the to entropy entanglement, of black given hole by states the synchronous that can be overlap used to between set an absolutetheir swept normalization areas. for the correlation function. 1/2 For definiteness, consider a square arrangment with sides L. For a square, Eq. (18) gives n An/AQ LAQ , 1/2 ⇡ ⇡ hence a natural unit of length `Q = AQ . This fixes an absolute normalization for the signal correlationp function, 3 `Q = 2ln2~Gc /p3. (42) q It could be that the appropriate fundamental unit to use in Eq. (39) is not `P , but the loop- normalized Planck length `Q (Eq. 42). However, since they are almost the same, the di↵erence does not matter for the purposes of planning experiments. The di↵erence would however be measurable in an actual experiment, which measures the absolute value of the Planck scale. V. ROTATIONAL FLUCTUATIONS AND COSMIC ACCELERATION The properties of the Planck scale rotational correlations are scale-free power laws, apart from the Planck scale itself. Of course systems in the real world are not scale free; systems including stars and their remnant black holes FERMILAB-PUB- Planck Scale Correlations in Interferometer Signals from Quantum Geometry: Rotational Fluctuations in a Sagnac Circuit Craig Hogan University of Chicago and Fermilab Center for Particle Astrophysics Theoretical estimates are reviewed of bounds on quantum geometrical information, and of the magnitude and character of associated exotic Planckian correlations of position. General principles are sketched that relate observed correlations in interferometer signals with exotic correlations in 3+1 dimensions, depending on quantum geometry theory and interferometer spatial architecture. A specific hypothesis is formulated in algebraic and graphical form for the quantum geometrical correlation in a Sagnac interferometer signal associated with emergence of local, nonrotating inertial frames. A specific prediction is made for the spectrum of observable Planckian signal correlations in a Sagnac interferometer, with a normalization suggested from the area quantization spectrum in Loop Quantum Gravity. I. INTRODUCTION The content of any quantum theory can be expressed as correlations betweenFERMILAB-PUB- observable quantities. The magnitude of quantum correlations has an inverse relation to the information in a system: they vanish in the limit of a classical Signalsystem, Correlations which has an in infinite a Sagnac amount Holometer of information. from Loop In standard Quantum physics, Gravity matter is quantum-mechanical, and geometrical relationships are classical. However, it is generally thought that a quantum system underlies space and time. If so, it should also produce correlations among observable quantities, beyond those of quantum matter. There is no current standard theory of correlations from quantum-geometrical degrees of freedom. Theoretical extrapolations from standard⌅ theory(⌧) suggestx(t) x( thatt + ⌧) quantum gravity limits the amount of information(1) available to a system of matter and geometry, and⌘h therefore, newit forms of correlation not predicted in standard theory. It is not known what form those correlations take. In this paper we consider the hypothesis that quantum-geometrical correlations take the form of exotic correlations ⌅(⌧)=2A ` /C (` /c)(dA/d⌧) (2) in theExample: spatial positions Exotic of world lines,0 P as0 traced P by trajectories of massive bodies, and that these can be distinguished fromcorrelation standard quantum in a square correlations by precise measurements of positions of bodies in space. We argue that such measurementsSagnac inteferometer carry significant, specific and quantitative information about the structure of new physics at the Planck scale. ⌅(⌧)=`P (2A0/C0 dA(⌧)/cd⌧) (3) We have previously considered one class of such measurements, those possible with Michelson interferometers[1–5]. Here, we consider a wider class of possible correlations.L In particular, it is argued that measurements sensitive to the phase of radiation that travels in a closed circuit, as in a Sagnac interferometer, may detect correlations due to the ⌅(⌧)/` =2A /C dA(⌧)/cd⌧ (4) imperfect emergence, in a laboratory-scaleP 0 system,0 of a local nonrotating frame from quantum relationships of Planck L 2L A(⌧) is definedscale inelements. the inertial rest frame as the area swept out by a line connecting the two points a path length c⌧ from the beamsplitterIn an interferometer, along the circuit,2A0/P a0 in= quantum-geometrical L/2 opposite directions correlation is ultimately measured as the time domain correlation (or its frequency domain transform) of an observable x(t), with dimension of length: I. NORMALIZATION FROM LOOP⌅( QUANTUM⌧) x(t)x( GRAVITYt + ⌧) . (1) 0 ⌘h it Discrete spectrumA variety of of area estimates states in reviewed LQG: below suggest that quantum geometry in a system of size c⌧ produces exotic correlations 36 in positionPerimeter P0 with = 5 x 10 magnitude = 80 m roughly 1/2 18 RMS fluctuation (2A0/P0) = 10 = 12 attometers 3 RMS angular rotation rate ~ 1 radian/ A10j4 years=8 ⇡ ~Gc ji(ji + 1) (5) -1 Peak frequency of fluctuation ~ P0 ~ 4 MHz i ⌅(⌧)/c⌧ `P (2) X p ⇡ 36 where j are half-integers i/2 for3 integers i. The Immirizi parameter is estimated from black hole entropy to be i where `P = ~G/c denotes the Planck length. They represent a measure of the (inverse of the) information density =ln2/⇡pin3. theFor large geometry. areas (i>> We1) refer we canto correlations write the eigenvalues of this [of magnitude the “main as sequence” “exotic” states] Planck as correlations because they are not produced inp standard theory. A A n2 (6) Experiments with interferometers—n in⇡ particular,Q the Fermilab Holometer[6]— have shown that it is possible to for integers measuren,where position correlations with Planck sensitivity. A signal is produced by light emerging from a system of mirrors in some spatial arrangment. Their multidimensional correlation a↵ects the observable data stream, as in Eq. (1). 3 AQ 2⇡ ~Gc (7) Such experiments can probe Planck scale⌘ quantum degrees of freedom. The signal depends on the detailed spatial layout of the apparatus, and the detailed character of the exotic 3+1 D position correlations. Thus, an experimental is a fundamental Planck area unit. The di↵erence between adjacent area states is program can literally map the Planck scale correlations[1–5]. In order to estimate the projectionAn+1 operatorsAn 2AQ forn. a given apparatus, we need a phenomenology(8) for a still-incomplete theory of quantum geometry. It is possible ⇡ to constrain the forms of correlation from the symmetries of the emergent Notice thatsystem, the spacing that grows is, the with arrangmentn. Thus for of larger mirrors areas, in theeach emergent area eigenstate geometry, is bigger and than make the lastquantitative by much more tests of specific hypotheses. than a single Planck area pixel. This property reflects the same blurring or loss of relative positional information on large scales that appears in wave equation solutions, holographic information constraints, and other proposed “infrared” departures from field theory. It explicitly captures an essential nonlocal Planckian feature not compatible with a metric-based field approach: the positional relationships of the geometry itself depend on separation, and degrade over large areas. Arguments give above suggest that the correlation function for phase variations around a planar circuit scales with the area divided by the circumference. Not surprisingly, the correlations are related to the shape of the circuit. Consider an area defined by a planar surface between several world lines, say, the centers of mass of mirrors in a Sagnac interferometer, all at rest. In classical language, we propagate this surface forward in time by extending along their world lines in the rest frame. In Quantum General Relativity, there is no independent clock ticking; the equivalent of a Planckian clock tick is defined relationally by a transition between quantum states of the spacetime geometry. Consider the light propagating between the mirrors around this area in two directions from the beam splitter. 1/2 For definiteness, consider a square arrangment with sides L. For a square we have n LAQ , hence a natural 1/2 ⇡ unit of length `P = `Q = AQ . This fixes an absolute normalization for the signal correlation function. In a Design Features of Experiments Universal spectrum Depends only on Planck scale Known dependence on layout of the interferometer Path must enclose an area to show rotational fluctuations No effect from radial propagation Cross Correlation Twists are the same for different instruments that measure the same 4-volume Null configurations can be used to control for environment Noise power is at high frequencies ~ free spectral range Optimal experiment measures superluminal correlation Time sampling and correlation at high frequency 37 An experiment that studies quantum geometry: theThank Fermilab YouHolometer McCuller Fermilab Seminar 77 38 Experimental Site Location A.S. Chou, FNAL, KICP Chicago, 31 4/12/2013 McCuller Fermilab Seminar 31 39 Construction McCuller Fermilab Seminar 35 40 The Fermilab Holometer as currently configured Two collocated Michelson interferometers with cross correlated signals ~ trillions of independent measurements sensitive to well below Planck spectral density 2 the predicted cHN power which is smaller by more than five orders of magnitude from the instantaneous sensi- tivity, the signals from the two interferometers are cross- correlated and averaged. This paper reports on 145 hours of data constituting N>2 108 independent spectra in each interferometer which⇥ are used to achieve a pN improvement in sensitivity to correlated sub-shot-noise spectral power. The broadband nature of the predicted cHN allows further sensitivity improvement by searching for signal power over 2000 resolved frequency bins. The success of the experiment requires low instrumen- tal correlation between the two interferometers and re- quires nearly complete independence of the two devices, despite sharing an experimental hall. Each interferom- eter is enclosed in its own vacuum system, and the in- jection and41 control systems are operated on separate op- tics tables and electronics racks. The digitizers for the FIG. 1: Schematic of the two co-located interferometers and associated data acquisition channels. The two devices are two instruments are isolated and independently synchro- optically and electrically isolated to eliminate cross-talk. nized to GPS. They communicate with the realtime spec- trum processing computer only through optical fiber (see Fig. 1). the device using a 1000 ppm transmission mirror. The in- Data acquisition and methodology — Because of the sertion of this input coupling mirror forms a Fabry-Perot contrast defect light leakage, each interferometer has 200 cavity with free spectral range FSR 3.8 MHz deter- mW output power which provides the desired shot-noise- ⇡ mined by the common arm length (L1 +L2)/2. The laser limited sensitivity. However, the large dynamic range is frequency-locked to the instantaneous cavity frequency between the DC power and the shot noise level presents via the Pound-Drever-Hall (PDH) technique [14, 15] to challenges for linear detection. The output power is split achieve a typical power build-up from the injected 1.1 W by a secondary beam splitter to divide it between two laser power to intracavity power PBS 2.2 kW. The cav- custom low transimpedance photoreceivers based on high ⇡ ity with 650 Hz transmission bandwidth also serves to fil- linearity, 2 mm InGaAs photodiodes. Each detector is ter higher frequency amplitude and phase noise present demonstrated to achieve high linearity with photocur- on the incident laser beam. rents up to 130 mA. A low gain DC amplification chan- To produce a linear response to di↵erential length nel samples the photocurrent and has flat response from perturbations X, each interferometer is operated at a DC-80 kHz. A high gain, transimpedance-based, AC- DARM o↵set of around 1 nm from a dark fringe. A dig- coupled radiofrequency (RF) channel has full gain be- ital control system monitors fluctuations in the output tween 900 kHz-5 MHz and operates up to 20 MHz. light and feeds back di↵erential signals to piezo-electric The two interferometers together thus have four RF actuated end mirror mounts to hold the fringe o↵set output streams digitized at 50 MHz sample rate with 15 to better than 0.5 A˚ RMS, thus maintaining a stable bits. These four channels along with an additional four operating point. At this fringe o↵set, around 50 ppm auxiliary monitor channels are Fast Fourier Transformed of signal-bearing interference light appears at the anti- in real time with 381 Hz frequency resolution. An upper- symmetric port, as measured relative to the intracav- triangular 8 8 cross-spectrum matrix is computed. To ity power. This value is chosen to balance the interfer- reduce the data⇥ rate, measurements of these 36 spectra ence fringe light with the the contrast defect light leakage are averaged over 700 sequential spectral measurements ✏cd 50 ppm. This non-interfering light carries no signal (around 1.8 s) before being stored. The remaining aver- ⇡ but contributes shot noise variance. aging is performed in o✏ine analysis. The shot-noise-limited displacement sensitivity is Isolation of the two interferometers is established by measurement; as described below, the auxiliary channels 2E PSDshot =( /4⇡)2 (2) are used to determine the transfer function of known X · P BS noise sources into the interferometer output, to reduce where E = hc/ is the energy per photon. Tak- the noise coupling, and to monitor the possible remain- ing account of detection ine ciencies and the degrada- ing noise leakage in situ. During the accumulation for tion due to the contrast defect light, the DARM sen- this result the auxiliary channels are set at various times sitivity given by the 2 kW of photon power is around to monitor the PDH laser phase noise, the laser intensity 18 2 (2.5 10 m/pHz) ; this value is confirmed by cali- noise, and loop antennas detecting the local RF environ- bration⇥ measurements as summarized below. To reach ment. Measured Spectrum in the Fermilab Holometer First viewhttp://arxiv.org/abs/1512.01216 of 2 interferometer cross 42 McCuller Fermilab Seminar 71 Holometer Integrated Cross Spectrum Result rules out a model of exotic shear correlation ButFully has no Integratedsensitivity to exotic Crossrotational correlation Spectrum arXiv:1512.01216 Model amplitude Sets weighting Function Shapes statistical power Fully Accumulated Over Bandwidth Done as integral To observe Narrowband vs. Broadband Contributions 43 McCuller Fermilab Seminar 72 10 that can be combined into a quantity that scales like ✏ijkxk`P , in the same way as the exotic correlation on each hypersurface. In a Sagnac interferometer, the path of light is a closed circuit. The light follows the same path in two directions and + the signal records their phase di↵erence at the beamsplitter. Let ⌧ and ⌧ denote a ne parameters along the path in the two directions. Denote the classical path of the interferometer in space, in the rest frame of the beamsplitter, + by xi (⌧) and xi (⌧), in the two directions around the circuit. Here i =1, 2, 3 again denotes the 3D spatial indices, + although below we will assume a planar apparatus for simplicity. The functions xi (⌧) and xi (⌧) can be visualized as the trajectories of “tracer photons” in each direction around the circuit; they map positions on the circuit in 3-space to points on an interval on the real line, ( C0, +C0), where C0 denotes the circumference (or perimeter) of the circuit, the origin maps to the beamsplitter, and ⌧ represents a time interval in the proper time of the beamsplitter. The + e↵ects of quantum geometry on the measured correlation ⌅(⌧) depends only on the classical path, defined by xi (⌧) and xi (⌧), the positions of a pair of tracer photons that begin and end their circuit at the same time. The tangent vector to the path in each direction is @xj±/@⌧. It will be convenient to express a general form for the functional dependence of ⌅(⌧) on the classical path in terms of the “swept-out area”, A(⌧), similar to the concept used in Kepler’s second law of planetary motion. Define Ai(⌧) as an oriented area in the inertial rest frame, swept out by lines between the beamsplitter and the tracers in the two directions (see Fig. 2). The areas from the beamsplitter to the two tracers change as: + + + dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧] (27) dA /d⌧ = ✏ [x (⌧)dx /d⌧]. (28) i ijk k j 19 These quantities are completelyThis operator determined takes discrete eigenvalues: by the geometrical configuration of the apparatus. In general a transverse swept area can also be defined from from the swept area between the two tracers, adding xˆ 2 l = l(l + 1)`2 l . (A5) additional cross terms: | | | i P | i The discrete eigenvalues correspond to classical values of separation R according to: 1 + + dA⇥/d⌧ = ✏ [dx /d⌧ R+2 dxl( l +/d 1)⌧`2][x l2(`2⌧) x (⌧)]. (A6) (29) i ijk 2 j ⌘ j P ⇡k P k where l are positive integers, quantum numbers of eigenstates of xˆ 2. Let l denote eigenvalues of position components projected in direction| | i. In a state l of separation number l,the This component correspondsi to shear, rather than rotation about the beamsplitter.| Wei therefore assume here that projected position operatorx ˆi can have eigenvalues in units of `P , the emergent rotational component just depends on a combination of the two terms6 directly related to rotation about l = l, l 1,..., l, (A7) the observer/beamsplitter world line, Eqs. (27) and (28). Thisi assumption is also testable: if present, the shear terms Thus any volume there is a maximum number of distinguishable events, such that the density of clocks and signals used towould measure be their detectable separation, GPS-style, ingiving a Michelson 2 notl + 1 exceed possible interferometer[15, the values. density In of an a eigenstate black hole[37]. 16], with which a In definite a spacetime is value not sensitiveof volume position of in to direction the rotationali, terms measured in radius aR Sagnacand duration device.⌧ in Planck Experiments units, the bound in both on the configurations number of events wouldis about distinguishR⌧.Sincec⌧ theR for modes, a and explore the full character of xˆi l, li = li`P l, li .⇡ (A8) covariantlytransverse defined volume exotic (such position as a causal fluctuations. diamond), this is the same amount of information| i as| thei holographic estimates from statistical gravity just discussed.The total Indeed, number this of argument position eigenstates has been used in a as 3-sphere a statistical is basis for general relativity[37].The Note exotic that the correlation area here combines amplitude spatial andin the temporal signal dimensions, is givenR byand the⌧. projection of the exotic transverse position correlation This argument is not airtight; for example, the measurement bounds might not applylR to unmeasured degrees of represented by the apparatus. The swept area provides this: relations between dA±(⌧)/d⌧, x±(⌧) and dx±/d⌧ combine freedom if position information is subject to subtle squeezing. However, it does(R use)= an exact(2l + formulation 1) = l (l + of 2) quantum = (R/` i)2, k j(A9) NQ3S R R P informationinto that a function applies specifically of ⌧ that to event depends positions. on It the suggests exotic that large correlation scale exoticl=1✏ijk Planckianxk`P correlationson each hypersurface, indeed but where the spatial indices First Holometer result rules out exotic shear correlationX at the appearhave in measurements all contracted. of relative positions of events and world lines, that could be measured by light reflecting o↵ of macroscopicPlanck objects.diffraction scale,where but the lastnot equality a purely applies rotational in the large l limit. correlation Thus, the number of quantum-geometrical position eigenstates The projection can bein visualized a volume scales geometrically holographically, as (Fig. the surface 2). The area in essential Planck units. assumptions This simple Planckian are just quantization the holographic closely scaling resembles (but is not exactly the same as) the spectrum of area states more rigorously derived within the framework of transverse position correlations,of loop quantum and gravity the (Eq. standard 17). projective properties of emergent space-time that relate R, t and Planck Di↵raction Limit on Directional Resolution ⌧. Assume for simplicityDirect a planar calculation apparatus, (e.g., ref.[63]) and also thereby leadsworld to theline suppress following of product directional of amplitudes indices for measurements for Ai. The of either swept of the area rates Signal of current Holometertransverse is sensitive components onlyx ˆj,with j = i: HolographicdA±/d entanglement⌧ are proportional of fields in emergent to separation geometry may from be governed the6 originbeamsplitter by a at Planckian any point bound on directional the path, and to the projection of the path to correlations associated with differential µ 2 information. An estimate of the directional spread follows directly from theli relativisticxˆj li 1 li wave1 xˆ equation,j li =(l @+µl@i)( l =l 0.i + For 1)` /2, (A10) on the transverse direction at that point. Denote the angle between thei!t light pathP tangent and position vectors by waves ofjitter frequency of separation! propagating in in athe beam— two aarm stationary, directions monochromatic, h | wave| stateih proportional | | i to e (x, y, z), transverselynot✓0(⌧ to). localized correlated This in anglex and rotationy determinesand extendedagain of forboth inthe anyz— armsi a. the Thene wave left mapping equationside of equation becomes between (A10)t canand be interpreted⌧ via dt/d as the⌧ =sin expected✓0, value and for the the sweptoperator area via 2 2 [@ + @ 2i!@ ] =0, xˆj li 1 li 1 xˆj (12) (A11) x y dAz ±/d⌧ =sin✓0dA±/dt| =ih R (⌧|)sin✓0, (30) the same equation used to model laserfor beams. components The minimum with j = transversei, in a state angularli spreadof definite is thusx ˆi. the For dil>>↵raction1 and limitli l, this corresponds to the expected 2 6 | i ⇡ for a Planckwhere frequencyR(⌧)= beam[14],x (⌧ setting) . Thevariance! contributionmP inc components/~: ofto position the exotic transverse signal to separation: correlation at each ⌧, apart from a constant o↵set fixed | i | ⇡ 2 2 by the boundary condition determined ✓ = ` /R by . closure of thexˆj l circuit,i 1 li is1 xˆj alsoxˆ fixedj . (13) by a projection onto the(A12) path, in this h P i P h | ih | i!h i case of transverse positionWe can variance rewrite this (Eqs. as a formula 26, A13) for the : variance of the wave functionx ˆ in any direction transverse to separation, Physically, the role played by sharp longitudinal boundary conditions (mirrors) in a laser cavity is played in emergent? 1/2 geometry by causal structure: the requirementgiven by that the right radial hand position side of and Eq. causal (A10) structure in the limit haveruled of l>> Planck out1, length for l l precisioni = 1 (or indeed for any value of l li << l ): 2 1/2 ± | | at all separations. The degrees of freedom can be divided(⌅(⌧) into+ constant) radial information, = x which describes2 sin ✓ causal0 = `P structure,R(⌧)sin✓0. (31) R(⌧xˆ) = R`P . (A13) and directional information, which describes two-dimensional angular orientationh and? transversei h ?i position. One factor R/`P of holographic information contentThis describes simple result causal, is longitudinal used here to degrees set the scale of freedom; of transverse the other position factor variance, of R/`P for example in Eqs. (32) and (41). The describesSince directional this degrees also scales of freedom. linearly By contrast, with inR a(⌧ full), resolution the swept 3D space area the rate two isdirectional simply dimensions related on to the signal correlation by same algebra, with j and in place of x and `P , yields a formula for the width of the wave function for transverse 2 ~ their own would comprise (R/`P ) degreescomponents of freedom.not ofruled standard out angular momentum, in the limit of large j : Directions in space emerge from the Planck scale in such a way that(⌅( directional⌧) + constant) resolution = of` thedA/cd emergent| |⌧. space on (32) 2 P any scale is given by Eq. (13). Field states cannot exceed the angular resolution of geometry,ˆj so they= j become~. strongly (A14) entangled with geometry at a separation where Planck di↵raction matches the resolution ofh matter?i | fields,| (m/R)2; that separationDefine is the given total by the swept idealized areaHowever, ChandrasekharA( this⌧) assimple radius, a sum formula Eq. of (8). does the Note not rotational appear that a blackin standard components hole of treatments this size⇡ stillin of the angular has a two momentum. directions, so that much larger entropy than field states of mass m The directional blurring scale can also beThe derived exotic from correlation standard (Eq. wave 35) equations may be applied detectable to quantum using an states apparatus of wave similar to the Fermilab Holometer[64], packets with Planck frequency[14, 16,reconfigured 17]. In this estimation,to measure wavepure rotational functions of modes. world The lines Holometer of bodies of achieves any mass the sensitivity needed to measure exotic spread at the same rate as those of particles of Planck mass. The scale can be estimated starting from the standard wave function (xi,t) of a massive body in classical position and time. In its classical rest frame, a massive body obeys a 3+1-dimensional version of the paraxial wave equation (Eq. 12), with propagation in time in place of radial distance, and rest mass in place of frequency: 2 2 2 [@ + @ + @ 2i(m/~)@t] =0. (14) x y z Again like a laser beam, spreading of this wave function with time is unavoidable; any solution of duration ⌧ has a mean variance of position at least as large as 2 2 (r(t + ⌧) r(t)) = (⌧) > ~⌧/2m. (15) h i 0 the standard Heisenberg position uncertainty[38–41] for a body measured at times separated by ⌧.Inthiswave picture, it is derived in exact analogy to di↵ractive spreading of a light beam. Here, the sharp preparatory boundary A reconfigured Fermilab Holometer could measure exotic rotational correlations Current results rule out a signal correlation at the predicted amplitude Explores exotic correlations of space and time for the first time First direct evidence that emergent quantum gravity obeys a symmetry (i.e., little or no “jitter”, or shear degrees of freedom) Suggests that causal structure represents an exact symmetry But in the current Michelson layout, the Holometer is not sensitive to rotational correlations (i.e., “twists” in inertial frame) It can be reconfigured to detect exotic rotation by bending one of the arms in both interferometers at right angles 45 10 that can be combined into a quantity that scales like ✏ijkxk`P , in the same way as the exotic correlation on each hypersurface. In a Sagnac interferometer, the path of light is a closed circuit. The light follows the same path in two directions and + the signal records their phase di↵erence at the beamsplitter. Let ⌧ and ⌧ denote a ne parameters along the path in the two directions. Denote the classical path of the interferometer in space, in the rest frame of the beamsplitter, + by xi (⌧) and xi (⌧), in the two directions around the circuit. Here i =1, 2, 3 again denotes the 3D spatial indices, + although below we will assume a planar apparatus for simplicity. The functions xi (⌧) and xi (⌧) can be visualized as the trajectories of “tracer photons” in each direction around the circuit; they map positions on the circuit in 3-space to points on an interval on the real line, ( C0, +C0), where C0 denotes the circumference (or perimeter) of the circuit, the origin maps to the beamsplitter, and ⌧ represents a time interval in the proper time of the beamsplitter. The + e↵ects of quantum geometry on the measured correlation ⌅(⌧) depends only on the classical path, defined by xi (⌧) and xi (⌧), the positions of a pair of tracer photons that begin and end their circuit at the same time. The tangent vector to the path in each direction is @xj±/@⌧. It will be convenient to express a general form for the functional dependence of ⌅(⌧) on the classical path in terms of the “swept-out area”, A(⌧), similar to the concept used in Kepler’s second law of planetary motion. Define Ai(⌧) as an oriented area in the inertial rest frame, swept out by lines between the beamsplitter and the tracers in the two directions (see Fig. 2). The areas from the beamsplitter to the two tracers change as: + + + dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧] (27) dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧]. (28) These quantities are completely determined by the geometrical configuration of the apparatus. In general a transverse swept area can also be defined from from the swept area between the two tracers, adding additional cross terms: 1 + + dA⇥/d⌧ = ✏ [dx /d⌧ + dx /d⌧][x (⌧) x (⌧)]. (29) i ijk 2 j j k k This component corresponds to shear, rather than rotation about the beamsplitter. We therefore assume here that the emergent rotational component just depends on a combination of the two terms6 directly related to rotation about the observer/beamsplitter world line, Eqs. (27) and (28). This assumption is also testable: if present, the shear terms Thus any volume there is a maximum number of distinguishable events, such that the density of clocks and signals usedwould to measure be their detectable separation, in GPS-style, a Michelson not exceed interferometer[15, the density of a black hole[37]. 16], which In a spacetime is not sensitive volume of to the rotational terms measured in radiusa SagnacR and duration device.⌧ in ExperimentsPlanck units, the bound in both on the configurations number of events would is about distinguishR⌧.Sincec⌧ theR for modes, a and explore the full character of covariantly defined volume (such as a causal diamond), this is the same amount of information as the holographic⇡ estimatestransverse from statistical exotic gravity position just discussed. fluctuations. Indeed, this argument has been used as a statistical basis for general relativity[37].The exotic Note that correlation the area here combines amplitude spatial in and the temporal signal dimensions, is givenR byand the⌧. projection of the exotic transverse position correlation This argument is not airtight; for example, the measurement bounds might not apply to unmeasured degrees of freedomrepresented if position information by the apparatus. is subject to subtle The squeezing. swept area However, provides it does use this: an exact relations formulation between of quantumdAi±(⌧)/d⌧, xk±(⌧) and dxj±/d⌧ combine informationinto a that function applies specifically of ⌧ that to event depends positions. on It the suggests exotic that correlation large scale exotic✏ijk Planckianxk`P correlationson each indeed hypersurface, but where the spatial indices appearhave in measurements all contracted. of relative positions of events and world lines, that could be measured by light reflecting o↵ of macroscopic objects. The projection can be visualized geometrically (Fig. 2). The essential assumptions are just the holographic scaling of transverse position correlations, and the standard projective properties of emergent space-time that relate R, t and Planck Di↵raction Limit on Directional Resolution ⌧. Assume for simplicity a planar apparatus, and thereby suppress directional indices for Ai. The swept area rates HolographicdA±/d⌧ entanglementare proportional of fields in emergent to separation geometry mayfrom be the governed origin by a at Planckian any point bound on on directional the path, and to the projection of the path µ information. An estimate of the directional spread follows directly from the relativistic wave equation, @µ@ = 0. For on the transverse direction at that point. Denote the angle between thei!t light path tangent and position vectors by waves of frequency ! propagating in a beam— a stationary, monochromatic wave state proportional to e (x, y, z), transversely✓0(⌧). localized This angle in x and determinesy and extended the in z— a thene wave mapping equation becomes between t and ⌧ via dt/d⌧ =sin✓0, and the swept area via 2 2 [@ + @ 2i!@z] =0, (12) currentx configurationy dA±/d⌧ =sin✓0dA±/dt = R(⌧)sin✓0, (30) the same equation used to model laser beams. The minimum transverse angular spread is thus the di↵raction limit 2 for awhere Planck frequencyR(⌧)= beam[14],xi(⌧) setting. The! contributionmP c /~: to the exotic signal correlation at each ⌧, apart from a constant o↵set fixed ⇡ by the boundary| condition| determined by closure of the circuit, is also fixed by a projection onto the path, in this ✓2 = ` /R . (13) case of transverse position varianceh P (Eqs.i P 26, A13) : Physically, the role played by sharp longitudinal boundary conditions (mirrors) in a laser cavity is played in emergent geometry by causal structure: the requirement that radial position and causal structure2 have1/2 Planck length precision at all separations. The degrees of freedom can be(⌅ divided(⌧) + into constant) radial information, = x whichR describes(⌧) sin ✓ causal0 = ` structure,P R(⌧)sin✓0. (31) and directional information, which describes two-dimensional angular orientationh and? transversei position. One factor R/`P of holographic information content describes causal, longitudinal degrees of freedom; the other factor of R/`P describesSince directional this also degrees scales of freedom. linearly By contrast, with R in( a⌧ full), the resolution swept 3D space area the rate two isdirectional simply dimensions related on to the signal correlation by 2 their own would comprise (R/`P ) degrees of freedom. Directions in space emerge from the Planck scale in such a way( that⌅(⌧ directional) + constant) resolution = of the`P dA/cd emergent⌧ space. on (32) any scale is given by Eq. (13). Field states cannot exceed the angular resolution of geometry, so they become strongly entangledDefine with the geometry total at swept a separation area whereA(⌧ Planck) as di a↵raction sum of matches the rotational the resolution of components matter fields, in(m/R the)2; two directions, so that that separation is given by the idealized Chandrasekhar radius, Eq. (8). Note that a black hole of this size⇡ still has a much larger entropy than field states of mass m The directional blurring scale can also be derived from standard wave equations applied to quantum states of wave packets with Planck frequency[14, 16, 17]. In this estimation, wave functions of world lines of bodies of any mass spread at the same rate as those of particles of Planck mass. The scale can be estimated starting from the standard wave function (xi,t) of a massive body in classical position and time. In its classical rest frame, a massive body obeys a 3+1-dimensional version of the paraxial wave equation (Eq. 12), with propagation in time in place of radial distance, and rest mass in place of frequency: 2 2 2 [@ + @ + @ 2i(m/~)@t] =0. (14) x y z Again like a laser beam, spreading of this wave function with time is unavoidable; any solution of duration ⌧ has a mean variance of position at least as large as 2 2 (r(t + ⌧) r(t)) = (⌧) > ~⌧/2m. (15) h i 0 the standard Heisenberg position uncertainty[38–41] for a body measured at times separated by ⌧.Inthiswave picture, it is derived in exact analogy to di↵ractive spreading of a light beam. Here, the sharp preparatory boundary 10 that can be combined into a quantity that scales like ✏ijkxk`P , in the same way as the exotic correlation on each hypersurface. In a Sagnac interferometer, the path of light is a closed circuit. The light follows the same path in two directions and + the signal records their phase di↵erence at the beamsplitter. Let ⌧ and ⌧ denote a ne parameters along the path in the two directions. Denote the classical path of the interferometer in space, in the rest frame of the beamsplitter, + by xi (⌧) and xi (⌧), in the two directions around the circuit. Here i =1, 2, 3 again denotes the 3D spatial indices, + although below we will assume a planar apparatus for simplicity. The functions xi (⌧) and xi (⌧) can be visualized as the trajectories of “tracer photons” in each direction around the circuit; they map positions on the circuit in 3-space to points on an interval on the real line, ( C0, +C0), where C0 denotes the circumference (or perimeter) of the circuit, the origin maps to the beamsplitter, and ⌧ represents a time interval in the proper time of the beamsplitter. The + e↵ects of quantum geometry on the measured correlation ⌅(⌧) depends only on the classical path, defined by xi (⌧) and xi (⌧), the positions of a pair of tracer photons that begin and end their circuit at the same time. The tangent vector to the path in each direction is @xj±/@⌧. It will be convenient to express a general form for the functional dependence of ⌅(⌧) on the classical path in terms of the “swept-out area”, A(⌧), similar to the concept used in Kepler’s second law of planetary motion. Define Ai(⌧) as an oriented area in the inertial rest frame, swept out by lines between the beamsplitter and the tracers in the two directions (see Fig. 2). The areas from the beamsplitter to the two tracers change as: + + + dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧] (27) dAi /d⌧ = ✏ijk[xk (⌧)dxj /d⌧]. (28) These quantities are completely determined by the geometrical configuration of the apparatus. In general a transverse swept area can also be defined from from the swept area between the two tracers, adding additional cross terms: 1 + + dA⇥/d⌧ = ✏ [dx /d⌧ + dx /d⌧][x (⌧) x (⌧)]. (29) i ijk 2 j j k k This component corresponds to shear, rather than rotation about the beamsplitter. We therefore assume here that the emergent rotational component just depends on a combination of the two terms6 directly related to rotation about the observer/beamsplitter world line, Eqs. (27) and (28). This assumption is also testable: if present, the shear terms Thus any volume there is a maximum number of distinguishable events, such that the density of clocks and signals usedwould to measure be their detectable separation, in GPS-style, a Michelson not exceed interferometer[15, the density of a black hole[37]. 16], which In a spacetime is not sensitive volume of to the rotational terms measured in radiusa SagnacR and duration device.⌧ in ExperimentsPlanck units, the bound in both on the configurations number of events would is about distinguishR⌧.Sincec⌧ theR for modes, a and explore the full character of covariantly defined volume (such as a causal diamond), this is the same amount of information as the holographic⇡ estimatestransverse from statistical exotic gravity position just discussed. fluctuations. Indeed, this argument has been used as a statistical basis for general relativity[37].The exotic Note that correlation the area here combines amplitude spatial in and the temporal signal dimensions, is givenR byand the⌧. projection of the exotic transverse position correlation This argument is not airtight; for example, the measurement bounds might not apply to unmeasured degrees of freedomrepresented if position information by the apparatus. is subject to subtle The squeezing. swept area However, provides it does use this: an exact relations formulation between of quantumdAi±(⌧)/d⌧, xk±(⌧) and dxj±/d⌧ combine informationinto a that function applies specifically of ⌧ that to event depends positions. on It the suggests exotic that correlation large scale exotic✏ijk Planckianxk`P correlationson each indeed hypersurface, but where the spatial indices appearhave in measurements all contracted. of relative positions of events and world lines, that could be measured by light reflecting o↵ of macroscopic objects. The projection can be visualized geometrically (Fig. 2). The essential assumptions are just the holographic scaling of transverse position correlations, and the standard projective properties of emergent space-time that relate R, t and Planck Di↵raction Limit on Directional Resolution ⌧. Assume for simplicity a planar apparatus, and thereby suppress directional indices for Ai. The swept area rates HolographicdA±/d⌧ entanglementare proportional of fields in emergent to separation geometry mayfrom be the governed origin by a at Planckian any point bound on on directional the path, and to the projection of the path µ information. An estimate of the directional spread follows directly from the relativistic wave equation, @µ@ = 0. For on the transverse direction at that point. Denote the angle between thei!t light path tangent and position vectors by waves of frequency ! propagating in a beam— a stationary, monochromatic wave state proportional to e (x, y, z), transversely✓0(⌧). localized This angle in x and determinesy and extended the in z— a thene wave mapping equation becomes between t and ⌧ via dt/d⌧ =sin✓0, and the swept area via [@2 + @2 2i!@ ] =0, (12) Bendx army to dAmeasurez ±/d⌧ =sin✓0dA±/dt = R(⌧)sin✓0, (30) the same equation used to model laser beams.rotational The minimum correlation transverse angular spread is thus the di↵raction limit 2 for awhere Planck frequencyR(⌧)= beam[14],xi(⌧) setting. The! contributionmP c /~: to the exotic signal correlation at each ⌧, apart from a constant o↵set fixed ⇡ by the boundary| condition| determined by closure of the circuit, is also fixed by a projection onto the path, in this ✓2 = ` /R . (13) case of transverse position varianceh P (Eqs.i P 26, A13) : Physically, the role played by sharp longitudinal boundary conditions (mirrors) in a laser cavity is played in emergent geometry by causal structure: the requirement that radial position and causal structure2 have1/2 Planck length precision at all separations. The degrees of freedom can be(⌅ divided(⌧) + into constant) radial information, = x whichR describes(⌧) sin ✓ causal0 = ` structure,P R(⌧)sin✓0. (31) and directional information, which describes two-dimensional angular orientationh and? transversei position. One factor R/`P of holographic information content describes causal, longitudinal degrees of freedom; the other factor of R/`P describesSince directional this also degrees scales of freedom. linearly By contrast, with R in( a⌧ full), the resolution swept 3D space area the rate two isdirectional simply dimensions related on to the signal correlation by 2 their own would comprise (R/`P ) degrees of freedom. Directions in space emerge from the Planck scale in such a way( that⌅(⌧ directional) + constant) resolution = of the`P dA/cd emergent⌧ space. on (32) any scale is given by Eq. (13). Field states cannot exceed the angular resolution of geometry, so they become strongly entangledDefine with the geometry total at swept a separation area whereA(⌧ Planck) as di a↵raction sum of matches the rotational the resolution of components matter fields, in(m/R the)2; two directions, so that that separation is given by the idealized Chandrasekhar radius, Eq. (8). Note that a black hole of this size⇡ still has a much larger entropy than field states of mass m The directional blurring scale can also be derived from standard wave equations applied to quantum states of wave packets with Planck frequency[14, 16, 17]. In this estimation, wave functions of world lines of bodies of any mass spread at the same rate as those of particles of Planck mass. The scale can be estimated starting from the standard wave function (xi,t) of a massive body in classical position and time. In its classical rest frame, a massive body obeys a 3+1-dimensional version of the paraxial wave equation (Eq. 12), with propagation in time in place of radial distance, and rest mass in place of frequency: 2 2 2 [@ + @ + @ 2i(m/~)@t] =0. (14) x y z Again like a laser beam, spreading of this wave function with time is unavoidable; any solution of duration ⌧ has a mean variance of position at least as large as 2 2 (r(t + ⌧) r(t)) = (⌧) > ~⌧/2m. (15) h i 0 the standard Heisenberg position uncertainty[38–41] for a body measured at times separated by ⌧.Inthiswave picture, it is derived in exact analogy to di↵ractive spreading of a light beam. Here, the sharp preparatory boundary Simplest reconfiguration to measure rotational correlations Current configuration: no neither path encloses an response to a pure rotation area “Half-Bent Michelson”: sensitive to exotic rotational paths enclose correlations overlapping (but not mean rotation) areas no correlated Null reconfiguration signal encloses an area 48 FERMILAB-PUB- Planck Scale Correlations in Interferometer Signals from Quantum Geometry: Rotational Fluctuations in a Sagnac Circuit Craig Hogan University of Chicago and Fermilab Center for Particle Astrophysics Theoretical estimates are reviewed of bounds on quantum geometrical information, and of the magnitude and character of associated exotic Planckian correlations of position. General principles are sketched that relate observed correlations in interferometer signals with exotic correlations in 3+1 dimensions, depending on quantum geometry theory and interferometer spatial architecture. A specific hypothesis is formulated in algebraic and graphical form for the quantum geometrical correlation in a Sagnac interferometer signal associated with emergence of local, nonrotating inertial frames. A specific prediction is made for the spectrum of observable Planckian signal correlations in a Sagnac interferometer, with a normalization suggested from the area quantization spectrum in Loop Quantum Gravity. I. INTRODUCTION FERMILAB-PUB- The content of any quantum theory can be expressed as correlations between observable quantities. The magnitude Signalof quantum Correlations correlations in has a Sagnac an inverse Holometer relation to from the information Loop Quantum in a system: Gravity they vanish in the limit of a classical system, which has an infinite amount of information. In standard physics, matter is quantum-mechanical, and geometrical relationships are classical. However, it is generally thought that a quantum system underlies space and time. If so, it should also produce correlations among observable quantities, beyond those of quantum matter. There is no current standard theory of correlations from quantum-geometrical degrees of freedom. Theoretical extrapolations from standard⌅ theory(⌧) suggestx(t) x( thatt + ⌧) quantumt gravity limits the amount of information(1) available to a system of matter and geometry, and⌘h therefore, newi forms of correlation not predicted in standard theory. It is not known what form those correlations take. In this paper we consider the hypothesis that quantum-geometrical correlations take the form of exotic correlations ⌅(⌧)=2A ` /C (` /c)(dA/d⌧) (2) in the spatial positions of world lines,0 P as traced0 P by trajectories of massive bodies, and that these can be distinguished from standard quantum correlations by precise measurements of positions of bodies in space. We argue that such measurementsExotic carry significant,correlation specificin a Half-Bent and quantitative Michelson: information same as about the structure of new physics at the Planck scale. Sagnac of same⌅(⌧)= size,`P (2 butA0 /Chalf0 thedA (fluctuation⌧)/cd⌧) power (3) We have previously considered one class of such measurements, those possible with Michelson interferometers[1–5]. Here, we consider a wider class of possible correlations. In particular, it is argued that measurements sensitive to the phase of radiation that travels in a closed circuit, as in a Sagnac interferometer, may detect correlations due to the ⌅(⌧)/` =2A /C dA(⌧)/cd⌧ (4) imperfect emergence, in a laboratory-scaleP 0 system,0 of a local nonrotating frame from quantum relationships of Planck L 2L A(⌧) is definedscale elements.in the inertial rest frame as the area swept out by a line connecting the two points a path length c⌧ from the beamsplitterIn an interferometer, along the circuit, a inquantum-geometricalL/4 opposite directions correlation is ultimately measured as the time domain correlation (or its frequency domain transform) of an observable x(t), with dimension of length: I. NORMALIZATION FROM LOOP⌅( QUANTUM⌧) x(t)x( GRAVITYt + ⌧) . (1) 0 ⌘h it Discrete spectrumA variety of of area estimates states in reviewed LQG: below suggest that quantum geometry in a system of size c⌧ produces exotic correlations in position with magnitude roughly A =8⇡ Gc 3 j (j + 1) (5) j ~ i i ⌅(⌧)/c⌧ ` (2) i P 36 ⇡ Perimeter P0 = 5 x 10 =X 80 mp 3 RMS fluctuation 1018 /21/2= 8 attometers where ji arewhere half-integers`P = i/~2G/c for integersdenotesRMSi. the Theangular Planck Immirizi rotation rate length. parameter ~ 1 radian/ They 104 years representis estimated a frommeasure black of hole the entropy (inverse to of be the) information density -1 =ln2/⇡pin3. the For large geometry. areas (i>> We refer1) wePeak tocan frequency correlations write theof fluctuation eigenvalues of ~ thisP0 ~ 4 [of magnitudeMHz the “main as sequence” “exotic” states] Planck as correlations because they are not p produced in standard theory. 49 A A n2 (6) Experiments with interferometers—n in⇡ particular,Q the Fermilab Holometer[6]— have shown that it is possible to measure position correlations with Planck sensitivity. A signal is produced by light emerging from a system of mirrors for integers n,where in some spatial arrangment. Their multidimensional correlation a↵ects the observable data stream, as in Eq. (1). Such experiments can probe Planck scale quantum3 degrees of freedom. The signal depends on the detailed spatial AQ 2⇡ ~Gc (7) layout of the apparatus, and the detailed⌘ character of the exotic 3+1 D position correlations. Thus, an experimental is a fundamentalprogram Planck can area literally unit. The map di the↵erence Planck between scale adjacent correlations[1–5]. area states is In order to estimate the projection operators for a given apparatus, we need a phenomenology for a still-incomplete A A 2A n. (8) theory of quantum geometry. It isn possible+1 n ⇡ to constrainQ the forms of correlation from the symmetries of the emergent system, that is, the arrangment of mirrors in the emergent geometry, and make quantitative tests of specific hypotheses. Notice that the spacing grows with n. Thus for larger areas, each area eigenstate is bigger than the last by much more than a single Planck area pixel. This property reflects the same blurring or loss of relative positional information on large scales that appears in wave equation solutions, holographic information constraints, and other proposed “infrared” departures from field theory. It explicitly captures an essential nonlocal Planckian feature not compatible with a metric-based field approach: the positional relationships of the geometry itself depend on separation, and degrade over large areas. Arguments give above suggest that the correlation function for phase variations around a planar circuit scales with the area divided by the circumference. Not surprisingly, the correlations are related to the shape of the circuit. Consider an area defined by a planar surface between several world lines, say, the centers of mass of mirrors in a Sagnac interferometer, all at rest. In classical language, we propagate this surface forward in time by extending along their world lines in the rest frame. In Quantum General Relativity, there is no independent clock ticking; the equivalent of a Planckian clock tick is defined relationally by a transition between quantum states of the spacetime geometry. Consider the light propagating between the mirrors around this area in two directions from the beam splitter. 1/2 For definiteness, consider a square arrangment with sides L. For a square we have n LAQ , hence a natural 1/2 ⇡ unit of length `P = `Q = AQ . This fixes an absolute normalization for the signal correlation function. In a Predicted rotational noise power spectral density compared with current Holometer sensitivity PSD Predicted rotational holographic noise (m2/Hz) spectrum in a Holometer reconfigured by bending one arm 1.× 10-41 measured Holometer noise level in 145 hours, 1MHz frequency bins, including systematics 5.× 10-42 2× 106 4× 106 6× 106 8× 106 1× 107 Frequency (Hz) 50 Completing a comprehensive study of exotic correlations Basic hypothesis: exotic Planck scale correlations of geometry affect world lines of bodies and phases of fields General arguments set the amplitude but not the spatial pattern Limited options for spatial patterns: shear, strain, rotation Each experiment constrains a symmetry of exotic correlations Current result supports a Planck scale symmetry of causality After the conclusive exclusion of shear by the current Holometer, the rotational mode is the only remaining possibility Complete constraints on independent covariances in space This reconfiguration of the Holometer will complete the survey It will either detect the effect or rule out the basic hypothesis It will either detect a Planck scale correlation, or provide evidence for Planck scale symmetry of inertia 51 17 This formula can be derived by summing the extra contributions from each straight photon path segment around a circuit. Let t denote the “extra” propagation time for a photon— the extra time to the next reflection, say— in a small interval of a straight path, due to the fact that the apparatus is rotating. It is related to the transverse rotation d x = R!0d⌧ in the same interval by ? cd t/d x =sin✓0, (39) ? using the notation shown in Fig (5). Combining with the swept area (Eq. 30), the net time lag around a circuit is c d t =2! A /c. (40) ˆ 0 0 The total Sagnac e↵ect is twice this amount because it is the total di↵erence between opposite directions. 17 The exotic e↵ect produces the tiny fluctuations around this value, due to the imperfect definition of the frame. The exotic correlationThis formula amplitude can be derived for a by Sagnac summing device the extra on any contributions scale (Eq. from 36), each which straight is a mean photon square path segment displacement, around is equal to thea classical circuit. Let displacement t denote the for “extra” an apparatus propagation rotating time for at a angular photon— velocity the extra! time0 = c/ to2 theP0, next times reflection, the Planck say— length. in a Once again,small the interval exotic phase of a straight displacement path, due resembles to the fact approximately that the apparatus the is value rotating. of displacement It is related to in the a transverse Planckian rotation random walk, but withd x symmetries= R!0d⌧ in andthe same projection interval factors by appropriate to rotation. Notice that both the exotic fluctuation A0/P0 ? / and classical lag !0A0 are defined entirely in terms of properties of the light path, which have a clear invariant / cd t/d x =sin✓0, (39) meaning. ? using the notation shown in Fig (5). Combining with the swept area (Eq. 30), the net time lag around a circuit is V. EXOTIC ROTATIONAL CORRELATIONS AND COSMIC ACCELERATION c d t =2! A /c. (40) ˆ 0 0 TheThe properties total Sagnac of the e↵ect Planck is twice scale this rotational amount because correlations it is the are total scale-free di↵erence power between laws, opposite apart directions.from the Planck scale itself. SystemsThe in exotic the real e↵ect world produces have the scales tiny fluctuations determined around by the this physics value, due of matter to the imperfect fields. The definition previous of the discussion frame. The considers the eexotic↵ects correlation of entanglement amplitude with for a geometrical Sagnac device states on any on scale field (Eq. degrees 36), which of freedom, is a mean square including displacement, the reconciliation is equal with holographicto the classical gravity, displacement and small for phase an apparatus displacements rotating measurable at angular velocity in interferometers.!0 = c/2P0, times The the latter Planck calculation length. Once is definite enoughagain, for the a precise exotic phase experimental displacement test. resembles We now approximately consider estimates the value of of displacement the complementary in a Planckian e↵ect: random how entanglement walk, but with symmetries and projection factors appropriate to rotation. Notice that both the exotic fluctuation A /P of field vacuum and geometry could break the scale invariance of gravity, and account for the value/ of0 the0 scalar cosmologicaland classical constant lag in!0A the0 are emergent defined entirelyclassical in field terms equations. of properties of the light path, which have a clear invariant meaning. / V. EXOTICThe ROTATIONAL Cosmological CORRELATIONS Constant and AND Cosmic COSMIC Acceleration ACCELERATION TheThe Einstein properties field of equations the Planck of scale general rotational relativity correlations can are be scale-free contracted power with laws, the apart metric from tensor the Planckgµ⌫ scaleto yield itself. a scalar relation,Systems in the real world have scales determined by the physics of matter fields. The previous discussion considers the e↵ects of entanglement with geometrical states on field degrees of freedom, including the reconciliation with holographic gravity, and small phase displacements measurable2⇡G in interferometers. The latter calculation is definite enough for a precise experimental test. We now⇤ consider= estimates(T +2 of), the complementary e↵ect: how entanglement (41) c4 R of field vacuum and geometry could break the scale invariance of gravity, and account for the value of the scalar cosmological constant in the emergent classical field equations.4 µ⌫ where ⇤ denotes the cosmological constant, R(c /16⇡G) denotes the Ricci curvature scalar R g Rµ⌫ in units 4 R ⌘ ⌘ of 16⇡G/c , Rµ⌫ denotes the Ricci curvature tensor, and T denotes the trace of the classical energy momentum tensor of matter, T . Within general relativity, there is no mathematical di↵erence between ⇤ and a component of the µ⌫ The CosmologicalThere’s Constant More: and a Cosmicpossible Acceleration connection between the matter field vacuum with Tµ⌫ proportional to cosmologicalgµ⌫ . In standard constant theory, and ⇤theis field a new vacuum constant of nature, unconnected to other properties of gravity or matter[28]. Its value is also mysterious and arbitrary in holographic theories of The Einstein field equations of general relativityTrace of can Einstein be contracted equations includes with the a “cosmological metric tensor constant”,gµ⌫ to the yield sum of a scalar gravity[43,relation, 58]. traces from matter and geometry: Cosmic data suggest that the expansion of the universe is accelerating[59–62]. This phenomenon can be interpreted 2⇡G in classical General Relativity as an e↵ect of ⇤.⇤ = In addition(T +2 to), the standard attractive gravity of normal(41) forms of matter expressed in T , the cosmological term producesc4 an additionalR repulsive acceleration between two bodies at separation r, 4 µ⌫ where ⇤ denotes the cosmological constant, InR standard(c /16 ⇡theoryG) denotes it is arbitrary, the Ricciunrelated curvature to other constants scalar R g Rµ⌫ in units of 16⇡G/c4, R denotes the Ricci curvatureR tensor,⌘ and T denotes the trace of the classical energy⌘ momentum tensor µ⌫ 2 of matter, Tµ⌫ . Within general relativity, thereIn holographic is nor¨ mathematical= Hgravity,⇤r, it fixes di ↵theerence total cosmic between information⇤ and in a Planck component units: of the (42) matter field vacuum with Tµ⌫ proportional to gµ⌫ . In standard theory, ⇤ is a new constant of nature, unconnected 2 proportionalto other to properties their separation of gravityr orin matter[28]. Newtonian Its coordinates, value is also where mysteriousH⇤ and⇤/ arbitrary3. In cosmological in holographic models theories where of matter gravity[43, 58]. ⌘ has T>0, H⇤ is the asymptotic value of the Hubble~ area of expansion cosmic event rate horizon in the far future, after other forms of matter have thinnedCosmic out due data to suggest the expansion. that the expansion In other ofmodels, the universe cosmic is accelerating[59–62]. acceleration is caused This by phenomenon exotic new can forms be interpreted of “dark energy” in classical General Relativity as an e↵ect of ⇤. In addition to the standard attractive gravity of normal forms of with T<0, whose gravity creates acceleration evenIn an emergent with ⇤ universe= 0. Those the cosmological models constant also introduce must be related new to fundamental the scales of mass,matter generally expressed unconnected in T , the cosmological with known term physics produces [61–63]. an additional repulsive acceleration between two bodies at separation r, entanglement of the two quantum subsystems: field vacuum and geometry 2 52 r¨ = H⇤r, (42) proportional to their separation r in Newtonian coordinates, where H2 ⇤/3. In cosmological models where matter ⇤ ⌘ has T>0, H⇤ is the asymptotic value of the Hubble expansion rate in the far future, after other forms of matter have thinned out due to the expansion. In other models, cosmic acceleration is caused by exotic new forms of “dark energy” with T<0, whose gravity creates acceleration even with ⇤ = 0. Those models also introduce new fundamental scales of mass, generally unconnected with known physics [61–63]. 17 V. EXOTIC ROTATIONAL CORRELATIONS AND COSMIC ACCELERATION The properties of the Planck scale rotational correlations are scale-free power laws, apart from the Planck scale itself. Systems in the real world have scales determined by the physics of matter fields. The previous discussion considers17 the e↵ects of entanglement with geometrical states on field degrees of freedom, including the reconciliation with This formulaholographic can be derived gravity, by andsumming small phase the extra displacements contributions measurable from each in interferometers. straight photon The path latter segment calculation around is definite a circuit. Let tenoughdenote for the a “extra” precise experimental propagation test. time We for now a photon— consider the estimates extra time of a complementary to the next reflection, e↵ect: how say— entanglement in a small interval ofof a field straight vacuum path, and due geometry to the fact could that break the the apparatus scale invariance is rotating. of gravity, It is related and account to the transversefor the value rotation of the scalar cosmological constant in the emergent classical field equations. d x = R!0d⌧ in the same interval by ? 11 cd t/d x =sin✓0, 17(39) The Cosmological? Constant and Cosmic Acceleration using theis determined notation shown by a choice in Fig of (5). world Combining line or timelike with the observer. swept area In the (Eq. emergent 30), the space, net time spatial lag structure around isa circuit defined is by This formula canintersecting be derived light by cones summing from the the ends extra of any contributions interval on the from world each line, called straight causal photon diamonds. path World segment lines, around intervals a circuit. Let t denote theThe “extra” Einstein propagation field equations time of for general a photon— relativity the can extra be contracted time to with the the next metric reflection, tensor g say—µ⌫ to yield in a a scalar of properrelation, time, and causal diamonds represent invariant emergent geometrical objects. In the emergent system, there small interval of ais straight no fixed backgroundpath, due to space the to fact define that an the inertialc apparatusd frame.t =2! From isA rotating./c. one moment It is torelated the next, to the the transverse situation depends rotation only(40) ˆ 0 0 d x = R!0d⌧ inon the previous same relationships. interval by The causal dependences of rotation2 around⇡G a world line are mapped out by the light cones ? emanating from each event. The total geometrical state of⇤ a= system(T includes+2 ), entanglement of all of the nested causal (41) The total Sagnac e↵ect is twice this amount because it is the totalc4 di↵erenceR between opposite directions. diamonds, and the total variance in positioncd t/d isx the=sin sum✓ of0, variances from the nested diamonds (see Figs. 2, ??(39)). The exotic e↵ect produces the tiny fluctuations around? this value, due to the imperfect definition of the frame. The The e↵ect can be thought of as a quantum di↵erential rotation4 of space: everything closer than R rotates relativeµ⌫ using the notationexotic correlation shownwhere in amplitude Fig⇤ denotes (5). for Combining the a Sagnac cosmological with device constant, the on swept any scale areaR (Eq.(c (Eq./16 36),⇡G 30),) which denotes the is net the a mean time Ricci square lagcurvature around displacement, scalar a circuitR g is isR equalµ⌫ in units to everything more4 distant than R, by a very smallR amount.⌘ It is helpful to visualize the fluctuations as⌘ a sum of of 16⇡G/c , Rµ⌫ denotes the Ricci curvature tensor, and T denotes the trace of the classical energy momentum tensor to the classicalcoherent displacement discrete transverse for an spatial apparatus displacements, rotating at or quantum-gravitationalangular velocity !0 = twistsc/2P0 of, times space, the associated Planck length. with discrete Once of matter, Tµ⌫ . Within general relativity, there is no mathematical di↵erence between ⇤ and a component of the again, thedisplacements exotic phase in time.displacement The amplitude resemblesc correspondsd approximatelyt =2!0A to0 a/c. Planck the value scale of di displacement↵raction bound in on a Planckian directional random resolution(40) walk, in matter field vacuum with Tµ⌫ˆproportional to gµ⌫ . In standard theory, ⇤ is a new constant of nature, unconnected but withspace, symmetries or to a discrete and projection Planck random factors walk appropriate in transverse to rotation. position. Notice On a that null trajectory both the exotic from a fluctuation distant sourceA to0/P an0 to other properties of gravity or matter[27]. Its value is also mysterious and arbitrary in holographic/ theories of and classicalobserver, lag for each! A Planckare defined step, space entirely experiences in terms a random of properties transverse of the displacement light path, from which the have classical a clear trajectory, invariant in The total Sagnac e↵ect isgravity[42, twice0 this0 57]. amount because it is the total di↵erence between opposite directions. meaning.which it shares/ common with other radial trajectories at the same R. The exotic correlation arises from the sum of The exotic e↵ect producesCosmic the tiny data fluctuations suggest that the around expansion this of value, the universe due to is the accelerating[58–61]. imperfect definition This phenomenon of the frame. can be The interpreted exotic correlationcoherent amplitude transverse for a Sagnac displacement device at on the any 2D boundaries scale (Eq. of 36), the nested which causal is a mean diamonds. square displacement, is equal In thein quantum classical states General of an Relativity interferometer, as an e↵ fieldsect of are⇤. entangled In addition with to geometry. the standard The attractive measured gravity radiation of normal depends forms of to the classical displacementmatter for expressed an apparatus in T , the rotating cosmological at angular term produces velocity an!0 additional= c/2P0, repulsive times the acceleration Planck length. between Once two bodies at on theV. phases EXOTIC of light incidentROTATIONAL on the beamsplitter, CORRELATIONS which combines AND geometricalCOSMIC ACCELERATION information from macroscopically again, the exoticdi phase↵erent displacementseparation paths throughr, resembles the space[47–49]. approximately The projection the value of geometrical of displacement quantum in states a Planckian is determined random by the walk, true but with symmetriescausal and structure projection of the factors space, the appropriate therefore by to the rotation. transverse components to the observer’s radial direction. The The properties of the Planck scale rotational correlations are scale-free2 power laws, apart from the Planck scale itself. relationship between the two is analyzed in next section to computer¨ = H⇤ ther, e↵ect on an interferometer signal. (42) Systems in the real world have scales determined by the physics of matter fields. The previous discussion considers the e↵ects of entanglementproportional to with their geometrical separation r statesin Newtonian on field coordinates, degrees of where freedom,H2 including⇤/3. In cosmological the reconciliation models where with matter V. ROTATIONAL FLUCTUATIONS AND COSMIC ACCELERATION⇤ ⌘ holographic gravity,has T> and0, smallH⇤ is phasethe asymptoticBehavior displacements of value Exotic of Rotational measurable the Hubble Fluctuations expansion in interferometers. in rate Large in the Systems far The future, latter after calculation other forms is ofdefinite matter have enough for a precisethinned experimental out due to the test. expansion. We now In considerother models, estimates cosmic of acceleration the complementary is caused by e exotic↵ect: new how forms entanglement of “dark energy” The properties of the Planck scale rotational correlations are scale-free power laws, apart from the Planck scale itself. of field vacuumLike zerowith and pointT< geometry fluctuations0, whose could gravity of a break field creates excitation the acceleration scale in invariance a vacuum, even with of this⇤ gravity,= quantum 0. Those and “motion” models account also represents for introduce the value a quantum new of fundamental the system scalar scales Systems incosmological the realwith world a constant nonzeroof have mass, width in scales generally the of emergent determined a wave unconnected function classical by with that the field known physics has equations. a physicszero of mean. matter [60–62]. In fields. this picture, While the we zero have point discussed refers to athe stationary e↵ect of geometrical entanglementdirection in the on classical field degrees nonrotating of freedom, frame. A including quantum relationship the small with phase matter displacements out to the cosmic in interferometers, horizon prepares we have not yet estimatedthe global state the e and↵ect defines of the zero matter rotation. vacuum A zero on net emergent rotation quasi-classicalstate is defined by geometry. the classical It geometry, could be but that nearby the matter vacuum determinesthere is a nonzero the value quantum ofCosmologicalThe the variance Cosmological scalar that cosmological Constant depends Constant from on distance constant Entanglement and from Cosmic in an the observer. of Acceleration classical Geometry The field with wave equations. the function Field width Vacuum is identified with the exotic correlations and fluctuations. Directional degrees of freedom are subject to the usualCosmic holographic Centrifugal bounds Acceleration on information, so the above estimates The Einstein field equations of general relativity can be contracted with the metric tensor gµ⌫ to yield a scalar of directional varianceThe Cosmological lead to an estimate Constant of the andexotic Cosmic directional Acceleration correlations and their associated fluctuations,18 as relation,shown in Figure (2). The exotic correlation and rotational displacement variance at separation R from an observer are about If matter and geometry are subsystems of a single quantum system, the standard distinction between matter (T ) they are significantly entangled. As shown above, that happens2⇡G in an energy dependent way for matter fields, at the The Einstein field equationsand geometry of general ( ) becomes relativity ambiguous can⇤ be= on scales contracted(T where+2 they with), are the significantly metric tensor entangled.gµ⌫ Asto shown yield above, a scalar that(41) happens R 4 2 relation,Chandrasekhar radius forin a an given energy particle dependent energy. way ford It⌅ matterR is(⌧) natural/d⌧ fields,c(d to atx the suggest/dR ChandrasekharR)(dR/d that⌧) thec` radius valueP . for of a⇤ givenis set particle by entanglement energy. It is(22) natural to suggest that the value of ⇤ in the emergent⇡ classicalh ?i description⇡ is set by entanglement of a special scale of the field of a special scale of the matter vacuum with geometry. 4 µ⌫ where ⇤Thedenotes rotational the cosmological displacement isconstant, coherent in2 all⇡RG directions(c /16⇡G from) denotes the observer the Ricci at curvaturea constant scalar radial separationR g Rµ defined⌫ in units by We propose to interpretvacuum cosmic with acceleration geometry. as anR emergent⌘ kinematic property of space, associated with⌘ rotational of 16⇡G/c4, R denotes the Ricci curvature⇤ = tensor,4 ( andT +2T denotes), the trace of the classical energy momentum(41) tensor fluctuations in theits inertial causalµ⌫ structure—We frame propose and that to their interpret is, on entanglement a cosmic 2+1-D accelerationc tube with of spacelike known asR an emergent 2-surfaces matter kinematic fields. swept In out property this in time interpretation, of by space, causal associated diamonds the with scale with rotational of matter,fixedT sizesµ⌫ .fluctuations WithinR = c⌧/ general2. in For the calculational relativity, inertial frame there simplicity, and is their no in mathematical entanglement this paper we with di focus↵erence known on results matter between in fields. nearly-flat⇤ and In this a component space interpretation, relevant of for the the scale whereof cosmic⇤ denotes acceleration the cosmological is determined constant, by knownR length(c4/16 scales⇡G) denotes already the built Ricci into curvature physics, scalar the PlanckR gµ scale⌫ R andin units the matterexperiments, field vacuumof cosmic but with the accelerationT formulationµ⌫ proportional is baseddetermined to ongµ causal⌫ . by In known diamonds standard length shouldtheory, scales generalize⇤ alreadyis a new built covariantly constant into physics, to of curved nature, the spaces. Planckµ unconnected⌫ scale A toy and the strong interaction4 scale. On cosmological scales,R ⌘ the average e↵ect is the same as a classical constant⌘ ⇤. The most of 16⇡G/c to, R otherµ⌫ denotesmodel properties ofstrong the 3D holographic Ricci of interaction gravity curvature or quantum scale. matter[28]. tensor, On positional cosmological and ItsT value states,denotes scales, is summarized also the the mysterious average trace in of Appendix e↵ theect and classicalis the arbitrary (A), same based energy as in a on holographic classical momentum the 3D constant spin theories tensor algebra,⇤. The of most ofdirect matter, evidencegravity[43,Tµ⌫ . for Withinprovides this 58]. mechanism andirect general example evidence relativity, of would a for consistent this come there mechanism holographic from is no interferometric would mathematical model come of from quantum measurementinterferometric di↵erence position between states measurement of Planck in⇤ 3D.and rotational We of a exotic use component classical rotational fluctuations space-time of correlations the on matteron laboratory vacuumCosmic scales. withfor data theTµ⌫ time suggestlaboratoryproportional projection, that scales, the toas expansion as ingµ discussed Eq.⌫ . In (22). standard of above. Since the universe the theory, states is⇤ accelerating[59–62]. usedis a are new in constant 3D instead This of of nature, 4D, phenomenon time unconnected evolution can be is still interpreted to other treated heuristically, both here and in the calculation of interferometer response below. propertiesThe startingin of gravityclassical point or General is matter[28]. to drawA simple Relativity a simple starting Its value as comparison an pointCosmic is e↵ also isect to mysteriousof compareAcceleration with⇤. In Eq. Eq. addition and (42): (42) and arbitrary with in to Rotational a the a classical classical standard in holographic Acceleration system system attractive rotating rotating theories gravity at aat rate of a of gravity[43,rate! normal. A! body,abody forms at 58]. separation of We can describe the e↵ect as statistical fluctuations in rotation of the inertial frame around each axis. An ordinary atCosmic separation datamatterr suggestfrom expressed the thatr axisfrom in theT of the, expansion rotation the axis cosmological of rotation experiences of the experiences universe term a produces centrifugal is a accelerating[59–62]. centrifugal an acceleration,additional acceleration, repulsive This phenomenon acceleration between can be interpreted two bodies at rotation about a body at theClassical origin correspondskinematic acceleration to transverse at distance components r from an axis motion of rotation:dx /dt of a body at x according separation r, i k in classical Generalto the Relativity classical relation, as an e↵ect of the cosmological constant ⇤. In addition to the standard attractive gravity of normal forms of matter expressed in T , the cosmological r¨ = ! 2 r. term r¨ produces = ! 2 =r. rotation an rate additional repulsive acceleration(43) (43) r¨ = H2 r, (42) between two bodies at separation r, Cosmic acceleration by cosmologicaldxi/dt =⇤ ✏ ijkconstantxk!j at, separation r : (23) Like cosmic acceleration (Eq. 42), it is independent of mass, always directed outwards, and is proportional to r. 2 2 proportionalwhere to! theiris the separation rotation rater in for Newtonian componentr¨ j coordinates,=. InH ther, exotic where fluctuations,H ⇤ each/3. In component cosmological of rotation models of where a 2-sphere matter(42) of The two outward accelerationsj agree if ! = H⇤. However,⇤ it is well known⇤ ⌘ that cosmic acceleration cannot simply has T>radius0, H⇤Rishas the fluctuations asymptotic with value of the Hubble expansion rate in the far future, after other forms of matter have be due to a classical centrifugal accelerationMean square from exotic rotation rotational on fluctuation a cosmic2 in volume scale. of size Classical R: rotation is not isotropic, proportionalthinned to their out separation due to the expansion.r in Newtonian In other coordinates, models, cosmic where accelerationH⇤ ⇤/ is3. caused In cosmological by exotic new models forms whereof “dark matter energy” and the centrifugal e↵ect from rotation at a rate ! = H⇤ 2would cause2 ⌘ large3 amplitude, large scale anisotropy of has T>0,withH⇤ isT< the0, asymptotic whose gravity value creates of the acceleration Hubble expansion even with!i (R rate⇤) = 0. inc ` Those theP R far. models future, also after introduce other forms new fundamental of matter have scales(24) cosmic radiation and expansion that is not observed. Cosmich accelerationi⇡ could however be associated with rotational thinned outof due mass, to generally the expansion. unconnected In other with models, known cosmic physics acceleration [61–63]. is caused-2/3 by exotic new forms of “dark energy” fluctuations on aAll much transverse smaller displacements scale. These onCosmic average a 2-sphere and rotational out are in selfaccelerations a large consistent scale match according time at R ~ or H to space ~ the 60 km classical average, ~ relation so they Eq. do 23. not That conflict is, each with T<0, whose gravity creates accelerationgeometrical even entanglement with ⇤ = length 0. Those for standard models strong alsointeractions introduce new fundamental scales ofwith mass, cosmic generally isotropycomponent unconnected data. of Even rotational with if rotation known fluctuation has physics zero determines [61–63]. mean and an entangled fluctuates motion in random across the directions, whole sphere. its fluctuations In the same always way that push outwards, sospins the of mean quantum square particles fluctuating sentidentify in opposite component cosmic acceleration directions creates ~ are rotational a entangled, net, mismatch secular in of this centrifugalfield case, vacuum the rotation acceleration. of all the Colloquially, enclosed world we can say that the universe shakes apart.The universe “shakes apart”: ~ 60 km twists by ~ 1 Fermi, at ~ 5000 Hz Of courseCosmological in Newtonian Constant physics, from an Entanglement acceleration entails of Planckian a force. Centrifugal Fluctuations acceleration with3 the can Strong refer Vacuum to a real force, Reprises the Dirac/Eddington/Zeldovich/Bjorken coincidence, m ~ H as one experiences in a spinning merry-go-round, or a fictional one, as in a noninertial rotatingpion 0 frame. They refer to the same e↵ect: because of the noninertial relationshipCosmic Centrifugal between Acceleration radial53 and tangential components of position, a body that moves at a velocity that maintains a constant direction in a rotating frame accelerates outwards in the inertial frame.This description In an emergent of global geometry, dynamics the definition is at the of anclassical emergent level. acceleration If matter is and bound geometry up with are the subsystems definition of of emergent a single quantumposition. system, Here, rather the standard than interpreting distinction cosmic between acceleration matter (T as) aand new geometry force, we ( interpret) becomes it as ambiguous an emergent on scales kinematical where e↵ect of exotic rotational correlations that describe Planckian deviations from inertialR motion— the mismatch between the local and cosmic inertial frames. The rotational fluctuations average to smaller values at large scales and long durations, but there may be a very small residual systematically outward-directed emergent radial acceleration. Clearly, quantum-geometrical fluctuations in rotation do not have such an e↵ect on all scales. If they did, everything would fly apart at the Planck rate. But there is a scale where the Planckian rotational acceleration matches the 1 observed cosmic acceleration. As it turns out, that scale coincides with the geometrical entanglement scale for ⇤QCD , the structural scale of the strong interaction vacuum. This heuristic model suggests an explanation for the value of the cosmological constant. Absolute Value of the Cosmological Constant, ⇤ As a rough estimate, start with the root-mean-square rate of Planckian rotational fluctuations at separation R 3/2 given by Eq. (24) in Planck units, !R = R . Centrifugal acceleration thus matches cosmic acceleration— that is, !R = H⇤— for fluctuations of spatial extent 2/3 R H . (44) ⇤ ⇡ ⇤ This is the Chandrasekhar-Planck entanglement radius (Eq. 8) for particle mass 1/2 1/3 m(R = R )=R H . (45) C ⇤ ⇤ ⇡ ⇤ 1/3 It has long been noticed that in Planck units, H0 approximately coincides with a proton or pion mass. A direct 1/3 physical relationship between cosmological ⇤ and a strong interaction scale— m⇡ H⇤ — was posited long ago by Zeldovich[64], and more recently by Bjorken and others[65–69]. In the picture here,⇡ the connection is established, and the scale is set, via entanglement of the matter vacuum with geometrical rotational states. We conjecture that on small scales, rotational fluctuations are “virtual” in the sense that a net outward acceleration is not generated in the emergent space, but that above a scale R⇤ connected with strong interactions, entanglement with the matter vacuum produces a universal secular radial acceleration. At a spatial scale of R⇤, rotational fluctuations on a light crossing Vacuum Cosmic with agrees Acceleration Rotational Model of Standard Acceleration log(equivalent particle energy/ Planck mass) - - - 60 40 20 20 40 60 0 0 (~ pion mass) (~ pion QCD scale QCD scale 10 log R ~ m R ~ ( system 20 M -2 size 30 / Planck ~ 60 km cosmic acceleration produces rms acceleration ~ Exotic rotation on QCD scale 40 R length ~ m ) -1 50 with dark energy density energy dark with to theory agree standard in UV cutoff needed 60 Cosmic scale Cosmic 54 19 1/3 It has long been noticed that in Planck units, H0 approximately coincides with a proton or pion mass. A direct 1/3 physical relationship between cosmological ⇤ and a strong interaction scale— for example, m⇡ H⇤ — was posited long ago by Zeldovich[64], and more recently by Bjorken and others[65–69]. In the picture here,⇡ the connection is established, and the scale is set, via entanglement of the field vacuum with exotic geometrical rotational states. We conjecture that on small scales, rotational fluctuations are “virtual” in the sense that a net outward acceleration is not19 generated in the emergent space, but that above a scale R⇤ connected with strong interactions, entanglement with the field vacuum produces a universal secular radial acceleration. At a spatial scale of R , rotational fluctuations on time, with transverse width about equal to the strong interaction length, gives an rms angular⇤ rotation rate about a light crossing time, with transverse width about equal to the strong interaction scale, gives an rms angular rotation equal to the Hubble constant. rate about equal to the Hubble constant. The entanglement conjecture suggests a more precise way to relate the value of ⇤ to the matter vacuum. The The entanglement conjecture suggests a more precise way to relate the value of ⇤ to a field vacuum scale. The e↵ectivee↵ective constant constant⇤ in⇤ in the the classical classical field field equation equation comes comes about about from from entanglement entanglement of of the the subsystems subsystems represented by by matter T and curvature . In the classical view the total information in the system is expressed in holographic terms, matter T and curvatureR . In the classical view the total information in the system is expressed in holographic terms, thethe area area of the of the event event horizon horizonR as as determined determined by by⇤.⇤. The The number number of of degrees degrees of of freedom freedom of of the the cosmic cosmic system can can be be estimatedestimated in standard in standard physics physics from from the the matter matter side side or or from from the the geometry geometry side. side. They They agree agree when when the the 3D information information densitydensity for for the the field field vacuum vacuum equals equals the the mean mean 3D 3D density density of of holographic holographic information information within within the the cosmic cosmic horizon[22, 58]. 58]. The number of gravitational degrees of freedom is one quarter of the area of the cosmic event horizon in Planck The number of2 gravitational degrees of freedom is one quarter of the area of the cosmic event horizon in Planck units, given by ⇡H⇤ .2 Dividing by the 3-volume gives the holographic information density units, given by ⇡H⇤ . Dividing by the 3-volume gives the holographic information density ⇤ =3=3HH⇤/4/.4. (46)(46) I I⇤ ⇤ TheThe density density of field of field modes modes per per 3D 3D volume volume with with a a UV UV cuto cuto↵↵atatkmaxkmaxisis Cosmic information ~ number3 3 of QCD33 field modes f (k(maxk )=)=kk 4⇡4⇡/3(2/3(2⇡⇡)). . (47)(47) I If max maxmax Holographic information in (2D) cosmic horizon equals (3D) information 18 A scalarA scalar field field therefore therefore matches matches cosmic cosmic information information (that (that is, is, ⇤== f(k(kmax)))) for for a a field field cuto cuto↵↵ atat in field modes up to: I⇤I IIf max One estimate[20] of H⇤ from current cosmological data[53,2 2 61]1/1 yields3/3 k kmax==k k⇤ (H(H⇤9⇡9⇡//2)2) . . (48)(48) max ⇤ ⌘⌘ ⇤ 1/2 61 One estimate of H from current cosmologicalH⇤ = data[22,⌦⇤ H0 62,=0 70].99 yields0.018 10 (51) One estimate of H⇤ from⇤ currentAdopt cosmological a current cosmic data[62, measurement 70] yields[22] in Planck± units: ⇥ in Planck units, where H0 denotes the current1/2 value of the Hubble constant61 and ⌦⇤ denotes the density parameter H⇤ = ⌦ H1/02 =0.99 0.018 10 61 (49) H = ⇤/3=⇤⌦ H =0.99 0.018 10 (49) of the cosmological constant. The⇤ value of the⇤ field0 cuto↵±scale± that⇥ ⇥ matches the observed cosmological information indensity Planck is units, thus where H denotes the current value of the Hubble constant and ⌦ denotes the density parameter of in Planck units, where H 0denotesThe corresponding the currentp cut value off closely of the matches Hubble the QCD constant vacuum and scale:⌦⇤ denotes the density parameter the cosmological constant.0 (This value and error estimate are not meant to be definitive,⇤ but are taken to illustrate of the cosmological constant. (The value and errors are not meant20 to be definitive, but are taken to illustrate typical typical values of current measurementsk near⇤ =1 the.65 state0.01 of the10 art; themP values= 201 chosen1.2 here MeV correspond, to h =0.738 0.024 (52) values of current measurements near the state of± the art;⇥ the values here± correspond to h =0.7380 0.024± and and ⌦⇤ =0.7 0.01 in standard cosmological notation.) The value of the field cuto↵ scale that matches this± observed ⌦⇤ =0which.7 is0.01[22].) remarkably± The close value to of the the field scale cuto⇤ ↵ scaledefined that matches by the strong the observed interactions[62]. cosmological The information corresponding density spatial cosmological± information densityMain uncertainty is in excellentQCD agreement comes from QCD theory is thuscoherence scale of the rotational entanglement is of the order of Entanglement of quantum geometry 20with field vacuum could explain the k⇤ =1.65 0.01 102 20 mP = 20139 1.2 MeV. (50) valuek of⇤ the=1 cosmological.65 ±0R.01 =constantk⇥10 =3 fromm.7P only=10 known 201 =± scales1 60km.2 MeV of. physics. (50) (53) ± ⇤ ⇥ ⇤ ⇥ ± This value is close to the characteristic scale ⇤QCD of the55 strong vacuum. Estimates of that scale, from theoretical Thisextrapolation valueThe is scale close of invariance to measurements the characteristic of gravity at higher is scale thus energies[71],⇤ brokenQCD of by the have the strong strong a fractional vacuum. interactions, systematic Estimates as shown uncertainty of that in Figure scale, comparable (6).from The theoretical those e↵ect of of the extrapolationthematter cosmological vacuum of measurements measurements. is imprinted at here Withinhigher not energies[71], current by a coupling, systematic have but a errors, by fractional a process the cosmic systematic related and matter to uncertainty ongoing information collapse comparable densities of wave those agree. function of of theThe cosmologicaldirectionally corresponding measurements. entangled spatial fields coherence Within and geometry. scale current of the systematic The rotational scale errors,of entanglement cosmic the acceleration cosmic is about and matteris set by information the imperfect densities emergence agree. of the The correspondinginertial frame— spatial the scale coherence where scale the directional of the rotational information entanglement content is of about the field vacuum runs up against the Planck 2 39 limit. In the cosmic system, the strongR interaction⇤ = k =3 length.7 10 in 3D= 60km maps. holographically onto the Planck length(51) in 2D. 2⇤ ⇥ 39 The nature of the entanglement canR be⇤ = viewedk⇤ =3 from.7 either10 = the 60km matter. or the geometry side. On the matter(51) side, it ⇥ 1 comesIn this from scenario, the distance the scale where invariance the directional of gravity is alignment broken by of the a field strong mode interactions. of mass m The= e⇤↵QCDect of, an the angle field vacuum (R⇤QCD) , Inis this imprinted scenario, here the not scale by a invariance coupling, but of gravity by quantum is broken1/ entanglement2 by the strong of fields interactions. with geometry. The e The↵ect scale of the of cosmic matter equals the Planck di↵raction angle (Eq. 13), R . On the geometry side, it has to do with localization and vacuumaccelerationinformation is imprinted is loss set here onby the not cosmic imperfect by a coupling, scale. emergence In standard but of by the a quantum process inertial related frame— mechanics, to the ongoing the scale matter where quantum wave the entanglement directional function has information anof matter uncertainty withcontentgiven geometry. of by the Eq. The field (15) scale vacuum for of a cosmic runs particle up acceleration against of mass them Planck isover set by limit. duration the In imperfect the⌧; cosmic this emergence is system, the spatial exotic of the width correlation inertial of the frame— maps wave the the function strong scale of a wherecorrelation the directional length in information 3D holographically content of onto the the field Planck vacuum length runs in up 2D. against the Planck limit. In the cosmic system, particle world line, for a causal diamond extending over ⌧. For the matter vacuum with m = ⇤QCD, that equals the the strongThe nature interaction of the length entanglement in 3D maps can holographically be viewed1 from onto either the the Planck matter length or the in 2D. geometry side. On the matter Chandrasekhar radius at cosmic duration, H⇤ . Virtual rotation a↵ects secular radial acceleration when localization Theside, nature it comes of the from entanglement the distance can where be the viewed directional from either separation the matter between or field the modes geometry of mass side.m On= the⇤QCD matter, an in space matches1 the size of a field state over a Hubble time, the timescale1/2 for information from infinity to be refreshed, side,angle it comes (R⇤QCD from) , the equals distance the Planck where di the↵raction directional angle (Eq.separation 13), R between . On field the modes geometry of side, mass itm has= to⇤QCD do with, an and the natural1 scale for the cosmic expansion to “notice” the mismatch1/2 of local and cosmic inertial frames. At the anglelocalizationradius (R⇤QCDR ,) and incoming, information equals rotational the Planck loss cosmic on di the↵raction information cosmic angle scale. aligns (Eq. In standard the 13), longR quantum term. On average the mechanics, geometry with the the cosmic side, matter it inertial has wave to frame, function do with leaving has an uncertainty⇤ given by Eq. (15) for a particle of mass m over duration ⌧; this is the spatial width of the wave localizationa residual and centrifugal information acceleration loss on the at cosmic larger radii. scale. In standard quantum mechanics, the matter wave function hasfunction an uncertainty of a particle given world by Eq. line, (15) for for a causal a particle diamond of mass extendingm over over duration⌧. For⌧ the; this field is vacuum the spatial with widthm = ⇤ ofQCD the, wave that Although the detailed underlying mechanism of entanglement1 has not been rigorously modeled, these heuristic argu- functionequalsments of the suggest a particleChandrasekhar a relation world line, radius of the for Standard at a cosmic causal Model duration, diamond to cosmicH extending⇤ . Virtual acceleration over rotation⌧. viaFor a basic the↵ects matter physical secular vacuum principles,radial acceleration with withoutm = ⇤ whenQCD additional, localization in space matches the size of a field state over a Hubble1 time, the timescale for information from infinity thatparameters, equals the Chandrasekhar scales, or fields radius (see Figure at cosmic 6). duration, It is clearlyH⇤ too. Virtual simplistic rotation to picture a↵ects cosmic secular acceleration radial acceleration as simply due whento localization the universe in “shaking space matches apart”, the but size there of a appears field state to be over a plausible a Hubble connection time, the between timescale two for measurable information phenomena from infinitythat to have be refreshed, no relationship and the in natural standard scale theory: for the the cosmic cosmological expansion constant, to “notice” and the exotic mismatch correlations of local in and interferometers. cosmic inertialSince frames. the rotational At the radius correlationsR⇤, incoming could be rotational confirmed cosmic in an experiment, information the aligns Sagnac the long experimental term average program with described the cosmichere inertial could frame, provide leaving a concrete a residual clue to centrifugal the link between acceleration Standard at larger Model radii. physics and cosmic acceleration. It is clearly too simplistic to picture cosmic acceleration as simply due to the universe “shaking apart”, but there appears to be a plausible connection between two independently measurable phenomena that have no relationship in standard theory: the cosmological constant,Coherent and exotic Fluctuations correlations of Cosmic in interferometers. Acceleration Since the rotational correlations A new and unique feature of this scenario is that cosmic acceleration is not constant, but fluctuates holographically. The systematic acceleration that we measure in cosmic data is a sum of fluctuating contribution from many small scale rotational fluctuations. In principle, there must be a locally measurable, radial component to exotic Planck motion that was not included in the scale-free analysis used for the Sagnac interferometer. It corresponds to a monopole expansion mode of spatial motion, unlike the pure tensor modes associated with gravitational waves, or pure rotational modes associated with exotic Planck fluctuations, without the entanglement of the matter vacuum. Unlike a standard cosmological constant, the radial component is not constant but fluctuates. The fluctuations are not scale free, but have a characteristic temporal frequency at about c/R⇤ 5000 Hz, and a spatial coherence scale of about R⇤ 60 km. In principle, these distinctive features could enable a⇡ direct experimental test of this scenario for emergent⇡ cosmic acceleration. The temporal variation and spatial coherence distinguish it from classical backgrounds. The e↵ect could be measured using correlated signals between spatially separate, ground-based interferometers. Normally in gravitational wave observatories, the important signal is the dark port of the Michelson interferometer, which records the arm-length di↵erence or strain mode associated with gravitational waves. For cosmic gravitational wave sources, this signal is correlated over large interferometer separations. Here, we are interested in another signal, the common-mode or bright port signal associated with a breathing mode, which measures the sum of the arms. It should have significant power only at low frequencies (below about 5000 Hz), and should be correlated spatially only Emergent Cosmological Constant Suppose quantum geometry entangles with the Standard Model field vacuum to produce a cosmological constant in the emergent space-time Mean rotation vanishes, consistent with statistical isotropy Not perfectly constant, but fluctuates: coherence scale is about 60 km Cosmic acceleration fluctuates below ~ 5000 Hz Radial fluctuations by ~ 1 Planck length— likely unobservable Cosmic acceleration timescale naturally occurs on astrophysical evolution timescale They both depend on similar combinations of fundamental constants, related to the Chandrasekhar mass for protons (Planck+ QCD scales) Explains “why now” 56 Summary Planck scale quantum geometry can lead to exotic correlations in displacements of bodies and phases of fields that grow with scale, in the same way as standard quantum correlations in extended systems. Even without a theory of quantum gravity, basic quantum principles suffice to predict the effect in the signal of an interferometer. The prediction can be tested with a reconfiguration of the Fermilab Holometer. Entanglement of these correlations with the Standard Model vacuum might explain the origin of the cosmological constant. 57 Extra slides 58 14 Define the total swept area A(⌧) as a sum of the rotational components in the two directions, so that + dA/d⌧ = dA /d⌧ + dA /d⌧ , (37) | | | | and the total enclosed area is 1 P0 A0 = d⌧dA/d⌧ (38) 2 ˆ⌧=0 when integrated over the total perimeter P0. Adding the contributions from the two directions in Eq. (36) then leads to the following formula for exotic correlation in a planar Sagnac interferometer of any shape: ⌅(⌧)/` = ⌅ /` dA(⌧)/cd⌧ , (39) P 0 P | | where the correlation at zero lag, ⌅ ⌅(⌧ = 0) = 2A ` /P , (40) 0 ⌘ 0 P 0 gives the total mean square phase variation from the exotic correlation in length units. This constant is determined by the requirement that the correlation integrates to zero for c⌧ >P0, so that the average rotation vanishes in the classical limit of zero frequency. A specific example of the predicted| | correlation function for a square configuration is shown in Fig. (5). It is interesting to compare the exotic correlations with the classical Sagnac e↵ect. For an apparatus of any shape, the phase shift in length units between light going in two directions around a circuit rotating with angular velocity ⌦0, relative to the local nonrotating inertial frame, is c t =4A0⌦0/c. (41) The exotic e↵ect produces the tiny fluctuations around this value, due to the imperfect definition of the frame. At the Planck scale, the two formulas agree. The exotic correlation amplitude for a Sagnac device on any scale (Eq. 40), which is a mean square displacement, is proportional to the classical displacement for an apparatus rotating at angular velocity ⌦0 = c/2P0, times the Planck length; that is, the exotic phase displacement resembles a Planckian random walk in transverse spatial position. The correlation can also be visualized as due to rotational fluctuations in the inertial frame, as discussed below. 2 The general formula (Eq. 39) respects the symmetries expected for an emergent geometry, and for a Sagnac apparatus. It appearsII. to NORMALIZATION have the correct spatial FROM scaling LOOP and transformation QUANTUM properties GRAVITY to describe the 1D time domain correlation for a planar interferometer of any shape. It also makes physical sense; for example, in sections of the circuit where light is not interacting with matter, the path is straight, so that dA (⌧)/cd⌧ is constant for Discretecorresponding spectrum intervals of area of states⌧. Thus in LQG: as expected, the correlation is constant over those intervals;| the phase| of radiation is not a↵ected by trajectories of emergent world lines it does not interact with. At reflections, the light path changes 3 directions and the correlation changesA valuej =8 discontinuously,⇡ ~Gc asji( expectedji + 1) if the matter fields associated with the (6) positions of the mirrors are directionallyExotic Correlation entangled Appears with in Loop emergenti Quantum geometry. Gravity The area quantization from Loop Quantum Gravity describedX p above (Eq. 19) has an exact normalization derived Some theories of quantum gravity have no fixed background space where fromji are the half-integers entropy of blacki/2 holefor integers states thati. can The be Immirizi used to set parameter an absolute is normalization estimated from for the black correlation hole entropy function. to be =ln2/⇡p3. For large areas (i>>Discrete1) wespectrum can of write spatial area the states, eigenvalues normalized [ofby black the hole “main entropy: sequence” states] as 1/2 For definiteness, consider a square arrangment with sides L. For a square, Eq. (18) gives n An/AQ LAQ , 1/2 ⇡ ⇡ 2 hence a natural unit of length `Q = AQ . This fixesA an absoluteA n normalization for the signal correlationp function, (7) n ⇡ Q for integers n,where 1/2 3 p `Q = AQ = 2ln2~Gc / 3 (42) q 3 AQ 2⇡ ~Gc (8) It could be that the appropriateSeparation fundamental of geometrical unit states to⌘ use grows in Eq.like dA (39) ~ A1/2 is ~ not R `P , but the loop- normalized Planck length `Q (Eq. 42). They are almost the same, but in any case experiments should be designed to reach this value. The is a fundamental Planck area unit. The di↵erence between adjacent2 area states is di↵erence would however be measurableDegrees of freedom in an actualscale holographically, experiment, like R which measures the absolute value of the Planck scale. Exotic variance ofA position correlationsA 2~ AR n. (9) n+1 n ⇡ Q Notice that the spacing grows with n. Thus for larger areas, each area eigenstate is bigger than the last by much more V. ROTATIONAL…``the fundamental FLUCTUATIONS quanta of geometry are one dimensional, AND polymer-like COSMIC excitations ACCELERATION over nothing, than a single Planck area pixel. Thisrather than property gravitons, the wavy reflects undulations the over a continuum same background.” blurring or loss of relative positional information 1408.4336, Ashketar, Reuter and Rovelli on large scales that appears in wave equation solutions, holographic59 information constraints, and other proposed The properties of the Planck scale rotational correlations are scale-free power laws, apart from the Planck scale “infrared” departures from field theory. It explicitly captures an essential nonlocal Planckian feature not compatible itself. Of course systems in the real world are not scale free; systems including stars and their remnant black holes with a metric-based field approach: the positional relationships of the geometry itself depend on separation, and degrade over large areas. Arguments give above suggest that the correlation function for phase variations around a planar circuit scales with the area divided by the circumference. Not surprisingly, the correlations are related to the shape of the circuit. Consider an area defined by a planar surface between several world lines, say, the centers of mass of mirrors in a Sagnac interferometer, all at rest. In classical language, we propagate this surface forward in time by extending along their world lines in the rest frame. In Quantum General Relativity, there is no independent clock ticking; the equivalent of a Planckian clock tick is defined relationally by a transition between quantum states of the spacetime geometry. Consider the light propagating between the mirrors around this area in two directions from the beam splitter. 1/2 For definiteness, consider a square arrangment with sides L. For a square we have n LAQ , hence a natural 1/2 ⇡ unit of length `P = `Q = AQ . This fixes an absolute normalization for the signal correlation function. In a square, propagation around opposite sides of the circuit occurs with a maximal separation, and contributes maximal correlation for a given area. 3 `Q = 2ln2~Gc /p3 (10) q The time propagation of the world lines that define the corners of the square surface defines a relationship between (local) linear propagation around the circuit, and (nonlocal) areal properties of the enclosed space. When the light comes full circuit to a measurement at the beamsplitter, the wave function of the apparatus “collapses” into a definite phase di↵erence; the light in the circuit is in a superposition along with di↵erent geometries, each associated with di↵erent optical phases. Thus a correlation is created by the discrete geometrical system of world lines across the whole apparatus; thus, the result depends both on the total area and on the time propagation, which in turn depend on changes in direction from interactions with world lines of matter that create the circuit. Although we have expressed it as a variance in arm length, the positional correlation can also interpreted as a variation in time. The relative spatial positions of reflection events at opposite sides of the circuit, including their time positions, have a quantum ambiguity inherited from the granularity of the Planckian quantum states of the enclosed space. The e↵ect is not associated with any locally defined function of the classical metric and its derivatives, which would be independent of the measurement being made. This estimate provides an exact normalization for the predicted Sagnac quantum geometrical correlation function that applies to any shape. It is not su ciently rigorous to prove that the correlations have a pure-rotational symmetry, although that appears plausible given the structure of the theory. It is useful as a baseline target for design of experiments. It suggests that the tools of Loop Quantum Gravity will be su cient to make adequately precise predictions for unique quantum-gravitational e↵ects that are realistically measurable in the laboratory. 2 II. NORMALIZATION FROM LOOP QUANTUM GRAVITY Discrete spectrumDiscrete of area area states: states spacing in LQG: increases with size 3 Aj =8⇡ ~Gc ji(ji + 1) (6) i X p where ji are half-integers i/2 for integers i. The Immirizi parameter is estimated from black hole entropy to be =ln2/⇡p3. For large areas (i>>1) we can write the eigenvalues [of the “main sequence” states] as A A n2 (7) n ⇡ Q for integers n,where 3 AQ 2⇡ ~Gc (8) ⌘ is a fundamental Planck area unit. The di↵erence between adjacent area states is Planck area Difference >> Planck area An+1 An 2AQn. (9) 60 ⇡ Notice that the spacing grows with n. Thus for larger areas, each area eigenstate is bigger than the last by much more than a single Planck area pixel. This property reflects the same blurring or loss of relative positional information on large scales that appears in wave equation solutions, holographic information constraints, and other proposed “infrared” departures from field theory. It explicitly captures an essential nonlocal Planckian feature not compatible with a metric-based field approach: the positional relationships of the geometry itself depend on separation, and degrade over large areas. Arguments give above suggest that the correlation function for phase variations around a planar circuit scales with the area divided by the circumference. Not surprisingly, the correlations are related to the shape of the circuit. Consider an area defined by a planar surface between several world lines, say, the centers of mass of mirrors in a Sagnac interferometer, all at rest. In classical language, we propagate this surface forward in time by extending along their world lines in the rest frame. In Quantum General Relativity, there is no independent clock ticking; the equivalent of a Planckian clock tick is defined relationally by a transition between quantum states of the spacetime geometry. Consider the light propagating between the mirrors around this area in two directions from the beam splitter. 1/2 For definiteness, consider a square arrangment with sides L. For a square we have n LAQ , hence a natural 1/2 ⇡ unit of length `P = `Q = AQ . This fixes an absolute normalization for the signal correlation function. In a square, propagation around opposite sides of the circuit occurs with a maximal separation, and contributes maximal correlation for a given area. 3 `Q = 2ln2~Gc /p3 (10) q The time propagation of the world lines that define the corners of the square surface defines a relationship between (local) linear propagation around the circuit, and (nonlocal) areal properties of the enclosed space. When the light comes full circuit to a measurement at the beamsplitter, the wave function of the apparatus “collapses” into a definite phase di↵erence; the light in the circuit is in a superposition along with di↵erent geometries, each associated with di↵erent optical phases. Thus a correlation is created by the discrete geometrical system of world lines across the whole apparatus; thus, the result depends both on the total area and on the time propagation, which in turn depend on changes in direction from interactions with world lines of matter that create the circuit. Although we have expressed it as a variance in arm length, the positional correlation can also interpreted as a variation in time. The relative spatial positions of reflection events at opposite sides of the circuit, including their time positions, have a quantum ambiguity inherited from the granularity of the Planckian quantum states of the enclosed space. The e↵ect is not associated with any locally defined function of the classical metric and its derivatives, which would be independent of the measurement being made. This estimate provides an exact normalization for the predicted Sagnac quantum geometrical correlation function that applies to any shape. It is not su ciently rigorous to prove that the correlations have a pure-rotational symmetry, although that appears plausible given the structure of the theory. It is useful as a baseline target for design of experiments. It suggests that the tools of Loop Quantum Gravity will be su cient to make adequately precise predictions for unique quantum-gravitational e↵ects that are realistically measurable in the laboratory.