Quantum Entanglement of Matter and Geometry in Large Systems
FFERMILAB-PUB-14-327-A
Quantum Entanglement of Matter and Geometry in Large Systems
Craig J. Hogan
University of Chicago and Fermilab
Standard quantum mechanics and gravity are used to estimate the mass and size of idealizedgravitating systems where position states of matter and geometry become indeterminate. It isproposed that well-known inconsistencies of standard quantum field theory with general relativityon macroscopic scales can be reconciled by nonstandard, nonlocal entanglement of field states withquantum states of geometry. Wave functions of particle world lines are used to estimate scales ofgeometrical entanglement and emergent locality. Simple models of entanglement predict coherentfluctuations in position of massive bodies, of Planck scale origin, measurable on a laboratory scale,and may account for the fact that the information density of long lived position states in StandardModel fields, determined by the strong interactions, is the same as that determined holographicallyby the cosmological constant.
I. INTRODUCTION
In general relativity, space-time geometry is a dynam-ical system. Einstein’s equations govern the evolution ofa 4-manifold, whose dynamical degrees of freedom carryenergy and information. The system is classical: theEinstein equations relate physical quantities with defi-nite values at definite, localized points.This geometrical system also interacts with matter.Of course, matter in the real world is a quantum sys-tem, not a classical one[1, 2]; unlike geometry, propertiesof matter are indeterminate and not spatially localized.In standard general relativity, matter is approximatedby a classical entity, the expectation value of its energy-momentum tensor.In reality, geometry and matter must join together asparts of a single quantum system, in order to couple con-sistently with each other. In quantum mechanics, subsys-tems of a whole are entangled[3]. However, the nature ofthe entanglement of matter and geometry is not known,because it depends on unknown quantum degrees of free-dom of the geometry.Thus in standard physics, any model of a whole physi-cal system has two qualitatively different dynamical sub-systems, quantum matter and classical geometry. Thecombined system is usually approximated in differentways, depending on the situation: when quantum effectsare important, the dynamical behavior of the geometryis ignored, while if geometrical dynamics are important,the quantum character of the matter is ignored. Theseapproximations are only consistent in some situations; inothers, they lead to paradoxes or ambiguities. This pa-per analyzes systems where such inconsistencies arise onlarge scales, and seeks to reconcile them.Well-known inconsistencies in the standard scheme oc-cur at the Planck length, which has been the focus of most candidate unified quantum theories (e.g., [4, 5]).However, these theories generally use standard approx-imations on length scales much larger than Planck, sothey do not directly address how geometrical quantumdegrees of freedom behave in large systems.In fact, as discussed here in several examples, standardtheory predicts indeterminate geometry in systems muchlarger than the Planck length, extending to arbitrarilylarge scales. Isolated systems with low mass have macro-scopic trajectories with the character of quantum wavefunctions rather than classical trajectories. As shownhere, such systems are impractical to create and mea-sure, but they are not paradoxical; they require only arelatively small total mass, isolated from disturbanceson gravitational timescales. The quantum character ofsuch systems, while exotic, does not require new funda-mental quantum degrees of freedom of the geometry: inthem, gravity behaves like any other force in nonrelativis-tic quantum mechanics.However, important clues to new physics come frominconsistencies between general relativity and standardquantum field theory on large scales. In standard theory,the mass of quantum field states can exceed the massof a black hole in large volumes[6]. Some new, nonlocalprinciple must prevent excitations of standard quantumfields, even at modest energies, from forming impossiblymassive macroscopic geometries.It is suggested here that this new principle might takethe form of a quantum entanglement between matterfields and geometry. Although it emerges from newPlanck scale physics, some forms of this entanglement cancreate new effects, even in nearly-flat space, distinct fromclassical modifications of general relativity[7], or macro-scopic quantum behavior in classical geometry[8–10]. Insome models, unique signatures of this entanglement maybe found in precise laboratory measurements of positionsof bodies or mechanical systems[11–15]. a r X i v : . [ g r- q c ] D ec Furthermore, it is suggested here that geometrical en-tanglement might explain the value of the cosmologicalconstant, a quantity with no current explanation in stan-dard theory[16, 17]. The spatial structure of long-livedworld line states in field theory connects the density of lo-calized position states in the field vacuum to an emergentglobal curvature on a vastly different scale. Statisticalarguments based on information equipartition are usedhere to account for a long known coincidence betweenthe value of the cosmological constant and the sponta-neous spatial localization scale of Standard Model fields,fixed by the strong interactions.
II. LIMITS OF STANDARD THEORY
The notion that quantum effects of gravity can in somecircumstances be important on macroscopic scales is bothcounterintuitive and generally unfamiliar. Nevertheless,it is a simple consequence of standard theory. To clarifythe implications, it is useful to outline briefly which as-pects of standard theory are well-tested, and which arenot.
A. Quantum and Gravitational Extremes ofPhysical Systems
The Planck mass and length relate the energy scale ofpure spacetime systems, such as black holes and gravita-tional waves, to Planck’s quantum of action ¯ h : m P ≡ (cid:112) ¯ hc/G = 1 . × GeV / c = 2 . × − kg , (1) l P ≡ ct P ≡ (cid:112) ¯ hG/c = 1 . × − m , (2)where G denotes the Newton constant of gravity and c denotes the speed of light. Planck units are defined sothat ¯ h = c = G = 1.In Planck units, the size of the most compact spacetimeconfiguration, a black hole, is given by the Schwarzschildradius for mass M , R S = 2 M. (3)The minimum size of a quantum wave packet is given ap-proximately by the wavelength in Einstein’s photoelectricrelation, λ = 2 π/m, (4)for particle energy m .All physical systems fall between these two relations.At the Planck scale, a black hole is the same size as asingle quantum of the same energy. In a standard ge-ometry, systems can exist with smaller mass, but theycannot have a smaller size (see Figure 1). B. Classical Theory
The standard complete theory of physics— space-timeand matter— follows from the Einstein-Hilbert action S = S G + S M . (5)The actions for geometry S G [ g µν ( x )] and matter S M [ g µν ( x ) , φ i ( x )] are functionals of the space-time 4-metric g µν ( x ) and matter fields φ i ( x ).The geometrical action is S G = (cid:90) d x √− g R ( x ) (6)where R ≡ R ( c / πG ), is the Ricci curvature scalar R ≡ g µν R µν in units of 16 πG/c , R µν denotes the Riccicurvature tensor (which depends on g µν ( x ) and its deriva-tives), and g ≡ Det[ g µν ].Similarly, the matter action is S M = (cid:90) d x √− g L ( x ) (7)where L ( x ) is the matter Lagrangian density that rep-resents all the non-geometrical degrees of freedom, anddefines the theory of particles and fields and their inter-actions. It is a function of fields φ i ( x µ ) and their deriva-tives.The field equations of relativity follow from the actionthrough the variational principle, δS/δg µν = 0 . (8)Here, δS represents the variation of S for variations inthe metric, δg µν . The variations are arbitrary withinthe volume of the system, but assume that δg µν ( x ) → x µ → ∞ . Thecoupling of the geometry to matter, via T µν , emergesfrom the variation of the S M term in Eq. (5); the matteraction is is related to standard 4D energy-momentumtensor in the field equations by the functional derivative T µν = δS M /δg µν . (9)The variation (Eq. 8) then leads to the field equations intheir standard form, R µν − g µν R + g µν Λ = 8 πGc T µν , (10)the trace of which can be writtenΛ = 2 πGc ( T + 2 R ) , (11)where Λ denotes the cosmological constant.The equations of motion for the fields themselves followfrom similar variation of their degrees of freedom, δφ i : δS/δφ i = 0 . (12)This variational formulation leads to equations of mo-tion for any degrees of freedom, whether or not they are“fundamental”. C. Theories of Emergent Geometry
The theory above is classical. For example, Eq. (11)is an exact relationship between scalar quantities, eachof which has definite value at each event. In reality, itis known that matter is actually a quantum system, so T ( x µ ) is only a classical approximation for a system inwhich both information and energy are not localized. Atsome level, the same must be true of both Λ and R .For matter, there is a consistent quantized theory offields φ i , the Standard Model, that agrees with micro-scopic experiments[2]. It is generally thought that theStandard Model Lagrangian L SM will eventually be writ-ten in terms of another, deeper field theory. However, adeeper field theory will not provide a quantum theory ofthe combined system of fields and geometry.The field approach to quantum geometry would beto decompose the metric ( g µν ) into classical space-timeeigenmodes of frequency, then quantize amplitudes of themodes. It is possible to quantize geometry as if it werea spin-2 component of L , that is, to quantize g µν in thesame way as φ i . Although this effective theory is consis-tent at low energies, it is well known to be inconsistent(and nonrenormalizable) at the Planck scale. Apparentlythis canonical quantization of the metric is not the rightquantum system to represent the true quantum geomet-rical degrees of freedom.Another possibility is that the classical space-time sub-system emerges as a macroscopic approximation of aquantum system, whose dynamical degrees of freedomare not known. The question before us, is whether thosenew degrees of freedom can have observable effects inmacroscopic systems.The theory of black holes provides arguments againststandard, extensive, field-like degrees of freedom for themetric. Although the degrees of freedom are not known,they can be counted: Information in a gravitational sys-tem scales holographically, as the area instead of thevolume[18–20]. Again, the dynamics of the whole sys-tem is apparently not derived from a Lagrangian density,and the quantum degrees of freedom of the whole systemare apparently not those of a local quantum field theory.The geometrical equations of motion can be derivedfrom an entirely different type of system, based not onvariation of a metric but on statistical behavior of newdegrees of freedom. The dynamics of the emergent classi-cal system can be based on the thermodynamic principlethat the system evolves to the state of maximum entropy,a macro state that corresponds to the largest number ofmicro states.Indeed the Einstein field equations can be derivedexplicitly from a variational principle based not onthe Einstein-Hilbert action, but on thermodynamicprinciples[21]. The system is defined in terms of themacro state, an emergent space-time. For any point,consider the past of a small spacelike 2-surface element P chosen, via the equivalence principle, so that expansionand shear vanish in the neighborhood of the point. The space-time system is defined by the “local Rindler hori-zons of P ”, a set of sheets in all null directions from P .According to standard thermodynamic principles, a sys-tem in equilibrium, the most probable macroscopic state,obeys δQ = T dS. (13)Here, δQ denotes a heat flow carried by matter acrossthe local Rindler horizon H , T denotes the Unruh tem-perature for the same horizon, and dS ∝ δ A denotes thevariation of entropy associated with the areal variation δ A of a piece of the horizon. For a boost vector χ µ , theheat flow is related to the boost energy current of matter T µν χ µ by a surface integral, δQ = (cid:90) H T µν χ µ d Σ ν . (14)For equation (13) to hold, it must be that8 πGT µν k µ k ν = R µν k µ k ν (15)for all null k µ , which requires the Einstein field equationsto hold.This derivation[21] accounts for the laws of black holethermodynamics and their generalizations, including theholographic property of gravitational information. In-deed, it “builds in” an information content proportionalto area in Planck units, and a classical causal structuredefined by null trajectories. It does not, however, buildin a quantum model for the matter, T µν , or the quantumdegrees of freedom of the geometry.If spacetime and matter emerge from this kind of deepstructure, their relationship differs fundamentally fromstandard theory on all scales. In particular, there is arelative lack of information on large spatial scales. Mat-ter and geometry subsystems in any region are not sep-arable, but remain significantly entangled even on muchlarger scales than Planck. The question is whether someobservable behavior might reveal properties that can betraced to new physical principles (Eq. 13, as opposedto the standard formulation, Eq. 5), and perhaps offerspecific clues to the nature of the degrees of freedom. D. Locality and Geometrical Entanglement
Properties of matter in the classical theory are localand determinate quantities. In reality, states of quan-tum matter are indeterminate and nonlocal. The classi-cal properties are averages of the real quantum ones.Several aspects of quantum nonlocality are not in-cluded in the standard theory. The system action is re-lated to local scalar quantities L and R by a spacetimeintegral over a classical manifold, not a quantum system.The system boundary is also defined using classical ge-ometry. The variation may thus be non locally modifiedfrom standard theory in the full quantum system.For example, the spatial scale of curvature associatedwith the amplitude of R is the curvature radius in Planckunits, τ G ≈ R − / . A small curvature focuses trajectorieson a large length scale, of order τ G , and an indeterminacyin curvature translates into a macroscopic orbital inde-terminacy. This inherent nonlocality is not captured inlocal quantum field theory, since the states of the systemare non-locally entangled on length scale τ G and timescales τ ≈ τ G .Another kind of nonstandard quantum relationship offields with geometry arises from the fact that the ex-pected matter density (cid:104) ρ (cid:105) V in field configurations canlead to unphysical geometries in large volumes V — inparticular, states denser than black holes— that can-not be consistently included in a path integral for thequantum states. This effect is not included in standardphysics. In this case, the scale where it becomes impor-tant can be estimated from the gravitational influence ofquantized fields encapsulated in L .A similar macroscopic scale can be derived fromPlanckian bounds on information. A statistical origin ofgravity implies modification of standard theory on largescales, with holographic information content: the infor-mation in a theory based on a Planckian surface integral(Eq. 14) scales with size L like L , whereas for a vol-ume integral for fields up to some mass scale m (Eq. 7)it scales like L m . This restriction again requires non-local entanglement between matter and geometry stateson macroscopic scales. As explained below, this couldhappen in a purely geometrical way (“directional entan-glement”) that is not explicitly dependent on m , and isexperimentally testable.To summarize, the standard and emergent views de-scribe different approaches to a whole system of matterand geometry. The standard theory is an excellent ap-proximation over a limited volume, the maximum sizeof which depends on the field energy scale. The qual-itatively different structure of the full Hilbert space inemergent, holographic systems becomes more prominentin larger volumes. It is not known how the matter andgeometry relate to each other in detail. III. QUANTUM SYSTEMS WITH STANDARDGRAVITY AND NON-RELATIVISTIC MATTER
Before turning to untested new physical principles in-volving relativistic fields in later sections, we first sur-vey some physical systems fully characterized with onlystandard non-relativistic quantum mechanics and grav-ity. Even in these systems, entanglement of matter andgeometry can be important on macroscopic scales.The simplest system in classical physics is a body inempty space with no forces acting on it. A body witha large mass, so that quantum uncertainty can be ne-glected, simply moves on a classical geodesic in free fall.On the other hand, it is a basic principle of quantummechanics that any motion has physical significance only with respect to an observable operator— in concreteterms, a measurement. For any physically meaningfulmeasurement of position, even one with an arbitrarilysmall uncertainty, a second body is needed, along withsome concrete way of comparing their positions, such asinteractions with intermediate waves or particles. Butthen of course, once there is a second body, the motionsof both bodies are not force free: there is always a forcebetween them, because any two bodies interact by grav-ity, or as we now understand it, space-time curvature.Thus, the simplest quantum system involving positionin space is actually not one but two bodies, interactingat least by gravity. Gravity introduces universal rela-tionships between mass, length and time evolution thatcombine with quantum mechanics to yield universal con-straints on position measurement in any whole system.Of course, in most familiar situations other interac-tions are much more important than gravity, and quan-tum effects are generally confined to microscopic scales.In general however systems dominated by gravity canhave indeterminate properties on arbitrarily large scales.We now survey a number of simple systems to quantifythe limits of classical approximations.
A. State of Two Bodies Bound by Gravity
Consider a non-relativistic “gravitational atom”, thatis, a quantum system of two bodies that interact onlyby Newtonian gravity. This exact solution serves as anexample and reference for a number of the other systemsconsidered below.Denote the masses of the bodies M and M . In theusual way, define a reduced mass M = M M / ( M + M ) . (16)For radial separation r , the Newtonian potential inPlanck units is U ( r ) = 2 M /r. (17)The Hamiltonian isˆ H = − (¯ h / M ) ∂ i ∂ i + U ( r ) , (18)where ∂ i ≡ ∂/∂x i , and we sum over i = 1 , , E n = − M /n , where n = 1 , , . . . is the principal quantumnumber. The wave function ψ ( r, θ, φ ) is a product ofradial and angular functions. The radial part is a productof a generalized Laguerre polynomial with n − e − r/r n , where r n = n/ M , (19)that gives the characteristic size of the atom (see Fig. 2).Because gravity is so weak, the gravitational Bohr ra-dius is much larger than a real atom of the same mass.Similar systems were considered in the early years ofquantum mechanics by Weyl and Eddington. Dirac de-veloped the “Large Numbers Hypothesis” in response tothe coincidence that the size of such an atom is compa-rable to the size of the actual universe, if the mass ofthe bodies is comparable to that of actual atoms. (Adifferent view of this coincidence is developed below).The angular eigenfunctions are spherical harmonicsthat correspond to angular momentum states with totalangular momentum quantum numbers given by integers l < n , and projections m along any axis given by integersin the range − l < m < l . The radial and angular vari-ations of the wave function correspond to relative radialand transverse orbital momentum of the two bodies.The classical form of this system is of course just twobodies orbiting each other. At zero angular momentum,two classical bodies can oscillate on radial orbits of ar-bitrary orientation and size, determined by the energy.Angular momentum can take a continuum of values upto the centrifugal limit, with no minimum. Each bodyfollows follows a deterministic trajectory.The quantum system is qualitatively different from theclassical one. The overall state is a superposition of dis-crete wave states for the relative positions of the bod-ies. A stationary (that is, stable) system is in a stateof definite energy, but not in a state of definite separa-tion, angular momentum or orientation: these propertiesare formally indeterminate. The spectra of some proper-ties, such as energy or angular momentum, are discrete.The positions of the two individual bodies are not inde-pendent, since they are entangled subsystems; once theposition and/or momentum of one body is measured, thestate of the other is changed. The stationary states havea characteristic size, with a maximum probability at aseparation fixed by the wave function scale, n/ M . An-gular momentum vanishes in the n = 1 state of minimumenergy. Indeed in the ground state the spatial orienta-tion of the axis between the two bodies is completelyindeterminate; the state is spherically symmetric, witha spherically symmetric potential. Unlike the classicalcase, here is no stable quantum system below a certainsize. Thus, there are profound physical differences be-tween the classical and quantum systems.Although we have used Newtonian gravity to describethe forces in this system, there is also a classical descrip-tion using geometry. The gravitational potential is re-placed by a curved space-time metric. The configurationof the potential (or the metric) depends on the configu-ration of the bodies. The metric can also be used as abasis for modeling a quantum system. In that case, thegeometry becomes indeterminate.In the quantum system, the structure and behavior ofthe potential and metric have the same indeterminatecharacter, and the same symmetry, as the trajectories ofthe bodies. The potential has the discrete spectrum ofstates corresponding to energy, E n = − M /n , while the curvature radius has a discrete spectrum correspond-ing to orbital timescale, τ n = n / M . (20)This non-relativistic approximation is appropriate for M <<
1. (Above the Planck mass, the radius is smallerthan a black hole of the same mass, so such states areunphysical.)The atom is not analytically solvable with more thantwo bodies, in either the classical or quantum systems.However, it is clear from scaling the above argumentsthat for a given total mass, the spatial size of the groundstate scales like the number of particles, so quantum ef-fects in principle operate on even larger spatial scales.However, the effect of each particle on the potential isless. A nearly-uniform matter distribution is consideredbelow, in the context of a perturbed cosmological solu-tion without gravity; that estimate gives the same scaleof indeterminate curvature as the atom.
B. Quantum Kinematic Uncertainty of PositionCompared at Two Times
Consider now a system where gravity and other forcescan be neglected. In this case evolution is governed sim-ply by quantum kinematics, so it can be formulated in ageneral way applicable to a wide variety of systems.As in the case of the atom, observables are representedby operators. Components of spatial position ˆ x i and mo-mentum ˆ p i of a system are described by conjugate oper-ators with commutator[ˆ x i , ˆ p i ] = i ¯ hδ ij . (21)These operators can refer to the position and momen-tum of a body, or to some other degrees of freedom of asystem, characterized by equations of motion.The state of the system can be described, for exam-ple, by a wavefunction that represents a complex ampli-tude for any configuration, e.g., ψ ( x i ). The wave func-tions obey the standard Heisenberg uncertainty relationof standard deviations, ∆ x i ∆ p i > ¯ hδ ij , that follows di-rectly from the commutator. The equations of motioncan also be used to derive other uncertainty relations forwave functions of other observable quantitates, such asobservables at different times. These relations character-ize the preparation and measurement of states.In the force-free case (potential U = some constant),the motion of a system of mass M is governed by sim-ple kinematics, ∂ ˆ x i /∂t = ˆ p i /M . The standard quantumuncertainty of position difference measured at two timesseparated by an interval τ is then[22–25]∆ x q ( τ ) ≡ (cid:104) (ˆ x ( t ) − ˆ x ( t + τ )) (cid:105)| t > hτ /M. (22)It may seem surprising at first that this uncertaintygrows with time, since intuitively it seems that uncer-tainty should get smaller with a longer average. Theexplanation is that position after a long time is suscepti-ble to momentum uncertainty. The minimal uncertaintycorresponds to states prepared in such a way that ∆ x q ( τ )gets equal uncertainty from position and momentum un-certainty after time τ . As a result, in any system thatevolves slowly and lasts a long time, the scale of quantumuncertainty gets surprisingly large. This result approxi-mately applies to any system over timescales short com-pared to its natural dynamical timescale, since it assumesonly force-free kinematics (see Fig. 3) . C. Cosmological Systems
A typical real gravitating system is composed of mas-sive bodies whose individual wave function widths aremuch smaller than the system size. The quantum ef-fects on their orbits can then be neglected. However, anisolated system with mass and size comparable with theground state of the gravitational atom, and dominatedby gravitational forces, displays quantum characteristics,such as wave-like states. This situation could actually ap-ply, for example, in the real universe in deep intergalacticspace, far from concentrations of matter. In the real uni-verse, such systems are also affected by new physics ofcosmic acceleration or dark energy not included in thismodel, as discussed below.
1. Quantum Kinematic Uncertainty of Cosmic Expansion
The force-free kinematic model can be used to esti-mate quantum indeterminacy in the cosmic expansion.Here, the physical meaning concerns the precision of thestandard classical geometry: below what scales of massand length must the expansion be regarded as a quantumsystem?The unperturbed classical system in this case consistsof uniformly expanding matter with total mass M ina macroscopic volume of radius L . The expansion onthis scale corresponds to motion at the Hubble velocity, v = ˙ L = HL , where H ≈ τ − G has approximately themagnitude of the scalar space-time curvature, R . As-suming that potential and kinetic energy approximatelymatch (that is, approximately flat spatial slices), mass isrelated to expansion rate by M ≈ L H / . (23)A simple perturbation of the expansion along one di-rection can be approximated by one degree of freedom,a coherent displacement of mean amplitude δx and cor-responding perturbation of velocity of mean amplitude δv . Momentum conservation requires that the perturba-tion be symmetric about the center of mass. The meanclassical displacement and velocity δx c , δv c are related bystandard kinematics, ∂ t δx c = δv c .For a perturbation encompassing the bulk of a volumewith total mass M , the momentum associated with the perturbation is about δp ≈ M δv . The standard treat-ment of quantum indeterminacy then applies in the sameway as for a single massive body or particle. The cosmicexpansion does not change the kinematic relations: theconjugate quantum operators for the perturbation vari-ables, δ ˆ x, δ ˆ p , obey the same operator algebra as ˆ x, ˆ p thatleads to the standard quantum kinematic uncertainty Eq.(22). We adopt the same notation, ∆ x q , to denote thewidth of the wave function for the displacement δx . Overa time interval of duration τ , Eq. (22) then gives an es-timate of the quantum uncertainty in the displacement,∆ x q ( τ ) > τ /M. (24)For a perturbation of size L , define a dimensionlessfractional amplitude of the perturbation in expansionrate, δ ≡ δv/ ( HL ) <<
1. From Eq. (24) we then findthat
L > (∆ x q /δx c ) − / ( τ H ) − / δ − / H − / . (25)Thus, the standard quantum kinematic uncertainty ∆ x q in the matter displacement exceeds the classical displace-ment δx c on small scales L , and for small amplitudes δ .This simple kinematic model system ignores all forces, in-cluding pressure and gravity, but approximately applieson scales between the Jeans length and the horizon size,over durations up to about τ ≈ /H .
2. Quantum-Classical boundary for cosmic expansion
The resulting quantum-classical boundary character-izes the standard quantum indeterminacy of a simplecosmological system. It is valid down to the intersectionwith the gravitational atom relation, where the systembecomes nonlinear and its self gravity is larger than themean background curvature.Since all the factors in Eq. (25) multiplying H − / exceed unity for a linear classical perturbation, we canset ∆ x q = δx c to derive the scale above which classicalkinematics dominates quantum uncertainty: L q = H − / = τ / G , (26)Below this scale, cosmic motion has a quantum charac-ter, as in the case of the gravitational atom. Position,motion and density are entangled on this scale; informa-tion on the state of the system is not spatially localizedbetter than this. For the current mean cosmic density,the boundary scale is macroscopic: H − / ≈
60 meters . (27)The corresponding mass scale is M ≈ H / , or about 10 GeV.In the kinematic regime, where gravity is ignored, thesebounds apply to any system, so we can also write thequantum-classical boundary in terms of curvature, L q = R − / , M = R / . (28)The length scale of quantum uncertainty grows at smallcurvature, while the mass slowly decreases.For small perturbations δ , motions are indeterminatefor any L , although indeterminacy in δ is very small forlarge L . The amplitude of a linear perturbation is inde-terminate if ∆ x q > δx c . Solving for δ with τ H < x q = δx c ), δ q ≡ (cid:104) δ (∆ x q = δx c ) (cid:105) / ≈ H ( LH ) − / . (29)In particular, the typical amplitude on the horizon scaleis of order H . In smaller regions, the amplitude is larger.As expected, it becomes nonlinear ( δ q = 1) on the scale L q .A distortion in expansion can create a density pertur-bation, and hence a mass perturbation, of order∆ M ≈ M δ q = ( LH ) / (30)Again, this is the entire mean mass for a region of size L = L q . Dashed lines in Fig (3) show the length scale ofthe uncertainty (Eq. 22), and the mass uncertainty (Eq.30), for a particular duration, τ = 1 /H .The kinematically derived uncertainty is valid overtimes shorter than the gravitational timescale τ G of asystem. As expected, for duration equal to the gravi-tational timescale (or curvature radius), the uncertaintyscale agrees with the size of the gravitational atom. Thekinematic model of the perturbed expansion applies onlarger length scales, and the atomic gravitational solu-tion is a better approximation on smaller scales. Thetwo relations together define a boundary of the classicalregime, as shown in Figure (3).
3. Quantum Uncertainty of Black Hole Position over anEvaporation Time
An exotic application of these ideas is the motion ofa black hole. The center of mass should behave in thesame way as any other massive body. This applicationis interesting because seemingly reasonable assumptions,for example about locality of position information, havebeen shown to lead to apparent paradoxes[26–28].Consider the motion of a black hole of mass M overa timescale of the order of the time it takes for itsmass to evaporate by Hawking radiation, τ evap ≈ M in Planck units. The position of the hole is indetermi-nate by ∆ x q ( τ ) ≈ ( τ evap /M ) / ≈ M , that is, by aboutthe size of its event horizon, R = 2 M (see Fig. 4). Onsuch a long timescale, the event horizon of any black holeis a quantum object; its location in space is indetermi-nate by an amount of order its size, implying that theoverall causal structure is also indeterminate. Note thatthis uncertainty in position comes not from the recoilfrom evaporated particles (which are emitted at a rateof about one of energy 1 /M per time M ), but from thefundamental quantum limits of defining a position for the hole— from a measurement of its position at two times,each time by a single quantum with energy of order 1 /M .Although this effect if of no importance for black holesin the real universe, the simple kinematic model of theblack hole motion appears to imply a more exotic phe-nomenon, a nonlocal effect of quantum physics on thecausal structure of a space-time.One way to explain why local physics is not changed—why the event horizon still appears sharply defined tonearby matter, over short time scales— is to positthat approximate locality emerges from entanglement ofnearby events. The locality and emergence refer to thecausal structure of the space-time itself. A measurementor interaction collapses the metric so that the positionstate of the hole is shared with nearby matter, when com-pared with position of other matter at the distance τ evap .The black hole is a good example to show this effect, be-cause it is composed entirely of space-time, but standardnonrelativistic kinematics still applies to the mean mo-tion of the hole. A similar argument applies to a systemof (say) two nearby black holes orbiting each other; theuncertainty discussed here applies not to their relativeposition, but to the center of mass of the pair relative toa coordinate system extending to a very large distance.This example shows that a definite classical metricdoes not apply even on the largest scales, where it isoften assumed. It supports a particular view of how en-tanglement can give rise to emergent locality in a thermo-dynamic view of geometry: geometrical states appear tobe entangled by proximity, independent of any propertiesof matter.
4. Inflationary perturbations
The nonrelativistic, kinematic approximation is not agood description of an inflationary period, where theclassical dynamics is dominated by a relativistic scalarfield[29]. However, the model gives about the same am-plitude that inflation does for horizon-scale fluctuationsin expansion rate, such as those that lead to tensor modesof cosmological fluctuations. This simple result suggeststhat the generic prediction of such fluctuations is moregeneral than the specific framework of inflation, or anyother specific model of the system: it is just a result ofquantum principles and gravity.
D. Practical and Fundamental Limits onMeasurements
These examples illustrate that quantum effects are notparticularly associated with small systems, and that theapproximately classical behavior of typical gravitationalsystems derives mainly from the much stronger interac-tions of other forces. It is interesting to ask whethera real apparatus can measure a quantum behavior of agravitational system.
1. Artificial Gravitational Atom
Apart from the technical challenge, what new could welearn about physics by actually making a gravitationalatom— a system of masses bound by gravity, close toits quantum ground state? In some sense such an ex-periment could test a new extremity of nature; it testsquantum binding by exchange of gravitons, as opposed tophotons in an ordinary atom. At laboratory density, thetransitions have typical wavelengths given by the orbitalperiod— about a light hour, far larger than the boundsystem size. (The corresponding ratio is also large fornormal atoms: a Bohr radius is much smaller than atransition wavelength. Of course the ratio is much big-ger in the gravitational case because of the tiny force.)The energy levels have a tiny separation correspondingto emission of single graviton quanta that are not de-tectable. Since radiative transitions are often the bestway to probe atomic quantum states, it is not clear whatprobe of quantum behavior is available in the case of thegravitational atom. We should therefore consider someof the practical, as well as fundamental limits on such anexperiment.In principle, advanced LIGO[30–32] can come close tothe quantum limit (Eq. 22) for mass M of tens of kilo-grams, the mass of the interferometer mirrors, and τ ofthe order of 0.01 seconds, the timescale of the measure-ment. However, this system is very far from a gravita-tional atom state. Creating a gravitational atom requiresexcluding all other sources of noise and eliminating non-gravitational accelerations, at least in one measurabledimension. In addition, it requires a small total systemmass, to make the quantum uncertainty in the wave func-tion of the position of the measured masses comparableto the system size.As a practical matter, the masses cannot be denserthan a solid material, and they must be electrically neu-tral. In a solid density system, ρ ≈ × − in Planckunits, the scales of time, mass and length are within reachof existing nanoscale laboratory setups. A gravitationalatom then has ≈ seconds orbital time, a mass about5 × − ≈ amu, and a size about ρ − / ≈ or about 10 − meters— about 1000 times bigger than anormal atom (see Figure 5).Such a freely falling nanoscale system will have a wavefunction dominated by gravitational dynamics. Two ma-jor challenges are gravitational isolation and electricalisolation. They are somewhat in conflict, because thegravitationally isolated environments of deep space arebathed in penetrating ionizing particles, and of course,heavy shielding causes gravitational disturbances.It is necessary to achieve total charge neutrality in thenear environment of the system. Material has to be com-pletely charge-free and current-free because of tiny polar-ization or induction effects; any non gravitational accel-eration must be smaller than about 1000 Bohr radii perhour squared. The best environments for charge neutral-ity are deep underground, shielded from cosmic rays, as in direct detection dark matter experiments.Another requirement is a nondestructive technique forpreparation and measurement. A system must be pre-pared and measured close to the ground state, which im-plies extraordinarily low acceleration over a long dura-tion, of order a free fall time— for solid density, of theorder of hours. Pendulum techniques or space experi-ments are preferred as they allow longer periods thanatomic fountains. The best isolation is achieved today onthe ground with pendulums, both in LIGO’s suspensions,and in torsion-pendulum gravitational experiments. Thelatter may have the potential to achieve the required longtime constants. On the other hand it is hard to imag-ine achieving sufficient gravitational isolation at such lowfrequencies in any near-earth environment.Designs for gravitational wave detectors in space[32,33] achieve good control over electrical and Newtoniangravitational forces over fairly long periods of time, ofthe order of 10 seconds. Some of these will be testedsoon on the LISA Pathfinder satellite, but only withmuch larger test masses— kilogram masses instead ofnano scale. Overall the prospects for such a measure-ment do not appear promising in the short term.
2. Direct Measurement of Cosmic Expansion andAcceleration
In large cosmic systems, gravity dominates by a largemargin and density is even lower than an artificial atom.In that case, quantum uncertainty sets a macroscopicminimum scale for classical cosmic expansion.It is interesting to ask, what are fundamental quantumbounds on experiments that directly measure cosmic ex-pansion and acceleration? The effect of the expansion oracceleration must exceed both the gravity of the appara-tus (say, two test masses), and the quantum uncertaintyin their position, over a realistic time interval.Suppose we wish to measure the cosmic expansion us-ing two bodies in a quantum state of minimal relativedisplacement uncertainty ∆ x q with separation L . Theuncertainty in their separation is less than the change inseparation due to cosmic expansion in time τ ,( τ /M ) / ≈ ∆ x q < τ HL, (31)if the reduced mass M of the bodies satisfies a bound, M > τ − H − L − . (32)A determinate classical trajectory requires multiple sam-ples in a single orbit, so τ H <
1. But for an experimentalmeasurement of the expansion, τ H must be very smallindeed— say, of order 10 − for a measurement taking ayear.We also require the measuring apparatus not to domi-nate the cosmic density, so M < H L . Combining thesegives a minimum size for an apparatus to measure the lo-cal effects of the cosmic expansion: L > H − / τ − / (33)For a duration τ of a year, this works out to a mini-mum size of about 10 meters. The corresponding massis about 10 − grams. A larger mass avoids the quantumuncertainty, but then the separation must be larger sothe apparatus does not dominate the dynamics.Now suppose we wish to measure the effects of cosmicacceleration directly, again by measuring the change inposition of two bodies of mass M over a time interval τ .The quantum uncertainty in the separation of the bodiesis less than the effect of cosmic acceleration over time τ if ( τ /M ) / ≈ ∆ x q < τ ˙ v ≈ τ H L. (34)The lower bound on mass is then M > τ − H − L − . (35)For a direct measurement over a period of time muchshorter than the acceleration timescale of 10 years, thestandard quantum limit leads to large mass and size forthe apparatus. Adding the density requirement M
In the systems considered above, the quantum degreesof freedom of the matter correspond to standard posi-tions and momenta in classical space. The relationshipsand scales are consequences of standard, well tested nonrelativistic quantum mechanics and gravity.We now turn to systems composed of quantum fieldsand a dynamical geometry, whose quantum degrees offreedom are not yet constrained by experiments. Somerelevant scales can be estimated from theoretical boundson relativistic quantum field states imposed by their grav-ity, and by gravitational information bounds originallymotivated by the theory of black holes.
A. Entanglement of Geometry with Fields
1. Quantum Field States Denser than Black Holes
Consider a system of fields in a cubic volume of size L . The field degrees of freedom are the amplitudes ofnormal modes at each wavelength λ = 2 π/ω . A systemof non-interacting fields in a 3-volume V = L , with afrequency or mass cutoff at m = ω/
2, has a total numberof modes ≈ L m (see Eq. 78). Each mode acts like aharmonic oscillator with energies E = m ( n + 1), wherethe occupation number of each mode is an integer n ≥ n = 0 has approximately zero gravi-tational density, to agree with cosmology[16]. The meandensity of the field system with mean occupation ¯ n up tomass m is then about ¯ ρ f ≈ ¯ nm , (39)independent of L . This extensive property is insensitiveto the details of the field theory Lagrangian[6, 16]. If aquantity of energy is introduced into any volume of space,it will thermalize and excite the field with ¯ n ≈ T ≈ m , with ¯ ρ f ≈ T .The paradox is that highly excited states of this fieldsystem in large volumes are unphysical, because theirdensities are incompatible with general relativity. Sincedensity couples to gravity (via Eq. 5 etc.), the system isonly consistent if the fields live in a geometry that corre-sponds to the same mean density of matter as its source.The Hilbert space of the fields includes impossible states.Specifically, consistency becomes impossible to achievein systems with large V and field modes with large m .In a sufficiently large volume, the energy density of moregeneral field states, such as thermal states with ¯ n ≈ m > ρ BH ≈ L − . (40)Some radical new principle must prohibit field states fromhaving more mass than a black hole of the same volume.0Local field theory must be left intact, yet somehow theallowable field states must non locally “know about” thevolume of a whole space-time system.Motivated by this conundrum, Cohen, Kaplan and Nel-son (CKN)[6] posited an IR cutoff to field states— a limitto the box size allowed in quantum field theory. Thishypothesis goes outside the effective local field theoryframework. It does not address the nature of the physicalrelationship of field states to quantum geometry directly,but it does allow an estimate of the effects on renormal-ization group flow and other measurable effects on thethe fields. CKN showed that this modification of fieldstates does not produce an observable effect in currentparticle experiments, which generally measure effects inmuch smaller volumes.In this model, the standard description of field statesis only valid up to a finite range, such that the sum of theenergies of field states in a volume does not have moreenergy than a black hole of the same size. The bound onthe spatial extent of field modes of mass m is about L < L G ( m ) = m − , (41)where we use L G to denote the scale of significant entan-glement with geometry. This relation is shown in Figure(6). Note that in this case, the mass m refers to particlemass, not system mass.This bound suggests that there is a maximum systemsize, above which standard local quantum field theory atscale m breaks down. Somehow, the degrees of freedomof the fields and geometry do not act like independentsubsystems, and this introduces a nonlocal behavior notdescribable by a Lagrangian density in large volumes.
2. Directional Entanglement of Fields with Geometry
The IR limit just discussed can be accounted for ina simple model of how field states relate to geometricalones, based on a Planckian bound on directional informa-tion. In this “directional entanglement” model[13], thecutoff has a purely geometrical origin. It gives the cor-rect IR cutoff scale independently of m or other assumedproperties of the fields, and makes some other uniquepredictions.In a model where the cutoff is due to entanglement,the field subsystem is entangled with the geometrical oneeven in the ground state. The structure of the field de-grees of freedom does not depend on the field excitation;it holds even for the field vacuum state, in a nearly-flatspatial geometry. Thus, the Hilbert space describes a setof consistent possibilities.In this model, the total information in a volume is bro-ken into radial information, which describes the causalstructure of a spacetime in terms of light cones aroundeach event on a world-line, and directional informa-tion, which describes two-dimensional angular orienta-tion. The directional information is bounded by a Planckdiffraction limit imposed by the quantum geometry. In this way, directions in space emerge from new physics atthe Planck scale in such a way that angular resolution offield states never exceeds the angular resolution of Planckfrequency radiation.This model does not impose an abrupt limit on thespatial extent of field states. Instead, states of fields areentangled with states of quantum geometry in a partic-ular way[13]. The transverse phases of field wave func-tions are convolved at separation L from an observer’sworld line with a quantum geometrical directional phase.The spread in direction ∆ θ P or transverse position ∆ x ⊥ is given by the Planck diffraction resolution limit, fromstandard wave optics, for states of extent L :∆ θ P ≈ ∆ x ⊥ /L ≈ L − / . (42)The amount of directional information— the number ofdistinguishable directions of propagation in the systemof fields— is bounded by the resolution of Planck wave-length states. At this angular resolution limit, directionalinformation is mainly geometrical, and is shared amongfield modes.The geometrical contribution to the transverse phaseinvalidates the standard count of independent field states.The number of field states of mass m is significantly re-duced beyond the separation L where the geometricaleffect on the transverse phase causes phase changes oforder unity in typical orientations. That happens when∆ x ⊥ ≈ m − , or L (∆ θ P = ( mL ) − ) ≈ m − ≈ L G . (43)Thus, transverse components of field phases are signifi-cantly affected by geometry at about the right scale L G to reconcile virtual field states with gravity (Eq. 41).However, in other respects the effects of directional en-tanglement are not the same as a simple volume cutoff.Since the effect on fields is purely transverse, it has neg-ligible effect on longitudinal phase, so the modificationsof standard field theory in particle experiments are likelyto be different in detail from those estimated by CKN.Because the geometrical effect is purely transverse, thereare no dispersive effects, even at frequencies up to thePlanck scale, of the kind that can be measured in astro-nomical observations (e.g., [34–36]).On large scales where geometry dominates, the num-ber of directional degrees of freedom in this model is L ,instead of ( Lm ) as in standard field theory. At the sametime, the number of radial degrees of freedom is still ≈ L , so the total information agrees with holographicinformation from gravitational theory. An exact calcula-tion of directional uncertainty[12] normalized to gravity,based on a quantum commutator of position analogousto angular-momentum algebra, yields the formal Planckuncertainty of direction at separation L : (cid:104) ∆ θ P (cid:105) ≡ (cid:104) ˆ x ⊥ (cid:105) /L = l P / √ πL. (44)As discussed below, directional entanglement may haveobservable consequences in position measurements ofmassive bodies.1
3. Comparison of Holographic Information Bounds
Two different quantum-classical boundaries can be de-rived from holographic arguments. The information infield states up to frequency m , in a volume of size L , isabout I f ≈ m L (45)whereas the holographic bound on total information is I H < L . (46)Combining these we get a bound on system size for thetotal holographic information not to exceed the informa-tion in fields: L < L I ( m ) ≈ m − . (47)In a volume of this size, the mean information density offields of mass < m matches that of geometry.The directional entanglement hypothesis posits thatonly angular information is affected by geometrical de-grees of freedom. The angular information in field statesup to mass m is I fθ ≈ m L , (48)so that when the radial part is included, I f ≈ mL I fθ ≈ m L as before. The angular information in geometry islimited by the Planck diffraction bound, I Hθ ≈ L, (49)so that when the radial part is included, I H ≈ I Hθ L ≈ L as before. Combining these we get a bound on systemsize for directional holographic information not to exceedthe directional information in fields, L < L G ≈ m − . (50)This bound is the same as that from field gravity orPlanck diffraction (Eq. 43), and is more restrictive thanEq. (47). A cosmological interpretation of these boundsis discussed below.
4. Standard Quantum Uncertainty Exceeds PlanckianDirectional Uncertainty Below a Planck Mass
Compare the standard quantum kinematic uncertainty∆ x ≈ τ /L over duration τ (Eq. 22) with the Planckiandirectional uncertainty of position,∆ x ⊥ ≈ L. (51)For the standard quantum uncertainty be less than theholographic one, we require M > τ /L. (52) For durations τ > L , M > . (53)That is, standard quantum uncertainty always dominatesPlanckian directional uncertainty for system masses lessthan the Planck mass[12].The Planck mass thus defines a kind of threshold oflocalization at all length scales. For larger masses, thestandard quantum uncertainty of position of the systemcan be dominated, in transverse directions, by the ge-ometrical uncertainty of the space it resides in. Forsmaller masses, standard quantum mechanics dominates,so classical geometry gives a good approximation to thetotal uncertainty. This helps explain why the stan-dard paradigm of classical geometry works so well forall particle physics and indeed, all precision experimentsto date. Standard quantum mechanics also dominatesin the regime of gravitational atoms, all of which have M <
1. It is possible however that a very sensitive,larger-mass apparatus may isolate dominant effects ofquantum-geometrical directional uncertainty on trans-verse or angular position.
5. Relation to Chandrasekhar mass
When the degenerate electrons supporting a whitedwarf star become relativistic, the system becomes un-stable to collapse; the formal radius in a hydrostatic equi-librium solution with gravity goes to zero. The mass atwhich this happens is given approximately by the Chan-drasekhar mass limit,
M < M C ≈ √ πm − (54)where m is the mass per electron, approximately equal toa nucleon mass. Approximately the same criterion givesthe maximum mass of any system composed of parti-cles with a number density of order λ − ≈ m , whetherthey are degenerate fermions or thermally populated bo-son modes; thus, Eq. (54) also gives approximately themaximum mass of a stable neutron star.It is no accident that this formula resembles the boundson field system extent, Eqs. (41) and (43). The Chan-drasekhar relation appears as a bound on system massrather than spatial extent, but in Planck units they areactually the same number. Thus, a neutron star radiusis almost as small as a black hole of the same mass,and about the same size as the entanglement length forneutron-mass field modes.Both bounds saturate the maximum allowable grav-ity of fields, but in different circumstances. The Chan-drasekhar bound invokes the gravitating effect of realparticles and no exotic physics, while the field systembound invokes virtual field states to motivate a new prop-erty of geometry.2 B. Effects of Geometrical Entanglement onObservables
The above discussions of geometrical entanglementbased on the gravity of virtual field configurations, oron Planck directional information capacity, focus on theeffects on field states. The flip side is the effect of mat-ter fields on the geometrical states. Directional entan-glement in particular leads to a departure from classicalgeometrical positions of massive bodies and atoms thatmay actually be possible to measure.
1. Holographic Noise in Laser Interferometers fromDirectional Entanglement
Geometrical entanglement can slightly alter positionstates of matter even in nearly-flat space, in systemswhere gravity is negligible. In the directional entangle-ment model, the nature of this alteration can be esti-mated precisely from a simple geometrical model of emer-gent locality[11–14].In this model, the Planckian bound on directional in-formation leads to a particular kind of geometrical en-tanglement in the position states of massive bodies andparticles. The geometrical part of the wave function hasa Planckian directional component that is shared in com-mon among all bodies in a small region of space, relativeto other regions. The emergence of a classical space col-lapses its uncertainty in a coherent way that exactly re-spects emergent causal structure. Again, this is not astandard quantum uncertainty: it depends only on posi-tion, not on the masses of bodies. Its effects cannot bedetected in a purely local or purely radial measurement,but when positions are compared over a large region inmore than one direction, the uncertainty manifests ascoherent random transverse or directional fluctuations.The model predicts that the transverse or directional po-sition of any two bodies at separation L fluctuates ontimescale L with amplitude given by Eq. (44), (cid:104) ∆ x ⊥ (cid:105) = l p L √ π . (55)Fluctuations with this character— a very small am-plitude transverse displacement on a light-crossing time,correlated within causally connected regions— appearas Planckian “holographic noise” in the signal of suit-ably configured laser interferometers. A blend of tech-nology from gravitational wave detectors and quantumoptics currently achieves[37] approximately the precisionrequired to detect or rule out the hypotheses leadingto Eq. (55). A detection of the predicted noise wouldconfirm a particular kind of directional entanglement be-tween fields and geometry.
2. Atom Interferometers and Clocks
Another promising technology for detecting geomet-rical fluctuations uses “direct measurement of the timeintervals between optical pulses, as registered by atomictransitions which serve as high stability oscillators.”[38]The basic set-up involves laser pulses that interact withwidely separated clouds of atoms. The atomic state issplit into a superposition that includes an excited com-ponent with a recoil velocity, then later recombined usinganother laser pulse that stimulates a recoil in the reversedirection. If the atomic system as a whole accelerates,it affects the phase of the atomic wavefunction that ismeasured in the readout. The atoms act as very preciseclocks; the laser pulses can be thought of as a way tocompare clocks on widely separated world-lines.For example, one proposed scheme for gravitationalwave detection[38] uses laser pulses on the two cloudsof atoms.The measured phases apply to momenta inthe same direction as the laser pulses. The interactionevents in the two clouds therefore have null separation.For a gravitational wave measurement, phases of eventsneed to be compared at spacelike separation, so the null-separated laser pulse interaction events need to be sup-plemented by ultra-stable atomic clocks associated witheach cloud. The laser pulses can be thought of as a wayto compare clocks on widely separated world-lines.Now consider basic limits on the stability of atomicclocks. An atomic state with a lifetime τ has a fre-quency width δν = τ − . If it is interrogated with a laserof frequency ν l , the fractional stability over time τ is δν/ν > ( τ ν l ) − . In practice, lasers have ν l < Hz.The best clocks based on single atoms[39], stabilized withatomic states that have lifetimes of order 100 seconds,achieve a stability over that timescale of about 10 − .Thus, the relation between measurement time and per-atom clock stability is fundamentally limited by the laserfrequency. Geometrical fluctuations of frequency f canbe measured, in a time 1 /f , with rms strain amplitudeof order h ≈ δL/L ≈ δν/ν ≈ f /ν l ≈ − ( f / h ≡ (cid:104) ( δg/g ) T T (cid:105) = (cid:90) df S f , (56)as an integral over fluctuation frequency of the spectraldensity S f . For a system of N atoms, the sensitivity overtime τ is at best S f ≈ N − ( δν/ν ) N =1 ≈ N − ( τ ν l ) − . (57)To probe geometrical entanglement requires S f ≈ N ≈ ( τ ν l ) − ≈ ( τ /
100 sec) − . (58)3Laser interferometers, by using very large numbers ofphotons, already achieve approximately Planck sensitiv-ity in these units. If suitable atomic-clock systems canbe built with very large numbers of atoms (more thanabout 10 ), they may be competitive with lasers, andmeasure properties of the space-time quantum state tothe precision required to detect directional entanglementat low frequencies. V. PARTICLE LOCALIZATION ANDGEOMETRICAL CURVATUREA. Paraxial Approximation for Wave Functions ofParticle World Lines in Field Theory
Some field states correspond to world lines of massiveparticles. Of course, they are not really world lines, butare wave functions with a nonzero width. We now outlinea simple paraxial analysis of the structure of these statesin standard field theory. These states are used in thefollowing sections to quantify the scales where fields andcurvature become entangled.To choose the simplest example of a relativistic field,consider the scalar Klein-Gordon wave equation( ∇ − ∂ t − m ) A = 0 , (59)where A ( (cid:126)x, t ) denotes the complex quantum amplitudeof a field at each point in space-time and m denotes theinvariant mass associated with the field. Solutions ofthis equation represent states that describe the evolutionof the field. Usually, attention centers on plane wavesolutions, which represent eigenstates of momentum de-localized in space. However, it is possible to rewrite itin a form that more closely describes the world lines ofmassive particles localized in space.Consider solutions of the form A = e iωt ψ ( (cid:126)x, t ) (60)where ω = m + p , p denotes the momentum operator,and ψ ( (cid:126)x, t ) denotes the time-varying spatial profile of thefield wave function. We can rewrite the wave equation as e iωt ( ∇ + p − iω∂ t − ∂ t ) ψ ( (cid:126)x, t ) = 0 (61)This form, which is still exact, is useful to describestates of the system that resemble a particle nearly atrest, over a long time interval. The approach resemblesthe paraxial approximation used to describe solutions ofthe wave equation close to a single propagation direction.Here, the propagation direction is time, and the states arefield solutions close to a classical force-free world line.The standard paraxial approximation is to assume thatmodes are mostly unidirectional— along the z axis, say—and to neglect second order variation in that direction.The equivalent procedure here neglects the last term inEq. (61): ∇ ψ + ( ∇ ψ ) − iω∂ t ψ = 0 , (62) where we have replaced each momemtum component p i by its quantum operator representation, i∂ i . Thus, thetime variation of the state is dominated by the expo-nential oscillation factor. It can be further simplified bychoosing a frame where the origin is a reference worldline defined by ∇ ψ = 0, that is, zero momentum, andsetting ω → m . In wave mechanics, this is the trajec-tory orthogonal to the wave surfaces of constant phase;it is the classical path defined by a path integral or Fer-mat principle. In this approximation the wave equationbecomes ∇ ψ − im∂ t ψ = 0 . (63)Thus in these limits, the wave equation approximates thenonrelativistic Schr¨dinger equation for a free particle. Itis paraxial in time: it has the same form (with one addedspatial dimension) as the usual paraxial wave equation,but describes states of a massive particle along the timeaxis instead of a massless particle along a particular spa-tial direction (see Fig. 7). That is, instead of describing awave function transverse to an axis of propagation, it de-scribes a wave function in three spatial dimensions closeto a timelike axis, the world line.Paraxial wave solutions are often used to describestates of a laser cavity. For states of fields resemblingspatially localized particles over long intervals of time,these solutions are better than the commonly adoptedplane waves to illuminate the physical properties of thestates. Instead of a line, the quantum trajectory of a par-ticle state resembles a narrow beam over some duration τ , beyond which it spreads at a faster rate. Dependingon the state preparation, it spreads at different rates; awider beam spreads more slowly. There is a well definedminimum width for a given duration determined by theanalog of spatial diffraction, that can be sketched as a“world tube” (see Fig. 7).The normal modes of the wave function include spa-tial patterns in three dimensions that extend over a τ -dependent diffraction-sized patch, just like the well-known two dimensional patterns of laser modes in acavity[40, 41]. The narrowest patch is given by the sim-plest, isotropic gaussian solutions of Eq. (63). The wavefunction can be written as ψ ( r, t ) = exp[ − i ( P + mr / q )] (64)where r = x + y + z , and the state is described by twocomplex parameters with the properties ∂ t q = 1 , ∂ t P = − i/q . They are related to the variance σ ( t ) of the spatialwave function and the radius of curvature of the constantphase surfaces R c by1 q = 1 R c − √ imσ . (65)The family of solutions for the width and curvature de-pends on a waist parameter σ : σ ( t ) = σ [1 + t m σ ] (66)4and R c ( t ) = t [1 + σ m t ] . (67)For these symmetric modes, the probability density as-sociated with this state in four dimensions is |(cid:104) ψ ∗ | ψ (cid:105)| ∝ e − r /σ ( t ) . (68)The actual solution depends on the preparation of thestate. The minimum width for a state of duration τ ,extending from t = − τ / t = + τ /
2, occurs for σ/σ = R c /t = √
2, or σ = σ min = τ / √ m. (69)This state of the field system— a spatially confinedwave function extended in time— is the field representa-tion of a particle world line. It is closer to reality thaneither a plane wave state, which is completely delocalizedin transverse directions, or a classical world line, whichcompletely neglects the diffraction limit of a time evolv-ing quantum wave. Normally, the same spatial states areassembled out of wave packets of plane waves, but thesedo not explicitly display the subtle correlations needed tomaintain locality over a long time. These solutions ap-ply generally to field theory in classical geometry, withoutgeometrical entanglement. B. Localization of Particle World Lines
The world line of a particle over a long duration τ hasa standard irreducible 3D spatial uncertainty σ min ( m, τ )much larger than the field wavelength m − . The min-imum width of the state over a time interval τ agreeswith the uncertainty of a force-free kinematic trajectoryof a body of mass M = m from non-relativistic quan-tum mechanics (Eq. 22). Of course this long-durationuncertainty is not locally observable because states ofnearby particles and bodies are typically entangled witheach other; local measurements do not show the smallin-common components of indeterminate momenta thatcause their states to spread.The Schr¨odinger equation for the nonrelativistic grav-itational atom (Eq. 18) included a gravitational inter-action between two massive bodies. The chronoparaxialapproximation of the relativistic equation (Eq. 63) ac-tually obeys the same wave equation for mass m as theposition wave function of a nonrelativistic body of mass M , but with no gravity.However, the field wave equation, even in the non-relativistic limit, represents a different physical system.When quantized, the field amplitude represents stateswith any number of particles of mass m . A particle stateis described by a creation operator acting on a modeof frequency m , that corresponds to its space-time wavefunction. In the chronoparaxial approximation (Eq. 63), the states are those of massive particles at rest. Theparaxial solutions represent relativistic field states corre-sponding to particle world lines, so they are good repre-sentations to study the emergence of locality in macro-scopic field systems.The field theory also includes vacuum states of the fieldsystem. A particle is defined as an excited state of thefield vacuum. The state of a particle localized to a worldline corresponds to a creation operator operating not on aplane wave state, but on a paraxial world line eigenstateof the vacuum. The localization is thereby affected bythe mass of the particle and the duration of the state.Quantized field world line states resemble quantizedlaser cavity solutions[23, 24]. Particles in a cavity eigen-mode are completely delocalized over the volume of thecavity. Similarly here, particles in a world line eigenstateare delocalized in the swept-out 4-volume. Transverse lo-calization in a cavity wave now appears as 3D localizationaround a world line.Typically, particles are prepared and measured instates that are microscopically localized. Their worldlines only have a relatively short duration before theyspread. These field states are not appropriate for com-parison with long-duration world lines of a macroscopicgeometry. The appropriate localization for a given dura-tion τ is given by Eq. (69).In the Standard Model, the Lagrangian is Lorentz in-variant. However, position eigenstates of massive parti-cles spontaneously break that symmetry of the vacuumsince a particle world line singles out a preferred frame.The state of matter is entangled on a cosmic scale with ageometrical rest frame that also defines a set of preferredworld lines, those of comoving observers. It thus seemsnatural to consider a connection between the localiza-tion scale of the field Standard Model vacuum, and themaximum duration of coherent cosmic world line states. C. Entanglement of Matter Vacuum withGeometrical Curvature
Now suppose that these field states entangle with ge-ometry. The spatial localization of a particle to a worldline implies a wave function that depends on system du-ration, unlike modes in the usual plane wave decompo-sition of field states. Geometrical entanglement affectsthe coherence of a state when σ min ( m, τ ) exceeds L G ( m ).That in turn implies some sort of entanglement on a muchlarger curvature radius scale, τ . What happens when theworld line of a massive particle lasts so long that its worldline state width, σ min ( m ), becomes larger than the scale L G ( m ) of geometrical entanglement?We now develop the idea that geometrical entangle-ment with the matter vacuum may affect the mean cur-vature of the emergent geometry in such a way that thiscannot happen. In a gravitating system that includesmatter fields and a geometry based on thermodynam-ics, the maximum density of positional information in5particle states instead gives rise to a small curvature ofemergent geometry— a very small but non vanishing de-parture from flatness.For laser cavity modes, the cavity size is about thesame as the radius of curvature of the wave fronts; sim-ilarly, for massive field states, the duration of a worldline corresponds to the space-time curvature radius ofa constant-time surface. It could be that the curvatureof long duration field states, whose spatial width corre-sponds to the extent of states at a localization scale ofsome mass m fixed by the field vacuum, determines theradius of curvature entanglement of emergent geometry.In relativity, a massive body or particle follows a time-like world-line. As seen above, in quantum mechanics,the path is indeterminate. The wave function in spaceand time depends on the preparation of the state. Somestates are famously indeterminate, such as those in EPR-type experiments. The best approximation to a classicalworld line is a state prepared with the minimal uncer-tainty discussed above (Eqs. 22, 69). It can be visualizedas a world tube or beam that represents a range of tra-jectories between two times. The width of the tube canbe derived from kinematics or from field evolution. Theminimum width of the tube grows with time interval, anddecreases with particle mass.These properties of standard quantum states are mod-ified if there is a new additional entanglement betweenmatter and space-time. The states of a particle of mass m are affected when the wave packet exceeds the entan-glement scale (Eq. 43). The corresponding elapsed timeinterval— when the world tube width equals the maxi-mum size of a quantum state— gives a space-time curva-ture radius. This amount of curvature affects the worldtube trajectory geometrically by about its own width.A geometrical curvature of this (tiny) magnitude— an“entangled curvature” scale associated with a particlemass scale— modifies the curved wavefront in the stan-dard quantum state, in a way that is shared in commonby all nearby particles and bodies. At this scale, the clas-sical approximation to the space-time used for the actionintegral of a path, or the boundaries used to computematter or geometry action in a 4-volume, are modifiedby geometrical entanglement.We now characterize this scale in two ways, based re-spectively on the modal structure of wave like states, andon state-counting or information.
1. Physical Estimate of Curvature Entanglement Scalefrom Particle Wavepacket States
Consider long-lived field states that correspond toworld lines of some mass scale m in the matter la-grangian. The evolution of these states, together withthe geometry, connects spatially localized wave functionsat different times, with time like separation.As usual, a localized particle state is prepared as awave packet, expressed as a superposition of modes. If geometrical entanglement enforces a maximum size L G to field states, there is a corresponding minimum amountof particle momentum spread, so the wave packet disas-sembles over some period of time. Geometrical entangle-ment prohibits the existence of some states that wouldbe allowed in a classical geometry— states with with alarge positional uncertainty, and a small momentum un-certainty. That prohibition leads to an estimate of emer-gent curvature.The minimum overall momentum uncertainty of a par-ticle of mass m that is maximally delocalized— that is,a wave packet with a size L G ( m ) given by Eq. (41), themaximum size of field states at this frequency — gives adecoherence time τ G for the wave packet to spread. Fromstandard quantum uncertainty (Eq. 22) connecting thewave function at two times separated by τ G , we have: L G ≈ ∆ x q = 2 τ G /m, (70)and hence τ G ≈ mL G . (71)Combining Eqs. (71) and (41) yields the estimated cur-vature entanglement scale, τ G ≈ m − . (72)Information about location of massive particle statesextends over a region of size L G , and duration τ G . Thiscurvature-entanglement radius for particle mass m is thesame as the spatial (not curvature) radius of a gravi-tational atom of mass M = m : the relation (Eq. 72)between particle mass and orbital time τ G is the same asthe relation between mass and physical size of a gravita-tional atom (Eq. 19). These relationships are illustratedin Figure (8).We conjecture that curvature entanglement of theemergent metric with massive particle states occurs witha curvature radius, or orbital duration, given by τ G .Gravitational acceleration from this value of curvaturemoves a particle in a free-fall time a distance equal tothe maximum size of the particle position wave function.The corresponding momentum is the minimum standardparticle momentum uncertainty. The standard assump-tion that field states and geometrical curvature states areindependent subsystems breaks down— the two systemsbecome entangled— when the width of the position wavefunction of a particle measured over a gravitational timeexceeds the spatial extent of a field mode state. At theentanglement curvature, the gravitational trajectory of aparticle in free fall— really, a field wave packet movingthrough the emerged curved space, following a geodesic—moves a distance about equal to the extent of its fieldstates.
2. Estimate from Information Equipartition
A value of entangled curvature is thus related to a fieldmass scale by particle localization. The same relationship6can also be interpreted in terms of positional informationor entropy. In a universe with an asymptotic horizon ra-dius H − , the relationship m ≈ H / can be derivedby equating the total positional information of the fieldsubsystem, ≈ m H − , with that of the geometrical sub-system, given by H − / VI. COSMIC ACCELERATION AND THESTANDARD MODEL
The acceleration of the cosmic expansion[42, 43]emerges from a still-unknown relationship of matter andgeometry as parts of a single system. Cosmic accelerationsuggests the existence of a fundamental scale in physicsvery different from those in the Standard Model.The simplest model accounting for cosmic accelerationis a non-zero value of Einstein’s cosmological constant,a parameter in the field equations of general relativity.Its value is essentially unexplainable within the standardframework of field theory and classical geometry[16, 17].A number of alternative models of cosmicacceleration[44, 45] have been proposed, such asvarious forms of modified gravity, and in some cases theylead to significant predicted deviations from standardbehavior, for example in the growth of cosmic structure. These models generally involve adding an extra empiri-cally derived scale ad hoc for dark energy related to thecurrent expansion rate, and in some cases a macroscopicinteraction or screening scale for new forces.For simplicity, the following discussion assumes thatcosmic acceleration behaves in the macroscopic limit likeEinstein’s cosmological constant, rather than a more ex-otic variant. This assumption is also motivated by inter-nal consistency and symmetries, as discussed below.
A. Value of the Cosmological Constant
The cosmological constant is related to the asymptoticcurvature by Eq. (11) in the limit of zero expected matterdensity, ¯ T →
0, and zero 3-curvature:Λ = 4 π R ¯ T → ≡ H , (73)where H Λ denotes the asymptotic expansion rate. Instandard theory, its value is arbitrary. In the entangledmatter/geometry system proposed here, the value of thecosmological constant could have a quantum-geometricalconnection with Standard Model fields. In particular,curvature entanglement could relate its value directly tothe particle scale of the long-lived position states fixedby confinement, Λ QCD . B. Cosmic Information Density
Cosmological data suggest that the universe has anevent horizon, which in thermodynamic gravity implies afinite total information content. Indeed, with a few rea-sonable assumptions, current data already provides anestimate of the absolute value of the total cosmic infor-mation and information density (in Planck units), withan accuracy of better than ten percent.The absolute scale of the cosmic expansion is set bythe Hubble rate H , parameterized by a dimensionlessvalue h , c/H = 0 . × h − m . (74)Values of h can be estimated in various ways, with dif-ferent assumptions and systematic errors. We adopt atypical current value[46], h = 0 . ± . h = 0 . ± . Λ = 0 . ± . H − of the event horizon in the asymp-totic future. The value of Einstein’s cosmological con-stant in these units is given by Λ = 3 H . Again assuminga standard (ΛCDM) cosmology, the cited measurementsgive a current estimated value, H − = Ω − / H − = 1 . ± . × . (75)This dimensionless number may represent a more funda-mental property of the universe than other combinationsof cosmological parameters— one that does not dependon cosmic epoch or history, akin to parameters of theStandard Model. It is directly related to the total cos-mic information I , one quarter of the area of the eventhorizon in Planck units, I Λ = πH − = 3 . ± . × . (76)Also assuming a flat 3-geometry, the mean density ofcosmic information n Λ per 3-volume is given by n Λ = 3 H Λ / . ± . × − . (77) C. Comparison of Cosmic and Field Information
We next estimate the field scale m I ( H Λ ) that gives thesame information density as the geometry with horizonradius 1 /H Λ . For a spin-zero field, the amount of infor-mation is given by the number of modes in a volume,which we count in the standard way as follows.For a field with a UV cutoff at | k max | = m , the volumeof phase space accessible to modes is V k = 4 πm /
3. In aspatial volume of size L in any direction, modes occur in k space with a mean spacing (2 π/L ) in that direction, soin the total field information density for a cubical volume V L = L is n f ( m ) = I f /V L = V k (2 π/L ) − V − L = m π/ π ) , (78)independent of L or the shape of the volume. (Of course,as argued above, the estimate of field information shouldonly actually hold for field-like states for volumes up toabout L G ( m ) where the geometrical entanglement be-comes important; this is information density for fieldsignoring geometrical entanglement.)Combining these relations, the overall cosmic informa-tion density is equal to the mean field information den-sity, n Λ = n f ( m I ), for a field cut off at mass give by m I = H Λ (2 π ) (9 / π ) . (79)This equation provides a more precise estimate than Eq.(72), for the specific case of a scalar field.For measured cosmological parameters, the particlemass where field information matches cosmic informationis m I ( H Λ ) = 1 . ± . × − = 201 ± . . (80) (This estimate would be a few percent higher if currentlocal measurements of H are used.) This value closelycorresponds to the mass scale Λ QCD where the strong in-teraction running coupling formally diverges, which setsthe position information density for the states of theStandard Model vacuum. The approximate agreementsuggests that measured by total information density inposition states, the scale of the cosmological constant isthe same as that of the Standard Model.
This coincidence motivates the hypothesis of an emer-gent, entangled geometrical system. A more exact ver-sion of this theory could provide an exact calculation torelate cosmological and microscopic scales.
D. Significance of the Strong Interaction Scale
What is the physical reason that we identify Λ
QCD asthe appropriate field scale m Λ for entangled curvature?Briefly it is that QCD defines the scale of spatial particlelocalization for the states of the Standard Model. Athigher energies, the character of particle states changes;particles are never in stationary eigenstates of position.Strongly interacting particles are confined in spatiallylocalized hadrons. The Lorentz invariance of the La-grangian is spontaneously broken by the rest frame de-fined by these massive hadronic states. The spatial sizeof the states, as well as the range of the strong interac-tions, is given approximately by the strong scale, Λ QCD .Thus in the rest frame, the fundamental particles are al-ways in collective quantum states that are only localizedto within about a length scale Λ − QCD . It is not possibleto prepare a state with a smaller rest-frame uncertaintyfor any particle position for an extended duration.Lorentz invariance is also broken by cosmology. In cos-mological space-times with matter, such as Friedman-Robertson-Walker models, world lines of matter, orisotropy of radiation, define a cosmic rest frame at anyposition. The proposal is that the total amount of posi-tional information in the field vacuum equals the amountof cosmic positional information. Equipartition betweenthe number of field and geometry states sets the valueof the effective, emergent cosmological constant and theoverall positional information in the cosmic system as awhole. In the most probable configurations of the entan-gled system, the total information encoded by the globalgeometry relates directly to the confinement scale.In a thermodynamic model of gravity (i.e. Eq. 13), thecausal structure of the space-time adjusts by curvature,in order to make this so. As described by Jacobson[21],“the gravitational lensing by matter energy distorts thecausal structure of spacetime so that the Einstein equa-tion holds.” Here we go one step further, and posit thatthe mysterious constant of integration in that theory isfixed by an additional entanglement with the field vac-uum, an effect not included in Jacobson’s emergent ge-ometry.The equipartition of field positional information den-8sity with geometry explains why the scale of the QCDvacuum is matched with the curvature scale of cosmicacceleration. The proposal is that this coincidence isnot accidental, but arises because QCD sets the maxi-mal positional information in all field states that definelong-lived timelike trajectories.The effective temperature of the emergent geometrymatches the heat flux and horizon area via Eq. (13). Thelost information may be thought of as the information offield states being absorbed into the space-time degreesof freedom through the horizon. From the field point ofview, the same effect appears as an entanglement withdirectional geometrical degrees of freedom.The properties of the Standard Model appear to beremarkably constant over cosmic time. The connectionmade here suggests that the same should be true of emer-gent cosmic acceleration, which would imply behaviorlike a cosmological constant with an equation of stateparameter w = − E. Astrophysical Coincidences
In various forms, conjectures about cosmic coinci-dences with elementary particle masses have a long his-tory, going back to Weyl, Eddington[48] and Dirac[49].For example, why should a “gravitational atom” of ap-proximately atomic mass have approximately a Hubblesize? Dirac’s answer[49] was that the constants of nature,including G , vary with time such that this is always thecase. Later, others, including Dicke[50] and Carter[51]had a different answer: they argued that the current ageof the universe— the time when we show up— is de-termined by the lifetimes of stars, which in turn is longbecause gravity is (and always has been) so weak.The relation m π ≈ H / between the pion massand the present Hubble scale was noted long ago byZeldovich[52, 53]. In this context, m π = 134 .
98 GeV =1 . × − is the Yukawa nuclear interaction scale, aproxy for Λ QCD . Others[54–58] have argued further thatthe strong interactions may have a direct physical rela-tionship with the now-observed rate of cosmic accelera-tion. That is also the point of view here.The arguments here suggest that this relationship ac-tually has a geometrical origin at the entangled interfaceof quantum geometry and matter, some aspect of whichcan be probed by laboratory experiments. The physicalmotivation comes from emergent gravity, and a specificconcrete physical interpretation: in this scenario, there isa cosmological constant, whose value is fixed to the mi-croscopic scale where the matter vacuum spontaneouslycondenses into massive hadronic particles.If this is indeed the case, the astrophysics of stars ex-plains the Dicke/Carter-type coincidence of the cosmicacceleration time with stellar lifetimes. That is, if thescale of cosmic acceleration is physically connected withthat of strong interactions by H Λ ≈ Λ QCD , its timescale automatically coincides with the typical lifetime of a Sun-like star.A typical stellar lifetime can be roughly estimated[59]by the time it takes to radiate at nuclear efficiency (aboutone percent of rest mass) at solar luminosity (about onepercent of Eddington luminosity), τ star ≈ α m − proton m − electron , (81)where α is the fine structure constant. Since this productof constants approximately coincides with Λ − QCD (at leastin the Standard Model), the coincidence H Λ ≈ Λ QCD naturally implies that τ star ≈ /H Λ .The coincidence τ star ≈ /H Λ (or τ galaxy ≈ /H Λ )seems so unnatural from a pure field theory perspectivethat Weinberg[16] was led to an explanation for the valueof H Λ based on a multiverse model (Carter’s “strong an-thropic principle”), but that appears not to be necessaryin the framework of entanglement sketched here. (Ofcourse, there may still be a multiverse where one or morephysical parameters “scan” across an ensemble[60], butit is not needed to explain this coincidence.) Note thatthe mass scales of stellar formation and structure bothscale approximately with the Chandrasekhar mass, whichaccounts in broad terms for why the baryons of the uni-verse tend to form stars at all, and can form relativisticremnants. VII. SUMMARY
In practice, systems much larger than the Plancklength and mass can usually be treated in the standardway, as a classical, deterministic, continuous space-time,in which quantized matter fields propagate. In this ap-proximation, the Planck scale enters only as a couplingconstant for a classical gravitational force.Depending on the size and mass of a system, the prepa-ration and measurement of its state, and particularly onthe duration of time over which its behavior is being con-sidered, quantum behavior of geometry can appear onany scale. Simple examples studied here of low densitygravitational systems involve just standard physics, yethave wave functions of position and coupled geometrywith a macroscopic scale of indeterminacy.In standard local field theory, Planck scale interactionsare highly suppressed in macroscopic systems. However,standard theory predicts states of fields far below thePlanck mass that produce unphysical geometries on largescales— that is, systems denser than black holes. Thus,quantum states of matter entangle with those of space-time in a nonlocal way, that cannot be captured by acanonical quantization of geometry.The form of this entanglement depends on unknownquantum-geometrical degrees of freedom. One form itmay take, directional entanglement, can be described onlarge scales in a purely geometrical way, without refer-ence to the particular properties of the fields. This en-tanglement can lead to new observable behavior, such as9tiny, rapid fluctuations in transverse positions of widelyseparated massive bodies, that may be studied directlyusing interferometers.In an emergent model of geometry, curvature on cos-mological scales may connect with known scales of par-ticle physics far below the Planck mass. The connectioncan be expressed in terms of information content in fieldand geometrical degrees of freedom. The total cosmicholographic information associated with the cosmologicalconstant is about the same as the positional informationin fields, as determined by the strong interactions. Thislong-known cosmological coincidence may be a natural consequence of geometrical entanglement.
Acknowledgments
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A310 , 347 (1983)[52] Y. B. Zeldovich, JETP Lett. , 316 (1967) [Pisma Zh.Eksp. Teor. Fiz. , 883 (1967)].[53] Ya. B. Zel’dovich Sov. Phys. Usp. , 381 (1968 )[54] R. Schutzhold, Phys. Rev. Lett. , 081302 (2002)[55] S. Carneiro, Int. J. Mod. Phys. D , 1669 (2003)[56] A. Randono, Gen. Rel. Grav. , 1909 (2010)[57] J. D. Bjorken, arXiv:1008.0033 [hep-ph]. (2010)[58] J. D. Bjorken, Phys. Rev. D , 043508 (2003)[59] C. J. Hogan, Rev. Mod. Phys. , 1149 (2000)[60] B. J. Carr, ed., “Universe or Multiverse?”, Cambridge,2009. l o g ( m a ss / P l a n c k m a ss ) - - - log ( radius or wavelength / Planck length )0 10 20 30 40 50 60 m o s t c o m p a c t qu a n t u m s y s t e m ( p a r t i c l e , m = / L ) m o s t c o m p a c t c l a s s i c a l g e o m e t r y ( b l a c k h o l e , M = R / )( G e n e r a l R e l a t i v i t y ) ( Q u a n t u m F i e l d T h e o r y ) Planck scale inconsistency
FIG. 1: Mass and length scales of extreme physical sys-tems in Planck units, derived from quantum theory and clas-sical general relativity. The Schwarzschild relation for blackhole mass versus radius (Eq. 3) defines the most compactgeometry in general relativity. The photoelectric relation be-tween particle energy and wavelength (Eq. 4) defines themost compact quantum system, a single quantum particle.All physical systems lie between these lines. In the up-per part of the figure, geometrical dynamics become moreimportant; in the lower part, quantum mechanics becomemore important. For the universe today, the Hubble timeis c/H ≈ . × m ≈ × , which sets the scale for thelabels and the boundary of the figure. The following figureselucidate in more detail the quantum-classical boundary inlarge systems that also have low curvature. l o g ( m a ss / P l a n c k m a ss ) - - - log ( radius / Planck length )0 10 20 30 40 50 60 g r a v i t a t i o n a l a t o m , M = r - / b l a c k h o l e i n d e t e r m i n a t e o r b i t s FIG. 2: Smallest mass of nearly-classical systems governedby quantum mechanics and nonrelativistic gravity, as shownby the gravitational atom ground state radius (Eq. 19). Insystems below this line, gravitational orbits are indetermi-nate, and locations of gravitating masses are entangled. Al-though new Planck scale physics is not needed to describesuch systems, their behavior is not captured by the standardapproximation, the use of the expectation value of the classi-cal mass as a source for gravity. l o g ( m a ss / P l a n c k m a ss ) - - - log ( radius / Planck length )0 10 20 30 40 50 60 g r a v i t a t i o n a l a t o m d e n s i t y = H qu a n t u m p o s i t i o n un c e r t a i n t y M = / L H e x p a n s i o n u n c e r t a i n t y M = L = H - Quantum-classical boundary for cosmic expansion: ~ 60 meters measure acceleration in 1 year
FIG. 3: Mass and length scales of quantum/classical bound-aries for cosmic expansion, derived from quantum mechanicsand nonrelativistic gravity. The figure shows the standardquantum position uncertainty over time (Eq. 22), mean cos-mic density (Eq. 23), the uncertainty in cosmic perturbationmass (Eq. 30), and the quantum limit on measurement ofcosmic acceleration in 1 year (Eq. 36). Dashed lines scalewith the expansion rate H , plotted here for the current value H = 10 − in Planck units. l o g ( m a ss / P l a n c k m a ss ) - - - log ( radius or duration / Planck length )0 10 20 30 40 50 60 b l a c k h o l e m a s s , M = R B H / ( p o s i t i o n un c e r t a i n t y ) > t e v ap / M b l a c k h o l e r a d i a t i o n t e v a p ~ M (position uncertainty over time t evap ) ~ R BH FIG. 4: Scales associated with black hole evaporation. Inaddition to the Schwarzschild relation M ( R ), plot shows theevaporation time τ evap ≈ M . The standard quantum uncer-tainty for the position of the hole (Eq. 22) over time τ evap ,or the spatial width of its world line, is about equal to theSchwarzschild radius. On this time scale and separation scale,the location of the event horizon, hence the causal structure,is an indeterminate quantity, not even approximately definedby classical dynamics. The scale here is arbitrary: the sameargument applies for any choice of τ evap . l o g ( m a ss / P l a n c k m a ss ) - - - log ( radius / Planck length )0 10 20 30 40 50 60 b l a c k h o l e qu a n t u m s o li d d e n s i t y p o s i t i o n un c e r t a i n t y i n o r b i t a l t i m e ( ~ h o u r s ) nanoscale system ~10 -7 m g r a v i t a t i o n a l a t o m FIG. 5: Scales associated with a laboratory-scale experimentthat could create a system with indeterminate gravitationalorbits— a nanoscale artificial gravitational atom. l o g ( p a r t i c l e m a ss / P l a n c k m a ss ) - - - log ( system size / Planck length )0 10 20 30 40 50 60 s i n g l e qu a n t u m Field states exceed black hole mass in same volumeField theory OKField modes exceed Planck angular resolution g e o m e t r i c a l e n t a n g l e m e n t , L = m - FIG. 6: Length scale where quantum field systems withparticle mass or UV cutoff > m become inconsistent withgravity by exceeding the mass of a black hole, or exceedingthe Planck diffraction bound on directional information (Eqs.41, 43). Shaded region shows range where standard quantumapproximations are little affected by these bounds. Abovethis region, it is conjectured that field states are significantlyentangled with geometrical states, reducing the number ofdegrees of freedom to a value consistent with holography. Forparticle states of mass m , directional entanglement modulatestransverse field phase from the classical background by aboutone radian (that is, a length ≈ m − ) at separation m − . time duration, spacetime curvature radius3D width around world line spatial cavity size, wavefront curvature radius 2D width around beam axis wavelength FIG. 7: Sketch of how the 3+1D volume swept out by aparaxial solution of the wave equation, which represents thequantum state of the world line of a spatially localized massiveparticle at rest, resembles the 2+1D spatial wave functionthat represents the quantum state of monochromatic light ina cavity. Dotted lines represent a range of typical world-linesfor a wave function extending over some macroscopic timeinterval. Solutions to Eq. (63) relate the curvature of theconstant-time surfaces, as measured by field phase, to the 3Dwidth of the position wave function. It is proposed that globalgeometrical curvature is entangled with the field at the radiuswhere the 3D width is about equal to the maximum extentof vacuum field states for the particle mass characteristic oflocalization in the field vacuum, that is, the QCD scale. l o g ( p a r t i c l e m a ss / P l a n c k m a ss ) - - - log ( system length scale / Planck length )0 10 20 30 40 50 60 d i r e c t i o n a l e n t a n g l e m e n t , L ~ m - d e n s i t y = H w o r l d li n e w i d t h f o r d u r a t i o n / H Standard Model localization (QCD) M = L = / H c u r v a t u r e r a d i u s , L ~ m - cosmic curvature FIG. 8: Solid lines show particle mass m as a function ofthe scale of significant field directional entanglement (Eqs.41, 43), and the radius of curvature entanglement (Eq. 72).Dashed lines show cosmological density, and world line spa-tial width for particles of mass m and H = H , which isclose to the asymptotic value of the expansion rate H Λ (Eq.75). Particle mass associated with QCD confinement scale,Λ QCD ≈ − , L G (Λ QCD ) ≈40