##### Document Text Contents

Page 1

Now Broadcasting in Planck Definition

Craig Hogan

University of Chicago and Fermilab

If reality has finite information content, space has finite fidelity. The quantum wave function that

encodes spatial relationships may be limited to information that can be transmitted in a “Planck

broadcast”, with a bandwidth given by the inverse of the Planck time, about 2 × 1043 bits per

second. Such a quantum system can resemble classical space-time on large scales, but locality

emerges only gradually and imperfectly. Massive bodies are never perfectly at rest, but very slightly

and slowly fluctuate in transverse position, with a spectrum of variation given by the Planck time.

This distinctive new kind of noise associated with quantum geometry would not have been noticed

up to now, but may be detectable in a new kind of experiment.

At the turn of the last century, Max Planck derived from first principles a universal formula for the spectrum of

radiation emitted by opaque matter. Planck’s radiation law solved a long-standing experimental mystery unexplained

by classical physics, and agreed exactly with measurements. It flowed from a simple, powerful and radically new

idea: that everything that happens in nature occurs in discrete minimum packages of action, or quanta. Planck’s

breakthrough started the quantum revolution in physics that defined much of twentieth century science and technology.

A few years after Planck’s triumph, Albert Einstein introduced his theory of relativity. While Planck’s theory

addressed the nature of matter, Einstein’s addressed the nature of space and time. It also solved long standing

mysteries, and flowed from a simple idea: that the laws of physics should not depend on how one moves. Einstein

extended his theory with another powerful idea— that local physics is the same in any freely falling frame— to

reveal that space and time form an active dynamical geometry, whose curvature creates the force of gravity. General

Relativity was revolutionary, but it is entirely classical: Einstein’s space-time is not a quantum system.

These two great theories of twentieth century physics have never been fully reconciled, because their core ideas

are incompatible. Relativity is based on the notion of locality, a concept not respected by quantum physics; indeed,

experiments with quantum systems prove that states in reality are not localized in space. The central role of mea-

surement in quantum physics flies against the relativistic notion that reality is independent of an observer. Perhaps

most fundamentally, relativity violates quantum precepts by assigning tangible reality to unobservable things, such

as events and paths in space-time.

This clash of ideas led to agonized epistemological debates in the early part of the century, most famously between

Bohr and Einstein. But most of physics has moved on. For all practical purposes so far, it works just fine to assume

a continuous classical space and put quantum matter into it. That is what the well-tested Standard Model of physics

does. It is a quantum theory, but only of matter, not of space-time.

Classical continuous space, as usually assumed, maps onto real numbers: it has an infinite information density.

Quantum theory suggests instead that the information content of the world is fundamentally limited. It is natural to

suppose that all spatial relationships are just another sort of observable relationship, to be derived from the quantum

theory of some system. Let us adopt a working hypothesis different from the usual one: information in spatial

position is limited by the broadcast capacity of exchanged information at a bandwidth given by a fundamental scale. It

is possible to work out some experimental consequences of this hypothesis even in the absence of a full theory, because

the fidelity of space is limited by its information capacity.

We have some clues to the amount of information involved. Planck’s formula came with a new constant of nature,

a fundamental unit for the quantum of action that we now call Planck’s constant, h̄. By combining his constant with

Newton’s constant of gravity G and the speed of light c, Planck obtained new “natural” units of length, time and

mass. In Einstein’s theory, G controls the dynamics of space and time. Planck’s units therefore set the natural scale

for the quantum mechanics of space-time itself, where information about location becomes fundamentally discrete.

Because gravity is weak, the Plank scale is very small; for example, the Planck time is tP =

√

h̄G/c5 = 5.4× 10� 44

seconds. And because that scale is so small, its quanta are very fine grained, and so far undetectable. No experiment

shows an identifiable quantum behavior of space and time. That is why for the last century, physicists have been able

to treat space and time like a definite, continuous, classical medium.

The lack of an experiment means that we have no guide to interpret mathematical ideas about blending quantum

mechanics and space-time. Physicists were forced into the strange world of quantum mechanics by experimental

measurements, such as radiation spectra from black bodies and gases. As Rabi said, “Physics is an experimental

science.” So, let us ask a very practical question: How can we build an experiment that directly reveals the discrete

character of space-time information at the Planck scale? The answer depends on the character of that information—

Page 2

2

the encoding of quantum geometry in macroscopic position states.

Consider how quantum systems work. In pre-quantum physics, all properties of a system, such as positions and

velocities of particles, have de�nite values, and change with time according to de�nite rules. In quantum physics, as

the state of a system evolves, relationships among its properties evolve according to de�nite rules| but in general,

individual properties do not have de�nite values, even in principle. Instead, the entire system is described by a wave

function of possibilities. Reality is that multitude of possibilities, a set of relationships. In general, de�nite, observable

outcomes are impossible to predict.

Any combined system is literally more than the sum of its parts; a composite system contains information that

cannot be separated into information about one subsystem or another. Information in a combined system generally

resides in the correlation between its parts, a property known as \quantum entanglement".

The quantum challenge to conventional notions of what is real were highlighted in a famous 1935 paper by Einstein,

Podolsky, and Rosen. Schr�odinger responded by introducing the idea of entanglement, as well as the provocative

thought-experiment with the uranium, the

ask of hydrocyanic acid, and the unlucky cat. As he noted, entanglement

comes with nonlocality: \Maximal knowledge of a total system does not necessarily include total knowledge of all its

parts, not even when these are fully separated from each other and at the moment are not in

uencing each other at

all."[1] Einstein referred to such behavior as \spooky action at a distance".

Although these ideas have created controversy over the years, it is an experimental fact that information in the real

world is not localized in space and time. A measurement in one place is correlated with, and a�ects the state of a

system everywhere.[2, 3] There are even real-world experiments that show examples of such quantum entanglement

between particles that never co-existed at the same time.[4] Although experiments that demonstrate such e�ects are

quite subtle to mount, entanglement, and the nonlocality that goes with it, are woven into the fabric of reality.

Einstein and others realized that quantum nonlocality is a big problem for relativity.[5] It seems to directly contradict

the foundational notion of space and time, that everything happens at a de�nite time and place. Even the most

advanced theory of space-time, General Relativity, is based on a metric that speci�es the intervals between events.

Quantum mechanics implies that intervals between events, or indeed any property of events themselves, can never be

exactly measured, even in principle. And some things that happen in the world| the quantum correlations created

by interactions that we interpret as the collapse of a wave function| are entirely delocalized in space and time.

Physics has advanced by working around the apparent paradox of delocalization. Often, it is not important to

know exactly where something happens, since many important properties of matter do not depend on location in

any particular place. For example, in quantum �eld theory, the quantized system is a mode of a �eld wave extended

in space and time| a delocalized state. This approximation leads to extraordinarily successful predictions for all

experiments on collisions of elementary particles at energies far below the Planck mass, such as those at Fermilab and

CERN.

Quantum delocalization inspires a view of the world made not so much of material as of information. This idea

may be extended to space and time as well as matter. Some properties of space and time that seem fundamental,

including localization, may actually emerge only as a macroscopic approximation, from the

ow of information in a

quantum system.

Even within the entirely classical framework of relativity and gravitation, theory has provided some hints about

the possibility of such emergence, and about the signi�cance of the Planck scale| how quantum mechanics blends

with the physics of space and time. The purest states of space-time, black holes, obey thermodynamic properties: for

example, in a system of black holes, the total area of event horizons always increases, like entropy. More generally,

the equations of general relativity can be derived from a purely statistical theory, by requiring that entropy is always

proportional to horizon area.[6] Similarly, Newton’s laws of motion and gravity can be derived from a statistical theory

based on entropy and coarse-graining, where information lives on surfaces and is associated with position of bodies.[7]

In these derivations, the dynamics of space-time, the equivalence principal, and concepts of inertia and momentum

for massive bodies, all arise as emergent properties.

These results are based on essentially classical statistical arguments, but they refer to quantum information. The

detailed quantum character of the underlying quantum degrees of freedom associated with position in space is not

known, but we can guess some of their properties. The precision and universality of light propagation, even across

cosmic distances[8], suggests that causal structure arises from a fundamental symmetry, even if locality is only approx-

imate. Gravitational thermodynamics also suggest an exact number for the distribution and amount of information

in the system: the information �ts on two-dimensional sheets, and the total information density is the area of a sheet

in Planck units. From these arguments, we do not know how this holographic information is encoded, but we know

how much of it there is, and something about how it maps onto space.

Taken together, these theoretical ideas hint that quantum mechanics limits the amount of information in space-time.

Presumably, such a limit must place some kind of limit on the �delity of space-time itself: not all classically described

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3

locations are physically different from each other.

Video buffs know that higher bandwidth gives you a better picture. Suppose that the bandwidth of information

transmission is limited by the Planck frequency: ωP ≡ t−1P ≡

√

c5/h̄G = 1.85 × 1043 Hertz. If this is best that

the cosmic Internet Service Provider can give us, we do not get a perfect picture. There is only a finite amount

of information in the positional relationships of material bodies. Perhaps a very careful experiment, that looks at

position closely in the right way, might be able to see a bit of blurring, a lack of sharpness and clarity, like a little extra

noise in the image or clipping in the cosmic sound track. The properties of the clipping may even reveal something

about the compression or encoding algorithm.

An effect like that would of course be interesting to physicists, who are essentially hobbyists of nature. We used

to say that physics is about discovering laws of nature, but these days we could just as well say that it is all about

figuring out how the system of the universe works— how its instructions are encoded, and what operating system it

runs on. An experiment would provide some useful clues.

Imagine then that the real world is the ultimate 4-dimensional video display. How good is it?

At first you might guess that the Planck bandwidth limit would simply create a system with Planck size pixels

everywhere; that is, a frame refresh rate given by the Planck time tP , and pixel (or voxel) size given by the Planck

length, ctP ≡

√

h̄G/c3 = 1.6× 10−35 meters, in each of the three space dimensions.

To achieve such a fine grained picture, we need a Planck bandwidth channel for every pixel— a Planck density of

information in four dimensions, or a Planck bandwidth in every three dimensional Planck volume. This value is the

amount of information in the standard model of quantum fields, if we include all frequencies up to the Planck scale.

But this guess does not agree with holographic emergence. Our radically different hypothesis is that space and time

are created from information propagating with Planck bandwidth. In such a “Planck broadcast”, space is not assumed

to exist a priori , but is a set of relationships that emerges from Planck-limited information processing. Instead of a

world densely packed with Planck size cells, as in field theory, perhaps positions in space and time only contain the

amount of information that can be carried on a Planck frequency carrier wave. In that case, a large spatial volume

has a much smaller density of information.

Imagine that someone sends broadcast video at the Planck frequency. That is only enough information to refresh

one pixel every Planck time. For a larger screen, the refresh rate and resolution get worse. How much worse?

If the broadcast encodes all of real space, it needs to encode all directions. Suppose that the video screen is a sphere

of with a radius L about our broadcast point, and has pixels of size ∆x. We encode the information on the screen to

refresh more slowly, with a refresh interval given by the time it takes light to get to the screen and back, τ = 2L/c—

the slowest acceptable rate for encoding a position at this distance. Then the minimum pixel size is given by setting

the total number of pixels 4πL2∆x−2 per time 2L/c equal to the Planck information rate t−1P , so the pixel size is

∆x =

√

2πLctP— very small, but still much larger than the Planck length.

Of course, nature is not really pixelated in little squares, but the same answer for the blurring scale emerges from

a more realistic physical model based on waves. Positions encoded by wave functions that have a cutoff or bandwidth

limit convey only a limited amount of transverse spatial information from one place to another.[9, 10]

Imagine a wave that passes through a a pair of narrow slits. The wave creates an interference pattern on a screen

at distance L that depends on (i.e., encodes) the transverse separation of the two slits. However, there is a resolution

limit: if the two slits have transverse separation much smaller than

∆x⊥ ≈

√

LctP , (1)

the interference pattern of radiation at frequency ωP is not distinguishable from that of a single slit. The resolution

limit from this point of view is a diffraction limit in wave mechanics, but it is really an information bound: the waves

simply do not have enough information to resolve smaller transverse distances than that. Notice that the distance to

the screen— and causal structure— can be defined with much higher precision ≈ ctP , by counting wave fronts. The

transverse resolution, the slit separation (Eq. 1), gets much poorer at large L.

The corresponding angular uncertainty,

∆θ = ∆x⊥/L ≈ ctP /L, (2)

gets smaller on large scales. Thus, angles get sharper at larger separation, so the notion of direction emerges more

clearly on larger scales. The total amount of angular information grows, but only linearly with L, more slowly than

it would for a display with Planck size pixels.

Overall there are about L/ctP degrees of freedom corresponding to radial separation, and L/ctP corresponding to

angle. For each Planck time in duration or radial separation, there are L/ctP angular degrees of freedom. The total

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4

amount of information is the number of directions times the duration, so it grows holographically, like (L/ctP )

2. The

density of information is constant on surfaces, but in 3D space it thins out with time and distance as it spreads.

This holographic scaling is just what is needed for the statistics of emergent gravity to work. If we invoke that idea

to set the scale of information density, the prediction for transverse mean square position uncertainty becomes very

precise[11]:

〈x̂2⊥〉 = LctP /

√

4π = (2.135× 10−18m)2(L/1m), (3)

with no free parameters. We don’t know the character of the actual quantum theory that controls geometry, but

this estimate of the transverse blurring scale is relatively robust, because it is just determined by the amount of

information.

Apparently, if space-time is a quantum system with limited information— a Planck broadcast— there should be

a new kind of quantum fuzziness of positions, not just for small particles, but for everything, even for large masses.

The blurring is larger for larger L: the position resolution gets worse at larger distances. In a laboratory size system,

it is much larger than the Planck length— about an attometer in scale, a billionth of a billionth of a meter.

There is vastly less information in this macroscopic quantum system than in standard theory— that is, a system

of quantum fields in classical space-time with a Planck cut off— but there is enough angular information to agree

with the apparent sharpness and classical behavior of space, as measured in experiments to date. If things could be

measured at separations on the Planck scale, the angular uncertainty would be huge; directions are not even really

well defined, and it essentially a 2D holographic system. On the scale explored by particle colliders, about 1016 times

larger than Planck length, things are already very close to classical; angular blurring is too small to detect with

particle experiments of limited precision, and in any case the particle masses are small so standard quantum effects

overwhelm the geometrical ones.

Indeed, the new Planck blurring is always negligible compared to standard quantum uncertainty (which does depend

on mass) for systems much smaller than about a Planck mass, mP =

p

h̄G/c = 2.176×10−8 kg.[11] In measurements

of small numbers of particles, the geometrical effects are not detectable. Unlike standard quantum effects, the Planck

information limit is only important for large masses.

At first, it also seems strange that the resolution depends on a macroscopic separation. Intuition suggests that the

state of affairs of matter and energy should not depend on how far away it is; after all, how can it “know” where we,

the Planck broadcaster or observer, are? According to Einstein, the laws of physics ought to be independent of the

location and motion of an observer.

A related worry is that an attometer scale uncertainty, while small, is really not all that small by the standards of

particle physics. That scale is now routinely resolved by particle colliders, like the Tevatron and the LHC. Yet there

is no sign in experiments of a new kind of fuzziness in space-time. Indeed, if we set L comparable to the size the

universe, we find that ∆x is actually on a scale you can see with your own eyes, of the order of 0.1mm, the width of

a hair. Space certainly doesn’t display any lack of sharpness on that scale when you look around.

These worries may be resolved by invoking entanglement. Space-time is the ultimate, universal entangled system.

Locality itself can emerge, via entanglement, as an approximate behavior on large scales.

Information is not localized in space, but resides in non localized correlations. The density of information can

depend on scale, and can be smaller for larger systems. The effective fidelity of space-time can change depending

on where something is relative to an observer. A measurement confined to a small volume does not know or care

about a transverse geometrical displacement relative to some distant place, so the uncertainty is not observable in

local measurements.

In quantum mechanics, measurements make projections— in Copenhagen language, they “collapse” the wave func-

tion. Until they are made, there is uncertainty given by the width of the wave function— in our case, the scale-

dependent blurring. In an emergent space-time, every world-line defines a particular projection of the wave function

associated with the structure of nested light cones (or “nested causal diamonds”) around it.

Thus, the quantum geometrical position information is entangled for bodies whose world-lines are close together.

If you measure the transverse position state of one massive body, you will find almost the same projection of the

geometrical state for any body nearby. That does not mean that any two bodies are in the same position; it only

means that their quantum deviation from the classical position is almost the same, relative to any arbitrary far away

point. The local relationships of the bodies in space are changed very little from standard quantum mechanics.

A classical space-time is the limiting case of a fully coherent system. The approach to the classical limit however

reveals slight departures from classical behavior that are not present in standard theory. Nonlocal projections of the

quantum state in different directions are slightly different, even on large scales.

As the system unfolds in time, the uncertainty leads to random variations— a new kind of noise in position

measurements to distant bodies in different directions. The positions of nearby bodies change together, carried along

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with the geometry, into the same new definite state. Local measurements are not affected by this collective change of

position— a new kind of “movement without motion”.

This interpretation of the angular uncertainty opens up a way to build an experiment that probes Planck scale

physics. The Planck broadcast model of quantum geometry predicts that positions fluctuate, with a power spectrum

of angular variations given approximately by the Planck time— that is, in an average over duration τ , the mean

square variation is

〈∆θ2〉� ≈ tP /τ. (4)

In an experiment of size L, the variations accumulate up to durations τ ≈ L/c, ultimately leading to variance in

position given by the overall uncertainty, Eq. (3). This prediction can be tested by making very sensitive measurements

of transverse positions of massive bodies. The measurement process must make a nonlocal comparison of position in

different directions.

An experiment designed to detect or rule out fluctuations with these properties, called the Fermilab Holome-

ter, is currently being developed.[12] It uses a technique based on laser interferometers like those used to measure

gravitational waves. The intensity of light emerging from a Michelson interferometer allows a precise and coherent

measurement of the positions of mirrors over an extended region of space, in this case, 40 meters in two directions.

The precision of such devices is extraordinary; they can detect variations in mean position differences on the order of

attometers, limited primarily by the quantum character of the laser light. In the Holometer, correlations are measured

between the signals of two adjacent, aligned interferometers. The correlations are sensitive to tiny, random in-common

motions that change very quickly, on timescales comparable to a light-crossing time, less than a microsecond. (On

longer timescales, entanglement-driven locality reduces the variation.) The effective speed of the motionless movement

is tiny— comparable to continental drift, only centimeters per year.

Because of quantum entanglement, the holographic noise created by the Planck broadcast information limit creates

tiny, rapid fluctuations in signals from the two adjacent interferometers that are coherent with each other, even if

there is no connection between the devices apart from proximity. Arguments like those outlined here, based on

information in holographic emergent space, can be used to make an exact prediction for the expected cross-correlated

noise spectrum, even without knowing details of the fundamental theory[11, 13].

Whether or not new Planck scale holographic noise is detected, the Holometer is interesting as an exploratory

experiment, because it tests the fidelity and coherence of nonlocal spatial relationships with Planck precision for the

first time. The outcome will either reveal a signature of new Planck scale physics, or experimentally prove a coherence

of macroscopic space greater than what is possible with a Planck broadcast. We don’t know what we will find.

Now Broadcasting in Planck Definition

Craig Hogan

University of Chicago and Fermilab

If reality has finite information content, space has finite fidelity. The quantum wave function that

encodes spatial relationships may be limited to information that can be transmitted in a “Planck

broadcast”, with a bandwidth given by the inverse of the Planck time, about 2 × 1043 bits per

second. Such a quantum system can resemble classical space-time on large scales, but locality

emerges only gradually and imperfectly. Massive bodies are never perfectly at rest, but very slightly

and slowly fluctuate in transverse position, with a spectrum of variation given by the Planck time.

This distinctive new kind of noise associated with quantum geometry would not have been noticed

up to now, but may be detectable in a new kind of experiment.

At the turn of the last century, Max Planck derived from first principles a universal formula for the spectrum of

radiation emitted by opaque matter. Planck’s radiation law solved a long-standing experimental mystery unexplained

by classical physics, and agreed exactly with measurements. It flowed from a simple, powerful and radically new

idea: that everything that happens in nature occurs in discrete minimum packages of action, or quanta. Planck’s

breakthrough started the quantum revolution in physics that defined much of twentieth century science and technology.

A few years after Planck’s triumph, Albert Einstein introduced his theory of relativity. While Planck’s theory

addressed the nature of matter, Einstein’s addressed the nature of space and time. It also solved long standing

mysteries, and flowed from a simple idea: that the laws of physics should not depend on how one moves. Einstein

extended his theory with another powerful idea— that local physics is the same in any freely falling frame— to

reveal that space and time form an active dynamical geometry, whose curvature creates the force of gravity. General

Relativity was revolutionary, but it is entirely classical: Einstein’s space-time is not a quantum system.

These two great theories of twentieth century physics have never been fully reconciled, because their core ideas

are incompatible. Relativity is based on the notion of locality, a concept not respected by quantum physics; indeed,

experiments with quantum systems prove that states in reality are not localized in space. The central role of mea-

surement in quantum physics flies against the relativistic notion that reality is independent of an observer. Perhaps

most fundamentally, relativity violates quantum precepts by assigning tangible reality to unobservable things, such

as events and paths in space-time.

This clash of ideas led to agonized epistemological debates in the early part of the century, most famously between

Bohr and Einstein. But most of physics has moved on. For all practical purposes so far, it works just fine to assume

a continuous classical space and put quantum matter into it. That is what the well-tested Standard Model of physics

does. It is a quantum theory, but only of matter, not of space-time.

Classical continuous space, as usually assumed, maps onto real numbers: it has an infinite information density.

Quantum theory suggests instead that the information content of the world is fundamentally limited. It is natural to

suppose that all spatial relationships are just another sort of observable relationship, to be derived from the quantum

theory of some system. Let us adopt a working hypothesis different from the usual one: information in spatial

position is limited by the broadcast capacity of exchanged information at a bandwidth given by a fundamental scale. It

is possible to work out some experimental consequences of this hypothesis even in the absence of a full theory, because

the fidelity of space is limited by its information capacity.

We have some clues to the amount of information involved. Planck’s formula came with a new constant of nature,

a fundamental unit for the quantum of action that we now call Planck’s constant, h̄. By combining his constant with

Newton’s constant of gravity G and the speed of light c, Planck obtained new “natural” units of length, time and

mass. In Einstein’s theory, G controls the dynamics of space and time. Planck’s units therefore set the natural scale

for the quantum mechanics of space-time itself, where information about location becomes fundamentally discrete.

Because gravity is weak, the Plank scale is very small; for example, the Planck time is tP =

√

h̄G/c5 = 5.4× 10� 44

seconds. And because that scale is so small, its quanta are very fine grained, and so far undetectable. No experiment

shows an identifiable quantum behavior of space and time. That is why for the last century, physicists have been able

to treat space and time like a definite, continuous, classical medium.

The lack of an experiment means that we have no guide to interpret mathematical ideas about blending quantum

mechanics and space-time. Physicists were forced into the strange world of quantum mechanics by experimental

measurements, such as radiation spectra from black bodies and gases. As Rabi said, “Physics is an experimental

science.” So, let us ask a very practical question: How can we build an experiment that directly reveals the discrete

character of space-time information at the Planck scale? The answer depends on the character of that information—

Page 2

2

the encoding of quantum geometry in macroscopic position states.

Consider how quantum systems work. In pre-quantum physics, all properties of a system, such as positions and

velocities of particles, have de�nite values, and change with time according to de�nite rules. In quantum physics, as

the state of a system evolves, relationships among its properties evolve according to de�nite rules| but in general,

individual properties do not have de�nite values, even in principle. Instead, the entire system is described by a wave

function of possibilities. Reality is that multitude of possibilities, a set of relationships. In general, de�nite, observable

outcomes are impossible to predict.

Any combined system is literally more than the sum of its parts; a composite system contains information that

cannot be separated into information about one subsystem or another. Information in a combined system generally

resides in the correlation between its parts, a property known as \quantum entanglement".

The quantum challenge to conventional notions of what is real were highlighted in a famous 1935 paper by Einstein,

Podolsky, and Rosen. Schr�odinger responded by introducing the idea of entanglement, as well as the provocative

thought-experiment with the uranium, the

ask of hydrocyanic acid, and the unlucky cat. As he noted, entanglement

comes with nonlocality: \Maximal knowledge of a total system does not necessarily include total knowledge of all its

parts, not even when these are fully separated from each other and at the moment are not in

uencing each other at

all."[1] Einstein referred to such behavior as \spooky action at a distance".

Although these ideas have created controversy over the years, it is an experimental fact that information in the real

world is not localized in space and time. A measurement in one place is correlated with, and a�ects the state of a

system everywhere.[2, 3] There are even real-world experiments that show examples of such quantum entanglement

between particles that never co-existed at the same time.[4] Although experiments that demonstrate such e�ects are

quite subtle to mount, entanglement, and the nonlocality that goes with it, are woven into the fabric of reality.

Einstein and others realized that quantum nonlocality is a big problem for relativity.[5] It seems to directly contradict

the foundational notion of space and time, that everything happens at a de�nite time and place. Even the most

advanced theory of space-time, General Relativity, is based on a metric that speci�es the intervals between events.

Quantum mechanics implies that intervals between events, or indeed any property of events themselves, can never be

exactly measured, even in principle. And some things that happen in the world| the quantum correlations created

by interactions that we interpret as the collapse of a wave function| are entirely delocalized in space and time.

Physics has advanced by working around the apparent paradox of delocalization. Often, it is not important to

know exactly where something happens, since many important properties of matter do not depend on location in

any particular place. For example, in quantum �eld theory, the quantized system is a mode of a �eld wave extended

in space and time| a delocalized state. This approximation leads to extraordinarily successful predictions for all

experiments on collisions of elementary particles at energies far below the Planck mass, such as those at Fermilab and

CERN.

Quantum delocalization inspires a view of the world made not so much of material as of information. This idea

may be extended to space and time as well as matter. Some properties of space and time that seem fundamental,

including localization, may actually emerge only as a macroscopic approximation, from the

ow of information in a

quantum system.

Even within the entirely classical framework of relativity and gravitation, theory has provided some hints about

the possibility of such emergence, and about the signi�cance of the Planck scale| how quantum mechanics blends

with the physics of space and time. The purest states of space-time, black holes, obey thermodynamic properties: for

example, in a system of black holes, the total area of event horizons always increases, like entropy. More generally,

the equations of general relativity can be derived from a purely statistical theory, by requiring that entropy is always

proportional to horizon area.[6] Similarly, Newton’s laws of motion and gravity can be derived from a statistical theory

based on entropy and coarse-graining, where information lives on surfaces and is associated with position of bodies.[7]

In these derivations, the dynamics of space-time, the equivalence principal, and concepts of inertia and momentum

for massive bodies, all arise as emergent properties.

These results are based on essentially classical statistical arguments, but they refer to quantum information. The

detailed quantum character of the underlying quantum degrees of freedom associated with position in space is not

known, but we can guess some of their properties. The precision and universality of light propagation, even across

cosmic distances[8], suggests that causal structure arises from a fundamental symmetry, even if locality is only approx-

imate. Gravitational thermodynamics also suggest an exact number for the distribution and amount of information

in the system: the information �ts on two-dimensional sheets, and the total information density is the area of a sheet

in Planck units. From these arguments, we do not know how this holographic information is encoded, but we know

how much of it there is, and something about how it maps onto space.

Taken together, these theoretical ideas hint that quantum mechanics limits the amount of information in space-time.

Presumably, such a limit must place some kind of limit on the �delity of space-time itself: not all classically described

Page 3

3

locations are physically different from each other.

Video buffs know that higher bandwidth gives you a better picture. Suppose that the bandwidth of information

transmission is limited by the Planck frequency: ωP ≡ t−1P ≡

√

c5/h̄G = 1.85 × 1043 Hertz. If this is best that

the cosmic Internet Service Provider can give us, we do not get a perfect picture. There is only a finite amount

of information in the positional relationships of material bodies. Perhaps a very careful experiment, that looks at

position closely in the right way, might be able to see a bit of blurring, a lack of sharpness and clarity, like a little extra

noise in the image or clipping in the cosmic sound track. The properties of the clipping may even reveal something

about the compression or encoding algorithm.

An effect like that would of course be interesting to physicists, who are essentially hobbyists of nature. We used

to say that physics is about discovering laws of nature, but these days we could just as well say that it is all about

figuring out how the system of the universe works— how its instructions are encoded, and what operating system it

runs on. An experiment would provide some useful clues.

Imagine then that the real world is the ultimate 4-dimensional video display. How good is it?

At first you might guess that the Planck bandwidth limit would simply create a system with Planck size pixels

everywhere; that is, a frame refresh rate given by the Planck time tP , and pixel (or voxel) size given by the Planck

length, ctP ≡

√

h̄G/c3 = 1.6× 10−35 meters, in each of the three space dimensions.

To achieve such a fine grained picture, we need a Planck bandwidth channel for every pixel— a Planck density of

information in four dimensions, or a Planck bandwidth in every three dimensional Planck volume. This value is the

amount of information in the standard model of quantum fields, if we include all frequencies up to the Planck scale.

But this guess does not agree with holographic emergence. Our radically different hypothesis is that space and time

are created from information propagating with Planck bandwidth. In such a “Planck broadcast”, space is not assumed

to exist a priori , but is a set of relationships that emerges from Planck-limited information processing. Instead of a

world densely packed with Planck size cells, as in field theory, perhaps positions in space and time only contain the

amount of information that can be carried on a Planck frequency carrier wave. In that case, a large spatial volume

has a much smaller density of information.

Imagine that someone sends broadcast video at the Planck frequency. That is only enough information to refresh

one pixel every Planck time. For a larger screen, the refresh rate and resolution get worse. How much worse?

If the broadcast encodes all of real space, it needs to encode all directions. Suppose that the video screen is a sphere

of with a radius L about our broadcast point, and has pixels of size ∆x. We encode the information on the screen to

refresh more slowly, with a refresh interval given by the time it takes light to get to the screen and back, τ = 2L/c—

the slowest acceptable rate for encoding a position at this distance. Then the minimum pixel size is given by setting

the total number of pixels 4πL2∆x−2 per time 2L/c equal to the Planck information rate t−1P , so the pixel size is

∆x =

√

2πLctP— very small, but still much larger than the Planck length.

Of course, nature is not really pixelated in little squares, but the same answer for the blurring scale emerges from

a more realistic physical model based on waves. Positions encoded by wave functions that have a cutoff or bandwidth

limit convey only a limited amount of transverse spatial information from one place to another.[9, 10]

Imagine a wave that passes through a a pair of narrow slits. The wave creates an interference pattern on a screen

at distance L that depends on (i.e., encodes) the transverse separation of the two slits. However, there is a resolution

limit: if the two slits have transverse separation much smaller than

∆x⊥ ≈

√

LctP , (1)

the interference pattern of radiation at frequency ωP is not distinguishable from that of a single slit. The resolution

limit from this point of view is a diffraction limit in wave mechanics, but it is really an information bound: the waves

simply do not have enough information to resolve smaller transverse distances than that. Notice that the distance to

the screen— and causal structure— can be defined with much higher precision ≈ ctP , by counting wave fronts. The

transverse resolution, the slit separation (Eq. 1), gets much poorer at large L.

The corresponding angular uncertainty,

∆θ = ∆x⊥/L ≈ ctP /L, (2)

gets smaller on large scales. Thus, angles get sharper at larger separation, so the notion of direction emerges more

clearly on larger scales. The total amount of angular information grows, but only linearly with L, more slowly than

it would for a display with Planck size pixels.

Overall there are about L/ctP degrees of freedom corresponding to radial separation, and L/ctP corresponding to

angle. For each Planck time in duration or radial separation, there are L/ctP angular degrees of freedom. The total

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amount of information is the number of directions times the duration, so it grows holographically, like (L/ctP )

2. The

density of information is constant on surfaces, but in 3D space it thins out with time and distance as it spreads.

This holographic scaling is just what is needed for the statistics of emergent gravity to work. If we invoke that idea

to set the scale of information density, the prediction for transverse mean square position uncertainty becomes very

precise[11]:

〈x̂2⊥〉 = LctP /

√

4π = (2.135× 10−18m)2(L/1m), (3)

with no free parameters. We don’t know the character of the actual quantum theory that controls geometry, but

this estimate of the transverse blurring scale is relatively robust, because it is just determined by the amount of

information.

Apparently, if space-time is a quantum system with limited information— a Planck broadcast— there should be

a new kind of quantum fuzziness of positions, not just for small particles, but for everything, even for large masses.

The blurring is larger for larger L: the position resolution gets worse at larger distances. In a laboratory size system,

it is much larger than the Planck length— about an attometer in scale, a billionth of a billionth of a meter.

There is vastly less information in this macroscopic quantum system than in standard theory— that is, a system

of quantum fields in classical space-time with a Planck cut off— but there is enough angular information to agree

with the apparent sharpness and classical behavior of space, as measured in experiments to date. If things could be

measured at separations on the Planck scale, the angular uncertainty would be huge; directions are not even really

well defined, and it essentially a 2D holographic system. On the scale explored by particle colliders, about 1016 times

larger than Planck length, things are already very close to classical; angular blurring is too small to detect with

particle experiments of limited precision, and in any case the particle masses are small so standard quantum effects

overwhelm the geometrical ones.

Indeed, the new Planck blurring is always negligible compared to standard quantum uncertainty (which does depend

on mass) for systems much smaller than about a Planck mass, mP =

p

h̄G/c = 2.176×10−8 kg.[11] In measurements

of small numbers of particles, the geometrical effects are not detectable. Unlike standard quantum effects, the Planck

information limit is only important for large masses.

At first, it also seems strange that the resolution depends on a macroscopic separation. Intuition suggests that the

state of affairs of matter and energy should not depend on how far away it is; after all, how can it “know” where we,

the Planck broadcaster or observer, are? According to Einstein, the laws of physics ought to be independent of the

location and motion of an observer.

A related worry is that an attometer scale uncertainty, while small, is really not all that small by the standards of

particle physics. That scale is now routinely resolved by particle colliders, like the Tevatron and the LHC. Yet there

is no sign in experiments of a new kind of fuzziness in space-time. Indeed, if we set L comparable to the size the

universe, we find that ∆x is actually on a scale you can see with your own eyes, of the order of 0.1mm, the width of

a hair. Space certainly doesn’t display any lack of sharpness on that scale when you look around.

These worries may be resolved by invoking entanglement. Space-time is the ultimate, universal entangled system.

Locality itself can emerge, via entanglement, as an approximate behavior on large scales.

Information is not localized in space, but resides in non localized correlations. The density of information can

depend on scale, and can be smaller for larger systems. The effective fidelity of space-time can change depending

on where something is relative to an observer. A measurement confined to a small volume does not know or care

about a transverse geometrical displacement relative to some distant place, so the uncertainty is not observable in

local measurements.

In quantum mechanics, measurements make projections— in Copenhagen language, they “collapse” the wave func-

tion. Until they are made, there is uncertainty given by the width of the wave function— in our case, the scale-

dependent blurring. In an emergent space-time, every world-line defines a particular projection of the wave function

associated with the structure of nested light cones (or “nested causal diamonds”) around it.

Thus, the quantum geometrical position information is entangled for bodies whose world-lines are close together.

If you measure the transverse position state of one massive body, you will find almost the same projection of the

geometrical state for any body nearby. That does not mean that any two bodies are in the same position; it only

means that their quantum deviation from the classical position is almost the same, relative to any arbitrary far away

point. The local relationships of the bodies in space are changed very little from standard quantum mechanics.

A classical space-time is the limiting case of a fully coherent system. The approach to the classical limit however

reveals slight departures from classical behavior that are not present in standard theory. Nonlocal projections of the

quantum state in different directions are slightly different, even on large scales.

As the system unfolds in time, the uncertainty leads to random variations— a new kind of noise in position

measurements to distant bodies in different directions. The positions of nearby bodies change together, carried along

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with the geometry, into the same new definite state. Local measurements are not affected by this collective change of

position— a new kind of “movement without motion”.

This interpretation of the angular uncertainty opens up a way to build an experiment that probes Planck scale

physics. The Planck broadcast model of quantum geometry predicts that positions fluctuate, with a power spectrum

of angular variations given approximately by the Planck time— that is, in an average over duration τ , the mean

square variation is

〈∆θ2〉� ≈ tP /τ. (4)

In an experiment of size L, the variations accumulate up to durations τ ≈ L/c, ultimately leading to variance in

position given by the overall uncertainty, Eq. (3). This prediction can be tested by making very sensitive measurements

of transverse positions of massive bodies. The measurement process must make a nonlocal comparison of position in

different directions.

An experiment designed to detect or rule out fluctuations with these properties, called the Fermilab Holome-

ter, is currently being developed.[12] It uses a technique based on laser interferometers like those used to measure

gravitational waves. The intensity of light emerging from a Michelson interferometer allows a precise and coherent

measurement of the positions of mirrors over an extended region of space, in this case, 40 meters in two directions.

The precision of such devices is extraordinary; they can detect variations in mean position differences on the order of

attometers, limited primarily by the quantum character of the laser light. In the Holometer, correlations are measured

between the signals of two adjacent, aligned interferometers. The correlations are sensitive to tiny, random in-common

motions that change very quickly, on timescales comparable to a light-crossing time, less than a microsecond. (On

longer timescales, entanglement-driven locality reduces the variation.) The effective speed of the motionless movement

is tiny— comparable to continental drift, only centimeters per year.

Because of quantum entanglement, the holographic noise created by the Planck broadcast information limit creates

tiny, rapid fluctuations in signals from the two adjacent interferometers that are coherent with each other, even if

there is no connection between the devices apart from proximity. Arguments like those outlined here, based on

information in holographic emergent space, can be used to make an exact prediction for the expected cross-correlated

noise spectrum, even without knowing details of the fundamental theory[11, 13].

Whether or not new Planck scale holographic noise is detected, the Holometer is interesting as an exploratory

experiment, because it tests the fidelity and coherence of nonlocal spatial relationships with Planck precision for the

first time. The outcome will either reveal a signature of new Planck scale physics, or experimentally prove a coherence

of macroscopic space greater than what is possible with a Planck broadcast. We don’t know what we will find.