COPYRIGHT 2004 SCIENTIFIC AMERICAN, INC

COPYRIGHT 2004 SCIENTIFIC AMERICAN, INC.

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TABLE OF CONTENTS

ScientificAmerican.comexclusive online issue no. 12E X T R E M E P H Y S I C S

Time travel, teleportation, parallel universes—in certain sectors of the physics community,notions once relegated to the realm of science fiction are now considered quite plausible. Indeed,by some accounts, the truth may be stranger than fiction. Consider the possibility that the uni-verse is a huge hologram or that matter is composed of tiny, vibrating strings. Perhaps space andtime are not continuous but instead come in discrete pieces. These are the wonderfully weirdways in which theorists are beginning to conceive of the world (or worlds!) around us.

In this exclusive online issue, leading authorities share their expertise on these cutting-edgeideas. Brian Greene untangles string theory; Max Tegmark reveals how astronomical observa-tions support the existence of parallel universes; other scholars tackle quantum teleportation,negative energy, the holographic principle and loop quantum gravity; and Gordon Kane ushers inthe dawn of physics beyond the Standard Model. —The Editors

Negative Energy, Wormholes and Warp Drive BY LAWRENCE H. FORD AND THOMAS A. ROMAN; SCIENTIFIC AMERICAN, JANUARY 2000The construction of wormholes and warp drive would require a very unusual form of energy. Unfortunately, the same laws of physics that allow the existence of this "negative energy" also appear to limit its behavior

Quantum Teleportation BY ANTON ZEILINGER; SCIENTIFIC AMERICAN, APRIL 2000The science-fiction dream of "beaming" objects from place to place is now a reality—at least for particles of light

Parallel Universes BY MAX TEGMARK; SCIENTIFIC AMERICAN, MAY 2003Not just a staple of science fiction, other universes are a direct implication of cosmological observations

Information in the Holographic Universe BY JACOB D. BEKENSTEIN; SCIENTIFIC AMERICAN, AUGUST 2003Theoretical results about black holes suggest that the universe could be like a gigantic hologram

The Future of String Theory: A Conversation with Brian Greene BY GEORGE MUSSER; SCIENTIFIC AMERICAN, NOVEMBER 2003The physicist and best-selling author demystifies the ultimate theories of space and time, the nature of genius, multiple universes, and more

Atoms of Space and Time BY LEE SMOLIN; SCIENTIFIC AMERICAN, JANUARY 2004We perceive space and time to be continuous, but if the amazing theory of loop quantum gravity is correct, they actually come in discrete pieces

The Dawn of Physics beyond the Standard Model BY GORDON KANE; SCIENTIFIC AMERICAN, JUNE 2003The Standard Model of particle physics is at a pivotal moment in its history: it is both at the height of its success and on the verge of being surpassed

1 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE FEBRUARY 2004COPYRIGHT 2004 SCIENTIFIC AMERICAN, INC.

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Can a region of space contain less than nothing?Common sense would say no; the most one coulddo is remove all matter and radiation and be left

with vacuum. But quantum physics has a proven ability toconfound intuition, and this case is no exception. A region ofspace, it turns out, can contain less than nothing. Its energyper unit volume—the energy density—can be less than zero.

Needless to say, the implications are bizarre. According toEinstein’s theory of gravity, general relativity, the presence ofmatter and energy warps the geometric fabric of space andtime. What we perceive as gravity is the space-time distortionproduced by normal, positive energy or mass. But when nega-tive energy or mass—so-called exotic matter—bends space-time, all sorts of amazing phenomena might become possible:traversable wormholes, which could act as tunnels to other-wise distant parts of the universe; warp drive, which would al-low for faster-than-light travel; and time machines, whichmight permit journeys into the past. Negative energy couldeven be used to make perpetual-motion machines or to de-stroy black holes. A Star Trek episode could not ask for more.

For physicists, these ramifications set off alarm bells. Thepotential paradoxes of backward time travel—such as killingyour grandfather before your father is conceived—have longbeen explored in science fiction, and the other consequencesof exotic matter are also problematic. They raise a questionof fundamental importance: Do the laws of physics that per-mit negative energy place any limits on its behavior? We andothers have discovered that nature imposes stringent con-

straints on the magnitude and duration of negative energy,which (unfortunately, some would say) appear to render theconstruction of wormholes and warp drives very unlikely.

Double Negative

Before proceeding further, we should draw the reader’s at-tention to what negative energy is not. It should not be

confused with antimatter, which has positive energy. Whenan electron and its antiparticle, a positron, collide, they anni-hilate. The end products are gamma rays, which carry posi-tive energy. If antiparticles were composed of negative ener-gy, such an interaction would result in a final energy of zero.One should also not confuse negative energy with the energyassociated with the cosmological constant, postulated in in-flationary models of the universe [see “Cosmological Anti-gravity,” by Lawrence M. Krauss; Scientific American,January 1999]. Such a constant represents negative pressurebut positive energy. (Some authors call this exotic matter; wereserve the term for negative energy densities.)

The concept of negative energy is not pure fantasy; some ofits effects have even been produced in the laboratory. They arisefrom Heisenberg’s uncertainty principle, which requires that theenergy density of any electric, magnetic or other field fluctuaterandomly. Even when the energy density is zero on average, asin a vacuum, it fluctuates. Thus, the quantum vacuum can nev-er remain empty in the classical sense of the term; it is a roilingsea of “virtual” particles spontaneously popping in and out of

The construction of wormholes and warp drive would require a very unusual form of energy. Unfortunately, the same laws ofphysics that allow the existence of this “negative energy” also appear to limit its behavior

by Lawrence H. Ford and Thomas A. Roman

Negative Energy,Wormholesand Warp Drive

2 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE FEBRUARY 2004

originally published January 2000


existence [see “Exploiting Zero-Point Energy,” by Philip Yam;Scientific American, December 1997]. In quantum theory,the usual notion of zero energy corresponds to the vacuum withall these fluctuations. So if one can somehow contrive to damp-en the undulations, the vacuum will have less energy than itnormally does—that is, less than zero energy.

As an example, researchers in quantum optics have createdspecial states of fields in which destructive quantum interfer-ence suppresses the vacuum fluctuations. These so-calledsqueezed vacuum states involve negative energy. More pre-cisely, they are associated with regions of alternating positiveand negative energy. The total energy averaged over all spaceremains positive; squeezing the vacuum creates negative en-ergy in one place at the price of extra positive energy else-where. A typical experiment involves laser beams passingthrough nonlinear optical materials [see “Squeezed Light,”by Richart E. Slusher and Bernard Yurke; Scientific Amer-ican, May 1988]. The intense laser light induces the materialto create pairs of light quanta, photons. These photons alter-nately enhance and suppress the vacuum fluctuations, lead-ing to regions of positive and negative energy, respectively.

Another method for producing negative energy introducesgeometric boundaries into a space. In 1948 Dutch physicistHendrik B. G. Casimir showed that two uncharged parallelmetal plates alter the vacuum fluctuations in such a way as toattract each other. The energy density between the plates waslater calculated to be negative. In effect, the plates reduce thefluctuations in the gap between them; this creates negative

energy and pressure, which pulls the plates together. The nar-rower the gap, the more negative the energy and pressure,and the stronger is the attractive force. The Casimir effect hasrecently been measured by Steve K. Lamoreaux of Los Alam-os National Laboratory and by Umar Mohideen of the Uni-versity of California at Riverside and his colleague AnushreeRoy. Similarly, in the 1970s Paul C. W. Davies and Stephen A.Fulling, then at King’s College at the University of London,predicted that a moving boundary, such as a moving mirror,could produce a flux of negative energy.

For both the Casimir effect and squeezed states, researchershave measured only the indirect effects of negative energy.Direct detection is more difficult but might be possible usingatomic spins, as Peter G. Grove, then at the British Home Of-fice, Adrian C. Ottewill, then at the University of Oxford,and one of us (Ford) suggested in 1992.

Gravity and Levity

The concept of negative energy arises in several areas ofmodern physics. It has an intimate link with black holes,

those mysterious objects whose gravitational field is sostrong that nothing can escape from within their boundary,the event horizon. In 1974 Stephen W. Hawking of the Uni-versity of Cambridge made his famous prediction that blackholes evaporate by emitting radiation [see “The QuantumMechanics of Black Holes,” by Stephen W. Hawking; Scien-tific American, January 1977]. A black hole radiates ener-

WORMHOLE acts as a tunnel between two different locationsin space. Light rays from A to B can enter one mouth of thewormhole, pass through the throat and exit at the other mouth—a journey that would take much longer if they had to go thelong way around. At the throat must be negative energy (blue),

whose gravitational field allows converging light rays to begindiverging. (This diagram is a two-dimensional representationof three-dimensional space. The mouths and throat of thewormhole are actually spheres.) Although not shown here, awormhole could also connect two different points in time.

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gy at a rate inversely proportional to the square of its mass.Although the evaporation rate is large only for subatomic-size black holes, it provides a crucial link between the laws ofblack holes and the laws of thermodynamics. The Hawkingradiation allows black holes to come into thermal equilibri-um with their environment.

At first glance, evaporation leads to a contradiction. Thehorizon is a one-way street; energy can only flow inward. Sohow can a black hole radiate energy outward? Because energymust be conserved, the production of positive energy—whichdistant observers see as the Hawking radiation—is accompa-nied by a flow of negative energy into the hole. Here the nega-tive energy is produced by the extreme space-time curvaturenear the hole, which disturbs the vacuum fluctuations. In thisway, negative energy is required for the consistency of the uni-fication of black hole physics with thermodynamics.

The black hole is not the only curved region of space-timewhere negative energy seems to play a role. Another is thewormhole—a hypothesized type of tunnel that connects oneregion of space and time to another. Physicists used to thinkthat wormholes exist only on the very finest length scales, bub-bling in and out of existence like virtual particles [see “Quan-tum Gravity,” by Bryce S. DeWitt; Scientific American, De-

cember 1983]. In the early1960s physicists Robert Fullerand John A. Wheeler showedthat larger wormholes wouldcollapse under their own gravi-ty so rapidly that even a beamof light would not have enoughtime to travel through them.

But in the late 1980s variousresearchers—notably MichaelS. Morris and Kip S. Thorne ofthe California Institute of Tech-nology and Matt Visser ofWashington University—foundotherwise. Certain wormholescould in fact be made largeenough for a person or space-ship. Someone might enter themouth of a wormhole stationedon Earth, walk a short distanceinside the wormhole and exitthe other mouth in, say, the An-dromeda galaxy. The catch isthat traversable wormholes re-quire negative energy. Becausenegative energy is gravitational-ly repulsive, it would preventthe wormhole from collapsing.

For a wormhole to betraversable, it ought to (at bareminimum) allow signals, in theform of light rays, to passthrough it. Light rays enteringone mouth of a wormhole areconverging, but to emerge fromthe other mouth, they must de-focus—in other words, theymust go from converging to di-verging somewhere in between[see illustration on page 3].

This defocusing requires negative energy. Whereas the curva-ture of space produced by the attractive gravitational field ofordinary matter acts like a converging lens, negative energyacts like a diverging lens.

No Dilithium Needed

Such space-time contortions would enable another stapleof science fiction as well: faster-than-light travel. In 1994

Miguel Alcubierre Moya, then at the University of Wales atCardiff, discovered a solution to Einstein’s equations that hasmany of the desired features of warp drive. It describes aspace-time bubble that transports a starship at arbitrarilyhigh speeds relative to observers outside the bubble. Calcula-tions show that negative energy is required.

Warp drive might appear to violate Einstein’s special theo-ry of relativity. But special relativity says that you cannot out-run a light signal in a fair race in which you and the signalfollow the same route. When space-time is warped, it mightbe possible to beat a light signal by taking a different route, ashortcut. The contraction of space-time in front of the bubbleand the expansion behind it create such a shortcut [see illus-tration above].

SPACE-TIME BUBBLE is the closest that modern physics comes to the “warp drive” of sci-ence fiction. It can convey a starship at arbitrarily high speeds. Space-time contracts at the frontof the bubble, reducing the distance to the destination, and expands at its rear, increasing the dis-tance from the origin (arrows). The ship itself stands still relative to the space immediatelyaround it; crew members do not experience any acceleration. Negative energy (blue) isrequired on the sides of the bubble.

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One problem with Alcubierre’s original model, pointed outby Sergei V. Krasnikov of the Central Astronomical Observa-tory at Pulkovo near St. Petersburg, is that the interior of thewarp bubble is causally disconnected from its forward edge. Astarship captain on the inside cannot steer the bubble or turn iton or off; some external agency must set it up ahead of time.To get around this problem, Krasnikov proposed a “superlu-minal subway,” a tube of modified space-time (not the same asa wormhole) connecting Earth and a distant star. Within thetube, superluminal travel in one direction is possible. Duringthe outbound journey at sublight speed, a spaceship crewwould create such a tube. On the return journey, they couldtravel through it at warp speed. Like warp bubbles, the sub-way involves negative energy. It has since been shown by KenD. Olum of Tufts University and by Visser, together with BruceBassett of Oxford and Stefano Liberati of the InternationalSchool for Advanced Studies in Trieste, that any scheme forfaster-than-light travel requires the use of negative energy.

If one can construct wormholes or warp drives, time travelmight become possible. The passage of time is relative; it de-pends on the observer’s velocity. A person who leaves Earthin a spaceship, travels at near lightspeed and returns willhave aged less than someone who remains on Earth. If thetraveler manages to outrun a light ray, perhaps by taking ashortcut through a wormhole or a warp bubble, he may re-turn before he left. Morris, Thorne and Ulvi Yurtsever, thenat Caltech, proposed a wormhole time machine in 1988, andtheir paper has stimulated much research on time travel overthe past decade. In 1992 Hawking proved that any construc-tion of a time machine in a finite region of space-time inher-ently requires negative energy.

Negative energy is so strange that one might think it mustviolate some law of physics. Before and after the creation ofequal amounts of negative and positive energy in previouslyempty space, the total energy is zero, so the law of conserva-tion of energy is obeyed. But there are many phenomena thatconserve energy yet never occur in the real world. A brokenglass does not reassemble itself, and heat does not sponta-neously flow from a colder to a hotter body. Such effects areforbidden by the second law of thermodynamics. This gener-al principle states that the degree of disorder of a system—itsentropy—cannot decrease on its own without an input of en-ergy. Thus, a refrigerator, which pumps heat from its cold in-terior to the warmer outside room, requires an external pow-er source. Similarly, the second law also forbids the completeconversion of heat into work.

Negative energy potentially conflicts with the second law.Imagine an exotic laser, which creates a steady outgoing beamof negative energy. Conservation of energy requires that a by-product be a steady stream of positive energy. One could di-rect the negative energy beam off to some distant corner ofthe universe, while employing the positive energy to performuseful work. This seemingly inexhaustible energy supplycould be used to make a perpetual-motion machine and there-by violate the second law. If the beam were directed at a glassof water, it could cool the water while using the extracted pos-

VIEW FROM THE BRIDGE of a faster-than-light starship asit heads in the direction of the Little Dipper (above) looksnothing like the star streaks typically depicted in science fic-tion. As the velocity increases (right), stars ahead of the ship(left column) appear ever closer to the direction of motion andturn bluer in color. Behind the ship (right column), stars shiftcloser to a position directly astern, redden and eventually dis-appear from view altogether. The light from stars directlyoverhead or underneath remains unaffected.

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itive energy to power a small motor—providing a refrigeratorwith no need for external power. These problems arise notfrom the existence of negative energy per se but from the un-restricted separation of negative and positive energy.

Unfettered negative energy would also have profound con-sequences for black holes. When a black hole forms by thecollapse of a dying star, general relativity predicts the forma-tion of a singularity, a region where the gravitational field be-comes infinitely strong. At this point, general relativity—andindeed all known laws of physics—are unable to say whathappens next. This inability is a profound failure of the cur-rent mathematical description of nature. So long as the singu-larity is hidden within an event horizon, however, the damageis limited. The description of nature everywhere outside of thehorizon is unaffected. For this reason, Roger Penrose of Ox-ford proposed the cosmic censorship hypothesis: there can beno naked singularities, which are unshielded by event horizons.

For special types of charged or rotating black holes—known as extreme black holes—even a small increase incharge or spin, or a decrease in mass, could in principle destroythe horizon and convert the hole into a naked singularity. At-tempts to charge up or spin up these black holes using ordi-nary matter seem to fail for a variety of reasons. One mightinstead envision producing a decrease in mass by shining abeam of negative energy down the hole, without altering itscharge or spin, thus subverting cosmic censorship. One mightcreate such a beam, for example, using a moving mirror. Inprinciple, it would require only a tiny amount of negative en-ergy to produce a dramatic change in the state of an extremeblack hole. Therefore, this might be the scenario in which neg-ative energy is the most likely to produce macroscopic effects.

Not Separate and Not Equal

Fortunately (or not, depending on your point of view), al-though quantum theory allows the existence of negative

energy, it also appears to place strong restrictions—known asquantum inequalities—on its magnitude and duration. Theseinequalities were first suggested by Ford in 1978. Over thepast decade they have been proved and refined by us andothers, including Éanna E. Flanagan of Cornell University,Michael J. Pfenning, then at Tufts, Christopher J. Fewsterand Simon P. Eveson of the University of York, and EdwardTeo of the National University of Singapore.

The inequalities bear some resemblance to the uncertaintyprinciple. They say that a beam of negative energy cannot bearbitrarily intense for an arbitrarily long time. The permissiblemagnitude of the negative energy is inversely related to its tem-poral or spatial extent. An intense pulse of negative energy canlast for a short time; a weak pulse can last longer. Furthermore,an initial negative energy pulse must be followed by a largerpulse of positive energy. The larger the magnitude of the nega-tive energy, the nearer must be its positive energy counterpart.These restrictions are independent of the details of how thenegative energy is produced. One can think of negative energyas an energy loan. Just as a debt is negative money that has tobe repaid, negative energy is an energy deficit. As we will dis-cuss below, the analogy goes even further.

In the Casimir effect, the negative energy density betweenthe plates can persist indefinitely, but large negative energydensities require a very small plate separation. The magnitudeof the negative energy density is inversely proportional to thefourth power of the plate separation. Just as a pulse with a

very negative energy density is limited in time, very negativeCasimir energy density must be confined between closelyspaced plates. According to the quantum inequalities, the ener-gy density in the gap can be made more negative than theCasimir value, but only temporarily. In effect, the more onetries to depress the energy density below the Casimir value, theshorter the time over which this situation can be maintained.

When applied to wormholes and warp drives, the quantuminequalities typically imply that such structures must either belimited to submicroscopic sizes, or if they are macroscopic thenegative energy must be confined to incredibly thin bands. In1996 we showed that a submicroscopic wormhole wouldhave a throat radius of no more than about 10–32 meter. Thisis only slightly larger than the Planck length, 10–35 meter, thesmallest distance that has definite meaning. We found that it ispossible to have models of wormholes of macroscopic size butonly at the price of confining the negative energy to an ex-tremely thin band around the throat. For example, in onemodel a throat radius of 1 meter requires the negative energyto be a band no thicker than 10–21 meter, a millionth the sizeof a proton. Visser has estimated that the negative energy re-quired for this size of wormhole has a magnitude equivalentto the total energy generated by 10 billion stars in one year.The situation does not improve much for larger wormholes.For the same model, the maximum allowed thickness of thenegative energy band is proportional to the cube root of thethroat radius. Even if the throat radius is increased to a size ofone light-year, the negative energy must still be confined to aregion smaller than a proton radius, and the total amount re-quired increases linearly with the throat size.

It seems that wormhole engineers face daunting problems.They must find a mechanism for confining large amounts ofnegative energy to extremely thin volumes. So-called cosmicstrings, hypothesized in some cosmological theories, involvevery large energy densities in long, narrow lines. But allknown physically reasonable cosmic-string models have pos-itive energy densities.

Warp drives are even more tightly constrained, as shown byPfenning and Allen Everett of Tufts, working with us. In Alcu-bierre’s model, a warp bubble traveling at 10 times lightspeed(warp factor 2, in the parlance of Star Trek: The Next Genera-tion) must have a wall thickness of no more than 10–32 meter.A bubble large enough to enclose a starship 200 meters acrosswould require a total amount of negative energy equal to 10billion times the mass of the observable universe. Similar con-straints apply to Krasnikov’s superluminal subway. A modifi-cation of Alcubierre’s model was recently constructed by ChrisVan Den Broeck of the Catholic University of Louvain in Bel-gium. It requires much less negative energy but places the star-ship in a curved space-time bottle whose neck is about 10–32

meter across, a difficult feat. These results would seem to make itrather unlikely that one could construct wormholes and warpdrives using negative energy generated by quantum effects.

Cosmic Flashing and Quantum Interest

The quantum inequalities prevent violations of the secondlaw. If one tries to use a pulse of negative energy to cool

a hot object, it will be quickly followed by a larger pulse ofpositive energy, which reheats the object. A weak pulse ofnegative energy could remain separated from its positivecounterpart for a longer time, but its effects would be indis-tinguishable from normal thermal fluctuations. Attempts to


capture or split off negative energy from positive energy alsoappear to fail. One might intercept an energy beam, say, byusing a box with a shutter. By closing the shutter, one mighthope to trap a pulse of negative energy before the offsettingpositive energy arrives. But the very act of closing the shuttercreates an energy flux that cancels out the negative energy itwas designed to trap [see illustration at right].

We have shown that there are similar restrictions on viola-tions of cosmic censorship. A pulse of negative energy inject-ed into a charged black hole might momentarily destroy thehorizon, exposing the singularity within. But the pulse mustbe followed by a pulse of positive energy, which would con-vert the naked singularity back into a black hole—a scenariowe have dubbed cosmic flashing. The best chance to observecosmic flashing would be to maximize the time separationbetween the negative and positive energy, allowing the nakedsingularity to last as long as possible. But then the magnitudeof the negative energy pulse would have to be very small, ac-cording to the quantum inequalities. The change in the massof the black hole caused by the negative energy pulse will getwashed out by the normal quantum fluctuations in the hole’smass, which are a natural consequence of the uncertaintyprinciple. The view of the naked singularity would thus beblurred, so a distant observer could not unambiguously veri-fy that cosmic censorship had been violated.

Recently we, and also Frans Pretorius, then at the Universi-ty of Victoria, and Fewster and Teo, have shown that thequantum inequalities lead to even stronger bounds on nega-tive energy. The positive pulse that necessarily follows an ini-tial negative pulse must do more than compensate for the neg-ative pulse; it must overcompensate. The amount of overcom-pensation increases with the time interval between the pulses.Therefore, the negative and positive pulses can never be madeto exactly cancel each other. The positive energy must alwaysdominate—an effect known as quantum interest. If negativeenergy is thought of as an energy loan, the loan must berepaid with interest. The longer the loan period or the largerthe loan amount, the greater is the interest. Furthermore,the larger the loan, the smaller is the maximum allowed loanperiod. Nature is a shrewd banker and always calls in itsdebts.

The concept of negative energy touches on many areas ofphysics: gravitation, quantum theory, thermodynamics.

The interweaving of so many different parts of physicsillustrates the tight logical structure of the laws of nature.On the one hand, negative energy seems to be required toreconcile black holes with thermodynamics. On the other,quantum physics prevents unrestricted production of negativeenergy, which would violate the second law of thermodynam-ics. Whether these restrictions are also features of somedeeper underlying theory, such as quantum gravity, remains tobe seen. Nature no doubt has more surprises in store.

The Authors

LAWRENCE H. FORD and THOMAS A. ROMANhave collaborated on negative energy issues for over adecade. Ford received his Ph.D. from Princeton Univer-sity in 1974, working under John Wheeler, one of thefounders of black hole physics. He is now a professor ofphysics at Tufts University and works on problems inboth general relativity and quantum theory, with a spe-cial interest in quantum fluctuations. His other pursuitsinclude hiking in the New England woods and gather-ing wild mushrooms. Roman received his Ph.D. in1981 from Syracuse University under Peter Bergmann,who collaborated with Albert Einstein on unified fieldtheory. Roman has been a frequent visitor at the TuftsInstitute of Cosmology during the past 10 years and iscurrently a professor of physics at Central ConnecticutState University. His interests include the implicationsof negative energy for a quantum theory of gravity. Hetends to avoid wild mushrooms.

Further Information

Black Holes and Time Warps: Einstein’s Outrageous Legacy. Kip S.Thorne. W. W. Norton, 1994.

Lorentzian Wormholes: From Einstein to Hawking. Matt Visser. Amer-ican Institute of Physics Press, 1996.

Quantum Field Theory Constrains Traversable Wormhole Geome-tries. L. H. Ford and T. A. Roman in Physical Review D, Vol. 53, No. 10,pages 5496–5507; May 15, 1996. Available at xxx.lanl.gov/abs/gr-qc/9510071on the World Wide Web.

The Unphysical Nature of Warp Drive. M. J. Pfenning and L. H. Ford inClassical and Quantum Gravity, Vol. 14, No. 7, pages 1743–1751; July 1997.Available at xxx.lanl.gov/abs/gr-qc/9702026 on the World Wide Web.

Paradox Lost. Paul Davies in New Scientist, Vol. 157, No. 2126, page 26;March 21, 1998.

Time Machines: Time Travel in Physics, Metaphysics, and Science Fic-tion. Second edition. Paul J. Nahin. AIP Press, Springer-Verlag, 1999.

The Quantum Interest Conjecture. L. H. Ford and T. A. Roman in Physi-cal Review D, Vol. 60, No. 10, Article No. 104018 (8 pages); November 15,1999. Available at xxx.lanl.gov/abs/gr-qc/9901074 on the World Wide Web.

ATTEMPT TO CIRCUMVENT the quantum laws that governnegative energy inevitably ends in disappointment. The experi-menter intends to detach a negative energy pulse from its com-pensating positive energy pulse. As the pulses approach a box(a), the experimenter tries to isolate the negative one by closingthe lid after it has entered (b). Yet the very act of closing the lidcreates a second positive energy pulse inside the box (c).

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The scene is a familiar one fromscience-fiction movies and TV:an intrepid band of explorers

enters a special chamber; lights pulse,sound effects warble, and our heroesshimmer out of existence to reappear onthe surface of a faraway planet. This isthe dream of teleportation—the abilityto travel from place to place withouthaving to pass through the tedious in-tervening miles accompanied by a phys-ical vehicle and airline-food rations. Al-though the teleportation of large objectsor humans still remains a fantasy, quan-tum teleportation has become a labora-tory reality for photons, the individualparticles of light.

Quantum teleportation exploits someof the most basic (and peculiar) featuresof quantum mechanics, a branch ofphysics invented in the first quarter of the20th century to explain processes thatoccur at the level of individual atoms.From the beginning, theorists realizedthat quantum physics led to a plethoraof new phenomena, some of which defycommon sense. Technological progressin the final quarter of the 20th centuryhas enabled researchers to conduct manyexperiments that not only demonstratefundamental, sometimes bizarre aspectsof quantum mechanics but, as in the caseof quantum teleportation, apply themto achieve previously inconceivable feats.

In science-fiction stories, teleportationoften permits travel that is instanta-neous, violating the speed limit set downby Albert Einstein, who concluded from

his theory of relativity that nothing cantravel faster than light [see “Faster ThanLight?” by Raymond Y. Chiao, Paul G.Kwiat and Aephraim M. Steinberg; Sci-entific American, August 1993]. Tele-portation is also less cumbersome thanthe more ordinary means of space trav-el. It is said that Gene Roddenberry, thecreator of Star Trek, conceived of the“transporter beam” as a way to save theexpense of simulating landings andtakeoffs on strange planets.

The procedure for teleportation in sci-ence fiction varies from story to storybut generally goes as follows: A devicescans the original object to extract allthe information needed to describe it. Atransmitter sends the information to thereceiving station, where it is used to ob-tain an exact replica of the original. Insome cases, the material that made upthe original is also transported to the re-ceiving station, perhaps as “energy” ofsome kind; in other cases, the replica ismade of atoms and molecules that werealready present at the receiving station.

Quantum mechanics seems to makesuch a teleportation scheme impossible inprinciple. Heisenberg’s uncertainty prin-ciple rules that one cannot know boththe precise position of an object and itsmomentum at the same time. Thus, onecannot perform a perfect scan of the ob-ject to be teleported; the location or ve-locity of every atom and electron wouldbe subject to errors. Heisenberg’s uncer-tainty principle also applies to other pairsof quantities, making it impossible to

measure the exact, total quantum state ofany object with certainty. Yet such mea-surements would be necessary to obtainall the information needed to describethe original exactly. (In Star Trek the“Heisenberg Compensator” somehowmiraculously overcomes that difficulty.)

A team of physicists overturned thisconventional wisdom in 1993, whenthey discovered a way to use quantummechanics itself for teleportation. Theteam—Charles H. Bennett of IBM;Gilles Brassard, Claude Crépeau andRichard Josza of the University of Mon-treal; Asher Peres of Technion–Israel In-stitute of Technology; and William K.Wootters of Williams College—foundthat a peculiar but fundamental featureof quantum mechanics, entanglement,can be used to circumvent the limita-tions imposed by Heisenberg’s uncer-tainty principle without violating it.

Entanglement

It is the year 2100. A friend who likesto dabble in physics and party tricks

has brought you a collection of pairs ofdice. He lets you roll them once, one pairat a time. You handle the first pair gin-gerly, remembering the fiasco with themicro–black hole last Christmas. Finally,you roll the two dice and get double 3.You roll the next pair. Double 6. Thenext: double 1. They always match.

The dice in this fable are behaving as ifthey were quantum entangled particles.Each die on its own is random and fair,

QUANTUM

by Anton Zeilinger

The science-fiction dream of “beaming” objects from place to place is now a reality—at least for particles of light


originally published April 2000


but its entangled partner somehow al-ways gives the correct matching out-come. Such behavior has been demon-strated and intensively studied with realentangled particles. In typical experi-ments, pairs of atoms, ions or photonsstand in for the dice, and properties suchas polarization stand in for the differentfaces of a die.

Consider the case of two photonswhose polarizations are entangled to berandom but identical. Beams of light andeven individual photons consist of oscil-lations of electromagnetic fields, and po-larization refers to the alignment of theelectric field oscillations [see illustrationabove]. Suppose that Alice has one ofthe entangled photons and Bob has itspartner. When Alice measures her pho-ton to see if it is horizontally or verticallypolarized, each outcome has a 50 per-cent chance. Bob’s photon has the sameprobabilities, but the entanglement en-sures that he will get exactly the same re-

sult as Alice. As soon as Alice gets the re-sult “horizontal,” say, she knows thatBob’s photon will also be horizontallypolarized. Before Alice’s measurementthe two photons do not have individualpolarizations; the entangled state speci-fies only that a measurement will findthat the two polarizations are equal.

An amazing aspect of this process isthat it doesn’t matter if Alice and Bob arefar away from each other; the processworks so long as their photons’ entangle-ment has been preserved. Even if Alice ison Alpha Centauri and Bob on Earth,their results will agree when they com-pare them. In every case, it is as if Bob’sphoton is magically influenced by Alice’sdistant measurement, and vice versa.

You might wonder if we can explainthe entanglement by imagining that eachparticle carries within it some recordedinstructions. Perhaps when we entanglethe two particles, we synchronize somehidden mechanism within them that de-

termines what results they will give whenthey are measured. This would explainaway the mysterious effect of Alice’smeasurement on Bob’s particle. In the1960s, however, Irish physicist John Bellproved a theorem that in certain situa-tions any such “hidden variables” expla-nation of quantum entanglement wouldhave to produce results different fromthose predicted by standard quantummechanics. Experiments have confirmedthe predictions of quantum mechanicsto a very high accuracy.

Austrian physicist Erwin Schrödinger,one of the co-inventors of quantum me-chanics, called entanglement “the essen-tial feature” of quantum physics. Entan-glement is often called the EPR effect andthe particles EPR pairs, after Einstein,Boris Podolsky and Nathan Rosen, whoin 1935 analyzed the effects of entangle-ment acting across large distances. Ein-stein talked of it as “spooky action at adistance.” If one tried to explain the re-

UNPOLARIZED LIGHT

a b

VERTICALPOLARIZING FILTER

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CRYSTAL SPLITSVERTICAL ANDHORIZONTAL

POLARIZATIONS

CALCITECRYSTAL

QUANTUM TELEPORTATION OF A PERSON (impossible in prac-tice but a good example to aid the imagination) would beginwith the person inside a measurement chamber (left) along-

side an equal mass of auxiliary material (green).The auxiliarymatter has previously been quantum-entangled with itscounterpart, which is at the faraway receiving station (right).

PREPARING FOR QUANTUM TELEPORTATION . . .

UNPOLARIZED LIGHT consists of photons that are polarizedin all directions (a). In polarized light the photons’ electric-fieldoscillations (arrows) are all aligned. A calcite crystal (b) splits alight beam in two, sending photons that are polarized parallelwith its axis into one beam and those that are perpendicular

into the other. Intermediate angles go into a quantum superposi-tion of both beams. Each such photon can be detected in onebeam or the other, with probability depending on the angle. Be-cause probabilities are involved, we cannot measure the un-known polarization of a single photon with certainty.

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sults in terms of signals traveling betweenthe photons, the signals would have totravel faster than the speed of light. Nat-urally, many people have wondered ifthis effect could be used to transmit in-formation faster than the speed of light.

Unfortunately, the quantum rulesmake that impossible. Each local mea-surement on a photon, considered inisolation, produces a completely ran-dom result and so can carry no informa-tion from the distant location. It tellsyou nothing more than what the distantmeasurement result probabilities wouldbe, depending on what was measuredthere. Nevertheless, we can put entan-glement to work in an ingenious way to

achieve quantum teleportation.

Putting Entangled Photons to Work

Alice and Bob anticipate that they willwant to teleport a photon in the fu-

ture. In preparation, they share an en-tangled auxiliary pair of photons, Alicetaking photon A and Bob photon B. In-stead of measuring them, they eachstore their photon without disturbingthe delicate entangled state [see upper il-lustration on next page].

In due course, Alice has a third pho-ton—call it photon X—that she wantsto teleport to Bob. She does not knowwhat photon X’s state is, but she wants

Bob to have a photon with that samepolarization. She cannot simply mea-sure the photon’s polarization and sendBob the result. In general, her measure-ment result would not be identical to thephoton’s original state. This is Heisen-berg’s uncertainty principle at work.

Instead, to teleport photon X, Alicemeasures it jointly with photon A, with-out determining their individual polariza-tions. She might find, for instance, thattheir polarizations are “perpendicular”to each other (she still does not know theabsolute polarization of either one, how-ever). Technically, the joint measurementof photon A and photon X is called aBell-state measurement. Alice’s measure-

JOINT MEASUREMENT carried out on the auxiliary matter andthe person (left) changes them to a random quantum stateand produces a vast amount of random (but significant)

data—two bits per elementary state. By “spooky action at adistance,” the measurement also instantly alters the quantumstate of the faraway counterpart matter (right). MORE>>>

. . . A QUANTUM MEASUREMENT ...

LASER BEAM

CRYSTAL

ENTANGLED PHOTON PAIRS are created when a laserbeam passes through a crystal such as beta barium borate. Thecrystal occasionally converts a single ultraviolet photon into twophotons of lower energy, one polarized vertically (on red cone),one polarized horizontally (on blue cone). If the photons hap-

pen to travel along the cone intersections (green), neither pho-ton has a definite polarization, but their relative polarizationsare complementary; they are then entangled. Colorized image(at right) is a photograph of down-converted light. Colors donot represent the color of the light.

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ment produces a subtle effect: it changesBob’s photon to correlate with a combi-nation of her measurement result and thestate that photon X originally had. Infact, Bob’s photon now carries her pho-ton X’s state, either exactly or modifiedin a simple way.

To complete the teleportation, Alicemust send a message to Bob—one thattravels by conventional means, such as atelephone call or a note on a scrap of pa-per. After he receives this message, if nec-essary Bob can transform his photon B,with the end result that it becomes an ex-

act replica of the originalphoton X. Which transfor-mation Bob must apply de-pends on the outcome ofAlice’s measurement. Thereare four possibilities, corre-sponding to four quantumrelations between her pho-tons A and X. A typicaltransformation that Bobmust apply to his photon isto alter its polarization by90 degrees, which he cando by sending it through acrystal with the appropri-ate optical properties.

Which of the four possi-ble results Alice obtains iscompletely random and in-dependent of photon X’soriginal state. Bob thereforedoes not know how to pro-cess his photon until helearns the result of Alice’smeasurement. One can saythat Bob’s photon instanta-neously contains all the in-formation from Alice’s orig-inal, transported there byquantum mechanics. Yet to

know how to read that information,Bob must wait for the classical informa-tion, consisting of two bits that can trav-el no faster than the speed of light.

Skeptics might complain that the onlything teleported is the photon’s polariza-tion state or, more generally, its quantumstate, not the photon “itself.” But be-cause a photon’s quantum state is itsdefining characteristic, teleporting itsstate is completely equivalent to teleport-ing the particle [see box on page 14].

Note that quantum teleportationdoes not result in two copies of photon

X. Classical information can be copiedany number of times, but perfect copy-ing of quantum information is impossi-ble, a result known as the no-cloningtheorem, which was proved by Woot-ters and Wojciech H. Zurek of LosAlamos National Laboratory in 1982.(If we could clone a quantum state, wecould use the clones to violate Heisen-berg’s principle.) Alice’s measurementactually entangles her photon A withphoton X, and photon X loses all mem-ory, one might say, of its original state.As a member of an entangled pair, ithas no individual polarization state.Thus, the original state of photon Xdisappears from Alice’s domain.

Circumventing Heisenberg

Furthermore, photon X’s state hasbeen transferred to Bob with neither

Alice nor Bob learning anything aboutwhat the state is. Alice’s measurementresult, being entirely random, tells themnothing about the state. This is how theprocess circumvents Heisenberg’s prin-ciple, which stops us from determiningthe complete quantum state of a particlebut does not preclude teleporting thecomplete state so long as we do not tryto see what the state is!

Also, the teleported quantum infor-mation does not travel materially fromAlice to Bob. All that travels materiallyis the message about Alice’s measure-ment result, which tells Bob how toprocess his photon but carries no infor-mation about photon X’s state itself.

In one out of four cases, Alice is luckywith her measurement, and Bob’s pho-ton immediately becomes an identicalreplica of Alice’s original. It might seemas if information has traveled instantly

MEASUREMENT DATA must be sent to the distant receivingstation by conventional means.This process is limited by the

speed of light, making it impossible to teleport the personfaster than the speed of light.

A

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ENTANGLEDPARTICLESOURCE

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From: [email protected]: [email protected]

Re: PhotonMessage: Use number 3

1 2 3 4

ALICE

BOB

1 2 3 4

IDEAL QUANTUM TELEPORTATION relies onAlice, the sender, and Bob, the receiver, sharing a pairof entangled particles A and B (green). Alice has aparticle that is in an unknown quantum state X(blue). Alice performs a Bell-state measurement onparticles A and X, producing one of four possible out-comes. She tells Bob about the result by ordinarymeans. Depending on Alice’s result, Bob leaves hisparticle unaltered (1) or rotates it (2, 3, 4). Either wayit ends up a perfect replica of the original particle X.

... TRANSMISSION OF RANDOM DATA ...

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from Alice to Bob, beating Einstein’sspeed limit. Yet this strange feature can-not be used to send information, becauseBob has no way of knowing that hisphoton is already an identical replica.Only when he learns the result of Alice’sBell-state measurement, transmitted tohim via classical means, can he exploitthe information in the teleported quan-tum state. Suppose he tries to guess inwhich cases teleportation was instantlysuccessful. He will be wrong 75 percentof the time, and he will not know whichguesses were correct. If he uses the pho-

tons based on such guesses, the resultswill be the same as if he had taken abeam of photons with random polariza-tions. In this way, Einstein’s relativityprevails; even the spooky instantaneousaction at a distance of quantum mechan-ics fails to send usable information fasterthan the speed of light.

It would seem that the theoreticalproposal described above laid out aclear blueprint for building a teleporter;on the contrary, it presented a great ex-perimental challenge. Producing entan-gled pairs of photons has become rou-

tine in physics experiments in the pastdecade, but carrying out a Bell-statemeasurement on two independent pho-tons had never been done before.

Building a Teleporter

Apowerful way to produce entangledpairs of photons is spontaneous

parametric down-conversion: a singlephoton passing through a special crystalsometimes generates two new photonsthat are entangled so that they willshow opposite polarization when mea-sured [see top illustration on page 10].

A much more difficult problem is toentangle two independent photons thatalready exist, as must occur during theoperation of a Bell-state analyzer. Thismeans that the two photons (A and X)somehow have to lose their private fea-tures. In 1997 my group (Dik Bouw-meester, Jian-Wei Pan, Klaus Mattle,Manfred Eibl and Harald Weinfurter),then at the University of Innsbruck, ap-plied a solution to this problem in ourteleportation experiment [see illustra-tion at left].

In our experiment, a brief pulse of ul-traviolet light from a laser passes througha crystal and creates the entangled pho-tons A and B. One travels to Alice, andthe other goes to Bob. A mirror reflectsthe ultraviolet pulse back through thecrystal again, where it may create an-other pair of photons, C and D. (Thesewill also be entangled, but we don’t usetheir entanglement.) Photon C goes to adetector, which alerts us that its partnerD is available to be teleported. PhotonD passes through a polarizer, which wecan orient in any conceivable way. Theresulting polarized photon is our pho-ton X, the one to be teleported, and

RECEIVER RE-CREATES THE TRAVELER, exact down to thequantum state of every atom and molecule, by adjusting the

counterpart matter’s state according to the random measure-ment data sent from the scanning station.

... RECONSTRUCTION OF THE TRAVELERD

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ENTANGLEDPARTICLESOURCE

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INNSBRUCK EXPERIMENT begins with a short pulse of ultraviolet laser light.Traveling left to right through a crystal, this pulse produces the entangled pair of photonsA and B, which travel to Alice and Bob, respectively. Reflected back through the crystal,the pulse creates two more photons, C and D. A polarizer prepares photon D in a specif-ic state, X. Photon C is detected, confirming that photon X has been sent to Alice. Alicecombines photons A and X with a beam splitter [see illustration on next page]. If she de-tects one photon in each detector (as occurs at most 25 percent of the time), she notifiesBob, who uses a polarizing beam splitter to verify that his photon has acquired X’s po-larization, thus demonstrating successful teleportation.


travels on to Alice. Once it passesthrough the polarizer, X is an indepen-dent photon, no longer entangled. Andalthough we know its polarization be-cause of how we set the polarizer, Alicedoes not. We reuse the same ultravioletpulse in this way to ensure that Alicehas photons A and X at the same time.

Now we arrive at the problem of per-forming the Bell-state measurement. Todo this, Alice combines her two photons(A and X) using a semireflecting mirror,a device that reflects half of the incidentlight. An individual photon has a 50–50chance of passing through or being re-flected. In quantum terms, the photongoes into a superposition of these twopossibilities [see illustration at right].

Now suppose that two photons strikethe mirror from opposite sides, withtheir paths aligned so that the reflectedpath of one photon lies along the trans-mitted path of the other, and vice versa.A detector waits at the end of each path.Ordinarily the two photons would be re-flected independently, and there wouldbe a 50 percent chance of them arrivingin separate detectors. If the photons areindistinguishable and arrive at the mir-ror at the same instant, however, quan-tum interference takes place: some possi-bilities cancel out and do not occur,whereas others reinforce and occur moreoften. When the photons interfere, theyhave only a 25 percent likelihood of end-ing up in separate detectors. Further-more, when that occurs it correspondsto detecting one of the four possible Bellstates of the two photons—the case thatwe called “lucky” earlier. The other 75percent of the time the two photonsboth end up in one detector, which cor-responds to the other three Bell statesbut does not discriminate among them.

When Alice simultaneously detectsone photon in each detector, Bob’s pho-ton instantly becomes a replica of Alice’soriginal photon X. We verified that thisteleportation occurred by showing thatBob’s photon had the polarization thatwe imposed on photon X. Our experi-ment was not perfect, but the correct po-larization was detected 80 percent of thetime (random photons would achieve 50percent). We demonstrated the proce-dure with a variety of polarizations: ver-tical, horizontal, linear at 45 degrees andeven a nonlinear kind of polarizationcalled circular polarization.

The most difficult aspect of our Bell-state analyzer is making photons A andX indistinguishable. Even the timing ofwhen the photons arrive could be used

to identify which photon is which, so itis important to “erase” the time infor-mation carried by the particles. In ourexperiment, we used a clever trick firstsuggested by Marek Zukowski of theUniversity of Gdansk: we send the pho-tons through very narrow bandwidthwavelength filters. This process makesthe wavelength of the photons very pre-cise, and by Heisenberg’s uncertainty re-lation it smears out the photons in time.

A mind-boggling case arises when theteleported photon was itself entangledwith another and thus did not have itsown individual polarization. In 1998my Innsbruck group demonstrated thisscenario by giving Alice photon D with-out polarizing it, so that it was still en-tangled with photon C. We showed thatwhen the teleportation succeeded, Bob’sphoton B ended up entangled with C.Thus, the entanglement with C had beentransmitted from A to B.

Piggyback States

Our experiment clearly demonstrat-ed teleportation, but it had a low

rate of success. Because we could identi-fy just one Bell state, we could teleportAlice’s photon only 25 percent of thetime—the occasions when that state oc-curred. No complete Bell-state analyzerexists for independent photons or forany two independently created quan-tum particles, so at present there is noexperimentally proven way to improveour scheme’s efficiency to 100 percent.

In 1994 a way to circumvent thisproblem was proposed by Sandu Popes-cu, then at the University of Cambridge.He suggested that the state to be tele-

ported could be a quantum state ridingpiggyback on Alice’s auxiliary photon A.Francesco De Martini’s group at theUniversity of Rome I “La Sapienza” suc-cessfully demonstrated this scheme in1997. The auxiliary pair of photons wasentangled according to the photons’ lo-cations: photon A was split, as by abeam splitter, and sent to two differentparts of Alice’s apparatus, with the twoalternatives linked by entanglement to asimilar splitting of Bob’s photon B. Thestate to be teleported was also carried byAlice’s photon A—its polarization state.With both roles played by one photon,detecting all four possible Bell states be-comes a standard single-particle mea-surement: detect Alice’s photon in oneof two possible locations with one oftwo possible polarizations. The draw-back of the scheme is that if Alice weregiven a separate unknown state X to beteleported she would somehow have totransfer the state onto the polarizationof her photon A, which no one knowshow to do in practice.

Polarization of a photon, the featureemployed by the Innsbruck and theRome experiments, is a discrete quanti-ty, in that any polarization state can beexpressed as a superposition of just twodiscrete states, such as vertical and hori-zontal polarization. The electromagnet-ic field associated with light also hascontinuous features that amount to su-perpositions of an infinite number ofbasic states. For example, a light beamcan be “squeezed,” meaning that one ofits properties is made extremely preciseor noise-free, at the expense of greaterrandomness in another property (à laHeisenberg). In 1998 Jeffrey Kimble’s

PHOTONBEAM SPLITTER

(SEMIREFLECTINGMIRROR)

DETECTOR

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BEAM SPLITTER, or semireflecting mirror (a), reflects half the light that hits it andtransmits the other half. An individual photon has a 50–50 chance of reflection or trans-mission. If two identical photons strike the beam splitter at the same time, one from eachside (b), the reflected and transmitted parts interfere, and the photons lose their individu-al identities. We will detect one photon in each detector 25 percent of the time, and it isthen impossible to say if both photons were reflected or both were transmitted. Only therelative property—that they went to different detectors—is measured.


group at the California Institute of Tech-nology teleported such a squeezed statefrom one beam of light to another, thusdemonstrating teleportation of a contin-uous feature.

Remarkable as all these experimentsare, they are a far cry from quantumteleportation of large objects. There aretwo essential problems: First, one needsan entangled pair of the same kind ofobjects. Second, the object to be tele-ported and the entangled pairs must besufficiently isolated from the environ-ment. If any information leaks to orfrom the environment through stray in-teractions, the objects’ quantum statesdegrade, a process called decoherence. Itis hard to imagine how we could achievesuch extreme isolation for a large pieceof equipment, let alone a living creaturethat breathes air and radiates heat. Butwho knows how fast developmentmight go in the future?

Certainly we could use existing tech-nology to teleport elementary states, likethose of the photons in our experiment,across distances of a few kilometers andmaybe even up to satellites. The technol-ogy to teleport states of individual atomsis at hand today: the group led by SergeHaroche at the École Normale Supé-rieure in Paris has demonstrated entan-glement of atoms. The entanglement ofmolecules and then their teleportationmay reasonably be expected within thenext decade. What happens beyond thatis anybody’s guess.

A more important application of tele-portation might very well be in the fieldof quantum computation, where theordinary notion of bits (0’s and 1’s) isgeneralized to quantum bits, or qubits,which can exist as superpositions and en-tanglements of 0’s and 1’s. Teleportationcould be used to transfer quantum infor-mation between quantum processors.Quantum teleporters can also serve asbasic components used to build a quan-tum computer [see box on page 16]. Thecartoon on the next page illustrates anintriguing situation in which a combina-tion of teleportation and quantum com-putation could occasionally yield an ad-vantage, almost as if one had receivedthe teleported information instantly in-stead of having to wait for it to arrive bynormal means.

Quantum mechanics is probably oneof the profoundest theories ever discov-ered. The problems that it poses for oureveryday intuition about the world ledEinstein to criticize quantum mechanicsvery strongly. He insisted that physics

Isn’t it an exaggeration to call this teleportation? After all, it is only a quantumstate that is teleported, not an actual object. This question raises the deeperphilosophical one of what we mean by identity.How do we know that an object—say,the car we find in our garage in the morning—is the same one we saw a whileago? When it has all the right features and properties.Quantum physics reinforcesthis point:particles of the same type in the same quantum state are indistinguish-able even in principle. If one could carefully swap all the iron atoms in the car withthose from a lump of ore and reproduce the atoms’ states exactly, the end resultwould be identical, at the deepest level, to the original car. Identity cannot meanmore than this:being the same in all properties.

Isn’t it more like “quantum faxing”? Faxing produces a copy that is easy to tellapart from the original, whereas a teleported object is indistinguishable even inprinciple.Moreover, in quantum teleportation the original must be destroyed.

Can we really hope to teleport a complicated object? There are many severe ob-stacles.First, the object has to be in a pure quantum state,and such states are veryfragile. Photons don’t interact with air much, so our experiments can be done inthe open,but experiments with atoms and larger objects must be done in a vacu-um to avoid collisions with gas molecules.Also, the larger an object becomes, theeasier it is to disturb its quantum state. A tiny lump of matter would be disturbedeven by thermal radiation from the walls of the apparatus.This is why we do notroutinely see quantum effects in our everyday world.

Quantum interference, an easier effect to produce than entanglement or tele-portation, has been demonstrated with buckyballs, spheres made of 60 carbonatoms. Such work will proceed to larger objects, perhaps even small viruses, butdon’t hold your breath for it to be repeated with full-size soccer balls!

Another problem is the Bell-state measurement.What would it mean to do a Bell-state measurement of a virus consisting of, say, 107 atoms? How would we extractthe 108 bits of information that such a measurement would generate? For an objectof just a few grams the numbers become impossible:1024 bits of data.

Would teleporting a person require quantum accuracy? Being in the samequantum state does not seem necessary for being the same person.We changeour states all the time and remain the same people—at least as far as we can tell!Conversely, identical twins or biological clones are not “the same people,” be-cause they have different memories. Does Heisenberg uncertainty prevent usfrom replicating a person precisely enough for her to think she was the same asthe original? Who knows. It is intriguing, however, that the quantum no-cloningtheorem prohibits us from making a perfect replica of a person.

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SKEPTICS CORNERTHE AUTHOR ANSWERS COMMON TELEPORTATION QUESTIONS

If we teleported a person’s body,would the mind be left behind?


THE QUANTUM ADVENTURES OF ALICE & BOB

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Intrepid explorer Alice discovers stable einsteinium crystals. Her competitor, the evil Zelda, also “discovers”the crystals. But Alice and her partner Bob (on Earth) have one advantage: QUANTUM COMPUTERS AND TELEPORTERS. Alice does some quantum data processing ...

... and teleports the output —”qubits” of data—to Bob. They are very lucky: the teleportation succeeds cleanly!

Alice sends a message to Bob by laser beam, telling himhis qubits have accurate data. Zelda laser beams her part-ner, Yuri, about the crystals.

Before the laser beam arrives on Earth,Bob feeds his qubits into a quantumsimulation of the economy.

Bob gets Alice’s message thathis qubits were accuratereplicas of hers!

Yuri gets Zelda’s messagebut can only now start hiscomputer simulation.

Bob invests his and Alice’s nest egg in einsteiniumfutures ahead of the crowd. Their success dependedon luck, one chance in four per qubit ...

… but they only had to get lucky once to strike itrich. Yuri and Zelda change to careers in the non-quantum service industry. THE END


should be an attempt to grasp a realitythat exists independently of its observa-tion. Yet he realized that we run intodeep problems when we try to assignsuch an independent physical reality tothe individual members of an entangledpair. His great counterpart, Danishphysicist Niels Bohr, insisted that onehas to take into account the whole sys-tem—in the case of an entangled pair,the arrangement of both particles to-gether. Einstein’s desideratum, the inde-pendent real state of each particle, is de-void of meaning for an entangled quan-tum system.

Quantum teleportation is a direct de-scendant of the scenarios debated byEinstein and Bohr. When we analyze theexperiment, we would run into all kindsof problems if we asked ourselves whatthe properties of the individual particlesreally are when they are entangled. Wehave to analyze carefully what it meansto “have” a polarization. We cannot es-cape the conclusion that all we can talkabout are certain experimental resultsobtained by measurements. In our po-larization measurement, a click of thedetector lets us construct a picture inour mind in which the photon actually“had” a certain polarization at the timeof measurement. Yet we must always re-member that this is just a made-up sto-ry. It is valid only if we talk about thatspecific experiment, and we should becautious in using it in other situations.

Indeed, following Bohr, I would arguethat we can understand quantum me-chanics if we realize that science is notdescribing how nature is but rather ex-presses what we can say about nature.This is where the current value of fun-damental experiments such as teleporta-tion lies: in helping us to reach a deeperunderstanding of our mysterious quan-tum world.

The Author

ANTON ZEILINGER is at the Institute for Ex-perimental Physics at the University of Vienna, hav-ing teleported there in 1999 after nine years at theUniversity of Innsbruck. He considers himself veryfortunate to have the privilege of working on ex-actly the mysteries and paradoxes of quantum me-chanics that drew him into physics nearly 40 yearsago. In his little free time, Zeilinger interacts withclassical music and with jazz, loves to ski, and col-lects antique maps.

Further Information

Quantum Information and Computation. Charles H. Bennett in Physics Today,Vol. 48, No. 10, pages 24–31; October 1995.

Experimental Quantum Teleportation. D. Bouwmeester, J. W. Pan, K. Mattle,M. Eibl, H. Weinfurter and A. Zeilinger in Nature, Vol. 390, pages 575–579; De-cember 11, 1997.

Quantum Information. Special issue of Physics World, Vol. 11, No. 3; March 1998.Quantum Theory: Weird and Wonderful. A. J. Leggett in Physics World, Vol. 12,No. 12, pages 73–77; December 1999.

More about quantum teleportation and related physics experiments is available at www.quantum.at on the World Wide Web.

QUANTUM COMPUTERS

Perhaps the most realistic application of quantum teleportation outside ofpure physics research is in the field of quantum computation.A conventional

digital computer works with bits, which take definite values of 0 or 1, but a quan-tum computer uses quantum bits, or qubits [see “Quantum Computing withMolecules,” by Neil Gershenfeld and Isaac L. Chuang; SCIENTIFIC AMERICAN, June1998]. Qubits can be in quantum superpositions of 0 and 1 just as a photon canbe in a superposition of horizontal and vertical polarization. Indeed, in sending asingle photon,the basic quantum teleporter transmits a single qubit of quantuminformation.

Superpositions of numbers may seem strange, but as the late Rolf Landauer ofIBM put it,“When we were little kids learning to count on our very sticky classicalfingers,we didn’t know about quantum mechanics and superposition.We gainedthe wrong intuition.We thought that information was classical.We thought thatwe could hold up three fingers, then four.We didn’t realize that there could be asuperposition of both.”

A quantum computer can work on a superposition of many different inputs atonce. For example, it could run an algorithm simultaneously on one million in-

puts, using only as many qubits as a conventionalcomputer would need bits to run the algorithmonce on a single input. Theorists have proved thatalgorithms running on quantum computers cansolve certain problems faster (that is, in fewer com-putational steps) than any known algorithm run-ning on a classical computer can.The problems in-clude finding items in a database and factoringlarge numbers, which is of great interest for break-ing secret codes.

So far only the most rudimentary elements ofquantum computers have been built: logic gates that can process one or twoqubits.The realization of even a small-scale quantum computer is still far away. Akey problem is transferring quantum data reliably between different logic gatesor processors, whether within a single quantum computer or across quantumnetworks.Quantum teleportation is one solution.

In addition, Daniel Gottesman of Microsoft and Isaac L. Chuang of IBM recentlyproved that a general-purpose quantum computer can be built out of three basiccomponents:entangled particles,quantum teleporters and gates that operate on asingle qubit at a time.This result provides a systematic way to construct two-qubitgates. The trick of building a two-qubit gate from a teleporter is to teleport twoqubits from the gate’s input to its output,using carefully modified entangled pairs.The entangled pairs are modified in just such a way that the gate’s output receivesthe appropriately processed qubits. Performing quantum logic on two unknownqubits is thus reduced to the tasks of preparing specific predefined entangledstates and teleporting.Admittedly, the complete Bell-state measurement neededto teleport with 100 percent success is itself a type of two-qubit processing. —A.Z.

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Parallel Universes

reading this article? A person who is not you but who lives ona planet called Earth, with misty mountains, fertile fields andsprawling cities, in a solar system with eight other planets? Thelife of this person has been identical to yours in every respect.But perhaps he or she now decides to put down this article with-out finishing it, while you read on.

The idea of such an alter ego seems strange and implausi-ble, but it looks as if we will just have to live with it, because itis supported by astronomical observations. The simplest andmost popular cosmological model today predicts that you havea twin in a galaxy about 10 to the 1028 meters from here. Thisdistance is so large that it is beyond astronomical, but that doesnot make your doppelgänger any less real. The estimate is de-rived from elementary probability and does not even assumespeculative modern physics, merely that space is infinite (or atleast sufficiently large) in size and almost uniformly filled withmatter, as observations indicate. In infinite space, even the mostunlikely events must take place somewhere. There are infinite-ly many other inhabited planets, including not just one but in-finitely many that have people with the same appearance, nameand memories as you, who play out every possible permutationof your life choices.

You will probably never see your other selves. The farthestyou can observe is the distance that light has been able to trav-el during the 14 billion years since the big bang expansion be-gan. The most distant visible objects are now about 4 × 1026

meters away—a distance that defines our observable universe,also called our Hubble volume, our horizon volume or simplyour universe. Likewise, the universes of your other selves arespheres of the same size centered on their planets. They are themost straightforward example of parallel universes. Each uni-verse is merely a small part of a larger “multiverse.”

By this very definition of “universe,” one might expect thenotion of a multiverse to be forever in the domain of meta-physics. Yet the borderline between physics and metaphysics isdefined by whether a theory is experimentally testable, not bywhether it is weird or involves unobservable entities. The fron-tiers of physics have gradually expanded to incorporate evermore abstract (and once metaphysical) concepts such as a roundEarth, invisible electromagnetic fields, time slowdown at highspeeds, quantum superpositions, curved space, and black holes.Over the past several years the concept of a multiverse has joinedthis list. It is grounded in well-tested theories such as relativityand quantum mechanics, and it fulfills both of the basic criteria

By Max Tegmark

Is there a copy of you

Not just a staple of science fiction,other universes are a direct implicationof cosmological observations


originally publishedMay 2003


of an empirical science: it makes predictions, and it can be fal-sified. Scientists have discussed as many as four distinct typesof parallel universes. The key question is not whether the mul-tiverse exists but rather how many levels it has.

Level I: Beyond Our Cosmic HorizonTHE PARALLEL UNIVERSES of your alter egos constitute theLevel I multiverse. It is the least controversial type. We all ac-cept the existence of things that we cannot see but could see ifwe moved to a different vantage point or merely waited, likepeople watching for ships to come over the horizon. Objectsbeyond the cosmic horizon have a similar status. The observ-able universe grows by a light-year every year as light from far-ther away has time to reach us. An infinity lies out there, wait-ing to be seen. You will probably die long before your alter egoscome into view, but in principle, and if cosmic expansion co-operates, your descendants could observe them through a suf-ficiently powerful telescope.

If anything, the Level I multiverse sounds trivially obvious.How could space not be infinite? Is there a sign somewhere say-ing “Space Ends Here—Mind the Gap”? If so, what lies beyondit? In fact, Einstein’s theory of gravity calls this intuition intoquestion. Space could be finite if it has a convex curvature oran unusual topology (that is, interconnectedness). A spherical,doughnut-shaped or pretzel-shaped universe would have a lim-ited volume and no edges. The cosmic microwave backgroundradiation allows sensitive tests of such scenarios [see “Is SpaceFinite?” by Jean-Pierre Luminet, Glenn D. Starkman and Jef-frey R. Weeks; Scientific American, April 1999]. So far,however, the evidence is against them. Infinite models fit thedata, and strong limits have been placed on the alternatives.

Another possibility is that space is infinite but matter is con-fined to a finite region around us—the historically popular “is-land universe” model. In a variant on this model, matter thinsout on large scales in a fractal pattern. In both cases, almost

all universes in the Level I multiverse would be empty and dead.But recent observations of the three-dimensional galaxy distri-bution and the microwave background have shown that thearrangement of matter gives way to dull uniformity on largescales, with no coherent structures larger than about 1024 me-ters. Assuming that this pattern continues, space beyond ourobservable universe teems with galaxies, stars and planets.

Observers living in Level I parallel universes experience thesame laws of physics as we do but with different initial condi-tions. According to current theories, processes early in the bigbang spread matter around with a degree of randomness, gen-erating all possible arrangements with nonzero probability. Cos-mologists assume that our universe, with an almost uniform dis-tribution of matter and initial density fluctuations of one part in100,000, is a fairly typical one (at least among those that con-tain observers). That assumption underlies the estimate thatyour closest identical copy is 10 to the 1028 meters away. About10 to the 1092 meters away, there should be a sphere of radius100 light-years identical to the one centered here, so all percep-tions that we have during the next century will be identical tothose of our counterparts over there. About 10 to the 10118 me-ters away should be an entire Hubble volume identical to ours.

These are extremely conservative estimates, derived simplyby counting all possible quantum states that a Hubble volumecan have if it is no hotter than 108 kelvins. One way to do thecalculation is to ask how many protons could be packed intoa Hubble volume at that temperature. The answer is 10118 pro-tons. Each of those particles may or may not, in fact, be present,which makes for 2 to the 10118 possible arrangements of pro-tons. A box containing that many Hubble volumes exhausts allthe possibilities. If you round off the numbers, such a box isabout 10 to the 10118 meters across. Beyond that box, univers-es—including ours—must repeat. Roughly the same numbercould be derived by using thermodynamic or quantum-gravita-tional estimates of the total information content of the universe.

Your nearest doppelgänger is most likely to be much clos-er than these numbers suggest, given the processes of planet for-mation and biological evolution that tip the odds in your favor.Astronomers suspect that our Hubble volume has at least 1020

habitable planets; some might well look like Earth.The Level I multiverse framework is used routinely to eval-

uate theories in modern cosmology, although this procedure israrely spelled out explicitly. For instance, consider how cos-mologists used the microwave background to rule out a finitespherical geometry. Hot and cold spots in microwave back-ground maps have a characteristic size that depends on the cur-vature of space, and the observed spots appear too small to beconsistent with a spherical shape. But it is important to be sta-tistically rigorous. The average spot size varies randomly fromone Hubble volume to another, so it is possible that our universeis fooling us—it could be spherical but happen to have abnor-mally small spots. When cosmologists say they have ruled outthe spherical model with 99.9 percent confidence, they reallymean that if this model were true, fewer than one in 1,000 Hub-ble volumes would show spots as small as those we observe. AL

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■ One of the many implications of recent cosmologicalobservations is that the concept of parallel universes isno mere metaphor. Space appears to be infinite in size. Ifso, then somewhere out there, everything that is possiblebecomes real, no matter how improbable it is. Beyond therange of our telescopes are other regions of space thatare identical to ours. Those regions are a type of paralleluniverse. Scientists can even calculate how distant theseuniverses are, on average.

■ And that is fairly solid physics. When cosmologists considertheories that are less well established, they conclude thatother universes can have entirely different properties andlaws of physics. The presence of those universes wouldexplain various strange aspects of our own. It could evenanswer fundamental questions about the nature of timeand the comprehensibility of the physical world.

Overview/Multiverses



How Far Away Is a Duplicate Universe? EXAMPLE UNIVERSEImagine a two-dimensional universe with space for four particles.Such a universe has 24, or 16, possible arrangements of matter. If more than 16 of these universes exist, they must begin torepeat. In this example, the distance to the nearest duplicate isroughly four times the diameter of each universe.

OUR UNIVERSEThe same argument applies to our universe, which has space for about 10118 subatomic particles. The number of possiblearrangements is therefore 2 to the 10118, or approximately 10 to the 10118. Multiplying by the diameter of the universegives an average distance to the nearest duplicate of 10 to the 10118 meters.

THE SIMPLEST TYPE of parallel universe is simply a region of spacethat is too far away for us to have seen yet. The farthest that wecan observe is currently about 4 × 1026 meters, or 42 billion light-years—the distance that light has been able to travel since the big

bang began. (The distance is greater than 14 billion light-yearsbecause cosmic expansion has lengthened distances.) Each of theLevel I parallel universes is basically the same as ours. All thedifferences stem from variations in the initial arrangement of matter.

LEVEL I MULTIVERSE

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OUR UNIVERSE


The lesson is that the multiverse theory can be tested andfalsified even though we cannot see the other universes. The keyis to predict what the ensemble of parallel universes is and tospecify a probability distribution, or what mathematicians calla “measure,” over that ensemble. Our universe should emergeas one of the most probable. If not—if, according to the multi-verse theory, we live in an improbable universe—then the the-ory is in trouble. As I will discuss later, this measure problemcan become quite challenging.

Level II: Other Postinflation BubblesI F THE LEVEL I MULTIVERSE was hard to stomach, tryimagining an infinite set of distinct Level I multiverses, someperhaps with different spacetime dimensionality and differentphysical constants. Those other multiverses—which constitutea Level II multiverse—are predicted by the currently populartheory of chaotic eternal inflation.

Inflation is an extension of the big bang theory and ties upmany of the loose ends of that theory, such as why the universeis so big, so uniform and so flat. A rapid stretching of space longago can explain all these and other attributes in one fell swoop[see “The Inflationary Universe,” by Alan H. Guth and Paul J.Steinhard; Scientific American, May 1984; and “The Self-Re-producing Inflationary Universe,” by Andrei Linde, November1994]. Such stretching is predicted by a wide class of theoriesof elementary particles, and all available evidence bears it out.The phrase “chaotic eternal” refers to what happens on the verylargest scales. Space as a whole is stretching and will continuedoing so forever, but some regions of space stop stretching andform distinct bubbles, like gas pockets in a loaf of rising bread.Infinitely many such bubbles emerge. Each is an embryonic Lev-el I multiverse: infinite in size and filled with matter deposited bythe energy field that drove inflation.

Those bubbles are more than infinitely far away from Earth,in the sense that you would never get there even if you traveledat the speed of light forever. The reason is that the space be-

tween our bubble and its neighbors is expanding faster than youcould travel through it. Your descendants will never see theirdoppelgängers elsewhere in Level II. For the same reason, if cos-mic expansion is accelerating, as observations now suggest,they might not see their alter egos even in Level I.

The Level II multiverse is far more diverse than the Level Imultiverse. The bubbles vary not only in their initial conditionsbut also in seemingly immutable aspects of nature. The prevail-ing view in physics today is that the dimensionality of spacetime,the qualities of elementary particles and many of the so-calledphysical constants are not built into physical laws but are theoutcome of processes known as symmetry breaking. For in-stance, theorists think that the space in our universe once hadnine dimensions, all on an equal footing. Early in cosmic histo-ry, three of them partook in the cosmic expansion and becamethe three dimensions we now observe. The other six are now un-observable, either because they have stayed microscopic with adoughnutlike topology or because all matter is confined to athree-dimensional surface (a membrane, or simply “brane”) inthe nine-dimensional space.

Thus, the original symmetry among the dimensions broke.The quantum fluctuations that drive chaotic inflation couldcause different symmetry breaking in different bubbles. Somemight become four-dimensional, others could contain only tworather than three generations of quarks, and still others mighthave a stronger cosmological constant than our universe does.

Another way to produce a Level II multiverse might bethrough a cycle of birth and destruction of universes. In a sci-entific context, this idea was introduced by physicist Richard C.Tolman in the 1930s and recently elaborated on by Paul J. Stein-hardt of Princeton University and Neil Turok of the Universityof Cambridge. The Steinhardt and Turok proposal and relatedmodels involve a second three-dimensional brane that is quiteliterally parallel to ours, merely offset in a higher dimension [see“Been There, Done That,” by George Musser; News Scan, Sci-entific American, March 2002]. This parallel universe is not M

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or infinite (center). (One caveat: some cosmologists speculate that thediscrepant point on the left of the graph is evidence for a finite volume.) Inaddition, WMAP and the 2dF Galaxy Redshift Survey have found that spaceon large scales is filled with matter uniformly (right), meaning that otheruniverses should look basically like ours.

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LEVEL II MULTIVERSE

Bubble Nucleation A QUANTUM FIELD known as the inflatoncauses space to expand rapidly. In the bulk ofspace, random fluctuations prevent the fieldfrom decaying away. But in certain regions,the field loses its strength and the expansionslows down. Those regions become bubbles.

EvidenceCOSMOLOGISTS INFER the presenceof Level II parallel universes byscrutinizing the properties of ouruniverse. These properties, includingthe strength of the forces of nature(right) and the number of observablespace and time dimensions ( far right), were established byrandom processes during the birth of our universe. Yet they haveexactly the values that sustain life.That suggests the existence of otheruniverses with other values.

ALL ATOMS ARE RADIOACTIVE

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A SOMEWHAT MORE ELABORATE type of parallel universe emergesfrom the theory of cosmological inflation. The idea is that our Level Imultiverse—namely, our universe and contiguous regions ofspace—is a bubble embedded in an even vaster but mostly empty

volume. Other bubbles exist out there, disconnected from ours.They nucleate like raindrops in a cloud. During nucleation,variations in quantum fields endow each bubble with propertiesthat distinguish it from other bubbles.

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really a separate universe, because it interacts with ours. But theensemble of universes—past, present and future—that thesebranes create would form a multiverse, arguably with a diver-sity similar to that produced by chaotic inflation. An idea pro-posed by physicist Lee Smolin of the Perimeter Institute in Wa-terloo, Ontario, involves yet another multiverse comparable indiversity to that of Level II but mutating and sprouting new uni-verses through black holes rather than through brane physics.

Although we cannot interact with other Level II parallel uni-verses, cosmologists can infer their presence indirectly, becausetheir existence can account for unexplained coincidences in ouruniverse. To give an analogy, suppose you check into a hotel,are assigned room 1967 and note that this is the year you wereborn. What a coincidence, you say. After a moment of reflec-tion, however, you conclude that this is not so surprising after all.The hotel has hundreds of rooms, and you would not have beenhaving these thoughts in the first place if you had been assignedone with a number that meant nothing to you. The lesson is thateven if you knew nothing about hotels, you could infer the ex-istence of other hotel rooms to explain the coincidence.

As a more pertinent example, consider the mass of the sun.The mass of a star determines its luminosity, and using basicphysics, one can compute that life as we know it on Earth ispossible only if the sun’s mass falls into the narrow range be-tween 1.6 × 1030 and 2.4 × 1030 kilograms. Otherwise Earth’sclimate would be colder than that of present-day Mars or hot-ter than that of present-day Venus. The measured solar massis 2.0 × 1030 kilograms. At first glance, this apparent coinci-dence of the habitable and observed mass values appears to bea wild stroke of luck. Stellar masses run from 1029 to 1032 kilo-grams, so if the sun acquired its mass at random, it had only asmall chance of falling into the habitable range. But just as inthe hotel example, one can explain this apparent coincidenceby postulating an ensemble (in this case, a number of planetarysystems) and a selection effect (the fact that we must find our-selves living on a habitable planet). Such observer-related se-lection effects are referred to as “anthropic,” and although the“A-word” is notorious for triggering controversy, physicistsbroadly agree that these selection effects cannot be neglectedwhen testing fundamental theories.

What applies to hotel rooms and planetary systems appliesto parallel universes. Most, if not all, of the attributes set bysymmetry breaking appear to be fine-tuned. Changing their val-ues by modest amounts would have resulted in a qualitativelydifferent universe—one in which we probably would not ex-ist. If protons were 0.2 percent heavier, they could decay intoneutrons, destabilizing atoms. If the electromagnetic force were4 percent weaker, there would be no hydrogen and no normalstars. If the weak interaction were much weaker, hydrogenwould not exist; if it were much stronger, supernovae wouldfail to seed interstellar space with heavy elements. If the cos-mological constant were much larger, the universe would haveblown itself apart before galaxies could form.

Although the degree of fine-tuning is still debated, these ex-amples suggest the existence of parallel universes with other val-

ues of the physical constants [see “Exploring Our Universe andOthers,” by Martin Rees; Scientific American, December1999]. The Level II multiverse theory predicts that physicistswill never be able to determine the values of these constantsfrom first principles. They will merely compute probability dis-tributions for what they should expect to find, taking selectioneffects into account. The result should be as generic as is con-sistent with our existence.

Level III: Quantum Many Worlds THE LEVEL I AND LEVEL I I multiverses involve parallelworlds that are far away, beyond the domain even of as-tronomers. But the next level of multiverse is right around you.It arises from the famous, and famously controversial, many-worlds interpretation of quantum mechanics—the idea thatrandom quantum processes cause the universe to branch intomultiple copies, one for each possible outcome.

In the early 20th century the theory of quantum mechanicsrevolutionized physics by explaining the atomic realm, whichdoes not abide by the classical rules of Newtonian mechanics.Despite the obvious successes of the theory, a heated debaterages about what it really means. The theory specifies the stateof the universe not in classical terms, such as the positions andvelocities of all particles, but in terms of a mathematical ob-ject called a wave function. According to the Schrödinger equa-tion, this state evolves over time in a fashion that mathemati-cians term “unitary,” meaning that the wave function rotatesin an abstract infinite-dimensional space called Hilbert space.Although quantum mechanics is often described as inherentlyrandom and uncertain, the wave function evolves in a deter-ministic way. There is nothing random or uncertain about it.

The sticky part is how to connect this wave function withwhat we observe. Many legitimate wave functions correspondto counterintuitive situations, such as a cat being dead and aliveat the same time in a so-called superposition. In the 1920sphysicists explained away this weirdness by postulating that thewave function “collapsed” into some definite classical outcomewhenever someone made an observation. This add-on had thevirtue of explaining observations, but it turned an elegant, uni-tary theory into a kludgy, nonunitary one. The intrinsic ran-domness commonly ascribed to quantum mechanics is the re-sult of this postulate.

Over the years many physicists have abandoned this viewin favor of one developed in 1957 by Princeton graduate stu-dent Hugh Everett III. He showed that the collapse postulateis unnecessary. Unadulterated quantum theory does not, in fact,pose any contradictions. Although it predicts that one classi-cal reality gradually splits into superpositions of many such re-alities, observers subjectively experience this splitting merely asa slight randomness, with probabilities in exact agreement withthose from the old collapse postulate. This superposition ofclassical worlds is the Level III multiverse.

Everett’s many-worlds interpretation has been bogglingminds inside and outside physics for more than four decades.But the theory becomes easier to grasp when one distinguishes AL

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QUANTUM MECHANICS PREDICTS a vast number of paralleluniverses by broadening the concept of “elsewhere.” Theseuniverses are located elsewhere, not in ordinary space but in anabstract realm of all possible states. Every conceivable way that

the world could be (within the scope of quantum mechanics)corresponds to a different universe. The parallel universes maketheir presence felt in laboratory experiments, such as waveinterference and quantum computation.

LEVEL III MULTIVERSE

Quantum DiceIMAGINE AN IDEAL DIE whose randomnessis purely quantum. When you roll it, thedie appears to land on a certain value atrandom. Quantum mechanics, however,predicts that it lands on all values atonce. One way to reconcile thesecontradictory views is to conclude thatthe die lands on different values indifferent universes. In one sixth of theuniverses, it lands on 1; in one sixth, on 2,and so on. Trapped within one universe,we can perceive only a fraction of the fullquantum reality.

Ergodicity ACCORDING TO THE PRINCIPLE of ergodicity, quantum paralleluniverses are equivalent to more prosaic types of parallel universes.A quantum universe splits over time into multiple universes (left).Yet those new universes are no different from parallel universes thatalready exist somewhere else in space—in, for example, other Level Iuniverses (right). The key idea is that parallel universes, of whatevertype, embody different ways that events could have unfolded.

The Nature of Time MOST PEOPLE THINK of time as a way to describechange. At one moment, matter has a certainarrangement; a moment later, it has another(left). The concept of multiverses suggests analternative view. If parallel universes contain allpossible arrangements of matter (right), thentime is simply a way to put those universes into asequence. The universes themselves are static;change is an illusion, albeit an interesting one.

==


between two ways of viewing a physical theory: the outsideview of a physicist studying its mathematical equations, like abird surveying a landscape from high above it, and the insideview of an observer living in the world described by the equa-tions, like a frog living in the landscape surveyed by the bird.

From the bird perspective, the Level III multiverse is simple.There is only one wave function. It evolves smoothly and de-terministically over time without any kind of splitting or par-allelism. The abstract quantum world described by this evolv-ing wave function contains within it a vast number of parallelclassical story lines, continuously splitting and merging, as wellas a number of quantum phenomena that lack a classical de-scription. From their frog perspective, observers perceive onlya tiny fraction of this full reality. They can view their own Lev-el I universe, but a process called decoherence—which mimicswave function collapse while preserving unitarity—preventsthem from seeing Level III parallel copies of themselves.

Whenever observers are asked a question, make a snap deci-sion and give an answer, quantum effects in their brains lead toa superposition of outcomes, such as “Continue reading the ar-ticle” and “Put down the article.” From the bird perspective, theact of making a decision causes a person to split into multiplecopies: one who keeps on reading and one who doesn’t. Fromtheir frog perspective, however, each of these alter egos is un-aware of the others and notices the branching merely as a slightrandomness: a certain probability of continuing to read or not.

As strange as this may sound, the exact same situation oc-curs even in the Level I multiverse. You have evidently decidedto keep on reading the article, but one of your alter egos in adistant galaxy put down the magazine after the first paragraph.The only difference between Level I and Level III is where yourdoppelgängers reside. In Level I they live elsewhere in good oldthree-dimensional space. In Level III they live on another quan-tum branch in infinite-dimensional Hilbert space.

The existence of Level III depends on one crucial assump-tion: that the time evolution of the wave function is unitary. Sofar experimenters have encountered no departures from unitar-ity. In the past few decades they have confirmed unitarity forever larger systems, including carbon 60 buckyball moleculesand kilometer-long optical fibers. On the theoretical side, thecase for unitarity has been bolstered by the discovery of deco-herence [see “100 Years of Quantum Mysteries,” by Max

Tegmark and John Archibald Wheeler; Scientific American,February 2001]. Some theorists who work on quantum gravityhave questioned unitarity; one concern is that evaporating blackholes might destroy information, which would be a nonunitaryprocess. But a recent breakthrough in string theory known asAdS/CFT correspondence suggests that even quantum gravity isunitary. If so, black holes do not destroy information but mere-ly transmit it elsewhere. [Editors’ note: An upcoming article willdiscuss this correspondence in greater detail.]

If physics is unitary, then the standard picture of how quan-tum fluctuations operated early in the big bang must change.These fluctuations did not generate initial conditions at ran-dom. Rather they generated a quantum superposition of allpossible initial conditions, which coexisted simultaneously. De-coherence then caused these initial conditions to behave clas-sically in separate quantum branches. Here is the crucial point:the distribution of outcomes on different quantum branchesin a given Hubble volume (Level III) is identical to the distrib-ution of outcomes in different Hubble volumes within a singlequantum branch (Level I). This property of the quantum fluc-tuations is known in statistical mechanics as ergodicity.

The same reasoning applies to Level II. The process of sym-metry breaking did not produce a unique outcome but rathera superposition of all outcomes, which rapidly went their sep-arate ways. So if physical constants, spacetime dimensionalityand so on can vary among parallel quantum branches at LevelIII, then they will also vary among parallel universes at Level II.

In other words, the Level III multiverse adds nothing newbeyond Level I and Level II, just more indistinguishable copiesof the same universes—the same old story lines playing outagain and again in other quantum branches. The passionate de-bate about Everett’s theory therefore seems to be ending in agrand anticlimax, with the discovery of less controversial mul-tiverses (Levels I and II) that are equally large.

Needless to say, the implications are profound, and physi-cists are only beginning to explore them. For instance, consid-er the ramifications of the answer to a long-standing question:Does the number of universes exponentially increase over time?The surprising answer is no. From the bird perspective, there isof course only one quantum universe. From the frog perspective,what matters is the number of universes that are distinguishableat a given instant—that is, the number of noticeably differentHubble volumes. Imagine moving planets to random new lo-cations, imagine having married someone else, and so on. At thequantum level, there are 10 to the 10118 universes with temper-atures below 108 kelvins. That is a vast number, but a finite one.

From the frog perspective, the evolution of the wave func-tion corresponds to a never-ending sliding from one of these 10to the 10118 states to another. Now you are in universe A, theone in which you are reading this sentence. Now you are in uni-verse B, the one in which you are reading this other sentence.Put differently, universe B has an observer identical to one inuniverse A, except with an extra instant of memories. All pos-sible states exist at every instant, so the passage of time may bein the eye of the beholder—an idea explored in Greg Egan’s

MAX TEGMARK wrote a four-dimensional version of the computergame Tetris while in college. In another universe, he went on to be-come a highly paid software developer. In our universe, however,he wound up as professor of physics and astronomy at the Uni-versity of Pennsylvania. Tegmark is an expert in analyzing thecosmic microwave background and galaxy clustering. Much of hiswork bears on the concept of parallel universes: evaluating evi-dence for infinite space and cosmological inflation; developing in-sights into quantum decoherence; and studying the possibilitythat the amplitude of microwave background fluctuations, the di-mensionality of spacetime and the fundamental laws of physicscan vary from place to place.

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1994 science-fiction novel Permutation City and developed byphysicist David Deutsch of the University of Oxford, indepen-dent physicist Julian Barbour, and others. The multiverseframework may thus prove essential to understanding the na-ture of time.

Level IV: Other Mathematical StructuresTHE INIT IAL CONDITIONS and physical constants in theLevel I, Level II and Level III multiverses can vary, but thefundamental laws that govern nature remain the same. Whystop there? Why not allow the laws themselves to vary? Howabout a universe that obeys the laws of classical physics, withno quantum effects? How about time that comes in discretesteps, as for computers, instead of being continuous? Howabout a universe that is simply an empty dodecahedron? In theLevel IV multiverse, all these alternative realities actually exist.

A hint that such a multiverse might not be just some beer-fueled speculation is the tight correspondence between theworlds of abstract reasoning and of observed reality. Equationsand, more generally, mathematical structures such as numbers,vectors and geometric objects describe the world with remark-able verisimilitude. In a famous 1959 lecture, physicist EugeneP. Wigner argued that “the enormous usefulness of mathemat-ics in the natural sciences is something bordering on the mys-terious.” Conversely, mathematical structures have an eerilyreal feel to them. They satisfy a central criterion of objective ex-istence: they are the same no matter who studies them. A the-orem is true regardless of whether it is proved by a human, acomputer or an intelligent dolphin. Contemplative alien civi-lizations would find the same mathematical structures as we

have. Accordingly, mathematicians commonly say that theydiscover mathematical structures rather than create them.

There are two tenable but diametrically opposed paradigmsfor understanding the correspondence between mathematicsand physics, a dichotomy that arguably goes as far back as Pla-to and Aristotle. According to the Aristotelian paradigm, phys-ical reality is fundamental and mathematical language is mere-ly a useful approximation. According to the Platonic paradigm,the mathematical structure is the true reality and observers per-ceive it imperfectly. In other words, the two paradigms disagreeon which is more basic, the frog perspective of the observer orthe bird perspective of the physical laws. The Aristotelian par-adigm prefers the frog perspective, whereas the Platonic para-digm prefers the bird perspective.

As children, long before we had even heard of mathemat-ics, we were all indoctrinated with the Aristotelian paradigm.The Platonic view is an acquired taste. Modern theoreticalphysicists tend to be Platonists, suspecting that mathematics de-scribes the universe so well because the universe is inherentlymathematical. Then all of physics is ultimately a mathematicsproblem: a mathematician with unlimited intelligence and re-sources could in principle compute the frog perspective—thatis, compute what self-aware observers the universe contains,what they perceive, and what languages they invent to describetheir perceptions to one another.

A mathematical structure is an abstract, immutable entityexisting outside of space and time. If history were a movie, thestructure would correspond not to a single frame of it but to theentire videotape. Consider, for example, a world made up ofpointlike particles moving around in three-dimensional space.SA

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AS MULTIVERSE THEORIES gain credence, the sticky issue of how tocompute probabilities in physics is growing from a minor nuisanceinto a major embarrassment. If there are indeed many identicalcopies of you, the traditional notion of determinism evaporates.You could not compute your own future even if you had completeknowledge of the entire state of the multiverse, because there is noway for you to determine which of these copies is you (they all feelthey are). All you can predict, therefore, are probabilities for whatyou would observe. If an outcome has a probability of, say, 50percent, it means that half the observers observe that outcome.

Unfortunately, it is not an easy task to compute what fractionof the infinitely many observers perceive what. The answerdepends on the order in which you count them. By analogy, thefraction of the integers that are even is 50 percent if you orderthem numerically (1, 2, 3, 4, . . . ) but approaches 100 percent if yousort them digit by digit, the way your word processor would (1, 10,100, 1,000, . . . ). When observers reside in disconnected universes,there is no obviously natural way in which to order them. Instead one must sample from the different universes with some statisticalweights referred to by mathematicians as a “measure.”

This problem crops up in a mild and treatable manner at Level I,

becomes severe at Level II, has caused much debate at Level III,and is horrendous at Level IV. At Level II, for instance, AlexanderVilenkin of Tufts University and others have published predictionsfor the probability distributions of various cosmologicalparameters. They have argued that different parallel universes thathave inflated by different amounts should be given statisticalweights proportional to their volume. On the other hand, anymathematician will tell you that 2 × 1 = 1, so there is no objectivesense in which an infinite universe that has expanded by a factor oftwo has gotten larger. Moreover, a finite universe with the topologyof a torus is equivalent to a perfectly periodic universe with infinitevolume, both from the mathematical bird perspective and from thefrog perspective of an observer within it. So why should its infinitelysmaller volume give it zero statistical weight? After all, even in theLevel I multiverse, Hubble volumes start repeating (albeit in arandom order, not periodically) after about 10 to the 10118 meters.

If you think that is bad, consider the problem of assigningstatistical weights to different mathematical structures at Level IV.The fact that our universe seems relatively simple has led manypeople to suggest that the correct measure somehow involvescomplexity. —M.T.

The Mystery of Probability:

What Are the Odds?



In four-dimensional spacetime—the bird perspective—theseparticle trajectories resemble a tangle of spaghetti. If the frogsees a particle moving with constant velocity, the bird sees astraight strand of uncooked spaghetti. If the frog sees a pair oforbiting particles, the bird sees two spaghetti strands inter-twined like a double helix. To the frog, the world is describedby Newton’s laws of motion and gravitation. To the bird, it isdescribed by the geometry of the pasta—a mathematical struc-ture. The frog itself is merely a thick bundle of pasta, whosehighly complex intertwining corresponds to a cluster of parti-cles that store and process information. Our universe is farmore complicated than this example, and scientists do not yetknow to what, if any, mathematical structure it corresponds.

The Platonic paradigm raises the question of why the uni-verse is the way it is. To an Aristotelian, this is a meaninglessquestion: the universe just is. But a Platonist cannot help butwonder why it could not have been different. If the universe isinherently mathematical, then why was only one of the manymathematical structures singled out to describe a universe? Afundamental asymmetry appears to be built into the very heartof reality.

As a way out of this conundrum, I have suggested that com-plete mathematical symmetry holds: that all mathematical struc-tures exist physically as well. Every mathematical structure cor-responds to a parallel universe. The elements of this multiverse

do not reside in the same space but exist outside of space andtime. Most of them are probably devoid of observers. This hy-pothesis can be viewed as a form of radical Platonism, assert-ing that the mathematical structures in Plato’s realm of ideas orthe “mindscape” of mathematician Rudy Rucker of San JoseState University exist in a physical sense. It is akin to what cos-mologist John D. Barrow of the University of Cambridge refersto as “π in the sky,” what the late Harvard University philoso-pher Robert Nozick called the principle of fecundity and whatthe late Princeton philosopher David K. Lewis called modal re-alism. Level IV brings closure to the hierarchy of multiverses, be-cause any self-consistent fundamental physical theory can bephrased as some kind of mathematical structure.

The Level IV multiverse hypothesis makes testable predic-tions. As with Level II, it involves an ensemble (in this case, thefull range of mathematical structures) and selection effects. Asmathematicians continue to categorize mathematical struc-tures, they should find that the structure describing our worldis the most generic one consistent with our observations. Sim-ilarly, our future observations should be the most generic onesthat are consistent with our past observations, and our past ob-servations should be the most generic ones that are consistentwith our existence.

Quantifying what “generic” means is a severe problem, andthis investigation is only now beginning. But one striking and C

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THE ULTIMATE TYPE of parallel universe opens up the full realm ofpossibility. Universes can differ not just in location, cosmologicalproperties or quantum state but also in the laws of physics. Existingoutside of space and time, they are almost impossible to visualize; thebest one can do is to think of them abstractly, as static sculpturesthat represent the mathematical structure of the physical laws that

govern them. For example, consider a simple universe: Earth, moonand sun, obeying Newton’s laws. To an objective observer, thisuniverse looks like a circular ring (Earth’s orbit smeared out in time)wrapped in a braid (the moon’s orbit around Earth). Other shapesembody other laws of physics (a, b, c, d). This paradigm solves variousproblems concerning the foundations of physics.

LEVEL IV MULTIVERSE

a b

c d

SUNEARTH’SORBIT



encouraging feature of mathematical structures is that the sym-metry and invariance properties that are responsible for thesimplicity and orderliness of our universe tend to be generic,more the rule than the exception. Mathematical structures tendto have them by default, and complicated additional axiomsmust be added to make them go away.

What Says Occam?THE SCIENTIFIC THEORIES of parallel universes, therefore,form a four-level hierarchy, in which universes become pro-gressively more different from ours. They might have differentinitial conditions (Level I); different physical constants and par-ticles (Level II); or different physical laws (Level IV). It is iron-ic that Level III is the one that has drawn the most fire in thepast decades, because it is the only one that adds no qualita-tively new types of universes.

In the coming decade, dramatically improved cosmologicalmeasurements of the microwave background and the large-scale matter distribution will support or refute Level I by fur-ther pinning down the curvature and topology of space. Thesemeasurements will also probe Level II by testing the theory ofchaotic eternal inflation. Progress in both astrophysics andhigh-energy physics should also clarify the extent to whichphysical constants are fine-tuned, thereby weakening orstrengthening the case for Level II.

If current efforts to build quantum computers succeed, theywill provide further evidence for Level III, as they would, inessence, be exploiting the parallelism of the Level III multiversefor parallel computation. Experimenters are also looking forevidence of unitarity violation, which would rule out Level III.Finally, success or failure in the grand challenge of modernphysics—unifying general relativity and quantum field theory—

will sway opinions on Level IV. Either we will find a mathe-matical structure that exactly matches our universe, or we willbump up against a limit to the unreasonable effectiveness ofmathematics and have to abandon that level.

So should you believe in parallel universes? The principalarguments against them are that they are wasteful and that theyare weird. The first argument is that multiverse theories are vul-nerable to Occam’s razor because they postulate the existenceof other worlds that we can never observe. Why should naturebe so wasteful and indulge in such opulence as an infinity of dif-ferent worlds? Yet this argument can be turned around to ar-gue for a multiverse. What precisely would nature be wasting?Certainly not space, mass or atoms—the uncontroversial Lev-el I multiverse already contains an infinite amount of all three,so who cares if nature wastes some more? The real issue hereis the apparent reduction in simplicity. A skeptic worries aboutall the information necessary to specify all those unseen worlds.

But an entire ensemble is often much simpler than one of itsmembers. This principle can be stated more formally using thenotion of algorithmic information content. The algorithmic in-formation content in a number is, roughly speaking, the lengthof the shortest computer program that will produce that num-ber as output. For example, consider the set of all integers.

Which is simpler, the whole set or just one number? Naively,you might think that a single number is simpler, but the entireset can be generated by quite a trivial computer program,whereas a single number can be hugely long. Therefore, thewhole set is actually simpler.

Similarly, the set of all solutions to Einstein’s field equationsis simpler than a specific solution. The former is described bya few equations, whereas the latter requires the specification ofvast amounts of initial data on some hypersurface. The lessonis that complexity increases when we restrict our attention toone particular element in an ensemble, thereby losing the sym-metry and simplicity that were inherent in the totality of all theelements taken together.

In this sense, the higher-level multiverses are simpler. Go-ing from our universe to the Level I multiverse eliminates theneed to specify initial conditions, upgrading to Level II elimi-nates the need to specify physical constants, and the Level IVmultiverse eliminates the need to specify anything at all. Theopulence of complexity is all in the subjective perceptions of ob-servers—the frog perspective. From the bird perspective, themultiverse could hardly be any simpler.

The complaint about weirdness is aesthetic rather than sci-entific, and it really makes sense only in the Aristotelian world-view. Yet what did we expect? When we ask a profound ques-tion about the nature of reality, do we not expect an answerthat sounds strange? Evolution provided us with intuition forthe everyday physics that had survival value for our distant an-cestors, so whenever we venture beyond the everyday world,we should expect it to seem bizarre.

A common feature of all four multiverse levels is that thesimplest and arguably most elegant theory involves parallel uni-verses by default. To deny the existence of those universes, oneneeds to complicate the theory by adding experimentally un-supported processes and ad hoc postulates: finite space, wavefunction collapse and ontological asymmetry. Our judgmenttherefore comes down to which we find more wasteful and in-elegant: many worlds or many words. Perhaps we will gradu-ally get used to the weird ways of our cosmos and find itsstrangeness to be part of its charm.

Why Is the CMB Fluctuation Level 10–5? Max Tegmark and Martin Rees inAstrophysical Journal, Vol. 499, No. 2, pages 526–532; June 1, 1998.Available online at arXiv.org/abs/astro-ph/9709058

Is “The Theory of Everything” Merely the Ultimate Ensemble Theory?Max Tegmark in Annals of Physics, Vol. 270, No.1, pages 1–51; November 20, 1998. Available online at arXiv.org/abs/gr-qc/9704009

Many Worlds in One. Jaume Garriga and Alexander Vilenkin in PhysicalReview, Vol. D64, No. 043511; July 26, 2001. Available online atarXiv.org/abs/gr-qc/0102010

Our Cosmic Habitat. Martin Rees. Princeton University Press, 2001.

Inflation, Quantum Cosmology and the Anthropic Principle. Andrei Lindein Science and Ultimate Reality: From Quantum to Cosmos. Edited by J. D.Barrow, P.C.W. Davies and C. L. Harper. Cambridge University Press, 2003.Available online at arXiv.org/abs/hep-th/0211048

The author’s Web site has more information atwww.hep.upenn.edu/~max/multiverse.html

M O R E T O E X P L O R E



Informationin the

HOLOGRAPHIC UNIVERSE

Theoretical results about

black holes suggest that

the universe could be like

a gigantic hologram

By Jacob D. Bekenstein

Illustrations by Alfred T. Kamajian


originally publishedAugust 2003


Yet if we have learned anything from engi-neering, biology and physics, information isjust as crucial an ingredient. The robot at theautomobile factory is supplied with metaland plastic but can make nothing usefulwithout copious instructions telling it whichpart to weld to what and so on. A ribosomein a cell in your body is supplied with aminoacid building blocks and is powered by en-ergy released by the conversion of ATP toADP, but it can synthesize no proteins with-out the information brought to it from theDNA in the cell’s nucleus. Likewise, a cen-tury of developments in physics has taughtus that information is a crucial player inphysical systems and processes. Indeed, acurrent trend, initiated by John A. Wheelerof Princeton University, is to regard thephysical world as made of information, withenergy and matter as incidentals.

This viewpoint invites a new look at ven-erable questions. The information storagecapacity of devices such as hard disk driveshas been increasing by leaps and bounds.When will such progress halt? What is theultimate information capacity of a devicethat weighs, say, less than a gram and can fitinside a cubic centimeter (roughly the size ofa computer chip)? How much information

Ask anybody whatthe physical worldis made of, and youare likely to be told“matter and energy.”



does it take to describe a whole universe?Could that description fit in a computer’smemory? Could we, as William Blakememorably penned, “see the world in agrain of sand,” or is that idea no morethan poetic license?

Remarkably, recent developments intheoretical physics answer some of thesequestions, and the answers might be im-portant clues to the ultimate theory of re-ality. By studying the mysterious proper-ties of black holes, physicists have de-duced absolute limits on how muchinformation a region of space or a quan-tity of matter and energy can hold. Relat-ed results suggest that our universe, whichwe perceive to have three spatial dimen-sions, might instead be “written” on atwo-dimensional surface, like a holo-gram. Our everyday perceptions of theworld as three-dimensional would thenbe either a profound illusion or merelyone of two alternative ways of viewing re-ality. A grain of sand may not encompassour world, but a flat screen might.

A Tale of Two EntropiesFORMAL INFORMATION theory orig-inated in seminal 1948 papers by Ameri-can applied mathematician Claude E.Shannon, who introduced today’s mostwidely used measure of information con-tent: entropy. Entropy had long been acentral concept of thermodynamics, thebranch of physics dealing with heat. Ther-modynamic entropy is popularly de-scribed as the disorder in a physical sys-tem. In 1877 Austrian physicist LudwigBoltzmann characterized it more precise-ly in terms of the number of distinct mi-

croscopic states that the particles com-posing a chunk of matter could be inwhile still looking like the same macro-scopic chunk of matter. For example, forthe air in the room around you, onewould count all the ways that the indi-vidual gas molecules could be distributedin the room and all the ways they couldbe moving.

When Shannon cast about for a wayto quantify the information contained in,say, a message, he was led by logic to aformula with the same form as Boltz-mann’s. The Shannon entropy of a mes-sage is the number of binary digits, or bits,needed to encode it. Shannon’s entropydoes not enlighten us about the value ofinformation, which is highly dependenton context. Yet as an objective measureof quantity of information, it has beenenormously useful in science and tech-nology. For instance, the design of everymodern communications device—fromcellular phones to modems to compact-disc players—relies on Shannon entropy.

Thermodynamic entropy and Shan-non entropy are conceptually equivalent:the number of arrangements that arecounted by Boltzmann entropy reflectsthe amount of Shannon information onewould need to implement any particulararrangement. The two entropies have twosalient differences, though. First, the ther-modynamic entropy used by a chemist ora refrigeration engineer is expressed inunits of energy divided by temperature,whereas the Shannon entropy used by acommunications engineer is in bits, es-sentially dimensionless. That difference ismerely a matter of convention.

Even when reduced to common units,however, typical values of the two en-tropies differ vastly in magnitude. A sili-con microchip carrying a gigabyte ofdata, for instance, has a Shannon entropyof about 1010 bits (one byte is eight bits),tremendously smaller than the chip’s ther-modynamic entropy, which is about 1023

bits at room temperature. This discrep-ancy occurs because the entropies arecomputed for different degrees of free-dom. A degree of freedom is any quanti-ty that can vary, such as a coordinatespecifying a particle’s location or onecomponent of its velocity. The Shannonentropy of the chip cares only about theoverall state of each tiny transistor etchedin the silicon crystal—the transistor is onor off; it is a 0 or a 1—a single binary de-gree of freedom. Thermodynamic en-tropy, in contrast, depends on the statesof all the billions of atoms (and theirroaming electrons) that make up eachtransistor. As miniaturization brings clos-er the day when each atom will store onebit of information for us, the useful Shan-non entropy of the state-of-the-art mi-crochip will edge closer in magnitude toits material’s thermodynamic entropy.When the two entropies are calculated forthe same degrees of freedom, they areequal.

What are the ultimate degrees of free-dom? Atoms, after all, are made of elec-trons and nuclei, nuclei are agglomera-tions of protons and neutrons, and thosein turn are composed of quarks. Manyphysicists today consider electrons andquarks to be excitations of superstrings,which they hypothesize to be the mostfundamental entities. But the vicissitudesof a century of revelations in physics warnus not to be dogmatic. There could bemore levels of structure in our universethan are dreamt of in today’s physics.

One cannot calculate the ultimate in-formation capacity of a chunk of matteror, equivalently, its true thermodynamicentropy, without knowing the nature ofthe ultimate constituents of matter or ofthe deepest level of structure, which Ishall refer to as level X. (This ambiguitycauses no problems in analyzing practi-cal thermodynamics, such as that of car

■ An astonishing theory called the holographic principle holds that the universeis like a hologram: just as a trick of light allows a fully three-dimensional imageto be recorded on a flat piece of film, our seemingly three-dimensional universecould be completely equivalent to alternative quantum fields and physical laws“painted” on a distant, vast surface.

■ The physics of black holes—immensely dense concentrations of mass—providesa hint that the principle might be true. Studies of black holes show that, althoughit defies common sense, the maximum entropy or information content of anyregion of space is defined not by its volume but by its surface area.

■ Physicists hope that this surprising finding is a clue to the ultimate theory of reality.

Overview/The World as a Hologram



engines, for example, because the quarkswithin the atoms can be ignored—theydo not change their states under the rel-atively benign conditions in the engine.)Given the dizzying progress in miniatur-ization, one can playfully contemplate aday when quarks will serve to store in-formation, one bit apiece perhaps. Howmuch information would then fit into ourone-centimeter cube? And how much ifwe harness superstrings or even deeper,yet undreamt of levels? Surprisingly, de-velopments in gravitation physics in thepast three decades have supplied someclear answers to what seem to be elusivequestions.

Black Hole ThermodynamicsA CENTRAL PLAYER in these develop-ments is the black hole. Black holes are aconsequence of general relativity, AlbertEinstein’s 1915 geometric theory of grav-itation. In this theory, gravitation arisesfrom the curvature of spacetime, whichmakes objects move as if they were pulledby a force. Conversely, the curvature is

caused by the presence of matter and en-ergy. According to Einstein’s equations, asufficiently dense concentration of matteror energy will curve spacetime so ex-tremely that it rends, forming a blackhole. The laws of relativity forbid any-thing that went into a black hole fromcoming out again, at least within the clas-sical (nonquantum) description of thephysics. The point of no return, called theevent horizon of the black hole, is of cru-cial importance. In the simplest case, thehorizon is a sphere, whose surface area islarger for more massive black holes.

It is impossible to determine what isinside a black hole. No detailed informa-tion can emerge across the horizon andescape into the outside world. In disap-

pearing forever into a black hole, howev-er, a piece of matter does leave sometraces. Its energy (we count any mass asenergy in accordance with Einstein’s E =mc2) is permanently reflected in an incre-ment in the black hole’s mass. If the mat-ter is captured while circling the hole, itsassociated angular momentum is addedto the black hole’s angular momentum.Both the mass and angular momentum ofa black hole are measurable from their ef-fects on spacetime around the hole. In thisway, the laws of conservation of energyand angular momentum are upheld byblack holes. Another fundamental law,the second law of thermodynamics, ap-pears to be violated.

The second law of thermodynamicssummarizes the familiar observation thatmost processes in nature are irreversible:a teacup falls from the table and shatters,but no one has ever seen shards jump upof their own accord and assemble into ateacup. The second law of thermody-namics forbids such inverse processes. Itstates that the entropy of an isolated phys-ical system can never decrease; at best, en-tropy remains constant, and usually it in-creases. This law is central to physicalchemistry and engineering; it is arguablythe physical law with the greatest impactoutside physics.

As first emphasized by Wheeler, whenmatter disappears into a black hole, its en-tropy is gone for good, and the secondlaw seems to be transcended, made irrel-evant. A clue to resolving this puzzle camein 1970, when Demetrious Christodou-lou, then a graduate student of Wheeler’sat Princeton, and Stephen W. Hawking ofthe University of Cambridge indepen-dently proved that in various processes,such as black hole mergers, the total areaof the event horizons never decreases. Theanalogy with the tendency of entropy toincrease led me to propose in 1972 that ablack hole has entropy proportional to

JACOB D. BEKENSTEIN has contributed to the foundation of black hole thermodynamics andto other aspects of the connections between information and gravitation. He is Polak Pro-fessor of Theoretical Physics at the Hebrew University of Jerusalem, a member of the IsraelAcademy of Sciences and Humanities, and a recipient of the Rothschild Prize. Bekensteindedicates this article to John Archibald Wheeler (his Ph.D. supervisor 30 years ago). Wheel-er belongs to the third generation of Ludwig Boltzmann’s students: Wheeler’s Ph.D. advis-er, Karl Herzfeld, was a student of Boltzmann’s student Friedrich Hasenöhrl.

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THE ENTROPY OF A BLACK HOLE is proportional to the area of its event horizon, the surface withinwhich even light cannot escape the gravity of the hole. Specifically, a hole with a horizon spanningA Planck areas has A⁄4 units of entropy. (The Planck area, approximately 10–66 square centimeter,is the fundamental quantum unit of area determined by the strength of gravity, the speed of lightand the size of quanta.) Considered as information, it is as if the entropy were written on theevent horizon, with each bit (each digital 1 or 0) corresponding to four Planck areas.

One Planck areaBlack holeevent horizon

One unit of entropy



the area of its horizon [see illustration onpreceding page]. I conjectured that whenmatter falls into a black hole, the increasein black hole entropy always compensatesor overcompensates for the “lost” en-tropy of the matter. More generally, thesum of black hole entropies and the ordi-nary entropy outside the black holes can-not decrease. This is the generalized sec-ond law—GSL for short.

The GSL has passed a large number ofstringent, if purely theoretical, tests.When a star collapses to form a blackhole, the black hole entropy greatly ex-ceeds the star’s entropy. In 1974 Hawk-ing demonstrated that a black hole spon-taneously emits thermal radiation, now

known as Hawking radiation, by a quan-tum process [see “The Quantum Me-chanics of Black Holes,” by Stephen W.Hawking; Scientific American, Janu-ary 1977]. The Christodoulou-Hawkingtheorem fails in the face of this phenom-enon (the mass of the black hole, andtherefore its horizon area, decreases), butthe GSL copes with it: the entropy of theemergent radiation more than compen-sates for the decrement in black hole en-tropy, so the GSL is preserved. In 1986Rafael D. Sorkin of Syracuse Universityexploited the horizon’s role in barring in-formation inside the black hole from in-fluencing affairs outside to show that theGSL (or something very similar to it) must

be valid for any conceivable process thatblack holes undergo. His deep argumentmakes it clear that the entropy enteringthe GSL is that calculated down to levelX, whatever that level may be.

Hawking’s radiation process allowedhim to determine the proportionality con-stant between black hole entropy andhorizon area: black hole entropy is pre-cisely one quarter of the event horizon’sarea measured in Planck areas. (ThePlanck length, about 10–33 centimeter, isthe fundamental length scale related togravity and quantum mechanics. ThePlanck area is its square.) Even in ther-modynamic terms, this is a vast quantityof entropy. The entropy of a black hole

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THE THERMODYNAMICS OF BLACK HOLES allows one todeduce limits on the density of entropy or informationin various circumstances.

The holographic bound defines how muchinformation can be contained in a specified region ofspace. It can be derived by considering a roughlyspherical distribution of matter that is contained withina surface of area A. The matter is induced to collapse toform a black hole (a). The black hole’s area must besmaller than A, so its entropy must be less than A⁄4

[see illustration on preceding page]. Because entropycannot decrease, one infers that the original distrib-ution of matter also must carry less than A⁄4 units ofentropy or information. This result—that the maximuminformation content of a region of space is fixed by itsarea—defies the commonsense expectation that thecapacity of a region should depend on its volume.

The universal entropy bound defines how muchinformation can be carried by a mass m of diameter d.It is derived by imagining that a capsule of matter isengulfed by a black hole not much wider than it (b). Theincrease in the black hole’s size places a limit on howmuch entropy the capsule could have contained. Thislimit is tighter than the holographic bound, exceptwhen the capsule is almost as dense as a black hole (in which case the two bounds are equivalent).

The holographic and universal information boundsare far beyond the data storage capacities of anycurrent technology, and they greatly exceed thedensity of information on chromosomes and thethermodynamic entropy of water (c). —J.D.B.

Surface area ABlack hole

Mass m is sucked into black hole

Mass M

Diameter d

LIMITS ON INFORMATION DENSITY

Mass M + m

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Library of Congress

Liter of water(thermodynamic entropy)

Universal entropy bound(for an object with density of water)

Holographic bound

Internet

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1060

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1050

1040

1030

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1010

110– 4 106 108



one centimeter in diameter would beabout 1066 bits, roughly equal to the ther-modynamic entropy of a cube of water 10billion kilometers on a side.

The World as a HologramTHE GSL ALLOWS US to set bounds onthe information capacity of any isolatedphysical system, limits that refer to the in-formation at all levels of structure downto level X. In 1980 I began studying thefirst such bound, called the universal en-tropy bound, which limits how much en-tropy can be carried by a specified massof a specified size [see box on oppositepage]. A related idea, the holographicbound, was devised in 1995 by LeonardSusskind of Stanford University. It lim-its how much entropy can be containedin matter and energy occupying a speci-fied volume of space.

In his work on the holographic bound,Susskind considered any approximatelyspherical isolated mass that is not itself ablack hole and that fits inside a closed sur-face of area A. If the mass can collapse toa black hole, that hole will end up with ahorizon area smaller than A. The blackhole entropy is therefore smaller than A⁄ 4.According to the GSL, the entropy of thesystem cannot decrease, so the mass’soriginal entropy cannot have been biggerthan A⁄ 4. It follows that the entropy of anisolated physical system with boundaryarea A is necessarily less than A⁄ 4. What ifthe mass does not spontaneously col-lapse? In 2000 I showed that a tiny blackhole can be used to convert the system toa black hole not much different from theone in Susskind’s argument. The bound is

therefore independent of the constitutionof the system or of the nature of level X.It just depends on the GSL.

We can now answer some of those elu-sive questions about the ultimate limits ofinformation storage. A device measuringa centimeter across could in principle holdup to 1066 bits—a mind-boggling amount.The visible universe contains at least10100

bits of entropy, which could in principlebe packed inside a sphere a tenth of alight-year across. Estimating the entropyof the universe is a difficult problem, how-ever, and much larger numbers, requiringa sphere almost as big as the universe it-self, are entirely plausible.

But it is another aspect of the holo-graphic bound that is truly astonishing.Namely, that the maximum possible en-tropy depends on the boundary area in-stead of the volume. Imagine that we arepiling up computer memory chips in a bigheap. The number of transistors—the to-tal data storage capacity—increases withthe volume of the heap. So, too, does thetotal thermodynamic entropy of all thechips. Remarkably, though, the theoreti-cal ultimate information capacity of thespace occupied by the heap increases onlywith the surface area. Because volume in-creases more rapidly than surface area, atsome point the entropy of all the chipswould exceed the holographic bound. Itwould seem that either the GSL or ourcommonsense ideas of entropy and infor-mation capacity must fail. In fact, whatfails is the pile itself: it would collapse un-der its own gravity and form a black holebefore that impasse was reached. There-after each additional memory chip would

increase the mass and surface area of theblack hole in a way that would continueto preserve the GSL.

This surprising result—that informa-tion capacity depends on surface area—

has a natural explanation if the holo-graphic principle (proposed in 1993 byNobelist Gerard ’t Hooft of the Univer-sity of Utrecht in the Netherlands andelaborated by Susskind) is true. In theeveryday world, a hologram is a specialkind of photograph that generates a fullthree-dimensional image when it is illu-minated in the right manner. All the in-formation describing the 3-D scene is en-coded into the pattern of light and darkareas on the two-dimensional piece offilm, ready to be regenerated. The holo-graphic principle contends that an ana-logue of this visual magic applies to thefull physical description of any system oc-cupying a 3-D region: it proposes that an-other physical theory defined only on the2-D boundary of the region completelydescribes the 3-D physics. If a 3-D systemcan be fully described by a physical theo-ry operating solely on its 2-D boundary,one would expect the information con-tent of the system not to exceed that of thedescription on the boundary.

A Universe Painted on Its BoundaryCAN WE APPLY the holographic prin-ciple to the universe at large? The realuniverse is a 4-D system: it has volumeand extends in time. If the physics of ouruniverse is holographic, there would bean alternative set of physical laws, oper-ating on a 3-D boundary of spacetime

THE INFORMATION CONTENT of a pile of computer chips increases in proportionwith the number of chips or, equivalently, the volume they occupy. That simplerule must break down for a large enough pile of chips because eventually theinformation would exceed the holographic bound, which depends on thesurface area, not the volume. The “breakdown” occurs when theimmense pile of chips collapses to form a black hole.



somewhere, that would be equivalent toour known 4-D physics. We do not yetknow of any such 3-D theory that worksin that way. Indeed, what surface shouldwe use as the boundary of the universe?One step toward realizing these ideas isto study models that are simpler than ourreal universe.

A class of concrete examples of theholographic principle at work involvesso-called anti–de Sitter spacetimes. Theoriginal de Sitter spacetime is a model uni-verse first obtained by Dutch astronomerWillem de Sitter in 1917 as a solution ofEinstein’s equations, including the repul-sive force known as the cosmological con-stant. De Sitter’s spacetime is empty, ex-pands at an accelerating rate and is veryhighly symmetrical. In 1997 astronomersstudying distant supernova explosionsconcluded that our universe now expandsin an accelerated fashion and will proba-bly become increasingly like a de Sitterspacetime in the future. Now, if the re-pulsion in Einstein’s equations is changedto attraction, de Sitter’s solution turnsinto the anti–de Sitter spacetime, whichhas equally as much symmetry. More im-portant for the holographic concept, itpossesses a boundary, which is located“at infinity” and is a lot like our everydayspacetime.

Using anti–de Sitter spacetime, the-orists have devised a concrete exampleof the holographic principle at work: auniverse described by superstring theoryfunctioning in an anti–de Sitter space-time is completely equivalent to a quan-tum field theory operating on the bound-ary of that spacetime [see box above].Thus, the full majesty of superstring the-ory in an anti–de Sitter universe is paint-ed on the boundary of the universe. JuanMaldacena, then at Harvard University,first conjectured such a relation in 1997for the 5-D anti–de Sitter case, and it waslater confirmed for many situations byEdward Witten of the Institute for Ad-vanced Study in Princeton, N.J., andSteven S. Gubser, Igor R. Klebanov andAlexander M. Polyakov of PrincetonUniversity. Examples of this holograph-ic correspondence are now known forspacetimes with a variety of dimensions.

This result means that two ostensiblyvery different theories—not even actingin spaces of the same dimension—areequivalent. Creatures living in one of theseuniverses would be incapable of deter-mining if they inhabited a 5-D universedescribed by string theory or a 4-D onedescribed by a quantum field theory ofpoint particles. (Of course, the structuresof their brains might give them an over-

whelming “commonsense” prejudice infavor of one description or another, injust the way that our brains construct aninnate perception that our universe hasthree spatial dimensions; see the illustra-tion on the opposite page.)

The holographic equivalence can al-low a difficult calculation in the 4-Dboundary spacetime, such as the behaviorof quarks and gluons, to be traded for an-other, easier calculation in the highly sym-metric, 5-D anti–de Sitter spacetime. Thecorrespondence works the other way,too. Witten has shown that a black holein anti–de Sitter spacetime corresponds tohot radiation in the alternative physicsoperating on the bounding spacetime.The entropy of the hole—a deeply myste-rious concept—equals the radiation’s en-tropy, which is quite mundane.

The Expanding UniverseHIGHLY SYMMETRIC and empty, the5-D anti–de Sitter universe is hardly likeour universe existing in 4-D, filled withmatter and radiation, and riddled with vi-olent events. Even if we approximate ourreal universe with one that has matter andradiation spread uniformly throughout,we get not an anti–de Sitter universe butrather a “Friedmann-Robertson-Walker”universe. Most cosmologists today concur

TWO UNIVERSES of different dimension andobeying disparate physical laws are renderedcompletely equivalent by the holographicprinciple. Theorists have demonstrated thisprinciple mathematically for a specific type offive-dimensional spacetime (“anti–de Sitter”)and its four-dimensional boundary. In effect, the5-D universe is recorded like a hologram on the 4-D surface at its periphery. Superstring theoryrules in the 5-D spacetime, but a so-calledconformal field theory of point particles operates on the 4-D hologram. A black hole in the 5-D spacetime is equivalent to hot radiationon the hologram—for example, the hole and theradiation have the same entropy even thoughthe physical origin of the entropy is completelydifferent for each case. Although these twodescriptions of the universe seem utterlyunalike, no experiment could distinguishbetween them, even in principle. —J.D.B.

5-Dimensional anti–de Sitter spacetime

Superstrings

Conformal fields Hot radiation

4-Dimensional flat spacetime(hologram)

Black hole

A HOLOGRAPHIC SPACETIME



that our universe resembles an FRW uni-verse, one that is infinite, has no boundaryand will go on expanding ad infinitum.

Does such a universe conform to theholographic principle or the holographicbound? Susskind’s argument based oncollapse to a black hole is of no help here.Indeed, the holographic bound deducedfrom black holes must break down in auniform expanding universe. The entropyof a region uniformly filled with matterand radiation is truly proportional to itsvolume. A sufficiently large region willtherefore violate the holographic bound.

In 1999 Raphael Bousso, then at Stan-ford, proposed a modified holographicbound, which has since been found towork even in situations where the boundswe discussed earlier cannot be applied.Bousso’s formulation starts with any suit-able 2-D surface; it may be closed like asphere or open like a sheet of paper. Onethen imagines a brief burst of light issuingsimultaneously and perpendicularly fromall over one side of the surface. The onlydemand is that the imaginary light raysare converging to start with. Light emit-ted from the inner surface of a sphericalshell, for instance, satisfies that require-ment. One then considers the entropy ofthe matter and radiation that these imag-inary rays traverse, up to the points wherethey start crossing. Bousso conjecturedthat this entropy cannot exceed the en-tropy represented by the initial surface—

one quarter of its area, measured inPlanck areas. This is a different way of tal-lying up the entropy than that used in theoriginal holographic bound. Bousso’sbound refers not to the entropy of a re-gion at one time but rather to the sum ofentropies of locales at a variety of times:those that are “illuminated” by the lightburst from the surface.

Bousso’s bound subsumes other en-tropy bounds while avoiding their limi-tations. Both the universal entropybound and the ’t Hooft-Susskind form ofthe holographic bound can be deducedfrom Bousso’s for any isolated systemthat is not evolving rapidly and whosegravitational field is not strong. Whenthese conditions are overstepped—as fora collapsing sphere of matter already in-side a black hole—these bounds eventu-

ally fail, whereas Bousso’s bound con-tinues to hold. Bousso has also shownthat his strategy can be used to locate the2-D surfaces on which holograms of theworld can be set up.

Augurs of a RevolutionRESEARCHERS HAVE proposed manyother entropy bounds. The proliferationof variations on the holographic motifmakes it clear that the subject has not yetreached the status of physical law. Butalthough the holographic way of think-ing is not yet fully understood, it seemsto be here to stay. And with it comes arealization that the fundamental belief,prevalent for 50 years, that field theoryis the ultimate language of physics mustgive way. Fields, such as the electromag-netic field, vary continuously from pointto point, and they thereby describe an in-finity of degrees of freedom. Superstring

theory also embraces an infinite numberof degrees of freedom. Holography re-stricts the number of degrees of freedomthat can be present inside a boundingsurface to a finite number; field theorywith its infinity cannot be the final story.Furthermore, even if the infinity is tamed,the mysterious dependence of informa-tion on surface area must be somehowaccommodated.

Holography may be a guide to a bettertheory. What is the fundamental theorylike? The chain of reasoning involvingholography suggests to some, notably LeeSmolin of the Perimeter Institute for The-oretical Physics in Waterloo, that such a fi-nal theory must be concerned not withfields, not even with spacetime, but ratherwith information exchange among physi-cal processes. If so, the vision of informa-tion as the stuff the world is made of willhave found a worthy embodiment.

Black Hole Thermodynamics. Jacob D. Bekenstein in Physics Today, Vol. 33, No. 1, pages 24–31; January 1980.

Black Holes and Time Warps: Einstein’s Outrageous Legacy. Kip S. Thorne. W. W. Norton, 1995.

Black Holes and the Information Paradox. Leonard Susskind in Scientific American, Vol. 276, No. 4, pages 52–57; April 1997.

The Universe in a Nutshell. Stephen Hawking. Bantam Books, 2001.

Three Roads to Quantum Gravity. Lee Smolin. Basic Books, 2002.


OUR INNATE PERCEPTIONthat the world is three-dimensional could be anextraordinary illusion.



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STRING THEORYof

THE FUTURE

A Conversation with Brian Greene

String theory used to get everyone all tied up in knots. Even itspractitioners fretted about how complicated it was, while other physicistsmocked its lack of experimental predictions. The rest of the world was largelyoblivious. Scientists could scarcely communicate just why string theory wasso exciting—why it could fulfill Albert Einstein’s dream of the ultimate unifiedtheory, how it could give insight into such deep questions as why the universeexists at all. But in the mid-1990s the theory started to click togetherconceptually. It made some testable, if qualified, predictions. The outsideworld began to pay attention. Woody Allen satirized the theory in a NewYorker column this past July—probably the first time anyone has used Calabi-Yau spaces to make a point about interoffice romance.

Few people can take more credit for demystifying string theory thanBrian Greene, a Columbia University physics professor and a major contributorto the theory. His 1999 book The Elegant Universe reached number four onthe New York Times best-seller list and was a finalist for the Pulitzer Prize.Greene is now host of a three-part Nova series on PBS and has just complet-ed a book on the nature of space and time. Scientific American staff edi-tor George Musser recently spoke with him over a plate of stringy spaghetti.Here is an abridged, edited version of that conversation.

Q&ABRIANGREENE


originally publishedNovember 2003



SCIENTIFIC AMERICAN: Sometimes when our read-ers hear the words “string theory” or “cosmology,” theythrow up their hands and say, “I’ll never understand it.”

BRIAN GREENE: I’ve definitely encountered a certainamount of intimidation at the outset when it comes toideas like string theory or cosmology. But what I havefound is that the basic interest is so widespread and sodeep in most people that I’ve spoken with, that there isa willingness to go a little bit further than you mightwith other subjects that are more easily taken in.

SA: I noticed that at several points in The Elegant Uni-verse, you first gave a rough idea of the physics conceptsand then the detailed version.

BG: I found that to be a useful way of going about it, es-pecially in the harder parts. It gives the reader permis-sion: If the rough idea is the level at which you want totake it in, that’s great; feel free to skip this next stuff. Ifnot, go for it. I like to say things more than one way. Ijust think that when it comes to abstract ideas, you needmany roads into them. From the scientific point of view,if you stick with one road, I think you really compro-mise your ability to make breakthroughs. I think that’s

really what breakthroughs are about. Everybody’s look-ing at a problem one way, and you come at it from theback. That different way of getting there somehow re-veals things that the other approach didn’t.

SA: What are some examples of that back-door approach?

BG: Well, probably the biggest ones are Ed Witten’sbreakthroughs. Ed [of the Institute for Advanced Studyin Princeton, N.J.] just walked up the mountain andlooked down and saw the connections that nobody elsesaw and in that way united the five string theories thatpreviously were thought to be completely distinct. It wasall out there; he just took a different perspective, andbang, it all came together. And that’s genius.

To me that suggests what a fundamental discoveryis. The universe in a sense guides us toward truths, be-

cause those truths are the things that govern what wesee. If we’re all being governed by what we see, we’re allbeing steered in the same direction. Therefore, the dif-ference between making a breakthrough and not oftencan be just a small element of perception, either true per-ception or mathematical perception, that puts things to-gether in a different way.

SA: Do you think that these discoveries would have beenmade without the intervention of genius?

BG: Well, it’s tough to say. In the case of string theory,I think so, because the pieces of the puzzle were reallybecoming clearer and clearer. It may have been five or10 years later, but I suspect it would have happened. Butwith general relativity, I don’t know. General relativity

is such a leap, such a monumental rethinking of space,time and gravity, that it’s not obvious to me how andwhen that would have happened without Einstein.

SA: Are there examples in string theory that you thinkare analogous to that huge leap?

BG: I think we’re still waiting for a leap of that magni-tude. String theory has been built up out of a lot of small-er ideas that a lot of people have contributed and beenslowly stitching together into an ever more impressivetheoretical edifice. But what idea sits at the top of that ed-ifice, we still don’t really know. When we do have thatidea, I believe that it will be like a beacon shining down;it will illuminate the edifice, and it will also, I believe, giveanswers to critical questions that remain unresolved.

SA: In the case of relativity, you had the equivalenceprinciple and general covariance in that beacon role. Inthe Standard Model, it’s gauge invariance. In The Ele-gant Universe you suggested the holographic principlecould be that principle for string theory [see also “In- N

OVA

The difference betweenmaking a breakthrough andnot can often be just a small

element of perception.

Q&ABRIANGREENE



formation in the Holographic Universe,” by Jacob D.Bekenstein; Scientific American, August]. What’syour thinking on that now?

BG: Well, the past few years have only seen the holo-graphic principle rise to a yet greater prominence and be-lievability. Back in the mid-’90s, shortly after the holo-graphic ideas were suggested, the supporting ideas wererather abstract and vague, all based upon features ofblack holes: Black hole entropy resides on the surface;therefore, maybe the degrees of freedom reside on thesurface; therefore, maybe that’s true of all regions thathave a horizon; maybe it’s true of cosmological horizons;maybe we’re living within a cosmological region whichhas its true degrees of freedom far away. Wonderfullystrange ideas, but the supporting evidence was meager.

But that changed with the work of Juan Maldacena[of the Institute for Advanced Study in Princeton, N.J.],in which he found an explicit example within string the-ory, where physics in the bulk—that is, in the arena thatwe consider to be real—would be exactly mirrored byphysics taking place on a bounding surface. There’d beno difference in terms of the ability of either descriptionto truly describe what’s going on, yet in detail the de-scriptions would be vastly different. One would be in fivedimensions, the other in four. So even the number of di-mensions seems not to be something which you can counton, because there can be alternative descriptions thatwould accurately reflect the physics you’re observing.

So to my mind, that makes the abstract ideas nowconcrete; it makes you believe the abstract ideas. Andeven if the details of string theory change, I think, asmany others do—not everyone, though—that the holo-graphic idea will persist and will guide us. Whether ittruly is the idea, I don’t know. I don’t think so. But Ithink that it could well be one of the key stepping-stonestowards finding the essential ideas of the theory. It stepsoutside the details of the theory and just says, Here’s avery general feature of a world that has quantum me-chanics and gravity.

SA: Let’s talk a bit about loop quantum gravity andsome of the other approaches. You’ve always describedstring theory as the only game in town when it comes toquantum gravity. Do you still feel that way?

BG: Well, I think it’s the most fun game in town! But tobe fair, the loop-quantum-gravity community has madetremendous progress. There are still many very basicquestions that I don’t feel have been answered, not to mysatisfaction. But it’s a viable approach, and it’s greatthere are such large numbers of extremely talented peo-ple working on it. My hope—and it has been one that Lee

Smolin [of the Perimeter Institute in Waterloo, Canada]has championed—is that ultimately we’re developing thesame theory from different angles. It’s far from impossi-ble that we’re going down our route to quantum gravi-ty, they’re going down their route to quantum gravity,

and we’re going to meet someplace. Because it turns outthat many of their strengths are our weaknesses. Manyof our strengths are their weaknesses.

One weakness of string theory is that it’s so-calledbackground-dependent. We need to assume an existingspacetime within which the strings move. You’d hope,though, that a true quantum theory of gravity wouldhave spacetime emerge from its fundamental equations.They [the loop-quantum gravity researchers], howev-er, do have a background-independent formulation intheir approach, where spacetime does emerge more fun-damentally from the theory itself. On the other hand,we are able to make very direct contact with Einstein’sgeneral relativity on large scales. We see it in our equa-tions. They have some difficulty making contact with or-dinary gravity. So naturally, you’d think maybe onecould put together the strengths of each.

SA: Has that effort been made?

BG: Slowly. There are very few people who are really wellversed in both theories. These are both two huge subjects,and you can spend your whole life, every moment of yourworking day, just in your own subject, and you still won’tknow everything that’s going on. But many people areheading down that path and starting to think along thoselines, and there have been some joint meetings.

SA: If you have this background dependence, what hopeis there to really understand, in a deep sense, what spaceand time are?

BG: Well, you can chip away at the problem. For in-stance, even with background dependence, we’velearned things like mirror symmetry—there can be twospacetimes, one physics. We’ve learned topologychange—that space can evolve in ways that we wouldn’thave thought possible before. We’ve learned that the

Relativity is a monumentalrethinking of space and time.We’re still waiting for anotherleap of that magnitude.



microworld might be governed by noncommutativegeometry, where the coordinates, unlike real numbers,depend upon the order in which you multiply them. Soyou can get hints. You can get isolated glimpses ofwhat’s truly going on down there. But I think withoutthe background-independent formalism, it’s going to behard to put the pieces together on their own.

SA: The mirror symmetry is incredibly profound, be-cause it divorces spacetime geometry from physics. Theconnection between the two was always the Einsteinianprogram.

BG: That’s right. Now, it doesn’t divorce them com-pletely. It simply says that you’re missing half of the sto-ry. Geometry is tightly tied to physics, but it’s a two-to-one map. It’s not physics and geometry. It’s physics andgeometry-geometry, and which geometry you want topick is up to you. Sometimes using one geometry givesyou more insight than the other. Again, different waysof looking at one and the same physical system: two dif-ferent geometries and one physics. And people havefound there are mathematical questions about certainphysical and geometrical systems that people couldn’tanswer using the one geometry. Bring in the mirrorgeometry that had previously gone unrealized, and, allof a sudden, profoundly difficult questions, when trans-lated, were mind-bogglingly simple.

SA: Can you describe noncommutative geometry?

BG: Since the time of Descartes, we’ve found it verypowerful to label points by their coordinates, either onEarth by their latitude and longitude or in three-space

by the three Cartesian coordinates, x, y and z, that youlearn in high school. And we’ve always imagined thatthose numbers are like ordinary numbers, which havethe property that, when you multiply them together—

which is often an operation you need to do in physics—

the answer doesn’t depend on the order of operation: 3times 5 is 5 times 3. What we seem to be finding is thatwhen you coordinatize space on very small scales, thenumbers involved are not like 3’s and 5’s, which don’t

depend upon the order in which they’re multiplied.There’s a new class of numbers that do depend on theorder of multiplication.

They’re actually not that new, because for a longtime we have known of an entity called the matrix. Sureas shooting, matrix multiplication depends upon the or-der of multiplication. A times B does not equal B timesA if A and B are matrices. String theory seems to indi-cate that points described by single numbers are re-placed by geometrical objects described by matrices. Onbig scales, it turns out that these matrices become moreand more diagonal, and diagonal matrices do have theproperty that they commute when you multiply. It does-n’t matter how you multiply A times B if they’re diago-nal matrices. But then if you venture into the mi-croworld, the off-diagonal entries in the matrices getbigger and bigger and bigger until way down in thedepths, they are playing a significant part.

Noncommutative geometry is a whole new field ofgeometry that some people have been developing foryears without necessarily an application of physics inmind. The French mathematician Alain Connes has thisbig thick book called Noncommutative Geometry. Eu-clid and Gauss and Riemann and all those wonderfulgeometers were working in the context of commutativegeometry, and now Connes and others are taking off

and developing the newer structure of noncommutativegeometry.

SA: It is baffling to me—maybe it should be baffling—

that you would have to label points with a matrix orsome nonpure number. What does that mean?

BG: The way to think about it is: There is no notion of apoint. A point is an approximation. If there is a point,you should label it by a number. But the claim is that, onsufficiently small scales, that language of points becomessuch a poor approximation that it just isn’t relevant.When we talk about points in geometry, we really talkabout how something can move through points. It’s the N

OVA

Q&ABRIANGREENE

The theory seems to be able to give rise to many

different universes, of whichours seems to be only one.



motion of objects that ultimately is what’s relevant. Theirmotion, it turns out, can be more complicated than justsliding back and forth. All those motions are captured bya matrix. So rather than labeling an object by what pointit’s passing through, you need to label its motion by thismatrix of degrees of freedom.

SA: What is your current thinking on anthropic andmultiverse-type ideas? You talked about it in The Ele-

gant Universe in the context of whether there is somelimit to the explanatory power of string theory.

BG: I and many others have never been too happy withany of these anthropic ideas, largely because it seems tome that at any point in the history of science, you cansay, “Okay, we’re done, we can’t go any further, and thefinal answer to every currently unsolved question is:‘Things are the way they are because had they not beenthis way, we wouldn’t have been here to ask the ques-tion.’ ” So it sort of feels like a cop-out. Maybe that’s thewrong word. Not necessarily like a cop-out; it feels a lit-tle dangerous to me, because maybe you just needed fivemore years of hard work and you would have answeredthose unresolved questions, rather than just chalkingthem up to, “That’s just how it is.” So that’s my con-cern: that one doesn’t stop looking by virtue of havingthis fallback position.

But you know, it’s definitely the case that the an-thropic ideas have become more developed. They’renow real proposals whereby you would have many uni-verses, and those many universes could all have differ-ent properties, and it very well could be that we’re sim-ply in this one because the properties are right for us tobe here, and we’re not in those others because we could-n’t survive there. It’s less of just a mental exercise.

SA: String theory, and modern physics generally, seemto be approaching a single logical structure that had tobe the way it is; the theory is the way it is because there’sno other way it could be. On the one hand, that wouldargue against an anthropic direction. But on the otherhand, there’s a flexibility in the theory that leads you toan anthropic direction.

BG: The flexibility may or may not truly be there. Thatreally could be an artifact of our lack of full under-standing. But were I to go by what we understand to-day, the theory seems to be able to give rise to many dif-ferent worlds, of which ours seems to be potentially one,but not even necessarily a very special one. So yes, thereis a tension with the goal of absolute, rigid inflexibility.

SA: If you had other grad students waiting in the wings,what would you steer them to?

BG: Well, the big questions are, I think, the ones thatwe’ve discussed. Can we understand where space andtime come from? Can we figure out the fundamental ideasof string theory or M-theory? Can we show that this fun-damental idea yields a unique theory with the unique so-lution, which happens to be the world as we know it? Isit possible to test these ideas through astronomical ob-servations or through accelerator-based experiment?

Can we even take a step further back and under-stand why quantum mechanics had to be part and par-cel of the world as we know it? How many of the thingsthat we rely on at a very deep level in any physical the-ory that has a chance of being right—such as space, time,quantum mechanics—are truly essential, and how manyof them can be relaxed and potentially still yield theworld that appears close to ours?

Could physics have taken a different path thatwould have been experimentally as successful but com-pletely different? I don’t know. But I think it’s a real in-teresting question to ask. How much of what we believeis truly fundamentally driven in a unique way by dataand mathematical consistency, and how much of itcould have gone one way or another, and we just hap-pened to go down one path because that’s what we hap-pened to discover? Could beings on another planet havecompletely different sets of laws that somehow workjust as well as ours?

NO

VA

The full transcript of this conversation, with comments on everything from television to the arrow of time, is available at www.sciam.com

On the Web . . .

IF YOU WERE A STRING, spacetime might look something like this:six extra dimensions curled into a so-called Calabi-Yau shape.



little more than 100 years ago most people—

and most scientists—thought of matter ascontinuous. Although since ancient timessome philosophers and scientists had specu-lated that if matter were broken up into smallenough bits, it might turn out to be made upof very tiny atoms, few thought the existence

of atoms could ever be proved. Today we have imaged individualatoms and have studied the particles that compose them. Thegranularity of matter is old news.

In recent decades, physicists and mathematicians have askedif space is also made of discrete pieces. Is it continuous, as welearn in school, or is it more like a piece of cloth, woven out ofindividual fibers? If we could probe to size scales that were smallenough, would we see “atoms” of space, irreducible pieces ofvolume that cannot be broken into anything smaller? And whatabout time: Does nature change continuously, or does the world D

USA

N P

ETR

ICIC

We perceive space andtime to be continuous,

but if the amazingtheory of loop quantumgravity is correct, they

actually come indiscrete pieces

By Lee Smolin43 SCIENTIFIC AMERICAN EXCLUSIVE ONLINE ISSUE FEBRUARY 2004

originally published January 2004


evolve in series of very tiny steps, acting more like a digitalcomputer?

The past 16 years have seen great progress on these ques-tions. A theory with the strange name of “loop quantum gravi-ty” predicts that space and time are indeed made of discretepieces. The picture revealed by calculations carried out withinthe framework of this theory is both simple and beautiful. Thetheory has deepened our understanding of puzzling phenome-na having to do with black holes and the big bang. Best of all, itis testable; it makes predictions for experiments that can be donein the near future that will enable us to detect the atoms of space,if they are really there.

QuantaMY COLLEAGUES AND I developed the theory of loop quan-tum gravity while struggling with a long-standing problem inphysics: Is it possible to develop a quantum theory of gravity?

To explain why this is an important question—and what it hasto do with the granularity of space and time—I must first say abit about quantum theory and the theory of gravity.

The theory of quantum mechanics was formulated in thefirst quarter of the 20th century, a development that was close-ly connected with the confirmation that matter is made of atoms.The equations of quantum mechanics require that certain quan-tities, such as the energy of an atom, can come only in specific,discrete units. Quantum theory successfully predicts the prop-erties and behavior of atoms and the elementary particles andforces that compose them. No theory in the history of sciencehas been more successful than quantum theory. It underlies ourunderstanding of chemistry, atomic and subatomic physics, elec-tronics and even biology.

In the same decades that quantum mechanics was being for-mulated, Albert Einstein constructed his general theory of rela-tivity, which is a theory of gravity. In his theory, the gravitational

AtomsofSpace and Time



force arises as a consequence of space andtime (which together form “spacetime”)being curved by the presence of matter. Aloose analogy is that of a bowling ballplaced on a rubber sheet along with amarble that is rolling around nearby. Theballs could represent the sun and theearth, and the sheet is space. The bowlingball creates a deep indentation in the rub-ber sheet, and the slope of this indentationcauses the marble to be deflected towardthe larger ball, as if some force—gravity—

were pulling it in that direction. Similar-ly, any piece of matter or concentration ofenergy distorts the geometry of spacetime,causing other particles and light rays to bedeflected toward it, a phenomenon wecall gravity.

Quantum theory and Einstein’s theo-ry of general relativity separately haveeach been fantastically well confirmed byexperiment—but no experiment has ex-plored the regime where both theoriespredict significant effects. The problem isthat quantum effects are most prominentat small size scales, whereas general rela-tivistic effects require large masses, so ittakes extraordinary circumstances tocombine both conditions.

Allied with this hole in the experi-mental data is a huge conceptual prob-lem: Einstein’s theory of general relativi-ty is thoroughly classical, or nonquan-tum. For physics as a whole to be logicallyconsistent, there has to be a theory thatsomehow unites quantum mechanics andgeneral relativity. This long-sought-aftertheory is called quantum gravity. Because

general relativity deals in the geometry ofspacetime, a quantum theory of gravitywill in addition be a quantum theory ofspacetime.

Physicists have developed a consider-able collection of mathematical proce-dures for turning a classical theory into aquantum one. Many theoretical physicistsand mathematicians have worked on ap-plying those standard techniques to gen-eral relativity. Early results were discour-aging. Calculations carried out in the1960s and 1970s seemed to show thatquantum theory and general relativitycould not be successfully combined. Con-sequently, something fundamentally newseemed to be required, such as addition-al postulates or principles not included in

quantum theory and general relativity, ornew particles or fields, or new entities ofsome kind. Perhaps with the right addi-tions or a new mathematical structure, aquantumlike theory could be developedthat would successfully approximate gen-eral relativity in the nonquantum regime.To avoid spoiling the successful predic-tions of quantum theory and general rel-ativity, the exotica contained in the fulltheory would remain hidden from exper-iment except in the extraordinary cir-cumstances where both quantum theoryand general relativity are expected to havelarge effects. Many different approachesalong these lines have been tried, withnames such as twistor theory, noncom-mutative geometry and supergravity.

An approach that is very popular withphysicists is string theory, which postu-lates that space has six or seven dimen-sions—all so far completely unobserved—

in addition to the three that we are famil-iar with. String theory also predicts theexistence of a great many new elementaryparticles and forces, for which there is sofar no observable evidence. Some re-searchers believe that string theory is sub-sumed in a theory called M-theory [see“The Theory Formerly Known as Strings,”by Michael J. Duff; Scientific Ameri-can, February 1998], but unfortunatelyno precise definition of this conjecturedtheory has ever been given. Thus, manyphysicists and mathematicians are con-vinced that alternatives must be studied.Our loop quantum gravity theory is thebest-developed alternative.

A Big LoopholeIN THE MID-1980S a few of us—in-cluding Abhay Ashtekar, now at Penn-sylvania State University, Ted Jacobson ofthe University of Maryland and CarloRovelli, now at the University of the Med-iterranean in Marseille—decided to reex-amine the question of whether quantummechanics could be combined consis-tently with general relativity using thestandard techniques. We knew that thenegative results from the 1970s had animportant loophole. Those calculationsassumed that the geometry of space iscontinuous and smooth, no matter howminutely we examine it, just as people D

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■ To understand the structure of space on the very smallest size scale, we must

turn to a quantum theory of gravity. Gravity is involved because Einstein’sgeneral theory of relativity reveals that gravity is caused by the warping ofspace and time.

■ By carefully combining the fundamental principles of quantum mechanics andgeneral relativity, physicists are led to the theory of “loop quantum gravity.” In this theory, the allowed quantum states of space turn out to be related todiagrams of lines and nodes called spin networks. Quantum spacetimecorresponds to similar diagrams called spin foams.

■ Loop quantum gravity predicts that space comes in discrete lumps, the smallestof which is about a cubic Planck length, or 10–99 cubic centimeter. Time proceeds indiscrete ticks of about a Planck time, or 10–43 second. The effects of this discretestructure might be seen in experiments in the near future.

Overview/Quantum Spacetime

SPACE IS WOVEN out of distinct threads.



had expected matter to be before the dis-covery of atoms. Some of our teachersand mentors had pointed out that if thisassumption was wrong, the old calcula-tions would not be reliable.

So we began searching for a way todo calculations without assuming thatspace is smooth and continuous. We in-sisted on not making any assumptionsbeyond the experimentally well testedprinciples of general relativity and quan-tum theory. In particular, we kept twokey principles of general relativity at theheart of our calculations.

The first is known as background in-dependence. This principle says that thegeometry of spacetime is not fixed. In-stead the geometry is an evolving, dy-namical quantity. To find the geometry,one has to solve certain equations that in-clude all the effects of matter and energy.Incidentally, string theory, as currentlyformulated, is not background indepen-dent; the equations describing the strings

are set up in a predetermined classical(that is, nonquantum) spacetime.

The second principle, known by theimposing name diffeomorphism invari-ance, is closely related to background in-dependence. This principle implies that,unlike theories prior to general relativity,one is free to choose any set of coordi-nates to map spacetime and express theequations. A point in spacetime is definedonly by what physically happens at it, notby its location according to some specialset of coordinates (no coordinates are spe-cial). Diffeomorphism invariance is verypowerful and is of fundamental impor-tance in general relativity.

By carefully combining these twoprinciples with the standard techniques ofquantum mechanics, we developed amathematical language that allowed us todo a computation to determine whetherspace is continuous or discrete. That cal-culation revealed, to our delight, thatspace is quantized. We had laid the foun-

dations of our theory of loop quantumgravity. The term “loop,” by the way,arises from how some computations inthe theory involve small loops markedout in spacetime.

The calculations have been redone bya number of physicists and mathemati-cians using a range of methods. Over theyears since, the study of loop quantumgravity has grown into a healthy field ofresearch, with many contributors aroundthe world; our combined efforts give usconfidence in the picture of spacetime Iwill describe.

Ours is a quantum theory of the struc-ture of spacetime at the smallest sizescales, so to explain how the theory workswe need to consider what it predicts for asmall region or volume. In dealing withquantum physics, it is essential to specifyprecisely what physical quantities are to be measured. To do so, we consider a region somewhere that is marked out by a boundary, B [see illustration below].

QUANTUM STATES OF VOLUME AND AREA

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A CENTRAL PREDICTION of the loop quantumgravity theory relates to volumes and areas.Consider a spherical shell that defines theboundary, B, of a region of space havingsome volume (above). According to classical(nonquantum) physics, the volume could be any positive realnumber. The loop quantum gravity theory says, however, thatthere is a nonzero absolute minimum volume (about one cubicPlanck length, or 10–99 cubic centimeter), and it restricts theset of larger volumes to a discrete series of numbers. Similarly,

there is a nonzero minimum area (about one square Plancklength, or 10–66 square centimeter) and a discrete series oflarger allowed areas. The discrete spectrum of allowed quantumareas (left) and volumes (center) is broadly similar to thediscrete quantum energy levels of a hydrogen atom (right).

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The boundary may be defined by somematter, such as a cast-iron shell, or it maybe defined by the geometry of spacetimeitself, as in the event horizon of a blackhole (a surface from within which evenlight cannot escape the black hole’s grav-itational clutches).

What happens if we measure the vol-ume of the region? What are the possibleoutcomes allowed by both quantum the-ory and diffeomorphism invariance? If

the geometry of space is continuous, theregion could be of any size and the mea-surement result could be any positive realnumber; in particular, it could be as closeas one wants to zero volume. But if thegeometry is granular, then the measure-ment result can come from just a discreteset of numbers and it cannot be smallerthan a certain minimum possible volume.The question is similar to asking howmuch energy electrons orbiting an atom-

ic nucleus have. Classical mechanics pre-dicts that that an electron can possess anyamount of energy, but quantum mechan-ics allows only specific energies (amountsin between those values do not occur).The difference is like that between themeasure of something that flows contin-uously, like the 19th-century conceptionof water, and something that can becounted, like the atoms in that water.

The theory of loop quantum gravity NAD

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VISUALIZING QUANTUM STATES OF VOLUMEDIAGRAMS CALLED SPIN NETWORKS are used by physicists who studyloop quantum gravity to represent quantum states of space at aminuscule scale. Some such diagrams correspond to polyhedra-shapedvolumes. For example, a cube (a) consists of a volume enclosed withinsix square faces. The corresponding spin network (b) has a dot, or node,representing the volume and six lines that represent the six faces. Thecomplete spin network has a number at the node to indicate the cube’svolume and a number on each line to indicate the area of thecorresponding face. Here the volume is eight cubic Planck lengths, andthe faces are each four square Planck lengths. (The rules of loopquantum gravity restrict the allowed volumes and areas to specificquantities: only certain combinations of numbers are allowed on thelines and nodes.)

If a pyramid sat on the cube’s top face (c), the line representingthat face in the spin network would connect the cube’s node to thepyramid’s node (d). The lines corresponding to the four exposed facesof the pyramid and the five exposed faces of the cube would stick outfrom their respective nodes. (The numbers have been omitted forsimplicity.)

In general, in a spin network, one quantum of area isrepresented by a single line (e), whereas an areacomposed of many quanta is represented by many lines( f ). Similarly, a quantum of volume is represented byone node ( g), whereas a larger volume takes manynodes ( h). If we have a region of space defined by aspherical shell, the volume inside the shell is given bythe sum of all the enclosed nodes and its surface area isgiven by the sum of all the lines that pierce it.

The spin networks are more fundamental than thepolyhedra: any arrangement of polyhedra can berepresented by a spin network in this fashion, but somevalid spin networks represent combinations of volumesand areas that cannot be drawn as polyhedra. Such spinnetworks would occur when space is curved by a stronggravitational field or in the course of quantumfluctuations of the geometry of space at the Planck scale.

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predicts that space is like atoms: there is adiscrete set of numbers that the volume-measuring experiment can return. Vol-ume comes in distinct pieces. Anotherquantity we can measure is the area of theboundary B. Again, calculations using thetheory return an unambiguous result: thearea of the surface is discrete as well. Inother words, space is not continuous. Itcomes only in specific quantum units ofarea and volume.

The possible values of volume andarea are measured in units of a quantitycalled the Planck length. This length is re-lated to the strength of gravity, the size ofquanta and the speed of light. It measuresthe scale at which the geometry of spaceis no longer continuous. The Plancklength is very small: 10–33 centimeter. Thesmallest possible nonzero area is about asquare Planck length, or 10–66 cm2. Thesmallest nonzero volume is approximate-ly a cubic Planck length, 10–99 cm3. Thus,the theory predicts that there are about1099 atoms of volume in every cubic cen-timeter of space. The quantum of volumeis so tiny that there are more such quantain a cubic centimeter than there are cubiccentimeters in the visible universe (1085).

Spin Networks WHAT ELSE DOES our theory tell usabout spacetime? To start with, what dothese quantum states of volume and arealook like? Is space made up of a lot of lit-tle cubes or spheres? The answer is no—

it’s not that simple. Nevertheless, we candraw diagrams that represent the quan-tum states of volume and area. To thoseof us working in this field, these diagramsare beautiful because of their connectionto an elegant branch of mathematics.

To see how these diagrams work,imagine that we have a lump of spaceshaped like a cube, as shown in the illus-tration on the opposite page. In our dia-grams, we would depict this cube as a dot,which represents the volume, with sixlines sticking out, each of which repre-sents one of the cube’s faces. We have towrite a number next to the dot to specifythe quantity of volume, and on each linewe write a number to specify the area ofthe face that the line represents.

Next, suppose we put a pyramid on

top of the cube. These two polyhedra,which share a common face, would be de-picted as two dots (two volumes) con-nected by one of the lines (the face thatjoins the two volumes). The cube has fiveother faces (five lines sticking out), andthe pyramid has four (four lines stickingout). It is clear how more complicatedarrangements involving polyhedra otherthan cubes and pyramids could be de-picted with these dot-and-line diagrams:each polyhedron of volume becomes adot, or node, and each flat face of a poly-hedron becomes a line, and the lines jointhe nodes in the way that the faces join thepolyhedra together. Mathematicians callthese line diagrams graphs.

Now in our theory, we throw awaythe drawings of polyhedra and just keepthe graphs. The mathematics that de-scribes the quantum states of volume andarea gives us a set of rules for how thenodes and lines can be connected andwhat numbers can go where in a diagram.Every quantum state corresponds to oneof these graphs, and every graph thatobeys the rules corresponds to a quantumstate. The graphs are a convenient short-hand for all the possible quantum statesof space. (The mathematics and other de-tails of the quantum states are too com-plicated to discuss here; the best we can

do is show some of the related diagrams.)The graphs are a better representation

of the quantum states than the polyhedraare. In particular, some graphs connect instrange ways that cannot be convertedinto a tidy picture of polyhedra. For ex-ample, whenever space is curved, thepolyhedra will not fit together properly inany drawing we could do, yet we can stilleasily draw a graph. Indeed, we can takea graph and from it calculate how muchspace is distorted. Because the distortionof space is what produces gravity, this ishow the diagrams form a quantum theo-ry of gravity.

For simplicity, we often draw thegraphs in two dimensions, but it is betterto imagine them filling three-dimensionalspace, because that is what they represent.Yet there is a conceptual trap here: thelines and nodes of a graph do not live atspecific locations in space. Each graph isdefined only by the way its pieces connecttogether and how they relate to well-de-fined boundaries such as boundary B. Thecontinuous, three-dimensional space thatyou are imagining the graphs occupy doesnot exist as a separate entity. All that ex-ist are the lines and nodes; they are space,and the way they connect defines thegeometry of space.

These graphs are called spin networksbecause the numbers on them are relatedto quantities called spins. Roger Penroseof the University of Oxford first proposedin the early 1970s that spin networksmight play a role in theories of quantumgravity. We were very pleased when wefound, in 1994, that precise calculationsconfirmed his intuition. Readers familiarwith Feynman diagrams should note thatour spin networks are not Feynman dia-grams, despite the superficial resemblance.Feynman diagrams represent quantuminteractions between particles, whichproceed from one quantum state to an-other. Our diagrams represent fixed quan-

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MATTER EXISTS at the nodes of the spin network.

LEE SMOLIN is a researcher at the Perimeter Institute for Theoretical Physics in Waterloo,Ontario, and an adjunct professor of physics at the University of Waterloo. He has a B.A. fromHampshire College and a Ph.D. from Harvard University and has been on the faculty of Yale,Syracuse and Pennsylvania State universities. In addition to his work on quantum gravity,he is interested in elementary particle physics, cosmology and the foundations of quan-tum theory. His 1997 book, The Life of the Cosmos (Oxford University Press), explored thephilosophical implications of developments in contemporary physics and cosmology.

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tum states of spatial volumes and areas.The individual nodes and edges of the

diagrams represent extremely small re-gions of space: a node is typically a vol-ume of about one cubic Planck length,and a line is typically an area of about onesquare Planck length. But in principle,nothing limits how big and complicated aspin network can be. If we could draw adetailed picture of the quantum state ofour universe—the geometry of its space,as curved and warped by the gravitationof galaxies and black holes and every-thing else—it would be a gargantuan spinnetwork of unimaginable complexity,with approximately 10184 nodes.

These spin networks describe thegeometry of space. But what about all thematter and energy contained in thatspace? How do we represent particles andfields occupying positions and regions ofspace? Particles, such as electrons, corre-spond to certain types of nodes, which arerepresented by adding more labels onnodes. Fields, such as the electromagnet-ic field, are represented by additional la-bels on the lines of the graph. We repre-sent particles and fields moving throughspace by these labels moving in discretesteps on the graphs.

Moves and FoamsPARTICLES AND FIELDS are not theonly things that move around. Accordingto general relativity, the geometry ofspace changes in time. The bends andcurves of space change as matter and en-ergy move, and waves can pass throughit like ripples on a lake [see “Ripples inSpace and Time,” by W. Wayt Gibbs;Scientific American, April 2002]. Inloop quantum gravity, these processesare represented by changes in the graphs.They evolve in time by a succession ofcertain “moves” in which the connectiv-ity of the graphs changes [see illustrationon next page].

When physicists describe phenomenaquantum-mechanically, they computeprobabilities for different processes. Wedo the same when we apply loop quan-tum gravity theory to describe phenome-na, whether it be particles and fields mov-ing on the spin networks or the geometryof space itself evolving in time. In partic-

ular, Thomas Thiemann of the PerimeterInstitute for Theoretical Physics in Wa-terloo, Ontario, has derived precise quan-tum probabilities for the spin networkmoves. With these the theory is com-pletely specified: we have a well-definedprocedure for computing the probabilityof any process that can occur in a worldthat obeys the rules of our theory. It re-mains only to do the computations andwork out predictions for what could beobserved in experiments of one kind oranother.

Einstein’s theories of special and gen-eral relativity join space and time togeth-er into the single, merged entity known asspacetime. The spin networks that repre-sent space in loop quantum gravity theo-ry accommodate the concept of spacetimeby becoming what we call spin “foams.”With the addition of another dimen-sion—time—the lines of the spin net-works grow to become two-dimension-al surfaces, and the nodes grow to be-come lines. Transitions where the spinnetworks change (the moves discussedearlier) are now represented by nodeswhere the lines meet in the foam. Thespin foam picture of spacetime was pro-posed by several people, including CarloRovelli, Mike Reisenberger (now of theUniversity of Montevideo), John Barrettof the University of Nottingham, LouisCrane of Kansas State University, JohnBaez of the University of California atRiverside and Fotini Markopoulou of the

Perimeter Institute for Theoretical Physics.In the spacetime way of looking at

things, a snapshot at a specific time is likea slice cutting across the spacetime. Tak-ing such a slice through a spin foam pro-duces a spin network. But it would bewrong to think of such a slice as movingcontinuously, like a smooth flow of time.Instead, just as space is defined by a spinnetwork’s discrete geometry, time is de-fined by the sequence of distinct movesthat rearrange the network, as shown inthe illustration on the opposite page. Inthis way time also becomes discrete. Timeflows not like a river but like the tickingof a clock, with “ticks” that are about aslong as the Planck time:10–43 second. Or,more precisely, time in our universe flowsby the ticking of innumerable clocks—ina sense, at every location in the spin foamwhere a quantum “move” takes place, aclock at that location has ticked once.

Predictions and TestsI HAVE OUTLINED what loop quantumgravity has to say about space and time atthe Planck scale, but we cannot verify thetheory directly by examining spacetime onthat scale. It is too small. So how can wetest the theory? An important test iswhether one can derive classical generalrelativity as an approximation to loopquantum gravity. In other words, if thespin networks are like the threads woveninto a piece of cloth, this is analogous toasking whether we can compute the rightelastic properties for a sheet of the mater-ial by averaging over thousands ofthreads. Similarly, when averaged overmany Planck lengths, do spin networksdescribe the geometry of space and its evo-lution in a way that agrees roughly withthe “smooth cloth” of Einstein’s classicaltheory? This is a difficult problem, but re-cently researchers have made progress forsome cases, for certain configurations ofthe material, so to speak. For example,long-wavelength gravitational wavespropagating on otherwise flat (uncurved)space can be described as excitations ofspecific quantum states described by theloop quantum gravity theory.

Another fruitful test is to see whatloop quantum gravity has to say aboutone of the long-standing mysteries of D

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TIME ADVANCES by the discrete ticks ofinnumerable clocks.



gravitational physics and quantum theo-ry: the thermodynamics of black holes, inparticular their entropy, which is relatedto disorder. Physicists have computedpredictions regarding black hole thermo-dynamics using a hybrid, approximatetheory in which matter is treated quan-

tum-mechanically but spacetime is not. Afull quantum theory of gravity, such asloop quantum gravity, should be able toreproduce these predictions. Specifically,in the 1970s Jacob D. Bekenstein, now atthe Hebrew University of Jerusalem, in-ferred that black holes must be ascribed

an entropy proportional to their surfacearea [see “Information in a HolographicUniverse,” by Jacob D. Bekenstein; Sci-entific American, August 2003]. Short-ly after, Stephen Hawking deduced thatblack holes, particularly small ones, mustemit radiation. These predictions are

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EVOLUTION OF GEOMETRY IN TIMECHANGES IN THE SHAPE of space—such as those occurring when matterand energy move around within it and when gravitational waves flowby—are represented by discrete rearrangements, or moves, of the spinnetwork. In a, a connected group of three volume quanta merge tobecome a single volume quantum; the reverse process can also occur. Inb, two volumes divide up space and connect to adjoining volumes in adifferent way. Represented as polyhedra, the two polyhedra wouldmerge on their common face and then split like a crystal cleaving on adifferent plane. These spin-network moves take place not only whenlarge-scale changes in the geometry of space occur but also incessantlyas quantum fluctuations at the Planck scale.

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ANOTHER WAY to represent moves is toadd the time dimension to a spinnetwork—the result is called a spin foam(c). The lines of the spin network becomeplanes, and the nodes become lines.Taking a slice through a spin foam at aparticular time yields a spin network;taking a series of slices at different timesproduces frames of a movie showing thespin network evolving in time (d). Butnotice that the evolution, which at firstglance appears to be smooth andcontinuous, is in fact discontinuous. Allthe spin networks that include theorange line ( first three frames shown)represent exactly the same geometry ofspace. The length of the orange line doesn’t matter—all that matters for thegeometry is how the lines are connected and what number labels each line. Thoseare what define how the quanta of volume and area are arranged and how big theyare. Thus, in d, the geometry remains constant during the first three frames, with 3quanta of volume and 6 quanta of surface area. Then the geometry changesdiscontinuously, becoming a single quantum of volume and 3 quanta of surfacearea, as shown in the last frame. In this way, time as defined by a spin foam evolvesby a series of abrupt, discrete moves, not by a continuous flow.

Although speaking of such sequences as frames of a movie is helpful forvisualization, the more correct way to understand the evolution of the geometry isas discrete ticks of a clock. At one tick the orange quantum of area is present; at thenext tick it is gone—in fact, the disappearance of the orange quantum of areadefines the tick. The difference in time from one tick to the next is approximately thePlanck time, 10–43 second. But time does not exist in between the ticks; there is no“in between,” in the same way that there is no water in between two adjacentmolecules of water.

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among the greatest results of theoreticalphysics in the past 30 years.

To do the calculation in loop quan-tum gravity, we pick the boundary B tobe the event horizon of a black hole.When we analyze the entropy of the rel-evant quantum states, we get preciselythe prediction of Bekenstein. Similarly,the theory reproduces Hawking’s predic-tion of black hole radiation. In fact, itmakes further predictions for the finestructure of Hawking radiation. If a mi-croscopic black hole is ever observed, thisprediction could be tested by studying thespectrum of radiation it emits. That maybe far off in time, however, because wehave no technology to make black holes,small or otherwise.

Indeed, any experimental test of loopquantum gravity would appear at first tobe an immense technological challenge.The problem is that the characteristic ef-fects described by the theory become sig-nificant only at the Planck scale, the verytiny size of the quanta of area and vol-ume. The Planck scale is 16 orders ofmagnitude below the scale probed in thehighest-energy particle accelerators cur-rently planned (higher energy is needed to

probe shorter distance scales). Because wecannot reach the Planck scale with an ac-celerator, many people have held out lit-tle hope for the confirmation of quantumgravity theories.

In the past several years, however, afew imaginative young researchers havethought up new ways to test the predic-tions of loop quantum gravity that can bedone now. These methods depend on thepropagation of light across the universe.When light moves through a medium, itswavelength suffers some distortions, lead-ing to effects such as bending in water andthe separation of different wavelengths,or colors. These effects also occur for lightand particles moving through the discretespace described by a spin network.

Unfortunately, the magnitude of theeffects is proportional to the ratio of thePlanck length to the wavelength. For vis-ible light, this ratio is smaller than 10–28;even for the most powerful cosmic raysever observed, it is about one billionth.For any radiation we can observe, the ef-fects of the granular structure of space arevery small. What the young researchersspotted is that these effects accumulatewhen light travels a long distance. And we

detect light and particles that come frombillions of light years away, from eventssuch as gamma-ray bursts [see “TheBrightest Explosions in the Universe,” byNeil Gehrels, Luigi Piro and Peter J. T.Leonard; Scientific American, Decem-ber 2002].

A gamma-ray burst spews out pho-tons in a range of energies in a very briefexplosion. Calculations in loop quantumgravity, by Rodolfo Gambini of the Uni-versity of the Republic in Uruguay, JorgePullin of Louisiana State University andothers, predict that photons of differentenergies should travel at slightly differentspeeds and therefore arrive at slightly dif-ferent times [see illustration above]. Wecan look for this effect in data from satel-lite observations of gamma-ray bursts. Sofar the precision is about a factor of 1,000below what is needed, but a new satelliteobservatory called GLAST, planned for2006, will have the precision required.

The reader may ask if this resultwould mean that Einstein’s theory of spe-cial relativity is wrong when it predicts auniversal speed of light. Several people,including Giovanni Amelino-Camelia ofthe University of Rome “La Sapienza” N

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AN EXPERIMENTAL TEST

RADIATION from distant cosmic explosions called gamma-raybursts might provide a way to test whether the theory of loopquantum gravity is correct. Gamma-ray bursts occur billions oflight-years away and emit a huge amount of gamma rays withina short span. According to loop quantum gravity, each photonoccupies a region of lines at each instant as it moves throughthe spin network that is space (in reality a very large number oflines, not just the five depicted here). The discrete nature of

space causes higher-energy gamma rays to travel slightlyfaster than lower-energy ones. The difference is tiny, but itseffect steadily accumulates during the rays’ billion-yearvoyage. If a burst’s gamma rays arrive at Earth at slightlydifferent times according to their energy, that would beevidence for loop quantum gravity. The GLAST satellite, which isscheduled to be launched in 2006, will have the requiredsensitivity for this experiment.

Gamma-ray burst

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and João Magueijo of Imperial CollegeLondon, as well as myself, have devel-oped modified versions of Einstein’s the-ory that will accommodate high-energyphotons traveling at different speeds. Ourtheories propose that the universal speedis the speed of very low energy photonsor, equivalently, long-wavelength light.

Another possible effect of discretespacetime involves very high energy cos-mic rays. More than 30 years ago re-searchers predicted that cosmic-ray pro-tons with an energy greater than 3 × 1019

electron volts would scatter off the cosmicmicrowave background that fills spaceand should therefore never reach theearth. Puzzlingly, a Japanese experimentcalled AGASA has detected more than 10cosmic rays with an energy over this lim-it. But it turns out that the discrete struc-ture of space can raise the energy requiredfor the scattering reaction, allowing high-er-energy cosmic-ray protons to reach theearth. If the AGASA observations holdup, and if no other explanation is found,then it may turn out that we have alreadydetected the discreteness of space.

The CosmosIN ADDITION to making predictionsabout specific phenomena such as high-energy cosmic rays, loop quantum gravi-ty has opened up a new window throughwhich we can study deep cosmologicalquestions such as those relating to the ori-gins of our universe. We can use the the-ory to study the earliest moments of timejust after the big bang. General relativitypredicts that there was a first moment oftime, but this conclusion ignores quan-tum physics (because general relativity isnot a quantum theory). Recent loop quan-tum gravity calculations by Martin Bo-jowald of the Max Planck Institute forGravitational Physics in Golm, Germany,indicate that the big bang is actually a bigbounce; before the bounce the universewas rapidly contracting. Theorists arenow hard at work developing predictionsfor the early universe that may be testablein future cosmological observations. It isnot impossible that in our lifetime wecould see evidence of the time before thebig bang.

A question of similar profundity con-

cerns the cosmological constant—a pos-itive or negative energy density that couldpermeate “empty” space. Recent obser-vations of distant supernovae and thecosmic microwave background stronglyindicate that this energy does exist and ispositive, which accelerates the universe’sexpansion [see “The Quintessential Uni-verse,” by Jeremiah P. Ostriker and PaulJ. Steinhardt; Scientific American,January 2001]. Loop quantum gravityhas no trouble incorporating the positiveenergy density. This fact was demon-strated in 1990, when Hideo Kodama ofKyoto University wrote down equationsdescribing an exact quantum state of auniverse having a positive cosmologicalconstant.

Many open questions remain to beanswered in loop quantum gravity. Someare technical matters that need to be clar-ified. We would also like to understandhow, if at all, special relativity must bemodified at extremely high energies. Sofar our speculations on this topic are notsolidly linked to loop quantum gravitycalculations. In addition, we would like toknow that classical general relativity is agood approximate description of the the-ory for distances much larger than thePlanck length, in all circumstances. (Atpresent we know only that the approxi-mation is good for certain states that de-

scribe rather weak gravitational wavespropagating on an otherwise flat space-time.) Finally, we would like to under-stand whether or not loop quantum grav-ity has anything to say about unification:Are the different forces, including gravi-ty, all aspects of a single, fundamentalforce? String theory is based on a partic-ular idea about unification, but we alsohave ideas for achieving unification withloop quantum gravity.

Loop quantum gravity occupies avery important place in the developmentof physics. It is arguably the quantum the-ory of general relativity, because it makesno extra assumptions beyond the basicprinciples of quantum theory and relativ-ity theory. The remarkable departure thatit makes—proposing a discontinuousspacetime described by spin networks andspin foams—emerges from the mathe-matics of the theory itself, rather than be-ing inserted as an ad hoc postulate.

Still, everything I have discussed istheoretical. It could be that in spite of allI have described here, space really is con-tinuous, no matter how small the scale weprobe. Then physicists would have toturn to more radical postulates, such asthose of string theory. Because this is sci-ence, in the end experiment will decide.The good news is that the decision maycome soon.

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HOW CLASSICAL REALITY arisesfrom quantum spacetime is stillbeing worked out.

Three Roads to Quantum Gravity. Lee Smolin. Basic Books, 2001.The Quantum of Area? John Baez in Nature, Vol. 421, pages 702–703; February 2003.How Far Are We from the Quantum Theory of Gravity? Lee Smolin. March 2003. Preprint availableat http://arxiv.org/hep-th/0303185Welcome to Quantum Gravity. Special section. Physics World, Vol. 16, No. 11, pages 27–50;November 2003.Loop Quantum Gravity. Lee Smolin. Online at www.edge.org/3rd–culture/smolin03/smolin03–index.html




PHYSICS BEYOND THE The Dawn of

By Gordon Kane


originally publishedJune 2003


STANDARD MODELThe Standard Modelof particle physics isat a pivotal momentin its history: it isboth at the height ofits success and onthe verge of beingsurpassed

A NEW ERA IN PARTICLE PHYSICS could soon be heralded by the detectionof supersymmetric particles at the Tevatron collider at Fermi NationalAccelerator Laboratory in Batavia, Ill. A quark and an antiquark (red andblue) smashing head-on would form two heavy supersymmetric particles(pale magenta). Those would decay into W and Z particles (orange) andtwo lighter supersymmetric particles (dark magenta). The W and Z wouldin turn decay into an electron, an antielectron and a muon (all green),which would all be detected, and an invisible antineutrino (gray).

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Today, centuries after the search began for the funda-mental constituents that make up all the complexity and beau-ty of the everyday world, we have an astonishingly simple an-swer—it takes just six particles: the electron, the up and thedown quarks, the gluon, the photon and the Higgs boson.Eleven additional particles suffice to describe all the esoteric phe-nomena studied by particle physicists [see box at right]. This isnot speculation akin to the ancient Greeks’ four elements ofearth, air, water and fire. Rather it is a conclusion embodied inthe most sophisticated mathematical theory of nature in histo-ry, the Standard Model of particle physics. Despite the word“model” in its name, the Standard Model is a comprehensivetheory that identifies the basic particles and specifies how theyinteract. Everything that happens in our world (except for theeffects of gravity) results from Standard Model particles inter-acting according to its rules and equations.

The Standard Model was formulated in the 1970s and ten-tatively established by experiments in the early 1980s. Nearlythree decades of exacting experiments have tested and verifiedthe theory in meticulous detail, confirming all of its predictions.In one respect, this success is rewarding because it confirms thatwe really understand, at a deeper level than ever before, how na-ture works. Paradoxically, the success has also been frustrating.Before the advent of the Standard Model, physicists had becomeused to experiments producing unexpected new particles or oth-er signposts to a new theory almost before the chalk dust hadsettled on the old one. They have been waiting 30 years for thatto happen with the Standard Model.

Their wait should soon be over. Experiments that achieve col-lisions that are higher in energy than ever before or that study cer-

■ The Standard Model of particle physics is the mostsuccessful theory of nature in history, but increasinglythere are signs that it must be extended by adding newparticles that play roles in high-energy reactions.

■ Major experiments are on the verge of providing directevidence of these new particles. After 30 years ofconsolidation, particle physics is entering a new era ofdiscovery. Many profound mysteries could be resolved bypost–Standard Model physics.

■ One element of the Standard Model—a particle called theHiggs boson—also remains to be observed. The Tevatroncollider at Fermilab could detect Higgs bosons within thenext few years.

Overview/A New Era

The Particles ALTHOUGH THE STANDARD MODEL needs to be extended, its particlessuffice to describe the everyday world (except for gravity) and almostall data collected by particle physicists.

In the Standard Model, thefundamental particles of ordinary matter are the electron, the up quark(u) and the down quark (d). Triplets of quarks bind together to formprotons (uud) and neutrons (udd), which in turn make up atomic nuclei( above). The electron and the up and the down quarks, together withthe electron-neutrino, form the first of three groups of particles calledgenerations. Each generation is identical in every respect except for themasses of the particles ( grid at right). The values of the neutrinomasses in the chart are speculative but chosen to be consistent withobservations.

The Standard Model describes three ofthe four known forces: electromagnetism, the weak force (which isinvolved in the formation of the chemical elements) and the strong force(which holds protons, neutrons and nuclei together). The forces aremediated by force particles: photons for electromagnetism, the W and Zbosons for the weak force, and gluons for the strong force. For gravity,gravitons are postulated, but the Standard Model does not includegravity. The Standard Model partially unifies the electromagnetic andweak forces—they are facets of one “electroweak” force at high energiesor, equivalently, at distances smaller than the diameter of protons.

One of the greatest successes of the Standard Model is that theforms of the forces—the detailed structure of the equations describingthem—are largely determined by general principles embodied in thetheory rather than being chosen in an ad hoc fashion to match acollection of empirical data. For electromagnetism, for example, thevalidity of relativistic quantum field theory (on which the StandardModel is based) and the existence of the electron imply that the photonmust also exist and interact in the way that it does—we finallyunderstand light. Similar arguments predicted the existence andproperties, later confirmed, of gluons and the W and Z particles.

THE STANDARD MODEL

MATTER PARTICLES (FERMIONS)

FORCE CARRIERS (BOSONS)

Atom

Nucleus

Gluon

Up quark Down quark

Electron

Proton



In addition to the particles describedabove, the Standard Model predicts the existence of the Higgsboson, which has not yet been directly detected by experiment.The Higgs interacts with the other particles in a special mannerthat gives them mass.

Might the Standard Model be superseded by atheory in which quarks and electrons are made up of morefundamental particles? Almost certainly not. Experiments haveprobed much more deeply than ever before without finding a hintof additional structure. More important, the Standard Model is aconsistent theory that makes sense if electrons and quarks arefundamental. There are no loose ends hinting at a deeperunderlying structure. Further, all the forces become similar at highenergies, particularly if supersymmetry is true [see box on nextpage]. If electrons and quarks are composite, this unificationfails: the forces do not become equal. Relativistic quantum fieldtheory views electrons and quarks as being pointlike—they arestructureless. In the future, they might be thought of as tinystrings or membranes (as in string theory), but they will still beelectrons and quarks, with all the known Standard Modelproperties of these objects at low energies.

FERMIONS* BOSONS

FirstGeneration

SecondGeneration

Mas

s (g

iga-

elec

tron

-vol

ts)

ThirdGeneration

MASSLESS BOSONS

Photon

Electron

Muon-neutrino

Muon

Charm quark

Tau

Top quarkZ

W

Higgs

Down quark

Gluon

Electron-neutrino

Tau-neutrino

Bottom quark

Strange quark

Up quark

103

102

101

100

10–1

10–2

10–3

10–4

10–10

10–11

10–12

The Rules of the GameTHE STANDARD MODEL describes thefundamental particles and how they interact.For a full understanding of nature, we alsoneed to know what rules to use to calculatethe results of the interactions. An examplethat helps to elucidate this point is Newton’slaw, F = ma. F is any force, m is the mass ofany particle, and a is the acceleration of theparticle induced by the force. Even if you knowthe particles and the forces acting on them,you cannot calculate how the particlesbehave unless you also know the rule F = ma.The modern version of the rules is relativisticquantum field theory, which was invented inthe first half of the 20th century. In thesecond half of the 20th century thedevelopment of the Standard Model taughtresearchers about the nature of the particlesand forces that were playing by the rules ofquantum field theory. The classical concept ofa force is also extended by the StandardModel: in addition to pushing and pulling onone another, when particles interact they canchange their identity and be created ordestroyed.

Feynman diagrams (a–g, at right), firstdevised by physicist Richard P. Feynman,serve as useful shorthand to describeinteractions in quantum field theory. Thestraight lines represent the trajectories ofmatter particles; the wavy lines representthose of force particles. Electromagnetism isproduced by the emission or absorption ofphotons by any charged particle, such as anelectron or a quark. In a, the incoming electronemits a photon and travels off in a newdirection. The strong force involves gluonsemitted (b) or absorbed by quarks. The weakforce involves W and Z particles (c, d), which areemitted or absorbed by both quarks andleptons (electrons, muons, taus and neutrinos).Notice how the W causes the electron tochange identity. Gluons (e) and Ws and Zs ( f)also self-interact, but photons do not.

Diagrams a through f are calledinteraction vertices. Forces are produced bycombining two or more vertices. For example,the electromagnetic force between an electron and a quark islargely generated by the transfer of a photon (g). Everything thathappens in our world, except for gravity, is the result ofcombinations of these vertices. —G.K.

DEEPER LEVELS?

THE SOURCE OF MASS

Electron

Electron

Electron

Quark

Quark

W boson

W boson

Z boson

W boson

Z boson

Electron-neutrino

Neutrino

Photon

Photon

Gluon

Gluon

a

b

c

d

e

f

g

* T h e f e r m i o n s a r e s u b d i v i d e d i n t o q u a r k s a n d l e p t o n s , w i t h l e p t o n si n c l u d i n g e l e c t r o n s , m u o n s , t a u s a n d t h r e e f o r m s o f n e u t r i n o .

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tain key phenomena with greater precision are on the verge ofgoing beyond the Standard Model. These results will not over-turn the Standard Model. Instead they will extend it by uncov-ering particles and forces not described by it. The most impor-tant experiment is occurring at the upgraded Tevatron colliderat Fermi National Accelerator Laboratory in Batavia, Ill., whichbegan taking data in 2001. It might produce directly the still elu-sive particles that complete the Standard Model (Higgs bosons)and those predicted by the most compelling extensions of the the-ory (the so-called superpartners of the known particles).

Significant information is also beginning to come from “Bfactories,” particle colliders running in California and Japanconfigured to create billions of b quarks (one of the 11 additionalparticles) and their antimatter equivalents to study a phen-omenon called CP violation. CP (charge-parity) is the symmetryrelating matter to antimatter, and CP violation means that anti-matter does not exactly mirror matter in its behavior. The amountof CP violation observed so far in particle decays can be accom-modated by the Standard Model, but we have reasons to expectmuch more CP violation than it can produce. Physics that goesbeyond the Standard Model can generate additional CP violation.

Physicists are also studying the precise electric and magnet-ic properties of particles. The Standard Model predicts that elec-trons and quarks behave as microscopic magnets with a specif-ic strength and that their behavior in an electric field is deter-mined purely by their electric charge. Most extensions of theStandard Model predict a slightly different magnetic strengthand electrical behavior. Experiments are beginning to collectdata with enough sensitivity to see the tiny effects predicted.

Looking beyond the earth, scientists studying solar neutri-nos and cosmic-ray neutrinos, ghostly particles that barely inter-act at all, have recently established that neutrinos have masses,a result long expected by theorists studying extensions of the Stan-dard Model [see “Solving the Solar Neutrino Problem,” by ArthurB. McDonald, Joshua R. Klein and David L. Wark; ScientificAmerican, April]. The next round of experiments will clarify theform of theory needed to explain the observed neutrino masses.

In addition, experiments are under way to detect mysteriousparticles that form the cold dark matter of the universe and toexamine protons at higher levels of sensitivity to learn whetherthey decay. Success in either project would be a landmark ofpost–Standard Model physics.

As all this research proceeds, it is ushering in a new, data-richera in particle physics. Joining the fray by about 2007 will be theLarge Hadron Collider (LHC), a machine 27 kilometers in cir-

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Strong Force

Weak Force

Electromagnetic Force

Forc

e St

reng

th

102 104 106 108 10101012 1014

1016

Interaction Energy (GeV)

STANDARD MODEL

Strong Force

Weak Force

Electromagnetic Force

Forc

e St

reng

th

102 104 106 108 10101012 1014

1016

MINIMAL SUPERSYMMETRIC STANDARD MODEL

Interaction Energy (GeV)

GORDON KANE, a particle theorist, is Victor Weisskopf CollegiateProfessor of Physics at the University of Michigan at Ann Arbor. Hiswork explores ways to test and extend the Standard Model of par-ticle physics. In particular he studies Higgs physics and the Stan-dard Model’s supersymmetric extension, with a focus on relatingtheory and experiment and on the implications of supersymme-try for particle physics and cosmology. His hobbies include play-ing squash, exploring the history of ideas, and seeking to under-stand why science flourishes in some cultures but not others.

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Evidence for Supersymmetry

THE MOST WIDELY FAVORED THEORY to supersede theStandard Model is the Minimal Supersymmetric StandardModel. In this model, every known particle species has asuperpartner particle that is related to it bysupersymmetry. Particles come in two broad classes:bosons (such as the force particles), which can gather enmasse in a single state, and fermions (such as quarks andleptons), which avoid having identical states. Thesuperpartner of a fermion is always a boson and viceversa.

Indirect evidence for supersymmetry comes from theextrapolation of interactions to high energies. In theStandard Model, the three forces become similar but notequal in strength (top). The existence of superpartnerschanges the extrapolation so that the forces all coincide atone energy (bottom)—a clue that they become unified ifsupersymmetry is true.



cumference now under construction at CERN, the European lab-oratory for particle physics near Geneva [see “The Large HadronCollider,” by Chris Llewellyn Smith; Scientific American,July 2000]. A 30-kilometer-long linear electron-positron collid-er that will complement the LHC’s results is in the design stages.

As the first hints of post–Standard Model physics areglimpsed, news reports often make it sound as if the Standard

Model has been found to be wrong, as if it were broken andready to be discarded, but that is not the right way to thinkabout it. Take the example of Maxwell’s equations, writtendown in the late 19th century to describe the electromagneticforce. In the early 20th century we learned that at atomic sizesa quantum version of Maxwell’s equations is needed. Later theStandard Model included these quantum Maxwell’s equationsas a subset of its equations. In neither case do we say Maxwell’sequations are wrong. They are extended. (And they are still usedto design innumerable electronic technologies.)

A Permanent EdificeS IMILARLY, THE STANDARD MODEL is here to stay. It isa full mathematical theory—a multiply connected and highlystable edifice. It will turn out to be one piece of a larger such ed-ifice, but it cannot be “wrong.” No part of the theory can failwithout a collapse of the entire structure. If the theory werewrong, many successful tests would be accidents. It will contin-ue to describe strong, weak and electromagnetic interactions atlow energies.

The Standard Model is very well tested. It predicted the ex-istence of the W and Z bosons, the gluon and two of the heav-ier quarks (the charm and the top quark). All these particles weresubsequently found, with precisely the predicted properties.

A second major test involves the electroweak mixing angle,a parameter that plays a role in describing the weak and elec-tromagnetic interactions. That mixing angle must have the samevalue for every electroweak process. If the Standard Model werewrong, the mixing angle could have one value for one process,a different value for another and so on. It is observed to have thesame value everywhere, to an accuracy of about 1 percent.

Third, the Large Electron-Positron (LEP) collider at CERN,which ran from 1989 to 2000, looked at about 20 million Zbosons. Essentially every one of them decayed in the manner ex-pected by the Standard Model, which predicted the number ofinstances of each kind of decay as well as details of the energies

and directions of the outgoing particles. These tests are but a fewof the many that have solidly confirmed the Standard Model.

In its full glory, the Standard Model has 17 particles andabout as many free parameters—quantities such as particle mass-es and strengths of interactions [see box on pages 55 and 56].These quantities can in principle take any value, and we learn thecorrect values only by making measurements. Armchair crit-

ics sometimes compare the Standard Model’s many parameterswith the epicycles on epicycles that medieval theorists used todescribe planetary orbits. They imagine that the Standard Mod-el has limited predictive power, or that its content is arbitrary,or that it can explain anything by adjusting of some parameter.

The opposite is actually true: once the masses and interac-tion strengths are measured in any process, they are fixed forthe whole theory and for any other experiment, leaving no free-dom at all. Moreover, the detailed forms of all the StandardModel’s equations are determined by the theory. Every parameter but the Higgs boson mass has been measured. Untilwe go beyond the Standard Model, the only thing that canchange with new results is the precision of our knowledge of theparameters, and as that improves it becomes harder, not easier,for all the experimental data to remain consistent, becausemeasured quantities must agree to higher levels of precision.

Adding further particles and interactions to extend the Stan-dard Model might seem to introduce a lot more freedom, butthis is not necessarily the case. The most widely favored exten-sion is the Minimal Supersymmetric Standard Model (MSSM).Supersymmetry assigns a superpartner particle to every parti-cle species. We know little about the masses of those superpart-ners, but their interactions are constrained by the supersymme-try. Once the masses are measured, the predictions of the MSSMwill be even more tightly constrained than the Standard Modelbecause of the mathematical relations of supersymmetry.

Ten MysteriesIF THE STANDARD MODEL works so well, why must it beextended? A big hint arises when we pursue the long-standinggoal of unifying the forces of nature. In the Standard Model, wecan extrapolate the forces and ask how they would behave atmuch higher energies. For example, what were the forces like inthe extremely high temperatures extant soon after the big bang?At low energies the strong force is about 30 times as powerfulas the weak force and more than 100 times as powerful as elec-

It is not surprising that there are questions that theStandard Model cannot answer—every successful theory in science has increased the number of answered questions but has left some unanswered.



tromagnetism. When we extrapolate, we find that the strengthsof these three forces become very similar but are never all exact-ly the same. If we extend the Standard Model to the MSSM, theforces become essentially identical at a specific high energy [seebox on page 57]. Even better, the gravitational force approachesthe same strength at a slightly higher energy, suggesting a con-nection between the Standard Model forces and gravity. Theseresults seem like strong clues in favor of the MSSM.

Other reasons for extending the Standard Model arise fromphenomena it cannot explain or cannot even accommodate:

1. All our theories today seem to imply that the universeshould contain a tremendous concentration of energy, evenin the emptiest regions of space. The gravitational effects ofthis so-called vacuum energy would have either quicklycurled up the universe long ago or expanded it to muchgreater size. The Standard Model cannot help us understandthis puzzle, called the cosmological constant problem.

2. The expansion of the universe was long believed to beslowing down because of the mutual gravitational attractionof all the matter in the universe. We now know that the ex-pansion is accelerating and that whatever causes the accel-eration (dubbed “dark energy”) cannot be Standard Modelphysics.

3. There is very good evidence that in the first fraction of asecond of the big bang the universe went through a stage ofextremely rapid expansion called inflation. The fields re-sponsible for inflation cannot be Standard Model ones.

4. If the universe began in the big bang as a huge burst of en-ergy, it should have evolved into equal parts matter and an-timatter (CP symmetry). But instead the stars and nebulaeare made of protons, neutrons and electrons and not their an-tiparticles (their antimatter equivalents). This matter asym-metry cannot be explained by the Standard Model.

5. About a quarter of the universe is invisible cold dark mat-ter that cannot be particles of the Standard Model.

6. In the Standard Model, interactions with the Higgs field(which is associated with the Higgs boson) cause particlesto have mass. The Standard Model cannot explain the veryspecial forms that the Higgs interactions must take.

7. Quantum corrections apparently make the calculatedHiggs boson mass huge, which in turn would make all parti-cle masses huge. That result cannot be avoided in the Stan-dard Model and thus causes a serious conceptual problem.

8. The Standard Model cannot include gravity, because itdoes not have the same structure as the other three forces.

9. The values of the masses of the quarks and leptons (such

as the electron and neutrinos) cannot be explained by theStandard Model.

10. The Standard Model has three “generations” of particles.The everyday world is made up entirely of first-generationparticles, and that generation appears to form a consistenttheory on its own. The Standard Model describes all three gen-erations, but it cannot explain why more than one exists.

In expressing these mysteries, when I say the Standard Mod-el cannot explain a given phenomenon, I do not mean that thetheory has not yet explained it but might do so one day. TheStandard Model is a highly constrained theory, and it cannotever explain the phenomena listed above. Possible explanationsdo exist. One reason the supersymmetric extension is attractiveto many physicists is that it can address all but the second andthe last three of these mysteries. String theory (in which parti-cles are represented by tiny, one-dimensional entities instead ofpoint objects) addresses the last three [see “The Theory FormerlyKnown as Strings,” by Michael J. Duff; Scientific American,February 1998]. The phenomena that the Standard Model can-not explain are clues to how it will be extended.

It is not surprising that there are questions that the StandardModel cannot answer—every successful theory in science has in-creased the number of answered questions but has left some unan-swered. And even though improved understanding has led tonew questions that could not be formulated earlier, the numberof unanswered fundamental questions has continued to decrease.

Some of these 10 mysteries demonstrate another reason whyparticle physics today is entering a new era. It has become clearthat many of the deepest problems in cosmology have their so-lutions in particle physics, so the fields have merged into “par-ticle cosmology.” Only from cosmological studies could welearn that the universe is matter (and not antimatter) or that theuniverse is about a quarter cold dark matter. Any theoretical un-derstanding of these phenomena must explain how they arise aspart of the evolution of the universe after the big bang. But cos-mology alone cannot tell us what particles make up cold darkmatter, or how the matter asymmetry is actually generated, orhow inflation originates. Understanding of the largest and thesmallest phenomena must come together.

The HiggsPHYSICISTS ARE TACKLING all these post–Standard Mod-el mysteries, but one essential aspect of the Standard Model alsoremains to be completed. To give mass to leptons, quarks, andW and Z bosons, the theory relies on the Higgs field, which hasnot yet been directly detected.

The Higgs is fundamentally unlike any other field. To un-derstand how it is different, consider the electromagnetic field.Electric charges give rise to electromagnetic fields such as thoseall around us (just turn on a radio to sense them). Electromag-netic fields carry energy. A region of space has its lowest possi-ble energy when the electromagnetic field vanishes throughoutit. Zero field is the natural state in the absence of charged parti-



cles. Surprisingly, the Standard Model requires that the lowestenergy occur when the Higgs field has a specific nonzero value.Consequently, a nonzero Higgs field permeates the universe, andparticles always interact with this field, traveling through it likepeople wading through water. The interaction gives them theirmass, their inertia.

Associated with the Higgs field is the Higgs boson. In theStandard Model, we cannot predict any particle masses fromfirst principles, including the mass of the Higgs boson itself.

One can, however, use other measured quantities to calculatesome masses, such as those of the W and Z bosons and the topquark. Those predictions are confirmed, giving assurance to theunderlying Higgs physics.

Physicists do already know something about the Higgs mass.Experimenters at the LEP collider measured about 20 quantitiesthat are related to one another by the Standard Model. All theparameters needed to calculate predictions for those quantitiesare already measured—except for the Higgs boson mass. One cantherefore work backward from the data and ask which Higgsmass gives the best fit to the 20 quantities. The answer is that theHiggs mass is less than about 200 giga-electron-volts (GeV). (Theproton mass is about 0.9 GeV; the top quark 174 GeV.) Thatthere is an answer at all is strong evidence that the Higgs exists.If the Higgs did not exist and the Standard Model were wrong,it would take a remarkable coincidence for the 20 quantities tobe related in the right way to be consistent with a specific Higgsmass. Our confidence in this procedure is bolstered because asimilar approach accurately predicted the top quark mass be-fore any top quarks had been detected directly.

LEP also conducted a direct search for Higgs particles, butit could search only up to a mass of about 115 GeV. At that veryupper limit of LEP’s reach, a small number of events involvedparticles that behaved as Higgs bosons should. But there werenot enough data to be sure a Higgs boson was actually discov-ered. Together the results suggest the Higgs mass lies between115 and 200 GeV.

LEP is now dismantled to make way for the construction ofthe LHC, which is scheduled to begin taking data in four years.In the meantime the search for the Higgs continues at the Teva-tron at Fermilab. If the Tevatron operates at its design intensityand energy and does not lose running time because of technicalor funding difficulties, it could confirm the 115-GeV Higgsboson in about two to three years. If the Higgs is heavier, it willtake longer for a clear signal to emerge from the background.The Tevatron will produce more than 10,000 Higgs bosonsaltogether if it runs as planned, and it could test whetherthe Higgs boson behaves as predicted. The LHC will be a“factory” for Higgs bosons, producing millions of them and

allowing extensive studies.There are also good arguments that some of the lighter su-

perpartner particles predicted by the MSSM have masses smallenough so that they could be produced at the Tevatron as well.Direct confirmation of supersymmetry could come in the nextfew years. The lightest superpartner is a prime candidate tomake up the cold dark matter of the universe—it could be di-rectly observed for the first time by the Tevatron. The LHC willproduce large numbers of superpartners if they exist, definitive-

ly testing whether supersymmetry is part of nature.

Effective TheoriesTO FULLY GRASP the relation of the Standard Model to therest of physics, and its strengths and limitations, it is useful tothink in terms of effective theories. An effective theory is a de-scription of an aspect of nature that has inputs that are, in prin-ciple at least, calculable using a deeper theory. For example, innuclear physics one takes the mass, charge and spin of the pro-ton as inputs. In the Standard Model, one can calculate thosequantities, using properties of quarks and gluons as inputs. Nu-clear physics is an effective theory of nuclei, whereas the Stan-dard Model is the effective theory of quarks and gluons.

From this point of view, every effective theory is open-end-ed and equally fundamental—that is, not truly fundamental atall. Will the ladder of effective theories continue? The MSSMsolves a number of problems the Standard Model does not solve,but it is also an effective theory because it has inputs as well. Itsinputs might be calculable in string theory.

Even from the perspective of effective theories, particle phys-ics may have special status. Particle physics might increase ourunderstanding of nature to the point where the theory can beformulated with no inputs. String theory or one of its cousinsmight allow the calculation of all inputs—not only the electronmass and such quantities but also the existence of spacetime andthe rules of quantum theory. But we are still an effective theoryor two away from achieving that goal.

The Particle Garden. Gordon Kane. Perseus Publishing, 1996.

The Rise of the Standard Model: A History of Particle Physics from1964 to 1979. Edited by Lillian Hoddeson, Laurie M. Brown, MichaelRiordan and Max Dresden. Cambridge University Press, 1997.

The Little Book of the Big Bang: A Cosmic Primer. Craig J. Hogan.Copernicus Books, 1998.

Supersymmetry: Unveiling the Ultimate Laws of Nature. Gordon Kane.Perseus Publishing, 2001.

An excellent collection of particle physics Web sites is listed atparticleadventure.org/particleadventure/other/othersites.html


Particle physics might increase our understanding of nature tothe point where the theory can be formulated with no inputs.



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