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Home , Planck length

QUANTUM AT THE PLANCK

Joseph Polchinski’ Institute for Theoretical University of California Santa Barbara, CA 93106-4030

ABSTRACT

I describe our understanding of physics near the Planck length, in particular the great progress of the last four years in theory.

*Supported by NSF Grants PHY9407194 and PHY97-22022.

@ 1998 Joseph Polchinski.

-209- selection. In section 4 I introduce the idea of duality, including weak-strong and electric-magnetic. I explain how supersymmetry gives information about strongly coupled systems. I then describe the consequences for , including string duality, the eleventh dimension, D-branes, and M-theory. In section 5 I develop an alternative theory of , only to find that “all roads lead to string theory.” In section 6 I explain how the new methods have solved some of the puzzles of . This in turn leads to the Maldacena dualities, which give detailed new information about supersymmetric gauge field - theories. In section 7 I discuss some of the ways that the new ideas might affect particle physics, through the unification of the couplings and the possibility of b) low energy string theory and large new dimensions. In section 8 I summarize and Fig. 2. a) A with one large dimension and one small one. We assume present the outlook. here that the small dimensions are nearly Planck sized; the possibility of larger dimensions will be considered later. b) The same spacetime as seen by a low 2 Beyond Four Dimensions energy observer.

Gravity is the dynamics of spacetime. It is very likely that at near the Planck scale (Lp = 1O-33 cm) it becomes evident that spacetime has more than the Suppose that this same symmetry breaking principle holds for the spacetime four dimensions that are visible to us. That is, spacetime is as shown in figure 2a, symmetries. The visible spacetime symmetry is SO(3, l), the Lorentz invariance with four large dimensions (including time) and some additional number of small of consisting of the boosts and rotations. A larger symmetry and highly curved spatial dimensions. A physicist who probes this spacetime with would be SO(d, 1) for d > 3, the Lorentz invariance of d + 1 spacetime dimensions. wavelengths long compared to the of the small dimensions sees only the large Figure 2 shows how this symmetry would be broken by the geometry of spacetime. ones, as in figure 2b. I will first give two reasons why this is a natural possibility So extra dimensions are cosmologically plausible, and are a natural extension to consider, and then explain why it is a good idea. of the familiar phenomenon of spontaneous symmetry breaking. In addition, they The first argument is cosmological. The universe is expanding, so the dimen- may be responsible for some of the physics that we see in nature. To see why sions that we see were once smaller and highly curved. It may have been that this is so, consider first the following cartoon version of grand unification. The initially there were more than four small dimensions, and that only the four that traceless 3 x 3 and 2 x 2 matrices for the strong and weak gauge interactions fit are evident to us began to expand. That is, we know of no reason that the initial into a 5 x 5 matrix, with room for an extra U(1) down the diagonal: expansion had to be isotropic. The second argument is based on symmetry breaking. Most of the symmetry in nature is spontaneously broken or otherwise hidden from us. For example, of the SU(3) x SU(2) x U(1) gauge symmetries, only a U(1) is visible. Similarly the flavor symmetry is partly broken, as are the symmetries in many condensed matter systems. This symmetry breaking is part of what makes physics so rich: Now let us try to do something similar, but for gravity and electromagnetism. if all of the symmetry of the underlying theory were unbroken, it would be much Gravity is described by a metric g,,,,, which is a 4 x 4 matrix, and electromagnetism easier to figure out what that theory is!

-211- x up to some energy scale, beyond which new physics appears. The new physics should have the effect of smearing out the interaction in spacetime and so softening the high energy behavior. One might imagine that this could be done in many ways, but in fact it is quite difficult to do without spoiling Lorentz invariance or causality; this is because Lorentz invariance requires that if the interaction is spread out in space it is also spread out in time. The solution to the short- distance problem of the weak interaction is not quite unique, but combined with two of the broad features of the weak interaction - its V - A structure and its universal coupling to different and leptons - a unique solution emerges. This is depicted in figure 3c, where the four-fermi interaction is resolved into the exchange of a vector boson. Moreover, this vector boson must be of a very specific b) kind, coming from a spontaneously broken gauge invariance. And indeed, this is the way that nature w0rks.t For gravity the discussion is much the same. The gravitational interaction is Fig. 4. a) Exchange of a between two elementary particles. b) The same depicted in figure 4a. As we have already noted in discussing figure 1, the gravita- interaction in string theory. The amplitude is given by the sum over histories, tional coupling Gx has units of length-squared and so the dimensionless coupling over all embeddings of the string world-sheet in spacetime. The world-sheet is is GNE*. This grows large at high energy and gives again a nonrenormalizable per- smooth: there is no distinguished point at which the interaction occurs (the cross turbation theory. $ Again the natural suspicion is that new short-distance physics section on the intermediate line is only for illustration). smears out the interaction, and again there is only one known way to do this. It involves a bigger step than in the case of the weak interaction: it requires that at the Planck length the graviton and other particles turn out to be not points but one-dimensional objects, loops of “string,” figure 5a. Their spacetime histories are then two-dimensional surfaces as shown in figure 4b. At first sight this is an odd idea. It is not obvious why it should work and not other possibilities. It may simply be that we have not been imaginative enough, but because UV problems are so hard to solve we should consider carefully this one solution that we have found. And in this case the idea becomes increasingly attractive as we consider it.

tit could also have been that the divergences are an artifact of perturbation theory but do not appear in the exact amplitudes. This is a logical possibility, a “nontrivial UV fixed point.” Al- though possible, it seems unlikely, and it is not what happens in the case of the weak interaction. SNote that the bad gravitational interaction of figure 4a is the same graph as the smeared-out Fig. 5. a) A closed loop of string. b) An open string, which appears in some weak interaction of figure 3c. However, its high energy behavior is worse because gravity couples to energy rather than charge. theories. c) The basic splitting-joining interaction.

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These have led to steady progress was the discovery by ‘t Hooft and Polyakov that grand unified theories predict on the questions that we need to answer, and to many new results and many magnetic monopoles. These monopoles are solitons, smooth classical field config- surprises. This progress is the subject of the rest of my lectures. urations. Thus they look rather different from the electric charges, which are the basic quanta: the latter are light, pointlike, and weakly coupled while monopoles 4 Duality in Field and String Theory are heavy, “fuzzy,” and (as a consequence of the Dirac quantization) strongly coupled. 4.1 Dualities In 1977 Montonen and Olive proposed that in certain supersymmetric unified theories the situation at strong coupling would be reversed: the electric objects One important idea in the recent developments is duality. This refers to the would be big, heavy, and strongly coupled and the magnetic objects small, light, equivalence between seemingly distinct physical systems. One starts with different and weakly coupled. The symmetry of the sourceless Maxwell’s equations would Hamiltonians, and even with different fields, but after solving the theory one finds then be extended to the interacting theory, with an inversion of the coupling con- that the spectra and the transition amplitudes are identical. Often this occurs stant. Thus electric-magnetic duality would be a special case of weak-strong du- because a quantum system has more than one , so that one gets back ality, with the magnetically charged fields being the dual variables for the strongly to the same quantum theory by “quantizing” either classical theory. coupled theory. This phenomenon is common in quantum field theories in two spacetime di- The evidence for this conjecture was circumstantial: no one could actually mensions. The duality of the Sine-Gordon and Thirring models is one example; find the dual magnetic variables. For this reason the reaction to this conjecture the high--low-temperature duality of the Ising model is another. The was skeptical for many years. In fact the evidence remains circumstantial, but in great surprise of the recent developments is that it is also common in quantum recent years it has become so much stronger that the existence of this duality is field theories in four dimensions, and in string theory. in little doubt. A particularly important phenomenon is weak-strong duality. I have empha- sized that perturbation theory does not converge. It gives the asymptotics as the coupling g goes to zero, but it misses important physics at finite coupling, and at 4.2 Supersymmetry and Strong Coupling large coupling it becomes more and more useless. In some cases, though, when g The key that makes it possible to discuss the strongly coupled theory is szlpersym- becomes very large there is a simple alternate description, a weakly coupled dual mety. One way to think about supersymmetry is in terms of extra dimensions theory with g’ = l/g. In one sense, as g + co the quantum fluctuations of the - but unlike the dimensions that we see, and unlike the small dimensions dis- original fields become very large (non-Gaussian), but one can find a dual set of cussed earlier, these dimensions are “fermionic.” In other words, the coordinates fields which become more and more classical. for ordinary dimensions are real numbers and so commute with each other: they Another important idea is electric-magnetic duality. A striking feature of are “bosonic;” the fermionic coordinates instead satisfy Maxwell’s equations is the symmetry of the left-hand side under E + B and B -+ -E. This symmetry suggests that there should be magnetic as well as 9,0j = -8j0i is) electric charges. This idea became more interesting with Dirac’s discovery of the

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iaIq!S!AU! uaaq SLEM~ STQuoyuaunp S!Q$LQM s! S!Q$ :o = g punow uoisuvdxa U'I?SF % II! hoaQ$ uoyvqntyad ‘Iyna!$.n?d UI 'g a%%?[s! %5a%%[ pw 21 I~IIIS SF % IIWIIS ‘SF 1~9~ we sum the squares of all of them. The indices m and n run over the D - 1 So this simple idea seems to be working quite well, but we said that we were spatial directions, and M and M’ are large masses, of order of the Planck scale. going to fail in our attempt to find an alternative to string theory. In fact we have The potential term is chosen as follows. We want to recover ordinary quantum failed because this is not an alternative: it is string theory. It is actually one piece mechanics at low energy. The potential is the sum of the squares of all of the of string theory, namely the Hamiltonian describing the low energy dynamics of components of all of the commutators of the matrices X,j, with a large coefficient. N DO-branes. This illustrates the following principle: that all good ideas are part It is therefore large unless all of these matrices commute. In states with energies of string theory. That sounds arrogant, but with all the recent progress in string below the Planck scale, the matrices will then commute to good approximation, theory, and a fuller understanding of the dualities and dynamical possibilities, so we do not see the new (16) and we recover the usual quantum string theory has extended its reach into more areas of mathematics and has mechanics. In particular, we can find a basis which diagonalizes all the commuting absorbed previous ideas for unification (including D = 11 ). Xy. Thus the effective coordinates are just the N diagonal elements XE of each We have discussed this model not just to introduce this principle, but because matrix in this basis, which is the right count for N particles in ordinary quantum the model is important for a number of other reasons. In fact, it is conjectured that mechanics: the X,7 behave like ordinary coordinates. it is not just a piece of string theory, but is actually a complete description. The The Hamiltonian (19) has interesting connections with other parts of physics. idea is that if we view any state in string theory from a very highly boosted frame, First, the commutator-squared term has the exact same structure as the four- it will be described by the Hamiltonian (19) with N large. Particle physicists are gluon interaction in Yang-Mills theory. This is no accident, as we will see later familiar with the idea that systems look different as one boosts them: the parton on. Second, there is a close connection to supersymmetry. In supersymmetric distributions evolve. The idea here is that the DO-branes are the partons for string quantum mechanics, one has operators satisfying the algebra (7,8). Again in gen- theory; in effect the string is a necklace of partons. This is the matrix theory idea eral there are several supersymmetry charges, and the number N of these Qs is of Banks, Fischler, Shenker, and Susskind (based on earlier ideas of Thorn), and significant. For small values of N, like 1, 2, or 4, there are many Hamiltonians at this point it seems very likely to be correct or at least a step in the correct with the symmetry. As N increases the symmetry becomes more constraining, direction. and N = 16 is the maximum number. For N = 16 there is only one invariant To put this in context, let us return to the illustration in figure 7 of the space Hamiltonian, and it is none other than our model (19). To be precise, super- of string vacua, and to the point made earlier that the perturbation theory does symmetry requires that the particles have spin, that the Hamiltonian also has a not define the theory for finite g. In fact, every indication is that the string spin-dependent piece, and that the spacetime dimension D be 10. In fact, su- description is useful only near the five cusps of the figure in which the string persymmetry is necessary for this idea to work. The vanishing of the potential coupling becomes weak. In the center of the parameter space, not only do we not for commuting configurations was needed, but we only considered the classical know the Hamiltonian but we do not know what degrees of freedom are supposed potential, not the quantum corrections. The latter vanish only if the theory is to appear in it. It is likely that they are not the one-dimensional objects that one supersymmetric. usually thinks of in string theory; it is more likely that they are the coordinate So this model has interesting connections, but let us return to the idea that we matrices of the D-branes. want a theory of gravity. The interactions among low energy particles come about as follows. We have argued that the potential forces the X, to be diagonal: the off-diagonal pieces are very massive. Still, virtual off-diagonal excitations induce interactions among the low-energy states. In fact, the leading effect, from one loop of the massive states, produces precisely the (super)gravity interaction among the low energy particles.

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poyq~EI.8 1'eD!sse13II! %!ST?aI3apUOU s! 11 XdoIlua aql ayL[ s! (s$!un yXre[d u!) ^eaIB uoz!loq $uaAa aq$‘Is[w!ysd UI ?+uvudpourJaq~30 SMEI aql put? saysqsaur aloq qcyq30 sMy aqluaafitaq L.%olw~ asOpS s! aIaq+ pq? puno3w~y SOL, aq+u~ This has been a source of great controversy. While most physicists would be in the near-horizon geometry of a black p-brane. Further, for p 5 3 the low pleased to see quantum mechanics replaced by something less weird, the particular energy physics of N Dp-branes is described by U(N) Yang-Mills theory with modification proposed by Hawking simply makes it uglier, and quite possibly N = 16 supersymmetries. That is, the gauge fields live on the D-branes, so inconsistent. But 20 years of people trying to find Hawking’s “mistake,” to identify that they constitute a field theory in p + 1 “spacetime” dimensions, where here the mechanism that preserves the purity of the quantum state, has only served spacetime is just the world-volume of the brane. For p = 0, this is the connection of to sharpen the paradox: because the quantum correlations are lost behind the matrix quantum mechanics to Yang-Mills theory that we have already mentioned horizon, either quantum mechanics is modified in Hawking’s way, or the locality below (19). of physics must break down in a way that is subtle enough not to infect most of The Maldacena duality then implies that various quantities in the gauge theory physics, yet act over long distances. can be calculated more easily in the dual black p-brane geometry. This method The duality conjecture above states that the black hole is equivalent to an is only useful for large N, because this is necessary to get a black hole which is ordinary quantum system, so that the laws of quantum evolution are unmodified. larger than string scale and so described by ordinary general relativity. Of course However, to resolve fully the paradox one must identify the associated nonlocality we have a particular interest in gauge theories in 3 + 1 dimensions, so let us focus in the spacetime physics. This is hard to do because the local properties of on p = 3. The Maldacena duality for p = 3 partly solves an old problem in spacetime are difficult to extract from the highly quantum D-brane system: this the strong interaction. In the mid-‘70s ‘t Hooft observed that Yang-Mills theory is related to the . This term refers to the property of a simplifies when the number of colors is large. This simplification was not enough hologram, that the full picture is contained in any one piece. It also has the to allow analytic calculation, but its form led ‘t Hooft to conjecture a duality further connotation that the quantum state of any system can be encoded in between large-N gauge theory and some unknown string theory. The Maldacena variables living on the boundary of that system, an idea that is suggested by the duality is a precise realization of this idea, for supersymmetric gauge theories.! entropy-area connection of the black hole. This is a key point where our ideas are For the strong interaction we need of course to understand nonsupersymmetric in still in flux. gauge theories. One can obtain a rough picture of these from the Maldacena duality, but a precise description seems still far off. It is notable, however, that 6.3 Black Holes and Gauge Theory string theory, which began as an attempt to describe the strong interaction, has now returned to its roots, but only by means of an excursion through black hole Dualities between two systems give information in each direction: for each sys- physics and other strange paths. tem there are some things that can be calculated much more easily in the dual description. In the previous subsection we used the Maldacena duality to make 6.4 Spacetime Topology Change statements about black holes. We can also use it in the other direction, to calculate properties of the D-brane theory. This subsection is not directly related to black holes, but deals with another exotic To take full advantage of this we must first make a generalization. We have question in quantum gravity. Gravity is due to the bending of spacetime. It is an said that D-branes can be points, strings, sheets, and so on: they can be extended old question, whether spacetime could not only bend but break: does its topology in p directions, where here p = 0, 1,2. Thus we refer to Dp-branes. The same is as well as its geometry evolve in time? true of black holes: the usual ones are local objects, but we can also have black Again, string theory provides the tools to answer this question. The answer strings - strings with event horizons - and so on. A black p-brane is extended §For p = 3 the near-horizon geometry is the product of an anti-de Sitter space and a sphere, in p directions and has a black hole geometry in the orthogonal directions. The while the supersymmetric gauge theory is conformally invariant (a conformal field theory), so full Maldacena duality is between the low energy physics of Dp-branes and strings this is also known as the AdS-CFT correspondence.

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(3 h Q---_____------_____---_____------22 E many ideas to explain this. There could be additional particles at the weak, intermediate, or unified scales, which change the running of the gauge couplings so as to raise the unification point. Or it may be that the gauge couplings actually do unify first, so that there is a normal grand unified theory over a small range of scales before the gravitational coupling unifies. These ideas focus on changing the behavior of the gauge couplings. Since these already unify to good approximation, it would be simpler to change the behavior of the gravitational coupling so that it meets the other three at a lower scale - to lower the Planck scale. Unfortunately this is not so easy. The energy-dependence of the gravitational coupling is just , which is not so easy to change.7 There is a way to change the dimensional analysis - that is, to change the dimension! We have discussed the possibility that at some scale we pass the Fig. 11. A Horava-Witten spacetime. The two planes represent 3 + 1 dimensional threshold to a new dimension. Suppose that this occurred below the unification walls, in which all the Standard Model particles live, while gravity moves in the scale. For both the gauge and the gravitational couplings the units change, so 4 + 1 dimensional bulk between the walls. In string theory there are six additional that both turn upward as in figure lob. This does not help; the couplings meet dimensions, which could be much smaller and are not shown. The wall is then no sooner. 9 + 1 dimensional in all, and the spacetime 10 + 1 dimensional. There is a more interesting possibility, which was first noticed in the strong coupling limit of the Es x Es heterotic string. Of the five string theories, this is the one whose weakly coupled physics looks most promising for unification. Its strong- the Standard Model lives in a brane and does not move in the full space of the coupling behavior, shown in figure 11, is interesting. A new dimension appears, compact dimensions, while gravity does do so. but it is not simply a circle. Rather, it is bounded by two walls. Moreover, all This new idea leads in turn to the possibility of radically changing the scales the gauge fields and the particles that carry gauge charges move only in the walls, of new physics in string theory. To see this, imagine lowering the threshold energy while gravity moves in the bulk. Consider now the unification of the couplings. (the kink) in figure 10~; this also lowers the string scale, which is where the grav- The dynamics of the gauge couplings, and their running, remains as in 3 + 1 itational coupling meets the other three. From a completely model-independent dimensions; however, the gravitational coupling has a kink at the threshold, so point of view, how low can we go? The string scale must be at least a TeV, or else the net effect can be as in figure 10~. If the threshold is at the correct scale, the we would already have seen string physics. The five-dimensional threshold must four couplings meet at a point. correspond to a radius of no more than a millimeter, or else Cavendish experiments As it stands this has no more predictive power than any of the other proposed would already have shown the four-dimensional inverse square law turning into a solutions. There is one more unknown parameter, the new threshold scale, and five-dimensional inverse cube. Remarkably, it is difficult to improve on these ex- one more prediction. However, it does illustrate that the new understanding of treme model-independent bounds. The large dimension in particular might seem string theory will lead to some very new ideas about the nature of unification. to imply a whole tower of new states at energies above 10e4 eV, but these are Figure 11 is only one example of a much more general idea now under study, that very weakly coupled (gravitational strength) and so would not be seen. It may be that construction of a full model, with a sensible cosmology, will raise these Tit was asked whether the gravitational couplinghas additional p-function type running. Al- scales, but that they will still lie lower than we used to imagine. though this could occur in principle, it doesnot do so becauseof a combinationof dimensional analysisand symmetryarguments. I had been somewhat skeptical about this idea, for a reason that is evident in

-223- 00 0 mological constant suggests a new phase of supersymmetry, whose phenomenology Two very recent reviews, including the Maldacena conjecture: at this point is completely unknown. Still, the discovery and precision study of . A. Sen, “Developments in ,” hep-ph/9810356. supersymmetry remains the best bet for testing all of these ideas. . J. Schwarz, “Beyond Gauge Theories,” hepth/9807195. In conclusion, the last few years have seen remarkable progress, and there is a For more on low energy string theory and millimeter dimensions, see the talk by real prospect of answering difficult and long-standing problems in the near future. Nima Arkani-Hamed at the SSI Topical Conference, and references therein.

References

Two texts on string theory:

l M. B. Green, J. H. Schwarz, and E. Witten, Szlperstring Theoy, Vols. 1 and 2 (Cambridge University Press, Cambridge, 1987).

l J. Polchinski, String Theory, I/ok. 1 and 2 (Cambridge University Press, Cambridge, 1998). An earlier version of these lectures, with extensive references:

l J. Polchinski, “String Duality,” Rev. Mod. Phys. 68, 1245 (1996), hep- th/9607050. The talk by Michael Peskin at the SSI Topical Conference gives more detail on some of the subjects in my lectures. Other popular accounts:

l G. P. Collins, “Quantum Black Holes Are Tied to D-Branes and Strings,” Physics Today 50, 19 (March 1997).

l E. Witten, “Duality, Spacetime, and Quantum Mechanics,” Physics Today 50, 28 (May 1997).

l B. G. Levi, “Strings May Tie Quantum Gravity to Quantum Chromodynam- its,” Physics Today 51, 20 (August 1998). A summer school covering many of the developments up to 1996:

l C. Efthimiou and B. Greene, editors, Fields, Strings, and Duality, TASK 1996 (Singapore: World Scientific, 1997). Lectures on matrix theory:

l T. Banks, “Matrix Theory,” Nucl. Phys. Proc. Suppl. 67, 180 (1998), hep-th/9710231.

l D. Bigatti and L. Susskind, “Review of Matrix Theory,” hepth/9712072. Lectures on developments in black hole quantum mechanics:

l G. T. Horowitz, ‘

l A. W. Peet, “The Bekenstein Formula and String Theory,” Classical and Quantum Gravity 15, 3291 (1998), hepth/9712253.

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