Edward Witten, Magic, Mystery, And Matrix

Magic, Mystery, andMatrixEdward Witten

1124 NOTICES OF THE AMS VOLUME 45, NUMBER 9

In the twentieth century, the quest for deeperunderstanding of the laws of nature haslargely revolved around the development oftwo great theories: namely, general relativityand quantum mechanics.

General relativity is, of course, Einstein’s theoryaccording to which gravitation results from the cur-vature of space and time; the mathematical frame-work is that of Riemannian geometry. While pre-viously spacetime was understood as a fixed arena,given ab initio, in which physics unfolds, in gen-eral relativity spacetime evolves dynamically, ac-cording to the Einstein equations. Part of the prob-lem of physics, according to this theory, is todetermine, given the initial conditions as input, howspacetime will develop in the future.

The influence of general relativity in twentieth-century mathematics has been clear enough. Learn-ing that Riemannian geometry is so central inphysics gave a big boost to its growth as a math-ematical subject; it developed into one of the mostfruitful branches of mathematics, with applica-tions in many other areas.

While in physics general relativity is used tounderstand the behavior of astronomical bodiesand the universe as a whole, quantum mechanicsis used primarily to understand atoms, molecules,and subatomic particles. Quantum theory has had

a much more complex history than general rela-tivity, and in some sense most of its influence onmathematics belongs to the twenty-first century.The quantum theory of particles—which is morecommonly called nonrelativistic quantum me-chanics—was put in its modern form by 1925 andhas greatly influenced the development of func-tional analysis, and other areas.

But the deeper part of quantum theory is thequantum theory of fields, which arises when onetries to combine quantum mechanics with specialrelativity (the precursor of general relativity, inwhich the speed of light is the same in every in-ertial frame but spacetime is still flat and given abinitio). This much more difficult theory, developedfrom the late 1920s to the present, encompassesmost of what we know of the laws of physics, ex-cept gravity. In its seventy years there have beenmany milestones, ranging from the theory of “an-timatter”, which emerged around 1930, to a moreprecise description of atoms, which quantum fieldtheory provided by 1950, to the “standard modelof particle physics” (governing the strong, weak,and electromagnetic interactions), which emergedby the early 1970s, to new predictions in our owntime that one hopes to test in present and futureaccelerators.

Quantum field theory is a very rich subject formathematics as well as physics. But its develop-ment in the last seventy years has been mainly byphysicists, and it is still largely out of reach as arigorous mathematical theory despite important ef-forts in constructive field theory. So most of its im-pact on mathematics has not yet been felt. Yet inmany active areas of mathematics, problems are

Edward Witten is professor of physics at the Institute forAdvanced Study. His e-mail address is [email protected].

This article is based on the Josiah Willard Gibbs Lecturegiven at the Joint Meetings in Baltimore in January 1998.

The author’s work is supported in part by NSF Grant PHY-9513835.

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OCTOBER 1998 NOTICES OF THE AMS 1125

studied that actuallyhave their most nat-ural setting in quan-tum field theory. Ex-amples includeDonaldson theory offour-manifolds, theJones polynomial ofknots and its general-izations, mirror sym-metry of complexmanifolds, elliptic co-homology, and manyaspects of the study ofaffine Lie algebras.

To a certain extentthese problems arestudied piecemeal,with difficulty in un-derstanding the rela-tions among them, be-cause their naturalhome in quantum fieldtheory is not now partof the mathematicaltheory. To make arough analogy (Figure1), one has here a vast mountain range, most ofwhich is still covered with fog. Only the loftiestpeaks, which reach above the clouds, are seen inthe mathematical theories of today, and thesesplendid peaks are studied in isolation, becauseabove the clouds they are isolated from one an-other. Still lost in the mist is the body of the range,with its quantum field theory bedrock and thegreat bulk of the mathematical treasures.

So there is one rather safe, though perhapsseemingly provocative, prediction about twenty-first century mathematics: trying to come to gripswith quantum field theory will be one of the mainthemes.

The 1/r2SingularityTo see a little further than this, we must discussquantum mechanics in a little more depth. The ori-gin and subsequent development of quantum me-chanics depended a lot on the “inverse square law”of gravity and electricity. The gravitational forcesbetween two masses M1 and M2 separated a dis-tance r is

−GM1M2

r2

(with G being Newton’s constant), and the electricalforce between two charges q1 and q2 separated adistance r is likewise

q1q2

r2 .

For elementary particles one typically hasq1q2 > GM1M2, which is why gravity can generallybe ignored on an atomic scale and below, but for

astronomical bodies typically GM1M2 > q1q2, sogravity dominates on large scales.

Obviously, the inverse square law means that theforce becomes infinite for r → 0. This singularitydid not cause great difficulties for Newton, since(for instance) the Moon was always at a safe dis-tance from the Earth, far from r = 0. However,once the electron and atomic nucleus were dis-covered almost a century ago, the 1/r2 singular-ity did become a severe problem. A simple calcu-lation based on nineteenth-century physics showedthat, because of the strong force at small r , the elec-tron should spiral into the nucleus in about 10−9

seconds. This was obviously not the case.To cure this problem, quantum mechanics was

invented. In quantum mechanics the position x andmomentum p of a particle do not commute, butobey Heisenberg’s relation

[p, x] = −i~,with ~ being Planck’s constant. This relation givesa sort of “fuzziness” to the electron and otherparticles. Because of this fuzziness, one never re-ally gets to r = 0, and the problem is averted.

What I have just described is nonrelativisticquantum mechanics—or quantum mechanics ofparticles, as I called it before. This theory was de-veloped by about 1925 and has long since beenmore or less assimilated mathematically. The wholetheory of elliptic operators on manifolds is a kindof mathematical counterpart of nonrelativisticquantum mechanics; group representation theoryis also a close cousin.

Figure 1.

witten.qxp 8/12/98 8:19 AM Page 1125


Quantum Field TheoryBut trying to allow for special relativity makesthings much more challenging. In special relativ-ity one cannot assume the “instantaneous actionat a distance” that is implicit in the inverse squarelaws of gravity and electricity. Instead, the forcemust be mediated by a field, and consistency ofthe whole setup requires that an uncertainty rela-tion analogous to the Heisenberg formula[p, x] = −i~ must be applied to the field. Thenthings become much more complicated and muchmore interesting. From the uncertainty relationone deduces that the field comes in “quanta”,which are observed as particles of a new kind—pho-tons in the case of the electromagnetic field. Themore familiar particles, like the electron, mustlikewise be reinterpreted as quanta of a field. Onesoon learns that, like classical electromagneticwaves, these quanta can be created and annihilated(Figure 2). This leads to the concept of antimatterand the prediction of matter-antimatter creationand annihilation. By this time one is living in aworld that is much more surprising and interest-ing and certainly much more challenging math-ematically.

Though quantum mechanics was invented be-cause of the 1/r2 singularity, it turned out thatonce special relativity was included, quantum me-chanics did not automatically cure all of the prob-lems associated with that singularity. Much of thedevelopment of physics since 1930 had to do withthe 1/r2 problem in the light of quantum me-

chanics plus special relativity. Among the mile-stones, some of which I alluded to before, were thefollowing:

•By about 1950, renormalization theory andquantum electrodynamics gave a much more pre-cise theory of electrons and atoms.

•In 1967–73 nonabelian gauge theories were in-corporated in the description of nature (giving theelectroweak part of the standard model) to over-come the problem of the 1/r2 singularity in thecase of the weak interactions.

•In 1973 asymptotic freedom of nonabeliangauge theories was discovered and used to over-come and fully tame the 1/r2 problem in the caseof the nuclear force. This also completed the con-struction of the standard model.

Those last developments set the stage for a newkind of interaction between quantum field theoryand geometry. Nonabelian gauge theories, sup-plemented in time by other ingredients that I havenot yet mentioned, notably supersymmetry andstring theory, led physicists to gradually ask newkinds of questions that involved geometrical con-cepts and techniques not previously used inphysics. In time it was realized that things couldbe turned around and that the quantum field the-ory methods could be used to draw inferencesabout geometry. And so it is that although quan-tum field theory is a rather old subject, its math-ematical influence is in many respects rather re-cent and still lies mainly in the future.

Quantum GravityThe developments that I have already mentioned,leading to the standard model of particle physics,put most of the known phenomena in physics, ex-cept gravity, more or less under one roof. Themain hurdle that remains is to include gravity, butthis involves problems of a quite different nature.At first sight, gravity presents us with just an-other instance of the familiar 1/r2 singularity.Gravity and electricity are indeed very similar inmany ways, but the relation between them is notnearly as straightforward as is suggested by the factthat in classical physics they are both governed byinverse square laws. Relativistically, for instance,the field equations of electromagnetism (Maxwell’sequations) are linear, while Einstein’s equations forthe gravitational field are highly nonlinear. Quan-tum fuzziness, springing from the uncertainty re-lation [p, x] = −i~ , is apparently not enough todeal with the 1/r2 singularity in the gravitationalforce. Overcoming this problem—combining quan-tum mechanics and gravity—is probably the mainobstacle to unifying the forces of nature.

Making sense of quantum gravity is essential aswell for addressing many commonplace questionsthat one might well ask with no special training inphysics. Astronomers, for example, see that the uni-verse is expanding today, and as far as we can tell

space

e−e+

Time

γ γ

Figure 2. A spacetime history with an electron e− and itsantiparticle, the positron e+ , annihilating into two photons orbasic units of light (labeled by γ). If the direction of time isreversed, one gets a process in which incoming light wavescombine to create a charged particle-antiparticle pair. Suchphenomena occur all the time in quantum field theory.



this expansion began in an explosion, oftencalled the big bang. But contemplation of thebig bang may seem to present paradoxes.What started the clocks? What was there be-fore the big bang? Gravity and quantum me-chanics were both important near the bigbang, so the answers must depend on howgravity and quantum mechanics work to-gether.

Physicists learned rather unexpectedly,beginning in the early 1970s, that the prob-lem of quantum gravity could be overcomeby introducing a new sort of fuzziness. Onereplaces “point particles” by “strings”. Ofcourse, the point particles and strings mustboth be treated quantum mechanically.Quantum effects are proportional to Planck’sconstant ~, and stringy effects are propor-tional to a new constant α′ (equal to ap-proximately (10−32 cm)2) that determinesthe size of strings. In this theory stringinessand quantum uncertainty both contribute tosmearing things out; together they tame the1/r2 singularity of gravity.

If string theory is correct, then α′ is justas fundamental in physics as ~, and its ef-fects are at least as interesting. The ~ and α′ de-formations both involve fundamental new toolsand ideas in geometry. About the ~ deformationwe have ample experience and fairly extensivenonrigorous knowledge concerning some of thegeometrical applications, though, as I explained be-fore, the mathematical development still lies largelyin the future. The α′ deformation is far more mys-terious and challenging even for physicists, as thebasic tools and concepts have not yet been un-earthed. Seeking to do so is perhaps the most ex-citing adventure in theoretical physics for the nextfew decades. The mathematical questions posedby the ~ deformation are at least beginning to beasked, though answers still lie mainly ahead, butthe equally exciting mathematical questions as-sociated with the α′ deformation are for the mostpart not yet even being asked. The reason for thisis simply that the basic prerequisite for under-standing what the α′ deformation is supposed tomean is a thorough familiarity with the ~ defor-mation, and this is not yet available mathematically.

The idea of replacing point particles by stringssounds so naive that it may be hard to believe thatit is truly fundamental. But in fact this naive-sound-ing step is probably as basic as introducing thecomplex numbers in mathematics. If the real andcomplex numbers are regarded as real vectorspaces, one has dimR(R) = 1, dimR(C) = 2. Theorbit of a point particle in spacetime (Figure 3) isone-dimensional and should be regarded as a realmanifold, while the orbit of a string in spacetimeis two-dimensional (over the reals) and should beregarded as a complex Riemann surface. Physics

without strings is roughly analogous to math-ematics without complex numbers.

The String TheoriesThe requirements of quantum mechanics plus spe-cial relativity are so tight that historically con-structing string theories was very difficult. Theconditions that must be obeyed are highly overde-termined. A vast effort went into the constructionof string theories, and by the time the dust clearedin 1984–85 it was found that there were five ofthem. They differ by very general properties of thestrings:

•In two theories (the Type IIA and Type IIBtheories, which differ by whether there is invari-ance under reversal of the orientation of spacetime)the strings are closed and oriented and are elec-trical insulators.

•In two theories (the heterotic superstringswith gauge groups SO(32) and E8 × E8) the stringsare closed, oriented, and superconducting.

•In the last case (Type I) the strings are unori-ented and insulating and can have boundaries, inwhich case they carry electric charges on theirboundaries.

Because there are so few string theories, the gen-eral framework of string theory makes certain gen-eral predictions that are out of reach without stringtheory:

1. Gravity. Each of the five string theories pre-dicts gravity (plus quantum mechanics): that is,these theories predict a structure that looks justlike general relativity at long distances, with cor-rections (unfortunately unmeasurably small in

(a) (b)

time time

space

Figure 3. The spacetime orbit of a point particle (a) or a string (b) is amanifold of real dimension 1 or 2.



practice) proportional to α′. This is very striking,since, as I have stressed, standard quantum fieldtheory makes gravity impossible. It is the singlemost important reason for the intensive study ofstring theory in the last generation.

2. Nonabelian Gauge Symmetry. The secondgeneral prediction is nonabelian gauge symmetry(again with unmeasurably small corrections pro-portional to α′), which is of course the bread andbutter of particle physics.

3. Supersymmetry. The last general predictionis “supersymmetry”, a subtle new kind of sym-metry of elementary particles. We do not yet knowif nature is supersymmetric, but there are hints (forinstance, from accurate measurements of the low-energy gauge couplings) that it is. There is a goodchance that we will know for certain from accel-erator experiments within about a decade. Thefact that we do not yet really know if it is rightmeans that supersymmetry (which historically wasdiscovered at least in part because of its role instring theory) is a genuine prediction of string the-ory, while gravity and nonabelian gauge symme-try (which were already known before they wereseen to be consequences of string theory) mightbetter be called postdictions.

To properly explain supersymmetry requiresassuming some familiarity with quantum field the-ory and so is beyond our scope here. But as a veryrough analogy, supersymmetric quantum theoryis to ordinary quantum theory as differential formson a manifold are to functions on a manifold. Avery large fraction of geometrical applications ofquantum field theory found in the eighties andnineties depend on supersymmetry. (Examples in-

clude the supersymmetric proofs ofthe positive energy theorem, theAtiyah-Singer index theorem, andthe Morse inequalities, and the quan-tum field theory approaches to el-liptic cohomology and to Donald-son theory.) This, along with itsbeauty and the boost its discoverywould give to string theory, is yet an-other reason to hope that super-symmetry will be found! Surely, if su-persymmetry is confirmed inaccelerators, mathematical attentionwill be focussed on this fruitfulbranch of quantum field theoryroughly as the discovery of generalrelativity focussed attention on Rie-mannian geometry.

Apart from the general predic-tions that I have stressed, string the-ory also leads in a simple way to el-egant and qualitatively correctmodels that combine quantum grav-ity and the other known forces in na-ture, recovering the main features ofthe standard model. To improve

these constructions further, the most vital need isprobably to understand the vanishing (or extremesmallness) of the cosmological constant (the en-ergy density of the vacuum) after supersymmetrybreaking. That remains out of reach.

Although physicists do not have any systematicunderstanding of the new geometrical ideas asso-ciated with the α′ deformation, powerful methodsusing two-dimensional conformal field theory areavailable for exploring some of the associated phe-nomena. In the late 1980s and early 1990s mucheffort in string theory was focussed on describingsome of these phenomena. An example is mirrorsymmetry, a relation between two spacetimes thatare different in classical geometry but are equiv-alent for α′ 6= 0. This symmetry has attracted muchmathematical interest because it has striking con-sequences, some of which can be extracted fromtheir natural conformal field theory setting andstated in isolation. Closely related is the phenom-enon of topology change. In general, in string the-ory the question, What is the topology of space-time? does not make sense, because in general forα′ 6= 0 classical ideas in geometry are not valid. Butin a suitable limit, upon varying a parameter, clas-sical ideas may be a good approximation. It wasfound that one can perfectly well have a family ofstring theory states depending on a real parame-ter t that interpolate between two different space-times in the following sense. For t →∞, classicalideas in geometry are a good approximation, andone observes a spacetime X. For t → −∞ , classi-cal geometry is again a good approximation andone observes a different (and perhaps topologically

M-theory

Type I

SO(32) heterotic

E8 × E8 heterotic

11-dimensional supergravity

Type IIA

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Figure 4. The five string theories—and a wild card, eleven-dimensionalsupergravity, that has proved to be important in getting a systematicunderstanding—are now understand as different limiting cases of a morecomprehensive (and little understood) theory known as M-theory. Thefigure is meant to suggest a family of physical situations that are possiblein M-theory. With some oversimplification one can think of the parametersin the figure as ~ and α′.



distinct) spacetime Y. Somewhere in between largepositive t and large negative t one passes througha “stringy” region in which classical geometry isnot a good description and the interpolation fromX to Y takes place.

M-TheoryWhile understanding the new geometrical ideasthat prevail for α′ 6= 0 remains in all likelihood aproblem for the next century, the problem haslately been recast in a much wider context. For yearsthe existence of five string theories, though it rep-resented a dramatic narrowing of the possibilitiesthat existed in prestring physics, posed a puzzle.It is rather strange to be told that there is a richnew framework for physics which unifies quantummechanics and gravity and that in this new frame-work there are five possible theories. If one ofthose theories describes our universe, who lives inthe other four worlds?

By learning something about what happenswhen α′ and ~ are both nonzero, we have learneda very satisfying answer to this question. The fivestring theories traditionally studied are differentlimiting cases of one richer and still little-under-stood theory. For ~ = 0 these theories are really dif-ferent, but with ~ and α′ both nonzero one can in-terpolate between them. The relation between them(Figure 4) is rather like the relation between the clas-sical spacetimes X and Y mentioned two para-graphs ago. These are distinct in classical geome-try—that is, for α′ = 0—but for α′ 6= 0 they are twodifferent limiting cases of a more subtle struc-ture.

The richer theory, which has as limiting casesthe five string theories studied in the last genera-tion, has come to be called M-theory, where Mstands for magic, mystery, or matrix, according totaste. The magic and mystery are clear enough,while “matrix” refers to a new noncommutativity,roughly analogous to [p, x] = −i~ but very differ-ent, that seems to enter the theory. Physicists andmathematicians are likely to spend much of thenext century trying to come to grips with this the-ory.

Reading Suggestions: For an introduction toquantum field theory and perturbative string the-ory, the reader might consult Quantum Fields andStrings: A Course for Mathematicians, P. Deligne,P. Etinghof, D. Freed, L. Jeffrey, D. Kazhdan, D. Mor-rison, and E. Witten, eds. (American MathematicalSociety, to appear). A recent string theory text (forphysicists) treating some of the nonperturbativeas well as perturbative developments is String The-ory, Vols. I and II, by Joseph Polchinski (CambridgeUniversity Press, 1998).


Edward Witten, Magic, Mystery, And Matrix

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