why every physicst should know string theory

8/18/2019 Why Every Physicst Should Know String Theory

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Physics Today

What every physicist should know about string theory

Edward Witten

Citation: Physics Today 68(11), 38 (2015); doi: 10.1063/PT.3.2980

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38 November 2015 Physics Today www.physicstoday.org

tring theory has, even among theoretical physi-

cists, the reputation of being mathematicallyintimidating. But many of its essential elementscan actually be described simply. This articleaims to answer a few basic questions. How does

string theory generalize standard quantum field the-ory? Why does string theory force us to unify generalrelativity with the other forces of nature, while stan-dard quantum field theory makes it so difficult toincorporate general relativity? Why are there no ul-traviolet divergences in string theory? And whathappens to Albert Einstein’s conception of spacetime?

Anyone who has studied physics is aware thatalthough physics—like history—does not preciselyrepeat itself, it does rhyme, with similar structuresappearing in different areas. For example, Einstein’s

gravitational waves are analogous to electromag-netic waves or to the water waves at the surface ofa pond. I will begin with one of nature’s rhymes: ananalogy between quantum gravity and the theory ofa single particle.

Even though we do not really understand it,quantum gravity is supposed to be some sort of the-

ory in which, at least from a macroscopic point of

view, we average, in a quantum mechanical sense,over all possible spacetime geometries. (We do notknow to what extent that description is valid micro-scopically.) The averaging is performed, in the sim-plest case, with a weight factor exp(iI /ħ), where I isthe Einstein–Hilbert action:

Here G is Newton’s constant, g is the determinant ofthe metric tensor, R is the curvature scalar, Λ is a cos-mological constant, and d4x is the spacetime volumeelement. We could add matter fields, but we do notseem to need them.

Let us try to make such a theory with one space-time dimension instead of four. The choices for aone-manifold are quite limited:

Moreover, the curvature scalar is identically zero inone dimension, and all that’s left of the Einstein–Hilbert action is the cosmological constant. How-

I d x g R= √ ( − 2 ).4 Λ1

16πG ∫

Edward Witten

Some of nature’s rhymes—the appearance of similar structures in differentareas of physics—underlie the way that string theory potentially unifiesgravity with the other forces of nature and eliminates the ultravioletdivergences that plague quantum gravity.

Edward Witten is the Charles Simonyi Professor ofmathematical physics at the Institute for Advanced Studyin Princeton, New Jersey.

L E W I S R O N A L D

What every physicist should know about


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ever, Einstein’s fundamental insights were not tiedto the specific Einstein–Hilbert action. Rather, theywere in the broader ideas that the spacetime geom-etry can vary dynamically and that the laws ofnature are generally covariant, or invariant underarbitrary diffeomorphisms (coordinate transforma-tions) of spacetime. (See the article by Michel Janssenand Jürgen Renn on page 30.) By applying thoseinsights, we can make a nontrivial quantum gravitytheory in one dimension provided we include mat-

ter fields.

Adding matterThe simplest matter fields are scalar fields XI , whereI = 1 , … , D. The standard general relativistic actionfor scalar fields is

where gtt is a 1 × 1 metric tensor, and the Λ term has been replaced with m2/2.

Let us introduce the canonical momentumPI = dXI /dt. The Einstein field equation—which is theequation of motion obtained by varying the action I

with respect to g—is just

We pick the gauge g tt = 1, so the equation isP2 + m2 = 0, with P2 = ∑I PI 2. Quantum mechanically(in units with ħ = 1), PI = −i∂/∂XI , and the meaning ofthe equation P2 + m2 = 0 is that the wavefunctionΨ(X), where X is the set of all XI , must be annihi-lated by the differential operator that correspondsto P2 + m2:

This is a familiar equation—the relativisticKlein–Gordon equation in D dimensions—but inEuclidean signature, in which time and space are onequal footing. To get a sensible physical interpreta-tion, we should reverse the kinetic energy of one ofthe scalar fields XI so that the action becomes

Now the wavefunction obeys a Klein–Gordon equa-tion in Lorentz signature:

So we have found an exactly soluble theoryof quantum gravity in one dimension that describes a spin-0 particle of mass m propagating in D-dimensional Minkowski spacetime. Actually, we canreplace Minkowski spacetime by any D-dimensionalspacetime M with a Lorentz (or Euclidean) signa-ture metric GIJ , the action being then

From here on, summation over repeated indices isimplied. The equation obeyed by the wavefunctionis now the massive Klein–Gordon equation incurved spacetime M:

where D represents covariant differentiation. Just to make things more familiar, let us go back

to the case of flat spacetime (I will work in Euclideansignature to avoid having to keep track of some fac-tors of i). Let us calculate the probability amplitude

for a particle to start at one point x in spacetime andend at another point y. We do so by evaluating aFeynman path integral in our quantum gravitymodel. The path integral is performed over all met-rics g(t) and scalar fields XI (t) on the one-manifold

with the condition that X(t) is equal to x at one endand to y at the other.

Part of the process of evaluating the path inte-gral in our quantum gravity model is to integrateover the metric on the one-manifold, modulo diffeo-morphisms. But up to diffeomorphism, the one-manifold has only one invariant, its total length τ ,which we will interpret as the elapsed proper time.In our gauge gtt = 1, a one-manifold of length τ is de-scribed by a parameter t that covers the range0 ≤ t ≤ τ. Now on the one-manifold, we have to inte-grate over all paths X(t) that start at x at t = 0 andend at y at t = τ. That is the basic Feynman integralof quantum mechanics with the Hamiltonian being

∂2

∂X2I ( (∑

I =1

D

− + ( ) = 0.m X2 Ψ

∂2

∂X20

∂2

∂X2i( (∑i=1D−1

− + ( ) = 0.m X2 Ψ

I dt g g G m= √ − .tt IJ 21

212 ∫

dXI

dt

dX J

dt ((

D

DXI D

DX J ( (− + ( ) = 0,G m XIJ 2 Ψ

x

y

I dt g g m= √ − ,tt2

212

12 ∫

dXI dt( ( [[ ∑I =1

D

∑I =1

D

g P m .tt I 2 2

+ = 0

12

12 ∫ { [ [ {dXidt( (dX0dt( ( ∑i=1

D−1I dt g g m= √ − + − .tt

2 22

www.physicstoday.org November 2015 Physics Today 39

x1

x3

x2

x4

y1τ1

τ3

τ2 τ4

y3

y2

y4

Figure 1. A graph with trivalent vertices. Thenatural path integral to consider is one in which thepositions x 1, … , x 4 of the four external particles arefixed, and the integration is over everything else.A convenient first step is to evaluate an integral inwhich the positions y 1, … , y 4 of the vertices are alsofixed. This Feynman diagram can generate an ultra-violet divergence in the limit that the proper-time

parameters τ 1, … , τ 4 in the loop all vanish.


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H = ½(P2 + m2). According to Feynman, the result isthe matrix element of exp(−τH )

But we have to remember to do the gravitationalpart of the path integral, which in the present con-

text means to integrate over τ.The integral over τ gives our final answer:

This formula is the output of the complete path in-tegral—an integral over metrics g(t) and paths X(t)with the given endpoints, modulo diffeomorphisms—in our quantum gravity model.

The function G(x , y) is the standard Feynmanpropagator in Euclidean signature, apart from aconvention-dependent normalization factor. More-over, an analogous derivation in Lorentz signature (for

both the spacetime M and the particle world line) givesthe correct Lorentz-signature Feynman propagator.

So we have interpreted a free particle in D-dimensional spacetime in terms of 1D quantumgravity. How can we include interactions? There isactually a perfectly natural way. There are not a lotof smooth one-manifolds, but there is a large supplyof singular one-manifolds in the form of graphs,such as the one in figure 1. Our quantum-gravity ac-tion makes sense on such a graph. We simply takethe same action that we used before, summed overall the line segments that make up the graph.

Now to do the quantum-gravity path integral,

we have to integrate over all metrics on the graph,up to diffeomorphism. The only invariants are thetotal lengths or proper times of each of the seg-ments. Some of the lines in figure 1 have been la-

beled by length or proper-time variables τi.The natural amplitude to compute is one in

which we hold fixed the positions x1 , … , x4 of thegraph’s four external particles and integrate over allthe τi and over the paths the particles follow on theline segments. To evaluate such an integral, it is con-

venient to first perform a computation in which wehold fixed the positions y1 , … , y4 of the vertices inthe graph. That means all endpoints of all segmentsare labeled. The computation that we have to per-form on each segment is the same as before andgives the Feynman propagator. The final integrationover y1 , … , y4 imposes momentum conservation ateach vertex. Thus we arrive at Feynman’s recipe forcomputing the amplitude associated with a Feyn-man graph—a Feynman propagator for each lineand an integration over all momenta subject to mo-mentum conservation.

A more perfect rhymeWe have arrived at one of nature’s rhymes. If we im-itate in one dimension what we would expect to doin four dimensions to describe quantum gravity, weend up with something that is certainly importantin physics—namely, ordinary quantum field theoryin a possibly curved spacetime. In our example infigure 1, the ordinary quantum field theory is scalarϕ3 theory because of the particular matter systemwe started with and because our graph had cubicvertices. Quartic vertices, for instance, would giveϕ4 theory, and a different matter system would givefields of different spins. Many or maybe all quantumfield theories in D dimensions can be derived in thatsense from quantum gravity in one dimension.

There is actually a much more perfect rhyme if

we repeat the procedure in two dimensions—that is,for a string instead of a particle. We immediately runinto the fact that a two-manifold Σ can be curved:

On a related note, 2D metrics are not all locallyequivalent under diffeomorphisms. A 2D metric ingeneral is a 2 × 2 symmetric matrix constructed fromthree functions:

A transformation of the 2D coordinates σ , generated

by

can remove only two functions, leaving the curva-ture scalar as an invariant.

All those complications suggest that the inte-gral over 2D metrics will not much resemble whatwe found in the 1D case. But now we notice the fol-lowing. The natural anolog of the action that weused in one dimension is the general relativistic ac-tion for scalar fields in two dimensions, namely

d pD

(2 )π Dτ

2 ∫ [ ]G x y ip y x p m( , ; ) = exp[ ·( − )] exp − ( + ) .τ2 2

d pD

(2 )π D2

p m2 2+ ∫ G x y d G x y ip y x( , )= ( , ; ) = exp[ ·( − )] .τ τ ∫ 0∞

g11 g12 g21 g22

g , g gab = = .21 12

→ + ( ), = 1, 2 ,σ σ σa a ah a

40 November 2015 Physics Today www.physicstoday.org

String theory

a b

τ1 τ3τ2

τ1ˆτ2ˆ

τ3ˆ

Figure 2. From lines to tubes. (a) A Feynman diagram with proper-

time parameters τ 1, τ 2, and τ 3 (top) can be turned into a correspondingRiemann surface (bottom) by slightly thickening all the lines in thediagram into tubes that join together smoothly. The Riemann surface isparameterized, up to coordinate and Weyl transformation, by complexvariables τ̂ 1, τ̂ 2, and τ̂ 3. (b) The same procedure can turn the one-loopFeynman diagram (top) into its string theory analog (bottom).


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But this is conformally invariant, that is, it is invari-

ant under a Weyl transformation of the metric gab → eϕ gab for any real function ϕ on Σ. This is trueonly in two dimensions (and only if there is no cos-mological constant, so we omit that term in going totwo dimensions). Requiring Weyl invariance as wellas diffeomorphism invariance is enough to makeany metric gab on Σ locally trivial (locally equivalentto δab), similar to what we said for one-manifolds.

Some very pretty 19th-century mathematicsnow comes into play. A two-manifold whose metricis given up to a Weyl transformation is called a Rie-mann surface. As in the 1D case, a Riemann surfacecan be characterized up to diffeomorphism by fi-nitely many parameters. There are two big differ-ences: The parameters are now complex rather than

real, and their range is restricted in a way that leavesno room for an ultraviolet divergence. I will returnto that last point later.

But first, let us take a look at the relation be-tween the 1D parameters and the 2D ones. A metricon the Feynman graph in figure 2a depends, up todiffeomorphism, on three real lengths or proper-time parameters τ1 , τ2 , and τ3. If the graph is “thick-ened” into a two-manifold, as suggested by thefigure, then a metric on that two-manifold depends,up to diffeomorphism and Weyl transformation, onthree complex parameters τ̂1 , τ̂2 , and τ̂3. Figure 2bgives another illustration of the relation betweena Feynman graph and a corresponding Riemannsurface.

We used 1D quantum gravity to describe quan-tum field theory in a possibly curved spacetime butnot to describe quantum gravity in spacetime. Thereason that we did not get quantum gravity inspacetime is that there is no correspondence be-tween operators and states in quantum mechanics.We considered the 1D quantum mechanics withaction

What turned out to be the external states in a Feyn-man diagram were just the states in that quantummechanics. But a deformation of the spacetime met-ric is represented not by a state but by an operator.

When we make a change δGIJ in the spacetime met-ric GIJ , the action changes by I → I + ∫ dt √‾ gO , whereO = ½ gttδGIJ ∂tXI ∂tX J is the operator that encodes achange in the spacetime metric. Technically, to com-pute the effect of the perturbation, we include in thepath integral a factor δI = ∫ dt √‾ gO , integrating overthe position at which the operator O is inserted.

A state would appear at the end of an externalline in the Feynman graph. But an operator O suchas the one describing a perturbation in the space-time metric appears at an interior point in the graph,as shown in figure 3a. Since states enter at ends ofexternal lines and operators are inserted at internalpoints, there is in general no simple relation be-tween operators and states.

But in conformal field theory, there is a corre-spondence between states and operators. The oper-ator O = ½ gabδGIJ ∂aXI ∂bX J that represents a fluc-tuation in the spacetime metric automaticallyrepresents a state in the quantum mechanics. Thatis why the theory describes quantum gravity inspacetime.

The operator–state correspondence arises froma 19th-century relation between two pictures thatare conformally equivalent. Figure 3b shows a two-manifold Σ with a marked point p at which an op-erator O is inserted. In figure 3c, the point p has beenremoved from Σ, and a Weyl transformation of themetric of Σ has converted what used to be a smallneighborhood of the point p to a semi-infinite tube.The tube is analogous to an external line of a Feyn-man graph, and what would be inserted at the endof it is a quantum string state. The relation betweenthe two pictures is the correspondence between op-erators and states.

To understand the Weyl transformation be-tween the two pictures, consider the metric of theplane (figure 4) in polar coordinates:

We think of inserting an operator at the point r = 0.I dt g g G m= √ − .tt IJ

212

12 ∫

dXI

dt

dX J

dt (( ds dr r d2 2 2 2= + .θ

I d g g G= √ .2 ab IJ σ∂XI

∂σa∂X J

∂σb ∫


p

p

a cb

Figure 3. States and operators. (a) A deformation of the spacetime metric corresponds to an operator O thatcan be inserted at some internal point p on a Feynman graph. By contrast, a state in the quantum mechanicswould be attached to the end of one of the outgoing lines of the graph. (b) A Riemann surface can also havean operator insertion. (c) If the marked point in panel b is deleted, the Riemann surface is conformally equiva-lent to one with an outgoing tube that is analogous to an external line of a Feynman graph. The operator O that was inserted at p is converted to a quantum state of the string that propagates on the tube.


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Now remove the point and make a Weyl transforma-tion by multiplying ds2 with 1/r2 to get a new metric

In terms of w = log r , −∞


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But to explain the idea without any extraneous tech-nicalities, we will consider, instead of parallelograms,only rectangles:

We label the height and base of the rectangle as sand s′, respectively.

Only the ratio u = s’/s is conformally invariant.Also, since what we call the “height” as opposed tothe “base” of a rectangle is arbitrary, we are free toexchange s and s′, which corresponds to u ↔ 1/u. Sowe can restrict ourselves to s′ ≥ s , and thus the rangeof u is 1 ≤ u < ∞.

The proper-time parameter τ of the particle cor-responds to u in string theory, with the key differ-ence being that for the particle, 0 ≤ τ < ∞, but for the

string, 1 ≤ u < ∞. So in the approximation of consid-ering only rectangles and not parallelograms, theone-loop cosmological constant in string theory is

There is no ultraviolet divergence, because thelower limit on the integral is 1 instead of 0. A morecomplete analysis with parallelograms shifts thelower bound on u from 1 to √‾3/2.

I have described a special case, but the conclusionis general. The stringy formulas generalize the fieldtheory formulas, but without the region that can giveultraviolet divergences in field theory. The infraredregion (τ → ∞ or u → ∞) lines up properly between

field theory and string theory, and that is why a stringtheory can imitate field theory in its predictions for behavior at low energies or long times and distances.

Emergent spacetimeMy final goal here is to explain, at least partly, in whatsense spacetime emerges from something deeper ifstring theory is correct. Let us focus on the followingfact. The spacetime M with its metric tensor GIJ (X)was encoded as the data that enabled us to defineone particular 2D conformal field theory. That is theonly way that spacetime entered the story.

In our construction, we could have used a dif-ferent 2D conformal field theory (subject to a fewgeneral rules that I will omit for the sake of brevity).

Now if GIJ (X) is slowly varying (the radius of curva-ture is everywhere large), the Lagrangian by whichwe described the 2D conformal field theory is weaklycoupled and useful. In that case, string theory matchesthe ordinary physics that we are familiar with. In thissituation, we may say that the theory has a semiclas-sical interpretation in terms of strings in spacetime—and it will reduce at low energies to an interpretationin terms of particles and fields in spacetime.

When we get away from a semiclassical, weak-coupling limit, the Lagrangian is not so useful and

the theory does not have any particular interpreta-

tion in terms of strings in spacetime. The potential breakdown of a simple spacetime interpretation hasmany nonclassical consequences, such as the abilityto make continuous transitions from one spacetimemanifold to another, or the fact that certain typesof singularities (but not black hole singularities) inclassical general relativity turn out to represent per-fectly smooth and harmless situations in string the-ory. An example of the nonclassical behavior ofstring theory is sketched in figure 6.

In general, a string theory comes with no par-ticular spacetime interpretation, but such an inter-pretation can emerge in a suitable limit, somewhatas classical mechanics sometimes arises as a limit ofquantum mechanics. From this point of view, space-time emerges from a seemingly more fundamentalconcept of 2D conformal field theory.

I have not given a complete explanation of thesense in which, in the context of string theory, space-time emerges from something deeper. A completelydifferent side of the story, beyond the scope of thepresent article, involves quantum mechanics andthe duality between gauge theory and gravity. (Seethe article by Igor Klebanov and Juan Maldacena,PHYSICS TODAY , January 2009, page 28.) However,what I have described is certainly one importantand relatively well-understood piece of the puzzle.It is at least a partial insight about how spacetime asconceived by Einstein can emerge from somethingdeeper, and thus hopefully is of interest in the pres-ent centennial year of general relativity.

Additional resources‣ B. Zwiebach, A First Course in String Theory , 2nded., Cambridge U. Press (2009).‣ J. Polchinski, String Theory, Volume 1: An Intro-duction to the Bosonic String , Cambridge U. Press(2005).‣ M. B. Green, J. S. Schwarz, E. Witten, SuperstringTheory, Volume 1: Introduction , Cambridge U. Press(1987). ■

s

s′

Λ1 = Tr exp(− ) .uH 12

du

u ∫ 1∞


M1 M2

M3

Figure 6. Schematic representation of a familyof two-dimensional conformal field theories (thegray region bounded by black lines) that depend ontwo parameters. For some values of the parameters,the theories have semiclassical interpretations interms of strings propagating in a spacetime M1, M2,or M3. Generically there is no such interpretation.However, one can make a continuous transitionfrom one possible classical spacetime to another,as indicated by the colored lines.


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why every physicst should know string theory

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