SPECIAL SECTION: SUPERSTRINGS – A QUEST FOR A UNIFIED THEORY String theory John H. Schwarz California Institute of Technology, Pasadena, CA 91125, USA energies, it is present in exactly the form proposed by This article presents the basic concepts of string the- Einstein. This is significant, because it is arising within ory followed by an overview of the peculiar history of how it arose. the framework of a consistent quantum theory. Ordinary quantum field theory does not allow gravity to exist; string theory requires it! The second general fact is that MANY of the major developments in fundamental phys- Yang–Mills gauge theories of the sort that comprise the ics of the past century arose from identifying and over- standard model, naturally arise in string theory. We do coming contradictions between existing ideas. For not understand why the specific SU(3) ´ SU(2) ´ U(1) example, the incompatibility of Maxwell’s equations gauge theory of the standard model should be preferred, and Galilean invariance led Einstein to propose the spe- but (anomaly-free) theories of this general type do arise cial theory of relativity. Similarly, the inconsistency of naturally at ordinary energies. The third general feature special relativity with Newtonian gravity led him to of string theory solutions is supersymmetry. The develop the general theory of relativity. More recently, mathematical consistency of string theory depends cru- the reconciliation of special relativity with quantum cially on supersymmetry, and it is very hard to find mechanics led to the development of quantum field the- consistent solutions (quantum vacua) that do not pre- ory. We are now facing another crisis of the same char- serve at least a portion of this supersymmetry. This pre- acter, namely general relativity appears to be diction of string theory differs from the other two incompatible with quantum field theory. Any straight- (general relativity and gauge theories) in that it really is forward attempt to ‘quantize’ general relativity leads to a prediction. It is a generic feature of string theory that a nonrenormalizable theory. In my opinion, this means has not yet been discovered experimentally. that the theory is inconsistent and needs to be modified at short distances or high energies. The way that string theory does this is to give up one of the basic assump- Supersymmetry tions of quantum field theory, the assumption that ele- mentary particles are mathematical points, and instead As we have just said, supersymmetry is the major pre- to develop a quantum field theory of one-dimensional diction of string theory that could appear at accessible extended objects, called strings. There are very few energies, that has not yet been discovered. A variety of consistent theories of this type, but superstring theory arguments, not specific to string theory, suggest that the shows great promise as a unified quantum theory of all characteristic energy scale associated with supersym- fundamental forces, including gravity. There is no real- metry breaking should be related to the electro-weak istic string theory of elementary particles that could scale, in other words in the range 100 GeV–1 TeV. The serve as a new standard model, since there is much that symmetry implies that all known elementary particles is not yet understood. But that, together with a deeper should have partner particles, whose masses are in this understanding of cosmology, is the goal. This is still a general range. This means that some of these superpart- work in progress. 1,2 ners should be observable at the CERN Large Hadron Even though string theory is not yet fully formu- Collider (LHC), which will begin operating in the mid- lated, and we cannot yet give a detailed description of dle part of this decade. There is even a chance that Fer- how the standard model of elementary particles should milab Tevatron experiments could find superparticles emerge at low energies, there are some general features earlier than that. of the theory that can be identified. These are features In most versions of phenomenological supersym- that seem to be quite generic irrespective of how vari- metry, there is a multiplicatively conserved quantum ous details are resolved. The first, and perhaps most number called R-parity. All known particles have even important, is that general relativity is necessarily incor- R-parity, whereas their superpartners have odd R-parity. porated in the theory. It gets modified at very short dis- This implies that the superparticles must be pair- tances/high energies but at ordinary distances and produced in particle collisions. It also implies that the lightest supersymmetry particle (or LSP) should be ab- e-mail: [email protected] solutely stable. It is not known with certainty which CURRENT SCIENCE, VOL. 81, NO. 12, 25 DECEMBER 2001 1547 SPECIAL SECTION: particle is the LSP, but one popular guess is that it is a Basic concepts of string theory ‘neutralino’. This is an electrically neutral fermion that is a quantum-mechanical mixture of the partners of the In conventional quantum field theory the elementary 0 photon, Z , and neutral Higgs particles. Such an LSP particles are mathematical points, whereas in perturbat- would interact very weakly, more or less like a neutrino. ive string theory the fundamental objects are one- It is of considerable interest, since it is an excellent dimensional loops (of zero thickness). Strings have a dark-matter candidate. Searches for dark-matter parti- characteristic length scale, which can be estimated by cles called WIMPS (weakly interacting massive parti- dimensional analysis. Since string theory is a relativistic cles) could discover the LSP some day, though current quantum theory that includes gravity, it must involve experiments might not have sufficient detector the fundamental constants c (the speed of light), ¬ volume to compensate for the exceedingly small cross- (Planck’s constant divided by 2p), and G (Newton’s sections. gravitational constant). From these one can form a There are three unrelated arguments that point to the length, known as the Planck length same mass range for superparticles. The one we have just been discussing, a neutralino LSP as an important h 3/2 component of dark matter, requires a mass of order l æ G ö -33 p = ç 3 ÷ = 1.6 ´10 cm. (1) 100 GeV. The precise number depends on the mixture è c ø that comprises the LSP, what its density is, and a num- ber of other details. A second argument is based on the Similarly, the Planck mass is famous hierarchy problem. This is the fact that standard model radiative corrections tend to renormalize the h 1/2 æ cö 19 2 Higgs mass to a very high scale. The way to prevent m p = ç ÷ = 1.2 ´10 GeV / c . (2) this is to extend the standard model to a supersymmetric è G ø standard model and to have the supersymmetry be broken at a scale comparable to the Higgs mass, and Experiments at energies far below the Planck energy hence to the electro-weak scale. The third argument that cannot resolve distances as short as the Planck length. gives an estimate of the susy-breaking scale is Thus, at such energies, strings can be accurately ap- grand unification. If one accepts the notion that the proximated by point particles. From the viewpoint of standard model gauge group is embedded in a larger string theory, this explains why quantum field theory gauge group such as SU(5) or SO(10), which is broken has been so successful. at a high mass scale, then the three standard model cou- As a string evolves in time it sweeps out a two- pling constants should unify at that mass scale. Given dimensional surface in space–time, which is called the the spectrum of particles, one can compute the evolu- world sheet of the string. This is the string counterpart tion of the couplings as a function of energy using re- of the world line for a point particle. In quantum field normalization group equations. One finds that if one theory, analysed in perturbation theory, contributions to only includes the standard model particles, this unifica- amplitudes are associated to Feynman diagrams, which tion fails quite badly. However, if one also includes all depict possible configurations of world lines. In particu- the supersymmetry particles required by the minimal lar, interactions correspond to junctions of world lines. supersymmetric extension of the standard model, then Similarly, pertubative string theory involves string the couplings do unify at an energy of about world sheets of various topologies. A particularly sig- 2 ´ 1016 GeV. For this agreement to take place, it is nificant fact is that these world sheets are generically necessary that the masses of the superparticles are less smooth. The existence of interaction is a consequence than a few TeV. of world-sheet topology rather than a local singularity There is further support for this picture, such as the on the world sheet. This difference from point-particle ease with which supersymmetric grand unification ex- theories has two important implications. First, in string plains the masses of the top and bottom quarks and theory the structure of interactions is uniquely deter- electro-weak symmetry breaking. Despite all these indi- mined by the free theory. There are no arbitrary interac- cations, we cannot be certain that this picture is correct tions to be chosen. Second, the ultraviolet divergences until it is demonstrated experimentally. One could sup- of point-particle theories can be traced to the fact that pose that all this is a giant coincidence, and the correct interactions are associated to world-line junctions at description of TeV-scale physics is based on something specific space–time points.