## String Theory

**-**

## String Theory

# STRING THEORY

String theory is a proposed unified theory of fundamental physics, incorporating both particle physics and gravity. It is based on the idea that the basic building blocks of nature are strings, one-dimensional objects of zero thickness, which form either closed loops or open curves (Figure 1). This theory has not yet been experimentally tested, but it has attracted the attention of theoretical physicists from a wide range of fields because it unifies many of the central concepts of physics and resolves a number of longstanding theoretical problems.

**FIGURE 1**

## A Brief History

Two of the central questions in physics are the nature of matter and the nature of gravity. In the twentieth century, very successful theories of each were discovered: the Standard Model of matter and the general theory of relativity. However, both theories are incomplete. The Standard Model is based on a complicated pattern of particles and forces, similar to the Periodic Table of the elements, and this pattern must be explained. General relativity, when combined with quantum mechanics, suffers from several problems and paradoxes when applied to very short distances or to black holes. Further, ultimately matter and gravity should not be described by two unrelated theories but should be understood in a unified way. String theory is believed to solve all of these problems.

The idea of building blocks that are one-dimensional, rather than zero-dimensional points, is rather novel, and it has had an odd history. It was first developed between 1968 and 1973 as a theory of the strong interaction: mesons such as the pion behave in some respects like open strings. This idea was superceded by the 1973 discovery of the true theory of the strong interaction, quantum chromodynamics, but a small handful of theorists regarded string theory as a compelling idea and continued to develop it. In the following years it was discovered that string theory is actually a theory of gravity, that it implies a symmetry between bosons and fermions (which was named supersymmetry), and that it is free of the unphysical infinities that plagued all previous theories of quantum gravity.

In 1984 a discovery by Michael Green and John Schwarz, known as anomaly cancellation, showed that string theory could also describe quarks, leptons, and gauge interactions. This led to a tremendous wave of research activity, often called the first superstring revolution, as theorists who had been pursuing other approaches to unification began to develop string theory. The discovery of Calabi-Yau compactification by Philip Candelas, Gary Horowitz, Andrew Strominger, and Edward Witten and of heterotic string by David Gross, Jeff Harvey, Lance Dixon, and Ryan Rohm strengthened the evidence for string theory. However, work at this time was limited to a certain approximation, known as perturbation theory, which applies only to small numbers of strings interacting weakly. In 1995 Witten, extending results of Chris Hull, Paul Townsend, and others, identified the principle of string duality, which governs the behavior of strongly interacting strings. The ensuing period, known as the second superstring revolution, has produced many further discoveries. One has been a new understanding of the quantum mechanics of black holes, resolving some long-standing puzzles. Another has been the understanding that string theory contains extended structures known as branes, which has led to new ideas for realistic models and for experimental and cosmological tests.

String theory is still an incomplete theory. It is widely believed that the description of the theory in terms of one-dimensional building blocks is not ultimately the simplest or most complete but rather is a stepping-stone toward a more fundamental principle, which is being actively sought by string theorists today. String theory is equivalently referred to as superstring theory, reflecting the central role of supersymmetry. Since 1995 the term M theory has also been used, reflecting the fact that the string picture is believed to be just a stepping-stone to a final theory. What the "M" stands for is deliberately left unspecified, reflecting the unknown nature of the final theory; "magic," "mother," "mystery," "membrane," and "matrix" have all been suggested.

## String Theory and Particle Physics

Experimentally, the electron, quarks, photon, and other Standard Model particles are all points: no experiment has revealed any substructure, down to the distance scale of 10^{-16} cm. The idea of string theory is that under sufficient magnification the Standard Model particles will be seen to be loops or segments of string. The magnification needed is very large, as the size of the string is expected to be of order 10^{-32} cm: this is the Planck length, the distance scale where gravity and quantum mechanics come together. This scale is far beyond the reach of particle accelerators, so that experimental tests of the theory will have to be indirect. (Recent ideas, to be discussed below, raise the possibility that the strings are larger and so more accessible to experiment.)

The string of string theory is much like a violin string: it vibrates, and this vibration can be decomposed into a sum of notes or harmonics. Depending upon which harmonics are excited, and to what degree, the string will behave like different kinds of particles, and all the particles found in the Standard Model can be obtained as different states of vibration of this one building block. In particular, one of the states of vibration of the closed string is a massless particle of spin two. This was a problem when string theory was supposed to describe the strong interaction, as there is no such hadron, but this particle has precisely the properties of the graviton, the particle associated with the gravitational field, and this is why string theory must incorporate gravity. In addition to vibrating, strings can break in two and join together (Figure 2). All of the basic processes of nature, such as an electron emitting a photon or a graviton, or a *Z* boson decaying into a quark-antiquark pair, arise from this one basic string process.

Before the first superstring revolution, many other ideas were explored for unifying the Standard Model and explaining its patterns. Three ideas of particular note are grand unification, supersymmetry, and extra dimensions. These can be thought of as new symmetry principles, meaning that Standard Model particles that appear to be different are really the same kind of particle but with, in some sense, a different orientation. Each of these ideas had some successes, and one of the attractive features of string theory is that it automatically incorporates all three of these enlarged symmetries.

## Extra Dimensions

The idea that space-time has more than the four visible dimensions is almost as old as general relativity. Gravity and electromagnetism are similar in that both forces fall off as the inverse square of the distance; they differ in that gravity couples to energy and momentum, and electromagnetism

**FIGURE 2**

couples to charge. Theodore Kaluza in 1919 and Oscar Klein in 1926 put forward the idea that elec tromagnetism would actually originate from grav ity if space-time were five-dimensional, with the fifth dimension too small to be seen directly (Figure 3). The gravitional field has a polarization (spin). If this polarization is fully aligned along the large dimensions, the five-dimensional graviton be haves like a four-dimensional graviton and pro duces the gravitational force. If it is partly aligned along the small dimension, then it behaves like a four-dimensional photon and produces the elec tromagnetic force. What is seen as electric charge is actually momentum that is directed along the small dimension. This elegant unification of the two then-known forces fascinated some of the greatest physicists of the early twentieth century. Albert Einstein and Wolfgang Pauli each spent sub stantial periods trying to develop it further.

String theory requires that space-time have ten dimensions (nine space and one time). Ultimately this originates from the mathematical structure of the supersymmetry algebra. There are actually five consistent string theories, known as types I, IIA, IIB, heterotic SO(32), and heterotic E_{8} × E_{8}. These differ primarily in the way that the supersymmetry acts on the states of the string; also, type I theory has both open and closed strings, while the other four have only closed strings.

**FIGURE 3**

In Kaluza-Klein theory, the single extra dimension forms a circle. The six extra dimensions of string theory can have a much more complex topology and geometry. Although we do not see these dimensions directly, their shape determines the physics that we do see—the spectrum of particles, and their masses and couplings. A relatively simple set of spaces known as Calabi-Yau manifolds, when combined with the E_{8} × E_{8} heterotic string, give a result very much like the grand unified supersymmetric Standard Model. One of the central problems in string theory is that there are many different Calabi-Yau manifolds, as well as other possible spaces, and to account for the precise details of the Standard Model requires knowing the precise shape of the extra dimensions.

The simplest estimate of the size of the extra dimensions in string theory gives the Planck length, which would put them far beyond direct experimental detection. As will be discussed below, more recent ideas raise the possibility that they are much larger and might have a variety of observable effects.

## String Theory and Quantum Gravity

General relativity and quantum mechanics are two of the central principles in physics, and each has been verified experimentally in great detail. General relativity is important at astronomical scales, but its effects are negligible in the microscopic regime of atomic and particle physics. Quantum mechanics is essential to microscopic physics, but its effects are negligible at astronomical scales. Thus, in ordinary circumstances one does not encounter general relativistic and quantum effects together.

However, general relativity and quantum mechanics conflict with one another, and this conflict will appear in certain extreme situations. At very short distances one encounters the problem of space-time foam. General relativity states that space-time is curved and that the effect of this curvature is gravity. Quantum mechanics states, roughly speaking, that nothing sits still (the uncertainty principle). Taken together, these imply that space-time does not sit still, its shape is constantly fluctuating. These fluctuations are totally negligible on astronomical scales and even on the scales of particle physics, but as one goes to very short distances they become more evident. At the Planck length of 10^{-32} cm they become so severe that shorter distances do not make sense at all—space-time, in a sense, tears itself to pieces. In the language of particle physics, this means that quantum gravity is not renormalizable: when the effects of virtual gravitons are included in quantum mechanical amplitudes, the result is infinite (Figure 4).

The problem of renormalization arose for the three particle interactions as well and in each case was an important clue to the correct theory. In the case of gravity, string theory removes the problem by changing the theory at distances below the Planck scale.

Other approaches to this problem are still under study, but to date string theory is the only known finite theory of quantum gravity. The problem of quantum foam means that distances smaller than the Planck scale cannot make sense, and so the historic progression toward ever-smaller constituents must end. In string theory, the size of a string represents a minimum length, the shortest distance that can be probed.

## Modern String Theory

*String Duality and D-branes*

As physics has progressed toward more unified theories based on more fundamental principles, there has been a growing expectation that there is

**FIGURE 4**

a unique theory that incorporates all of the laws of physics. The existence of five different string theories was therefore a puzzle (though the situation is much better than quantum field theory, where there is an infinite number of theories that are characterized by different symmetries, particles, masses, and couplings). In 1995 it was understood, through string duality, that these are all part of a single theory. Essentially, they are different phases (Figure 5), related to each other much like the liquid, solid, and gas phases of water. In the case of water, one varies the pressure and temperature to change one phase into another. In the case of string theory, one varies the shape and size of the extra dimensions, and in certain regimes the theory behaves like one or the other of the string theories. There are also new phases, most notably a phase known as D = 11 supergravity where a new space-time dimension, the eleventh, appears.

In addition to the strings themselves, string theory contains a variety of higher-dimensional objects known as branes. These were discussed before 1995, but in the context of string duality it became clear that they play a central role. A particular class known as D-branes (Figure 6) were shown by Joseph Polchinski to have the special property that strings can end on them, and play an important role in understanding many of the phases of the theory.

**FIGURE 5**

**FIGURE 6**

## Black Hole Entropy and Information

Black holes are among the most extreme objects in physics, and they present another situation where the laws of relativity and quantum physics come together. In the early 1970s it was found that black holes satisfy laws parallel to those of thermodynamics. In particular, Jakob Bekenstein and Stephen Hawking argued that they have an entropy, implying a microscopic structure of states. The nature of these states was mysterious until 1996, when Strominger and Cumrun Vafa showed that they were accounted for by string theory. In particular, for certain charged black holes, D-branes give a precise construction of the microscopic states.

Hawking also discovered that black holes radiate and eventually disappear and that this leads to a paradox. The particles produced in the decay do not depend on what initially forms the black hole, so information is lost in the process of black hole formation and decay; this information loss is inconsistent with the laws of quantum mechanics. Hawking argued the quantum mechanics must therefore be modified. This claim inspired much further work that showed that if quantum mechanics were not modified then the principle that physical processes are local in space-time must break down in a subtle way (there are other alternatives, but these are generally regarded as less likely). This issue is not yet decided, but string duality relates black holes to ordinary systems that do satisfy the laws of quantum mechanics, and this suggests in fact it is the principle of space-time locality, not quantum mechanics, that must be modified.

## Braneworlds, Large Extra Dimensions, and Low-energy Strings

The existence of branes of various dimensionalities suggests that we might actually live on a three-brane, a brane with three space dimensions (plus time, of course). String theory still requires nine space dimensions, or ten in the D = 11 supergravity phase, so our brane would be embedded in this higher-dimensional space. The open strings that attach to the D-brane can give rise to all the particles of the Standard Model except the graviton. Thus there are string models in which everything seen in nature except for gravity is attached to a brane, while the gravitational field lines spread out in all the dimensions. This is different from the previous extra-dimensional ideas, where there are no branes, and all particles live in the full set of extra dimensions.

For extra dimensions without branes, dimensional analysis indicates that both the size of the dimensions and the string scale are near the Planck length and so remote from experiment. The situation with branes is more complicated. The size of the dimensions is not fixed by theory, and it could be very much larger. It is then important to consider the experimental limits on the size. In 1998, Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali argued that, although particle accelerators probe physics down to 10^{-16} cm, the extra dimensions could be much larger than this. If everything but gravity were attached to a brane, the extra dimensions would not be seen readily at accelerators but only in gravitational experiments. They would show up as a change in the gravitational force law, from the inverse square law that comes from the field lines spreading in three space dimensions, to a different behavior. Since the inverse square law is tested only down to 0.1 cm, the extra dimensions could be as large as this.

The large extra dimensions might be seen at a particle accelerator in an indirect way. If two particles moving along the brane collide with enough energy to leave the brane and move into the extra dimensions, they become undetectable, and the result is an event in which energy seems to disappear. This process could also occur in such environments as the core of a supernova, and the observation of neutrinos from supernovae actually gives a more stringent upper limit than the force law experiments, around 10^{-4} cm. Arkani-Hamed, Dimopoulos, and Dvali also showed that in theories with large extra dimensions the string size is larger than the Planck length, so that string physics, and gravitational effects like microscopic black holes, might be seen at particle accelerators. There is no definite prediction yet for the size of the extra dimensions, and many theorists still expect that they are too small to be observed, but it is an exciting new possibility that is under theoretical and experimental study.

## The Future of Theory and Experiment

The existence of branes of all dimensions raises the issue of whether the one-dimensional strings are truly fundamental. So also does the string-duality phase diagram, which shows that strings exist only in certain phases. These and other arguments have led string theorists to believe that the defining principle of the theory, the analog of the equivalence principle of general relativity and the uncertainty principle of quantum mechanics, has yet to be found. Many ideas are under investigation. A common theme is that physics is expected to be nonlocal, even before the Planck length. Two concepts being considered are the holographic principle, which is connected with the black hole information problem, and noncommutative geometry, which is an extension of the uncertainty principle involving only lengths and not momenta.

Experimentally, string theory does not yet make firm predictions. The part of the theory that is most likely to be accessible to accelerators is supersymmetry; large extra dimensions are a striking but less probable signature. String theory may eventually make distinctive predictions for cosmology. Finally, experience shows that as the theory is understood better, unexpected new possibilities are found.

*See also:*Particle Physics, Elementary; Planck Scale; Grand Unification; Unified Theories

## Bibliography

Arkani-Hamed, N.; Dimopoulos, S.; and Dvali, G. "The Universe's Unseen Dimensions." *Scientific American*** 283** (2), 62–70 (2000).

Duff, M. "The Theory Formerly Known as Strings." *Scientific American*** 278** (2), 64–69 (1998).

Greene, B. R. *The Elegant Universe: Superstrings, Hidden Dimensions and the Quest of the Ultimate Theory* (Norton, New York,1999).

Horowitz, G. T., and Teukolsky, S. A. "Black Holes." *Reviews of Modern Physics*** 71** (2), S180–S186 (1999).

Mukerjee, M. "Explaining Everything." *Scientific American*** 274** (1), 88–94 (1996).

Schwarz, J. H., and Seiberg, N. "String Theory, Supersymmetry, Unification, and All That." *Reviews of Modern Physics***71** (2), S112–S120 (1999).

Susskind, L. "Black Holes and the Information Paradox." *Scientific American*** 276** (4), 52–57 (1997).

Weinberg, S. "A Unified Physics by 2050?" *Scientific American***281** (6), 68–75 (1999).

*Joseph Polchinski*

## String Theory

# STRING THEORY

Physicists believe there to be four fundamental forces. Three of these—the electromagnetic, the strong force, and the weak force—are amalgamated in the standard model of elementary particle physics, a family of quantum field theories that has enjoyed stupendous empirical success. Gravity, the fourth and feeblest fundamental force, is the subject of a stupendously successful *nonquantum* field theory, Einstein's general theory of relativity (GTR). Desiring to fit all of fundamental theoretical physics into a quantum mechanical framework, and suspecting that GTR would break down at tiny ("Planck scale," i.e., 10^{−33} cm) distances where quantum effects become significant, physicists have been searching for a quantum theory of gravity since the 1930s. In the last quarter of the twentieth century, string theory became the predominant approach to quantizing gravity, as well as to forging a unified picture of the four fundamental forces. A minority approach to quantizing gravity is the program of loop quantum gravity, which promises no grand unification. Both attempts to quantize gravity portend a science of nature radically different from the Newtonian one that frames much of classical philosophical discourse. They also present gratifying instances of working physicists actively concerned with recognizably philosophical questions about space, time, and theoretical virtue.

## The Standard Model

String theory would quantize gravity by treating the gravitational force as other forces are treated. In the standard model, pointlike elementary particles, quarks, and leptons constitute matter. Each particle is characterized by invariants, such as mass, spin, charge, and the like. The matter-constituting particles have half-integer multiples of spin, which makes them fermions. Beside fermions, the standard model posits gauge bosons, "messenger particles" or carriers of the interaction, for each force in its ambit. Bosons are distinguished from fermions by having whole-integer multiples of spin. As early as 1934, preliminary work on the sort of coupling with matter required by a quantum theory suggested that, if the gravitational force had a gauge boson, it must be a mass 0 spin 2 particle, dubbed the *graviton*. No such particle is predicted by the standard model.

According to string theory, the elementary particles of the standard model are not the ultimate constituents of nature. Filamentary objects—strings—are. Different vibrational modes of these strings correspond to the different masses (charges, spins) of elementary particles. The standard model is recovered, and fundamental physics unified, in a string theory incorporating vibrational modes corresponding to every species of particle in the standard particle zoo (and so incorporating the strong, weak, and electromagnetic forces), as well as to the graviton (and so incorporating gravity).

## The Early Years of String Theory

String theory evolved from attempts, undertaken within the standard model in the 1970s, to model the strong nuclear force in terms of a band between particles. As a theory of the strong nuclear force, these attempts suffered in comparison to quantum chromodynamics. They also predicted the existence of a particle that had never been detected: a mass 0 spin 2 particle. In 1974, John Schwartz and Joël Scherk proposed to promote this empirical embarrassment to a theoretical resource: The undetected particle, they suggested, was in fact the graviton! (Further evidence that string theory encompasses gravity comes in the form of a consistency constraint on the background spacetime in which string theoretic calculations are carried out, which consistency constraint resembles the equations of GTR.)

String theory evolved piecemeal in the 1970s and 1980s, roughly by adapting perturbative approximation techniques developed for the standard model's point particles to stringy entities. One benefit of the adaptation was the suppression of infinities that arise in perturbative calculations for point particles. In the standard model, these infinities call for the expedient of renormalization, the barelyprincipled subtraction of other infinities to yield finite outcomes. Perturbative string theories require no such expedient. Worries that they harbored inconsistencies all their own, called *anomalies*, were allayed by Schwartz's and Michael Green's 1984 argument that string theories were anomaly-free—a result that galvanized research in the field.

By the early 1990s there were five different consistent realizations of perturbative string theory. These realizations shared some noteworthy features. First, their equations were consistent only in ten space-time dimensions. To accord with the appearance that space is three-dimensional, the extra six dimensions are supposed to be Planck-scale and compactified ("rolled up"). (The usual analogy invokes the surface of a cylinder, which is a two-dimensional object: one dimension runs along the length of the cylinder; the other is "rolled up" around its circumference. Supposing the rolled-up dimension to be small enough, a cylinder looks like a one-dimensional object, a line.) Details of the geometries of these extra dimensions influence the physics string theory predicts. These details are adjustable; only with certain choices of the geometries can string theory mimic the standard model.

The initial string theories dealt only with bosons. So that they might incorporate fermions as well, *supersymmetry* was imposed. That is, the equations of string theory were required to be invariant under half-integer changes in spin. Thus the theory predicts for every particle in the standard zoo that it has a supersymmetric partner. For the (spin 1/2) electron, a spin 0 "selectron;" for the (spin-1) photon, a spin 1/2 "photino," and so on. Of these supersymmetric partners, none are observable using present technologies. But there is hope of detecting the lightest, the neutralino, with the Large Hadron Collider, slated to come on-line at CERN in 2007.

Parameters describing, for example, coupling strengths or the volume of the compactified extra dimensions appear in string theories. This means that each string theory can be thought of as a member of a family of related string theories, obtained from the first by varying the values of these parameters. A *duality* is said to obtain between theories so related. In the mid-1990s, Ed Witten and others uncovered evidence of dualities connecting pairs in the set of five consistent perturbative string theories. This embolded Witten to propose that the existing, approximate, string theories were all approximations to a single underlying exact theory he dubbed "M-theory." Although the equations of M-theory are unknown, it is believed that they hold in an eleven-dimensional spacetime, and have eleven-dimensional supergravity (ironically enough, a leading contender for the title "theory of everything" which string theory dislodged in the early 1980s) as their low-energy limit. In addition to strings, M-theory boasts higher-dimensional supersymmetric objects—membranes—some theorists have put to cosmological use, for example, by maintaining that the three spatial dimensions of this world are a three-brane moving through an eleven-dimensional universe harboring other worlds such as this one.

Most predictions of fledgling programs in quantum gravity are experimentally inaccessible, and liable to stay that way. But a nonempirical circumstance is widely believed to confirm string theory. In black hole thermodynamics (developed by Stephen Hawking, Jacob Bekenstein, and others), black holes are attributed properties, such as temperature and entropy, that obey thermodynamic laws. (For instance, entropy, identified as the surface area of a black hole's future event horizon, never decreases.) For certain black holes known as extremal black holes, string theoretic calculations exactly reproduce the Bekenstein entropy formula. Although there has never been an observation confirming (or disconfirming) black hole thermodynamics, the recovery of the black hole entropy formula is widely held to be evidence that string theory is on the right track.

More empirical tests have been proposed, none strong. For example, if the extra dimensions posited by string theory are large enough, new mechanisms for the production of microscopic black holes could be unleashed at energies attainable in the Large Hadron Collider. But string theory is not required to posit large extra-dimensions. So the failure of microscopic black holes to appear would not force the abandonment of string theory.

Despite its successes, there are causes for complaint about string theory. It is not an exact theory yet. Its predictions might seem unduly sensitive to the discretionary matter of the geometry of the extra dimensions. In addition to predicting the existence of the standard particles and the graviton, it predicts the existence of infinitely many particles, including supersymmetric particles, humans have not seen (yet). It requires seven extra spatial dimensions humans have not seen (yet). And as formulated *at present*, it takes place in a fixed space-time background.

## String Theory and Loop Quantum Gravity

The game of background-independent M-theory is afoot; some (e.g., Smolin 2001) hope that its pursuit will reveal connections between string theory and its main rival, loop quantum gravity. Background-independence is the rallying cry of the (much less populated) loop quantum gravity camp. Largely trained as general relativists, adherents of this approach take the fundamental moral of GTR to be that space-time is not a setting in which physics happens but is itself a dynamical object, malleable in response to the matter and energy filling it. Whereas string theory seeks a quantum theory of gravity on the model of early twenty-first century quantum theory of other forces—a model that adds a graviton to a particle zoo revealed by approximations carried out in a fixed spacetime background—loop quantum gravity seeks a quantum theory of gravity by quantizing gravity: that is, by casting GTR as a classical theory in Hamiltonian form, and following a canonical procedure for quantizing such theories. Insofar as GTR's variables determine the geometry of space-time, should the quantization procedure succeed, space-time itself would be the commodity quantized.

The quantization procedure is complicated by the fact that GTR is a constrained Hamiltonian system: its canonical momenta are not independent. Instead, they satisfy constraint equations that must be reflected in the final quantum theory. The origin of these constraint equations is the diffeomorphism invariance of GTR, that is, if one starts with a solution to the equations of GTR and smoothly reassigns the dynamical fields comprising that solution to the manifold on which they are defined, one winds up with a solution to the equations of GTR. Adherents of loop quantum gravity take diffeomorphism invariance to express the background independence of GTR.

Loop quantum gravity exploits a Hamiltonian formulation of GTR due to Abhay Ashtekar, a physicist at Syracuse University. Its quantization is set in a Hilbert space spanned by spin-network states: graphs whose edges are labeled by integer multiples of 1/2. Not set in some background space, these spin-network states are supposed to be the constituents from which space is built. Defined on their Hilbert space are area and volume (but not length) operators that have discrete spectra. A free parameter in the theory can be adjusted so that this quantization occurs at the Planck scale. On these grounds, its adherents claim loop quantum gravity to be a background-independent exact theory that quantizes space. Like string theory, loop quantum gravity finds quasi-confirmation in its accord with black hole thermodynamics: for all black holes, loop quantum gravity reproduces the Bekenstein entropy within a factor of 4.

Despite its successes, there are causes for complaint about loop quantum gravity. It does not incorporate the predictions of the standard model. So whereas it may be a quantum theory of gravity, it is not a theory of everything. More telling, loop quantum gravity as yet fails to reflect the full diffeomorphism invariance of GTR in a way that is both consistent and has GTR as its classical limit. The sticking point is the classical Hamiltonian constraint, related to diffeomorphisms that can be interpreted as time translations. Until this constraint is wrangled, loop quantum gravity lacks a dynamics: it consists of a space of possible instantaneous spacetime geometries, without an account of their time development. Given loop quantum gravity's ideology of background-independence, this is disappointing.

There is no established philosophy of quantum gravity. But there is much to provoke the philosopher. What, according to string theory or loop quantum gravity, is the nature of space(-time)? How many dimensions has it? (These questions are complicated by dualities between string theories revealed by varying the volumes of their compactified geometries, as well as by the holographic hypothesis, according to which physics in the interior of a region—an n-dimensional space—is dual to physics on that region's boundary—an (n-1)-dimensional space.) The search for quantum gravity was set off by no glaring empirical shortcoming in existing theories, and has reached theories for which no empirical evidence is readily forthcoming. In the absence of empirical adequacy, other theoretical virtues occupy center stage: the ideal of unification, the capacity to reproduce the results, or preserve the insights, of other theories (even unconfirmed ones); the susceptibility of puzzles posed in one theoretical framework to solution techniques available in another. The nature of these virtues, and how best to pursue them, are often live questions for quantum gravity researchers. Their work holds interest for the methodologist and the metaphysician alike.

** See also ** Atomism; Philosophy of Physics; Quantum Mechanics; Relativity Theory.

## Bibliography

Callendar, Craig, and Nicholas Huggett, eds. *Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity*. Cambridge, U.K.: Cambridge University Press: 2001.

Greene, Brian. *The Elegant Universe*. New York: Vintage, 2000.

Polchinski, Joseph. *String Theory*. Cambridge, U.K.: Cambridge University Press, 1999.

Rovelli, Carlo. "Quantum Spacetime: What Do We Know?" Cornell University on-line arXiv, 1999. Available from http://lanl.arxiv.org/PS_cache/gr-qc/pdf/9903/9903045.pdf.

Rovelli, Carlo. "Strings, Loops and Others: A Critical Survey of the Present Approaches to Quantum Gravity." Cornell University on-line arXiv, 1998. Available from http://lanl.arxiv.org/PS_cache/gr-qc/pdf/9803/9803024.pdf.

Smolin, Lee. *Three Roads to Quantum Gravity*. New York: Basic Books, 2001.

Stachel, John. "Early History of Quantum Gravity." In *Black Holes, Gravitational Radiation and the Universe*, edited by B. R. Iyer and B. Bhawal. Dordrecht, Netherlands: Kluwer, 1999.

Wald, Robert M. *Quantum Theory in Curved Spacetime and Black Hole Thermodynamics*. Chicago: University of Chicago Press, 1994.

*Laura Ruetsche (2005)*

## String Theory

# String Theory

String theory, also called *superstring theory,* is, generally speaking, any physico-mathematical framework that describes fundamental physical reality in terms of superstrings *Strings* in this context should not be confused with *cosmic strings,* which are one-dimensional (string-like) regions of cosmic extent containing vacuum energy different from that of the true vacuum. The superstrings of string theory, in contrast, are extremely tiny loops, or possibly segments, that have been suggested as the most fundamental of all physical entities, and as the source of all other fields and particles.

Before the 1980s, the most fundamental entities were most often considered to be *particles,* which are zero-dimensional objects, but it has become clear that particle models do not provide a rich and flexible enough basis for fundamental quantum field theories; strings are much more suitable. More specifically, string theory provides promising candidates for an adequate quantum theory of gravity and, at the same time, for theories of the total unification of all four fundamental physical interactions (gravity, electromagnetism, and the strong and weak nuclear forces. Grand Unified Theory (GUT) will provide unification of the three nongravitational interactions.

Quantum mechanics (along with its extension to quantum field theory) and Albert Einstein's (1879–1955) theory of gravitation are two important pillars of contemporary physics. And yet, as they are presently formulated, they are deeply incompatible with one another. As of 2002, constructing a complete and adequate quantum theory of gravity has evaded the best efforts of theoreticians. Exciting and surprising work on superstrings since about 1984, however, has moved science much closer to achieving quantization of the gravitational field, thus resolving and healing this incompatibility. It is already clear that the leading string theory candidates yield general relativity as their low-energy limit. Essentially, this means that string theory, if successful, will become not only the quantum theory of gravity, but also the quantum theory of space and time, with crucial applications to early-universe cosmology.

It also appears likely that some version of string theory will at the same time unify all four fundamental physical interactions, including gravity, thus bringing to successful completion the much heralded quest for unification that motivated the physicist James Clark Maxwell (1831–1879), Einstein, and so many others. In order to accomplish this unification, the strings must manifest *supersymmetry* —they must be *superstrings.* Consider that all fundamental particles have either half-integral spin (1/2, 3/2, . . . ) or integral spin (0, 1, 2, . . . ). The half-integral spin particles are called *fermions,* and constitute the building blocks of matter; protons, neutrons, electrons, and quarks are all fermions. The integral spin particles are called *bosons,* and are the force-carriers between the fermions, mediating the electromagnetic, gravitational, and strong and weak interactions. *Photons, W massive bosons, Z massive bosons, gluons,* and *gravitons* are the bosons that mediate the electromagnetic, weak, strong, and gravitational interactions, respectively.

Fermions and bosons satisfy different statistics and symmetries, and have to be treated differently in standard quantum field theory. The first seriously considered string theories—studied for purposes other than those for which newer superstring theories are studied—were bosonic strings, which only incorporated the symmetries and statistics of bosons. Obviously, if a theory is going to unify all particles and fields, it will have to incorporate the symmetries of both fermions and bosons within the same framework; it will have be *supersymmetric,* and the strings will therefore have be superstrings.

Where would the superstring description of reality be needed? Certainly, it would provide a detailed and physically complete explanation of all the characteristics and parameters of material reality, including their deep interconnections and their origins in the vibrations and interactions of the fundamental superstrings. It would, at the same time, provide an adequate description of material reality at temperatures higher than $1032 K, where the general relativistic description of space, time, and mass-energy breaks down. There was a time in the very early universe, immediately after the Big Bang, when those temperatures obtained and during which the physics of the universe was that of a single unified fully quantitized superforce. This era is referred to as the *Planck era,* after the German theoretical physicist Max Planck (1858–1947). In fact, it is only in such terms that the Big Bang itself, as well as the emergence or origin of space, time, and matter, can really be characterized.

Superstring theories resolve a number of difficult anomalies and divergences in quantum theory. But they also lead to some features that are, at first sight, puzzling. One of these is that they almost always require higher dimensions—for example ten or twenty-six—rather than the three spatial dimensions and one time dimension that characterize the low-energy world. How then can these superstring theories be reconciled with reality as we know it? The answer is straightforward but surprising. At very high energies or temperatures, such as immediately after the Big Bang, reality will be ten dimensional or twenty-six dimensional, as described by superstring theory. But, as the universe exits the Planck era, and enters the classical domain where gravity is adequately described by Einstein's general relativity and is no longer unified with the other interactions, the extra dimensions compactify (curl up into infinitesimal knots) leaving only the four-dimensional spacetime with which we are familiar. Of course, if this is true, scientists should find some evidence of these extra curled-up dimensions. Such relics of the supersymmetric past would constitute powerful confirmation of superstring theories. This is an active area of research.

## Relevance to theology

The relevance of string theory for the relationship between science and theology is clear, particularly in light of its applications to very early universe cosmology. First, a fully adequate string theory would give a complete unification and explanation of the laws of nature at the level of physics. In so doing, it would fill out the description of one of the most fundamental and pervasive sets of relationships through which God creatively acts in the universe. Secondly, it would give a much better description of the physics of the earliest phase of the universe's evolution, doing away with the initial singularity and helping scientists to speak more precisely about the origin of space and time, of all the laws of physics, and possibly of mass-energy. This would certainly help to delineate the limits of scientific explanation more compellingly. It is extremely unlikely, for instance, that the ultimately successful string theory will entail the existence of a unique universe or that it will explain why there is something rather than absolutely nothing, or that it will account for why there is this type of order, as specified by the string theory, rather than some other order. A clear appreciation of such limitations would enhance the understanding of the interactions, possible and desirable, between religion and science.

*See also* Cosmology; Grand Unified Theory; Gravitation; Physics, Quantum; Superstrings

*Bibliography*

davies, paul. "the new physics: a synthesis." in the new physics, ed. paul davies. cambridge, uk: cambridge university press, 1989.

green, michael b; schwarz, john h.; and witten, edward. superstring theory, vol. 1. cambridge, uk, and new york: cambridge university press, 1987.

greene, brian. the elegant universe: superstrings, hidden dimensions, and the quest for the ultimate theory. new york and london: norton, 1999.

kaku, michio. introduction to superstrings and m-theory, 2nd edition. new york, berlin, and heidelberg, germany: springer-verlag, 1998.

polchinski, joseph. string theory, volume 1. cambridge, uk: cambridge university press, 1998.

salam, abdus. "overview of particle physics." in the new physics, ed. paul davies. cambridge, uk: cambridge university press, 1989.

william r. stoeger

## String Theory

# String Theory

String theory (also termed superstring theory) is a mathematical attempt to describe all fundamental forces and particles as manifestations of a single, underlying entity, the string. The basic premise underlying string theory is that the fundamental constituents are strings with a size in the range of the Planck length, about 10^{-35} meters, which vibrate in different ways depending on the specific constituent (electron, proton, photon, neutrino, etc.) being considered.

String theory’s predictions are consistent with all known experimental data, and it is felt by some physicists to be a candidate for the long-sought theory of everything (TOE; i.e., a single theory describing all fundamental physical phenomena); however, string theory has proved difficult to subject to definitive experimental tests, and therefore actually remains a speculative hypothesis.

Physics as defined before the twentieth century, classical physics, envisioned the fundamental particles of matter as tiny, solid spheres. Quantum physics, which originated during the first 40 years of the twentieth century, envisioned the fundamental particles simultaneously as particles and as waves; both mental pictures were necessary to make sense out of certain experimental results.

String theory, the first forms of which were developed in the late 1960s and early 1970s, proposes that the fundamental unit of everything is the string. It is pictured as a bit of taut wire or string on the order of 10^{-33} cm in length (a factor of 10^{-20} smaller than a proton). These strings may be open, like a guitar string, or form loops like rubber bands; also, they may merge with other strings or divide into substrings.

Beginning a few years after its initial formulation, string theory stagnated for a decade because of mathematical difficulties, but exploded in the 1980s with the discovery that the theory actually possesses a highly desirable mathematical feature termed E(8)×E(8) symmetry. Physicists found that string theory had the ability to describe all the known elementary particles and their interactions. String theorists won several major theoretical victories in the 1990s, and intense efforts to extend string theory continue in the 2000s. String theory is not the product of a single mind, like the theory of relativity, but has been produced by scores of physicists refining each other’s ideas in stages.

Like the waves or particles of traditional quantum mechanics, of which string theory is an extension or refinement, strings are not objects like those found in the everyday world. A string-theory string is not made of any substance in the way that a guitar string, say, may be made of steel; nor is it stretched between anchor-points. If string theory is correct, a fundamental string simply *is*.

String theory proposes that the string is the fundamental building block of all physical reality. It makes this proposition work, mathematically, by asserting that the universe works not merely in the four dimensions of traditional physics—three spatial dimensions plus time—but in 10 or 11 dimensions, 6 or 7 of which are hidden from human senses because they curled up to subatomic size. Experimental proof of the existence of these extra dimensions has not yet been produced. Even the possibility of 26 dimensions is being considered in certain realms of string theory.

Although the strings of string theory are not actual strings or wires, the string concept is nevertheless a useful mental picture. Just as a taut string in the everyday world is capable of vibrating in many modes and thus of producing a number of distinct notes (harmonics), the vibrations of an elementary string manifest, the theory proposes, as different particles: photon, electron, quark, and so forth.

The string concept also resolves the problem of the point particle in traditional quantum physics. This arises during the mathematical description of collisions between particles, during which particles are treated as mathematical points having zero diameter. Because the fields of force associated with particles, such as the electric field that produces repulsion or attraction of charges, go by 1/*r,* where *r* is the distance to the particle, the force associated with a zero-diameter particle goes to infinity during a collision as *r* → 0. The infinities in the point-particle theory have troubled quantum physicists’ efforts to describe particle interactions for decades, but in the mathematics of string theory they do not occur at all.

In the Standard Model, quantum physicists’ systematically list of all the fundamental particles and their properties. The graviton (the particle that mediates the gravitational force) is tacked on as an afterthought because it is hypothesized to exist, not because the equations of the Standard Model explicitly predict its existence. In string theory, however, a particle having all the properties required of the graviton is predicted as a natural consequence of the mathematical system.

In fact, when the existence of this particle was calculated by early string-theory workers, they did not recognize that it might be the graviton. This happened because it had not occurred to them that their new theory might be powerful enough to resolve the biggest problem of modern fundamental physics: the split between general relativity (the large-scale theory of space, time, and gravity) and quantum mechanics (the small-scale theory of particles and of all forces except gravity).

String theory—or, rather, the string theories, as a variety of different versions of string theory have been put forward—thus not only predict all the particles and forces catalogued by the Standard Model, but may offer a theory of quantum gravity, a long-sought goal of physics.

Doubt lingers, however, as to whether string theory may be too flexible to fulfill its promise. If it cannot be cast into a form specific enough to be tested against actual data, then its mathematical beauty may be a valuable tool for exploring new ideas, but it will fail to constitute an all-embracing theory of the real world, a theory of everything. In the early years of the twenty-first century, excitement and skepticism about string theory both continue to run high in the world of professional physics.

*See also* Cosmology; Relativity, general; Relativity, special.

## Resources

### BOOKS

Falk, Dan. *Universe on a T-shirt: The Quest for the Theory of Everything*. New York: Arcade Publishing, 2004.

Mohapatra, Rabindra N. *Unification and Supersymmetry: The Frontiers of Quark-Lepton Physics*. New York: Springer-Verlag, 2003.

Oyibo, Gabriel A. *Grand Unified Theorem: Representation of the Unified Field Theory or the Theory of Everything*. New York: Nova Science Publishers, 2001.

Rae, Alastair I.M. *Quantum Physics: A Beginner’s Guide*. Oxford, UK: Oneworld, 2005.

### OTHER

Cambridge University. “Cambridge Cosmology.” <http://www.damtp.cam.ac.uk/user/gr/public/cos_home.html> (accessed October 30, 2006).

Schwarz, Patricia. “The Official String Theory Website.”2002. <http://superstringtheory.com/> (accessed October 30, 2006).

Larry Gilman

## String Theory

# String theory

String theory (also termed "superstring" theory) is a mathematical attempt to describe all fundamental forces and particles as manifestations of a single, underlying entity, the "string."

String theory's predictions are consistent with all known experimental data, and it is felt by some physicists to be a candidate for the long-sought "theory of every thing" (i.e., a single theory describing all fundamental physical phenomena); however, string theory has proved difficult to subject to definitive experimental tests, and therefore actually remains a speculative hypothesis.

**Physics** as defined before the twentieth century, "classical" physics, envisioned the fundamental particles of **matter** as tiny, solid spheres. Quantum physics, which originated during the first 40 years of the twentieth century, envisioned the fundamental particles simultaneously as particles and as waves; both mental pictures were necessary to make sense out of certain experimental results.

String theory, the first forms of which were developed in the late 1960s, proposes that the fundamental unit of everything is the "string," pictured as a bit of taut wire or string on the order of 10-33 cm in length (a **factor** of 10-20 smaller than a **proton** ). These strings may be "open," like a guitar string, or form loops like rubber bands; also, they may merge with other strings or divide into substrings.

Beginning a few years after its initial formulation, string theory stagnated for a decade because of mathematical difficulties, but exploded in the 1980s with the discovery that the theory actually possesses a highly desirable mathematical feature termed E(8)×E(8) **symmetry** . Several major theoretical victories were won by string theorists in the 1990s, and intense efforts to extend string theory continue today. String theory is not the product of a single mind, like the theory of relativity, but has been produced by scores of physicists refining each other's ideas in stages.

Like the "waves" or "particles" of traditional **quantum mechanics** , of which string theory is an extension or refinement, "strings" are not objects like those found in the everyday world. A string-theory string is not made of any substance in the way that a guitar string, say, may be made of **steel** ; nor is it stretched between anchor-points. If string theory is right, a fundamental string simply *is*.

Not only does string theory propose that the string is the fundamental building block of all physical reality, it makes this proposition work, mathematically, by asserting that the Universe works not merely in the four dimensions of traditional physics—three spatial dimensions plus time—but in 10 or 11 dimensions, 6 or 7 of which are "hidden" from our senses because they "curled up" to subatomic size. Experimental proof of the existence of these extra dimensions has not yet been produced.

Although the "strings" of string theory are not actual strings or wires, the "string" concept is nevertheless a useful mental picture. Just as a taut string in the everyday world is capable of vibrating in many modes and thus of producing a number of distinct notes (**harmonics** ), the vibrations of an elementary string manifest, the theory proposes, as different particles: **photon** , **electron** , quark, and so forth.

The string concept also resolves the problem of the "point particle" in traditional quantum physics. This arises during the mathematical description of collisions between particles, during which particles are treated as mathematical points having zero diameter. Because the fields of **force** associated with particles, such as the electric field that produces repulsion or attraction of charges, go by 1/*r*, where *r* is the distance to the particle, the force associated with a zero-diameter particle goes to **infinity** during a collision as *r* 0. The infinities in the point-particle theory have troubled quantum physicists' efforts to describe particle interactions for decades, but in the **mathematics** of string theory they do not occur at all.

In the **Standard Model** , quantum physicists' systematic list of all the fundamental particles and their properties, the graviton (the particle that mediates the gravitational force) is tacked on as an afterthought because it is hypothesized to exist, not because the equations of the Standard Model explicitly predict its existence; in string theory, however, a particle having all the properties required of the graviton is predicted as a natural consequence of the mathematical system.

In fact, when the existence of this particle was calculated by early string-theory workers, they did not recognize that it might be the graviton, for it had not occurred to them that their new theory might be powerful enough to resolve *the* biggest problem of modern fundamental physics, the split between general relativity (the large-scale theory of space, time, and gravity) and quantum mechanics (the small-scale theory of particles and of all forces except gravity).

String theory—or, rather, the string theories, as a variety of different versions of string theory have been put forward—thus not only predict all the particles and forces catalogued by the Standard Model, but may offer a theory of "quantum gravity," a long-sought goal of physics.

Doubt lingers, however, as to whether string theory may be too flexible to fulfill its promise. If it cannot be cast into a form specific enough to be tested against actual data, then its mathematical beauty may be a valuable tool for exploring new ideas, but it will fail to constitute an all-embracing theory of the real world, a "theory of everything." Excitement and skepticism about string theory both, therefore, continue to run high in the world of professional physics.

See also Cosmology; Relativity, general; Relativity, special.

## Resources

### books

Barnett, Michael R., Henry Möhry, and Helen R. Quinn. *The**Charm of Strange Quarks: Mysteries and Revolutions of Particle Physics.* New York: Springer-Verlag, 2000.

Kaku, Michio, and Jennifer Thompson. *Beyond Einstein.* New York: Anchor Books, 1995.

### periodicals

Taubes, Gary, "String Theorists Find a Rosetta Stone." *Science* Vol. 285, No. 5427 (23 July 1999): 512-517.

### other

Cambridge University. "Cambridge Cosmology." [cited February 14, 2003]. <http://www.damtp.cam.ac.uk/user/gr/public/cos_home.html>.

Schwarz, Patricia. "The Official String Theory Website." 2002 [cited February 13, 2003]. <http://superstringtheory.com/>.

Larry Gilman