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String Theory
 
 
 

String theory

 
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Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory
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Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory

String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that are the basis of the Standard Model of particle physics. For this reason, string theories are able to avoid problems associated with the presence of pointlike particles in a physical theory. Studies of string theories have revealed that they require not just strings, but also higher-dimensional objects.

The basic idea is that the fundamental constituents of reality are strings of energy of the Planck length (about 10-35 m) which vibrate at resonant specific frequencies[1]. Another key claim of the theory is that no measurable differences can be detected between strings that wrap around dimensions smaller than themselves and those that move along larger dimensions (i.e., physical processes in a dimension of size R match those in a dimension of size 1/R). Singularities are avoided because the observed consequences of "big crunches" never reach zero size. In fact, should the universe begin a "big crunch" sort of process, string theory dictates that the universe could never be smaller than the size of a string, at which point it would actually begin expanding.

Interest in string theory is driven largely by the hope that it will prove to be a theory of everything. It is a possible solution of the quantum gravity problem, and in addition to gravity it can naturally describe interactions similar to electromagnetism and the other forces of nature. Superstring theories include fermions, the building blocks of matter, and incorporate supersymmetry. It is not yet known whether string theory will be able to describe a universe with the precise collection of forces and matter that is observed, nor how much freedom to choose those details that the theory will allow. String theory as a whole has not yet made falsifiable predictions that would allow it to be experimentally tested, though various special corners of the theory are accessible to planned observations and experiments. Hence critics of string theory occasionally remark that the theory "... is not even wrong," quoting a quip attributed to Wolfgang Pauli.

Work on string theory has led to advances in mathematics, mainly in algebraic geometry. String theory has also led to other theories, supersymmetric gauge theories, which will be tested at the new Large Hadron Collider experiment.

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History

String theory
Related topics
See also

String theory was originally invented to explain peculiarities of hadron (subatomic particle which experiences the strong nuclear force) behavior. In particle-accelerator experiments, physicists observed that the spin of a hadron is never larger than a certain multiple of the square of its energy. No simple model of the hadron, such as picturing it as a set of smaller particles held together by spring-like forces, was able to explain these relationships. In 1968, theoretical physicist Gabriele Veneziano was trying to understand the strong nuclear force when he made a startling discovery. Veneziano found that a 200-year-old formula created by Swiss mathematician Leonhard Euler (the Euler beta function) perfectly matched modern data on the strong force. Veneziano applied the Euler beta function to the strong force, but no one could explain why it worked.

In 1970, Yoichiro Nambu, Holger Bech Nielsen, and Leonard Susskind presented a physical explanation for Euler's strictly theoretical formula. By representing nuclear forces as vibrating, one-dimensional strings, these physicists showed how Euler's function accurately described those forces. But even after physicists understood the physical explanation for Veneziano's insight, the string description of the strong force made many predictions that directly contradicted experimental findings. The scientific community soon lost interest in string theory, and the standard model, with its particles and fields, remained unthreatened.

Then, in 1974, John Schwarz and Joel Scherk, and independently Tamiaki Yoneya, studied the messenger-like patterns of string vibration and found that their properties exactly matched those of the gravitational force's hypothetical messenger particle — the graviton. Schwarz and Scherk argued that string theory had failed to catch on because physicists had underestimated its scope. This led to the development of bosonic string theory, which is still the version first taught to many students. The original need for a viable theory of hadrons has been fulfilled by quantum chromodynamics, the theory of quarks and their interactions. It is now hoped that string theory or some descendant of it will provide a fundamental understanding of the quarks themselves.

Bosonic string theory is formulated in terms of the Polyakov action, a mathematical quantity which can be used to predict how strings move through space and time. By applying the ideas of quantum mechanics to the Polyakov action — a procedure known as quantization — one can deduce that each string can vibrate in many different ways, and that each vibrational state appears to be a different particle. The mass the particle has, and the fashion with which it can interact, are determined by the way the string vibrates — in essence, by the "note" which the string sounds. The scale of notes, each corresponding to a different kind of particle, is termed the "spectrum" of the theory.

These early models included both open strings, which have two distinct endpoints, and closed strings, where the endpoints are joined to make a complete loop. The two types of string behave in slightly different ways, yielding two spectra. Not all modern string theories use both types; some incorporate only the closed variety.

However, the bosonic theory has problems. Most importantly, the theory has a fundamental instability, believed to result in the decay of space-time itself. Additionally, as the name implies, the spectrum of particles contains only bosons, particles like the photon which obey particular rules of behavior. While bosons are a critical ingredient of the Universe, they are not its only constituents. Investigating how a string theory may include fermions in its spectrum led to supersymmetry, a mathematical relation between bosons and fermions which is now an independent area of study. String theories which include fermionic vibrations are now known as superstring theories; several different kinds have been described.

Roughly between 1984 and 1986, physicists realized that string theory could describe all elementary particles and interactions between them, and hundreds of them started to work on string theory as the most promising idea to unify theories of physics. This first superstring revolution was started by a discovery of anomaly cancellation in type I string theory by Michael Green and John Schwarz in 1984. The anomaly is cancelled due to the Green-Schwarz mechanism. Several other ground-breaking discoveries, such as the heterotic string, were made in 1985.

Edward Witten
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Edward Witten

In the 1990s, Edward Witten and others found strong evidence that the different superstring theories were different limits of a new 11-dimensional theory called M-theory. These discoveries sparked the second superstring revolution. When Witten named it M-theory, he did not specify what the "M" stood for, presumably because he did not feel he had the right to name a theory which he had not been able to fully describe. Guessing what the "M" stands for has become a kind of game among theoretical physicists. The "M" sometimes is said to stand for Mystery, or Magic, or Mother. More serious suggestions include Matrix or Membrane. Sheldon Glashow has noted that the "M" might be an upside down "W", standing for Witten. Others have suggested that the "M" in M-theory should stand for Missing, Monstrous or even Murky. According to Witten himself, as quoted in the PBS documentary based on Brian Greene's The Elegant Universe, the "M" in M-theory stands for "magic, mystery, or matrix according to taste."

Many recent developments in the field relate to D-branes, objects which physicists discovered must also be included in any theory which includes open strings of the super string theory.

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Basic properties

The term 'string theory' properly refers to both the 26-dimensional bosonic string theories and to the 10-dimensional superstring theories created by adding supersymmetry. Nowadays, 'string theory' usually refers to the supersymmetric variant while the earlier is given its full name, 'bosonic string theory'.

String Theories
Type Spacetime dimensions
 		 Details
Bosonic 26 Only bosons, no fermions means only forces, no matter, with both open and closed strings; major flaw: a particle with imaginary mass, called the tachyon
I 10 Supersymmetry between forces and matter, with both open and closed strings, no tachyon, group symmetry is SO(32)
IIA 10 Supersymmetry between forces and matter, with closed strings only, no tachyon, massless fermions spin both ways (nonchiral)
IIB 10 Supersymmetry between forces and matter, with closed strings only, no tachyon, massless fermions only spin one way (chiral)
HO 10 Supersymmetry between forces and matter, with closed strings only, no tachyon, heterotic, meaning right moving and left moving strings differ, group symmetry is SO(32)
HE 10 Supersymmetry between forces and matter, with closed strings only, no tachyon, heterotic, meaning right moving and left moving strings differ, group symmetry is E8×E8

Note that in the type IIA and type IIB string theories closed strings are allowed to move everywhere throughout the ten-dimensional space-time (called the bulk), while open strings have their ends attached to D-branes, which are membranes of lower dimensionality (their dimension is odd - 1,3,5,7 or 9 - in type IIA and even - 0,2,4,6 or 8 - in type IIB, including the time direction).

While understanding the details of string and superstring theories requires considerable mathematical sophistication, some qualitative properties of quantum strings can be understood in a fairly intuitive fashion. For example, quantum strings have tension, much like regular strings made of twine; this tension is considered a fundamental parameter of the theory. The tension of a quantum string is closely related to its size. Consider a closed loop of string, left to move through space without external forces. Its tension will tend to contract it into a smaller and smaller loop. Classical intuition suggests that it might shrink to a single point, but this would violate Heisenberg's uncertainty principle. The characteristic size of the string loop will be a balance between the tension force, acting to make it small, and the uncertainty effect, which keeps it "stretched". Consequently, the minimum size of a string must be related to the string tension.

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Worldsheet

Imagine a point-like particle. If we draw a graph which depicts the progress of the particle as time passes by, the particle will draw a line in space-time. This line is called the particle's worldline. Now imagine a similar graph depicting the progress of a string as time passes by; the string (a one-dimensional object - a small line - by itself) will draw a surface (a two-dimensional manifold), known as the worldsheet. The different string modes (representing different particles, such as photon or graviton) are surface waves on this manifold.

A closed string looks like a small loop, so its worldsheet will look like a pipe, or - more generally - as a Riemannian manifold (a two-dimensional oriented surface) with no boundaries (i.e. no edge). An open string looks like a short line, so its worldsheet will look like a strip, or - more generally - as a Riemannian manifold with a boundary.

Strings can split and connect. This is reflected by the form of their worldsheet (more accurately, by its topology). For example, if a closed string splits, its worldsheet will look like a single pipe splitting (or connected) to two pipes (see drawing at the top of this page). If a closed string splits and its two parts later reconnects, its worldsheet will look like a single pipe splitting to two and then reconnecting, which also looks like torus connected to two pipes (one representing the ingoing string, and the other - the outgoing one). An open string doing the same thing will have its wolrdsheet looking like a ring connected to two strips.

Note that the process of a string splitting (or strings connecting) is a global process of the worldsheet, not a local one: locally, the worldsheet looks the same everywhere, and it is not possible to determine unambiguously at which point on the worldsheet the splitting occurs. Therefor these processes are an integral part of the theory, and are described by the same dynamics that controls the string modes.

In some string theories (namely closed strings in Type I and string in some version of the bosonic string), strings can split and reconnect in an opposite orientation (as in a Möbius strip or a Klein bottle). These theories are called unoriented. Formally, the worldsheet in these theories is an unoriented surface).

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Dualities

Before the 1990s, string theorists believed there were five distinct superstring theories: type I, types IIA and IIB, and the two heterotic string theories (SO(32) and E8×E8). The thinking was that out of these five candidate theories, only one was the actual correct theory of everything, and that theory was the theory whose low energy limit, with ten dimensions spacetime compactified down to four, matched the physics observed in our world today. But now it is known that this naive picture was wrong, and that the five superstring theories are connected to one another as if they are each a special case of some more fundamental theory, of which there is only one. These theories are related by transformations that are called dualities. If two theories are related by a duality transformation, it means that the first theory can be transformed in some way so that it ends up looking just like the second theory. The two theories are then said to be dual to one another under that kind of transformation. Put differently, the two theories are two different mathematical descriptions of the same phenomena.

These dualities link quantities that were also thought to be separate. Large and small distance scales, strong and weak coupling strengths – these quantities have always marked very distinct limits of behavior of a physical system, in both classical field theory and quantum particle physics. But strings can obscure the difference between large and small, strong and weak, and this is how these five very different theories end up being related.

Suppose we're in ten spacetime dimensions, which means we have nine space and one time. Take one of those nine space dimensions and make it a circle of radius R, so that traveling in that direction for a distance L = 2πR takes you around the circle and brings you back to where you started. A particle traveling around this circle will have a quantized momentum around the circle, because its momentum is linked to its wavelength (see Wave-particle duality), and 2πR must be a multiple of that. In fact, the particle momentum around the circle - and the contribution to its energy - is of the form n/R (in standard units, for an integer n), so that at large R there will be many more states compared to small R (for a given maximum energy). A string, in addition to traveling around the circle, may also wrap around it. The number of times the string winds around the circle is called the winding number, and that is also quantized (as it must be an integer). Winding around the circle requires energy, because the string must be streched against its tension, so it contributes an amount of energy of the form wR/L_{st}^2, where Lst is the string length and w is the winding number (an integer). Now (for a given maximum energy) there will be many different states (with different momenta) at large R, but there will also be many different states (with different windings) at small R. In fact, a theory with large R and a theory with small R are equivalent, where the role of momentum in the first is played by the winding in the second, and vice versa. Mathematically, taking R to L_{st}^2/R and switching n and w will yield the same equations. So exchanging momentum and winding modes of the string exchanges a large distance scale with a small distance scale.

This type of duality is called T-duality. T-duality relates type IIA superstring theory to type IIB superstring theory. That means if we take type IIA and Type IIB theory and compactify them both on a circle, then switching the momentum and winding modes, and switching the distance scale, changes one theory into the other. The same is also true for the two heterotic theories. T-duality also relates type I superstring theory to both type IIA and type IIB superstring theories with certain boundary conditions (termed orientifold).

Formally, the location of the string on the circle is described by two fields living on it, one which is left-moving and another which is right-moving. The movement of the string center (and hence its momentum) is related to the sum of the fields, while the string stretch (and hence its winding number) is related to their difference. T-duality can be formally described by taking the left-moving field to minus itself, so that the sum and the difference are interchanged, leading to switching of momentum and winding.

On the other hand, every force has a coupling constant, which is a measure of its strength, and determines the chances of one particle to emit or receive another particle. For electromagnetism, the coupling constant is proportional to the square of the electric charge. When physicists study the quantum behavior of electromagnetism, they can't solve the whole theory exactly, because every particle may emit and receive many other particles, which may also do the same, endlessly. So events of emission and reception are considered as perturbations and are dealt with by a series of approximations, first assuming there is only one such event, then correcting the result for allowing two such events, etc (this method is called Perturbation theory. This is a reasonable approximation only if the coupling constant is small, which is the case for electromagnetism. But if the coupling constant gets large, that method of calculation breaks down, and the little pieces become worthless as an approximation to the real physics.

This also can happen in string theory. String theories have a coupling constant. But unlike in particle theories, the string coupling constant is not just a number, but depends on one of the oscillation modes of the string, called the dilaton. Exchanging the dilaton field with minus itself exchanges a very large coupling constant with a very small one. This symmetry is called S-duality. If two string theories are related by S-duality, then one theory with a strong coupling constant is the same as the other theory with weak coupling constant. The theory with strong coupling cannot be understood by means of perturbation theory, but the theory with weak coupling can. So if the two theories are related by S-duality, then we just need to understand the weak theory, and that is equivalent to understanding the strong theory.

Superstring theories related by S-duality are: type I superstring theory with heterotic SO(32) superstring theory, and type IIB theory with itself.

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Extra dimensions

Calabi-Yau manifold (an artist's impression)
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Calabi-Yau manifold (an artist's impression)

One intriguing feature of string theory is that it predicts the number of dimensions which the universe should possess. Nothing in Maxwell's theory of electromagnetism or Einstein's theory of relativity makes this kind of prediction; these theories require physicists to insert the number of dimensions "by hand". The first person to add a fifth dimension to Einstein's four was the German mathematician Theodor Kaluza in 1919. The reason for the unobservability of the fifth dimension (its compactness) was suggested by the Swedish physicist Oskar Klein in 1926.

Instead, string theory allows one to compute the number of spacetime dimensions from first principles. Technically, this happens because for a different number of dimensions, the theory has a gauge anomaly. This can be understood by noting that in a consistent theory which includes a photon (technically, a particle carring a force related to an unbroken gauge symmetry), it must be massless. The mass of the photon which is predicted by string theory depends on the energy of the string mode which represents the photon. This energy includes a contribution from Casimir effect, namely from quantum fluctuations in the string. The size of this contribution depends on the number of dimensions since for a larger number of dimensions, there are more possible fluctuations in the string position. Therefore, the photon will be massless - and the theory consistent - only for a particular number of dimensions.

The only problem is that when the calculation is done, the universe's dimensionality is not four as one may expect (three axes of space and one of time), but twenty-six. More precisely, bosonic string theories are 26-dimensional, while superstring and M-theories turn out to involve 10 or 11 dimensions. In bosonic string theories, the 26 dimensions come from the Polyakov equation (see technical details in the preprint "Quantum Geometry of Bosonic Strings - Revisited"). However, these results appear to contradict the observed four dimensional space-time.

Two different ways have been proposed to solve this apparent contradiction. The first is to compactify the extra dimensions; i.e., the 6 or 7 extra dimensions are so small as to be undetectable in our phenomenal experience. The 6-dimensional model's resolution is achieved with Calabi-Yau spaces. In 7 dimensions, they are termed G2 manifolds. Essentially these extra dimensions are compactified by causing them to loop back upon themselves.

A standard analogy for this is to consider multidimensional space as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. Indeed, think of a ball small enough to enter the hose but not too small. Throwing such a ball inside the hose, the ball would move more or less in one dimension; in any experiment we make by throwing such balls in the hose, the only important movement will be one-dimensional, that is, along the hose. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling inside it would move in two dimensions (and a fly flying in it would move in three dimensions). This "extra dimension" is only visible within a relatively close range to the hose, or if one "throws in" small enough objects. Similarly, the extra compact dimensions are only visible at extremely small distances, or by experimenting with particles with extremely small wave lengths (of the order of the compact dimension's radius), which in quantum mechanics means very high energies (see wave-particle duality).

Another possibility is that we are stuck in a 3+1 dimensional (i.e. three spatial dimensions plus the time dimension) subspace of the full universe. This subspace is supposed to be a D-brane, hence this is known as a braneworld theory.

In either case, gravity acting in the hidden dimensions affects other non-gravitational forces such as electromagnetism. In principle, therefore, it is possible to deduce the nature of those extra dimensions by requiring consistency with the standard model, but this is not yet a practical possibility. It is also possible to extract information regarding the hidden dimensions by precision tests of gravity, but so far these have only put upper limitations on the size of such hidden dimensions.

Image:Question_dropshade.png
Unsolved problems in physics: Is string theory, superstring theory, or M-theory, or some other variant on this theme, a step on the road to a "theory of everything," or just a blind alley?


 

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Aspects of quantum field theory

Many first principles in quantum field theory are explained, or get further insight, in string theory:

  • Emission and absorption: one of the most basic building blocks of quantum field theory, is the notion that particles (such as electrons) can emit and absorb other particles (such as photons). Thus, an electron may just "split" into an electron plus a photon, with a certain probability (which is, roughly, the coupling constant). This is described in string theory as one string spliting into two. As is explained above (under worldsheet), this process is an integral part of the theory. The mode on the original string also "splits" between its two parts, resulting in two strings which possibly have different modes, representing two different particles.
  • Coupling constant: in quantum field theory this is, roughly, the probability for one particle to emit or absorb another particle, the latter typically being a gauge boson (a particle carrying a force). In string theory, the coupling constant is no longer a constant, but is rather determined by the abundance of strings in a particular mode, the dilaton. Strings in this mode couple to the worldsheet curvature of other strings, so their abundance through space-time determines the measure by which an average string worldsheet will be curved. This determines its probabilty to split or connect to other strings: the more a worldsheet is curved, it has a higher chance of splitting and reconnecting.
  • Spin: each particle in quantum field theory has a particuar spin s, which is an internal angular momentum. Classically, the particle rotates in a fixed frequency, but this cannot be understood if particles are point-like. In string theory spin is understood by the rotation of the string; For example, a photon with well-defined spin components (i.e. in circular polarization) looks like a tiny straight line revolving around its center.
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Problems

String theory remains to be verified. No version of string theory has yet made a prediction which differs from those made by other theories — at least, not in a way that could be checked by a currently feasible experiment. In this sense, string theory is still in a "larval stage": it is properly a mathematical theory but is not yet a physical theory. It possesses many features of mathematical interest and may yet become supremely important in our understanding of the universe, but it requires further developments before it is accepted or falsified. Since string theory may not be tested in the foreseeable future, some scientists[2] have asked if it even deserves to be called a scientific theory: it is not yet falsifiable in the sense of Popper.

It is by no means the only theory currently being developed which suffers from this difficulty; any new development can pass through a stage of uncertainty before it becomes conclusively accepted or rejected. As Richard Feynman noted in The Character of Physical Law, the key test of a scientific theory is whether its consequences agree with the measurements taken in experiments. It does not matter who invented the theory, "what his name is", or even how aesthetically appealing the theory may be — "if it disagrees with experiment, it's wrong." (Of course, there are subsidiary issues: something may have gone wrong with the experiment, or perhaps the person computing the consequences of the theory made a mistake. All these possibilities must be checked, which may take a considerable time.) These developments may be in the theory itself, such as new methods of performing calculations and deriving predictions, or they may be advances in experimental science, which make formerly ungraspable quantities measurable.

On a more mathematical level, another problem is that, like quantum field theory, much of string theory is still only formulated perturbatively (i.e., as a series of approximations rather than as an exact solution). Although nonperturbative techniques have progressed considerably — including conjectured complete definitions in space-times satisfying certain asymptotics — a full nonperturbative definition of the theory is still lacking.

Another problem is the theory describes not just one but some 10500 universes, all of which can have different physical laws and constants.[3]

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Testing the theory

Since the influence of quantum effects upon gravity only become significant at distances many orders of magnitude smaller than human beings have the technology to observe (or at roughly the Planck length, about 10-35 meters), string theory, or any other candidate theory of quantum gravity, will be very difficult to test experimentally. Eventually, scientists may be able to test string theory by observing cosmological phenomena which may be sensitive to string physics.

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String theory and cosmic strings

In the early 2000s, string theorists revived interest in an older concept, the cosmic string. Originally discussed in the 1980s, cosmic strings are a different type of object than the entities of superstring theories. For several years, cosmic strings were a popular model for explaining various cosmological phenomena, such as the way galaxies formed in the early Universe. However, further experiments — and in particular the detailed measurements of the cosmic microwave background — failed to support the cosmic-string model's predictions, and the cosmic string fell out of vogue. If such objects did exist, they must be few and far between. Several years later, it was pointed out that the expanding Universe could have stretched a "fundamental" string (the sort which superstring theory considers) until it was of intergalactic size. Such a stretched string would exhibit many of the properties of the old "cosmic" string variety, making the older calculations useful again. Furthermore, modern superstring theories offer other objects which could feasibly resemble cosmic strings, such as highly elongated one-dimensional D-branes (known as "D-strings"). As theorist Tom Kibble remarks, "string theory cosmologists have discovered cosmic strings lurking everywhere in the undergrowth". Older proposals for detecting cosmic strings could now be used to investigate superstring theory. For example, astronomers have also detected a few cases of what might be string-induced gravitational lensing.

Superstrings, D-strings or other stringy objects stretched to intergalactic scales would radiate gravitational waves, which could presumably be detected using experiments like LIGO. They might also cause slight irregularities in the cosmic microwave background, too subtle to have been detected yet but possibly within the realm of future observability.

While intriguing, these cosmological proposals fall short in one respect: testing a theory requires that the test be capable, at least in principle, of falsifying the theory. For example, if observing the Sun during a solar eclipse had not shown that the Sun's gravity deflected light, Einstein's general relativity theory would have been proven wrong. Not finding cosmic strings would not demonstrate that string theory is fundamentally wrong — merely that the particular idea of highly stretched strings acting "cosmic" is in error. While many measurements could in principle be made that would suggest that string theory is on the right track, scientists have not at present devised a stringent "test".

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Popular culture

The book The Elegant Universe by Brian Greene, Professor of Physics at Columbia University, was adapted into a three-hour documentary for Nova and also shown on British television. It was also shown by Discovery Channel on Indian television.

String theory is also a series of books based in the Star Trek: Voyager universe.

In the TV series Angel, the character of Winifred Burkle (aka Fred) puts forward a theory about String Theory & Alternate Dimensions to the Physics Institute following her own experience of being trapped in one such delicate alternate dimension for five years. The episode which this is referenced to is "Supersymmetry".

A theory named string theory was used in the science fiction television series Quantum Leap. In the series it relates to a theory of time travel. It views a person's life as a string that moves from one end to the other. However, if it were possible to roll up this string into a ball it would be possible to leap from one section to another. This was the explanation given to the time travelling occurring in the series.

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See also

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References and further reading

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Footnote

  1. ^ To imagine the Planck length: you can stretch along the diameter of an atom the same number of strings as the number of atoms you can line up to Proxima Centauri (the nearest star to Earth after the Sun). The tension of a string (8.9×1042 newtons) is about 1041 times the tension of an average piano string (735 newtons). The graviton (the proposed messenger particle of the gravitational force), for example, is predicted by the theory to be a string with wave amplitude zero.
  2. ^ Prominent critics include Philip Anderson ("string theory is the first science in hundreds of years to be pursued in pre-Baconian fashion, without any adequate experimental guidance", New York Times, 4 January 2005), Sheldon Glashow ("there ain't no experiment that could be done nor is there any observation that could be made that would say, `You guys are wrong.' The theory is safe, permanently safe", NOVA interview), Lawrence Krauss ("String theory [is] yet to have any real successes in explaining or predicting anything measurable", New York Times, 8 November 2005), Peter Woit (see his blog, article and forthcoming book) and Carlo Rovelli (see his Dialog on Quantum Gravity).
  3. ^ "Is string theory in trouble?" Amanda Gefter NewScientist.com news service 17 December 2005
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Popular books and articles

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Textbooks

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External links