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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. Study of string theories has revealed that they require not just strings but other objects, variously including points, membranes, and higher-dimensional objects.

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 is able to describe a universe with the precise collection of forces and matter that is observed, nor how much freedom to choose those details 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.

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

History

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 unveiled the physics beneath 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 studied the messenger-like patterns of string vibration and found that their properties exactly matched those of the gravitational force’s hypothetical messenger particle -- 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.

In the 1990s, Edward Witten and others found strong evidence that the different superstring theories were different limits of an unknown 11-dimensional theory called M-theory. These discoveries sparked the second superstring revolution. (Several meanings of the "M" have been proposed; physicists joke that the true meaning will only be chosen when the theory is finally understood.)

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.

Basic properties

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

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.

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?

Extra dimensions

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 Lorentz invariance can only be satisfied in a certain number of dimensions. This is roughly like saying that if an observer measures the distance between two points, then rotates by some angle and measures again, the observed distance only stays the same if the universe has 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 CERN preprint "Quantum Geometry of Bosonic Strings - Revisited").

However, these models appear to contradict observed phenomena. Physicists usually solve this problem in one of two different ways. 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. This is akin to the 4 macroscopic dimensions we are accustomed to dealing with every day. If, however, one approaches the hose, one discovers that it contains a second dimension, its circumference. This "extra dimension" is only visible within a relatively close range to the hose, just as the extra dimensions of the Calabi-Yau space are only visible at extremely small distances, and thus are not easily detected.

(Of course, everyday garden hoses exist in three spatial dimensions, but for the purpose of the analogy, its thickness is neglected and only motion on the surface of the hose is considered. A point on the hose's surface can be specified by two numbers, a distance along the hose and a distance along the circumference, just as points on the Earth's surface can be uniquely specified by latitude and longitude. In either case, the object has two spatial dimensions. Like the Earth, garden hoses have an interior, a region that requires an extra dimension; however, unlike the Earth, a Calabi-Yau space has no interior.)

Another possibility is that we are stuck in a 3+1 dimensional subspace of the full universe, where the "3+1" reminds us that time is a different kind of dimension than space. Because it involves mathematical objects called D-branes, this is known as a braneworld theory.

In either case, gravity acting in the hidden dimensions produces 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.

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 possesses many features of mathematical interest, and it 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[1] 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.

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.

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".

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.


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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.
. 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). - Statistics. 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". Main article: Liverpool F.C. 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. See also: List of Liverpool FC players.

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. See Also: List of Liverpool FC players. 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. Liverpool's reserve team plays its home games at the Racecourse Ground, the home stadium of Wrexham A.F.C.. 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. As of 31 January, 2006. Superstrings, D-strings or other stringy objects stretched to intergalactic scales would radiate gravitational waves, which could presumably be detected using experiments like LIGO. ² Title shared.

For example, astronomers have also detected a few cases of what might be string-induced gravitational lensing. ¹ More than any other English club. Older proposals for detecting cosmic strings could now be used to investigate superstring theory. Treatment of the old stadium requires sensitivity as a number of deceased fans have had their ashes officially scattered on the pitch over the years. As theorist Tom Kibble remarks, "string theory cosmologists have discovered cosmic strings lurking everywhere in the undergrowth". The old stadium will become a public plaza surrounded by apartments, offices, bars, restaurants and a hotel, and possibly including a memorial garden. 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"). At that time the club was hoping to start construction in summer 2005 and open the ground in 2007, but finance has not yet been obtained and the likely completion date is not currently known.

Such a stretched string would exhibit many of the properties of the old "cosmic" string variety, making the older calculations useful again. Despite pressure from Governmental and funding bodies, Liverpool refused to share the new ground with their local rivals Everton and "final" talks on a groundshare failed in January 2005. 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. On July 30, 2004, Liverpool City Council granted the club planning permission to build a new 60,000 seat stadium, nearby at Stanley Park. If such objects did exist, they must be few and far between. Signed on a free-transfer from Manchester City until the end of the season, when an extension to his contract will be considered. 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. On 27th January 2006, Benítez pleased the vast majority of Liverpool FC fans and re-signed the prominent former striker Robbie Fowler.

For several years, cosmic strings were a popular model for explaining various cosmological phenomena, such as the way galaxies formed in the early Universe. The game was decided by a single goal scored by São Paulo midfielder Mineiro. Originally discussed in the 1980s, cosmic strings are a different type of object than the entities of superstring theories. Liverpool's final opponents were to be São Paulo. In the early 2000s, string theorists revived interest in an older concept, the cosmic string. Goals coming from Peter Crouch (2) and Steven Gerrard. Eventually, scientists may be able to test string theory by observing cosmological phenomena which may be sensitive to string physics. The semi-final was played against Deportivo Saprissa, which Liverpool won 3-0.

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. Liverpool joined the competition at the semi-final stage, courtesy of a bye. 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. As the UEFA Champions League 2005 winners, Liverpool earned the right to represent UEFA in the FIFA Club World Championship 2005 held in Japan. 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. Liverpool trailed 0-3 at half time, but three goals in the second half forced extra time and then a penalty shoot out, which Liverpool won. 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. in the semis, the Reds met Italian side AC Milan in the final.

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. After eliminating hot favourites Juventus in the quarter finals and English rivals Chelsea F.C. Since string theory may not be tested in the foreseeable future, some scientists[1] have asked if it even deserves to be called a scientific theory: it is not yet falsifiable in the sense of Popper. However, the season had a glorious ending at the European Cup final in Istanbul. In this sense, string theory is still in a "larval stage": it possesses many features of mathematical interest, and it may yet become supremely important in our understanding of the Universe, but it requires further developments before it is accepted or falsified. Spaniard Rafael Benítez took over from Houllier, but erratic league form quickly put paid to their Premiership title hopes, eventually finishing fifth. 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. Houllier guided Liverpool to another League Cup triumph in 2003, Houllier and Liverpool parted by mutual consent at the end of the 2003-04 season after failing to bring the Premiership title to Anfield and not looking like making any further progress, despite having spent large sums of money.

String theory remains to be verified. Liverpool finished Premiership runners-up in 2001-02, their best league finish since 1991, but suffered a humiliating defeat in the League Cup Third Round at the hands of Grimsby Town. 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. Houllier at last won the faith of the club's fans, who were delighted at this triumph. In either case, gravity acting in the hidden dimensions produces other non-gravitational forces such as electromagnetism. The likes of Michael Owen, Robbie Fowler, Steven Gerrard and Sami Hyypia contributed to an excellent season in which the Reds finished third in the Premiership (thus qualifying for the Champions League) as well as completing a unique treble of the FA Cup, League Cup and UEFA Cup, followed by the FA Community Shield and European Super Cup later in 2001. Because it involves mathematical objects called D-branes, this is known as a braneworld theory. 2000-01 was Liverpool's best season for many years.

Another possibility is that we are stuck in a 3+1 dimensional subspace of the full universe, where the "3+1" reminds us that time is a different kind of dimension than space. They would have qualified for the Champions League had it not been for a 1-0 defeat at Bradford on the final day of the season. Like the Earth, garden hoses have an interior, a region that requires an extra dimension; however, unlike the Earth, a Calabi-Yau space has no interior.). But 1999-2000 was a bit better, as Liverpool finished fourth and qualified for the UEFA Cup. In either case, the object has two spatial dimensions. The Reds finished eighth in the Premiership - their lowest finish since 1994 - and failed to qualify for the UEFA Cup despite Robbie Fowler's return from injury and Michael Owen's continued impressive showings. A point on the hose's surface can be specified by two numbers, a distance along the hose and a distance along the circumference, just as points on the Earth's surface can be uniquely specified by latitude and longitude. Evans quit in November after failing to settle into the partnership.

(Of course, everyday garden hoses exist in three spatial dimensions, but for the purpose of the analogy, its thickness is neglected and only motion on the surface of the hose is considered. The Liverpool directors brought in Frenchman Gérard Houllier to work alongside Roy Evans for the 1998-99 season after Houllier had gained fame as the Technical Director of France's World Cup winning side in 1998. This "extra dimension" is only visible within a relatively close range to the hose, just as the extra dimensions of the Calabi-Yau space are only visible at extremely small distances, and thus are not easily detected. 1997-98 saw Liverpool finish third in the Premiership, but the big news of the season was the emergence of brilliant young striker Michael Owen - the Premiership's equal top scorer with 18 goals who became the youngest full England international at the time and also won the PFA Young Player of the Year Award. If, however, one approaches the hose, one discovers that it contains a second dimension, its circumference. They had been hoping to win the Cup Winners Cup for the first time, but defeat at the hands of Paris St Germain in the semi finals put paid to these hopes. This is akin to the 4 macroscopic dimensions we are accustomed to dealing with every day. Liverpool led the Premiership at several stages during the 1996-97 season, but in the end they finished fourth and had to settle for a UEFA Cup place.

If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. The club's youth system had produced two excellent players in Robbie Fowler and Steve McManaman, and including Jamie Redknapp and Stan Collymore (all of whom went on to play for England), looked to be the most potent attacking force in the Premier League in the mid-late 1990s. A standard analogy for this is to consider multidimensional space as a garden hose. Still, Liverpool were recognised as a top team once again, and were starting to attract top players once more. Essentially these extra dimensions are compactified by causing them to loop back upon themselves. But they finished third in the Premiership behind champions Manchester United and runners-up Newcastle, and lost 1-0 to Man United in the FA Cup final. In 7 dimensions, they are termed G2 manifolds. Liverpool continued to progress the following season, being within a shout of the domestic double right up to the final few weeks of the season.

The 6-dimensional model's resolution is achieved with Calabi-Yau spaces. Boot room veteran Roy Evans took over from Souness, and in his first full season (1994-95) they finished fourth in the Premiership and won the League Cup. 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. However, there was a promising young striker emerging in the shape of 18-year-old Robbie Fowler. Physicists usually solve this problem in one of two different ways. Ian Rush was now the only player remaining from the Paisley era, and the current squad was mostly made up of unfashionable players who just a few seasons earlier would never have pulled on a red jersey. However, these models appear to contradict observed phenomena. But his only success as manager at Liverpool came with the FA Cup in 1992, and he was finally ousted in January 1994 after a shock FA Cup defeat at home to Bristol City and three years of reckless spending on players who often failed to live up to expectations.

(see technical details in the CERN preprint "Quantum Geometry of Bosonic Strings - Revisited"). Souness had captained Liverpool during the glorious 1980s before taking the Rangers manager's job in 1986, and had brought a host of titles to Ibrox. In bosonic string theories, the 26 dimensions come from the Polyakov equation . Long serving coach Ronnie Moran spent two months in temporary charge of the team, but by the time his successor Graeme Souness arrived the league title and F.A Cup had been whisked away by Arsenal and Tottenham respectively. More precisely, bosonic string theories are 26-dimensional, while superstring and M-theories turn out to involve 10 or 11 dimensions. His Liverpool side looked well placed to win the domestic double, but he quit the club and blamed the pressures of management for his decision to leave. 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. Kenny Dalglish stunned the football world by resigning as Liverpool manager in February 1991.

This is roughly like saying that if an observer measures the distance between two points, then rotates by some angle and measures again, the observed distance only stays the same if the universe has a particular number of dimensions. Liverpool have had some glorious moments during the years that followed the 1990 title glory, but life at Anfield has never been the same without the championship trophy in the club's boardroom. Technically, this happens because Lorentz invariance can only be satisfied in a certain number of dimensions. He helped them win their 18th league title in 1989-90, but nobody could have guessed at the time that it would be their last championship success to date. Instead, string theory allows one to compute the number of spacetime dimensions from first principles. Ian Rush had rejoined Liverpool after one miserable season with Juventus, and scored twice in the 1989 FA Cup final. The reason for the unobservability of the fifth dimension (its compactness) was suggested by the Swedish physicist Oskar Klein in 1926. A 96th fan died nearly four years later having never regained consciousness.

The first person to add a fifth dimension to Einstein's four was the German mathematician Theodor Kaluza in 1919. 94 fans died that day and 95th fan died in hospital from his injuries four days later. 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". Hundreds of Liverpool fans were trampled on the terraces at Hillsborough. One intriguing feature of string theory is that it predicts the number of dimensions which the universe should possess. But the season was overshadowed by the Hillsborough Stadium tragedy which had struck the club in the F.A Cup semi-final. Consequently, the minimum size of a string must be related to the string tension. A last minute goal from Arsenal's Michael Thomas (who later joined Liverpool) gave the league title to the visitors because they had a superior goal difference.

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". They lifted the F.A Cup with another victory over Everton in the final, but the league title slipped out of their grasp in the last minute of their last game of the season at home to Arsenal. Classical intuition suggests that it might shrink to a single point, but this would violate Heisenberg's uncertainty principle. Liverpool came close to repeating the double once more in 1988-89. Its tension will tend to contract it into a smaller and smaller loop. Wimbledon, meanwhile, had been First Division members for just two seasons and had only joined the league 11 years earlier. Consider a closed loop of string, left to move through space without external forces. Liverpool had by this stage been one of England's strongest sides for more than 20 years.

The tension of a quantum string is closely related to its size. The downside of the season was a shock 1-0 defeat at the hands of Wimbledon in the F.A Cup final. For example, quantum strings have tension, much like regular strings made of twine; this tension is considered a fundamental parameter of the theory. New winger John Barnes was credited with the Player of the Year Award. 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. There were fears that Liverpool's glory days were over when striker Ian Rush was sold to Juventus in a £3.2million deal, but his £750,000 successor John Aldridge silenced the critics by topping the club's goalscoring charts in the 1987-88 season and inspiring the Reds to another championship success - this time achieved with just two defeats all season. Nowadays, 'string theory' usually refers to the supersymmetric variant while the earlier is given its full name, 'bosonic string theory'. 1986-87 was trophyless as Dalglish's men finished league runners-up to Everton and lost to Arsenal in the League Cup final.

The term 'string theory' properly refers to both the 26-dimensional bosonic string theories and to the 10-dimensional superstring theories discovered by adding supersymmetry. They also beat neighbours Everton 3-1 in the F.A Cup final to become only the third team to win the league championship/F.A Cup double in the 20th century. 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. His first season in charge could not have been better, as the Reds fought off competition from Everton, West Ham United and Manchester United to win the league title. (Several meanings of the "M" have been proposed; physicists joke that the true meaning will only be chosen when the theory is finally understood.). Fagan handed over the reins to striker Kenny Dalglish, who had established himself as a world class player and now wanted to prove himself as a top manager. These discoveries sparked the second superstring revolution. The sequel of the tragedy was a 5-year ban on English clubs in European football, while Liverpool had to serve an extra year once all other English clubs were readmitted.

In the 1990s, Edward Witten and others found strong evidence that the different superstring theories were different limits of an unknown 11-dimensional theory called M-theory. Some Liverpool fans were later convicted on manslaughter charges relating to the tragedy. Several other ground-breaking discoveries, such as the heterotic string, were made in 1985. But the disappointment of the defeat was irrelevant, as 39 spectators - nearly all Juventus fans - were crushed to death during crowd trouble before the kick off. The anomaly is cancelled due to the Green-Schwarz mechanism. The Reds had a rare trophyless season as they lost 1-0 to Juventus in the European Cup Final at Heysel Stadium in Brussels. 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. Having lost to neighbours Everton in the race for the league title, Fagan decided to retire and wanted to go out on a high with the European Cup.

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. Fagan's second season as manager was his last - and it had a traumatic ending. String theories which include fermionic vibrations are now known as superstring theories; several different kinds have been described. Veteran coach Joe Fagan moved up to the manager's seat on Paisley's retirement, and his first season at the helm saw Liverpool become the first English club to win three major trophies in a single season - the league title, the League Cup and the European Cup. 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. They included Graeme Souness, Ian Rush, Alan Hansen and arguably the greatest player to ever wear Liverpool colours, Kenny Dalglish who is also a Celtic legend. While bosons are a critical ingredient of the Universe, they are not its only constituents. Under Paisley, a new era of stars emerged.

Additionally, as the name implies, the spectrum of particles contains only bosons, particles like the photon which obey particular rules of behavior. He won a total of 21 trophies, including three European Cups and three successive league titles, during his spell as manager and ended his career on a high with the league championship and League Cup double. Most importantly, the theory has a fundamental instability, believed to result in the decay of space-time itself. Bob Paisley was manager of Liverpool F.C from 1974 until 1983, and during those nine years he became one of the greatest managers ever to take charge of an English club. However, the bosonic theory has problems. Shankly ignored these pleas and joined the club's fans on The Kop as a spectator, while he handed over his managerial duties to Bob Paisley. Not all modern string theories use both types; some incorporate only the closed variety. The club's players and fans tried to persuade him to carry on, and a local factory even threatened to go on strike.

The two types of string behave in slightly different ways, yielding two spectra. They won the FA Cup again a year later, but Shankly stunned the footballing world soon afterwards by announcing his retirement. 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. Liverpool made their first impact on the European game in 1973 with a UEFA Cup triumph, as well as winning another league title that season. The scale of notes, each corresponding to a different kind of particle, is termed the "spectrum" of the theory. By now Shankly was one of the most highly rated managers in the game, and his squad contained some of the finest players in England - Roger Hunt, Ian St John and Ron Yeats to name but a few. 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. A year later Liverpool regained their championship crown.

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. They failed to retain the championship trophy the following season but compensation came in the form of a first-ever FA Cup. 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. In 1964, Liverpool lifted the league championship. It is now hoped that string theory or some descendant of it will provide a fundamental understanding of the quarks themselves.). In his third season as manager they won the Second Division championship and were promoted to the top flight where they have remained ever since, never finishing below eighth in the league. (The original need for a viable theory of hadrons has been fulfilled by quantum chromodynamics, the theory of quarks and their interactions. Bill Shankly became Liverpool manager in 1959 and over the next 15 years he transformed them into one of the finest club sides in Europe.

This led to the development of bosonic string theory, which is still the version first taught to many students. Over a decade of mediocrity was to follow. Schwarz and Scherk argued that string theory had failed to catch on because physicists had underestimated its scope. In 1946-7, the first season after World War 2, Liverpool were surprise league champions. Then, in 1974, John Schwarz and Joel Scherk studied the messenger-like patterns of string vibration and found that their properties exactly matched those of the gravitational force’s hypothetical messenger particle -- graviton. In 1922, and again in 1923, captained by England full back Ephraim Longworth, Liverpool won the league. The scientific community soon lost interest in string theory, and the standard model, with its particles and fields, remained unthreatened. On April 25, 1914, Liverpool made their first appearance in the FA Cup final at Crystal Palace but lost 1-0 to Burnley.

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. In 1901 Scottish international Alex Raisbeck was the first Liverpool captain to collect the Football League Championship, and Liverpool were league champions again in 1906. By representing nuclear forces as vibrating, one-dimensional strings, these physicists showed how Euler’s function accurately described those forces. Main article: History of Liverpool F.C.. In 1970, Yoichiro Nambu, Holger Bech Nielsen, and Leonard Susskind unveiled the physics beneath Euler’s strictly theoretical formula. . Veneziano applied the Euler beta function to the strong force, but no one could explain why it worked. Helens.

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. Liverpool Football Club is not to be confused with Liverpool (Rugby) Football Club which was formed in 1858 and is now known as Liverpool St. In 1968, theoretical physicist Gabriele Veneziano was trying to understand the strong nuclear force when he made a startling discovery. After Heysel, English clubs were withdrawn from European competition for a period of five years, six in the case of Liverpool. 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. The club was present at two of the biggest tragedies in European football — at Heysel in 1985 and Hillsborough in 1989. 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. Liverpool were elected to the Football League alongside Woolwich Arsenal two years later.

String theory was originally invented to explain peculiarities of hadron (subatomic particle which experiences the strong nuclear force) behavior. refused to recognise the team as Everton). . (after the F.A. String theory has also led to insight into supersymmetric gauge theories, which will be tested at the new Large Hadron Collider experiment. but was changed to Liverpool F.C. Work on string theory has led to advances in mathematics, mainly in algebraic geometry. The original name was to be Everton F.C.

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. Houlding decided to form his own team after Everton left Anfield in an argument over rent. It is not yet known whether string theory is able to describe a universe with the precise collection of forces and matter that is observed, nor how much freedom to choose those details the theory will allow. The club was founded on March 15, 1892 by John Houlding, the owner of Anfield. Superstring theories include fermions, the building blocks of matter, and incorporate supersymmetry. The club's home ground is the 45,362 capacity Anfield stadium, which is about three miles from the centre of Liverpool. 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. Since being founded in 1892, Liverpool is the only English team to have won the Champions league five times, 18 Football League titles as well as being the English club to have won the most number of UEFA Cups, European Supercups and English League Cups.

Interest in string theory is driven largely by the hope that it will prove to be a theory of everything. Liverpool Football Club (usually known simply as Liverpool) is an English football club based in Liverpool, Merseyside, they are the current champions of Europe and the most successful English football team to date. Study of string theories has revealed that they require not just strings but other objects, variously including points, membranes, and higher-dimensional objects. Calum Woods. For this reason, string theories are able to avoid problems associated with the presence of pointlike particles in a physical theory. Paul Willis. 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. James Smith.

David Roberts. Conal Platt. Lee Peltier. Michael Nardiello.

Jordon Holmes. Adam Hammill. Danny Guthrie. James Frayne.

Paul Barrett. Charlie Barnett. Paul Anderson. Ramon Calliste.

Danny O'Donnell. Godwin Antwi. Miki Roque. Jack Hobbs.

Zak Whitbread (on loan to Millwall). Darren Potter (on loan to Southampton). Florent Sinama-Pongolle (on loan to Blackburn Rovers). David Raven (on loan to Tranmere Rovers).

Neil Mellor (on loan to Wigan Athletic). Carl Medjani (on loan to Metz). Anthony Le Tallec (on loan to Sunderland). Chris Kirkland (on loan to West Brom).

Robbie Foy (on loan to Wrexham). Salif Diao (on loan to Portsmouth). Bruno Cheyrou (on loan to Bordeaux). Mark González (on loan to Real Sociedad).

First broadcast of BBC TV's Match of the Day: on the 22nd August 1964, playing against Arsenal; the BBC 2 broadcast was presented by Kenneth Wolstenholme [2]. The six English players he has brought in are strikers Peter Crouch and Robbie Fowler, goalkeepers Scott Carson and David Martin, and youth players Jack Hobbs and Paul Anderson. As well as having a Spanish manager, there are 5 Spaniards in the current squad and 10 players in total brought to Liverpool directly from La Liga. has a strong Spanish influence.

Under Benitez, today's Liverpool F.C. The late Pope John Paul II, who played as a goalkeeper in Poland like current No.2 keeper Jerzy Dudek, also professed a fondness for Liverpool, when he met the Polish keeper. Celebrity fans of the club include the late DJ John Peel, Clive Owen, Ian McCulloch, Elvis Costello, Cherie Blair, and former World Heavyweight boxing Champion Mike Tyson. Other popular chants include "Fields of Anfield Road" (to the tune of "The Fields of Athenry"), "Poor Scouser Tommy", "Liverbird Upon My Chest", "We've Won It Five Times", and "Ring of Fire".

Liverpool fans, singing "You'll Never Walk Alone," were featured in the Pink Floyd song, "Fearless.". [1]. Claims that the singing of You'll Never Walk Alone was started by fans of other clubs before those of Liverpool are widely dismissed as untrue. Other German, Dutch and Northern Irish clubs have also adopted the song.

The song has since gained popularity among the fans of other clubs, most notably Celtic, Ajax and Milan. The song "You'll Never Walk Alone", originally from the Rodgers and Hammerstein musical Carousel and famously recorded by Liverpool musicians Gerry & The Pacemakers, is the anthem of Liverpool FC (and is included in their crest) and has been sung since the early 1960s. Liverpool's longest distance goal was scored by Xabi Alonso, 65 yards from goal. 1994-5, 4 minutes, 32 seconds.

Quickest Premiership hattrick: Robbie Fowler vs Arsenal F.C. 3 consecutive hat tricks: Jack Balmer 1946-7 (his only hat tricks). Most career hat tricks: Gordon Hodgson 17 (1926-36). Robbie Fowler, 1993/94.

Ian Rush, 1983/84. John Evans, 1954/55. Andy McGuigan, 1901/02. These are:

    .

    Only four people have scored 5 goals in one match. on September 12th 1989. Biggest league win: 9-0 v Crystal Palace F.C. Ian Rush holds the record in Liverpool FC for most goals in all competitions for one season - he scored 47 goals in the 1983/84 season.

    on September 17th 1974. Biggest win: 11-0 v Strømsgodset I.F. Only 14 first team players were used in the 1965/66 season, when Liverpool won the League. First European match: on August 17th 1964 they played against KR Reykjavik, Iceland, for the European Cup, and won 5-0 away.

    Roger Hunt has the most league goals for one season - in the 1961/62 season he scored 41 goals. on 11 December 1954 in Football League Division 2. Biggest defeat: 0-8 v Huddersfield in 1935 and 1-9 Birmingham City F.C. In 1910 Liverpool won the first match at Old Trafford beating Manchester United 4-3.

    December 1909 Newcastle United led 5-2 at Anfield, but the reds rallied to win 6-5. Rovers won 1-0, but six goals were disallowed during this game. on September 5th 1896. Liverpool played against Blackburn Rovers F.C.

    First Honour: in the 1893/94 season they became the Second Division champions. on 2nd September 1893 (Division 2). League debut: 2-0 v Middlesbrough Ironopolis F.C. FA Cup debut: September 1892 4-0 v Nantwich.

    Their side did not have one English player. They won 8-0. Liverpool's first ever competitive game was in the Lancashire League against Higher Walton. 1957, 1969, 1970, 1971, 1973, 1974, 1975, 1976, 1977, 1979, 1981, 1982, 1984, 1985, 1990, 2000.

    Reserves Division One Winners 16

      . 1986. Super Cup Winners 1
        . 1977, 2001, 2005.

        European Super Cup Winners 3¹

          . 1964², 1965², 1966, 1974, 1976, 1977², 1979, 1980, 1982, 1986², 1988, 1989, 1990, 2001. FA Community Shield Winners 14
            . 1981, 1982, 1983, 1984, 1995, 2001, 2003.

            League Cup Winners 7¹

              . 1996. FA Youth Cup Winners 1
                . 1965, 1974, 1986, 1989, 1992, 2001.

                FA Cup Winners 6

                  . 1973, 1976, 2001. UEFA Cup Winners 3¹
                    . AC Milan.

                    2005 3-3 (3-2 in penalty shootout) vs. AS Roma. 1984 1-1 (4-2 in penalty shootout) vs. Real Madrid.

                    1981 1-0 vs. Club Brugge. 1978 1-0 vs. Borussia Mönchengladbach.

                    1977 3-1 vs. UEFA Champions League 5¹

                      . 1893. Lancashire League Champions 1
                        .

                        1894, 1896, 1905, 1962. Football League Second Division Champions 4

                          . 1901, 1906, 1922, 1923, 1947, 1964, 1966, 1973, 1976, 1977, 1979, 1980, 1982, 1983, 1984, 1986, 1988, 1990. Football League titles 18¹
                            .