String theory

The Birth and Brief History of String Theory:

1943: The S-matrix approach to strong interactions

The S-matrix (or Scattering Matrix) theory was a proposed principle of particle interactions. The approach was started by Werner Heisenberg in 1943, who was building on work by John Archibald Wheeler, from the previous decade. 

Why should theoretical physicists have any business in knowing the inner workings of hadrons? The advocates of the S-matrix wanted to keep the physics as close to the experimental data as possible. They didn't want to speculate too much about the inner workings of the proton. The input of this S-matrix is some specified collection of particles that are about to collide. These are some composite particles (hadrons), such as protons, neutrons, mesons or even some atomic nuclei. Each of these particles, has properties, such as: momentum, spin and electric charge. When these particles enter the S-matrix, they disappear. However, groups of particles will come out of the S-matrix. These are the byproducts of the collision, which also have their own properties. But what exactly was happening in the S-matrix? Physicists were afraid to look inside to see. What were it's underlying mechanisms? The S-matrix went from the initial, right to the final state particles. The S-matrix would be a table of quantum mechanical probabilities. For each input, the S-matrix, will give you the probability of the output. The S-matrix would make these quantum mechanical predictions based on the direction and momentum of the incoming and outgoing particles. 

The S-matrix approach, was proposed to replace the standard techniques of quantum field theory, as it could answer some phenomenon at strong coupling. One of the reasons that the S-matrix approach was abandoned was because of the obscurity of it's mathematical methods. The S-matrix, did not rely on local notions of space and time. This makes it difficult to formulate a physical theory. Space and time were avoided and were replaced with some of the S-matrix's mathematical properties. Instead, the infinite past was related to the infinite future. This was done in one step. It could not be reduced to slices of time or to intermediate steps. On a related note, Werner Heisenberg believed that space and time broke down at the nuclear scale. The proton and neutron, are not point like particles, like the electron. This was beginning to be suspected at the time. If only had they known that the application of the S-matrix theory to strongly interacting particles would lead to the birth of string theory! That being said, the S-matrix was completely abandoned in the 1970s, when it was discovered that Quantum chromodynamics is the proper theory of strongly interacting particles. 

1961: Chew-Frautschi plots

Geoffrey Chew and Steven Frautschi, in 1961, will propose that mesons, when plotted with their angular momentum, against the squared of their masses, will fall into straight line trajectories. These are called Regge trajectories.

It is going to be discovered by theorists that these two ways particles were exchanging, included each other and could be used together to describe the amplitude as a whole.

1968: Veneziano amplitude/ Dual resonance model

In 1968, Gabriele Veneziano, is going to construct a scattering amplitude (the Veneziano amplitude) with the property of Dolen-Horn-Schmid duality and that will also follow the Regge trajectories. 

This is going to be the most successful S-matrix approach. The idea was that there was a consistent expansion that began with stable particles on straight line Regge trajectories. This property of Dolen-Horn-Schmid duality will prove to be what leads Veneziano, to construct his self-consistent scattering amplitude. 

Veneziano, was working in Israel at the time. He was interested in figuring out the mathematics of the S-matrix theory. Nobody knew the mathematical expression that satisfied the S-matrix's rules. This search is what led to his proposal of the Veneziano amplitude. The Veneziano amplitude was not an elegant mathematical expression. It is a table of probabilities. It was a formula that tells us what happens when two hadrons collide. When Hector Rubinstein, showed this formula to Leonard Susskind, he made the connection. He believed that the formula had something to do with quantum mechanical harmonic oscillators. A harmonic oscillator is anything in physics that can vibrate, or swing back and forth repeatedly. 

It should be noted that despite this being a kind of celebrated victory of the S-matrix theory over quantum field theory, and there being a flurry of papers being published that attempted to apply the Beta function to the data that was coming out of atom smashers all over the world, there was still a problem with Chew's S-matrix: unitarity. Unitarity, in this sense, is the conservation of probability. 

1969-70: A theory of strings

In 1969 and in 1970, Yoichiro Nambu, Holger Bech Nielsen and Leonard Susskind are going to recognize that the dual resonance model, could have a description as a spacetime theory of strings. Susskind, Nambu and Nielsen are going to reveal the unknown physics that lurked behind Veneziano's discovery. The idea was that nuclear forces, could be represented, as vibrating, 1-dimensional strings. Indeed, this was the proposal that the Veneziano model was a theory of strings, thus, is considered to be the birth of string theory. 

Yoichiro Nambu, who is a bit older than Leonard Susskind, had been doing the same calculations in Chicago.  Nambu was born in Japan, and after World War II, came to the University of Chicago. Nambu, had for some time, been one of the most eminent theoretical physicists in the world. Nambu had a reputation as one who saw things long before everyone else.  Nambu was, renowned for his quiet, well-mannered, but always penetrating style. Nambu lets the merit of his work speak for itself. We can also say that it was clearly Nambu who wrote down the basic equations of string theory. Nambu wanted to make sense of the hundreds of hadron particles that were being discovered, that could, of course, not be fundamental. Nambu thought that this was indicative of some kind of underlying structure. Nambu's idea was that the hadron consisted of some sort of vibrating string. Each mode of string vibration corresponded to a kind of subatomic particle. This was quite the seminal insight, that the Euler Beta function found by Veneziano and Suzuki could be explained by vibrating strings. Nambu was doing this work along with Tetsuo Gotto from Nihon University.

Susskind thought of the Veneziano amplitude and began to visualize elastic strings. The string was stretched out between two quarks and it could vibrate. These strings could have different patterns of vibration. Susskind proposed that Veneziano's S-matrix formula described two rubber bands on a collision course. 

Holger Nielsen, a physicist from Denmark, was also thinking about similar ideas. However, he had a different angle on the elastic strings.

Bosonic string theory

This was the earliest version of string theory, that had a considerable number of problems: The theory only contained bosons in its spectrum. These are the particles that mediate fundamental interactions as excitations of their corresponding fields, that have an integer value spin. However, if string theory were to be a consistent theory of nature, it would also have to include fermions, these are the matter particles with a ½ integer value spin. There was also the presence of a particle called a tachyon. This is a hypothetical particle that moves faster than the speed of light. Of course, this is not allowed by special relativity, thus, this was an inconsistency with the theory. The theory came about in the late 1960s and is 26 dimensional. It was shown by Claud Lovelace that these earliest versions of string theory had problems, unless, they were formulated in 26 dimensions. 

This was the original version of string theory, developed in the late 1960s and it was known as bosonic string theory. Let us recall that there are two kinds of subatomic particles: fermions, which have a spin of ½ integer, which are typically matter particles, like the quarks and electron, and bosons, which have an integer value spin of 1, like the photon. Bosons are typically messenger particles. Supersymmetry, is a kind of symmetry that posits a relationship between these two families of particles. However, supersymmetry had yet to be incorporated into string theory at this time and thus the theory only contained bosonic patterns of vibration. If string theory were to be a theory of everything, it would have to be able to account for both bosonic and fermionic vibrations. 

The idea that these three theorists proposed was one that would include particles with spin. These Neveu-Schwarz-Ramond strings eventually became the superstring theory. 

Although the string picture was an interesting and pleasing idea, it was soon after shown to fail as a theory of the strong nuclear force. Many of string theory's early predictions were proved incorrect by many 1970s observations. Also, quantum chromodynamics, which models quarks and gluons, as point particles was showing to have overwhelming success at describing the strong nuclear force. String theory was, thus, dismissed as a theory of the strong nuclear force. However, a dedicated team of researchers still continued to pursue the idea. 

John H. Schwarz said that "the mathematical structure of string theory was so beautiful and had so many miraculous properties that it had to be pointing toward something deep."

There was also a problem with string theory: string theory contained additional messenger particles that didn't appear to be relevant to the strong nuclear force at all. In 1974, John H. Schwarz and Joel Scherk, will study the patterns of this string vibration. What they found was that it was a perfect match to the hypothetical force mediating particle for gravity: the graviton. This was a revelation, that string theory could not only describe the strong nuclear force, however, gravity as well.

This was the even more alarming discovery that was yet to be had, upon examination of the messenger-like particles that appeared in string theory. These particles appeared to have no relevance to the strong nuclear force. However, in 1974, John H. Schwarz and Joel Scherk, will propose that this particle’s properties matched precisely those of the hypothetical force mediating particle for gravity: the graviton. These particles of gravity have never been observed in experiment. However, the properties of this particle can be predicted using the techniques of quantum field theory. At any rate, this was an alarming proposal, that string theory was not merely a theory of the strong nuclear force, however, it was also a quantum theory of gravity. 

The conflict between general relativity and quantum mechanics:

There are two foundational pillars upon which all of modern physics rests. The first of these pillar’s is Albert Einstein’s general theory of relativity. General relativity is applicable to understanding the physics of very large length scales. These are the stars, the galaxies, the cluster of the galaxies and even the Universe as a whole. General relativity is Einstein’s formulation of gravity. General relativity shows that space and time communicate gravity through their curvature. The second pillar of modern physics is quantum mechanics. Quantum mechanics is applicable at the smallest of length scales. This includes molecules and atoms all the way down to the subatomic particles such as the quarks and electrons. Quantum mechanics is an unfamiliar framework that governs the universe with features such as the uncertainty principle (that there are properties such as momentum and position that cannot both be known simultaneously with complete precision) and wave-particle duality (that objects manifest both wave-like and particle-like properties at the quantum scale). At any rate, both of these theories have been tested and experimentally verified to a remarkable degree of accuracy. So what is the problem? The problem is that there is a fundamental incompatibility or antagonism between these two theoretical frameworks that cannot be overlooked. Indeed, in most situations either general relativity or quantum mechanics can be used. However, what is something is both very small and very massive? An example of this phenomenon would be the gravitational singularity at a black hole or at the beginning of the universe at the Big Bang. This is a gargantuan mass compressed into an infinitesimal scale. The issue is this: when physicists actually attempt to apply both general relativity and quantum mechanics to understand this phenomenon, it does not work. These two theories cannot be harmoniously integrated into one another as would be hoped and expected. Nonsensical answers rise from the predictions that arise from an attempted synthesis of general relativity and quantum mechanics. These are called infinities, that arise in a nonrenormalizable theory of quantum gravity. An infinity is a nonsensical answer that emerges from the calculations of a theory of quantum gravity. There have been attempts to incorporate gravity into quantum mechanics. However, they have failed, since, at distances shorter than the Planck length, there are violent fluctuations in the very fabric of space. It is becoming increasingly more obvious that a deeper level of understanding is necessary.

The discovery was made by John H. Schwarz and Joel Scherk in 1974. The boson like pattern of string vibration is going to be analysed by these theorists. What they are going to speculate is a vibrational mode of the string corresponds to the properties of the graviton. The graviton is the hypothetical force mediating particle for gravitation, in the same way the photon mediates electromagnetism. The graviton would have to have no mass (since the range of gravity is infinite) and should be a tensor boson with a spin of 2 (since the source of gravitation is the stress energy tensor). The graviton would act as the hypothetical force messenger particle or force carrier for the gravitational interaction. One of the reasons that this was such a huge discovery, was that, string theory, prior to this proposal by Schwarz and Scherk, was believed to be a theory of hadrons. Hadrons are the particles like the proton and neutron. We now know that protons and neutrons are baryons (hadrons composed of 3 quarks) held together by gluons, the force mediating particle for the strong nuclear force. However, string theory did not suffice as a theory of the strong nuclear force, however, it contained a particle in its spectrum with no mass and a spin of 2: the graviton. This was quite the proposal: that string theory was a theory of quantum gravity, that could, potentially, reconcile general relativity with quantum mechanics, a longstanding problem in theoretical physics. According to John H. Schwarz and Joel Scherk, this was the reason that string theory failed to catch on: it’s scope was underestimated. It was not a theory of hadrons, however, of quantum gravity. To make sense of the extra dimensions that would appear in these string theories, Schwarz and Scherk reintroduced Kaluza-Klein theory: where the extra dimensions are compactified or curled up on very small geometries, to be undetectable to physical observation or even high energy experiment. 

Sadly, this work was not initially met by the mainstream theoretical physics community with enthusiasm. In fact, John H. Schwarz said; "our work was universally ignored." Since string theory failed at first at describing the strong nuclear force, why should it succeed at an even grander goal (being a theory of quantum gravity)? There were even further conflicts between string theory and quantum mechanics that came about in the 1970s and 1980s. String theory at that point appeared to be just another attempt at bringing together quantum mechanics with gravity. 

This is, until 1984! Michael Green and John H. Schwarz showed that the conflicts between string theory and the quantum theory could be resolved. This paper was the result of more than a decade of intense research. String theory, at this point, was proposed to have the capacity to describe all four of the fundamental forces: gravity, electromagnetism and the strong and weak nuclear forces. This remarkable work by John H. Schwarz and Michael Green in 1984 led to string theory being center stage in high energy physics. 

Indeed, string theory, will be taken serious again in 1984, after another remarkable proposal by John. H. Schwarz and Michael Green. Indeed, the proposal that string theory could provide the resolution to this conundrum between general relativity and quantum mechanics came in 1984, to Michael Green and John H. Schwarz. This was the so-called “first superstring revolution”. String theory, at this point was shown to have the capacity to be able to describe all four of the fundamental interactions as well: gravity, electromagnetism and the strong and weak nuclear forces. It could also describe matter. 

This period of research actually started by work done by Edward Witten and Luis Alvarez Gaume, who proved a year earlier, in 1983, that quantum field theories, have gravitational anomalies. Their research led to the conclusion that type I string theories were not consistent theories of gravity. However, John H. Schwarz and Michael Green, are going to show that these anomalies cancel (or are avoided) in some particular versions of string theory. What they did was changed the gauge group to SO(32) in type I string theory, and the anomaly was avoided. The dimension of the gauge group had to be 496. This was a necessary condition for the superstring to make sense as a theory of everything. This kind of satisfactory theory of everything, must account the four fundamental interactions of nature. As we can see, mathematically, 4 dimensions, is not big enough to account for these forces. This is the demand for a powerful theory, such as string theory. The original Riemannian metric tensor of Einstein, however, can be raised to incorporate other forces, and hence, other dimensions. The physical laws depend on the geometry of these hidden, extra dimensions. The theme appears to be that the laws of physics become simpler in higher dimensions. Are the symmetries of the subatomic world, hidden in extra dimensional space?

Superstring theory

The problem was that supersymmetry could be incorporated into string theory in five distinct ways. These were the five ten-dimensional versions of superstring theory that existed in the 1980s: type I, type IIA, type IIB, heterotic E 8 x E 8 and heterotic SO(32). 

Superstring theory came about in the 1980s, to attempt to solve the issue of why string theory only has bosons and not fermions. Supersymmetry, was incorporated into string theory in the 1980s, to include these fermionic vibrations. Supersymmetry is a principle that a theory might have. It is a transformation, or, a relationship between these two families of particles. Each boson will have a superpartner fermion and vice-versa. Superstring theory also removed the presence of the tachyon. The theory came about in the 1980s and is 10 dimensional and there were 5 different versions:

• Type I

Has both open and closed strings. This is the only version of superstring theory that has open strings. It is also the only theory where strings are unorientated or both orientations are equivalent. Type I string theory involves both open and closed strings. 

• Type IIA

Has only oriented closed strings. Type IIA superstring theory is non-chiral or parity conserving. This means that massless fermions spin both ways. Type IIA string theory involves closed strings with left-right symmetric vibrational patterns. 

• Type IIB

Has only oriented closed strings. Type IIB superstring theory is chiral or is not parity conserving. This means that massless fermions can only spin one way. Type IIB string theory involves closed strings with left-right asymmetric vibrational patterns. 

• Heterotic SO (32) and Heterotic E 8 x E 8

Has only closed strings. Heterotic strings are a kind of hybrid of a bosonic string and a type I superstring, where the right moving and left moving strings differ. The two kinds depend on their two different gauge groups: SO(32) and E 8 x E 8. Heterotic SO(32) string theory and Heterotic E 8 x E 8 string theory involve closed strings. These strings’ right moving vibrations resemble the pattern of the type II string. The left moving vibrations of the heterotic strings resemble the pattern of the bosonic string. The two flavors of heterotic string theory only differ in subtle ways.

M-theory

M-theory came about in 1995 and is 11 dimensional. M-theory is a conjectured theory that would unify the known versions of superstring theory. M-theory is a proposed unification of superstring theory with 11-dimensional supergravity. Witten, and others, proposed that the 5 superstring theories are actually different ways of looking at one theory. In fact, the fact that there were 5 different versions of string theory was an embarrassment for physicists, since, we live in one universe, not five. This unification required one more dimension of space, bringing the grand total to: 11. This proposal sparked a period of research known as the "second superstring revolution." However, the formulation of M-theory is still incomplete.

M-theory was proposed by Edward Witten in 1995, at the University of Southern California, based on theoretical observations that the different versions of superstring theory were actually related by mathematical dualities: S-duality and T-duality. There was also a proposed relationship between superstring theory and a field theory known as 11-dimensional supergravity.

There is no final or definitive reason as to what the "M" stands for, however, it was stated by Witten, that it could stand for "magic, mystery of matrix". He even stated "murky", as the theory is not yet well along (he later regretted suggesting that one).

1987 - Bergshoeff, Sezgin and Townsend: showed that 11-dimensional supergravity allowed for 2-dimensional branes! The name of their paper was Supermembranes and eleven-dimensional supergravity. They say "Now that we have become accustomed to the notion that strings should replace particles, it is natural to investigate the properties of higher-dimensional extended objects, in particular of membranes since they are the simplest extended objects, and they might describe strings in an appropriate limit. " 

1987 (shortly after) - Duff, Howe, Inami and Stelle: showed how one dimension in 11-dimensional supergravity could curl up into a circle:

• A membrane would wrap around a circular dimension.

• If the circle has a small enough radius: the entire construction looks just like type IIA 10-dimensional strings!

The name of their paper was SUPERSTRINGS IN D = 10 FROM SUPERMEMBRANES IN D = 11.

Their abstract declares that the Type IIA superstring in 10 dimensions is derived from the supermembrane in 11 dimensions by a simultaneous dimensional reduction of the world-volume and the space-time. 

S-duality

In string theory, strings are strongly interacting if they combine and decay frequently. Strings are weakly interacting if they combine and decay, less frequently. In S-duality, a collection of strongly interacting strings can be viewed as being a collection of weakly interacting strings. They are equivalent. Type I string theory is related to the Heterotic SO (32) string theory by S-duality and Type IIB string theory is related to itself by S-duality.

S-duality is a strong-weak duality. This is when a strongly coupled string theory is dual, or shown to be physically identical to a weakly coupled string theory. A coupling constant is strong if it is more than 1 and weak if it is less than 1. The coupling constant of string theory is a positive number that will tell us the probability of a collection of strings joining or splitting. This is the most basic process in string theory. Type I string theory is related to be heterotic SO(32) string theory by S-duality. Type IIB string theory is also related to itself by S-duality. 

The N=4 supersymmetric Yang-Mills theory describes particles like the quarks and gluons. The first kind of S-duality relationship was proposed by Claus Montonen and David Olive. They proposed that the N=4 supersymmetric Yang-Mills theory with the coupling constant "g" is equivalent to the same theory with the coupling constant "1/g". This result was known as Montonen-Olive duality. This was generalized to the S-duality relationship by some theorists in the 1990s. 

• Ashoke Sen, studied 4-dimensional heterotic strings, using S-duality. 

• Chris Hull and Paul Townsend, showed that type IIB string theory, with a large coupling constant is equivalent to itself, by S-duality, to the same theory with a small coupling constant.

T-duality

T-duality, will consider strings propagating around the geometry of different extra curricular dimensions and show them to be equivalent. Strings propagating on spacetime geometries of different sizes can be physcially equivalent. 

Let’s say you have two different circular extra dimensions. One with the radius: r, and one with the radius: 1/r. Now, let’s say that there are strings that are propagating along these differing geometries. The string will have two properties as it does so:

• Momentum, as it propagates along this extra curricular dimension.

• Winding number, the number of times the strings winds counterclockwise around this circular extra dimension.

In T-duality, these properties will be identified with each other, in differing string pictures. In other words: momentum and winding number will be swapped moving from the first string picture to the second. 

Type IIA and Type IIB string theory are related to each other by T-duality and Heterotic SO (32) and Heterotic E 8 x E 8 are also related to each other by T-duality.

T-duality is another kind of relationship that can exist between string theories. Here we have to consider strings propagating on some kind of circular extra dimension, such as a cylinder. These strings can either be wrapped or unwrapped. The number of times that the string wraps around this circular dimension is known as the winding number. In T-duality, large circular radiuses of these extra dimensions, each have their own corresponding smaller circular radius extra dimension. They are equivalent descriptions. It should be noted, however, that moving from one geometric description to the other, the momentum and winding number of the string is swapped. Type IIA and type IIB string theory are related by T-duality. Both flavors of heterotic string theory are also related to each other by T-duality. 

M-theory is also interesting because, not only does it contain 1-dimensional strings, however, it also contains higher dimensional membranes or “branes”. A brane is any extended object that occurs in string theory. A 1-brane is a string. A 2-brane is a membrane. A 3-brane is has three extended dimensions. We can also say that a p-brane, has “p” spatial dimensions.