“The moment you encounter string theory and realize that almost all of the major developments in physics over the last hundred years emerge – and emerge with such elegance – from such a simple starting point, you realize that this incredibly compelling theory is in a class of its own”[i].

Over the past 100 years, physicists have made remarkable inroads into our understanding of the fundamental details of nature.  Since the discovery of the atomic nucleus by Rutherford in 1931, the fields of particle physics and quantum mechanics have allowed us to build models of the most fundamental particles and forces, and make predictions based on these to incredible degrees of accuracy.  The current embodiment of this work is called the Standard Theory. Despite its success, there are a number of short-comings and gaps in the Standard Model.  Whilst extraordinarily accurate, some believe that it is an approximation for a more complete theory.  The leading candidate to usurp the Standard Model is String Theory.  This report will discuss the issues that have driven the development of this alternate theory, its details and how it does and doesn’t address the problems of describing the physical world around us.

The Standard Model

In order to discuss string theory, we need to examine the existing description of the universe.  The Standard Model identifies several fundamental particles and forces that can be used to describe all matter and physical reactions readily encountered.  These particles are believed to be zero-dimensional point-like structures, that have no further internal constituents.  The standard model identifies two classes of particles, broken into three tiers.  These particles are outlined below:  

Table 1: Particles in the standard model The Standard Model also identifies four forces, although whilst gravity is acknowledged, it is not included in the standard model.  The forces mediate through messenger particles, with varying strengths, as summarized below:  

Table 2: Forces in the standard model (and gravity) In the Standard Model, the electro-magnetic force and the weak force are believed to be two different manifestations of the same force (the electro-weak force).  The strong force, is also believed to be the same force, although theories incorporating this notion are referred to as Grand United Theories, and have not been subject to the same rigor as the Standard Model[iv].

What’s wrong with the Standard Model?

The most striking feature of the Standard Model is the absence of gravity, which has resisted attempts to be shown as being related to the other three forces.  Whilst this may appear a significant drawback, the extremely feeble nature of gravity means that it has essentially no influence on particle reactions in observable phenomenon. There are still two downsides to this exclusion.  Firstly, the Standard Model cannot be described as a complete model, because of its non-incorporation of gravity.  This signifies it as a model that can be expanded and improved. More significantly, when attempts to incorporate gravity are attempted, the equations and models break down, and would seem to indicate that these two theories are incompatible.  This has serious consequences. The two key achievements in twentieth century physics are Einstein’s General Theory of Relativity, of which his Special Theory is a subset, and Quantum Mechanics, a thorough description of behavior at atomic levels and below.

General Relativity

Einstein’s Special Theory discussed the relationship between the observation of distance and time in non-accelerating reference frames – from this came the relationship between mass and energy – the famous E=MC².  General relativity expanded this to include accelerating reference frames.  Via his equivalence principle, Einstein showed that accelerated motion and being within a gravity field are indistinguishable, and further showed that gravity is the result of the warping of space-time by the presence of mass.  Mass curves space, but in the absence of matter or energy, General Relativity predicts flat space. General Relativity has been an outstanding success in describing gravitational phenomenon – which is essentially the interaction of massive objects.

Quantum Mechanics

The heart of quantum mechanics is the notion that any physical system may be viewed as having several possible states, with each state having a certain probability[v].  This overrides the notion of having complete knowledge about a system, and states that any attempt to measure a system will influence the system – the greater the degree of measurement, the greater the influence.  Heisenberg’s Uncertainty principle states that the uncertainty in the measurement of the position of a particle varies inversely as the uncertainty in the measurement of its momentum  - ΔxΔp ≥ h – where h is Planck’s constant.  A constraint of quantum mechanics is that certain quantities (e.g. energy, angular momentum, light) can only exist in certain discrete amounts[vi].  These amount are called quanta. Mass and energy interchangeable via E=MC².  As an extension of Heisenberg’s Uncertainty principle – the more we know about the time an event occurred, the less we know about its energy level.  A consequence of this is the ability for energy to produce mass (specifically a particle-anti-particle pair) over short periods of time.  However, this energy ‘debt’ must be quickly repaid, so the particles annihilate.  Such particles are referred to as virtual particles.  The shorter the time period, the greater the uncertainty, the greater the mass that can be created.  So as we look at the an empty volume of space over shorter and shorter time periods, the supposedly empty space actually contains larger and larger quantities of matter, being created by these fluctuations.  Because quantum mechanics deals in probabilities, the empty space is really just a manifestation of the average energy of the space, which over time evens out to nothing.

The conflict of general relativity and quantum mechanics

If you are a scientist and you are dealing with very massive things (stars, galaxies) you would use General Relativity.  If the problem involves very small things (atoms, sub-atomic particles) you would use quantum mechanics.  Very rarely do the very massive and the very small interact.  In the absence of such overlaps, both these theories have been enormously successful.   However, there are a few scenarios where the very massive meets the very small.  Because the interaction of volume and mass is density, quantum mechanics and General Relativity intersect in very dense situations (where there is a lot of mass in a small volume).  When this occurs, the theories breakdown.  Specifically, the quantum mechanical equations (which work in probabilities) yield answers that are infinity[vii].  The concept of an event having infinite probability is unacceptable (probabilities must fall in the range of zero and one). Although the overlap is rare, there are two signficant examples.  Firstly, black holes, the densest objects possible may provide such conditions, and are becoming more prominent in out understanding of the mechanisms powering high energy phenomenon in the universe.  More signficantly, during the very early stages of the universe, just after then big bang, such densities were almost certainly achieved, and thus a theory that allows us to understand these extreme conditions is sought[viii]. The key element of the conflict revolves around the concept of empty space.  A key tenet of General Relativity is that empty space is flat.  However, on smaller and smaller time and distance scales, the Heisenberg Uncertainty principle says that the space actually does contain mass.  So rather than a flat expanse of space, there are quantum fluctuations that distort flat space.  This issue does not relate solely to empty space, but rather sees any space-time curvature subject to distortions The reason this conflict has not been such a great issue is the distance scale involved.  Quantum mechanics are a governed by the Planck constant.  When this number is integrated with the force of gravity (which is vastly weaker than the other three known forces), a distance scale at which quantum fluctuations interfere with General Relativity is derived – the so called Planck distance.  At 10^-33 cm, this is an enormously small distance, some 10¹⁵ orders of magnitude beyond distance scales currently probable with the most powerful particle accelerators. However, for the two key theories that describe the universe to be incompatible is a major issue to scientists, and has been described as the major problem in physics today.

Fundamentals of string theory

Before we discuss string theory, we should call it by its real name – supersymmetric string theory.  We will discuss supersymmetry later, but should be aware that string theory is just an accepted abbreviation, and only describes one aspect of the theory.  Also, there are a number of string theory models (five to be exact – see Table 3), and so when discussing string theory, we are referring to a class of models sharing similar properties.  The difficulty of having five theories, and a possible solution, will also be discussed further. The first steps towards the conecpts of strings were based on a chance association between an obscure mathematical function – the Euler beta-function[ix], and the properties of particles interacting via the strong force.  Extending this association, Nambu, Nielsen and Susskind[x] demonstrated that if point-particles were replaced with one dimensional strings, an exact agreement with Euler’s function could be achieved.  Although string theory did not ultimately offer a useable description of the strong force in this context, it triggered investigation into the use of strings as the basis of matter, leading to the development of string theory. The key driver of string theory is the belief that assumptions made about the point-like nature of the fundamental particles in the universe are the key barrier to allowing the standard model to integrate General Relativity and quantum mechanics.  The basic notion is fairly straight-forward (one of the few aspects that is).  Rather than being zero-dimensional points, the basic particles are one dimensional loops, referred to as strings.  But more importantly, they are all instances of the same base string.  Whereas the standard model treats electrons and quarks as being composed of different types of matter, string theory says that all particles are made of the same matter, extending this to include both fermions (matter particles) and bosons (force particles). If all particles are made of the same stuff, why do they have different properties?  String theory states that a particle’s properties are the result of the vibrational pattern of the string.  Different vibrational patterns will yield an electron or a gluon or a quark.  As discussed, mass and energy are exchangeable under Einstein’s Special Theory of Relativity.  The more energy a particle possesses, the greater its mass[xi].  Thus a string that vibrates more energetically will have a greater mass.  And so a quark string would vibrate more energetically than an electron string.  By following more complex paths, physicists have shown that the vibrational pattern of a string can be used to determine other properties, such as its force interactions.  Strings that resonate in other ways may manifest themselves as bosons, offering a truly united model of the universe. Strings are small, much smaller than current probing technology, and so not accessible to direct physical scrutiny (one of the key barriers to confirming their existence).  However, models have been able to provide some parameters to quantify strings. As shown in Table 2, the graviton is the predicted messenger particle for gravitation. One of the predictions of early string theory was a particle whose properties exactly matched those predicted of the graviton.  This was a healthy boost for string theory.  By accommodating the graviton, and therefore the gravitational force, string theory offered the potential to present a truly unified theory of the universe – the so-called ‘Theory of Everything’. In these string models of the graviton, scientists determined that an inverse relationship existed between the force of a graviton and the tension of the underlying string – the weaker the force, the greater the tension[xii].  As the graviton is particularly weak (see  Table 2), this represents an incredibly high tension level – 10³⁹ tons[xiii].  Adding to our list of Planck constants, this value is termed the Planck tension. The basic string of most string theories is a loop (although some string theories allow for open strings).  Applying this tension to a loop causes and incredibly large contraction, and therefore makes strings very small.  Applying the Planck tension to a string yields a string with a length of 10^-33 cm – the previously identified Planck length. As tension is a contractional force, the energy required to achieve a particular vibration pattern is dependant on the tension of the string.  The greater the tension, the greater the energy required.  And so, tension and vibrational pattern are the two parameters needed to determine the energy of a particular string. Because quantum mechanics affects all physical processes, the vibrational energies of strings are similarly affected, and must manifest themselves in minimal energy amounts – discrete quanta.  Therefore string vibrations have a minimal amount, which is proportional to the tension of the string and the frequency of the vibrational pattern.  All vibration energy levels must be whole multiples of this amount, which in-turn defines the amplitude of the string’s vibrational energy[xiv]. However, the high tension of strings means that a high energy level must exist for even this minimal energy vibration to occur, and quantum mechanics forcing discrete multiples of these energy levels, indicates that strings are likely to be very heavy.  In fact, this minimal energy level turns out to be related to the various Planck constants, and is defined as the Planck energy, some 1.22 × 10¹⁹ GeV[xv].  Once again, applying Einstein’s E=MC², this give a mass of 10²² electron volts[xvi] – 10¹⁹ times the mass of a proton (and not surprisingly, given a Planck designator – the Planck mass). At this point we hit an obvious snag – how do we make a proton out of particles whose minimum weight is so much greater.  Once again, quantum mechanics comes to the rescue.  Heisenberg’s uncertainty principle says we cannot know position and momentum, so all particles are subject to quantum vibrations[xvii].  In the case of strings, these quantum fluctuations may actually counter the string vibrations, therefore offsetting the vibrational energy of the string, and reducing its energy and associated mass.  In instances where this energy offset is large enough (approaching the Planck energy), the vibrational energy will be so low that low mass particles are produced.  Extending this further, if the quantum fluctuation energies exactly match the Planck energy, then the mass of the string will be zero – as would be the expected case in zero mass particles such as the photon and the graviton. Although we are able to explain how strings can exists as low energy (compared to the Planck energy) particles, strings and the interaction with quantum fluctuations should make an infinite number of string manifestations.  However, the standard model identifies only a handful of particles that make up the universe we see, and the forces to govern it.  Because of the large energies involved, such particles may have existed, but decayed into smaller particles over time.  Because the energy of the Planck energy vastly exceeds the capability of current particle accelerators, experimenters have been unable to scratch more than the surface of the potential string manifestations. Another compelling motivation for pursuing string theory is that it appears to offer a theory that provides not only a description of the universe, but reasons for why it is the way it exists.  Under the standard model, electron’s negative electrical charge are simply a property discovered by experimentation.  String theory, on the other hand, holds forth the possibility of understanding what about an electron gives it its negative electrically charged.

Table 3: String theories  

Supersymmetry

As mentioned in the introduction, string theory is a shortening of the term supersymmetric string theory.  We have already discussed the string component, so we shall now discuss supersymmetry. All particles have a property called spin, which is analogous to the rotation of the earth around its axis[xix].  Fermions (the particles of mass) and bosons (the particles of force) are actually defined by their spin.  Fermions have what is called ‘half spin’ (leptons and quarks have spins of ½), whilst bosons have ‘whole spin’ (their spin being a multiple of a whole number, including zero[xx]).  Although more esoteric than mass or charge, because spin differentiates matter and force[xxi], it is a key physical property. Within nature, scientists have observed a number of symmetries, whereby physical processes are unaffected by changes in observation.  As an example, charge symmetry dictates that if all particles are replace by their anti-particles, equivalent physical reactions will still occur[xxii].  There are several compelling reasons to believe that a symmetry between bosons and fermions exists (that is, each boson has a matching fermion and vice versa).  These ‘super-partners’ would differ in spin by ½[xxiii]. The known bosons and fermions do not form the supersymmetrical partners, so if supersymmetry is a feature of the universe, then these particles remain to be detected.  Prediction place their masses higher than those achievable through current particle accelerators, perhaps explaining their non-discovery. When particles exists as super-partners, they have quantum fluctuations that tend to cancel one and other out.  This has implications for a number of quantum mechanical processes that would otherwise require extremely precise tuning (to a precision of roughly 10^-15[xxiv]).  Further, as discussed previously, Grand United Theories propose that the three non-gravitational forces are the one force at sufficiently high energy levels (10²⁸ ° K).  However, this is only achieved if supersymmetry is incorporated. Within the domain of string theory, like mass and other properties, spin is a function of a string’s vibrational pattern.  The first formulation of string theory, which did not incorporate supersymmetry, was called ‘bosonic string theory’.  Although important at the time, it suffered from two major setbacks.  Firstly, it was not able to describe fermions (hence the name) and therefore could not incorporate matter into its structure.  Secondly, one of its predictions was a particle with negative mass – the so-called tachyon.  A negative mass implies a speed always greater than the speed of light.  Although this is not specifically disallowed in Special Relativity, it is a logically inconvenient result for a theory to produce.  By incorporating supersymmetry, subsequent string theories (see Table 3) have avoided this complication, and addressed other issues. Although supersymmetry has been incorporated into point-particle models, it is none-the-less an essential feature of string theory.

Extra dimensions

The existence of unknown particles is not a major mental hurdle, as the discovery or creation of new particles has been a feature of particle physics over the last century.  However, we are very grounded in the notion that the universe consists of three spatial dimensions, as well as one time dimension (which Einstein united in his Relativity Theories).  However, there are no physical reasons for this upper limit on the number of dimensions, beyond our own perceptions. One of the principle claims of string theory is that beyond the three spatial dimensions we are familiar with, several smaller dimensions also exist.  The five contemporary string theories all propose six further spatial dimensions, giving ten in total (including one for time).  Beyond predicting these extra dimensions, string theory actually requires them to exist. Although these dimensions are beyond the realms of normal and experimental detection, there is no reason that if they are small enough, these dimensions cannot exist.  And because strings are so small (10^-33 cm), they would be able to interact with these dimensions even if they too were incredibly small.  This interaction would take the form of vibrating through that dimension, and because a particle’s properties are defined by the vibrational pattern of the underlying string, these dimensions could potentially be crucial for defining the properties of particles.   Quantum mechanical probability calculations must yield a result between one and zero.  String theory calculations in this field yielded negative probabilities until these extra dimensions were added, with the negative results only being cancelled out when nine dimensions are incorporated[xxv]. Although we cannot at this stage probe these postulated dimensions, some theoretical descriptions have been offered.  A class a six-dimensional geometrical shapes, called Calabi/Yau spaces have been proposed as the form that the ‘hidden’ extra six dimensions may take.  An example of a  Calabi/Yau shape is shown in figure 1[xxvi].  Although there are tens of thousands of possible Calabi/Yau shapes, string theorists are looking for shapes that through their structure offer insights into how these extra dimensions influence the physical characteristics of string particles.  As an example, investigation has been made into a class of Calabi/Yau shapes that have three holes, as a possible explanation as to why fermions exist in three families (see Table 1) [xxvii].

How does string theory resolve the General Relativity/quantum mechanical divide?

Although string theory may appeal on the grounds of elegance, and it offers a quantum field theory that incorporates gravity, how does it help to resolve conflicts between General Relativity and quantum mechanics?  One of the reasons general relativity breaks down at the sub-Planck level, is that particle interactions when using zero dimension point particles effectively occur at a distance of zero.  The concept of zero distance derives from the progressive reduction in size of particles below the Planck length.  If we add energy to say an electron, its wavelength will decrease, heading through the sub-Planck barrier.  However, in string theory, adding energy to a string does not allow it to break this barrier, but rather causes the string to increase in size. This presents a finite barrier to reactions taking place beyond the Planck distance, where General Relativity comes unstuck.  Reactions cannot occur at distances shorter than the plank length, because the fundamental constituents of the universe cannot go below this size.  This is a product of their one-dimensional nature, rather than the zero-dimensional point properties of the standard model.  So instead of a continual reduction in the scale of the universe ad infinitum, the physical properties of its fundamental units places a minimum level on distance scale.

Testing string theory

It has been estimated[xxviii] that to see a string directly, a particle accelerator the size of the universe would be required.  As this is not feasible, string theorists must look to other avenues for experimental proof of the existence of strings. As supersymmetry is a key component of string theory, the discovery of new supersymmetric particles would be a significant boost.  Although the masses of the undiscovered particles are not properly known, it is hoped that new energy levels achievable by particle accelerators (particularly the Large Hadron Collider being built at CERN in Switzerland) may be sufficient to produce one or more of these particles.  However, identification would not be conclusive proof of the accuracy of string theory (and non-discovery would not necessarily be dismissive). Other potentials avenues include:

• identifying a reaction that is predicted by string theory, but not by the standard model such as proton decay or obscure fractionally charged particles
• using a prediction from string theory to address problems in other areas of science , such as dark matter, neutrino mass or vacuum energy

On the second approach some progress has been made, with string theory making important contributions to the relationship between quantum theory (and associated quantum microstates) and entropy properties potentially displayed by black holes[xxix].  Although not conclusive (and relying on advanced areas of string theory to be discussed shortly), it is a forerunner to the types of applications of string theory that may provide experimental grounding to its theoretical structure.

Future directions in string theory

String theory is incredibly complex, and not only are scientists limited to partial and approximate answers, but indeed they are confined by only being able to develop approximate and partial equations on which to structure their theories.  The refinement of these equations over time has lead the greater confidence in tackling some of the outstanding issues in string theory. One of the key question marks is the existence of five separate string theories (see Table 3), and the means of determining which one is correct.  However, a 1995 address by leading string theorist Edward Witten proposed that rather than being competitive theories, the five string theories were special cases of a overarching theory, now referred to as M-Theory. M-Theory is not a developed or understood theory, but rather a set of principles that can unite the five current string theories (and potentially incorporate a sixth).  The chief tool is the concept of dualities, whereby properties of one theory can be transformed into those of another, linking quantities that were thought to be separate.  Two such dualities are T-duality, which is related to distances, and S-duality which is related to coupling constants (how strong an interaction is – this will not be expanded upon).  T-duality is driven by the one-dimensional nature of strings, which can not only traverse a dimension, but if the dimension is small enough, wrap around it (a process known a winding).  Any resultant measurement of the dimension can then be taken from the perspective of either traveling around the dimension, or the winding of that dimension.  These measurements yield inversely proportional answers (related to the radius of the dimension).  T-duality relates Type IIA and IIB string theory[xxx]. Another feature of M-theory is that rather than occupying the ten dimensions (nine spatial, one time) proposed by each of the five string theories, the integration of these theories produces a theory with eleven dimensions (an additional spatial dimension).  Although still an emerging field, it is believed that an as yet undiscovered duality allows the ten dimensional string theories to be related to the eleven dimensional M-theory[xxxi]. String theorists have also determined that string theories are able to contain objects that have more than the one dimension possessed by strings.  Such objects are predicted for all nine spatial dimensions proposed by string theory (termed P-branes), with special localized open string referred to as D-branes.  Expanding into M-theory, these objects are referred to as M-branes.  The influence of these higher dimensional objects is one of the most studied areas of contemporary string theory. With these advances, string theory, as well as seeking experimental evidence to support it, will most likely be applied to some major areas as physics, such as:

• the fabric of space time – is Einstein’s description truly accurate
• the origin of the universe – the implication of strings on the earliest moments and formation of the universe
• the unification of kinematics and dynamics[xxxii]

Conclusion

String theory, now over thirty years old, has achieved its longevity because it continues to offer insights into physical processes, and presents a potentially superior framework for formulating our understanding of the universe.  Although its primary fame lies in its resolution of the breakdown of general relativity at Planck distance scales, it offers the basis for a more thorough and complete description of the universe, including a quantum field theory of gravitation. Despite the almost overwhelming obstacles to achieving experimental verification, and the limited number of scenarios where its implementation offers superior results to the twin pillars of General Relativity and Quantum Mechanics, it holds out the tantalizing offer of a theory that will encapsulate our complete understanding of nature – a Theory Of Everything.

Bibliography

Kane, G. “The Particle Garden”, Perseus, 1996 Han, M. Y. “Quarks and Gluons: A Century of Particle Charges”, World Scientific, 1999 Greene, B. “The Elegant Universe”, Vintage Press, 1999 Sartori, L. “Understanding Relativity”, University of California Press, 1996