THE DEVELOPMENT OF STRING THEORY

KY LE R . ULR I CH

DU KE UNI VE R SIT YD EP AR TME NT O F E LE C TR IC AL AND CO MPU TE R EN GINE E RI NGPH YS IC S 1 0 5 – AS TR O PHY SI CS 2 5 AP RI L 2 01 1

ABSTRACT

This paper focuses on the historical context of superstring theory, the need for its development, and the current prospects of the theory. String theory was preceded by a strong desire to unify all fundamental forces into one compact theory. From Newtonian gravity to Einstein’s theories of special and general relativity to the development of the stochastic quantum mechanics, physics has always sought to unify all knowledge into one theory. However, this “Theory of Everything” still eludes physicists today. In a mathematical sense, string theory is the most promising solution, but can a theory that is incomplete and can’t be proven even be considered a theory? This paper illustrates how string theory began through discussing the desire to unify the fundamental forces, introducing why string theory might be necessary, and attempts to demystify the prospects of strings, membranes, and extra dimensions. Finally, the hopes of a future M-theory will be introduced as the prospect of a theory that solves the many issues of superstring theory.

KEYWORDS

String Theory, Fundamental Forces, Special Theory of Relativity, General Theory of Relativity, Quantum Mechanics, Beta Function, Kaluza-Klein Theory, Membrane, Spatial Dimensions, M-Theory

1. INTRODUCTION

String theory is derived from the desire to unify all forces of nature into one elegant

model and has therefore been dubbed as the “Theory of Everything.” The essential idea behind

string theory is that everything in nature, forces and matter alike, are made of tiny vibrating

strings of energy. All forces and matter come from different vibrations of the same basic strings.

However, the theorized strings are so small that they may never be able to be actually detected.

For this reason, string theory is a controversial field in physics. Even if string theory is found to

explain nature, we can’t guarantee that it is the underlying principle that governs nature if we

can’t provide any experimental evidence of its existence. So why should we believe that

something like string theory actually exists and defines everything in our universe in one elegant

theory?

2. DESIRE FOR UNIFICATION OF FUNDAMENTAL FORCES

The answer comes from historical attempts to unify the fundamental forces of nature.

These attempts began with Isaac Newton, who determined a gravitational force existed between

two massive objects m1 and m2 that obeyed Newton’s Universal Law of Gravitation

F12=GN m1 m2

|r12|2

where r12 is the distance between the two objects and GN is Newton’s constant, measured to be

6.67x10-11 m3 kg-1 s-2. With this, Newton was able to successfully unify the principles behind

planetary orbits with local interactions between Earth and a mass. This was a dramatic departure

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from beliefs of Newton’s time, but Newton’s gravity has provided stunningly accurate

experimental results. His equations were the basis behind the calculations that sent the first men

to the moon. However, Newton did not understand how his newly found gravity actually

worked. What caused gravity? For about three hundred years, this question was left

unanswered.

Albert Einstein’s special theory of relativity was a crucial step in answering this question.

The special theory of relativity is centered on the concept of invariance. Consider the inherent

invariance in a three dimensional Pythagorean Theorem

d s2=d x2+d y2+d z2

Such invariance implies that the distance ds is invariant when the path is rotated is space.

Einstein included time in this metric, such that

d s2=−c2 d t2+d x2+d y2+d z2

Just like the space metric being invariant under spatial rotations, Einstein’s relativity proposed

that space-time is invariant under rotations in space-time, called Lorentz transformations. Such a

transformation allows for us to see how observers with different constant velocities view their

surroundings differently. This forces the speed of light to be the same in any possible reference

frame and no object can move faster than the speed of light.

Einstein proposing that nothing can go faster than the speed of light directly opposed

Newton’s theory of gravity in the sense that Newton’s gravity was an instantaneous force

between two distant objects. A simple thought experiment can be used to describe Einstein’s

logic. It takes light approximately eight minutes to travel the 93 million miles from the sun to

the Earth. If the sun suddenly vanished, would Earth immediately abandon its orbit for a

tangential path at the instant of the sun’s disappearance? Or would the Earth continue to orbit

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the sun until information traveling at the speed of light reached the Earth? Einstein did not think

that gravity would outrun light. Therefore, Einstein faced the challenge of finding a theory that

does not break his cosmic speed limit (Greene).

The solution was a four dimensional fabric of space-time, which is warped and curved

due to the presence of matter as seen in Figure 1. This warping is what we feel as gravity.

Essentially, freely falling objects follow the straightest possible world-line, or path, in space-

time, called the geodesic. For objects on Earth, the geodesic happens to be curved toward the

center of the Earth. Also, changes in the fabric of space-time (i.e. the hypothesized removal of

the Sun’s mass) would propagate as a wave in space-time at the speed of light. Gravity is thus

seen to travel at the speed of light and a new picture of gravity known as the theory of general

relativity emerged. General relativity involves very complex mathematics, so Einstein’s

equations are simplified as

Gij=−8 πG

c4 T ij

where Gij is a tensor that determines the curvature of space and T ij is a tensor that describes the

mass-energy density of the objects in the space-time. This is a remarkable development, because

we now have a theory that states a freely falling object in space will follow a geodesic, which is

determined by the massive objects in our Universe.

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Figure 1: The curved space-time around a planet allows for an object to orbit the planet by following the shortest distance in space-time.

However, this did not satisfy Einstein, who wished to unify gravity with the only other

known force of the time: electromagnetism. James Maxwell had already developed the relation

between electricity and magnetism. Maxwell was determined to define a mathematical

relationship between the two different phenomena and was able to come up with the four

extremely elegant Maxwell equations, which are (in differential form):

∇ ∙ E= ρϵ 0∇ ∙ B=0∇× E=−∂B

∂ t∇× B=μ0 × J+μ0ϵ 0

∂ E∂ t

These four simple formulas are able to describe all interactions between electricity and

magnetism. With the development of general relativity, gravity and electromagnetism were both

postulated to propagate at the speed of light. Perhaps the same underlying principles drive both

forces. Einstein immersed himself in attempting to discover this underlying principle.

At this point only two fundamental forces had been observed: gravity and

electromagnetism. These were the only two forces that Einstein was trying to unite and he

encountered much difficulty because electromagnetic forces are typically 1036 times greater than

gravitational forces. Additionally, Niels Bohr and other physicists began to develop theories

concerning the composition of the atomic nucleus, and soon thereafter it was realized that the

forces of gravity and electromagnetism were not at all sufficient. The strong and weak nuclear

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forces were discovered to explain interactions within the nucleus of an atom. The strong nuclear

force is the force that binds protons and neutrons together within the nucleus as well as the force

that holds quarks and gluons1 together in order to compose subatomic particles such as protons

and neutrons. The weak nuclear force is associated with radioactive decay of nuclear particles

by changing quarks from one flavor to another. Einstein never tried to unify all four forces and

thus was never able to come up with a grander unification theorem.

Quantum mechanics quickly came about as a means to unify electromagnetic, strong, and

weak forces. It was able to describe the microscopic realm with great success and is surprisingly

accurate. There have never been experimental results that quantum mechanics cannot justify

mathematically, thus it is considered to be an extremely robust theory for extremely small

masses. However, this new theory describing the behavior of particles was unsettling to the way

in which we typically think of the universe. Essentially, quantum mechanics operates on the

principle that the position and momentum of a particle is not predictable – we can just calculate

the probability of the outcome of an experiment. The way this is done is by calculating the

probability amplitude of a system to be in a given state. The probability amplitude is the square

of the wave function ψ(x,t), which is a solution to the time dependent Schrodinger equation:

H Ψ=i h ∂ Ψ∂ t

where the Hamiltonian operator H is

H=−h2

2m (∇x1

2 +∇x2

2 +…+∇xN

2 )+V (x1 , x2 ,… xN)

1 A quark is an elementary particle that fundamentally constitutes matter. Quarks combine to form hadrons, which are composite particles such as protons or neutrons. Quarks come in six different flavors: up, down, charm, strange, top, and bottom. For example, a proton is made of two up quarks and one down quark. Gluons are elementary particles responsible for the color force, or source responsible for the strong interaction force, between quarks. Gluons hold together protons and neutrons in a nucleus.

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and x1, x2 , … xN is the position of each particle, V is the time-independent potential energy, and

∇2 is the Laplace operator, which is

∇2= ∂2

∂ x2 + ∂2

∂ y2 + ∂2

∂ z2

in a Cartesian three-dimensional space. If we consider the Schrodinger equation for two

particles, the wave equation will satisfy

Ψ ( x1 , x2 )=± Ψ (x2 , x1)

If the positive solution is observed, the two particles are known to be bosons, which are

subatomic particles that have integer spin (i.e. photons, W and Z bosons and gluons). These

bosons can occupy the same quantum state at the same time. For the negative solution, the

particles are fermions, which are subatomic particles that have half-integer spin (i.e. quarks and

leptons, the elementary particles that comprise the composite fermions of protons, neutrons, and

electrons). Two fermions cannot occupy the same quantum state due to Pauli repulsion. Pauli

repulsion is what explains the structure and stability of atoms and matter. Without fermions no

matter would exist in stable, predictable structures! This is a huge advantage of quantum

mechanics since the theory practically predicts nature to be as we know it – at least in a

probabilistic sense.

However, Einstein was not satisfied with quantum mechanics and has been famously

quoted as saying “God does not roll dice.” Einstein continued to believe that a concrete

unification theory exists that does not require a certain outcome having a probability of

occurring. But Einstein was still eluded by gravity being completely obscured by the other three

forces at the quantum level. When Einstein died, no other scientist was actively trying to unify

gravity with quantum mechanics. Physics was split into two groups – the realm of general

relativity for large masses and that of quantum mechanics for atomic level analysis. 7

So can general relativity fit in with quantum mechanics in any model? Without a unified

theory we cannot understand parts of our universe, such as black holes. Despite the lack of a

unified theory, the existence of black holes suggest that gravity can provide forces strong enough

to compare with the other fundamental forces. Karl Schwarzchild provided an exact solution to

the Einstein field equations of general relativity in order to define the event horizon of a non-

rotating black hole (Weisstein). The Schwarzchild radius defining the event horizon was found

to be

r Sch=2GM

c2

A dense enough mass could be seen to warp space time enough so that not even light can escape

once it is within the Schwarzchild radius. Black holes raise an interesting question: Do we use

general relativity (intended for extremely heavy masses) or quantum mechanics (intended for

matter on the atomic scale) to describe the singularity? Combining the two theories produces

nonsensical solutions for the singularity in a black hole. Although it is impossible to reconcile

the two laws of physics in order to describe the singularity, we can only imagine that there is

some underlying physical principle that governs all the forces. After all, gravity in a black hole

overcomes and interacts with the electromagnetic force, ultimately preventing photons escaping

the event horizon.

At this point it is obvious that there is a lot of evidence pointing towards the possibility of

a single theory existing that unifies the four fundamental forces. String theory is the most

popular option that accomplishes this unification. However, consequences of the string theory

result in the unavoidable creation of multiple dimensions2 and even the possibility of parallel

universes. You can imagine the amount of resistance this theory has encountered due to these

2 We still aren’t sure exactly how many, but current theories predict that 10, 11, or even 26 dimensions could exist.8

consequences. Despite the resistance, though, string theory is, mathematically, one of the most

robust theories of unification in existence today. In order to completely understand the

consequences of the theory, we will discuss how string theory progressed since it was founded in

1968.

3. HISTORY OF STRING THEORY

All of modern science is based on two conflicting theories: the general theory of relativity

to describe interactions between large masses over large distances in our universe, and quantum

mechanics to describe interactions between small masses over small distances. The development

of string theory began with a model of the strong nuclear force.

In the 1960’s, particle accelerators at the European Organization for Nuclear Research

(CERN) and other places were focusing on strong interacting processes (Vecchia). It was

generally understood that field theory was not useful in describing the strong interactions. In

order to model the results of these new hadron colliders, a scattering matrix (S matrix) was

created to represent the scattering process of the collision. At low energies, the experimental

results were known to be governed by an exchange of resonances in the direct channel. At high

energies, the experimental results were governed by the exchange of Regge poles in the

transverse channel. These were two completely different models for different operational

energies. Anyone could agree that having a model for each limiting case is troublesome; there

must be an overarching unification that each model can be derived from.

In 1968 an Italian theoretical physicist, Gabriele Veneziano, had a revelation in modeling

the scattering amplitudes of the strong interaction data. While observing Euler’s Beta function

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β (s , t )=Γ (−α (s )) Γ (−α(t ))

Γ (−α (s )−α(s))

where

Γ ( z )=∫0

∞

t z−1 e−t dt

with linearly rising Regge trajectories (Kiritsis)

α (s )=α 0+α ' s

Veneziano realized that these equations had the necessary features to explain the properties of

the strong force. His model was successful for all energies. In retrospect, this was the beginning

of string theory.

Physicists could understand how Euler’s Beta function described the strong force

mathematically but the underlying physical principles that drove the Beta function fit was not

understood. Leonard Susskind, a theoretical physicist at Stanford University, was considering

the implications of this Beta function on the strong nuclear force. He conceptually derived the

phenomenon that long strings that could wiggle and vibrate would be able to physically describe

the equations perfectly. However, Nature Physics refused to publish Susskind’s paper.

Physicists continued to describe microscopic particles as point particle building blocks of matter

instead of strings.

Soon thereafter discoveries were made showing that forces in nature could be described

as particles. The exchange of these messenger particles is what we see as a force between

objects. Physicists confirmed that these particles exist for strong, electromagnetic, and weak

forces where the mediating particles are gluons, photons, and vector bosons respectively. As

usual, there is no experimental evidence of a mediating particle for gravity. In this theory,

gravity is hypothesized to be mediated by a graviton, which has never been evidenced and is

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hard to incorporate into the theory. These mediating particles, combined with the discovery of

quarks, led to the development of the standard model of particle physics. This standard model

explains three fundamental forces but has an obvious omission of gravity.

Despite this obvious omission in the standard model, string theory was still not taken as a

serious alternative until the 1980’s. String theory had many problems, to include the existence of

a tachyon particle that must travel faster than the speed of light, the disturbing implications of the

required extra dimensions, and several anomalies in the equations. In 1984, John Schwarz, an

American theoretical physicist, and Michael Green, an English theorist, was able to show that

string theory is entirely self-consistent. They were able to get rid of all the anomalies in the

equations and essentially unified all four forces for a particular example. This was dubbed as

“the first theory of everything” and popularized string theory.

Each of the strings in these equations is unimaginably small. The different ways that

strings vibrate give particles their unique properties such as mass and charge. This theory of

everything is essentially a triumph of mathematics and is a fulfillment of Einstein’s dream of

uniting all of the fundamental forces. However, without any experiment or observation ever

being able to detect the existence of strings, many physicists are not too fond of string theory.

Nonetheless, string theory proposes some undeniable consequences, such as the creation of

multiple dimensions and even parallel universes. The next part of this paper discusses some

obvious questions: how can multiple dimensions exist, and why are they a necessary component

of string theory?

4. MULTIPLE DIMENSIONS

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This paper has already encountered one version of extra dimensions: Einstein’s five-

dimensional curved space-time in his general theory of relativity. Very few people, if any, can

envision the concept of extra dimensions. But if they exist, there needs to be some way to think

about the construct of multiple dimensions. One of the undeniable consequences of string theory

is that extra dimensions are necessary in order to satisfy the complex field equations. At this

point we are going to go on a tangent in order to explain the existence of theories that incorporate

extra dimensions and how they describe that extra dimension. We will then move to describing

the additional spatial dimensions of string theory, which requires at least six extra dimensions.

4.1 KALUZA-KLEIN THEORY

Gunnar Nordstrom, a Finnish theoretical physicist, was the first to suggest an additional

spatial dimension in 1914 in order to unify electromagnetism and general relativity. However,

Theodor Kaluza, a German mathematician and physicist, is credited with this discovery. The

basic idea is that an extra fifth dimension exists and five-dimensional space-time splits into what

is observed in Einstein’s general theory of relativity and Maxwell’s equations. Oskar Klein

proposed that this fourth spatial dimension is small and curled up to the point where we just can’t

see it. You can imagine this dimension to be curled into a circle with an extremely small radius.

A particle can thus move along this dimension and return to its starting location after a short

distance. This distance is called the size of the dimension. A decomposition from such a higher

dimensional space to a lower dimensional space can be seen in Figure 2.

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Figure 2: The M x C space is compacted to its prominent dimension after going through a Kaluza-Klein

decomposition.

In this model, electromagnetism is explained as a manifestation of the curvature of the

fourth spatial dimension, just as gravity is explained by Einstein’s general theory of relativity to

be a manifestation of the first three spatial dimensions. In the same way that gravity produces a

ripple in space-time, this allows for electromagnetism to exist as a ripple in the fourth spatial

dimension (Darling).

4.2 SPATIAL DIMENSIONS IN STRING THEORY

Similar to the Kaluza-Klein theory, the original string theory proposed that six extra

spatial dimensions exist in a 10-dimensional space-time. We are only able to observe four of

these dimensions because the other six spatial dimensions are curled up in a compact space. The

way strings are free to move and vibrate is defined by the geometries of these small dimensions

(Pierre). Strings are capable of having only certain vibrational modes due to these geometries

and each vibrational mode can be characterized by a set of quantum numbers (i.e. mass, spin,

etc.). Each mode is therefore defined by a unique set of quantum numbers, ultimately producing

what we know as a distinct fundamental particle. All fundamental particles, including the force

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mediating particles from the standard model (graviton included), can be modeled by a resonating

string in a higher dimensional space. Such vibrations ultimately produce what we see in nature,

such as protons and electrons.

Since string theory has been founded, several different theories have all stemmed off of

the same principle of strings. The reason for this split is due to many fundamental definitions.

As a theorist, you have the ability to say whether you want open strings or closed strings. You

can also produce theories in which no matter exists (only bosons exist). Most theories request

fermions to exist since fermions are what comprise matter. However, these theories require

supersymmetry to be true.

Supersymmetry requires that for every type of boson there exists a corresponding

fermion. These two particles are called superpartners and have the same mass and quantum

numbers except for a spin differing by half a unit. Supersymmetry is a central prediction of

string theory if matter is to exist. It is also important to note that supersymmetry is important in

solving the hierarchy problem in the standard model. Since supersymmetry is imperative to both

string theory and the standard model, there has been a large search to find evidence of

superpartners. Both CERN in Switzerland and Fermilab in Illinois have been monitoring high

speed collisions in hopes of finding evidence of supersymmetry. Theories that implement

supersymmetry are often referred to as superstring theory. String theory is generally used to

describe the entire field, so don’t be confused if the two terms are often used interchangeably –

superstring theory just omits the original bosonic theory, which is described in more detail later.

The existence of supersymmetry won’t prove that superstring theory is correct, but will

show evidence that we are on the correct path. Regardless, supersymmetry allows for the

derivation of various different superstring theories. There are five different theories in total.

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Some of these theories use open ended strings, some use closed loops, and some even have 26

dimensions instead of 10. Each of these theories is mathematically valid, but which one

describes our universe? These divisions are what have caused a lot of physicists to be reluctant

to enter the field of string theory. A solution to this problem did not seem possible, at least until

1995. Before we discuss this proposed solution, it is necessary to describe all the different

versions of string theory. And before we describe the different versions of string theory, it is

necessary to go into more depth on why exactly 10 or 26 dimensions are required for string

theory as well as what comprises a D-brane.

Strings, both open and closed, travel in a flat d-dimensional space-time. As strings move

through space-time, they sweep out a surface that is known as the string world-sheet, which has

one spatial dimension (σ) and one time dimension (τ). Such a world-sheet can be imagined as

seen in Figure 3.

Figure 3: A simplified view of a world-sheet for both open and closed strings moving through both time and space.

The boundary condition of a strings’ world-sheet is defined as

S= 14 πα '

∫ dσdτ √h[hmn ∂m Xa ∂n Xb ηab+α' R(2)Φ ]

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where hmn is the metric on the world-sheet, X(σ , τ ) is the strings coordinates, R(2) is the curvature

of the world-sheet, Φ is the dilation, and α is associated with the linearly rising Reggie

trajectories (previously discussed). Using this equation under conformal invariance, Euler’s Beta

function has a vanishing dilation field when d = 26. For superstring theories, superconformal

invariance is used to solve this equation instead and it is found that d = 10. Oscillations in this

space-time satisfy Lorentz invariance in all dimensions. Such possible oscillations can therefore

be classified by the space-time itself, producing the possibility for mass, spin, etc. If open strings

are used, massless oscillations with spin 1 are found. Closed strings are kindof like a product of

two open strings, producing a symmetric tensor with spin 2. A spin 2 mode resembles a

propagating gravitational field in general relativity. Because of this, most superstring theories

view closed strings as gravitons and the incorporation of gravity in this theory can be seen as a

type of quantum gravity.

If a string is open, it needs to have boundary conditions in order to define how its

endpoints interact with the manifold of space-time. There are two possibilities for such

conditions: Neumann or Dirichlet boundary conditions. Neumann boundary conditions specify

the values that the derivative of a solution has to be equal to at the boundaries of the domain.

Strings bound by Neumann conditions have one end bound to a manifold while the other end is

free to move about. No momentum leaves the string. Dirichlet boundary conditions, on the

other hand, specify the values that the solution has to be equal to at the boundaries of the domain.

Strings bound by Dirichlet conditions have both endpoints fixed to, but free to move about, the

same manifold. This manifold is known as a D-brane. Often a D-brane is referenced as a Dp-

brane where the p is equal to the number of spatial dimensions that the manifold consists of.

Figure 4 shows how strings can be bound to a D2-brane. In the 9 spatial dimensions of

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superstring theory, strings are only allowed to move on membranes up to a D9-brane. The value

of p in superstring theory can range from -1 to 9. When p = -1, all space and time coordinates

are fixed. This string is called an instanton because it only exists at that one place and time. A

D0-brane is called a D-particle because it is completely spatially fixed. In the same manner, a

D1-brane is called a D-string. All Dp-branes merely restrict all but p spatial dimensions to a

fixed value. D-branes are actually dynamic objects that fluctuate and have the ability to move

around. Such membranes are merely a construct of how strings are able to interact in their

higher dimensional spaces. Nevertheless, it is starting to become apparent that string theory has

many more complexities than just the existence of vibrating strings.

Figure 4: Strings moving on a D2-brane. The left one is Neumann-bound and the right one is Dirichlet-bound.

We are now able to discuss the different versions of string theory and understand why

they are different from each other. Here is a quick overview of all five theories plus the original

Bosonic theory (Schwarz):

1. The original bosonic theory included only bosons and no fermions. This means that

only forces could exist with no matter. Strings are both open and closed. This theory

posed major flaws concerning no mass and the need for a particle called the tachyon

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that can travel faster than the speed of light. In order to solve the field equations

under conformal invariance, there are 26 space-time dimensions.

2. Type I string theory assumes supersymmetry between forces and matter. It assumes

both open and closed strings exist. There are 10 space-time dimensions.

3. Type IIA string theory assumes supersymmetry between forces and matter. Only

closed strings exist, which are bound to D-branes. Massless fermions are non-chiral,

meaning a fermion is the same as its mirror image. Group symmetry is SO(32).

There are 10 space-time dimensions.

4. Type IIB string theory assumes supersymmetry between forces and matter. Only

closed strings exist, which are bound to D-branes. Massless fermions are chiral.

There are 10 space-time dimensions

5. Type Herotic-O string theory assumes supersymmetry between forces and matter.

Only closed strings exist. Group symmetry is SO(32). There are 10 space-time

dimensions.

6. Type Herotic-E string theory assumes supersymmetry between forces and matter.

Only closed strings exist. Group symmetry is E8xE8. There are 10 space-time

dimensions.

From this, we can see six distinct theories. Only five could describe our universe,

though, because the bosonic theory does not incorporate particles with mass. But when you take

the 10 space-time dimensions of any theory and try to compact them down to where only our

four space-time dimensions have appreciable magnitude, the number of theories grows even

larger because there are many ways you can mathematically justify this compactification. We

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can now appreciate the need to only have one superstring theory. So, how did promise for the

existence of a unified superstring theory develop?

In 1995 string theorists gathered for an annual conference at the University of Southern

California. Edward Witten, a theoretical physicist from the Institute for Advanced Study in

Princeton, came to the conference and astounded everyone there. Prior to the conference, Witten

was attempting to get rid of some of the five string theories and wound up providing a

completely new perspective on string theory. He argued that there was really just five ways of

looking at the same theory. This new perspective has become known as M-theory.

5. M-THEORY

Since the string theory conference in 1995, mathematicians have been able to show how

the existing superstring theories are related. The way this is possible is by performing a

transformation on one theory that results in another theory. These transformations are known as

dualities and the two related theories are said to be dual to one another. Essentially all five

superstring theories are expected to be special limits of a more fundamental theory. This more

fundamental theory, which is still unknown, has become known as M-theory. We will discuss

the two known dualities (T- and S-duality) and give an explanation on what string theorists

expect M-theory to be (Schwarz).

One possible duality symmetry obscures large and small distance scales. This is known

as T-duality, a process of compacting some of the extra space-time dimensions of superstring

theory. Let’s assume one of the 9 spatial dimensions is a circle of radius R. Therefore, traveling

a distance of L=2 πR will result in traversing the entire dimension and returning to the initial

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location. A string has two possibilities: 1) it can travel around this circle with a quantized

momentum that contributes to the energy of the string; or, 2) it can wrap around the circle in a

quantized manner called the wrapping number. These momentum and winding modes can be

interchanged through transformations so long as the radius R of the circle is replaced with Lst2 /R

where Lst is the length of the string. When this is done, a small distance scale R can be replaced

with a large distance scale if Lst ≫R. This duality relates Type IIA and Type IIB superstring

theories as well as Heterotic-O and Heterotic-E superstring theories. So, if you were to take

Type IIA and Type IIB theories, compact one spatial dimension on each, transform the

momentum and winding modes, and finally expand the spatial dimension, you switch the two

theories!

The other transformation is known as S-duality. This duality revolves around the

coupling constant of string theory. Every force has a coupling constant – for the gravitational

force the coupling constant is Newton’s constant and for electromagnetism the coupling constant

is proportional to the square of the electric charge. All forces act the same with different

coupling constants; the only difference is the strength of the force. String theory also has a

coupling constant that is related to the oscillation modes of the string, called the dilaton. If a

dilaton field is replaced with the opposite of itself, this exchanges a large coupling force with a

small one. If a string theory with a strong coupling constant is the same as another theory with a

small coupling constant, then they are related by S-duality. Therefore, if we understand the weak

theory, then we can equivalently understand the strong theory. Superstring theories related by S-

duality are Type I theory with Heterotic-O theory and Type IIB theory with itself.

We now have two conclusions: the relation between large and small distances is fluidic,

changing based on the method of measuring distance; and the strong coupling limit of a string

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theory can describe the weak coupling limit of a different theory. These conclusions could be

considered out of the ordinary, but if we reconsider Einstein’s theory of gravity, we realize that

gravity measures how the size of objects and interaction magnitudes are measured in space-time.

With this, T- and S- duality can be seen as a reasonable extension for a quantum theory of

gravity.

M-theory, in its simplest explanation, describes current attempts to find a transformation

that can finally derive a more fundamental theory such that the known superstring theories are

merely special limits. In essence, M-theory proposes the possibility of expanding closed strings

in a compact circles into a membrane with a topology of a torus. This creates an eleventh

dimension in M-theory in which there are hopes for such a transformation to exist. However,

this involves expanding the concept of D-branes into more general p-branes and M-branes. It is

important to note that M-theory has yet to be derived and most explanations of M-theory are

merely speculations of what we expect for the composition of M-theory. Because these are

merely speculations, we will not go into the details of M-theory in this paper. Rather, it is

sufficient, but not quite satisfying, to know that there are large hopes that we will soon have a

grand Theory of Everything.

6. CONCLUSION

Going back to ancient philosophers, humans have been attempting to understand our

surroundings. What are objects comprised of? How do we describe the forces between objects?

Is there a theory that can explain everything? Surprisingly, we are still answering these same

questions today. We have come a long way in understanding the composition of objects on a

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subatomic scale and understanding the four fundamental forces that govern our universe:

electromagnetism, strong and weak forces, and gravity. Today, we find ourselves understanding

modern physics through two theories that cannot be combined. General relativity describes

interactions between large masses over large distances while quantum mechanics describes

interactions between small particles on subatomic scales. String theory is considered to be a

potential unification theory for all the fundamental forces, thus allowing it to be called (some

would argue arrogantly called) “The Theory of Everything”. Whether or not string theory is the

basis of everything as we know it is an unanswered question, and it may remain unanswered for

a long time. There are several flaws in superstring theories, but while theorists and

mathematicians are working out the kinks, we can continue to believe that there is a promising

theory out there that unifies everything together, known as the Theory of M.

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BIBLIOGRAPHY

Brown, Julian. Superstrings - A Theory of Everything? New York: Cambridge University Press,

1988.

Darling, David. The Internet Encyclopedia of Science. n.d. 16 April 2011

<http://www.daviddarling.info/encyclopedia/K/Kaluza-Klein_theory.html>.

Kiritsis, Elias. String Theory in a Nutshell. Princeton: Princeton University Press, 2007.

NOVA's The Elegant Universe. Dir. Brian Greene. 2003.

Pierre, J. SUPERSTRINGS! Tutorial. n.d. 9 April 2011

<http://www.sukidog.com/jpierre/strings/tutor.htm>.

Schwarz, Patricia. The Official String Theory Web Site. n.d. 8 April 2011

<http://www.superstringtheory.com>.

Vecchia, P. Di. "The Birth of String Theory." Lecture Notes in Physics (2008): 59-118.

Weisstein, Eric. Wolfram Research. 2007. 20 April 2011

<http://scienceworld.wolfram.com/physics/SchwarzschildRadius.html>.

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