The Development of string theory
Kyle r. Ulrich
Duke University
Department of electrical and computer engineering
physics 105 – astrophysics
25 april 2011
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 and that obeyed Newton’s Universal Law of Gravitation
where is the distance between the two objects and 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 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
Such invariance implies that the distance ds is invariant when the path is rotated is space. Einstein included time in this metric, such that
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 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
where is a tensor that determines the curvature of space and 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.
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):
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 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 gluons[1] 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:
where the Hamiltonian operator H is
and is the position of each particle, V is the time-independent potential energy, and is the Laplace operator, which is
in a Cartesian three-dimensional space. If we consider the Schrodinger equation for two particles, the wave equation will satisfy
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.
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
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 theoriesproduces 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 dimensions[2] and even the possibility of parallel universes. You can imagine the amount of resistance this theory has encountered due to these 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
where
with linearly rising Regge trajectories (Kiritsis)
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 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
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.