FUNDAMENTALS OF STRING THEORY

-By Dibyasree Nandy Department of Instrumentation

ScienceJadavpur University

The Fundamental ForcesFour fundamental forces have been recognized to exist in nature:1) Gravitational force-First described accurately by Sir Isaac Newton. Later,

it underwent a profound reformulation in Albert Einstein’s theory of General Relativity, in which gravitational forces arise from the curvature of the spacetime arena of Special Relativity. However, Einstein’s General Relativity is a classical theory of gravitation, not a quantum one.

2) Electromagnetic force-Described by Maxwell’s equations. The E.M. theory is formulated as a classical theory of electromagnetic fields, which is fully consistent with Special Relativity.

3) Weak force-Responsible for the process of nuclear beta decay, in which a neutron decays into a proton, an electron and an anti-neutrino. In general, processes that involve neutrinos are mediated by weak forces.

4) Strong force-Nowadays known as the Colour Force. This force is at play in holding together the constituents of sub-nuclear particles. These constituents are known as quarks. Quarks are held together so tightly by the strong force, that they cannot be seen in isolation.

Unification of interactionsThe Weinberg-Salam Model of electroweak interactions (late 1960s) put

together electromagnetism and the weak force into a unified framework. This unified model was necessary for a predictive and consistent theory of the weak interactions. The theory was initially formulated with 4 massless particles that carry the force: W+, W-, Z0 and the photon. A process of symmetry-breaking gave mass to the 1st 3. These 3 particles are the carriers of the weak force. The photon is the carrier of the E.M. force.

Quantization procedure has been carried out in classical electrodynamics to explain phenomena at the microscopic level. The resulting theory is called Quantum Electrodynamics (QED), which is the quantum version of classical electrodynamics, in which the photon appears as the quantum of the E.M. field. Quantization procedure is also successful in the case of the strong colour force, and the resulting theory has been called Quantum Chromodynamics (QCD). The carriers of the strong force are 8 massless particles called gluons.

The Electroweak Theory along with QCD form the Standard Model of Particle Physics. But there has been no unification of the weak force and the strong force.

Standard Model of Particle Physics

Number of particles in the Standard Model: 8 gluons, W+, W-, Z0, photon- together these add up to 12 bosons. The electron, muon, tauon, electron-neutrino, muon-neutrino, tauon-

neutrino, and the antiparticles of each of these 6 leptons-together these add up to 12 leptons.

6 quarks:u,d,c,s,t,b, each having 3 colours, adding upto 18 quarks, the antiparticles of each of the 18 quarks-forming a total of 36 quarks.

The12 leptons and the 36 quarks form a total of 48 fermions. Therefore, the total number of particles described by this model = 60

( 12 bosons+48 fermions).Despite the large number of particles it describes, the Standard Model is very

powerful. However, it has 2 significant shortcomings:1) It does not incorporate gravity.2) It has about 20 parameters that cannot be calculated within its

framework.A more complete version of the Standard Model has to include

supersymmetry.

SupersymmetryAll elementary particles fall into 2 categories-bosons and fermions.

The amount of angular momentum they carry is commonly described as spin. A boson has an integer spin, whereas a fermion has a half-integer spin. Supersymmetry is a symmetry that is a means of amalgamating these two kinds of particles into a common description. According to this symmetry, for each boson, there has to be a corresponding fermion. Since all matter particles are fermions and all force carriers are bosons, this symmetry unifies matter and forces.

String theory as a unified theory of Physics

String theory is an excellent candidate for a unified theory of all forces in nature. It is a rather impressive prototype of a complete theory of physics, in which all forces and all particles are unified. Moreover, it include gravitation, and may be viewed as a quantum theory of gravity.

In string theory, each particle is identified as a particular vibrational mode of an elementary microscopic string. Just as a violin string can vibrate in different modes and each mode corresponds to a different sound, the modes of vibration of a fundamental string can be recognized as the different particles we know. One of the vibrational states of strings is the Graviton, the quantum of the gravitational field. Since there is just one type of string, and all particles arise from string vibrations, all particles are naturally incorporated into a single theory.

String theory is rather unique since it does not have adjustable dimensionless parameters, unlike the Standard Model. It has only one parameter, the string length. Another sign of the uniqueness of this theory is the fact that the dimensionality of spacetime is fixed. Our physical spacetime is 4-dimensional. In the Standard Model, this information is used to build the theory, it is not derived. I n string theory, however, the number of spacetime dimensions emerges from a calculation. The answer is not four, but rather, ten.

The Early days of String TheoryThe String theory has a very bizarre history. The subject originated

in an attempt to solve a completely different problem from that which it’s being used for today. The string theory was originally developed, around 1968-70, in an attempt to understand the strong nuclear force. Earlier, strings were thought to be a description of the force that was holding the quarks together, like bits of elastic.

Bosonic string theory and Superstring theory

The Bosonic string theory was the original version of the string theory,with 26 spacetime dimensions, but it fell short as a viable physical model since it predicted only the existence of bosons.

The superstring theory, also called the supersymmetric string theory, is the version of string theory that incorporates fermions and supersymmetry. The fundamental constituents of this theory are strings of the Planck length (=10^-33 cm). Each string has a harmonic. Different harmonics determine the different fundamental particles.

BranesIn string theory, a brane is a physical object that generalizes the

notion of a point particle to higher dimensions. For eg, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. The word ‘brane’ is derived from the word ‘membrane’.

Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. A p-brane sweeps out a (p+1) dimensional volume in spacetime, called its worldvolume.

In string theory, a string may be open ( forming a segment with 2 endpoints) or closed ( forming a closed loop). D-branes ( the letter ‘D’ refers to the Dirichlet boundary condition) are an important class of branes that arise when one considers open strings. As an open string propagates through spacetime, its endpoints are required to lie on a D-brane.

Cont…….It’s like living on the computer screen and being able to move only

on that two-dimensional surface. The computer itself exists in three space dimensions, but we can move only on a two-dimensional subspace made by the screen, so the spacetime experienced by us would look like three ( two of space and one of time) rather than four. That’s the sort of idea that builds up the braneworld higher dimensional theory. Our observed four dimensional spacetime is like a subspace of some bigger space that we cannot see because all matter and forces are constrained to move mainly on our subspace, or brane.

Kaluza-Klein TheoryThe Kaluza-Klein theory of gravitation and electromagnetism built around

the idea of a fifth dimension beyond the usual four of space and time. It is considered to be an important precursor to string theory.

Kaluza and Klein proposed that you could have one extra spatial dimension, but if that dimension was curled up into a circle with a very small radius, smaller than any other scale that was around, you would not notice it. Kaluza and Klein showed, using general relativity, that this extra dimension would still have an effect on the space around us. In particular,they showed that the effect of gravity in that very small fifth dimension would actually appear to us, from our larger-scale perspective, as electromagnetism.

This theory had many fascinating consequences. The radius of this extra dimension would be related to the electric charge of particles. A charged particle would move in one direction in the fifth dimension, and an oppositely charged particle would move in the other. And as the momentum of the 2 oppositely charged particles would cancel out when they collide, they would no longer move in the fifth dimension, making their combined charge neutral.

Kaluza-Klein Compactification and T-Duality

The original Kaluza-Klein theory spoke of an extra fifth dimension having circular topology. This property was referred to as cylindricity. In string theory, the Kaluza-Klein compactification of the extra dimensions has one important difference from the original theory from the 1920s. A closed string can get wound several times around a rolled-up dimension. When a string does this, the string oscillations have a winding mode as well as momentum. The winding modes add a symmetry to the theory that was not present in the original version. This symmetry is called T duality. T-duality is an equivalence of 2 physical theories. It basically relates 2 theories having different spacetime geometries. A theory with a rolled-up dimension of radius R turns out to be equivalent to a theory with a rolled-up dimension of radius (L^2)/R, where L is the string length. Because of this duality, momentum in one description takes discrete values and is equal to the number of times the string winds around the rolled-up extra dimension.

Calabi-Yau manifoldThe Calabi-Yau manifold is a special kind of topological space which is used to describe the manner in which the extra spacetime dimensions are curled up.

String theory in cosmologyIn the T duality, when the extra dimension radius R satisfies the

condition R=string length L, the string theory has an enhanced symmetry with some massive particles becoming massless. This is called the self dual point. This duality symmetry has led to a proposal for pre-Big Bang cosmology where the universe starts out flat, cold and very large instead of curved, hot and very small. This early universe is unstable and starts to collapse and contract until it reaches the self dual point, where it heats up and starts to expand to give the expanding Universe we observe today.

Inflation versus the Giant Brane Collision

Cosmic inflation is the exponential expansion of space in the early universe. A current alternative model to inflation is the giant brane collision or the Big Splat. This model starts out with a static 5-dimensional spacetime that is close to being perfectly supersymmetric. The 4 space dimensions are bounded by two 3-dimensional walls or three branes, and one of these 3-dimensional walls make up the space we live on. The other brane is hidden from our perception. According to this theory, there is a third 3-dimensional brane loose between the 2 bounding branes of the 4-dimensional bulk, and when this brane collides with the brane we live on, the energy from the collision heats up our brane and the Big Bang occurs in our visible universe. This proposal is quite new, and it remains to be seen whether it will survive careful scrutiny.

ConclusionFrom the previous slides, we can conclude that each point in space,

or what we thought was a point in space, is, in fact, a tiny six-dimensional ball about 10^-33 cm across, which are too small to be detected. Even the most sophisticated particle accelerators in use today, have been unable to penetrate distances less than 10^-16cm.

So we no longer think of the world as made up of particles at all, but as made up of little strings that wriggle about.

Bibliography Wikipedia A First Course in String Theory-Barton Zwiebach The Grand Design- Stephen Hawking and Leonard Mlodinow Superstrings-The Theory of Everything?-Edited by Davies and

Brown

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