String Theory

STRING THEORY arises from attempts to answer: 1) What is the world made of? 2) How & Why does the world change in time?

Answers: 1) MATTER! E.g. Cars, Houses, Buildings, Planets… All matter has Mass

2) Forms of matter exert Forces on each other and they move under the influence of these forces

F = ma For example: the attraction which causes an apple to fall off a tree.

GMm F= 2 r

F = ma So far: 2 Questions 2 Answers MATTER ; FORCE

Matter comes in millions of different forms: Apples, Oranges… Cars - Hondas BMWs... Garden Hoses, Grass, Trees... Earth, Sun, Moon… Galaxies, Planets, Stars...

All this diversity in fact comes from only a few hundred atoms: Hydrogen, Oxygen, Nitrogen...

• Each atom has a definite mass. • Each atom has an approximate size of about:

1 10 m = m 10000000000 −10

• By going from the large to the tiny, we reduce the diversity of millions to around a hundred! • Things simplify further if we look deeper into the atom: Protons:

+

Neutrons: Electrons:

Electrons orbit the nucleus which is composed of Protons and Neutrons

-

• The few hundred atoms reduce to only a handful of fundamental objects • Our Millions of Matter objects have been reduced to just 3 - Great!

Unfortunately, the story starts to get more complicated:

Proton & Neutron +

Quarks

Colours

up down

red

strange charm

blue

bottom top

green

• There are 3 ‘generations’ of these quarks & each quark can have one of 3 different ‘colours’ • Likewise there are 3 ‘generations’ of electron-like objects called leptons:

Leptons

e electron

ν e electron-neutrino

µ muon

ν µ muon-neutrino

τ

ντ

tau tau-neutrino

The idea of String Theory is that this diversity of quarks and leptons comes from 1 String

H

O p

e n

udsctb

1 String

eµτ

What about Forces? “Hammer Force”

“Fuel Force”

“Astronomical Force”

All this diversity of forces comes from: Gravity & Electromagnetism (& its 2 cousins)

The Physics of particles is governed by “Quantum Mechanics” -Which brings about a Radical Change in our concepts of space. E.g. Cannot say electron is here and going nowhere. Every position is assigned a number telling us how likely the electron is to be at that point.

x

Particle physics reduces both Matter & Force to one kind of Entity i.e. particles. electrons Matter protons photons Forces gravitons Force of Gravity

NEED OF STRING THEORY 

The standard model was designed within a framework known as Quantum Field Theory (QFT), which gives us the tools to build theories consistent both with quantum mechanics and the special theory of relativity .



With these tools, theories were built which describe with great success three of the four known interactions in Nature: Electromagnetism, and the Strong and Weak nuclear forces.



But unfortunately the fourth interaction, gravity, beautifully described by Einstein's General Relativity (GR), does not seem to fit into this scheme.



Whenever one tries to apply the rules of QFT to GR one gets results which make no sense.



For instance, the force between two gravitons (the particles that mediate gravitational interactions), becomes infinite and we do not know how to get rid of these infinities to get physically sensible results.



In string theory , the known elementary particles are no longer described as dimensionless mathematical point objects but rather as extended one dimensional objects.



The size of the individual strings is so small that any experiment ,we could possibly perform on an elementary particle would not reveal its string like nature—it would look just like the point particle we expect.



In String Theory, the myriad of particle types is replaced by a single fundamental building block, a `string'. These strings can be closed, like loops, or open, like a hair.



the string is free to vibrate, and different vibrational modes of the string represent the different particle types, since different modes are seen as different masses or spins.



One mode of vibration, or `note', makes the string appear as an electron, another as a photon .



There is even a mode describing the graviton, the particle carrying the force of gravity, which is an important reason why String Theory has received so much attention.



So, the first great achievement of String Theory was to give a consistent theory of quantum gravity, which resembles GR at macroscopic distances.



Moreover String Theory also possesses the necessary degrees of freedom to describe the other interactions! At this point a great hope was created that String Theory would be able to unify all the known forces and particles together into a single `Theory of Everything' .

ORIGIN 

Work on string theory is made difficult by the very complex mathematics involved, and the large number of forms that the theories can take depending on the arrangement of space and energy.



String theory was originally developed and explored during the late 1960s and early 1970s to explain some peculiarities of the behavior of hadrons (subatomic particles such as the proton and neutron which experience the strong nuclear force). In particular, Yoichiro Nambu (and later Lenny Susskind and Holger Nielsen) realized in 1970 that the dual resonance model of strong interactions could be explained by a quantum-mechanical model of strings.



In the mid 1990s, Joseph Polchinski discovered that the theory requires the inclusion of higher-dimensional objects, called Dbranes. These added an additional rich mathematical structure to the theory, and opened up many possibilities for constructing realistic cosmological models in the theory.



In 1995, at the annual conference of string theorists at the University of Southern California (USC), Edward Witten gave his famous speech on string theory that essentially united the five string theories that existed at the time, and giving birth to a new 11-dimensional theory called M-theory. This sparked the second superstring revolution.



Recently, the discovery of the string theory landscape, which suggests that string theory has an exponentially large number of different vacua, led to discussions of what string theory might eventually be expected to predict, and to the worry that the answer might continue to be nothing.

Deep inside proton: In addition to Electricity/Magnetism we need to know about Strong & Weak force.

The mediators of strong forces are gluons.

Forces

E.M. Strong Weak

γ,g

photons & gluons

But gravity does not fit nicely into the Quantum Mechanical world. Applying Q.M. to gravitation gives infinities.

String Theory combines all forces Strong,Weak, Gravitation, Electromagnetism

& all matter electrons,protons, neutrons…

into one object… …a Matter e-, p+, n

Force γ, gluon, graviton

How does a string accomplish this? •All particles have energy. In classical physics: I.

p2 K .E = 2m

;

also Rest energy

 2 4 p2  E = m c + + ... = 2m  

p 2c 2 + m 2c 4

E = mc 2

II. Photons have frequency (related to colour of light) E = hf Planck’s constant

•Strings can vibrate in different ways

For example: Guitar string Different vibration

Different sound!

•Fundamental string Different vibration

electron

photon

Different particles!

graviton

MATHEMATICAL PROOF 

Pythagoras could be called the first known string theorist .



Pythagoras realized that vibrating Lyre strings of equal tensions but different lengths would produce harmonious notes if the ratio of the lengths of the two strings were a whole number.



According to himwave equation for a string with a tension T and a mass per unit length m. If the string is described in coordinates as in the drawing below, where x is the distance along the string and y is the height of the string, as the string oscillates in time t,

then the equation of motion is the onedimensional wave equation

where vw is the wave velocity along the string. When solving the equations of motion, we need to know the "boundary conditions" of the string. Let's suppose that the string is fixed at each end and has an unstretched length L. The general solution to this equation can be written as a sum of "normal modes", here labeled by the integer n, such that

According to Einstein's theory, a relativistic equation has to use coordinates that have the proper Lorentz transformation properties. But then we have a problem, because a string oscillates in space and time, and as it oscillates, it sweeps out a two-dimensional surface in spacetime that we call a world sheet (compared with the world line of a particle). In the nonrelativistic string, there was a clear difference between the space coordinate along the string, and the time coordinate. But in a relativistic string theory, we wind up having to consider the world sheet of the string as a twodimensional spacetime of its own, where the division between space and time depends upon the observer. The classical equation can be written as

The general solution to the relativistic string equations of motion looks very similar to the classical nonrelativistic case above. The transverse space coordinates can be expanded in normal modes as: The string solution above is unlike a guitar string in that it isn't tied down at either end and so travels freely through spacetime as it oscillates. The string above is an open string, with ends that are floppy. For a closed string, the boundary conditions are periodic, and the resulting oscillating solution looks like two open string oscillations moving in the opposite direction around the string. These two types of closed string modes are called right-movers and leftmovers, and this difference will be important later in the supersymmetric heterotic string theory

WHY NO EXPERIMENTAL PROOF??? Work

on string theory is made difficult by the very complex mathematics involved, and the large number of forms that the theories can take depending on the arrangement of space and energy.

TYPES OF STRING THEORY 

At one time, string theorists believed there were five distinct superstring theories: type I, types IIA and IIB, and heterotic SO(32) and E8XE8 string theories.



The thinking was that out of these five candidate theories, only one was the actual correct Theory of Everything, and that theory was the theory whose low energy limit, with ten dimensions spacetime compactified down to four, matched the physics observed in our world today. But now it is known that this naive picture was wrong, and that the the five superstring theories are connected to one another as if they are each a special case of some more fundamental theory, of which there is only one



Type

Open (plus closed) Open (plus closed)

Closed

Closed

Orie nted ?

Details

Yes

Scalar tachyon, massless antisymmetric tensor, graviton and dilaton

No

Scalar tachyon, massless graviton and dilaton

Yes

Scalar tachyon, massless vector boson, antisymmetric tensor, graviton and dilaton

No

Scalar tachyon, massless graviton and dilaton

•In ordinary circumstances, we can understand the world by applying different forces according to the problem at hand. •A unified Theory of matter & forces is not just pretty - it is necessary for us to understand: 1. Physics of the Early Universe 2. Black Holes

E.g. •Universe is expanding •Every galaxy is facing away from every other

Going back in time, • All matter was crunched up in a very energetic state within a short distance. • Under these conditions all forces were equally important. • Hence String Theory becomes necessary for us to understand what − 42 10 s after the happened at Big Bang.

•And it makes a startling prediction:

The world contains… …Extra Dimensions!

M-THEORY 

Apart from the fact that instead of one there are five different, healthy theories of strings (three superstrings and two heterotic strings) there was another difficulty in studying these theories: we did not have tools to explore the theory over all possible values of the parameters in the theory.



Each theory was like a large planet of which we only knew a small island somewhere on the planet. But over the last four years, techniques were developed to explore the theories more thoroughly, in other words, to travel around the seas in each of those planets and find new islands.

And only then it was realized that those five string theories are actually islands on the same planet, not different ones! Thus there is an underlying theory of which all string theories are only different aspects. This was called M-theory. The M might stand for Mother of all theories or Mystery, because the planet we call M-theory is still largely unexplored. 

There is still a third possibility for the M in M-theory. One of the islands that was found on the M-theory planet corresponds to a theory that lives not in 10 but in 11 dimensions. This seems to be telling us that M-theory should be viewed as an 11 dimensional theory that looks 10 dimensional at some points in its space of parameters. Such a theory could have as a fundamental object a Membrane, as opposed to a string. Like a drinking straw seen at a distance, the membranes would look like strings when we curl the 11th dimension into a small circle. 

Black Hole  Recently,

a remarkable idea called holography has gained prominence in string theory .  It says that a system containing gravity can be described by degrees of freedom living on its boundary rather than in the bulk.  black holes are extreme geometrical objects with fascinating mathematical properties that have posed serious challenges to the foundations of classical and quantum physics.

What is black hole according to string theory  In

string theory at large distance scales, solutions to the Einstein equation are only modified by very small corrections.

 Two

important thermodynamic quantities are temperature and entropy related to temperature and entropy .

.

Temperature was revealed to be calculable from the average kinetic energy of a system of identical particles, and entropy was understood in terms of the number of quantum states available to the particles in that system

For an ideal gas, this quantity can be calculated from basic quantum principles to be :

The Bekenstein-Hawking entropy of a black hole is one fourth of the area of the event horizon (in units where Planck's constant=GN=1). Until string theory, there was no clear idea how this task could be accomplished. String theory has provided at least a partial answer to this question in terms of Dbranes.

Bearing that in mind, let's start with the simplest charged black p-brane solution known, which is a charged black hole in four spacetime dimensions, described by the metric

If the charge and mass are equal in magnitude (in units where c=GN=1) then we have an extreme black hole, with area 4pQ2, and therefore with entropy pQ2.

This extreme black hole is a special object because when M=Q, a condition for unbroken supersymmetry is satisfied that is called the BPS condition. This BPS condition results in the cancellation of quantum corrections to the effective action for string theory, so that precise answers can be found by simple calculations at lowest order in perturbation theory The above black hole can be thought of as a zero-brane. In the previous section we learned that string theories contain objects called p-branes and D-branes. A natural generalization of a black hole is a black p-brane. And there are also BPS black p-branes .