The String Theory

5/24/2018 The String Theory

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I ts the 21st centu ry! Time tofeed your mind

BasicsSo what is string theory? For that matter, what theheck are elementary particles? If this all soundstotally confusing, try this section first.

ExperimentWhat progress are physicists making towardsexperimental tests of string theory predictions?

MathematicsWhat kinds of math do string theorists use andwhy? And how has string theory changedmathematics?

Black HolesPersonal safety issues aside, when black holes aretied up in strings, they get even more interesting.

Cosmology

Was there a String Bang before the Big Bang, ordid the Universe simply unwind?

History

Find out how string theory outlasted the VietnamWar, Mrs. Thatcher and grunge music, in ourTimeline section

PeopleSo who are the people who work on stringtheory? Check them out in our People section.

TheatreNow playing in the String Theatre: a Real Audiophysics colloquium by Prof. John Schwarz.

BookstoreLooking for books on string theory or other topicsin modern theoretical physics?

BlogDiscuss string theory and alternative theories ofquantum gravity in our new blog.

It all started when Isaac Newton invented calculus to describe the motions of falling objects and orbitingplanets...basic/advanced

Here are the past successes of theoretical physics, from electromagnetism to particle physics and generalrelativity...basic/advanced

Why is it that the theory that works for describing gravity is so poorly compatible with the theory thatworks for describing elementary particles?basic/advanced

So what is string theorymade of? How does a string theory differ from a particle theory? Get the scoophere!basic/advanced

Let us count the ways...basic/advanced

What is the meaning of duality and what does this tell us about the relationships between string theories?basic/advanced
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If different string theories are related, then maybe they represent different limits of a bigger, morefundamental theory...basic/advanced

Theoretical physicists today still use a core technology that was developed in the 18 thcentury out of the

calculus pioneered by Isaac Newton and Gottfried von Leibniz.

Isaac Newton derived his three Laws of Motion through close, almost obsessive observation andexperimentation, as well as mathematical reasoning. The relationship he discovered between force andacceleration, which he expressed in his own arcane notation of fluxions, has had the most impact on theworld in the differential notation used by his professional rival, Wilhelm von Leibniz, as the familiardifferential equationfrom freshman physics:

2

2

d xF ma m

dt

After Newton accused Leibniz of plagiarism in the discovery of calculus, Leibnizvastly more convenientand intuitive differential and integral notation failed to become popular in England, and so the majority ofadvances in the development of calculus in the next century took place in France and Germany.

At the University of Basel, the multitalented Leonhard Euler began to develop the calculus of variationsthat was to become the most important tool in the tool kit of the theoretical physicist. The calculus ofvariations was useful for finding curves that were the maximal or minimal length given some set ofconditions.

Joseph-Louis Lagrange took Eulers results and applied them to Newtonian mechanics. The general

principle that emerged from the work of Euler and Lagrange is now called the Principle of Least Action,which could be called the core technology of modern theoretical physics.

In the Principal of Least Action, the differential equations of motion of a given physical system arederived by minimizing the actionof the system in question. For a finite system of objects, the action Sis

an integral over time of a function called the Lagrange function or Lagrangian ,L q dq dt , which

depends on the set of generalized coordinates and velocities ,q dq dt of the system in question.

,f

i

t

tS L q q dt

The differential equations that describe the motion of the system are found by demanding that the actionbe at its minimum (or maximum) value, where the functional differential of the action vanishes:

0S

This condition gives rise to the Euler-Lagrange equations

0, 1,...,n n

d L Ln N

dt q q

which, when applied to the Lagrangian of the system in question, gives the equations of motion for thesystem.
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As an example, take the system of a single massive particle with space coordinate x (in zero gravity). TheLagrangian is just the kinetic energy, and the action is the energy integrated over time:

21

,2

L x x mx

The Euler-Lagrange equations that minimize the action just reproduce Newtons equation of motion for a

free particle with no external forces:

0d L L d

mxdt x x dt

The set of mathematical methods described above are collectively known as the Lagrangian formalismof mechanics. In 1834, Dublin mathematician William Rowan Hamilton applied his work oncharacteristic functions in optics to Newtonian mechanics, and what is now called the Hamiltonianformalism of mechanicswas born.

The idea that Hamilton borrowed from optics was the concept of a function whose value remains constantalong any path in the configuration space of the system, unless the final and initial points are varied. Thisfunction in mechanics is now called the Hamiltonianand represents the total energy of the system. TheHamiltonian formalism is related to the Lagrangian formalism by a transformation, called a Legendre

transformation, from coordinates and velocities ,q dq dt to coordinates and momenta ,q p :

1

,N

n n nnn

LP H p q L

q

The equations of motions are derived from the Hamiltonian through the Hamiltonian equivalent of theEuler-Lagrange equations:

,n nn n

H Hp q

q p

For a massive particle in zero gravity moving in one dimension, the Hamiltonian is just the kineticenergy, which in terms of momentum, not velocity, is just:

2

2

pH

m

If the coordinate q is just the position of the particle along the x axis then the equations of motionbecome:

, 0p d

x p mxm dt

which is equivalent to the answer derived from the Lagrangian formalism.

Classical mechanics would have had a brief history if only the motion of finite objects such as

cannonballs and planets could be studied. But the Lagrangian formalism and the method of differentialequations proved well adaptable to the study of continuous media, including the flows of fluids andvibrations of continuous n-dimensional objects such as one-dimensional strings and two-dimensionalmembranes.


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The Lagrangian formalism is extended to continuous systems by the use of a Lagrangian densityintegrated over time and the D-dimensional spatial volume of the system, instead of a Lagrange functionintegrated just over time. The generalized coordinates q are now the fields q(x) distributed over space,and we have made a transition from classical mechanics to classical field theory. The action is nowwritten:

1

,

a D

a

q

S q x d xx

S

Here the coordinate ax refers to both time and space, and repetition implies a sum over all D + 1dimensions of space and time.

For continuous media the Euler-Lagrange equations become

0a

ax q q

S S

with functional differentiation of the Lagrange density replacing ordinary differentiation of the Lagrangefunction.

What is the meaning of the abstract symbol q(x)? This type of function in physics that depends on spaceand time is called a field, and the physics of fields is called, of course, field theory.

The first important classical field theorywas Newtons Law of Gravitation, where the gravitational forcebetween two particles of masses m1and m2can be written as:

1212 1 2 12 1 2 12 12

12

, ,r

F r x x rNG m m r r

The gravitation force F can be seen as deriving from a gravitational field G, which if we set x1 = 0and x2= x, can be written as:

2 12

, ,r

F G r Nk

m G k m Gr

Newtons Law of Gravitation was the beginning of classical field theory. But the greatest achievement ofclassical field theory came 200 years later and gave birth to the modern era of telecommunications.

Physicists and mathematicians in the 19thcentury were intensely occupied with understanding electricityand magnetism. In the late 19thcentury, James Clerk Maxwell found unified equations of motion of theelectric and magnetic fields, now known as Maxwells equations. The Maxwell equations in the absenceof any charges or currents are:

10

BE E

c t

10

EB B

c t

Maxwell discovered that there exist electromagnetic traveling wave solutions to these equations, whichcan be rewritten as


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2 22 2

2 2 2 2

1 1,

B EB E

c t c t

and in 1873 he postulated that these electromagnetic waves solved the ongoing question as to the natureof light.

The greatest year in classical field theory came in 1884 when Heinrich Hertz generated and studied the

first radio waves in his laboratory. Hertz confirmed Maxwells prediction and changed the world, andphysics, forever.

Maxwells theoretical unification of electricity and magnetism was engineered into the modern humanpower to communicate across space at the speed of light. This was a stunning and powerful achievementfor theoretical physics, one that shaped the face of coming 20th century as the century of globaltelecommunications.

But this was just the beginning. In the century that was just arriving, the power of theoretical physicswould grow to question the very nature of reality, space and time, and the technological consequences

would be even bigger.

The sense of achievement and closure for theoretical physics that came with the brilliant success of theclassical field theory of electromagnetism was short lived. The new technology invented out of themathematical unification of electricity with magnetism produced copious data about the nature of matterand light that snapped all of the mathematical threads that physicists had just succeeded in tying down.

And after this new data was unraveled and understood and explained using mathematics, the unifiedworldview of classical theoretical physics became split into two very different views of the universe -- the

particle view and the geometric view.

Particles and waves

The first sign of trouble was when J.J. Thomson discovered the electronin 1897. Experimentalists beganto see data that suggested a model of the atom with negatively charged particles orbiting around a

positively charged core. But according to Maxwells equations, such a system should be physicallyunstable. Classical field theory was unable to explain or describethe emerging data on atomic structure.

Another big mystery that came out of Maxwells equations was the thermal behavior of light. Hot objects,like a hot coal, glow by emitting light and that light is observed to consist of a distribution of waves of

different frequencies. But physicists who tried to explain the observed distribution of frequencies usinglight waves as described by Maxwells equations met with continued failure.

Then as the new 20thcentury was beginning, a young German physicist, in an act of despairover thegaps in the understanding of thermal radiation, made a guess called the Quantum Hypothesis, whichexplained the observed thermal spectrum of light as coming from a collection of identical discrete quantaof energy. His formula worked, but he didnt know why.

This was the beginning of the idea known as particle-wave duality, and the field of quantum

mechanics.


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Einstein used Plancks idea to explain the newly-observed photoelectric effect. Einstein proposed thatlight was emitted or absorbed by an excited electron in discrete quanta called photons whose energy was

proportional to the frequency of the light according to the relation

E hv ,

where h is a number called Plancks constant, determined by measurement to be 6.6 x 10-34 joule

seconds.

If a light wave could behave like a particle, then could a particle behave like a wave of some kind? In1923, French aristocrat Louis de Broglie put forward the idea that an electron traveling with somemomentum p could act like a continuous wave with wavelength according to the relation

Bhp

When the dust was settled, the new quantum theory described a given physical system not in terms of the

path of a particle or the strength of a field, but as the probability amplitudefor a given system to be in agiven quantum state. This probability amplitude is the square of a function called the wave function

,x t , which is a solution to the Schrodinger equation

,H it

1 2

22 2 2

1 2, , ,2 N

x x x NH V x x xm

Solutions to Schrdinger equation for more then one identical particle have an interesting symmetry. Forexample, lets consider a two particle system and exchange the two particles. The wave function willobey the relation

1 2 2 1, ,x x x x

In the plus case, the two particles are what we call bosons. Two bosons canoccupy the same quantumstate at the same time.

In the minus case, the two particles are what we call fermions. Two fermions cannotoccupy the same

quantum state at the same time. This effect is called Pauli repulsion, and Pauli repulsion explains thestructure of the periodic table of elements and the stability of atoms, and hence of all matter.

Relativity and geometry

The radical new idea of the quantum physics of atoms and light marked one direction of departure fromthe comforting sureness of 19thcentury classical field theory. The other big surprise of the 20 th centurycame with the astounding observation in an experiment by Michelson and Morley that the speed of lightwas independent of the motion of the observer.

Now normally one would think that is a person were capable of throwing a javelin at 5 miles per hourwhile standing still, that same person, when running across the ground at 10 miles per hour, would becapable of making the javelin travel across the ground at a speed of 15 miles per hour.


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But according to the data from the Michelson-Morley experiment, if one uses a laser instead of a javelin,then whether the person is sanding still or running 60 miles per hour or in a rocket traveling near thespeed of light - the light from the laser still travels the same speed!

This was an astounding result! How could it be explained using physics? Einstein came up with apowerful, simple theory, called the Special Theory of Relativity. Einstein used the geometric notion of ametric. The most familiar metric is just the Pythagorean Rule, which in three space dimensions in

differential form looks like

2 2 2 2ds dx dy dz

This formula has the special property that it is invariant under rotations. In other words, the length of astraight line does not change when you rotate the line in space. In the Special Theory of Relativity theidea of a metric is extended to include time, with a very crucial minus sign:

2 2 2 2 2 2ds c dt dx dy dz

Like the space metric, the space-time is invariant under rotations in space. But now there is a new twist -the space-time metric is also invariant under a kind of rotation of space and time called a Lorentztransformation, and this transformation tells us how different observers who are moving with someconstant velocity relative to one another see the world.

And under a Lorentz transformation, the speed of light always stays the same, which is consistent withthe shocking Michelson-Morley experiment.

Einsteins next target of revision was Newtons Universal Law of Gravitation. In Newtons formula thegravitational force 12F between two planets of masses m1and m2as depending on the inverse square of

the distance12

r between the planets

1 212 2

12r

NG m mF

NG is called Newtons constantand is measured to be 6.7x10-8cm3/(gm sec2).

Newtons Law was extremely successful at explaining the observed motions of the planets around theSun, and of the moon around the Earth, and easily extendible through the techniques of classical fieldtheory to continuous systems.

However, there was no hint in Newtons theory as to how a gravitational field would change in time,especially not in a manner that was consistent with the new understanding in Special Relativity thatnothing can travel faster than the speed of light.

Einstein took a very bold step, and reached out to some radical new mathematics called non-Euclideangeometry, where the Pythagorean rule is generalized to include metrics with coefficients that depend onthe space-time coordinates in the form

2ds g x dx dx

where repeated indices imply a sum over all space and time directions in the chosen coordinate system.Einstein extended the idea of Lorentz invariance to general coordinate invariance, proposing that the


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values of physical observables should be independent of a choice of coordinate system used to chartpoints in space-time. He called this new theory the General Theory of Relativity.

In Einsteins new theory, space-time can have curvature, like the surface of a beach ball has curvature,compared to the flat top of a table, which doesnt. The curvature is a function of the metric gab and its firstand second derivatives. In the Einstein equation

1 82

NR g R G T

the space-time curvature (represented by R and R ) is determined by the total energy and momentum

T of the stuffin the space-time like the planets, stars, radiation, interstellar dust and gas, black holes,

etc.

The Einstein equation is not strictly a departure from classical field theory, and the Einstein equation canbe derived as the solution to Euler-Lagrange equations that represent the stationary point, or extremum, ofthe action

34

16 N

cS R gd x

G

Two views of the world

Using quantum mechanics, the typical questions that can be answered concern the types of quantum statesand allowed transitions in a system that features one or more particles that has some type of potential

energy represented by the potential V x . A typical method of working is to take some given V x and

use the Schrdinger equation find the wave function, the energies of the quantum states of the system, andthe allowed transitions between those states.

In general relativity, things are very different. One performs calculations that compute the evolution andstructure of an entire universe at a time. A typical way of working is to propose some particular collectionof energy and matter in the universe, to provide the T . Given a particular T , the Einstein equation

turns into a system of second order nonlinear differential equations whose solutions give us the metric ofspace-time, g , which holds all the information about the structure and evolution of a universe with that

given T .

Given the difference in the fundamental questions and methodologies used in quantum mechanics and ingeneral relativity, it seems hardy surprising that uniting quantum physics with gravity, for a theory ofquantum gravity, would prove to be a very tough challenge.

Once special relativity was on firm observational and theoretical footing, it was appreciated that theSchrdinger equation of quantum mechanics was not Lorentz invariant, therefore quantum mechanics asit was so successfully developed in the 1920s was not a reliable description of nature when the systemcontained particles that would move at or near the speed of light.

The problem is that the Schrdinger equation is first order in time derivatives but second order in spatialderivatives. The Klein-Gordon equation is second order in both time and space and has solutionsrepresenting particles with spin 0:


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2 2 2 0m c

Dirac came up with square rootof Klein-Gordon equation using matrices called gamma matrices, andthe solutions turned out to be particles of spin 1/2:

p 0,mc p p i

, 2

where the matrixmn

h is the metric of flat space-time. But the problem with relativistic quantum

mechanics is that the solutions of the Dirac and Klein-Gordon equation have instabilities that turn out torepresent the creation and annihilation of virtual particles from essentially empty space.

Further understanding led to the development of relativistic quantum field theory, beginning withquantum electrodynamics, or QED for short, pioneered by Feynman, Schwinger and Tomonaga in the

1940s. In quantum field theory, the behaviors and properties of elementary particles can calculated usinga series of diagrams, called Feynman diagrams, that properly account for the creation and annihilation ofvirtual particles.

The set of the Feynman diagrams for the scattering of two electrons looks like

+ + + ...

The straight black lines represent electrons. The green wavy line represents a photon, or in classicalterms, the electromagnetic field between the two electrons that makes them repel one another. Each small

black loop represents a photon creating an electron and a positron, which then annihilate one another andproduce a photon, in what is called a virtual process. The full scattering amplitude is the sum of allcontributions from all possible loops of photons, electrons, positrons, and other available particles.

The quantum loop calculation comes with a very big problem. In order to properly account for all virtualprocesses in the loops, one must integrate over all possible values of momentum, from zero momentum toinfinite momentum. But these loop integrals for an particle of spin J in D dimensions take theapproximate form

4 8J DloopI p d p

If the quantity 4J + D 8 is negative, then the integral behaves fine for infinite momentum (or zerowavelength, by the de Broglie relation.) If this quantity is zero or positive, then the integral takes aninfinite value, and the whole theory threatens to make no sense because the calculations just give infiniteanswers.

The world that we see has D = 4, and the photon has spin J = 1. So for the case of electron-electronscattering, these loop integrals can still take infinite values. But the integrals go to infinity very slowly,like the logarithm of momentum, and it turns out that in this case, the theory can be renormalized so that


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the infinities can be absorbed into a redefinition of a small number of parameters in the theory, such asthe mass and charge of the electron.

Quantum electrodynamics was a renormalizable theory, and by the 19402, this was regarded as a solvedrelativistic quantum theory. But the other known particle forces - the weak nuclear force that makesradioactivity, the strong nuclear force that hold neurons and protons together, and the gravitational forcethat holds us on the earth - werent so quickly conquered by theoretical physics.

In the 1960s, particle physicists reached towards something called a dual resonance model in an attemptto describe the strong nuclear force. The dual model was never that successful at describing particles, butit was understood by 1970 that the dual models were actually quantum theories of relativistic vibratingstrings and displayed very intriguing mathematical behavior. Dual models came to be called string theoryas a result.

But in 1971, a new type of quantum field theory came on the scene that explained the weak nuclear forceby uniting it with electromagnetism into electroweak theory, and it was shown to be renormalizable. Thensimilar wisdom was applied to the strong nuclear force to yield quantum chromodynamics, or QCD, andthis theory was also renormalizable.

Which left one force - gravity - that couldnt be turned into a renormalizable field theory no matter howhard anyone tried. One big problem was that classical gravitational waves carry spin J = 2, so one shouldassume that a graviton, the quantum particle that carries the gravitational force, has spin J = 2. But for J =2, 4J 8 + D = D, and so for D = 4, the loop integral for the gravitational force would become infinitelike the fourth power of momentum, as the momentum in the loop became infinite.

And that was just hard cheese for particle physicists, and for many years the best people worked onquantum gravity to no avail.

But the string theory that was once proposed for the strong interactions contained a massless particle withspin J = 2.

In 1974 the question finally was asked: could string theory be a theory of quantum gravity?

The possible advantage of string theory is that the analog of a Feynman diagram in string theory is a two-dimensional smooth surface, and the loop integrals over such a smooth surface lack the zero-distance,infinite momentum problems of the integrals over particle loops.

In string theory infinite momentum does not even mean zero distance, because for strings, the relationshipbetween distance and momentum is roughly like

pL

p

The parameter a(pronounced alpha prime) is related to the string tension, the fundamental parameter ofstring theory, by the relation

1

2stringT

The above relation implies a minimum observable length for a quantum string theory of

min 2L


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The zero-distance behavior which is so problematic in quantum field theory becomes irrelevant in stringtheories, and this makes string theory very attractive as a theory of quantum gravity.

If string theory is a theory of quantum gravity, then this minimum length scale should be at least the sizeof the Planck length, which is the length scale made by the combination of Newtons constant, the speedof light and Plancks constant

33

3 1.6 10N

P

GL cmc

although as we shall see later, the question of length scales in string theory is complicated by stringduality, which can relate two theories with seemingly different length scales.

Pythagoras could be called the first known string theorist. Pythagoras, an excellent lyre player, figuredout the first known string physics - the harmonic relationship. Pythagoras realized that vibrating Lyrestrings of equal tensions but different lengths would produce harmonious notes (i.e. middle C and high C)if the ratio of the lengths of the two strings were a whole number.

Pythagoras discovered this by looking and listening. Today that information is more precisely encodedinto mathematics, namely the wave equation for a string with a tension T and a mass per unit length m. Ifthe string is described in coordinates as in the drawing below, where x is the distance along the string andy is the height of the string, as the string oscillates in time t,

then the equation of motion is the one-dimensional wave equation

2 2 222 2 2

, , ,w

y x t y x t y x tTv

t x x

wherew

v is the wave velocity along the string.

When solving the equations of motion, we need to know the boundary conditionsof the string. Letssuppose that the string is fixed at each end and has an unstretched length L. The general solution to thisequation can be written as a sum of normal modes, here labeled by the integer n, such that

1

, cos sin sinw wn nn

n v t n v t n xy x t a b

L L L

The condition for a normal mode is that the wavelength be some integral fraction of twice the stringlength, or


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2n

Ln

The frequency of the normal mode is then

2w

n

nvf

L

The normal modes are what we hear as notes. Notice that the string wave velocityw

v increases as the

tension of the string is increased, and so the normal frequency of the string increases as well. This is whya guitar string makes a higher note when it is tightened.

But thats for a nonrelativistic string, one with a wave velocity much smaller than the speed of light. Howdo we write the equation for a relativistic string?

According to Einsteins theory, a relativistic equation has to use coordinates that have the proper Lorentztransformation 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 space-time 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 sheetof the string as a two-dimensional space-time of its own, where the division between space and timedepends upon the observer.

The classical equation can be written as

2 2

22 2

, ,X Xc

where and are coordinates on the string world sheet representing space and time along the string,and the parameter 2c is the ratio of the string tension to the string mass per unit length.

These equations of motion can be derived from Euler-Lagrange equations from an action based on thestring world sheet

1

4mn

m nS d d hh X X

The space-time coordinatesm

X of the string in this picture are also fieldsm

X in a two-dimension field

theory defined on the surface that a string sweeps out as it travels in space. The partial derivatives arewith respect to the coordinates s and t on the world sheet and

mnh is the two-dimensional metric defined

on the string world sheet.

The general solution to the relativistic string equations of motion looks very similar to the classicalnonrelativistic case above. The transverse space coordinates can be expanded in normal modes as

0

,

12 cos sin cos

i i i

i

n

n

X x x

n c n c ni i

n L L L


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The string solution above is unlike a guitar string in that it isn t tied down at either end and so travelsfreely through space-time 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 liketwo open string oscillations moving in the opposite direction around the string. These two types of closedstring modes are called right-movers and left-movers, and this difference will be important later in thesupersymmetric heterotic string theory.

This is classical string. When we add quantum mechanics by making the string momentum and positionobey quantum commutation relations, the oscillator mode coefficients have the commutation relations

,m n m nm

The quantized string oscillator modes wind up giving representations of the Poincar group, throughwhich quantum states of mass and spin are classified in a relativistic quantum field theory.

So this is where the elementary particle arise in string theory. Particles in a string theory are like the

harmonic notes played on a string with a fixed tension

1

2stringT

The parameter is called the string parameter and the square root of this number represents theapproximate distance scale at which string effects should become observable.

In the generic quantum string theory, there are quantum states with negative norm, also known as ghosts.This happens because of the minus sign in the space-time metric, which implies that

0 0,m n m nm

So there ends up being extra unphysical states in the string spectrum.

In 26 space-time dimensions, these extra unphysical states wind up disappearing from the spectrum.Therefore bosonic string quantum mechanics is only consistent if the dimension of space-time is 26.

By looking at the quantum mechanics of the relativistic string normal modes, one can deduce that thequantum modes of the string look just like the particles we see in space-time, with mass that depends on

the spin according to the formula2

JJ M

Remember that boundary conditions are important for string behavior. Strings can be open, with ends thattravel at the speed of light, or closed, with their ends joined in a ring.

One of the particle states of a closed string has zero mass and two units of spin, the same mass and spin asa graviton, the particle that is supposed to be the carrier of the gravitational force.

The bosonic string world sheet action


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1

4mn

m nS d d hh X X

for a string propagating in flat 26-dimensional space-time with coordinates ,X can give rise tofour different quantum mechanically consistent string theories, depending on the choice of boundaryconditions used to solve the equations of motion. The choices are divided into two categories:

A. Are the strings open(with free ends) or closed(with ends joined together in a loop)?

B. Are the strings orientable (you can tell which direction youre traveling along the string) orunorientable(you canttell which direction youre traveling along the string)?

There are four different combinations of options, giving rise to the four bosonic string theories shown inthe table below. Notice in the table that open string theories also contain closed strings. Why is this?Because an open string can sometimes join its two free ends and become a closed string and then breakapart again into an open string. In pure closed string theory, the analog of that process does not occur.

The bosonic string theories are all unstable because the lowest excitation mode, or the ground state, is atachyonwith 2 1M . The massless particle spectrum always includes the graviton, so gravity isalways a part of any bosonic string theory. The vector boson is similar to the photon of electromagnetismor the gauge fields of any Yang-Mills theory. The antisymmetric tensor field carries a force that isdifficult to describe in this short space. The strings act as a source of this field.

Bosonic strings, d = 26

Type Oriented? Details

Open (plus closed) Yes Scalar tachyon, massless antisymmetric tensor, graviton anddilaton

Open (plus closed) No Scalar tachyon, massless graviton and dilaton

Closed YesScalar tachyon, massless vector boson, antisymmetric tensor,graviton and dilaton

Closed No Scalar tachyon, massless graviton and dilaton

Its just as well that bosonic string theory is unstable, because its not a realistic theory to begin with. Thereal world has stable matter made from fermions that satisfy the Pauli Exclusion Principle where twoidentical particles cannot be in the same quantum state at the same time.

Adding fermions to string theory introduces a new set of negative norm states or ghosts, to add to theghost states that come from the bosonic sector described on the previous page. String theorists learnedthat all of these bad ghost states decouple from the spectrum when two conditions are satisfied: thenumber of space-time dimensions is 10, and theory is supersymmetric, so that there are equal numbersof bosons and fermions in the spectrum.

Fermions have more complicated boundary conditions than bosons, so unraveling the different possibleconsistent superstring theories took researchers quite a bit of doing. The simplest way to examine a

superstring theory is to go to what is called superspace. In superspace, in addition to the normalcommuting coordinates X , a set of anticommuting coordinates A are added. In superstring theoriesindex A runs from 1 to 2 (an additional spinor index is not shown). The anticommutation relations of thecoordinates are


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0A B A B

The options of open vs closed, and oriented Vs unoriented boundary conditions are still present, but thereare also choices involving fermions that distinguish one superstring theory from another. The superspacecoordinates 1 and 2 behave like particles with spin 1/2 and zero mass, which can only spin two ways -with the spin axis in the same or opposite direction as the momentum. This property is called handedness.So 1 and 2 can have either the same or the opposite handedness.

The resulting consistent string theories can be described in terms of the massless particle spectrum andthe resulting number of space-time supersymmetry charges, denoted by the letter N in the table below.

None of the theories below suffer from the tachyonproblem that plagues bosonic string theories. All ofthe theories below contain gravity.

Superstrings, d = 10

TypeOpen or

closed?Oriented? N Details

IOpen (plusclosed)

No 1Graviton, no tachyon, SO(32) gauge symmetry,charges are attached to the ends of the strings

IIA Closed No 2Graviton, no tachyon, only a U(1) gaugesymmetry

IIB Closed Yes 2 Graviton, no tachyon, no gauge symmetry

Heterotic E8XE8 Closed Yes 1 Graviton, no tachyon, E8XE8 gauge symmetry

Heterotic SO(32) Closed Yes 1 Graviton, no tachyon, SO(32) gauge symmetry

A supersymmetric theory has a fermionic partner for every bosonic particle. The superpartner of agraviton is called a gravitino and has spin 3/2. All of the theories above contain gravitons and gravitinos.

For open superstrings, the choices turn out to be restricted by conditions too complicated to explainhere. It turns out that the only consistent theory has unoriented strings, with 1 and 2 having the samehandedness, with an SO(32) gauge symmetry included by attaching little charges to the ends of the openstring. These charges are called Chan Paton factors. The resulting theory is called Type I.

Closed string oscillations can be separated into modes that propagate around the string in different

directions, sometimes called left movers and right movers. If1

and2

have opposite handedness, thenthey also have opposite momentum, and hence travel in opposite directions. Therefore they provide a wayto tell which direction one is traveling around the string. Therefore these strings are oriented. This iscalled Type IIAsuperstring theory.

Because 1 and 2 have opposite handedness, this theory winds up being too symmetric for real life.Every fermion has a partner of the opposite handedness, which is not what is observed in our world,where the neutrino comes in a left-handed version but not a right-handed version. The real world seems to

be chiral, which means having a preferred handedness for massless fermions. But Type IIA superstringtheory is a nonchiraltheory. There is also no way to add a gauge symmetry to Type IIA superstrings, sohere also the theory fails as a model of the real world.

If 1 and 2 have the same handedness, and the string is oriented, then we get Type IIBsuperstringtheory. This theory is chiral, and so there will be massless fermions that dont have partners of theopposite handedness, as is observed in our world today. However, there is no way to add a gauge


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symmetry to the Type IIB theory. So there isnt a way to include any of the known forces other thangravity.

If 1 and 2 have the same handedness, but the string is unoriented, that turns out to just give the closedstring part of the Type I theory.

This seems to have exhausted all of the obvious options. But theres actually something crazy that can be

done with a closed string that yields two more important superstring theories.

The left-moving and right-moving modes of a string can be separated and treated as different theories. In1984 it was realized that consistent string theories could be built by combining a bosonic string theorymoving in one direction along the string, with a supersymmetric string theory with a single 1 moving inthe opposite direction. These theories are called heteroticsuperstring theories.

That sounds crazy - because bosonic strings live in 26 dimensions but supersymmetric string theories livein 10 dimensions. But the extra 16 dimensions of the bosonic side of the theory arent really space-timedimensions. Heterotic string theories are supersymmetric string theories living in ten space-timedimensions.

The two types of heterotictheories that are possible come from the two types of gauge symmetry thatgive rise to quantum mechanically consistent theories. The first is SO(32) and the second is the moreexotic combination called E8XE8.The E8XE8heterotic theory was previously regarded as the only stringtheory that could give realistic physics, until the mid-1990s, when additional possibilities based on theother theories were identified.

A new picture of string theory

At one time, string theorists believed there were five distinct superstring theories: type I, types IIAandIIB, and heterotic SO(32) and E8XE8string theories. The thinking was that out of these five candidatetheories, only one was the actual correct Theory of Everything, and that theory was the theory whoselow energy limit, with ten dimensions space-time compactified down to four, matched the physicsobserved in our world today. The other theories would be nothing more than rejected string theories,mathematical constructs not blessed by Nature with existence.

But now it is known that this naive picture was wrong, and that the five superstring theories areconnected to one another as if they are each a special case of some more fundamental theory, of which

there is only one. In the mid-nineties it was learned that superstring theories are related by dualitytransformations known as T duality and S duality. These dualities link the quantities of large and smalldistance, and strong and weak coupling, limits that have always been identified as distinct limits of a

physical system in both classical and quantum physics. These duality relationships between stringtheories have sparked a radical shift in our understanding of string theory, and have led to the reasonableexpectation that all five superstring theories- type I, types IIA and IIB, and heterotic SO(32) and E8XE8- are special limits of a more fundamental theory.

T duality

The duality symmetry that obscures our ability to distinguish between large and small distance scales is

called T-duality, and comes about from the compactification of extra space dimensions in a tendimensional superstring theory. Lets take the X9 direction in flat ten-dimensional space-time, andcompactify it into a circle of radius R, so that


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9 9 2x x R

A particle traveling around this circle will have its momentum quantized in integer multiples of 1/R, and aparticle in the nth quantized momentum state will contribute to the total mass squared of the particle as

22

2n

nm

R

A string can travel around the circle, too, and the contribution to the string mass squared is the same asabove.

But a closed string can also wrap around the circle, something a particle cannot do. The number oftimes the string winds around the circle is called the winding number, denoted as wbelow, and wis alsoquantized in integer units. Tension is energy per unit length, and the wrapped string has energy from

being stretched around the circular dimension. The winding contribution Ewto the string energy is equalto the string tension Tstringtimes the total length of the wrapped string, which is the circumference of thecircle multiplied by the number of times wthat the string is wrapped around the circle.

1, 2

2string w string

wRT E wRT

where

2sL

tells us the length scale Lsof string theory.

The total mass squared for each mode of the closed string is

2 2 2

2

2 2

22

n w Rm N

R

N N nw

The integers N and are the number of oscillation modes excited on a closed string in the right-movingand left-moving directions around the string.

The above formula is invariant under the exchange

,R n wR

In other words, we can exchange compactification radius R with radius R if we exchange the windingmodes with the quantized momentum modes.

This mode exchange is the basis of the duality known as T-duality. Notice that if the compactificationradius R is much smaller than the string scaleLs, then the compactification radius after the winding and

momentum modes are exchanged is much larger than the string scale Ls. So T-duality obscures thedifference between compactified dimensions that are much bigger than the string scale, and those that aremuch smaller than the string scale.

T-duality relates type IIA superstring theory to type IIB superstring theory, and it relates heterotic


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SO(32) superstring theory to heterotic E8XE8superstring theory. Notice that a duality relationshipbetween IIA and IIB theory is very unexpected, because type IIA theory has massless fermions of bothchiralities, making it a non-chiral theory, whereas type IIB theory is a chiral theory and has masslessfermions with only a single chirality.

T-duality is something unique to string physics. Its something point particles cannot do, because theydont have winding modes. If string theory is a correct theory of Nature, then this implies that on some

deep level, the separation between large vs. small distance scales in physics is not a fixed separation but afluid one, dependent upon the type of probe we use to measure distance, and how we count the states ofthe probe.

This sounds like it goes against all traditional physics, but this is indeed a reasonable outcome for aquantum theory of gravity, because gravity comes from the metric tensor field that tells us the distances

between events in space-time.

Strong and weak coupling

What is a coupling constant? This is some number that tells us how strong an interaction is. Newton sconstant GN, which appears in both Newtons law of gravity and the Einstein equation, is the couplingconstant for gravitational interactions. For electromagnetism, the coupling constant is related to theelectric charge through the fine structure constant

22 2 1

137QED

eg

hc

In both particle physics and string theory, usually the scattering amplitudes and other quantities have tobe computed as an expansion in powers of the coupling constant or loop expansion parameter, whichweve called g2below:

2 2 40 1 2A g A g A g A

At low energies in electromagnetism, the dimensionless coupling constant is very small compared tounity, and the higher powers in become too small to matter. The first few terms in the series make agood approximation to the real answer, which often cant be calculated at all because the mathematicaltechnology doesnt exist to solve the whole theory at once.

If the coupling constant gets very large compared to unity, perturbation theory becomes useless, becausehigher powers of the expansion parameter are bigger, not smaller, than lower powers. This is called astrongly coupled theory. Coupling constants in quantum field theory end up depending on energy

because of quantum vacuum effects. A quantum field theory can be weakly coupled at low energies andstrongly coupled at high energies, as is true with the fine structure constant in QED, or stronglycoupled at low energies and weakly coupled at high energies, as is true with the coupling constant forquark and gluon interactions in QCD.

Some quantities in a theory cannot be calculated at all using perturbation theory, especially not for weakcoupling. For example, the amplitude below cannot be expanded around the value g2 = 0

2 2expNP

A g c g


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because the amplitude is singular there. This is typical of a tunneling transition, which is forbidden byenergy conservation in classical physics and hence has no expansion around a classical limit.

String theories feature two kinds of perturbative expansions: an expansion in powers of the stringparameter in the conformal field theory on the two-dimensional string worldsheet, and a quantumloop expansion for string scattering amplitudes in d-dimensional space-time. But unlike in particletheories, the string quantum loop expansion parameter is not just a number, but depends on one of the

dynamic modes of the string, called the dilatonfield x

22 xstg e

This relationship between the dilaton and the string loop expansion parameter is important inunderstanding the duality relation known as S-duality. S-duality can be examined most easily in type IIBstring theory, because this theory happens to be S-dual to itself. The low energy limit of type IIB theory(meaning the lowest nontrivial order in the string parameter ) is a type IIB supergavity field theory,

which features a complex scalar field x whose real part is the axion field x and whose imaginary

part is the exponential of the dilaton field x :

ie

This field theory is invariant under a global transformation by the group SL(2, R) (broken by quantum

effects down to SL(2,Z)), with the field x transforming as

, 1a b

ad bcc d

If there is no contribution from the axion field, then the expectation value of the field x is given bythe dilaton alone. Because the dilaton is identified with gst, the SL(2, Z) transformation with b = -1, c = 1

st

ig

1 1st

st

gg

tells us that the theory at coupling gstis the same as the theory at coupling 1/gst!

This transformation is called S-duality. If two string theories are related by S-duality, then one theorywith a strong coupling constant is the same as the other theory with weak coupling constant. Type IIBsuperstring theory is S-dual to itself, so the strong and weak coupling limits are the same. Thisduality allows an understanding of the strong coupling limit of the theory that would not be possible byany other means.

Something more surprising is that type I superstring theory is S-dual to heterotic SO(32) superstringtheory. This is surprising because type I theories contain open and closed strings, where as heterotic

theories contain only open strings. Whats the explanation? At very strong coupling, heterotic SO(32)string theory has excitations that are open strings, but these open strings are highly unstable in the weakly

coupled limit of the theory, which is the limit in which heterotic string theory is commonly understood.


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More than just strings

To understand the presence of objects in string theory that are not strings, but higher dimensionalobjects, or even points, it helps to know the formulation of Maxwells equations in the language ofdifferential forms, because this is what tells us that the sources of charge in the Maxwell equationsare zero-dimensional objects. Gauge field strengths that are p+2-forms turn out to have sourcesthat are p-dimensional objects. We call these p-branes.

In the regular Maxwell equations in d=4 space-time dimension, the electric and magentic fields arepacked together into the field strength F, which satisfies the equation F=dA, dis the exteriorderivative, and Ais the vector potential, a one-form. The two-form*Fis the dual of Frelative to thespace-time volume four-form (The subscripts on F, etc., below are just to indicate the degree ofthe differential form.)

The charge sources enter through the equation d*F=*J, where *J is the three-form dual to thecurrent four-vector J=(,j). In the rest frame of the charge density , J=(,0), so *Jis times the

volume element for three-dimensional space. In a three-dimensional space, a surface that can belocalized in three dimensions (has codimension three) must be a zero-dimensional surface, alsoknown as a point.

This is the math that tells us that the Maxwell equations couple electrically to sources that arepoints, or zero-branes, as zero-dimensional objects are now called in string theory. (For magneticcouplings, the roles of Fand *Fare interchanged, but that wont be covered here.) This same mathworks for two-forms in any space-time dimension, so we know that Maxwells equations couple topoint charges in any space-time dimension.

Superstring theories contain electromagnetism, but they also contain field strengths that are three-forms, four-forms and on up. These field strengths obey equations just like the Maxwell equations,

and their sources can be analyzed in the same manner as above.

Suppose we start in d space-time dimensions with a vector potential Athat is a p+1-form. Then Fisa p+2-form, is a d-form (because its the volume element of d-dimensional space-time), *Fis a(d-p-2)-form, and d*Fis a (d-p-1)-form. (Once again, the subscripts are just to indicate the degreeof the differential form.)


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The equations of motion tell us that the source term *Jis also a (d-p-1)-form. In the rest frame ofan isolated source, *Jis proportional to a volume element of a (d-1-p)-dimensional subspace of (d-1)-dimensional space. The codimension of the source is therefore (d-p-1), and since space hasdimension d-1, the charges that serve as sources must be objects with p dimensions, known asp-branes. So a (p+2)-form field strength couples to sources that are p-branes. This little facthas turned out to be extremely important in string theory.

Superstring theories are theories with gravity, so these p-dimensional localizations of charge mustlead to space-time curvature. A p-brane space-time whose metric solves the equations of motion fora (p+2)-form field strength in d space-time dimensions can be described using p space coordinates{yi} along the p-brane and (d-1-p) space coordinates {xa} orthogonal to the p-brane.

The isometries of this space-time consist of translations (shifting the coordinate by a constant) andLorentz transformations in the (p+1)-dimensional world volume, plus spatial rotations in the (d-1-p)-dimensional space orthogonal to the p-brane.

Theres a problem with adding gravity, however. Most p-brane space-times turn out to be unstable.Supersymmetry stabilizes p-branes, but only for the certain values of p and d. Two of the mostimportant p-branes in string theory are the two-brane in d=11and the five-brane in d=10.

Since were talking about a space-time metric, were obviously in the low energy limit of stringtheory. But p-branes can be protected from quantum corrections by supersymmetry, if they satisfyan equality between mass and charge known as the BPS condition. These branes are then knownas BPS branes.

From p-branes to D-branes

A special class of p-branes in string theory are called D branes. Roughly speaking, a D brane is a p-brane where the ends of open strings are localized on the brane.

D-branes were discovered by investigating T-duality for open strings. Open strings dont havewinding modes around compact dimensions, so one might think that open strings behave likeparticles in the presence of circular dimensions. However, the stringiness of open strings in thepresence of compact dimensions exhibits itself in a more subtle manner, and the T-dual of an openstring theory is anything but uninteresting.

The normal open string boundary conditions in the string oscillator expansion comes from therequirement that there be no momentum exiting or entering through the ends of an open string.This translates into what are called Neumann boundary conditionsat the ends of the string at(=0) and (=):


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Suppose d-1-p of the space dimensions are compactified on a torus with radius R, and p of thespace dimensions are left noncompact as before. In the T-dual of this string theory, the boundaryconditions in those d-1-p directions are changed from Neumann to Dirichlet boundary conditions

This T-dual theory has strings with ends localized in d-1-p directions. So the T-dual of open stringscompactified on a torus of radius R is open strings with their ends fixed to static p-branes,which we then call D-branes.

D branes have been very important in understanding string theory in general (see below) but also ofcrucial importance in understanding black holes in string theory, especially in counting the quantumstates that lead toblack hole entropy.

How many dimensions?

Before string theory won the full attention of the theoretical physics community, the most popularunified theory was an eleven dimensional theory of supergravity, which is supersymmetry combinedwith gravity. The eleven-dimensional space-time was to be compactified on a small 7-dimensionalsphere, leaving four space-time dimensions visible to observers at large distances.

This theory didnt work as a unified theory of particle physics, because an eleven-dimensional

quantum field theory based on point particles is not renormalizable. Also, chiral fermions cannot bedefined in space-time with an odd number of dimensions. But this eleven dimensional theory wouldnot die. It eventually came back to life in the strong coupling limit of superstring theory in tendimensions.

The theory currently known as M

Technically speaking, M theoryis the unknown eleven-dimensional theory whose low energy limit isthe supergravity theory in eleven dimensions discussed above. However, many people have taken toalso using M theoryto label the unknown theory believed to be the fundamental theory from whichthe known superstring theories emerge as special limits.

We still dont know the fundamental M theory, but a lot has been learned about the eleven-dimensional M theory and how it relates to superstrings in ten space-time dimensions.

Recall that one of the p-brane space-times that are stabilized by supersymmetry is a two-brane ineleven space-time dimensions. This object is called the M2 branefor short.

Type IIA superstring theory has a stable one-brane solution called the fundamental string. If wetake M theory with the tenth space dimension compactified into a circle of radius R, and wrap one ofthe dimensions of the M2 brane around that circle, then the result is the fundamental string of thetype IIA theory. When the M2 brane is not around that circle, then the result is the two-dimensionalD-brane, the D2 brane, of the type IIA theory.

If the two theories are identified, the type IIA coupling constant turns out to be proportional to theradius R of the compactified tenth dimension in the M theory. So the weakly coupled limit of type IIAsuperstring theory, which is the usual ten-dimensional theory, is also an expansion around small R.The strong coupling limit of type IIA theory is where R becomes very large, and the extra dimension
http://www.superstringtheory.com/blackh/blackh5.htmlhttp://www.superstringtheory.com/blackh/blackh5.html


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of space-time is revealed. So type IIA superstring theory lives in ten space-time dimensions in theweak coupling limit, but eleven space-time dimensions in the strongly coupled limit.

We still dont know what the fundamental theory behind string theory is, but judging fromall of these relationships, it must be a very interesting and rich theory, one where distance scales,coupling strengths and even the number of dimensions in space-time are not fixed concepts but fluidentities that shift with our point of view.

The language of physics is mathematics. In order to study physics seriously, one needs to learnmathematics that took generations of brilliant people centuries to work out. Algebra, forexample, was cutting-edge mathematics when it was being developed in Baghdad in the 9thcentury. But today its just the first step along the journey.

Algebra

Algebra provides the first exposure to the use of variables and constants, and experiencemanipulating and solving linear equations of the form y = ax + b and quadratic equationsof the form y = ax2+bx+c.

Geometry

Geometry at this level is two-dimensional Euclidean geometry, Courses focus on learning toreason geometrically, to use concepts like symmetry, similarity and congruence, tounderstand the properties of geometric shapes in a flat, two-dimensional space.

Trigonometry

Trigonometry begins with the study of right triangles and the Pythagorean theorem. Thetrigonometric functions sin, cos, tan and their inverses are introduced and clever identitiesbetween them are explored.

Calculus (single variable)

Calculus begins with the definition of an abstract functions of a single variable, andintroduces the ordinary derivative of that function as the tangent to that curve at a givenpoint along the curve. Integration is derived from looking at the area under a curve, whichis then shown to be the inverse of differentiation.

Calculus (multivariable)

Multivariable calculus introduces functions of several variables f(x, y, z...), and studentslearn to take partial and total derivatives. The ideas of directional derivative, integrationalong a path and integration over a surface are developed in two and three dimensionalEuclidean space.

Analytic Geometry

Analytic geometry is the marriage of algebra with geometry. Geometric objects such asconic sections, planes and spheres are studied by the means of algebraic equations.Vectors in Cartesian, polar and spherical coordinates are introduced.

Linear Algebra

In linear algebra, students learn to solve systems of linear equations of the form ai1x1+ ai2x2+ ... + ainxn= ciand express them in terms of matrices and vectors. The properties ofabstract matrices, such as inverse, determinant, characteristic equation, and of certaintypes of matrices, such as symmetric, antisymmetric, unitary or Hermitian, are explored.

Ordinary Differential Equations

This is where the physics begins! Much of physics is about deriving and solving differentialequations. The most important differential equation to learn, and the one most studied inundergraduate physics, is the harmonic oscillator equation, ax+ bx+ cx = f(t), where xmeans the time derivative of x(t).


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Partial Differential Equations

For doing physics in more than one dimension, it becomes necessary to use partialderivatives and hence partial differential equations. The first partial differential equationsstudents learn are the linear, separable ones that were derived and solved in the 18th and19th centuries by people like Laplace, Green, Fourier, Legendre, and Bessel.

Methods of approximation

Most of the problems in physics cant be solved exactly in closed form. Therefore we haveto learn technology for making clever approximations, such as power series expansions,saddle point integration, and small (or large) perturbations.

Probability and statistics

Probability became of major importance in physics when quantum mechanics entered thescene. A course on probability begins by studying coin flips, and the counting ofdistinguishable vs. indistinguishable objects. The concepts of mean and variance aredeveloped and applied in the cases of Poisson and Gaussian statistics.

Here are some of the topics in mathematics that a person who wants to learn advanced topics intheoretical physics, especially string theory, should become familiar with.

Real analysis

In real analysis, students learn abstract properties of real functions as mappings,isomorphism, fixed points, and basic topology such as sets, neighborhoods, invariants andhomeomorphisms.

Complex analysis

Complex analysis is an important foundation for learning string theory. Functions of a

complex variable, complex manifolds, holomorphic functions, harmonic forms, Khlermanifolds, Riemann surfaces and Teichmuller spaces are topics one needs to becomefamiliar with in order to study string theory.

Group theory

Modern particle physics could not have progressed without an understanding of symmetriesand group transformations. Group theory usually begins with the group of permutations onN objects, and other finite groups. Concepts such as representations, irreducibility, classesand characters.

Differential geometry

Einsteins General Theory of Relativity turned non-Euclidean geometry from a controversialadvance in mathematics into a component of graduate physics education. Differentialgeometry begins with the study of differentiable manifolds, coordinate systems, vectorsand tensors. Students should learn about metrics and covariant derivatives, and how tocalculate curvature in coordinate and non-coordinate bases.

Lie groups

A Lie group is a group defined as a set of mappings on a differentiable manifold. Lie groupshave been especially important in modern physics. The study of Lie groups combinestechniques from group theory and basic differential geometry to develop the concepts ofLie derivatives, Killing vectors, Lie algebras and matrix representations.

Differential forms

The mathematics of differential forms, developed by Elie Cartan at the beginning of the20th century, has been powerful technology for understanding Hamiltonian dynamics,relativity and gauge field theory. Students begin with antisymmetric tensors, then developthe concepts of exterior product, exterior derivative, orientability, volume elements, and


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integrability conditions.

Homology

Homology concerns regions and boundaries of spaces. For example, the boundary of a two-dimensional circular disk is a one-dimensional circle. But a one-dimensional circle has noedges, and hence no boundary. In homology this case is generalized to The boundary of aboundary is zero.Students learn about simplexes, complexes, chains, and homologygroups.

Cohomology

Cohomology and homology are related, as one might suspect from the names. Cohomologyis the study of the relationship between closed and exact differential forms defined on somemanifold M. Students explore the generalization of Stokestheorem, de Rham cohomology,the de Rahm complex, de Rahms theorem and cohomology groups.

Homotopy

Lightly speaking, homotopy is the study of the hole in the donut. Homotopy is important instring theory because closed strings can wind around donut holes and get stuck, withphysical consequences. Students learn about paths and loops, homotopic maps of loops,

contractibility, the fundamental group, higher homotopy groups, and the Bott periodicitytheorem.

Fiber bundles

Fiber bundles comprise an area of mathematics that studies spaces defined on other spacesthrough the use of a projection map of some kind. For example, in electromagnetism thereis a U(1) vector potential associated with every point of the space-time manifold. Thereforeone could study electromagnetism abstractly as a U(1) fiber bundle over some space-timemanifold M. Concepts developed include tangent bundles, principal bundles, Hopf maps,covariant derivatives, curvature, and the connection to gauge field theories in physics.

Characteristic classes

The subject of characteristic classes applies cohomology to fiber bundles to understand thebarriers to untwisting a fiber bundle into what is known as a trivial bundle. This is usefulbecause it can reduce complex physical problems to math problems that are alreadysolved. The Chern class is particularly relevant to string theory.

Index theorems

In physics we are often interested in knowing about the space of zero eigenvalues of adifferential operator. The index of such an operator is related to the dimension of thatspace of zero eigenvalues. The subject of index theorems and characteristic classes isconcerned with

Supersymmetry and supergravity

The mathematics behind supersymmetry starts with two concepts: graded Lie algebras,and Grassmann numbers. A graded algebra is one that uses both commutation and anti-commutation relations. Grassmann numbers are anti-commuting numbers, so that x timesy = y times x. The mathematical technology needed to work in supersymmetry includesan understanding of graded Lie algebras, spinors in arbitrary space-time dimensions,covariant derivatives of spinors, torsion, Killing spinors, and Grassmann multiplication,derivation and integration, and Khler potentials.

These are topics in mathematics at the current cutting edge of superstring research.K-theory

Cohomology is a powerful mathematical technology for classifying differential forms. In the


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1960s, work by Sir Michael Atiyah, Isadore Singer, Alexandre Grothendieck, and FriedrichHirzebruch generalized coholomogy from differential forms to vector bundles, a subject thatis now known as K-theory.

Witten has argued that K-theory is relevant to string theory for classifying D-branecharges. D-brane objects in string theory carry a type of charge called Ramond-Ramondcharge. Ramond-Ramond fields are differential forms, and their charges should be classifiedby ordinary cohomology. But gauge fields propagate on D-branes, and gauge fields giverise to vector bundles. This suggests that D-brane charge classification requires ageneralization of cohomology to vector bundles -- hence K-theory.

Overview of K-theory Applied to Strings by Edward Witten

D-branes and K-theory by Edward Witten

Noncommutative geometry (NCG for short)

Geometry was originally developed to describe physical space that we can see andmeasure. After modern mathematics was freed from Euclids Fifth Axiom by Gauss and

Bolyai, Riemann added to modern geometry the abstract notion of a manifold M with pointsthat are labeled by local coordinates that are real numbers, with some metric tensor thatdetermines an extremal length between two points on the manifold.

Much of the progress in 20th century physics was in applying this modern notion ofgeometry to space-time, or to quantum gauge field theory.

In the quest to develop a notion of quantum geometry, as far back as 1947, people weretrying to quantize space-time so that the coordinates would not be ordinary real numbers,but somehow elevated to quantum operators obeying some nontrivial quantumcommutation relations. Hence the term noncommutative geometry,or NCG for short.

The current interest in NCG among physicists of the 21st century has been stimulated bywork by French mathematician Alain Connes.

Two Lectures on D-Geometry and Noncommutative Geometry by Michael R. DouglasNoncommutative Geometry and Matrix Theory: Compactification on Tori by Alain Connes,Michael R. Douglas, Albert Schwarz

String Theory and Noncommutative Geometry by Edward Witten and Nathan Seiberg.

Non-commutative spaces in physics and mathematics by Daniela Bigatti

Noncommutative Geometry for Pedestrians by J. Madore

Isaac Newton made a Bible-based estimate of a few thousand years. Einstein believed in asteady state, ageless Universe. Since then, data collected from the Universe puts theprobable answer somewhere in the middle.basic/advanced

The Einstein equation predicts several possible ways for the Universe to evolve in time and space.What are these models and how do they compare with observation?basic/advanced
http://arxiv.org/abs/hep-th/0007175http://arxiv.org/abs/hep-th/9810188http://arxiv.org/abs/hep-th/9901146http://arxiv.org/abs/hep-th/9711162http://arxiv.org/abs/hep-th/9908142http://arxiv.org/abs/hep-th/0006012http://arxiv.org/abs/gr-qc/9906059http://www.superstringtheory.com/cosmo/cosmo1.htmlhttp://www.superstringtheory.com/cosmo/cosmo1a.htmlhttp://www.superstringtheory.com/cosmo/cosmo1a.htmlhttp://www.superstringtheory.com/cosmo/cosmo2.htmlhttp://www.superstringtheory.com/cosmo/cosmo2.htmlhttp://www.superstringtheory.com/cosmo/cosmo2a.htmlhttp://www.superstringtheory.com/cosmo/cosmo3.htmlhttp://www.superstringtheory.com/cosmo/cosmo2.htmlhttp://www.superstringtheory.com/cosmo/cosmo1.htmlhttp://www.superstringtheory.com/cosmo/cosmo3.htmlhttp://www.superstringtheory.com/cosmo/cosmo2.htmlhttp://www.superstringtheory.com/cosmo/cosmo1.htmlhttp://www.superstringtheory.com/cosmo/cosmo3.htmlhttp://www.superstringtheory.com/cosmo/cosmo2.htmlhttp://www.superstringtheory.com/cosmo/cosmo1.htmlhttp://www.superstringtheory.com/cosmo/cosmo3.htmlhttp://www.superstringtheory.com/cosmo/cosmo2.htmlhttp://www.superstringtheory.com/cosmo/cosmo1.htmlhttp://www.superstringtheory.com/cosmo/cosmo3.htmlhttp://www.superstringtheory.com/cosmo/cosmo2.htmlhttp://www.superstringtheory.com/cosmo/cosmo1.htmlhttp://www.superstringtheory.com/cosmo/cosmo2a.htmlhttp://www.superstringtheory.com/cosmo/cosmo2.htmlhttp://www.superstringtheory.com/cosmo/cosmo1a.htmlhttp://www.superstringtheory.com/cosmo/cosmo1.htmlhttp://arxiv.org/abs/gr-qc/9906059http://arxiv.org/abs/hep-th/0006012http://arxiv.org/abs/hep-th/9908142http://arxiv.org/abs/hep-th/9711162http://arxiv.org/abs/hep-th/9901146http://arxiv.org/abs/hep-th/9810188http://arxiv.org/abs/hep-th/0007175


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Take a tour through the chain of physical events that cosmologists believe occurred whilethe expanding Universe we observe today was very small and very young.Take the trip

Theres a lot of compelling evidence for the Big Bang, but what preceded it? The most accepted

model is called Inflation, but its not the kind of inflation that Alan Greenspan need fear.basic/advanced

What happens when the early universe is gummed up with string? And are any of these scenariostestable in the near future?basic/advanced

First ingredient: quantum mechanics

In the early 20th century, it was realized that the stability of atomic matter could not be explainedusing the Maxwell equations of classical electrodynamics. This triumph belonged to quantummechanics. The hydrogen atom was stable because the possible energy states of the electronin the atom are quantizedby the rule

where n is an integer, and is (approximately) the electron mass.

So when the electron changes energy for some reason, say by absorbing or emittingelectromagnetic radiation, it can only absorb or emit light of a wavelength corresponding to thedifference in quantized energy states of the electron. The collection of wavelengths of light emittedby hydrogen gas is called the emission spectrum of hydrogen, and there is a correspondingspectrum for absorption. One of the great successes of quantum mechanics was the calculation ofthe wavelengths in the observed hydrogen spectrum.

Second ingredient: relativity

The other great revolution that started the 20th century was the space-time revolution of specialand general relativity. In special relativity, when a source of light of wavelength emis moving away

from an observer at some velocity v, the observer sees the light at some other wavelength obs,determined by the principle that the speed of light is the same for all observers. The fractionaldifference between emand obsis called the red shift, denoted by the letter z, and is computedfrom the relative velocity v between the source and observer by

where c is the speed of light. If the source and observer are moving towards one another, the redshift becomes a blue shift and is given is given by taking v -> -v in above.

Conclusion: the Universe is expanding

Stars are made mostly out of hydrogen and helium, and the emission spectrum of the hydrogenatoms in a star in a far away galaxy ought to be the same as that of hydrogen atoms in a tube of
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gas in a laboratory on Earth. But thats not what Edwin Hubble found when he compared theemission spectra of different stars and galaxies. Hubble found that the emission wavelengths of thehydrogen gas were red shifted by an amount proportional to their distance from our solar system.Hubbles Lawrelates the red shift z to the distance D through

where the empirical constant H0is called Hubbles constant.

Hubbles observation suggested that the stars and galaxies in the Universe are hurtling away fromone another with a velocity that increases with distance, as if the whole Universe was expanding,like in a big explosion. When physicists extrapolated that motion backwards in time, it suggestedthat the Universe started out very hot and dense and somehow exploded into the huge cold placethat we see today. Hubbles Law was an empirical observation that demanded, and received, veryintense attention from modern theoretical physics after it was first proposed in 1924.

The equation of motion

When physicists want to study a given system, they turn to the equations of motion for that system.According to the theory of general relativity, the correct equation of motion for describing a Universeis the Einstein equation

relating the curvature of the space-time in a given Universe to the distribution of energy andmomentum in that Universe. The energy-momentum tensor Tincludes all of the energy from allnongravitational sources such as matter, electromagnetism or even quantum vacuum energy as we

shall see later.

The standard cosmological solution to the Einstein equation is written in the form of the Friedman-Robertson-Walker metric

The function a(t) is called the scale factor, because it tells us the size of the Universe. The scalefactor a(t) and the constant k are both determined by the particular type of matter and/or radiation

present in the Universe. This will be described in the next section.For any value of a(t) or k, the gravitational red shift z of light due to the changing size of theUniverse satisfies

where tobsis the time in the Universe that the light is being observed and temis the time when thelight was first emitted.

The Hubble parameter H(t) gives the relative rate of change in the scale factor a(t) by


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The observed Hubble constant is just the current value of the dynamically evolving Hubbleparameter. The uncertainties of the currently observed value of the Hubble constant have beenlumped into the parameter h0.

How old?

A quick approximation for the age of the Universe can be approximated by the inverse of the Hubbleconstant. The calculated age turns out to be

Current best estimates of h0are

so the Universe is most likely somewhere between 12 and 16 billion years old, at least accordingto this method of estimation.

But recall that according to relativity, time is relative. We can guess the amount of time likely tohave elapsed since the time when time was a meaningful quantity that could be measured. But wecant say anything about any processes that might have occurred before the notion of time madesense. In some sense, quantum gravity could be an eternal stage of the Universe, and the Big Bangcould be regarded as the end of eternity and the beginning of time itself.

The starting point of a theoretical exploration of cosmology is the Einstein equation

with a metric of the form

The space-time being modeled by this equation can be neatly separated into time and space, so wecan talk of this space-time as representing the evolution of space in time.

The space part of this space-time is homogeneous (looks the same at any point in a given direction)and isotro

The String Theory

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