University of LjubljanaFaculty of Mathematics and Physics

Department of Physics

STRINGS

Tine PorentaMentor: prof. dr. Norma Mankoč-Borštnik

Ljubljana, March 2007

AbstractString theory is a model of fundamental physics whose building blocks are one-dimensionalextended objects called strings, rather than the zero-dimensional point particles that formthe basis for the Standard Model of particle physics. String theorists are attempting toadjust the Standard Model by removing the assumption in quantum mechanics that par-ticles are point-like. By removing this assumption and replacing the point-like particleswith strings, it is hoped that string theory will develop into a sensible quantum theory ofgravity. Moreover, string theory appears to be able to ’unify’ the known natural forces(gravitational, electromagnetic, weak nuclear and strong nuclear) by describing them withthe same set of equations. This seminar will be divided into three main topics: firstly stan-dard model is described, secondly string theory is represented and finally the mathematicsbehind is revealed.

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Contents

1 Introduction 31.1 Standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Strings 62.1 The begining of String theory . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Assumed propertires of strings . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Benefits of strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Free particle → Free string 103.1 Action principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Boundary conditions determine type of strings . . . . . . . . . . . . . . . . 13

4 Conclusion 144.1 David Gross’s thinking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Einstein’s anecdote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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Chapter 1

Introduction

1.1 Standard modelThe Standard Model successfully describes fermions interacting with corresponding bosonsthrough charges (electromagnetic, weak and colour). The Lagrangian of each set of bosonsis invariant under a transformation called a gauge transformation, so these bosons arereferred to as gauge bosons. The bosons in the Standard Model are:

• Photons, which mediate the electromagnetic interaction.

• W and Z bosons, which mediate the weak nuclear force.

• Eight species of gluons, which mediate the strong nuclear force.

• The Higgs bosons, which induce spontaneous symmetry breaking of the electroweakgauge group and are responsible for the existence of mass.

Charges of fermions and bosons are described by using a unitary group called a ’gaugegroup’. Therefore, the Standard Model is often referred to as SU(3) × SU(2) × U(1),where SU(3) is the gauge group for colour charge, SU(2) for weak charge and U(1) forelectromagnetic charge. The Higgs boson is the only boson in the theory which is notrepresented as vector field. The Higgs has never been observed in experiments, and findingit is one of major goals of experimental particle physics today. Gravitons, the bosonsbelieved to mediate the gravitational interaction, are not accounted for in the StandardModel.

There are three different Families of fermions in the Standard Model. The protons andneutrons are made up of two of these: the up quark and down quark, bound togetherby the colour force. Together with the electron (bound to the nucleus in atoms by theelectromagnetic force), those fermions constitute the vast majority of everyday matter.All of the fundamental fermions in the Standard Model are given in Table 1.

The two additional families differ from the first one and among themselves only (asfar as known) in their masses. For example, the electron and muon both have half-integerspin, unit electric charge and do not participate in the colour interaction, but the muon isabout 200 times more massive than electron.

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1.2 ProblemsThe Standard Model plus gravity has fairly simple structure: the four interactions demon-strate local invariance principle. In spite of its impressive success, this theory is surely nota complete theory.

1. It is too arbitrary: why does this particular pattern of gauge fields and multipletsexist, and what determines the parameters in the Lagrangian?

2. While the three interactions (electromagnetic, weak, colour) can be treated as quan-tum field, thus is not the case for gravity.

3. Even on a classical level theory breaks at the singularities of general relativity.

4. The theory is in certain sense unnatural: some of the parameters in the Lagrangianare much smaller than one would expect to be.1

1In particle physics, the hierarchy problem is the question why the weak force is 1032 times strongerthan gravity. Both of these forces involve constants of nature, Fermi’s constant for the weak force andNewton’s constant for gravity. Furthermore if the Standard Model is used to calculate the quantumcorrections to Fermi’s constant, it appears that Fermi’s constant is unnaturally large and should be closerto Newton’s constant unless there is a delicate cancellation between the bare value of Fermi’s constantand the quantum corrections to it.

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Figure 1.1: Table 1. Famous picture of Standard model from CERN.

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Chapter 2

Strings

2.1 The begining of String theoryString theory grew out of a model that Gabriel Veneziano wrote down in 1968 to describethe world of nuclear particles and their interactions. Despite much initial excitement, themodel failed. It was abandoned several years later in a favor of quantum chromodinamics,which describes nuclear particles in terms of quarks. Quarks are confident inside a protonor a neutron, as if they were tied together by elastic strings. In retrospect, the originalstring theory had captured those stringy aspects of nuclear world. Only later was revivedas candidate for combining general relativity and quantum theory.

The basic idea is that elementary particles are not point like but rather infinitely thinone-dimensional objects - strings. The large zoo of elementary particles, each with itsown characteristic properties, reflects the many possible vibration patterns of a string.The answer, how this theory can describe such complicated world of particles and theirinteractions, can be found in quantum string ’magic’.

Figure 2.1: Diverse modes of a vibration can be induced in any string. Quantum mechanicsallows the waves to be interpreted as particles. If loops of a string about 10−35 m longare fundamental constituents of matter, then their vibrational energies are masses of theelementary particles such as electrons, quarks and photons.

2.2 Assumed propertires of stringsFirst, quantum strings have finite size. Were it not for quantum effect, a violin stringcould be cut in half, cut in half again and so on all the way down, finally becoming amassless point like particle. But the Heisenberg uncertainty principle eventually intrudesand prevents the lightest strings from being sliced smaller than Planck length, which is

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1.61624 10−35 meter. This irreducible quantum of length plays a crucial role in almost everyaspect of string theory, putting a finite limit on quantities that otherwise could becomeeither zero or infinite.

Second, quantum strings may have angular momentum even if they lack mass. Inclassical physics, the formula for angular momentum multiplies together velocity mass anddistance; hence, massless object have no angular momentum. But quantum fluctuationschange the situation. A tiny string can acquire up to 2 h of angular momentum withoutgaining any mass. This feature is very welcome because it precisely matches the propertiesof carriers of all known fundamental forces (photon, graviton,. . . ).

Third, quantum strings demand the existence of extra space dimensions. Whereasa classical violin string will vibrate no matter what the properties of space and time are,quantum string is more finicky. The equations describing the vibration become inconsistentunless spacetime either is highly curved(in contradiction with observations) or contains sixextra spatial dimensions.

Figure 2.2: Reducing dimensions of a space can be achieved by pasting its edges togetherand shrinking it. For example, a two-dimensional sheet of rubber is first curled into acylinder, and the curled dimension is then shrunk. When thin enough, the cylinder looklike a (one-dimensional) line. Twisting around this length of "hose" and sticking its edgetogether, one gets a doughnut shape. The radius of the doughnut can be shrunk until itis small enough to approximate a point - a zero dimensional space. Such changes couldexplain why the extra dimensions of spacetime that string theory says must exist are tosmall too be detectable.

Fourth, physical constant (gravitational G, αe = e2

4πε0hc, αW and αC , which appear in

physics and determine the properties of matter) no longer have arbitrary, fixed values. Theyoccur in string theories as fields, rather like the electromagnetic field, that can adjust theirvalues dynamically. These fields may have taken different values in different cosmologicalepochs or in remote regions of space, and even today physical ’constants’ may vary bya small amount. One such field, called dilaton, is the master key to string theory; itdetermines the overall strength of all interactions. The dilaton fascinates string theoristsbecause its value can be reinterpreted as the size of an extra dimension of space, giving agrand total of 11 spacetime dimensions.

2.3 Benefits of stringsIt looks like that string approach is the only one way known to spread out the gravitationalinteraction and cut off the divergence without spoiling the consistency of the theory. Sowhat do we find if we pursue the string theory?

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1. Every consistent string theory contains a massless spin-2 state called graviton1, whoseinteractions reduce at low energy to general relativity.

2. It looks like that string theory offers the way to consistent theory of quantum gravity.

3. String theories lead to gauge groups large enough to include the Standard model.

4. String theory requires a definite number of spacetime dimensions. The field equa-tions have solutions with four large flat and six small curved dimensions, with four-dimensional physics that resembles the Standard Model(There is approksimatly 10500

scenarios).

5. Consistent string theories require spacetime supersymmetry 2.

6. From only one parameter all the rest follow.

This is a remarkable list from simple supposition of one dimensional objects. First twopoints alone would be of great interest. The next three points come striking. And the lasttighten the structure and make it less arbitrary.

1In physics, the graviton is a hypothetical elementary particle that mediates the force of gravity in theframework of quantum field theory. If it exists, the graviton must be massless (because the gravitationalforce has unlimited range) and must have a spin of 2 (because gravity is a second-rank tensor field).

2In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry that interchanges bosonsand fermions. In supersymmetric theories, every fundamental fermion has a bosonic superpartner andvice versa. A supersymmetric quantum field theory tames quantum mechanical dynamics and sometimesallows the theory to be solved. If supersymmetry is applied to the Standard Model of particle physics, thehierarchy problem can be solved.

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Figure 2.3: String Theory predicts the existence of more than 3 space dimensions and 1time dimension we are all familiar with. According to string theory, there are additionaldimensions that we are unfamiliar with because they are curled up into tiny complicatedshapes that can only be seen on tiny scales. If we could shrink to this tiny, Planck-sizedscale, we could see that at every 3D point in space, we can also explore 6 additionaldimensions. This picture shows an array of Calabi-Yau spaces which are projections ofthese higher dimensions into the more familiar dimensions we are aware of.[?]

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Chapter 3

Free particle → Free string

3.1 Action principlesWe want to study the classical and quantum mechanics of one-dimensional object, a string.The string moves in D flat spacetime dimensions with metric ηµν = diag(+,−,−, . . . ,−).

Figure 3.1: World-line for point particle.

It is useful to begin with review of classic mechanics of a zero-dimensional object, arelativistic point particle. The best way to describe the motion of a particle is to introducea parameter τ along the particle’s world-line and describe the motion in spacetime by Dfunctions Xµ(τ). The parameterization is arbitrary: a different parameterization of thesame path is physically equivalent, and all physical quantities must be independent of thischoice. That is, for any monotonic function τ ′(τ) the two paths X ′µ and Xµ are the same,where:

X ′µ(τ ′(τ)) = Xµ(τ). (3.1)

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The simplest Poincaré-invariant action for free particle that does not depend on parame-terization would be proportional to the proper time along world-line:

Spp = −mc∫

dτ(XµXµ)1/2, (3.2)

where a dot denotes τ -derivate. The variation of the action, after integration by parts, is

δSpp = −mc∫

dτ uµδXµ (3.3)

where uµ = Xµ(XνXν)−1/2 is the normalized D-velocity. The equation of motion uµ = 0

thus describes free motion. The normalization constant m is the particle’s mass, as can bechecked by looking at the nonrelativistic limit.

The action can be put in another useful form by introducing an additional field onthe world-line, an independent world-line metric γττ (τ). Usual is more convenient to workwith the η(τ) = γ(τ)1/2, which is defined to be positive. Then

S ′pp = −1

2

∫dτ(η−1XµXµ + ηm2c2), (3.4)

where η2 = XµXµ/m2c2.

Figure 3.2: World-sheet for an open string

A one-dimensional object will sweep out a two-dimensional worldsheet1, which canbe described in terms of two parameters Xµ(xa) = Xµ(τ, σ). As in the case of particle,

1In physics, a world line of an object (approximated as a point in space, e.g. a particle or observer) is

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we insist that physical quantities such as the action depend only on the embedding inspacetime and not on the parameterization, which is necessary in a consistent relativisticquantum theory. The simplest invariant action, the Nambu-Goto action, is proportionalto the area of the world-sheet. We first define the induced metric hab where indices a, b, . . .run over values xa = (τ, σ):

hab = ∂aXµ∂bXµ (3.5)

The Nambu-Goto action is

SNG =∫

MdτdσLNG, (3.6)

LNG = − 1

2πα′(dethab)

1/2, (3.7)

where M denotes the world-sheet. α′ is some constant, which has units of spacetime-length-squared, is the Regge slope, which significance we will not meet in our seminar.

The Nambu-Goto action is analogous to the point particle action Spp, with derivatesin the square root. Again we can simplify it by introducing an independent world-sheetmetric γab(τ, σ). We will take γ = |detγab| and γab to have Lorentzian signature (+,−).The action is called the Brink-Di Vecchia-Howe-Desser-Zumino action or Polyakov action:

Sp[X, γ] = − 1

4πα′

∫M

dτdσγ1/2γab ∂aXµ∂bXµ (3.8)

3.2 SymmetriesLet us now consider the symmetries of the action, transformations of Xν(τ, σ) such thatSp(X

′, γ′) = Sp(X, γ). Sp has the following symmetries:

1. D-dimensional Poincaré invariance:

X ′µ(τ, σ) = ΛµνX

ν(τ, σ) + aµ (3.9)γ′ab(τ, σ) = γab(τ, σ) (3.10)

with Λµν a Lorentz transformation and aµ a translation.

2. Two-dimensional coordinate invariance, often called diffeormorphism (diff) invari-ance. For new coordinates x′a(xa) = (τ ′(τ, σ), σ′(τ, σ)), the transformation is:

X ′µ(τ ′, σ′) = Xµ(τ, σ) (3.11)∂x′c

∂xa

∂x′d∂xb

γ′cd(τ′, σ′) = γab(τ, σ) (3.12)

the sequence of spacetime events corresponding to the history of the object. A world line is a special typeof curve in spacetime. Below an equivalent definition will be explained: A world line is a time-like curvein spacetime. Each point of a world line is an event that can be labeled with the time and the spatialposition of the object at that time. A world line traces out the path of a single point in spacetime. Aworld sheet is the analogous two-dimensional surface traced out by a one-dimensional line (like a string)traveling through spacetime. The worldsheet of an open string (with loose ends) is a strip; that of a closedstring (a loop) is a cylinder.

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3.3 Boundary conditions determine type of stringsVarying Xµ gives the equaton of motion:

∂aγ1/2γab∂bX

µ = γ1/2∇2Xµ = 0 (3.13)

For world-sheet with boundary there is also a surface term in the variation of the action.To be specific, take the coordinate region to be

−∞ < τ < ∞, 0 ≤ σ ≤ l. (3.14)

We will take τ as a time variable and σ as spatial, so this is a single string propagatingwithout sources. Then

δSp =1

2πα′

∫ ∞

−∞dτ

∫ σ

0dσγ1/2δXµ∇2Xµ − 1

2πα′

∫ ∞

−∞dτγ1/2δXµ∂σXµ

∣∣∣∣σ=l

σ=0.[?] (3.15)

The boundary term vanishes if:

• we take Neumann boundary conditions on Xµ:

∂σXµ(τ, 0) = ∂σXµ(τ, l) = 0, (3.16)

or stated more covariantly,

na∂aXµ = 0 on ∂M, (3.17)

where na is the normal on the boundary ∂M . The ends of the open string movefreely in spacetime.

• or if we impose

Xµ(τ, 0) = Xµ(τ, l), ∂σXµ(τ, 0) = ∂σXµ(τ, l), γab(τ, 0) = γab(τ, l) (3.18)

These fields are periodic. There is no boundary; the endpoints are joints to form aclosed loop.

The open string boundary and closed string boundary are the only possibilities consistentwith D-dimensional Poincaré invariance and the equation of motion.

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Chapter 4

Conclusion

4.1 David Gross’s thinkingGross is one of the leading lights of string theory, and yet he is the first to admit thatphysicists really don’t know what string theory is. ’It’s as if we’ve stumbled in the darkinto what we thought was a two-bedroom apartment and now we’re discovering is a 19-room mansion,’ he says. ’At least. Maybe it’s got a thousand rooms, and we’re justbeginning our journey.’[?]

4.2 Einstein’s anecdoteTo conclude with, Einstein was once asked about some experiments that contradicted histheory of relativity. Someone asked him, ’So professor Einstein, what would you say, ifthese turned out to be true and your theory of relativity was wrong?’ And Einstein said,’Oh, I would have been very disappointed that God didn’t take advantage of this beautifulidea.’

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Bibliography

[1] B. Green: The fabric of cosmos, Random House, (2004)

[2] J. Polchinski: String theory, Volume 1, Cambridge University Press (1998)

[3] B. Zweibach: A first course in string theory (T)(369s) (2003)

[4] Scientific america, Explaining everything, January 1996

[5] Scientific america, The theory formerly known as strings, 2003

[6] Scientific america, The time before time, May 2004

[7] Scientific america, The string theory landscape, September 2004

[8] David Gross Viewpoints on String Theory, http://www.pbs.org/wgbh/nova/elegant/view-gross.html, March 2007

[9] Jeff Bryant Higher Dimensions from String Theory, http://members.wri.com/-jeffb/visualization/stringtheory.shtml, March 2007

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