McMahon D. String Theory Demystified

Подождите немного. Документ загружается.

String Theory Demystiﬁ ed

of gravity, are not attached to D-branes. They can travel or “leak off” a D-brane,

so we don’t see as many of them. This explains what until now has been a great

mystery, why electromagnetism (and the other known forces) is so much stronger

than gravity.

So this picture of the universe has a three-dimensional brane (or 3D-brane)

embedded in a higher-dimensional space-time called the bulk. Since we interact

with the physical world primarily through electromagnetic forces (light, chemical

reactions, etc.), which are mediated by particles that are really strings stuck to the

brane, we experience the world as having three spatial dimensions. Gravity is

mediated by strings that can leave the brane and travel off into the bulk, so we see

it as a much weaker force. If we could probe the bulk somehow, we would see that

gravity is actually comparable in strength.

HIGHER DIMENSIONS

We live in a world with three spatial dimensions. In a nutshell this means that there

are three distinct directions through which movement is possible: up-down, left-

right, and forward-backward. In addition, we have the ﬂ ow of time (forward only as

far as we know). Mathematically, this gives us the relativistic description of

coordinates

(, , ,

)

xyzt

It is possible to imagine a world where one of the spatial directions or dimensions

have been removed (say up-down). Such a two-dimensional world was described by

Edward Abbott in his classic Flatland. What if instead, we added dimensions? This

idea is actually pretty useful in physics, because it provides a pathway toward

unifying different physical theories. This kind of thinking was originally put forward

by two physicists named Kaluza and Klein in the 1920s. Their idea was to bring

gravity and electromagnetism into a single theoretical framework by imagining that

these two theories were four-dimensional limits of a ﬁ ve-dimensional supertheory.

This idea did not work out, because back then people did not know about quantum

ﬁ eld theory and so did not have a complete picture of particle interactions, and did

not know that the fully correct description of electromagnetic interactions is provided

by quantum electrodynamics. But this idea has a lot of appeal and reemerged in

string theory.

Kaluza and Klein had to explain why we don’t see the higher dimension, and hit

upon the idea of compactiﬁ cation—a procedure where we make the higher

dimensions so small they are not detectable at lower energy (i.e., on the kind of

energy scales that we live in). If they are small enough, the extra dimensions can’t

be noticed or detected scientiﬁ cally without the existence of the appropriate

technology. If they are so small that they are on the Planck scale, we might not be

able to see them at all. This concept is illustrated in Fig. 1.8.

CHAPTER 1 Introduction

String theory requires the existence of extra spatial dimensions for technical

reasons that we will discuss in later chapters. An interesting side effect of these

extra dimensions is that another mystery of particle physics is done away with.

Experimentalists have worked out that there are three families of particles. For

example, when considering leptons, there is the electron and its corresponding

neutrino. But there are also the “heavy electrons” known as the muon and the tau,

together with their corresponding neutrinos, that are really just duplicates of the

electron. The same situation exists for the quarks. Why are there three particle

families? And why are there the types of particle interactions that we see? It turns

out that higher spatial dimensions together with string theory may provide an

answer.

The way that you compactify the extra dimensions (the topology) determines the

numbers and types of particles seen in the universe. In string theory this results

from the way that the strings can wrap around the compactiﬁ ed dimensions,

determining what vibrational modes are possible in the string and hence what types

of particles are possible.

One important compactiﬁ ed manifold that we will see is called the Calabi-Yau

manifold. A Calabi-Yau manifold that compactiﬁ es six spatial dimensions and

leaves three spatial dimensions “macroscopic” plus time gives a ten-dimensional

universe as required by most of the string theories. A key aspect of Calabi-Yau

manifolds is that they break symmetries. Thus another mystery of particle physics

is explained, so-called spontaneous symmetry breaking (see Quantum Field Theory

Demystiﬁ ed for a description of symmetry breaking).

Close up, we see the full

dimension of a cylinder

If the radius of the cylinder

is very small, from far away

the cylinder appears one-

dimensional, as a line

Figure 1.8 Compactiﬁ cation explains why we may not be aware of extra

spatial dimensions even if they exist. If the radius of a cylinder is very

small, from far away it looks like a line.

String Theory Demystiﬁ ed

Summary

Quantum mechanics and general relativity were the major developments in

theoretical physics in the twentieth century. Unifying them into a single theoretical

framework has proven extremely challenging, if not impossible. This is because the

resulting quantum theories are plagued by inﬁ nities that result from the fact that

interactions take place at a single mathematical point (zero distance scale). By

spreading out the interactions, string theory offers the hope of developing not only

a uniﬁ ed theory of particle physics, but a ﬁ nite theory of quantum gravity.

Quiz

1. If

=+−>480JD

and

p → ⬁

then

(a) the loop integral is convergent.

(b) the loop integral diverges.

2. The scale of the Planck length and Planck mass tell us that quantum gravity

(a) operates on small-distance and high-energy scales.

(b) is nonsensical.

(d) operates on large-distance and small-energy scales.

3. Perturbation theory is possible in quantum electrodynamics because

(a)

(b)

= 1

(c)

(d) Perturbation theory is not possible in quantum electrodynamics

4. The quantum uncertainty relations are modiﬁ ed in string theory as

(a)

∆

∼



(b)

∆

′

∆

∼



(c)

∆

−

′

∆

∼



(d)

∆

′

∆

∼



CHAPTER 1 Introduction

5. The minimum distance scale in string theory is about

(a)

min

∼

′

(b)

Smin

∼ 2

(c)

min

∼ 0

(d)

min

6. The topology of compactiﬁ ed dimensions

(a) determines the types of particles seen in the universe.

(b) has no impact on particle interactions.

7. Heterotic string theory has the gauge group

(a)

(b)

SU()32

(c)

(d)

SO()16

8. String theory explains the difference between electromagnetism and gravity as

(a) String theory provides the uniﬁ cation energy of gravity and electromagnetism.

(b) Gravitons are not connected to the brane, so can leak off into the bulk

making gravity appear much weaker than electromagnetism.

more prominent.

9. Bosonic string theory is not realisitic because

(a) it includes 26 space-time dimensions.

(b) it does not allow Calabi-Yau compactiﬁ cation.

(d) it lacks a

symmetry group.

10. In string theory particle decay is explained by

(a) a string splitting apart into multiple daughter strings.

(b) it remains poorly understood.

(d) strong vibrational modes that decouple the string.

This page intentionally left blank

The Classical String I:

Equations of Motion

When you studied classical mechanics and quantum ﬁ eld theory, you learned about

the action and deriving the equations of motion from the Euler-Lagrange equations.

This can be done in the case of the string, and it can be done relativistically. If we

are going to consider a uniﬁ ed theory of physics, this is a good place to start—

ensuring that we understand how to describe the dynamics of strings in a manner

that is fully consistent with relativity before moving on to introduce the quantum

theory.

When we quantize our strings, our ﬁ rst foray into a fully relativistic, quantum

theory will be an instructive but unrealistic case, the bosonic string. As the name

implies, we are going to look at a theory consisting exclusively of bosons—that is,

states with integral spin. We know that this cannot be a realistic theory because in

CHAPTER 2

String Theory Demystiﬁ ed

the actual universe while force-carrying particles are indeed bosons, fundamental

matter particles (like electrons) have half-integral spin, that is, they are fermions.

So a theory that describes a world consisting entirely of bosons does not describe

the real universe.

Nonetheless, we start here because it is an easier way to approach string theory

and we can learn the nuts and bolts in a slightly simpler context. We are going to

approach bosonic strings in three steps. In this chapter, we will develop the theory

of classical, relativistic strings starting with the action principle and deriving the

equations of motion. In Chap. 3, we will learn about the stress-energy tensor and

conserved currents, speciﬁ cally conserved worldsheet currents. Finally, in the last

chapter of this part of the book, we will quantize the strings using a procedure of

ﬁ rst quantization (i.e., ﬁ rst quantization of point particles gives single-particle

states). In the end you have a quantized relativistic theory.

To this end, we begin our journey into the world of classical relativistic point

particle moving in space-time to illustrate the techniques used.

The Relativistic Point Particle

The task at hand is to describe the motion of a free (relativistic) point particle in

space-time. One way to approach the problem is by using an action principle.

Before we do that, let’s set up the arena in which the particle moves. Let its motion

be deﬁ ned with respect to space-time coordinates

where

is the timelike

coordinate (i.e.,

Xct

) and

where

i ≠

are the spacelike coordinates (say x, y,

and z). While you are probably used to lowercase letters like

to represent

coordinates, in string theory uppercase letters are used, so we will stick to that

convention.

Anticipating the fact that string theory takes place in a higher-dimensional arena,

rather than the usual one time dimension and three spatial dimensions we are used

to, we consider motion in a D-dimensional space-time. There is one time dimension

but now we allow for the possibility of

dD=−

spatial dimensions. We reserve 0

to index the time dimension hence our coordinates range over

= 0, ,... d

Now, the motion or trajectory of a particle is described such that the coordinates

are parameterized by

, which parameterizes the world-line of the particle. That is,

this is the time given by a clock that is moving or carried along with the particle

itself. We can emphasize this parameterization by writing the coordinates as

functions of the proper time:

µµ

= ()

(2.1)

CHAPTER 2 Equations of Motion

To describe distance measurements, we are going to need a metric, that is, a function

which allows us to deﬁ ne the distance between two points. Here we will stick with

special relativity and use the ﬂ at space Minkowski metric which is usually denoted

. You may recall that the time and spatial components of the metric have

different sign; the choice used is referred to as the signature of the metric. In string

theory, it is convenient to place the negative sign with the time component, so in the

case of

d = 3

spatial dimensions we can write the Minkowski metric as a matrix

µν

−

⎛

⎝

⎜

⎞

⎠

⎟

1000

0100

0010

0001

(2.2)

More compactly, we can write

µν

=−+++(,,,)

. Generalizing to D-dimensional

Minkowski space-time, we simply associate a plus sign with products of spatial

coordinates. So the Lorentz invariant length squared of a vector is

() ()() ()XXXXX X

µν

202122

==−+++$

(2.3)

Inﬁ nitesimal lengths or distances are described by

ds dX dX dX dX dX

d202122

=− = − −

µν

()() ()$

(2.4)

We include the minus sign out in front of the metric in Eq. (2.4) to ensure that

ds dX dX=−

µν

is real for timelike trajectories. With these notations in hand,

we are ready to describe the trajectory of a free relativistic particle using the action

principle.

The action principle tells us that the relativistic motion of a free particle is

proportional to the invariant length of the particles trajectory. That is,

Sds=−

∫

(2.5)

First let’s ﬁ gure out what the constant of proportionality is.

EXAMPLE 2.1

Given that the action of a free, non-relativistic particle is

Sdtmv

12=

∫

(/) ,

where

m is the mass of the particle and v is the particle velocity, determine the nature of

the constant in Eq. (2.5).

String Theory Demystiﬁ ed

SOLUTION

For simplicity, we consider motion in one spatial dimension. Now

Sds

dt dx

=−

=− −

∫

Now recall the binomial theorem. This tells us that

±≈±xx

Hence,

−≈−vv

Therefore

Sdtv

dt v dt dt v

=− −

≈− −

⎛

⎝

⎜

⎞

⎠

⎟

=− +

∫

∫∫

ααα

∫∫

Comparison of the second term in this expression with

Sdtmv

12=

∫

(/ )

tells us

that

must be the mass of the particle.

We can also determine the units of

and deduce that it is the mass of the

particle from dimensional analysis. First, what are the units of action? Recall from

your studies of quantum theory that the units of action from Planck’s constant

are

mass times length squared per time:

[]? =

(2.6)

CHAPTER 2 Equations of Motion

Now let’s look at

Sds=−

∫

. From the integral, we have length L, so we have

= []

So it must be the case that

[]

We can obtain this result using the mass of the particle together with the speed of

light c, which is of course a length over time. That is,

⇒=

[]

In units where

c ==? 1

, which are commonly used in particle physics and string

theory, the action is dimensionless. Hence mass is inverse length and

⇒==

[]

(2.7)

Now let’s see how to write down the action and obtain the equations of motion

from it. We start with the deﬁ nition of inﬁ nitesimal length given in Eq. (2.4). This

gives the action as

S m dX dX=− −

∫

µν

(2.8)

Let’s rewrite the integrand:

−=−

⎛

⎝

⎜

⎞

⎠

⎟

=−

ηη

τη

µν

dX dX

τττ

τη

µν

dXX=−

