McMahon D. String Theory Demystified

Подождите немного. Документ загружается.

String Theory Demystiﬁ ed

The Virasoro constraint becomes



XX XX

ii−−

′

=±

′

()

(4.32)

So we have a relation between light-cone coordinates and the transverse

coordinates. The mode expansion of the worldsheet coordinates for an open string

is given by

Xx p i

µµ µ

=+ +

≠

−

∑

cos

In particular

Xx p i

in−− −

−

≠

−

=+ +

∑

cos

We can then solve for the nonzero modes of

−

in terms of the transverse oscillators.

These are

αααδ

−

=−∞

∞

−

=−

⎛

⎝

⎜

∑∑

⎞⎞

⎠

⎟

(4.33)

In the case of the zeroth mode, we can derive an expression for the hamiltonian

== +

′

⎛

⎝

⎜

⎞

⎠

⎟

−

≠

∑

αα

(4.34)

Deﬁ ne a conjugate momentum

. The system is quantized by imposing

commutation relations on the transverse components of position and momentum:

[, ] ,[ (), ( )] ( )xp i X P i

i j ij i j ij

′

=−

′

δσσδδσσ

(4.35)

The mass-shell condition becomes

Mpp p Na

=−=−

+−

−

∑

()

(4.36)

The number operator is given by

−

∞

−

∑∑

αα

(4.37)

CHAPTER 4 String Quantization

Normal ordering leads to

αα αα

−

=−∞

∞

−

=−∞

∞

∑∑∑

−

∞

∑∑

−

The regularization trick can be applied to make the second term ﬁ nite. We ﬁ nd

−

=−

−

∞

∑

Again taking

a =

, one ﬁ nds

D = 26

In the previous two chapters, we constructed a relativistic theory of the string, the

classical theory. In this chapter we have introduced the simplest possible quantum

extension of the classical theory. This is a theory that consists only of bosons.

While the theory is not realistic since it does not include fermionic states, it is easier

to deal with and introduces important concepts and methods that will play a role in

the full quantum theory. The classical theory was quantized using two different

methods. The ﬁ rst method, called covariant quantization, is a straightforward

approach that imposes commutation relations on the

and their conjugate

momenta. This leads to negative norm states. The Virasoro constraints are imposed

to rid the theory of these states. When this is done, we ﬁ nd that the theory must have

26 space-time dimensions. We concluded the chapter with a different approach,

known as light-cone quantization.

Quiz

1. Explicitly calculate the commutators

[ (,), ( ,)]∂

′

∂

′

+−

µν

στ σ τ

and

[ (,), (,)]∂

′

∂

′

−+

µν

στ σ τ

2. Consider the closed string and explicitly calculate

[, ]xp

µν

3. Consider the ﬁ rst excited state of the closed string

εαα

µν

() ,kk

−−11

. Using the

condition satisﬁ ed by physical states

, in particular

ψψ

ﬁ nd

µν

Summary

String Theory Demystiﬁ ed

4. Let

−1

be the state of an open string and suppose that the normal

ordering constant a is undetermined. What is the mass of this state?

5. In the light-cone gauge, use

[, ] ,[ (), ( )] (

)

xp i X P i

i j ij i j ij

′

=−

′

δσσδδσσ

to ﬁ nd a commutation relation for the transverse modes,

[, ]

αα

6. Consider light-cone quantization for the closed string case. What additional

commutation relation do you think should be imposed for the modes?

Conformal Field

Theory Part I

In this chapter we study conformal ﬁ eld theory, an area of quantum ﬁ eld theory that

relies heavily on complex variables. In this chapter, we will introduce some of the

basic concepts of conformal ﬁ eld theory. The topic will be expanded and utilized in

many areas in the rest of the book. In the next chapter, we discuss other aspects of

conformal ﬁ eld theory along with BRST quantization.

Conformal ﬁ eld theory is an important tool used in the analysis of perturbative

string theory, so it plays a central role in our task at hand (understanding the

physics of quantized strings). In particular, since the worldsheet can be described

using two coordinates

)

τσ

two-dimensional conformal ﬁ eld theory is used. The

theory of complex variables plays an important role in the study of theoretical

physics, and string theory is no exception. If you are not familiar with complex

variables you should take time out to study the topic before proceeding any

further. You can do so using my book Complex Variables Demystiﬁ ed, also

published by McGraw-Hill.

CHAPTER 5

String Theory Demystiﬁ ed

In the theory of complex variables, a conformal transformation is one that maps

a region of the complex plane to a new more convenient region while preserving

angles, but not lengths. For example, you can map the unit disk to the upper-half

plane using a conformal transformation.

The notion that angles are preserved is a geometric interpretation and leads us to

the notion of a conformal transformation of space-time coordinates. Let us consider

a transformation of the space-time coordinates such that

xx→

′

. In general, the

metric

µν

()

is found to transform in the following way:

µν

αβ

′

∂

′

∂

′

() ()

(5.1)

Now consider a function of the space-time coordinates given by

Ω(

)

. If the

metric transforms in the following way:

gx xgx

µν µν

′

=( ) () ()Ω

(5.2)

Then we have a conformal transformation of the metric. Notice that

Ω()x

acts

as a scaling factor, hence it will preserve angles but not lengths. If a metric is

related to the ﬂ at Minkowski metric as

gxx

µν µν

=Ω() ()

we say that the metric

is conformally ﬂ at.

To see how a conformal transformation preserves angles, consider two tangent

vectors

uv and

. Using the metric, the angle between them is given by

cos

(, )

(, )(, )

uu vv

Now we apply the transformation given by Eq. (5.2) and ﬁ nd

cos

(, )

(, ) (, )

()(, )

(

→

′

′′

xguv

uu vv

uvΩ

Ω

xxg xg

gg)(, ) ()(, )

(, )

(, )(, )uu vv

uu vvΩ

Hence a conformal transformation preserves angles.

A conformal ﬁ eld theory is a quantum ﬁ eld theory that is invariant under

conformal transformations. These theories are Euclidean quantum ﬁ eld theories,

meaning that a Euclidean metric is used. The symmetry group of such a theory will

contain Euclidean symmetries (we will review those in a moment) along with local

conformal transformations. It turns out that two-dimensional conformal ﬁ eld

theories are of particular use. Two-dimensional conformal ﬁ eld theories have an

inﬁ nite number of conserved charges.

CHAPTER 5 Conformal Field Theory Part I

There are two important properties of conformal ﬁ eld theories. These can be

summed up by saying that a conformal ﬁ eld theory is scale invariant. This manifests

itself in two ways:

• Conformal ﬁ eld theories have no length scale.

• Conformal ﬁ eld theories have no mass scale.

To see why this is important, we can consider the quantum ﬁ eld theory of a scalar

ﬁ eld. Let

()x

be a scalar ﬁ eld in d space-time dimensions. Consider a rescaling of

the coordinates which we write as a scale transformation:

′

=xx

(5.3)

Under a scale transformation, a scalar ﬁ eld

()x

transforms using the classical

scaling dimension

∆= −()/d 22

as follows:

φφλλφλφ

() ( ) () ()xxx x

→

′

−∆

−+

For a ﬁ eld theory to be scale invariant, we require that the action be invariant

under this transformation. This is, in fact, true when we consider a free, massless

scalar ﬁ eld. The action in this case is

Sdx

=∂∂

∫

φφ

Under the transformation

′

, it is clear that

dx dx

′

, that is,

dx dxdx dx dxdx

′

==⋅

−

()()( )( )()(

λλ λ λλλ

01 1 01

……))()…dx dx

−

Now recall that

∂

is shorthand for

∂∂/(

)

. So we pick up a copy of

under a

scale transformation:

∂

→

∂

∂xxx

µµµ

λλ

()

So we have

∂∂→

⎛

⎝

⎜

⎞

⎠

⎟

∂

′

⎛

⎝

⎜

⎞

⎠

⎟

∂

′

⎛

⎝

⎜

⎞

µµ

φφ

µµ

11 1

() ()

⎠⎠

⎟

∂

⎛

⎝

⎜

⎞

⎠

⎟

∂

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

−+ −+

λφ λφ

⎟⎟

∂∂= ∂∂

−+ −

λφφλφφ

µµ

dd2

String Theory Demystiﬁ ed

where we used

φλ λ φ

() ( ) ()

(/)

xx x

→

′

−+21

. It follows that the action is unchanged

under a scale transformation:

′

∂

′

∂

′

=∂∂=∂∂

∫∫ ∫

′

−

Sdx dx dx

ddddd

µµµ

φφ λ λ φφ φ

µµµµ

= S

The problem is that a scale transformation does not change a ﬁ xed quantity like

mass. Let’s consider a free scalar ﬁ eld with mass m. The action is

Sdx m

=∂∂−

∫

()

φφ φ

This action is not invariant under a scale transformation because

mm m

d22 2 22 22

′

=≠

−+

φλφφ

Often, quantum theory breaks scale invariance. A prototypical example is

theory

for a massless ﬁ eld. The classical action is scale invariant:

Sdx

=∂∂−

⎛

⎝

⎜

⎞

⎠

⎟

∫

φφ φ

The problem is, that renormalization introduces a ﬁ xed mass term to the theory.

As a result scale invariance is broken. Conformal ﬁ eld theories provide a way out of

this conﬂ ict by giving us quantum ﬁ eld theories that are scale and mass invariant.

The Role of Conformal Field Theory

in String Theory

A question you should be asking yourself is why are conformal ﬁ eld theories

important in string theory? It turns out that two-dimensional conformal ﬁ eld theories

are very important in the study of worldsheet dynamics.

A string has internal degrees of freedom determined by its vibrational modes.

The different vibrational modes of the string are interpreted as particles in the

theory. That is, the different ways that the string vibrates against the background

space-time determine what kind of particle the string is seen to exist as. So in one

vibrational mode, the string is an electron, while in another, the string is a quark, for

example. Yet a third vibrational mode is a photon.

CHAPTER 5 Conformal Field Theory Part I

As we have seen already, the vibrational modes of the string can be studied by

examining the worldsheet, which is the two-dimensional surface. It turns out that

when studying the worldsheet, the vibrational modes of the string are described by

a conformal ﬁ eld theory.

If the string is closed, we have two vibrational modes (left movers and right

movers) moving around the string independently. Each of these can be described by

a conformal ﬁ eld theory. Since the modes have “direction” we call the theories that

describe these two independent modes chiral conformal ﬁ eld theories. This will be

important for open strings as well.

A Euclidean metric is simply a metric that resembles the distance measure from

ordinary geometry. Let’s try to clarify this point considering the simpliﬁ ed case

of one-time dimension and one-space dimension. In special relativity, we

distinguish between space and time with the use of a change of sign so that if we

are using the signature

(, )−+

ds dt dx

222

=− +

. So the two-dimensional Minkowski

metric would be

µν

−

⎛

⎝

⎜

⎞

⎠

⎟

What we’re after with a Euclidean metric is describing things in a way that we could using

ordinary geometry. In the x-y plane, the inﬁ nitesimal measure of distance is given by

dr dx dy

. This tells us that a Euclidean metric is one for which all quantities have

the same sign. We can rewrite the Minkowski metric in this way by using what is known

as a Wick rotation. Simply put, we make a transformation on the time coordinate by

letting

t→

−

. Then

dt idt→−

and it follows that

ds idt dx dt dx

2222

=−− + = +()

which is exactly what we want. In order to describe the worldsheet with coordinates

(, )

τσ

using a Euclidean metric, we make a Wick rotation

ττ

→

−i

In terms of the worldsheet coordinates

(, )

τσ

, the metric is

ds d d

222

=− +

τσ

So we see that making a Wick rotation

→−i

changes this to

ds d d

22 2

τσ

which is a Euclidean metric. Utilizing the Euclidean metric enables us to use

conformal ﬁ eld theory on the string.

Wick Rotations

String Theory Demystiﬁ ed

Complex Coordinates

A consequence of a Wick rotation is that the light-cone coordinates

)

+−

are

replaced with complex coordinates

(, )zz

. The description of the worldsheet is

transformed into complex variables by deﬁ ning complex coordinates

(, )zz

which

are functions of the real variables

(, )

τσ

. One way this can be done is as

follows:

zi zi=+ =−

τσ τσ

(5.4)

Let’s use this deﬁ nition to work out a few basic quantities and show how this

simpliﬁ es analysis. Keep the Polyakov action in the back of your mind. Using the

Euclidean metric the Polyakov action is written as

SddXXXX

′

∂∂ +∂∂

∫

πα

τσ

τµ σ

σµ

()

(5.5)

We’re going to ﬁ nd out that going to complex variables will simplify the form of

Eq. (5.5).

To transform coordinates we need to know how to compute derivatives with

respect to the coordinates

zz and

. This is easy enough. First we invert the coordinates

Eq. (5.4):

τσ

−zz zz

i22

(5.6)

It follows that

∂

=−

ττ σ σ

zz z i z i

and so

∂

−

∂

∂zz z i

στ σ τσ

⎛⎛

⎝

⎜

⎞

⎠

⎟

(5.7)

The shorthand notation

∂=∂= ∂−∂

i12/( )

τσ

is usually used. It is also easy to

see that

∂

=∂ =∂=

∂

−

∂

∂zzz i

στ σ τ

∂

⎛

⎝

⎞

⎠

∂+∂

i i

τσ

()

(5.8)

CHAPTER 5 Conformal Field Theory Part I

where we’ve introduced the abbreviation

∂=∂

. Now given that after a Wick

rotation the metric for the

(, )

τσ

coordinate system is written as

αβ

⎛

⎝

⎜

⎞

⎠

⎟

We can write down the metric in the new complex coordinates using Eq. (5.1).

We have

gg g g g

zz zz zz z z z z

=∂ ∂ +∂ ∂ +∂ ∂ +∂ ∂

ττ τσ στ σσ

==∂ ∂ +∂ ∂

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

zz z z

ττ σσ

⎛

⎝

⎜

⎞

⎠

⎟

=−=

(5.9)

Similarly,

= 0

. On the other hand

ggg g g

zz zz zz z z z z

=∂ ∂ +∂ ∂ +∂ ∂ +∂ ∂

ττ τσ στ σσ

==∂ ∂ +∂ ∂

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

−

zz z z

ττ σσ

⎛

⎝

⎜

⎞

⎠

⎟

=+==

(5.10)

In matrix form

µν

⎛

⎝

⎜

⎞

⎠

⎟

012

12 0

(5.11)

The inverse metric, written with raised indices has components given by

gg gg

zz zz zz zz

== ==02

(5.12)

The corresponding matrix is

µν

⎛

⎝

⎜

⎞

⎠

⎟