McMahon D. String Theory Demystified

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String Theory Demystiﬁ ed

modes

µν

(and for the closed string)

, what we have done is promote them to

operators. By extension, since the

στ

(, )

are deﬁ ned in terms of

, the

στ

)

are now to be thought of as operators as well. Therefore the next task in our program

is to determine the state space of the system, that is the states upon which the

and

by extension the

στ

)

act. This procedure is really pretty similar to what you’re

used to from your previous studies of quantum theory.

The ﬁ rst item to notice is that the commutation relations have some similarity to

the harmonic oscillator you learned about in ﬁ rst semester quantum mechanics.

Temporarily dispensing with our convention of setting

? =

, recall that we can

deﬁ ne the creation and annihilation operators for the harmonic oscillator as

ˆˆˆˆˆˆ

†

p=−

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

22??

These operators satisfy the commutation relation:

[

]

†

aa =1

(4.11)

The hamiltonian of the system is given by

ˆˆ

†

Haa=+

⎛

⎝

⎞

⎠

We introduce the number operator

ˆˆ

†

Naa=

which has eigenstates

Nn nn=

(4.12)

where

n = 012, , ,...

. A system consisting of an inﬁ nite collection of harmonic

oscillators is called a fock space.

Continuing, the number operator and its eigenstates allow us to write down the

quantized energy levels of the harmonic oscillator, which are given by

ˆˆ

Hn N n n n E n

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

=??

ωω

You should also recall that the system has a ground state, which is the lowest

possible energy state. This is denoted by

A comparison of Eqs. (4.9) and (4.11) indicates that we have a similar system in

the case of the string. This should not be surprising, since what else would you

CHAPTER 4 String Quantization

expect for a vibrating string, except a system that resembles a harmonic oscillator.

Let’s continue forward. Since

[, ]

αα η δ

µν µν

mn mn

+ 0

, we can write

αα αα η

µν µν µν

mn m m

m,,

⎡

⎣

⎤

⎦

⎡

⎣

⎤

⎦

−

(4.13)

At this point, it is important to take a step back and recognize a key fact. The

commutation relation not only bears some resemblance to the harmonic oscillator

of quantum mechanics, but there is a very important difference. Notice that the

presence of the metric

µν

means that we can have negative commutators. This is

the case for the time components. That is, since

=−

, it follows that

αα

−

⎡

⎣

⎤

⎦

=−

This is going to turn out to be important because it can lead to negative norm

states.

Now, in analogy with the harmonic oscillator from ordinary quantum mechanics,

we deﬁ ne a number operator. These are given in terms of the modes as

mmm

=⋅

−

αα

where we take

m ≥

. The eigenstates of the number operator satisfy

Ni ii

mm mm

The total number operator is deﬁ ned by summing over all possible

=⋅

−

αα

==⋅

∞

−

∞

∑∑

αα

(4.14)

Following a procedure used in elementary quantum mechanics, we can use

mmm

=⋅

−

αα

together with

[, ][, ]

αα αα η

µν µν

µν

mn m m

m==

−

to show that the

µµ

−mm

and

are raising and lowering operators, respectively. This follows from

Ni i

mmm mmmm

mm mm

αααα

αα α

αα

()

=⋅

()

=⋅ −

()

−

()

(

−−

−

=−=−

()

mm m mm

mm m m m m m m

imi

ii m i i m i

αα

αα α

)

()

String Theory Demystiﬁ ed

Hence,

acts like a lowering operator. A similar exercise shows that the

negative frequency mode

−

≥

m,1

acts like a raising operator:

Niimi

mmm m mm

αα

−−

()

The lowering operator

destroys the vacuum or ground state, which is the state

with

= 0

00=

(4.15)

We can construct higher-energy states using the raising operators which are the

negative modes,

−

≥

, that is,

ααααα

µµµµµ

−−−−−11112

000,,,

and so on. A

string state also carries momentum, so we can label a state by

. Considering

the ground state, supposing that the string carries momentum

, the momentum

operator acts as

pkkk

µµ

00,,=

(4.16)

Earlier we remarked that since the Minkowski metric

µν

appears in the

commutation relations, negative norm states can exist. We can demonstrate this

explicitly as follows. Consider the ﬁ rst excited state with momentum

, that is,

−1

0, k

. Using

()

†

αα

−

we ﬁ nd the norm of this state to be

ααα

−−

==−

00 01,, ,kk k

(4.17)

We can rid the theory of the negative norm states by applying the Virasoro

constraints. The classical expressions for the Virasoro constraints are

mmnn

=⋅ =⋅

−−

∑∑

αα αα

(4.18)

In the quantum theory, the Virasoro constraints are promoted to Virasoro

operators. However, since the modes must satisfy the given commutation relations,

some care must be applied when writing the Virasoro operators as derived from the

classical expressions. The technique of normal ordering is used. This will ensure

that the eigenvalues of the Virasoro operators will be ﬁ nite. The prescription of

normal ordering is simple:

• Move all lowering operators (positive frequency modes) to the right.

• Move all raising operators (negative frequency modes) to the left.

CHAPTER 4 String Quantization

A normal ordered product is denoted using two colons, that is

†

. In the case

of the Virasoro operator, we write

mmnn

=⋅

−

∑

αα

will lower the eigenvalue of the number operator by m. Looking at the

commutation relation Eq. (4.10), you can see that

αα

µµ

mn n−

and

commute when

m ≠

. This means that when

m ≠ 0

we can simply move raising and lowering

operators where we want in the expression for the Virasoro operator because no

extra terms will be added from the commutator. Normal ordering

gives

=+ ⋅

−

∞

∑

ααα

(4.19)

Now, to get this result, note that

αα αα αα αα

−

=−∞

∞

−

∞

⋅= ⋅+ ⋅+ ⋅

∑∑

n −−

∞

−

∞

−

∑

∑∑

=⋅+ ⋅+

αα α α η

∞

−

∞

∑∑∑

=⋅+ ⋅+

αα α α

To get from the ﬁ rst to the second line, the commutator for the modes was used

together with the fact that

−

⋅

represents the dot product in D space-time

dimensions. To get the normally ordered result we have thrown out the sum

∞

At ﬁ rst glance, you might think that this sum is inﬁ nite, but regularization can be

used to compute its ﬁ nite value. To see how this works, recall the geometric series:

∑

−1

Taking the derivative, we have

ar anr

∑∑

−

=−

−11

()

Now let

a = 2

πε

and multiply

∞

a−

, then

ene

da e

() ()

()

−−

−

∑∑

−

==−

String Theory Demystiﬁ ed

It follows that

DnD D

//()/221122

∞

=−=−/

. Another undetermined constant

piece is missing from the difference between the general expression of

and the

normal ordered expression of

. This is a normal ordering constant which is

denoted by a. Therefore in any calculation

is replaced by

−

where

is a

constant.

The point of all this is to write down the commutation relations for the Virasoro

operators. Using Eq. (4.10), one ﬁ nds that

[, ]( ) ( )

LL mnL

m n mn mn

=− + −

(4.20)

We call this commutation relation the Virasoro algebra with central extension.

The central charge is the space-time dimension D which has shown up in the second

term on the right-hand side. This is also the number of free scalar ﬁ elds on the

worldsheet. It is clear that if

m =±0

the central extension term will vanish. This

singles out

LL L

10 1

, , and

−

which form a closed subalgebra. We call this the SL

(2, R) algebra.

The Virasoro operators can be used to eliminate unphysical states (i.e., negative

norm states) from the theory by requiring that the expectation value of

−

vanishes for a physical state

. That is, we impose the constraint

ψδψ

−=

for

m ≥

. The term

takes care of the fact that we only need the normal

ordering constant a in the case of

. To eliminate negative norm states, speciﬁ c

conditions must be put on a and D, which is the origin of the “extra dimensions”

in string theory. In particular, it can be shown that negative norm states can be

eliminated if

aD==126

(4.21)

The reason that

a =1

is chosen is a bit beyond the scope we want to cover in this

book, see the references if interested in the proof.

We can proceed further to obtain a mass operator. First recall that Einstein’s

equation tells us

pp m m pp

+= ⇒=−

CHAPTER 4 String Quantization

To obtain a formula for the mass operator in string theory, which is denoted by

, we use the condition on physical states. Taking

nn00

12=+⋅

∞

−

ααα

, using

−

with

a =1

, we arrive at the condition

()La

−=⇒+⋅−

⎛

⎝

⎜

⎞

⎠

⎟

⇒

−

∞

∑

ψαααψ

10+⋅−=

−

∞

∑

αα

The ﬁ rst term in this expression is nothing other than the mass squared:

(/ )12

αα

=−

′

where

′

12/( )T

. So, in bosonic string theory the “mass

shell” condition becomes

′

−

()

(4.22)

where N is the total number operator. The term

sets the energy scale of the

theory, it is taken to be on the order of the Planck mass. This is the origin of the high

energy scale of string theory.

It can be shown that

′

−

⎛

⎝

⎜

⎞

⎠

⎟

Notice that setting

a =1

forces us to take

D = 26

. The number operator acts on

the ground state as

N 00=

Hence the mass of the ground state is

′

−

⎛

⎝

⎜

⎞

⎠

⎟

=−

′

−

=−

′

ααα

So the ground state of bosonic string theory in the open string case has negative

mass. This means that the ground state is a Tachyon. This is an unphysical state

which travels faster than the speed of light. Consistency of bosonic string theory

requires that we choose

a =

, so the Tachyon cannot be removed from the theory.

String Theory Demystiﬁ ed

It will turn out that the introduction of supersymmetry (that is the introduction of

fermionic states) into the theory will get rid of the Tachyon, giving us a realistic

string theory. We will see that this also changes the number of space-time

dimensions.

Now let’s consider the mass of the ﬁ rst excited state. The ﬁ rst excited state is

−

where i is a spatial index. Here,

iD=−1

, ...,

, and the state transforms

as a vector in space-time. You will recall from your studies of quantum ﬁ eld theory

that a vector is a spin-1 particle that in general has

D −

components, the fact that this

state has

D − 2

components implies that it is a massless state. An example of a massless

vector is the photon, it only has transverse components of spin. This explains why

there are

D − 2

components rather than

D −

. The mass of the state is

ii2

126

−−

′

−

⎛

⎝

⎜

⎞

⎠

⎟

′

−

⎛

⎝

⎜

⎞

⎠⎠

⎟

−

In order for the state to be massless,

26 24− D /

must vanish, once again setting

the number of space-time dimensions D to 26. Physicists refer to the bosonic string

theory with

aD==126,

as critical and call

D = 26

the critical dimension.

CLOSED STRING SPECTRUM

In the case of the closed string, things are a little more complicated than what

you’re used to from the harmonic oscillator in ordinary quantum theory due to the

fact that we have a second commutation relation in addition to

αα ηδ

µν

mn mn

⎡

⎣

⎤

⎦

+ 0

that must be satisﬁ ed, namely,

αα ηδ

µν

mn mn

⎡

⎣

⎤

⎦

+ 0

. What this is going to mean

is that we need to deﬁ ne two number operators. These are deﬁ ned by inﬁ nite sums

over the modes:

Rmm

Lmm

−

∞

−

∞

∑∑

αα αα

(4.23)

Together with the momentum operator

, the number operators

and

serve to characterize the state of a closed string. Let us denote a state by

nk,

. As

in the open string case, the momentum operator will act according to

pkkk

µµ

00,,=

(4.24)

Therefore, the state

0, k

of the string carries momentum

. Turning our

attention to the number operators, ﬁ rst let’s specify the action of the raising and

CHAPTER 4 String Quantization

lowering (creation and annihilation) operators. This follows what we did in the

open string case as well. Keeping

m ≥

, we deﬁ ne these as follows:

•

is a lowering operator.

•

−

is a raising operator.

Similar roles are played by

and

−

. We deﬁ ne the ground state by

00,

which is often abbreviated by

. Then, the lowering operators satisfy a relation

familiar from ordinary quantum mechanics:

αα

µµ

00 00==

(4.25)

Note also that we can deﬁ ne the ground state with momentum

by writing

The raising operators

−m

and

−n

act to raise the eigenvalues of the number

operators

and

by m and n, respectively. So if

im(,

)

and

im(, )

are

integers:

imi mk

(, )(, ),

µµ αα

()()

−

≥

∞

−

∏∏

k0,

The number operators

and

act on this state as follows:

N i mi m k mi m i mi m

(,)(,), (,)(,)(,),

µµ µµµ

(,)(,), (,)(,)

N i mi m k mi m i mi

µµ µµ

∑

= ((, ),

∑

Once again, we will have negative norm states due to the presence of the metric

in the commutation relations. This situation can be dealt with in the same manner

used in the open string case. So we won’t go into great detail and simply state the

results, noting that in the following we take

m ≥

. We proceed by introducing

normal ordered Virasoro operators, but this time must include

as well:

mmnn

=⋅ =⋅

−−

∑∑

:: ::

αα αα

(4.26)

These operators satisfy commutation relations called the Virasoro algebra:

[, ]( ) ( )

[, ]

LL mnL

m n mn mn

=− + −

(() ( )

mnL

mn mn

−+−

(4.27)

String Theory Demystiﬁ ed

If the following relations are satisﬁ ed by a state

() ()

La La

mm mm

−=−=

δψ δψ

(4.28)

then the state

is a physical state. That is, if

m > 0

, then the Virasoro operator

annihilates the physical state

ψψ

==00,

. The condition satisﬁ ed when

m = 0

is the “mass shell” condition,

() ,()La La

00−=−=

ψψ

. Once again, it

can be shown that negative norm states can be avoided in the theory provided that

aD==12

. The Virasoro operators

and

can be written in terms of the

number operators as follows:

pp N L

pp N

LR00

=+=+

ππ

The sum and difference of

and

annihilate the physical states:

()()LL a LL

00 00

20 0+− = − =

ψψ

The constraint

()LL

0−=

is called level matching. Using the Einstein

relations, we can arrive at an expression for a mass operator:

Mpp NN

2=− =

′

+−

()

(4.29)

The ground state of the closed bosonic string is

0, k

, found when

This is a tachyon that satisﬁ es

=−

′

The next case to consider is

==1

, which is the ﬁ rst excited state. Here

we get hints that string theory is a uniﬁ ed theory. The ﬁ rst excited states are massless,

. They are derived from the ground state in the following way:

εαα

µν

() ,kk

−−11

The object

µν

()k

is a tensor which can be decomposed into symmetric

µν

()

(

)

and antisymmetric

µν

()

(

)

parts (see Relativity Demystiﬁ ed if you aren’t sure about

CHAPTER 4 String Quantization

this). The symmetric part corresponds to a massless spin-2 particle which is the

graviton. The linearized metric

ρσ ρσ ρσ

ηε

=+()

satisﬁ es the linearized Einstein

equations

∂∂ = ∂ =

ρσ

µν

εε

() , ()xx00 and

. By taking the Fourier transform of

µν

()

(

)

, it can be shown that these equations are satisﬁ ed.

The trace

(

)

, which deﬁ nes a scalar, is also important. This corresponds to a

massless scalar particle called the dilaton.

We now turn our attention to a different method of quantization. We began the

chapter with a discussion of covariant quantization. This method is a straightforward

application of the imposition of commutation relations. We took this approach ﬁ rst

because it may seem familiar from your studies of ordinary quantum mechanics. In

addition, it preserves the Lorentz invariance of the theory. Physicists say that this

approach is “manifestly” Lorentz invariant, colloquially meaning that the Lorentz

invariance is obvious. The technique has the disadvantage in that negative norm

states appear. Although this is a problem, it is instructive to go through the process

of eliminating the negative norm states.

Another approach is possible which avoids the negative norm states at the cost

of losing manifest Lorentz invariance. This is called light-cone quantization. We

brieﬂ y discuss it here, considering the open string case.

We begin by using light-cone coordinates, which were introduced in Chap. 2 in

Eq. (2.34):

−

(4.30)

The remaining coordinates

are transverse coordinates. The center of mass

coordinate

and momentum

are also written as light-cone coordinates. In the

light-cone gauge, we choose

Xx p

++ +

=+C

(4.31)

which leads to

=≠0 for 0

, that is, the modes are zero for

. The Virasoro

constraints will lead us to a description based on transverse oscillators. We have the

freedom to set

, giving the center of mass position as

µµ

Light-Cone Quantization