McMahon D. String Theory Demystified

Подождите немного. Документ загружается.

116

String Theory Demystiﬁ ed

BRST Operators and Introductory Remarks

We begin by considering our old friend, Lie algebra. This is the algebra that is

obeyed by the familiar spin angular momentum operators of ordinary quantum

mechanics. In general, let some physical theory contain a gauge symmetry with

operators

. These operators satisfy the Lie algebra:

[, ]KK fK

ij ij

(6.1)

where the f

are called the structure constants of the theory (note we are using

the Einstein summation convention, so repeated indices are summed over). The

structure constants satisfy

ff f f f f

++=0

(6.2)

The BRST quantization procedure begins with the following. We introduce two

ghost ﬁ elds that are denoted by

and

which satisfy an anticommutation relation

given by

{, }cb

(6.3)

Furthermore

{, } {, }cc bb

==0

. Here the

are ghost ﬁ elds and the

are

“ghost momenta.”

Notice that since an anticommutation relation is satisﬁ ed by the ghost ﬁ elds,

these ﬁ elds are fermionic. Now, recall that a ﬁ eld

(, )zz

has conformal dimension

(, )hh

provided that it transforms under some conformal transformation

zwz→ ()

as follows:

φϕ

(, ) ( , )zz

∂

⎛

⎝

⎜

⎞

⎠

⎟

∂

⎛

⎝

⎜

⎞

⎠

⎟

(6.4)

The ﬁ elds bcand

are chosen such that they have conformal dimension 2 and

⫺1, as we will see later.

There are two operators that are constructed out of the ghost ﬁ elds and the

The ﬁ rst of these is the BRST operator which is given by

QcK fccb

iij

ki j

=−

(6.5)

CHAPTER 6 BRST Quantization

117

It is assumed that

QQ=

†

. We say that the BRST operator is nilpotent of degree

two. This means that if we square the operator we get zero:

(6.6)

Notice that Eq. (6.6) can also be expressed as

{, }QQ= 0

. We label the BRST

operator with a Q to imply that this is a conserved charge of the system, we often

call this the BRST charge. A second operator that is composed solely of ghost ﬁ elds

is called the ghost number operator U. This is given by

Ucb

(6.7)

(Again note the Einstein summation convention is being used.) This operator has

integer eigenvalues. If the dimension of the Lie algebra is n, then the eigenvalues of

U are the integers

0,...,n

. A state

has ghost number m if

ψψ

EXAMPLE 6.1

Show that Q raises the ghost number by 1.

SOLUTION

Using the anticommutation relations for the ghost ﬁ elds [Eq. (6.3)], notice that

UcKcbcKccbK

cK c c

==−

(

)

=−

∑∑

bbK

cK cK cb cK cKU

∑

=+ =+

Now consider some state

with ghost number m, that is

ψψ

. Then

UQ U cK f ccb

Uc K c b

iij

ki j

ψψ

(

)

=−

⎛

⎝

⎜

⎞

⎠

⎟

=−

∑∑

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟

=+−

fccb

cK cKU f

ki j

cbccb

∑

⎛

⎝

⎜

⎞

⎠

⎟

Use Uc K c K c K U

(

)

118

String Theory Demystiﬁ ed

=+− −

(

)

⎛

⎝

⎜

⎞

⎠

⎟

∑

cK cKU f c cb cb

iij

δ ψψ

[Apply Eq. (6.3)]

=+−cK cKU f

iij

kki j

kij

ki r

cc b f c cbc b+

⎛

⎝

⎜

⎞

⎠

⎟

∑

(Use the properties of

iij

ki j

cK cKU f ccb

)

=+−

kkij

ki r

fc c cbb+−

(

)

⎛

⎝

⎜

⎞

⎠

⎟

∑

δψ

[Applyy Eq. (6.3) again]

=+ + −QcKU fc c c

iij

ki r

(

)

⎛

⎝

⎜

⎞

⎠

⎟

∑

[Use Eq. (6.5) to wrrite Q

Q cKU fccb fcc cb

iij

ki j

kij

ki j r

]

=+ + −

∑

⎛

⎝

⎜

⎞

⎠

⎟

(Kroneker delta again)

= Q

+++ − −

(

)

∑

cKU f cc b f cc bc

iij

ki j

kij

ki j

⎛

⎝

⎜

⎞

⎠

⎟

[Yet again use Eq. (6.3)]

=+Qc

iij

ki j r

KU f cc c b

QQU m

−

⎛

⎝

⎜

⎞

⎠

⎟

=+ =+

∑

() )(QQUm

ψψψ

(

)

(

)

Apply

Hence the BRST operator raises the ghost number by 1.

BRST-Invariant States

A state

is called BRST invariant if it is annihilated by the BRST operator

[Eq. (6.5)]:

ψψ

=⇒0 is BRST invariant

(6.8)

BRST-invariant states are the physical states of the theory. Since Q

= 0, it follows

that any state

ψχ

=≠Q 0

is BRST invariant, since

ψχ

CHAPTER 6 BRST Quantization

119

Call

ψχ

= Q

a null state. Suppose that

is an arbitrary physical state so

that

. Notice that

φψφχ φχ

(

)

(

)

=QQ0

Hence, any amplitude taken between a physical state and a null state vanishes.

This implies a result, which is somewhat akin to the idea of a phase factor in ordinary

quantum mechanics. Two states

ab ea

, =

represent the same physical state

in quantum mechanics since the resulting amplitudes are the same. Likewise, since

any inner product between a physical state and a null state vanishes, adding a null

state

ψχ

= Q

to a physical state

generates a new state which is physically

equivalent to

′

φφ χ

because the second term on the right does not contribute to any inner product and hence

does not change the physical predictions of the theory, which are the amplitudes. Since

Q raises the ghost number of a state by 1, if the ghost number of

is m, then the ghost

number of

m −1. Important special cases are states

with ghost number 0. If

the ghost number is 0, then

= 0

Looking at the form of U, namely, Eq. (6.7), we see that this implies that

= 0

Furthermore, a state annihilated by the ghost ﬁ elds

cannot be annihilated by the

ghost ﬁ elds

. Since

ucb bcnbc

==−=−

∑∑ ∑

we see that

Unbc n bc

ψψψψ

=−

⎛

⎝

⎜

⎞

⎠

⎟

=−

∑∑

Hence if b

= 0, the c

cannot annihilate the state since this would result

in a contradiction. When

= 0

, it immediately follows that U

= 0, but

the above result shows that if

= 0

also, we would have

ψψ

=≠0

a contradiction.

The key result for states with zero ghost number is the following. Looking at Q

in Eq. (6.5), it is clear that if

= 0

then

QcK

ψψ

∑

120

String Theory Demystiﬁ ed

This implies that for a state with zero ghost number

= 0

(6.9)

So, we can say that a state which is BRST invariant with zero ghost number is

also invariant under the symmetry described by the generators

. Furthermore, if

a state has ghost number zero, this tells us that the state is not a ghost state, hence

we avoid negative probabilities.

BRST in String Theory-CFT

We will take a look at the BRST formalism in string theory by brieﬂ y considering

two approaches. The derivation of this approach is based on the use of path integrals,

which we are purposely avoiding due to the level of this text. So some results will

simply be stated, the reader who is interested in their derivation is encouraged to

check the references at the back of the book.

The application of BRST quantization to string theory can be done easily using

conformal ﬁ eld theory. The advantage of this approach is that the critical dimension

D = 26 arises in a straightforward manner. We work in the conformal gauge where

we take

αβ αβ

= . In this case the energy-momentum tensor has a holomorphic

component

(

)

and an antiholomorphic component

()

where

()

was given

in Eq. (5.33) as

()=

=−∞

∞

∑

In Example 5.5 we worked out the OPE of

TzT w

zz ww

() ( )

and found

TzT w

zz ww

ww w ww

() ( )

()

(

−

∂

)

−

Where for simplicity of notation we have omitted the multiplying

factor. The ghost

ﬁ elds are introduced as functions of a complex variable z as follows. We deﬁ ne

bzcw

()( ) ()()=

−

Next we write down an energy-momentum tensor

()

for the ghost ﬁ elds. This

is given by

T z bz cz bzcz

gh z z

() () () ()()=− ∂ −∂2

CHAPTER 6 BRST Quantization

121

The conformal dimension [Eq. (6.4)] of the ghost ﬁ elds follows from this

deﬁ nition. Using the ghost energy-momentum tensor together with the energy-

momentum tensor for the string we can arrive at the BRST current:

jz cz T z T z czT z c

zz gh zz

() () () () () ()=+

⎛

⎝

⎜

⎞

⎠

⎟

(() ()()zczbz

∂

The BRST charge is given by

jz=

∫

()

Now, the central charge (i.e., the critical space-time dimension) comes from the

leading term in the OPE of the energy-momentum tensor, which is

()

−

The presence of this extra term is called the conformal anomaly since it prevents

the algebra from closing. So we would like to get rid of it. This is done by considering

a total energy-momentum tensor, which is the sum of the string energy-momentum

tensor and the ghost energy-momentum tensor, that is,

TTz Tz

zz gh

=+() ()

. It can be

shown that the OPE of the ghost energy-momentum tensor is

TzTw

gh gh

gh w gh

() ( )

()

(

−

∂

)

−

Taking the leading term in this expression to be of the form

−−(/)/( )Dzw2

, we

see that the ghost ﬁ elds contribute a central charge of ⫺26 which precisely cancels

the conformal anomaly that arises from the matter energy-momentum tensor. This

result actually follows from the nilpotency requirement (i.e.,

0= ) of the BRST

charge.

BRST Transformations

Next we look at BRST quantization by considering a set of BRST transformations

which are derived using a path integral approach. This is a bit beyond the level of

the discussion used in the book, so we simply state the results. Working in light-

cone coordinates, we deﬁ ne a ghost ﬁ eld c and an antighost ﬁ eld b where c has

122

String Theory Demystiﬁ ed

components

+−

, and b has components

++ −−

and

. We also introduce an energy-

momentum tensor for the ghost ﬁ elds

±±

with components given by

Tibc bc

Tibc

++ ++ +

+++

−− −− −

−

=∂+∂

()

(

−−−−

−

bc)

The BRST transformations, using a small anticommuting operator

are

δε

µµ

Xic c X

cic cc

=∂+∂

=± ∂ + ∂

−

±+

−

±±

()

==± +

±± ±±

iT T

()

The action for the ghost ﬁ elds is

S dbcbc

=∂+∂

∫

++ −

−− +

−2

()

From which the following equations of motion follow:

∂=∂=∂=∂=

−++ +−− −

−

bbcc0

We can write down modal expansions of the ghost ﬁ elds. These are given by

cce cce

bbe

+−+−−−

−

∑∑

() ()

τσ τσ

iin

bbe

() ()

τσ τσ

−−

∑∑

The modes satisfy the following anticommutation relation:

{,}

mn mn

with

{,}{,}bb cc

mn mn

==0

. Virasoro operators are deﬁ ned for the ghost ﬁ elds using

the modes. Using normal-ordered expansions, these are

LmnbcLmnb

mn n

=− =−

+− +

∑

(): : ():

nnn

−

∑

CHAPTER 6 BRST Quantization

123

To write down the total Virasoro operator for the “real” ﬁ elds + ghost ﬁ elds, we

form a sum of the respective operators. That is

LLLa

tot

=+−

where the last term on the right is the normal-ordering constant for

m = 0

. It can be

shown that the commutation relation for the total Virasoro operator is of the form:

LL mnL Am

tot

,() ()

⎡

⎣

⎤

⎦

=− +

Notice that the presence of the term

Am()

on the right keeps us from obtaining a

relation that preserves the classical Virasoro algebra. As such, this term is called an

anomaly. The anomaly is determined in terms of two unknown constants which you

might guess by now are D and a. It has the form

mm m m am() ( ) ( )=−+−+

13 2

To make the anomaly vanish, we take

Da==26 1,

which is consistent with the

other results obtained so far in the book for bosonic string theory.

The BRST current is given by

jcT cT c

=+ +∂

The BRST charge is given by the mode expansion:

QcL mnccb c

nn mnmn

mnn

=+− −

−−−

∑∑

(): :

Using tedious algebra one can show that

QLLmnLcc

tot

⎡

⎣

⎤

⎦

−−

(

)

≈

+−−

∑

,()

26()D −

Hence the requirement that Q

forces us to take

D = 26.

Going back to the original BRST approach outlined in Eqs. (6.1) to (6.5), using

the classical algebra for the Virasoro operators:

[,]( )LL mnL

mn mn

=−

124

String Theory Demystiﬁ ed

We can identify the structure constants as

fmn

mnk

=−

()

To see how the physical spectrum can be constructed in string theory, we consider

the open string case. The states are built up from the ghost vacuum state. Let’s call

the ghost vacuum state

. This state is annihilated by all positive ghost modes.

Let

n >

, then

χχ

==0

The zero modes of the ghost ﬁ elds are a special case. They can be used to build

the physical states of the theory. Using the anticommutation relations [Eq. (6.3)],

the zero modes satisfy

{,}bc

Using Eq. (6.3) it should also be obvious that

. We also require that

for physical states

. Now we can construct a two-state system from

the zero modes of the ghost states. The basis states are denoted by

↑↓,

. The

ghost states act as

0↓= ↑=

↑=↓ ↓=↑

We choose

↓

as the ghost vacuum state. To get the total state of the system, we take

the tensor product of this state with the momentum state

k to give

↓,k

. To

generate a physical state, we act on it with the BRST charge Q. It can be shown that

Qk L c k↓= − ↓,( ),

The requirement that

Qk↓=,0

gives the mass-shell condition

10−=

which

describes the same Tachyon state we found in Chap. 4. Higher states can be

generated. We will have mode operators for each of the three ﬁ elds: the

στ

)

plus the two ghost ﬁ elds. To get the ﬁ rst excited state, we act with

−− −11 1

,,cband

as follows:

ςα ξ ξ

⋅+ +

(

)

↓

−− −11121

CHAPTER 6 BRST Quantization

125

The

and

are constants, but

is a vector with 26 components. In

Chap. 4, we found that the first excited state was massless, so we expect the

state to have physical degrees of freedom in the transverse directions. That

is, it should have 24 independent components. To get rid of the extra

parameters, we create a physical state with the requirement that

. It

can be shown that

Qpcpcpk

ψςξα

=+⋅+⋅

(

)

↓

−−

0121

() ,

The requirement that Q

= 0 enforces constraints on the parameters. With

this general prescription, there are 26 positive norm states and 2 negative norm

states.

We can eliminate the negative norm states by introducing some constraints. The

ﬁ rst constraint is to take

p ⋅=

and

. This rids the theory of the negative

norm states. Also note that

0= , which tells us that this is a massless state. We

also have two zero-norm states:

kk ck

−−

↓↓

,,and

These states are orthogonal to the physical states. Eliminating them gives us a

state with 24 degrees of freedom, as expected for a massless state in 26 space-time

dimensions.

No-Ghost Theorem

The no-ghost theorem is simply a statement of the results we have seen in Chap. 4

and here, namely, that if the number of space-time dimensions is given by

D = 26

then negative norm states are eliminated from the theory.

Summary

In this chapter we introduced the BRST formalism and illustrated how it can be used

to quantize strings. This is a more sophisticated approach than covariant quantization

or light-cone quantization. It takes a middle ground, preserving manifest Lorentz

invariance while living with ghost states. The approach makes the appearance of the

critical D = 26 dimension simple to understand.