McMahon D. String Theory Demystified

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String Theory Demystiﬁ ed

This can only be true if

sin k

= 0

, which means that k must be an integer. Denoting

it by n, a simple exercise shows that Eq. (2.47) can be written as

Xx p i

µµ µ

=+ +

≠

−

∑

cos( )

(2.53)

Closed Strings

In the case of closed strings, the boundary condition becomes one of periodicity,

namely

µµ

τσ τσ π

(, ) (, )=+2

(2.54)

This condition restricts the solutions to those for which the wave number k takes on

integral values. Hence,

τσ τ σ

(, ) ( )

(

=+ ++

−

≠

∑

)

n 0

(2.55)

τσ

τσ τ σ

(, ) ( )

()

=+ −+

−−

≠≠

∑

(2.56)

where n is an integer. In addition, periodicity enforces the condition that

µµ

(2.57)

for closed strings. We saw that if this condition was satisﬁ ed in the case of open

strings, there was no winding of the string permitted. In the case of closed strings,

however, the situation is a little bit more involved if we allow for the possibility

where the ambient space-time includes a compact extra dimension (then

does not hold). Then, we can consider, for example, the situation where the closed

string is compactiﬁ ed on a circle of radius R.

Using Eq. (2.57), we sum Eqs. (2.55) and (2.56) to obtain the complete solution,

focusing on the momentum term–and imposing the periodicity condition of Eq. (2.54).

This gives the total solution which can be written as

XRX RW

µµ

τσ π τσ π

(, ) (, )+= +22

(2.58)

CHAPTER 2 Equations of Motion

We call W the winding number, which literally tells us how many times the string

has wound around the compact dimension (so W must be an integer). As we let

σπ

→+2

, notice that the momentum terms change as

CC CC

ss ss

pp p

22 22

222

µµµ

τσπ τσπ τσ

()()()++ + −− = + +

222

ppp

µµµ

τσ

() ( )−+ −

That is, the total solution changes as

XX pp

µµ µµ

τσ π τσ

(, ) (, ) ( )+= + −

(2.59)

Therefore, we call

(

)

µµ

−

the winding contribution.

Finally, we consider open strings with ﬁ xed endpoints. The boundary condition is



(2.60)

(Dirichlet boundary condition). Using Eqs. (2.48) and (2.49), we have



Xpe

µµµτ

τσ α

(, )== +

−

≠

∑

(2.61)



Xpe

µµµτ

τσ α

(, )== +

−

≠

∑

(2.62)

So, Eq. (2.60) implies that

µµ

⇒=−

(2.63)

and

αα

µµ

+=0

(2.64)

If a dimension is noncompact for an open string, then

µµ

. To simultaneously

satisfy Eq. (2.63), the total momentum of the string must vanish. In the next

example, we consider the case where both endpoints are ﬁ xed.

Open Strings with Fixed Endpoints

String Theory Demystiﬁ ed

EXAMPLE 2.4

What is the length of a string that has both endpoints ﬁ xed?

SOLUTION

If both endpoints are ﬁ xed, then we must also satisfy the boundary condition



µµ

τσ π τσ π

(, ) (, )=+ ==0

In this case, we have



Xpe

µµµτπ

τσ π α

(, )

()

== +

−+

≠

∑



Xpe

µµµτπ

τσ π α

(, )

()

== +

−−

≠

∑

The boundary condition can only be satisﬁ ed if k is an integer. The overall solution

in this case can be written as

Xx p

µµ µ

=+ −

≠

−

∑

2sin()

(2.65)

Here we applied the conditions in Eqs. (2.63) and (2.64). This expression includes

the winding term

= C

(2.66)

Now let’s compute the string coordinates at the endpoints. We have

Xxw

µµ

τπ π

(,)

(, )

0 =

Hence, the length of the string is

XXw

µµ

τπ τ π

(, ) (,)−=0

CHAPTER 2 Equations of Motion

In going from ordinary classical mechanics to quantum theory, we follow Dirac and

use the correspondence between the Poisson brackets and commutators. In string

theory, we consider equal

Poisson brackets as our starting point when we quantize

the modes on the strings. Later, we will discuss hamiltonians and the Virasoro algebra,

an important concept in string theory, but for now we simply introduce the important

Poisson brackets which will allow us to quantize the string. First we note that

{(,),(,)} ( )XX

µν

τσ τσ δσ σ η



′

=−

′

(2.67)

or, in terms of momentum

{(,), (,)} ( )PX

µν

τσ τσ δσ σ η

′

=−

′

(2.68)

The Fourier expansion can be used to derive Poisson brackets for the modes. These

are

αα δ η

µν

σµν

mn mn

im,

{}

=−

(2.69)

The Poisson brackets will be the starting point to quantize the theory.

Quiz

1. Consider the lagrangian given in the Eq. (2.10) which includes the auxiliary

ﬁ eld. Use the Euler-Lagrange equations to derive the equation of motion.

2. Start with the Nambu-Goto lagrangian

XX X X=

′

−⋅

′



22 2

()

and consider

the gauge choice which gives the ﬂ at metric

. Using the constraints



XX XX

00+

′

=⋅

′

, ﬁ nd the equations of motion.

3. Consider the Polyakov action. A Weyl transformation is one of the form

heh

αβ

φτσ

αβ

→

(,)

and

X = 0

. Determine the form of the Polyakov action

under a Weyl transformation [hint:

heh

αβ φ τ σ α

→

− (,)

4. Using the Polyakov action, deﬁ ne the energy-momentum tensor on the

worldsheet by varying the action with respect to the intrinsic metric

αβ

=−

−

Poisson Brackets

String Theory Demystiﬁ ed

Using the fact that

αβ

−=− −hhhh

ﬁ nd an expression for the

induced metric so that it can be eliminated from the action. This should

allow you to recover the Nambu-Goto action.

5. Consider an open string with free endpoints. Find the variation of the center

of mass position of the string with

by calculating

στσ

∫

(, )

6. Find the conserved momentum of the open string with free endpoints by

calculating

PTdX

στσ

∫



(, )

where the dot represents differentiation with respect to

The Classical String II:

Symmetries and

Worldsheet Currents

In the last chapter we introduced some basic notions of classical string theory,

including the equations of motion and boundary conditions. In this chapter we

will expand our discussion of the classical string, discussing symmetries and

introducing the energy-momentum tensor and conserved currents. This will ﬁ nish

the groundwork we need for the classical string, and in the next chapter we will

quantize the string.

CHAPTER 3

String Theory Demystiﬁ ed

The Energy-Momentum Tensor

Let’s quickly review a few things before getting started. Recall that the intrinsic

distance on the worldsheet can be determined using the induced metric

αβ

. This is

given by

ds h d d

αβ

σσ

(3.1)

where

στσσ

==,

are the coordinates which parameterize points on the

worldsheet. A set of functions

στ

(, )

describe the shape of the worldsheet and

the motion of the string with respect to the background space-time, where

=−01

, , ..., D

for a D-dimensional space-time. To ﬁ nd the dynamics of the

string, we can minimize the Polyakov action [Eq. (2.27)]:

dhhXX

=− − ∂ ∂

∫

ση

αβ

µν

det( )

(3.2)

Minimizing

(by minimizing the area of the worldsheet) gives us the equations

of motion for the

στ

(, )

, and hence the dynamics of the string. In the quiz at the

end of Chap. 2 in Prob. 4, you were invited to show that the Polyakov and Nambu-

Goto actions were equivalent by considering the energy-momentum or stress-energy

tensor

αβ

which is given by

αβ

=−

−

(3.3)

In this book we’ll go mostly by the name energy-momentum tensor. In a nutshell,

the energy-momentum tensor describes the density and ﬂ ux of energy and

momentum in space-time. You should be familiar with the basics of what

αβ

from some exposure to or study of quantum ﬁ eld theory, so we’re just going to go

with that and describe how it works in string theory. When working out the solution

to Prob. 4 in Chap. 2, you should have found that

TXX hhXX

αβ α

µν αβ

ρσ

µν

ηη

=∂ ∂ − ∂ ∂

()

(3.4)

The ﬁ rst property that we will establish for the energy-momentum tensor is that it

has zero trace. We can calculate the trace using the induced metric:

Tr T T h T()

αβ

CHAPTER 3 Symmetries and Worldsheet Currents

The trace is easy to calculate:

hT h X X hh h X X

αβ

µν

αβ

ρσ

ηη

=∂∂ − ∂∂

νν

αβ

µν

ρσ

µν

ηη

()

=∂∂ −∂∂hXX hXX (because hh

hXX hXX

αβ

αβ α

αβ

µν

αβ

=∂∂ −∂∂

ηηρασβ

µν

(relabel dummy indices, →→

So we’ve established our ﬁ rst fact to ﬁ le away about the energy-momentum tensor

for the string. It’s traceless:

= 0

(3.5)

In a moment we will learn the physical reason why the energy-momentum tensor is

traceless.

In this section we will list some of the symmetries of the Polyakov action (and

hence of the bosonic string in Minkowski space-time). There are three symmetry

groups of the Polyakov action. These include

• Poincaré transformations

• Reparameterizations of the worldsheet coordinates

• Weyl transformations

The concept of symmetries is so important we will take a momentary digression

to discuss the topic before describing the symmetries of the Polyakov action.

Symmetries in physics can be global symmetries or local symmetries. These are

deﬁ ned as follows:

• A global symmetry is one that holds at all points in space-time. The

parameters of the transformation will not depend on space-time.

• A local symmetry is one that acts differently at different space-time points.

In this case, the parameters of the transformation will be functions of the

space-time coordinates.

You should recall from your studies of classical mechanics and quantum ﬁ eld

theory that a symmetry in physics leads to a conservation law. The formal statement

Symmetries of the Polyakov Action

String Theory Demystiﬁ ed

of this fact is called Noether’s theorem. Let’s quickly review the most famous

example of a conserved quantity, the conservation of electric charge.

The electromagnetic ﬁ eld tensor

µν

is deﬁ ned in terms of the 4-vector

potential via

FAA

µν µ ν ν µ

=∂ −∂

The Maxwell equations with source terms are written as

∂=

µν ν

Now it follows from the deﬁ nition of

µν

that

∂=

J 0

because

∂=∂∂−∂∂⇒∂∂

=∂∂∂ −∂∂∂

µν

νµ

µν

νµ

µν

νµ

FAAF

ννµ

νµ

µν

νµ

AA=∂∂ ∂ −∂ ∂ ∂ (partial derivatives commute)

(relabel dumm=∂∂∂ −∂∂∂

νµ

µν

νµ

µν

AA yy indices)

= 0

Hence

is a conserved quantity. This is a fact expressed in the famous continuity

equation which tells us that

∂

+∇⋅ =

where

is the charge density and

is the current density. It follows that charge is

conserved. The charge Q is of course deﬁ ned using

Qdx=

∫

Using the continuity equation and taking the surface of integration S to be at inﬁ nity

we get

dx J J dA

∂

=− ∇⋅ =− ⋅ =

∫∫ ∫

III

Hence, charge is conserved.

We demonstrated that charge is conserved starting with the equation of motion

for the electromagnetic ﬁ eld. More formally, we can determine what the conserved

CHAPTER 3 Symmetries and Worldsheet Currents

quantities are and relate them to symmetries by looking at the action S, or more

particularly the lagrangian. This is where Noether’s theorem comes into play—

symmetries in the lagrangian lead to conserved quantities.

You can understand Noether’s theorem with a simple one-dimensional example.

Consider a particle whose motion is described by a lagrangian

)



where



The momentum of the particle is given by

∂



The Euler-Lagrange equations are the equations of motion for this system:

∂

−

∂



Now suppose that the lagrangian is invariant under a symmetry. That is, the form

of the lagrangian does not change under a one parameter coordinate transformation

tst→ (

)

qt qs() ()→

Saying that the lagrangian is invariant under this symmetry means that

Lqs qs[(), ()]



= 0

The symmetry of the lagrangian can be written out explicitly using the chain rule as

Lqs qs

[(), ()]



∂

= 0

Now let’s get to the central idea. Noether’s theorem tells us that

is a conserved quantity, that is,

= 0