physics - introduction to string theory

Lecture 2Partition function as the trace of an operator Recall that the trace Tr of an operatorÙ in a finite dimensional Hilbert space isequal to the sum of the diagonal entries of a ma

&-, stands for some measure on the space of paths, . is a parameter (usually

very small, Planck constant) and%

is a scalar function The Minkowski partition function

of the theory is an integral

Example 1.1 Recall the integral expression for theX -function:

in (1.5), we obtain the Gauss integral:

Although in the substitution abovev

is a positive real number, one can show thatformula (1.6) make sense, as a Riemann integral, for any complexv

with Re

O When Re

to define a probability measure on1

It is called the Gaussian measure Let us compute

is equal to the number of ways to arrangec

› objects in pairs This gives us

9X be the number of its verticesand ¦§ MX be the number of its edges We have Ÿ

where the sum is taken over the set of labeled trivalent graphs Let­e 9® be the number

of labelled trivalent graphs which define the same unlabelled graph when we forgetabout the labelling We can write ¯

number of symmetries of the graph This gives the Feynman rules to compute the

contribution of this graph to the coefficient atƒ

9ƒ given by the formula (1.7)

Recall that the Principle of Stationary Phase says that the main contributions to the

and¿À iQ has no critical points

Proof We use induction on•

The assertion is obvious for•

Thus if ¿À iQ has finitely many critical pointsQ

is the space-dimension, and o

is the time-dimension A QFT in dimensionO

HVo

is thequantum mechanics In this case, we take

Q be the space of smooth maps+
P

such that

Q ]Gw

can be interpreted as the “probability amplitude” that a particle in the positionQ at themoment of time moves to the position at the time

Let us compute it for the action defined by the Lagrangian (1.9) with

Weshall assume that the potential function%

QÅç ]Gw

Q Fix a positive integer ­ and subdivide thetime interval *

Q over this space to get a number î Now we candefine (1.10) as the limit of integralsî when­ goes to infinity However, this limitmay not exist One of the reasons could be that î contains a factor ï

ë for someconstant ï with ð ïÍð

jño Then we can get the limit by redefining î , replacing itwithï

Q This is exactly what we are going to

do Also, when we restrict the functional to the finite-dimensional space

`ofpiecewise linear paths, we shall allow ourselves to replace the integralò

in (1.10) We should immediately warn the reader that the described method of giving

a value to the path integral is not the only possible

Q to define a certain Hermitian operator in the Hilbertspace

Recall that for any manifold 

with some Lebesgue measure D}

thespace

-integrable This implies thatÙ

M¿ is well-defined Using the Cauchy-Schwarzinequality, one can easily checks that

 where(

is a denselinear subspace of (note the analogy with rational maps in algebraic geometry) Forsuch operatorsÙ

we can define the adjoint operator as follows Let(

denote thedomain of definition ofÙ

The adjoint operatorÙ

will be defined on the set

We shall always assume thatÙ

cannot be extended to a linear operator on

a larger set than(

Notice thatÙ

cannot be bounded on(

since otherwise wecan extend it to the whole by continuity On the other hand, a self-adjoint operatorÙú

 are called unbounded linear operators.

Example 1.3 Let us consider the space

is defined for all

Property (N) says that the total probability amplitude of a particle to move fromQ

to somewhere is equal to 1 Notice that property (N) implies that the operatorÙ

É isunitary In fact,

Q to be the kernel of a Schmidt operator Indeed, it does not belong to the space

.The way about this is as follows (see [Rauch])

First let us recall the notion of the Fourier transform in 1

It is a linear operatordefined on the Schwartz spaceBÛ

@#D

GFž’ªQš&½ 9Q ç(v) D

'&JI

c‚uKD M&

Q is the propagator for the Schr¨odinger tion

Supose ¿À 9Q RNBÛ Let us find the solution in

B using the Fourier transform Using property (iii), we get E

¿À 9Q Of course, we have still to show the existence

of a solution We skip the check that formula (1.13) gives a solution in B Thisdefines us a linear operator (the propagator)

does not belong

toBÛ A way about it is to consider this function as a distribution and extend theForier transform to distributions

Recall that a distribution is a continuous linear functional on the spaceï

so the two definitions agree

For example letÙ

Since the Fourier transform is an example of an operator defined on B with

Proof Assume first that Re

O This proves the lemma

Now we can use the lemma to set

Q is the kernel of the operator

&Àð /^k

Let&ba be a normalized eigenfunction of an operatorÙ

with an eigenvaluec Physicistsdenote it by ð

left-hand-side is called the correlation•

-function In the example above

Compute the coefficient atƒ

S

S iQæF

(you have to give the meaning of the right-hand-side)

Lecture 2

Partition function as the trace of

an operator

Recall that the trace Tr of an operatorÙ

in a finite dimensional Hilbert space isequal to the sum of the diagonal entries of a matrix ofÙ

with respect to any basis If

we choose an orthonormal basis

where Sp is the spectrum ofÙ

(the set of eigenvalues) and D

ic is equal to thedimension of the eigensubspace corresponding to the eigenvaluec Notice that

is abounded operator First we try to generalize the definition of a trace by using (2.1).One chooses a basis

if the series convergent If the convergence is absolute, then this definition does notdepend on the choice of a basis In this caseÙ

is called a trace-class operator For

example, one can show that Tr Gn(o

IfËe 9Q is its kernel, then

Now for any Ù

such that has a basis of eigenvectors ofÙ

one can define the

MZ is an analytic function for Re MZ

w and it can be analytically extended

to an open subset containingO In this case we define

This obviously agrees with (2.3) when%

is finite-dimensional Also it is easy to seethat for any positive numberc

w‚m with the usual measureD

descended to the factor Note that in thismeasure the length of

We use the action

for some integer•

(equal to the degree of the map oforiented manifolds) Let Map

 be the set of maps corresponding to the same•

It is clear that each R Map

 can be uniquely written in the form

Comparing this with (2.6), we see that

Remark 2.1 If we repeat the computations for the Euclidean partition function

(replac-ing with’G ) we get

This is our first encounter with the theory of modular forms

There is another way to compute the partition function for the action

Here the measure(+* 9

is defined up to some multiplicative constantï In fact we will

be defining the correlation functions by the formula

This agrees with the computations in example 2.3 if we switch from the Minkowskipartition function to the Euclidean one.

Here is another application of the Gaussian integral for quadratic functionals sider the action functional defined by some Lagrangian

‘ÙÚ1

We knowthat its stationary points are classical solutions Write

The value of the action functional on the classical solution is

Note that if we compute the product using the zeta function of the operator F

 Âtm ;˜

Now recall that the kernel of a Hilbert-Schmidt unitary operator can be written inthe form

 provided that this sum converges for Re MZ O and has

a meromorphic continuation to the whole complex plane with no pole at Z

c‘u

m+˜

HVJKJJ m^˜

 and˜

 are linearindependent vectors in1

.(i) Compute the trace of the Laplace operator

is induced by the standard volume form on1

.(ii) Compute the Euclidian partition function>

for the maps from

1°ˆ c‘u3}

, where is any lift of

to a smooth map1|U1

.(iii) LetX

Lecture 3

Quantum mechanics

The quantum mechanics is aO

Hdodimension QFT

Let us recall the main postulates of quantum mechanics A quantum state of a

system is a line in a separable Hilbert space It can be represented (not uniquely) by

a vector/

of norm 1 To each observable quantity (like position, momentum or energy)one associates a self-adjoint operatorn in (an observable) A measurement of an

observablen depends on the given state/

and is not given precisely but instead there

is a probability that the value belongs to a subset GF

 ,where  ²±

 is the orthogonal projector operator to the subspace W

& is interpreted as the probabilityamplitude that the state & changes to the state/

Its absolute value is the probablity

of this event Note that by Cauchy-Schwarz inequality, this number is always less orequal to 1 and it is equal to 1 if and only if the two states are equal (as lines in theHilbert space)

The expectation value ofn in the state/

is the position operator corresponding

to the measurement of the coordinate Q It is defined by

M¿

Qš¿ This is an bounded self-adjoint linear operator We know from Lecture 1 thatð

Sfa (although27

they do not belong to the space but rather to the space of distributions) We have

In Heisenberg’s picture, the states do not change with time but the observables evolve

according to the law

of a classical mechanical system Another approach is via quantization of a classical

mechanical system Recall that the latter is defined by a Lagrangian

ÅÙÛ  1which, in its turn, defines an action functional on the space

D @Map

A critical point of this functional defines a motion of the mechanical system The

equations for a critical point are called the Euler-Lagrange equations If one chooses

W is a positive-definite quadratic form, we can use

to define a Riemannianmetric

on

A critical path becomes a geodesic

Another way to define the classical mechanics is via a Hamiltonian function which

is a function on the cotangent bundleÙK

.Recall that any non-degenerate quadratic form

on a vector space %

defines aquadratic formd

4`

Gº is equal to the maximum (ifd

O ) or the minimum(ifd

that its second differential is non-degenerate This is called the Legendre transform of

¿ By definition, for anyºAR

,Leg '¿ hº

º FÓ¿À where9

is the implicit function of º defined by º

Gº is a multivalued function, so the Legendre transform

is defined only locally in a neighborhood of an extremum point of the functionº F

¿À

We shall apply the Legendre transform to the Lagrangian function

We denotethe local coordinates in the cotangent bundleÙp

in the tangent bundleÙ

and can be identified with a basis Ҧ

of the path  to the base

(i.e the composition with the projection mapÙKÚá

)

is the path describing the equation of the motion The difference between theEuler-Lagrange equations and Hamilton’s equations is the following The first equation

is a second order ordinary differential equation onÙÛ

and the second one is a firstorder ODE onÙ&p

which has a nice interpretation in terms of vector fields

Recall that a (smooth) vector field on a smooth manifold¾ is a (smooth) section

F of its tangent bundleÙ

G¾ Given a smooth mape
Å*

, ¾ ,and a vector fieldF we say that satisfies the differential equation defined byF (or is

interpre-Recall that a symplectic form on a smooth manifold¾ is a smooth closed 2-form

is a 1-form on¾ , i.e., asection of the cotangent bundle Ù

So we see that the ODE corresponding to the Hamiltonian vector fieldÈ G. defined

by. is the vector from the right-hand-side of Hamilton’s equations We have

The Poisson bracket defines a structure of Lie algebra onï

´

in.The flow

Under the quantization the Hamiltonian function of the mechanical system becomes aself-adjoint operator. , called the Hamiltonian operator of the quantized system We

Hamil-For example, when a mechanical system is given on the configuration spaceÙ

) is called the position (resp momentum) operator

Let us give an example of quantization of a classical mechanical system given by a

harmonic oscillator It is given by the Lagrangian

viewed as the total energy of the system.

The corresponding Newton equation is

are the annihilation and the creation operators We shall see shortly the reason for

these names They are obviously adjoint to each other Using the commutator relation

So we are interested in the representation of the Lie algebra in

.Suppose we have an eigenvector/

of. with eigenvaluec and norm 1 Sincev

isadjoint tov

norm one with such a property is called a vacuum vector.

Denote a vacuum vector byð

/0J

This shows that v

is a new eigenvector with eigenvaluec FP.Ԙ Since eigenvaluesare bounded from below, we get thatv

One can show that the closure of the subspace of

spanned by the vectors ð

•=is

an irreducible representation of the Lie algebra

The existence of a vacuum vector is proved by a direct computation We solve thedifferential equation

is a Hermite polynomial of degree•

It is known also that the orthonormal system offunctions.

3.1 Consider the quantum mechanical system defined by the harmonic oscillator Find

the wave function of the moment operator 

3.3 Compute the Legendre transform of the function¿À iQ

(the dispersion of an observablen at the state /

) Prove the Heisenberg’s Uncertainty Principle

Lecture 4

The Dirichlet action

Now we shall move to QFT of dimension larger than 1, i.e jÂ6Ã

zjzo, for example,

Recall that any QFT is defined by an action functional on the space of paths In

a one-dimensional theory we defined by a Lagrangian

ÙTá1

The pull-backof

dj othis is not true anymore sinceÌÍ M& is a function

Recall that the jet bundle of order r of a fiber bundle¦ over a manifold¾ is avector bundle î M¦ whose local sections are local sections of¦ together with theirpartial derivatives up to orderr Let

of¦ can be uniquely extended

to a section  ofî 9¦ such that

L ŒŽŒŽŒ L &À iQ

Here we assume that the equality takes place when we evaluate the left-hand side on&

We also assume here that

is of the first order

Let us consider an example, which will be very much relevant to the string theory.First, a little of linear algebra Let %

be two vector spaces equipped with degenerate bilinear formsÊ

non-and, respectively We can define a symmetric bilinearform on the space of linear maps Lin

in© Let. be the matrix of

in the first basis andg be the matrix of

in the second basis Letn be the matrix of

¿ with respect to the bases, ando is the same for& Then the matrix of¿ is equal to

4`

defines an inner product on% 

So, our inner product on%  à

be a metric on

and

be a metric on

.Define the LagrangianÞ

'&

ð &Àð D}32

Here D}32

is the volume form defined by the metricÊ

and the adjointD

of Disdefined with respect to the metricsÊ

and.The corresponding action

Observe the following properties of the Dirichlet action:

(A1) (isometry invariance) For any diffeomorphismº

»{â The matrix is the matrix of the metric& It can be viewed as themetric on the image of

under the map& (called the world-sheet) Then

is chosen to be critical, the

action has a simple geometric meaning It is equal to the twice the area of the world sheet&Å in the metric induced from the metric of

In physics the latter action is called the Nambu-Goto action and the Dirichlet action

is called the Brink-DiVecchia-Howe-Desse-Zumino action, or the Polyakov action for

short

Remark 4.1 In the case when the metric on 

is Lorentzian, we have to replace ð ðwithð…F