Feynman Integral on Lie Groups

Cinvestav

Master Degree Thesis in Physics

Geometric Aspects of the FeynmanPath Integral on Lie Groups

Author:Carlos Jonathan Ramrez Valdez

Thesis Advisor:Dr. Hector Hugo Garca Compean

Physics Department

December 15, 2014

To my family and friends

ACKNOWLEDGEMENTS

I would like to express my gratitud to my advisor Dr. Garca Compean. Besides myadvisor, I would like to thank my thesis committee: Dr. Perez Lorenzana and Dr.Cordero Elizalde.

Similarly, thanks to Conacyt for the economic support.

iii

ABSTRACT

The problem of quantizing a classical system in physics can be formulated in termsof the A-model on a complexification of the phase space. This perspective creates arelation between the quantization of a classical system and the A-model. In otherwords, if we find the associated A-model then we can get the quantized version of thesystem.

With this in mind we are interested in the associated A-model to phase spacewith structure G/H where G is a Lie group and H is a subgroup. The reachedA-model in this work was a twisted topological A-model, with two A-branes beingthese a Lagrangian A-brane and a coisotropic A-brane (that fills all the complexifiedphase space). In other words we did a generalization of the correspondence establishbetween quantum mechanics and a non-lineal type A -model with a complexificationof G/H as target, through the algorithm developed by Edward Witten.

Finally, we do an example, tacking SL(2,R)/U(1) as the phase space and weobtain the corresponding A-model.

v

RESUMEN

El problema de cuantizar un sistema clasico en fsica puede se reformulado en terminosde un modelo tipo A en la complexificacion del espacio fase. Esta perspectiva creauna relacion entre la cuantizacion de un sistema clasico y el modelo A. En otraspalabras, si encontramos el modelo A asociado, entonces hemos obtenido la versioncuantizada del sistema.

Con esto en mente, estamos interesados en el modelo A asociado a un espaciofase con estructura G/H, donde G es un grupo de Lie y H un subgrupo. El modeloA alcanzado en este trabajo fue un modelo A topologico torcido, con dos A-branassiendo estas, una Lagrangiana y una coisotropica (la cual llena todo el espacio fase).En otras palabras hemos realizado una generalizacion de la correspondencia estable-cida entre la mecanica cuantica y el modelo- no lineal con una complexificacionde G/H como espacio objetivo, a traves del algoritmo desarrollado por Edward Witten.

Finalmente, hacemos un ejemplo, donde tomamos SL(2,R)/U(1) como el espaciofase, y obtenemos el correspondiente modelo A.

vii

CONTENTS

1 Introduction 1

2 Path Integral on Group Manifolds 5

2.1 Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Path Integral in Curved Spaces . . . . . . . . . . . . . . . . . 6

2.2 Example SU(2) Lie Group . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Finite times path integral . . . . . . . . . . . . . . . . . . . . 10

3 Quantization 15

3.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Morse Function . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Lagrangian and Coisotropic submanifolds . . . . . . . . . . . . 17

3.2 Quantization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Brane Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.1 A new perspective on quantization . . . . . . . . . . . . . . . 19

4 Path Integral Quantization of G/H 21

4.1 Basic Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.1.1 Analytic Continuation . . . . . . . . . . . . . . . . . . . . . . 22

4.1.2 Simplest Cycle of Integration . . . . . . . . . . . . . . . . . . 22

4.1.3 Another Cycle of Integration . . . . . . . . . . . . . . . . . . . 23

4.2 Sigma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3 Boundary conditions of the A-model . . . . . . . . . . . . . . . . . . 28

ix

CONTENTS

4.4 Example SL(2,R)/U(1) . . . . . . . . . . . . . . . . . . . . . . . . . 29

Conclusions 35

Bibliography 36

x

CHAPTER 1

INTRODUCTION

The quantization of a classical system is a very important topic in physics, if we takeany classical system, it does not exist a unique quantum version of it. Some problemsto quantize a physical system will be explained in the section 3.2. Many authorsare making efforts for a better understanding of the meaning of quantization, one ofthese efforts is called brane quantization [1].

This approach to quantization relies on the equivalence between two structures,a symplectic manifold, (M,), that describes completely a classical system, andan A-model of a complexification of M . The so reached quantization of (M,) isprecisely the space of strings with boundary conditions on two A-branes, Lagrangianthe first and coisotropic the other. The Lagrangian A-brane is supported on M , whilethe symplectic structure defines the coisotropic A-brane. With this in mind wewant to inquire in the case when the phase space M is a coset G/H. And with thisextend the scope of the theory in [2].

An interesting way to introduce this approach is to start from the Feynmansformalism of the quantum mechanics [2]. That is the reason to review it in chapter 2.We also clarify this formalism by quantizing the Lie group SU(2). After doing this,we delve in some needed definitions and describe the problem of quantizing a physicalsystem in chapter 3.

Subsequently, in chapter 4 we developed the main theory, i.e. we applied themethod given in [2] to cosets G/H. The procedure is briefly described as follows, we

1

Chapter 1. Introduction

take a complexification of the coset phase space. In doing this, we have to carefullychoose the integration cycles. Since, if we do not pick them up correctly then thereconvergence problems could exist. We can choose different cycles of integration,but we are particularly interested in one that has a direct interpretation as a twodimensional quantum field theory in which the target is the complexification of ourphase space. The output theory would be a topologically twisted A-model with afermionic symmetry and a coisotropic brane as the cycle of integration.

In our work we are interested in the case when the target is a complex coset G/Hwhere G is a Lie group. With this in mind, we first establish the structure of thepath integral in curved spaces [3][4]. The elements that appear in the exponentialof the path integral for these spaces are a term padq

a and an effective Hamilto-nian, where pa and q

a are the usual canonical coordinates while the last term arisefrom the ordering problem that we will explain in section 3.2. Fortunately, the termof our interest, padq

a, is the same independently of the choice of the order of operators.

Once we clarify the starting point, we can follow the theory developed in [2]. Wedo a complexification of our phase-space (and as consequence the complexificationof our integration space) duplicating our degrees of freedom. The main differencebetween this work and [2] is the quotient structure of the initial phase space. As

in [5] we suppose that the complexification G/H of the quotient G/H has to be aKahler manifold, this implies that we have a compatible metric with a complexstructure J , and a Kahler form A.

As mentioned, the complexification of the phase space, implies a complexificationof our integration space, but this is not necessarily our final integration cycle becausethe exponential of the path integral could be divergent, then, we have to buildour integration cycles properly. As a first approach we construct a simple cycle ofintegration. This is achieved by constructing a new submanifold (G/H)0 with thesame structure of G/H and a with different integration cycle. This new constructioncould arise from a different classical system, but it reaches the same complexified space.

On the other hand a different integration cycle can be constructed through theMorse flow equation. Through this we can narrow the divergence term on the path in-tegral, and also add a new evolution parameter. Then, the associated action becomesinto a non-linear sigma model. Next step is to analyse the corresponding boundaryconditions for the coset structure to relate the holomorphic and anti-holomorphic partsof the bosonic and fermionic fields. With this relations we reach a Kazama Suzuki

2

model with type A boundary conditions [6]. In our case this conditions correspond toa coisotropic D-brane wrapping around our integration cycle.

The final theory reached in this work is a topological twisted A-model withfermionic symmetry [5] constructed from a coset theory G/H.

Finally, we apply the theory to an example, we take the Lie group SL(2,R) andthe subgroup U(1) to form the coset the coset G/H = SL(2,R)/U(1). Next, weobtain the cycles of integration via the Morse flow, and we build the Lagrangian andthe coisotropic A-branes.

3

Chapter 1. Introduction

4

CHAPTER 2

PATH INTEGRAL ON GROUPMANIFOLDS

In this chapter we review the Feynman path integral formalism of Quantum Mechanicsin curved spaces. Also, we quantize a group manifold as an example, being thisSU(2).

2.1 Path Integral

The year 1925 can be seen as the beginning of modern quantum mechanics markedby the two almost simultaneously published papers of Heisenberg and Schrodinger.In the former it is proposed the formalism of matrix mechanics, while the latterproposes the formalism of wave mechanics. Schrodinger was the first in showing thatthe two formulations are physically equivalent. Both of these approaches were com-bined heuristically by Dirac into a more general formulation of quantum mechanics.The mathematically rigorous development of this general formulation of quantummechanics was carried out by von Neumann.

The general formulation of quantum mechanics developed by Dirac and vonNeumann lay on the Hamiltonian formalism of classical mechanics. The Lagrangianformalism did not have place in this general formulation of quantum mechanics,except in the derivation of Schrodingers wave equation from the Hamilton-Jacobiequation.

5

Chapter 2. Path Integral on Group Manifolds

The first hint of the possible importance of the Lagrangian in quantum mechanicswas given by Dirac, he remarked that the quantum amplitude qt|qt0 corresponds tothe classical quantity e

i~ tt0Ldt

. This remark led Feynman in 1941 to a new formulationof quantum mechanics.

Feynman and Hibbs gave a heuristic introduction to this new approach to Quan-tum Mechanics, whereas Schulman provided a more rigorous introduction to theFeynmans path integral on configuration space and considered a number of applica-tions of the method in different fields of physics.

This new approach can not lead us to new discoveries that the ordinary quantummechanics could not reach. However, the Feynmans formalism brings new ways toresolve some issues in easier manner and performs a better understanding of theconnection of quantum mechanics and classical mechanics [7].

2.1.1 Path Integral in Curved Spaces

We know that the Feynmans path integral of Quantum Mechanics is deduced as anintegral over the phase space of the system and has the following form

Dp(t)Dq(t)e i~padqaH(pa,qa)dt, (2.1)

where pa and qa are the usual canonical coordinates of the phase space and H(pa, q

a)the corresponding Hamiltonian, but this is for a flat space, and we want the Feynmanspath integral over a phase space of a Riemannian manifold M , because we want toextend the scope of the the theory in [2].

First, let g be a metric on a manifold M . If we try to construct the Feynmanspath integral from the formulation of operators of the quantum mechanics, we arriveto the ordering problem. This is the arbitrariness in choosing the order of operatorsin the Hamiltonian. Then, the path integral depends on this choice of order. Takinginto account this fact, the Feynmans path integral in curved spaces can be writing as

Dp(t)Dq(t)e i~padqaHeff (p,q)dt, (2.2)

6

2.2. Example SU(2) Lie Group

where the dependence on the ordering in Quantum Mechanics arises, as effectiveHamiltonians.

We will introduce the effective Hamiltonians for the most used orderings, whichare the Weyl, symmetric-rule (SR) and anti-standard (AS) orderings,

Heff (p, q, q) = H

(p,

1

2(q + q)

)+ VWeyl

(1

2(q + q)

),

Heff (p, q, q) =

1

2(H(p, q) +H(p, q)) + VSR(q),

Heff (p, q, q) = H(p, q) + VAS(q),

(2.3)

where q and q are consecutive elements of the partition that is used to calculate thepath integral, and the potential corrections V given by

VWeyl =~2

8m

(gabdac

cbd R

),

VSR =~2

8m

[abg

ab 2g1/4LBg1/4],

VAS =~2

6mR,

(2.4)

where R is the scalar of curvature, abc are the Christoffel symbols and LB1 is the

Laplace-Beltrami operator.

2.2 Example SU(2) Lie Group

In this section we will quantize the Lie Group SU(2) with the Feynmans path integralof Quantum Mechanics.

SU(2) is the set of special unitary matrices of order 22. This set is isomorphic toS3, so we will consider a parametrization of the 3-sphere in R4. This parametrizationis given by the euler angles, as follows

1LBf |g|1/2i|g|1/2gijjf , where f is a scalar function, gij the metric and g his determinant

7


u1 = cos

(

2

)cos

(+

2

), (2.5)

u2 = cos

(

2

)sin

(+

2

), [0, pi] (2.6)

u3 = sin

(

2

)cos

(

2

), [0, 2pi] (2.7)

u4 = sin

(

2

)sin

(

2

), [0, 2pi] (2.8)

with this parametrization the Lagrangian of a free particle over SU(2) is given by

L =1

2I u2 =

1

2I(u1

2 + u22 + u3

2 + u42), (2.9)

where I is the inertial moment.

On the other hand, the isomorphism between SU(2) and S3 is given by thefollowing relation

(u1, u2, u3, u4) (u1 + iu2 iu3 u4iu3 + u4 u1 iu2

), (2.10)

in terms of the euler angles an element of SU(2) has the following form(cos

2ei+2 i sin

2ei2

i sin 2ei

2 cos

2ei

+2

), (2.11)

now, considering two points on the manifold Ua and Ub, we define a product inthe Lie Group through an invariant of the group given by ua ub 12Tr

(UaU

1b

).

Since, SU(2) is a group, there exist an element U in the group such that U =UaU

1b , so we obtain the following result

1

2Tr(UaU

1b

)=

1

2Tr (U) , (2.12)

hence if we take the trace of (2.11),

1

2Tr (U) = cos

2cos

+

2. (2.13)

8


In addition, the inverse of eq. (2.11) is given by(cos

2ei

+2 i sin

2ei2

i sin 2ei

2 cos

2ei+2

), (2.14)

when considering the product in the relation (2.12), we obtain the following equations:

cos = cos a cos b sin a sin b cos (a b) , (2.15)

ei =eia

sin [sin a cos b cos a sin b cos (a b) i sin b sin (a b)] ,

(2.16)

ei+2 =

1

cos 2

[cos

a2

cosb2ei2

(ab+ab) + sina2

sinb2ei2

(aba+b)]. (2.17)

Furthermore, we can bring U to a diagonal form by doing a similarity transfor-mation, we will obtain a diagonal matrix that transforms Ua into Ub by a rotationaround of his principle axis. Considering them as initial and final states, respectively,the transformation have the meaning of the evolution of the system by a rotationaround this principle axis. The diagonal matrix has the form(

ei2 0

0 ei2

). (2.18)

An easy way to see this, is that the characteristic polynomial is of degree two, soit has two real or complex roots. The eigenvalues have to be the same for any , so, ifwe take = pi

2and = pi

2the matrix U gives rise to a rotation matrix of dimension

2 2 and we know that there does not exist any real eigenvalues that diagonalizesuch a matrix. Then, the only option is to have complex eigenvalues. Hence, theroots have to be complex conjugate of each other. In addition, the determinant ofthe matrix has to be one, this implies that the only option for the matrix is to havethe form of equation (2.18).

By the properties of the trace, we know that the traces of the (2.11) and (2.18)have the same value. Thus we obtain

cos

2= cos

2cos

+

2, (2.19)

this equation enables us to relate the inner product with only one angle, thatcorresponds to the parameter of the diagonal matrix.

9


2.2.1 Finite times path integral

The path integral is constructed through slices of time. That implies that the mostimportant part is to find his form at very small times and build the complete pathintegral by the method that we wish. Then for very small time, we propose as usualthe semiclassical approximation [8]

Kj = eiSj , (2.20)

where Sj is the action of the system and Kj is the kernel (propagator) between uj1and uj at short times. For the free particle case we can calculate Sj by integratingthe equation of motion,

Sj =I

2(uj uj1)2 , (2.21)

using (2.19) with = 2

and the fact that |u| = 1 we rewrite the action as

Sj =I

(1 cos j) . (2.22)

As we know, we have to include the ordering correction in the action [3][8], so theKernel at short times has to be,

Kj = exp i

(I

(1 cos j) +

8I

), (2.23)

we just found the kernel for SU(2) at short times. Now, we could try to calculatethe path integral with the kernel (2.23) but it does not have a practical form, and sowe need to do some changes before proceeding it.

Using the identity [3][8]

eu cos 12u

k=k=

exp

[ik + u 1

2u

(k2 1

4

)], (2.24)

with = , u =iI

we can change the cosine term in the exponential as follows,

Kj = ei( I+

8I )ei

I

cos j

i2Iei8I

kj=

eikj cos ei2I (k2j 14),

10


For our purpose it is useful to split the sum as follows

Kj i2Iei8I

l=0

exp

[ i

2I

(l2j

1

4

)] exp

[ i

2I

((lj + 1)

2 +1

4

)] lm=l

eimj ,

(2.25)the difference of the exponential with the indexes lj can be simplify with the followingproperty [8][3]

eax ebx = (a b)xe(a+b)x2s=1

[1 +

(a b)2 x24s2pi2

], (2.26)

then

l=0

exp

[ i

2I

(l2j

1

4

)] exp

[ i

2I

((lj + 1)

2 +1

4

)]

=i

2I(2lj + 1) exp

[ i

2I

(lj +

1

2

)2][4I

(2lj + 1) sin

((2lj + 1)

4I

)]

i2I

(2lj + 1) exp

[ i

2I

(lj +

1

2

)2].

where we have made an approximation considering that 0.

Besides, using the matrix representation with weight l and the addition anglestheorem on SU(2), we can expand the summation over mj as follows

ljmj=lj

eimj =lj

mj=lj

ljnj=lj

eimj(j1j)einj(j1j)P ljmjnj (cos j1)Pljnjmj

(cos j) ,

(2.27)so, the kernel reaches the form

Kj (i

2I

)2 l=0

ljmj=lj

ljnj=lj

(2lj + 1) ei2Ilj(lj+1)eimj(j1j)einj(j1j)

P ljmjnj (cos j1)Pljnjmj

(cos j) .

11


Now we are ready to integrate the kernel, so we consider the finite time pathintegral [4],

K (tf , xf ; ti, xf ) = limN

AN

dx1 . . . dxN1e

i~Nn=1

m2

(xnxn1

)V (xn+xn12 ), (2.28)

where AN =(

m2pi~

)N2 , and for our case we have I = m|u|2 = m, ~ = 1 and

dxj = sin jdjdjdj. Replacing our kernel in eq. (2.28), we get

K (ua, ub, T ) = limN

(I

2pii

)N+12 N

j=1

djeiN+1l=1 Sl , (2.29)

where T (N + 1) and dj sin jdjdjdj.

Then, we have


(I

2pii

)N+12 N

j=1

dj

N+1l=1

Kl, (2.30)

additionally, we know the following normalization relation

pi0

dj sin j

2pi0

dj

2pi2pi

djei(mjj+njj)ei(mj+1j+nj+1j)

P lnjmj j (cos j)Pmj+1nlj+1j+1(cos j) =1

2lj + 1lj lj+1mjmj+1njnj+1 ,

(2.31)

with the purpose of simplifying the calculations, we define the following new variables

lj=0

ljmlj

ljnlj

j (2lj + 1) e i2I lj(lj+1)K+sj ei(msj+nsj)P lsnsmsKsj ei(msj+nsj)P lsmsns .

(2.32)

With those eq. (2.31) can be rewritten asdjK

jjKj+1j =

1

2lj + 1ljlj+1lmlm+1lnln+1. (2.33)

12


Thus, the kernel eq. (2.30) is given by


(I

2pii

)N+12 N

j=1

dj

N+1l=1

jjK

jj1K

jj , (2.34)

the kernel is now in a very manageable form. In order togain a better understandingof his structure let us analyse two successive elements of the product in the integrand.

j

j+1jj+1K

jj1K

jjKj+1j K

j+1j+1d

=

j

j+1jj+1K

jj1K

j+1j+1

[KjjK

j+1j d

]=

j

j+1jj+1K

jj1K

j+1j+1

[1

2lj + 1ljlj+1lmlm+1lnln+1

]=

j2j

1

2lj + 1Kjj1K

jj+1,

(2.35)

then, we do over the above procedure until we obtain (doing the integration N times),

1N+11

(1

2lj + 1

)NK10 K

1N+1, (2.36)

finally, we obtain the kernel for two arbitrary points in the group, in other words wequantized the SU(2) group. The final form of the kernel is then

K (ua, ub, T )

= Al=0

(2l + 1) eiT2Il(l+1)

lm=l

ln=l

ei(ma+na)ei(mb+nb)P lmn (cos a)Plnm (cos b) .

(2.37)We have reached the quantized version of SU(2), from here we can directly obtain

the quantization of SU(2)/U(1). Equation (2.37) can be reduce to the quantizationof SU(2)/U(1) through the identification of a = b, thus obtaining the followingresult

K (ua, ub, T ) = Al=0

(2l + 1) eiT2Il(l+1)

ln=l

ein(ba)P ln (cos a)Pln (cos b) . (2.38)

13


We know that SU(2) = S3 which is compact and connected. Similarly, SU(2)/U(1) =S2 is compact and connected. These are the principal reasons for the validity ofeq. (2.38) as quantization for the coset SU(2)/U(1).

14

CHAPTER 3

QUANTIZATION

In this chapter, we will introduce some fundamentals, like the notion of Lagrangianmanifolds and coisotropic manifolds. The aim is to establish the essential tools tobuild the A-model.

3.1 Fundamentals

3.1.1 Morse Function

Let Z be a m-dimensional manifold with local coordinates wi i = 1, ...,m, andh : Z R the Morse function with non-degenerated critical points.

If p is a critical point of h then ih(p) = 0, and the matrix ijh(p) is invertiblesince h is non-degenerated.

The number of negative eigenvalues of ijh(p) is defined as the Morse index of hat p and is denoted by ip.

Let gijdwidwj be the metric on Z and introduce a time coordinate s through the

following equation called Morse theory flow equation

dwi

ds= gij h

wj. (3.1)

15

Chapter 3. Quantization

The flow equation eq. (3.1) implies that the Morse function h is always decreasingalong any non-constant flow.

In the neighborhood of a critical point p, with the Hessian matrix of h diagonal-ized, we can find a system of normal coordinates wi centring at p, such that in thesecoordinates the Morse function h can be expand as h = h0 + eiw

i +O(w3) and themetric as gij = ij +O(w2), where h0 = h(0) and the eis are the eigenvalues of theHessian matrix at p.

Then, the flow equation to a first order in w will become

dwi

ds= eiwi (3.2)

with solutions

wi = rieeis (3.3)

for these solutions, we can ask ourselves at what limit s reaches the critical pointp = 0, i.e. wi = 0 for any i. The answer depends on the sign of the eigenvalues ei. Fora non-trivial solution the limits at which s reaches a critical point are s ands . Each solution leaves some free parameters. Without lost of generality wenow focus in the solutions s , then we have ip indeterminate values becausewe have to fix ri = 0 for each ei > 0.

Thus the solutions at p = 0 form a family of dimension ip, now we define thesubspace Cp of Z as the set of all values of h at s = 0 with flows that begin ins .

Since h is a decreasing function, its maximum value is h(p) and this value alsobelongs to Cp because it is the value of the trivial flow. Hence, the maximum valueof Cp is h(p).

Now consider the case when Z is a complex manifold, with complex dimension nwith a function h = ReS where S is holomorphic. Then, the local form of h nearto a non-degenerated critical point p is h = h0 + Re

ni=1(z

i)2, with local complexcoordinates zi. Since Re zi = xiyi the stable and unstable directions for h are paired,then the subspace Cp of each p is middle dimensional. In this situation the Cp gen-erates a basis that classifies cycles on which h is bounded above and goes to at.

16

3.2. Quantization Problem

For the case where the critical points are not isolated, we have to do some changes.Let C be a component of the set of critical points, this set also will be a submanifoldof Z with complex dimension r, and suppose that Z has complex dimension n. Thereexist n r dimensions normal to C, then the local form of h is h = h0 +Re

nr1 (z

i)2,we have n r negative eigenvalues. This implies that CC has a real dimension,2r + (n r) = n+ r, so for obtain a middle dimension space of solutions, we have tofix r extra parameters, obtaining a new subset V C, and as a consequence we reacha new space of solutions CV of middle dimension.

3.1.2 Lagrangian and Coisotropic submanifolds

Let M be a manifold with a non-degenerated symplectic structure , then if we havea submanifold X M of middle dimension such that |X = 0 the submanifold X iscalled Lagrangian submanifold.

In the case where the |X = 0 where X is the ortogonal complement of X iscalled coisotropic submanifold. A coisotropic submanifold is a submanifold that islocally defined by first-class constraints.

3.2 Quantization Problem

In this section we will define the motivation of this work. We will establish some ofthe problems of quantization [1].

Usually, the passage from classical to quantum mechanics is through the corre-spondence principle. This is a recipe that establishes the bridge between the twoworlds. Consider for example the Hamiltonian formulation of classical mechanicsin which one of its big achievements is the introduction of canonical transforma-tions, (x, p) (X,P ), that preserve the Poisson brackets {x, p} = {X,P}. Here therecipe consist in replacing the last with commutators. Let a and b be two classicalobservables, Oa and Ob their associated operators, then the recipe is summarized asfollows

{a, b} 1i~

[Oa,Ob], (3.4)however, this simple description has some problems even for a basic phase spacei.e. the plain space R2n. For example, for a classical system, two different sets ofcoordinates related by a canonical transformation are equivalent, but the associated

17


quantum systems for each set of coordinates can be different, and these could not beconnected through an unitary transformation U , i.e. a change of basis between thedifferent associated Hilbert spaces.

Another difficulty is the so called ordering problem, let f(x, p) be an observableon the phase space and his associated operator Of . The function f could have acombination of products of x and p that generates an ambiguity in the choosing ofhis operator Of . This problem is avoided only if the function f is linear or a linearcombination of powers of only one variable. In this case we have a simple relationeq. (3.4) given by

[Of ,Og] = i~O{f,g}. (3.5)The two mentioned problems are related because a well defined observable function

does not remain invariant under canonical transformations, so these different choiceslead to different quantizations.

Now, consider a phase space M of dimension 2n. Still, if we can identify locallyM with R2n, we can not quantize the manifold since the quantization of R2n is notunique. Even, if we have a prequantum line bundle, there does not exist a recipe toinstruct us how to do the quantization of M , because we do not know how to gluethe local quantizations of the charts of M , so we need this information as a prerequisite.

Then, we can not quantize M without more information than the symplecticstructure and a prequantum line bundle, regrettably the additional structure requiredit is unknown. Hence, there does not exist a general theory of quantization.

In practice, the quantization is an ambiguous notion. An example in which weknow what quantization means is the R2n space with an affine structure (that isequivalent to establish the meaning of our linear observable functions). This structuredoes not need extra information to be quantized. The usual procedure is to splitthe linear functions in coordinates and momenta and then regard these to act asmultiplication and differentiation respectively. The resulting Hilbert space admitsan straightforward action of the symplectic group Sp(2n,R), or instead, of his dou-ble cover, without any dependence on the splitting between coordinates and momenta.

Another important example is the quantization of a cotangent bundle M = T Ufor a manifold U with a given symplectic structure. This structure can be quantize ina direct way through half-densities on U . A similarly procedure exist for quantization

18

3.3. Brane Quantization

of Kahler manifolds in which the basis of the method is the selection of holomorphicsections of an appropriate line bundle.

Now suppose that we known how to quantize a manifold M with a defined actiongroup G, besides some slight restrictions one can define a quantization of the sym-plectic quotient M//G by consider the G-invariant part of the quantization of M .

These different ways to quantize a theory can be combined, but one does notnecessarily reach equivalent quantizations. Hence if M is constructed to be cotangentbundle, or a Kahler manifold or a symplectic quotient, we have not the certainty tofind equivalent quantizations.

3.3 Brane Quantization

3.3.1 A new perspective on quantization

Let us start with a symplectic manifold, (M,), endowed with a prequantum line L asin the geometric quantization, this line bundle is complex with an unitary connectionof curvature .

Let M be a complexification of M with an antiholomorphic involution , such thatM is a component of fixed points. Here, acts as a restriction through of = and the line bundle L is extended over M with a connection Re.

The approach of ref. [1] to quantization is based on the A-model associated withthe imaginary part of the symplectic form M = Im. The A-model is obtained by

twisting a supersymmetric -model with target M . The assumption of having a goodA-model refers to have a model where M admits a complete hyper-Kahler metriccompatible with the symplectic structure .

The A-model admits some types of A-branes. Some of the most familiar areLagrangian A-branes that are supported on Lagrangian submanifolds with middledimension, but they also can admit A-branes whose support has a dimension greaterthan the half dimension of the manifold. The support of such A-branes are thecoisotropic submanifolds.

19


In particular, the choice of the line bundle L M with curvature Im de-termines a canonical coisotropic brane in the A-model with all M as support. Thequantization of (M,) is achieved by standing that the associated Hilbert space His the vector space of (Bcc,Bl) strings, where Bcc refers to a canonical coisotropicA-brane and Bl to a Lagrangian A-brane. This definition gives a vector space de-pending on the choice of Bl. Aldi and Zaslow shown in examples [9] that the explicitconstruction of this vector space is similar to quantization. Finally, the resultingHilbert space H, depends only on the choices of M ,L, and Bl.

Only one thing remains unsolved, we need a natural way to quantize a large classof functions on M , i.e. to define the operators on H. The functions on M that canbe naturally quantized are holomorphic functions on M that have a suitable behaviorat infinity. In some cases these holomorphic functions could be only the functions ofpolynomial growth.

20

CHAPTER 4

PATH INTEGRAL QUANTIZATION OFG/H

4.1 Basic Construction

In essence, the quantization of a physical system means to establish a clear relationbetween the classical system described on a phase-space M (with a symplectic form that is a closed two form) and a quantum system described by a Hilbert space H.

Consider a phase space with structure G/H where G is a Lie Group and H asubgroup of G, G/H has not necessarily a group structure. In particular, we will beinterested in the case when G/H is a Kahler space.

The Hilbert space H that we can associate to the quotient G/H will be finite-dimensional if and only if G/H has a finite volume. For our case we associate the1-loop n-point correlation function U1U2 . . . Un to the observables U1, U2, . . . , Un.We know from chapter 2 eq. (2.2) that the corresponding Feynman path integral forcurved manifolds is given by

U1U2 . . . Un =UDp(t)Dq(t) ei

(pldqlHeffdt) u1(t1)u2(t2) . . . un(tn). (4.1)

where U is the space of cycles of integration, pl(t) and ql(t) are cyclic variables ofperiod 2pi over the phase space of G/H and ui(t) are observables. For our purpose

21

Chapter 4. Path Integral Quantization of G/H

it is unimportant the value of the effective Hamiltonian, so hereinafter we fix Heff = 0.

Consider a phase space G/H with canonical coordinates (pl, ql) and a symplectic

form f given by f = db where b is an arbitrary connection, this canonical coordinatescan be viewed as the embedding map of the 1-sphere to the phase space, i.e. S1 G/H.

4.1.1 Analytic Continuation

Here we complexify our quotient G/H to get a complex manifold G/H with anextended closed holomorphic (2, 0) symplectic form W = A + if and require theexistence of an antiholomorphic involution such that (W ) = W . That we caninterpret as the minimal requirement to develop the theory [5][2][10][11]. Schematically

G/H G/H,f W,

the definition of implies the constraint ReW |G/H = 0.

Hereinafter we will denote the real components of our coordinates on G/H as g.We know that W is closed, then, it can be written as W = d, where = dg

.For convenience we write down = c + ib. This implies that dc = A and db = f .Thus the path integral is rewritten as

Dg(t) e

dg u1(t1)u2(t2) . . . un(tn), (4.2)

where is a subset of the complexification of U . That has to be a middle dimensionspace of integration cycles, where eq. (4.2) converges. Let us write the complexificationof U as U .

4.1.2 Simplest Cycle of Integration

Next, we will construct some cycles of integration, with the purpose of to have

middle dimension we take a middle dimensional sub-manifold (G/H)0 G/H and as the free loop space over (G/H)0, i.e. parametrizes maps from S

1 to (G/H)0.

We have to handle carefully the real part of the exponent of the path integral(4.2), h = Re

dg

, because it is unbounded by above. With the purpose to avoid

22

4.1. Basic Construction

this problem we choose h to vanish identically over (G/H)0. This implies that anyvariation of h have to be zero i.e.

h =

Ag

dg = 0. (4.3)

Then, A|(G/H)0 = 0, i.e. (G/H)0 is a Lagrangian submanifold with respect to A. Thiscondition makes h constant over all connected components of G/H0. If we wouldrequire c to be pure gauge then we have to fix h = 0.

If the imaginary part of the exponent on eq. (4.2) is non-degenerated, then, we havebasically the same properties between G/H and (G/H)0, and the path integral over(G/H)0 will match with the Feynman path integral associated with the quantizationof G/H. In other words the quantization of G/H and (G/H)0 correspond to different

cycles of integration in the same complexified space G/H.

4.1.3 Another Cycle of Integration

In order to obtain integration cycles with h bounded by above, we use the Morseflow eq. (3.1). In our case the Morse function will be the real part of the exponent

dg which is an holomorphic function on U ,

h = Re

dg

=

cdg

. (4.4)

the Morse flow equation will be

g(s, t)

s= h

g. (4.5)

now, we have to find the space of critical points of the Morse function that are givenby

h =

Ag

dg = 0. (4.6)

since A is non degenerated we obtain dg = 0, and so the set of critical points is

the set of constant curves, i.e. there exist a one-to-one relation between the constants

curves and the points on G/H. Then the set of critical points can be identified with

G/H. Let us name this copy as G/H. Later on, we chose a middle dimensional

space V G/H and we consider all the solutions in the half-line that starts in V .

23


As we know, initially we have a parametrization g(t) that is constant over V , such

that g : S1 G/H. Now we will introduce a new parameter s through the Morseequation, such that g(t) g(t, s), i.e. g will be a function of two parametersg : S1 R {0} G/H. Let us to denote the new domain by C.

The flow equation will not be automatically a differential equation over the newdomain C. For this, we need to define a metric on U . We choose dgdg to bethis metric and also as the metric for G/H, defined by

|g|2 =

ggdt. (4.7)

with this metric we reach the following flow equation,

g(s, t)

s= A g

(s, t)

t. (4.8)

The condition at s over g(s, t) is the approaching to some element lyingon the subspace V of the set of critical points. For s = 0 the only condition is thatg(0, t) have to be regular.

Now we choose I = A such that I

I

= (the space of metrics that

have this property is always not empty and contractible). With this condition we

obtain an almost complex structure I on G/H, if is an Hermitian metric1.

If I is an integrable complex structure then the map g : C G/H is holomor-phic. If it is not, then g is known as an pseudoholomorphic map.

The theory is conformally invariant then we do a convenient conformal mappinggiven by z = ew, where w = s+ it, this maps the cylinder C to a unit disc |z| 1 thatwe will denote by D. The map may not extend continuously at z = 0 but in this caseconverges to the limit g(t) V and is independent of t. Then it is continuous at z = 0.

On the other hand, we can ask ourselves if I coincides with the complex structuregiven by the complexification of G/H. We denote this complex structure as J . IfI = J then has to be (1, 1) with respect to J . Also, we know that A is real, then it is(2, 0)

(0, 2) and so have to be (2, 0)

(0, 2). But this contradicts the hypothesis

1If is Kahler, then I represents a complex structure.

24

4.2. Sigma Model

of a positive definite metric that stands for a metric type (1, 1) with respect to J .Hence I and J have to be different complex structures.

4.2 Sigma Model

We have a complexified path integral eq. (4.2), but we want to define the integrationonly over the disk D. For this we insert a Lagrangian multiplier T that restrictsthe integration over the coordinates related through the Morse flow equation (4.8).

T is a one-form on D with values in the pullback of the cotangent bundle of G/H,obeying T = ?I

T.

First we rewrite the condition on eq. (4.8) as follows

U = dg + ?I dg, (4.9)

where ?ds = dt and ?dt = ds, then we insert the restriction in the path integral oneq. (4.2) as a Dirac delta,

Dg(s, t)(U) e

dg u1(t1) . . . un(tn)OV (0), (4.10)

where

(U) = |det(U/g)|1DT ei

D TU , (4.11)

and OV (0) is a closed operator that impose the constraint that the point z = 0 ismapped to s = 0 or s = (the critical set given by the Lagrangian submanifold).

Now, we want to introduce in the path integral fermions and with differentfermionic quantum number +1, 1, respectively, with kinetic energy that is thelinearization of the equation U = 0. This term added has as a consequence thecancellation of the bosonic determinant up to a sign, in our case this sign can betaken as +1 since CV is connected.

Therefore are zero-forms that take values in the pullback of the complexified

tangent bundle TC(G/H) of G/H, and are (anti)holomorphic one-forms with

values in the pullback of the complexified cotangent bundle T C(G/H). Also obey = ?I

. Then the fermionic action can be written as

D

D. (4.12)

25


in this way the integral reaches the following structureDgDDDT ei

D(T?U+D)e

dg

nk

uk(tk), (4.13)

In the case of a symmetric Kahler coset group we know from [10][11][12] that thefermionic symmetry transformations of the twisted topological A-model are

g = i(g+ + g),

+ = = 0,

+ = g1zg,

= +(zg)g1,

(4.14)

where is the fermionic susy parameter and g is the embedding map of to G, are the projectors of complexified tangent space TC of G/H , into the splitting Tdue to the almost complex structure J . = and = , where areWeyl fermions with values on TC. The set of equations eq. (4.14) can be rewritten as

g = ,

= T,

= T = 0,

(4.15)

where we used the exponential map g = egX (X are the generators of G) with the

purpose of introduce the coordinates g of the target G/H, and also the followingrelations

= i(+ + ),

= + + ,

T = (z + +z) g.

(4.16)

The action is invariant by exact terms, then, let us add the following exact term

(

2

D

?T)

=

2

D

T ?T + 4

D

R (4.17)

thus the complete integral on the disc D yields

D

( 2

T ?T + iT ?U i D + 4

R ), (4.18)

26

4.2. Sigma Model

this integral is precisely a Gaussian integration over the Lagrangian multiplier T , andafter an elementary computation we find.

DgDD e

D( 12U?UiD+ 4R

)e

dg

nk

uk(tk)OV (0).(4.19)

Then, the bosonic kinetic energy is

1

2

D

U ?U =

D

dsdt

[1

2

(sg

sg + tg

tg)

+2

A sg

tg

],

(4.20)let us rewrite the RHS of eq. (4.20) with the following identity

(sg

sg + tg

tg)

= ijTr((g1ig)(g1jg)

), (4.21)

where g is the embedding map of D to G/H and the latin indexes refers to i = (t, s).

We note also thatD

dsdt A sgtg

=

D

A =

D

dc =

D

c = h. (4.22)

then, we get a bosonic kinetic energy given by

1

2

D

ijTr(g1ig)(g1jg) +2

h (4.23)

now, by adding the term that comes from padqa on eq. (4.2) to the term proportional

to h, and fixing = 2 we obtain

h+D

(c+ ib) = h+ h+ iD

b = i

D

b. (4.24)

Thus, without consider the terms with fermions, the exponential on eq. (4.19) isfound to be

12

D

ijTr(g1ig)(g1jg) + iD

b. (4.25)

this is the WZW action in an open non-linear twisted sigma model of type A.

27


If we do not fix the = 2 we rather obtain

12

D

ijTr(g1ig)(g1jg) D

A+ i

D

b (4.26)

where = 1 2.

Now we introduce the associated coisotropic D-brane L defined as G/H with aconnection b and a curvature f . Therefore the action, eq. (4.26), is a WZW actiondescribing open strings atached to a D-brane, or more precisely, to an A-brane. Inthe next section we will argue that this is precisely a coisotropic A-brane BG[6].

If we choose 1 2

= i there also holds another symmetry given by

A = ,b = .

here we can ask ourselves, whether it is possible to reach an N = 2 superconformal

field theory from a coset G/H. With the purpose to clarify this point we introducethe following theorem [10].

Theorem 1. When rank G = rank H, a necessary and sufficient condition for amodel on a coset G/H to have N = 2 superconformal symmetry is that G/H iskahlerian.

4.3 Boundary conditions of the A-model

D-branes can be studied by using the techniques of perturbative string theory. Theycan also be described in terms of boundary conditions of open strings, or as boundarystates in the closed string sector. The concept of a boundary state describes howclosed strings are emitted or absorbed on the D-brane world volume.

For the study of bosonic D-branes on cosets G/H we require certain boundaryconditions [5] for the bosonic and fermionic fields given by

I(z) = RI(z),

(z) = R(z),

28

4.4. Example SL(2,R)/U(1)

where R is an endomorphism of TX|Y , with Y being a submanifold that containssome or all the components of g(D) given by

R = idNY (|Y F )1(|Y + F ), (4.27)where TX|Y = NY TY and F = db, i.e. the curvature of a gauge field b. Also Rsatisfies the following conditions

RR = , (4.28)

RRA = A, (4.29)

RRR

f

= f, (4.30)

where f are the structure constants of the group.

The above conditions are known as A-type boundary conditions of the D-brane [5].

Now consider the cycle that we define in our target space G/H through the Morseflow, we want to study the D-branes wrapping around this cycle.

Let the cycle be n-dimensional then we have a basis constituted by n tangential

vectors to the cycle and dimG/H n normal vectors. The boundary conditions typeA in eq. (4.28) imply that the metric is block diagonal, i.e.

= + (4.31)

where refers to the parallel components and to the normal components of themetric along the cycle. In contrast, eq. (4.29) implies that

A| = 0 or A| = 0. (4.32)The vanishing of A and his non-degeneracy implies that the dimension of the

cycle has to be 12G/H. So we have a Lagrangian submanifold of G/H.

In the case when Y = G/H we do not have perpendicular components for

neither for A. Then we have an A-brane supported on G/H, that correspond to acoisotropic A-brane with a connection b.

4.4 Example SL(2,R)/U(1)As an example, we consider the coset SL(2,R)/U(1) as the symplectic structure thatwe want to quantize. The complexification of SL(2,R)/U(1) is SL(2,C)/U(1)C, and

29


can be rewritten as

G/H = SL(2,C)/U(1)C =SU(2) SU(2)U(1) U(1) =

SU(2)

U(1)

SU(2)

U(1)= S2 S2, (4.33)

The complex symplectic structure of G/H is

W = A + if , (4.34)

where A is a Kahler form on G/H and f = ImW is the analytic continuation to

G/H of the original real symplectic structure f on G/H.

Let g be the local coordinates on G/H. They are given in terms of the coordi-nates of S2 and S2. Namely g = (, , , ).

A is precisely the Kahler form given for each S2 as

A = A A, (4.35)

where

A = 2idz dz

(1 + zz)2; A = 2i

dz dz(1 + z z)2

, (4.36)

in terms of the coordinates g(, , , ),

z =x1 + ix2

1 x3 =sin ei

1 cos , (4.37)

with x1 = sin cos, x2 = sin sin, x3 = cos . Moreover,

z =x1 + ix2

1 x3 =sin ei

1 cos . (4.38)

Eq. 4.37 represents a conformal transformation between the 2-sphere and the flatplane with compactification points.

A from eq. (4.35) vanishes when one restricts to the diagonal submanifoldS2diag = {S2 S2|z = z(and z = z)}.

30


Figure 4.1: Diagram of G/H = S2 S2 with the submanifolds represented by lines.

For S2diag = and = and A|S2diag = 0. Thus S2diag is a Lagrangian submanifold

with respect to the symplectic structure A, see fig. 4.1.

On the other hand f restricted to S2diag has to be non-degenerated. That means

f |S2diag = f, (4.39)then

f =1

4sin d d+ 1

4sin d d, (4.40)

and

f |S2diag = f =1

2sin d d. (4.41)

Moreover, remember than = c + ib and f = db. After some calculations, it iseasy to see that c is given by

c = 2iz

1 + zzdz + 4i

z

1 + zzdz 2i

z

1 + z zdz 4i z

1 + z zdz, (4.42)

it is immediate to check that on the S2diag c vanishes, that is

c|S2diag = 0, (4.43)thus the Morse function reads

h =

c. (4.44)

31


The field b can be regarded as an abelian gauge field (connection one-form) on the

entire space G/H = S2 S2. Then as mentioned before db = f . Eq. 4.40 impliesthat

b =1

4(1 cos)d

=1

4

x1dx2 x2dx1x3 1 ,

sub-indexes correspond to the two posible open sets U which cover S2. Thatmeans

U+ U = S2 and U+ U = S1, (4.45)thus the abelian gauge field

b =1

4(1 cos )d+ 1

4(1 cos )d, (4.46)

wich on the diagonal S2diag with = and = we get

b|S2diag =1

2(1 cos )d. (4.47)

On S2 S2 we take the product metric

= g + g (4.48)

where g is the (Kahler) metric on S2 and g is the corresponding metric on S2.

The metric on S2 S2 is given by

=

1 0 0 00 sin2 0 00 0 1 0

0 0 0 sin2

, (4.49)then, the Morse flow is described by

dg

ds= h

g. (4.50)

these are four equations given by

32


d

ds= h

;

d

ds= sin2 h

d

ds= h

;

d

ds= sin2 h

Now, we can construct the A-branes for our example. First we recognize thesubmanifolds, we have a Lagrangian submanifold, this being S2, and the coisotropicsubmanifold has to be S2 S2. Thus, by eq. (4.27) the corresponding boundaryconditions are [6]

Lagrangian A-brane,

R = idNS2 (g f |S2diag)1(g + f |S2diag),(4.51)

Coisotropic A-brane,

R = ( f |S2S2)1( + f |S2S2),(4.52)

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34

CONCLUSIONS

We inquired in the brane quantization theory doing an approach to the quantizationof a classical system with a phase space with structure of a coset G/H where G is aLie group. We arrived to a twisted topological A-model with boundary conditionsthat defines a Lagrangian A-brane and a coisotropic A-brane.

To reach that theory we begin with a particle moving in a quotient G/H. then

we complexified the phase space to G/H, doing this we arrived to a modification ofour starting path integral in which we can construct different cycles of integration,particularly we use the Morse flow equation to construct our integration cycles.

Once we had our integration cycles in G/H we took care of the group structureand established the supersymmetric transformations of the fields[5]. We also estab-lished the corresponding boundary conditions associated to the A-branes wrapping inour integration cycles in a consistent way, arriving to a Kazama Suzuki model withboundary conditions type A, where this A-branes corresponding to Lagrangian andcoisotropic submanifolds with a gauge potential b, and curvature f .

Therefore, with this work the theory developed by Edward Witten in [2] is ex-tended to Lie groups in particular to cosets.

We provided an example to clarify the theory by taking the coset SL(2,R)/U(1)as our initial phase space to obtain the corresponding integration cycles and theLagrangian and coisotropic A-branes.

35


The prospects are, add a nonzero effective Hamiltonian to the theory and analysemore examples like the Quaternion-Kahler symmetric spaces.

36

BIBLIOGRAPHY

[1] Sergei Gukov and Edward Witten. Branes and quantization. Advances inTheoretical and Mathematical Physics, 13(5):14451518, 2009.

[2] Edward Witten. A new look of the path integral of quantum mechanics.arXiv:1009.6032v1, September 2010.

[3] A. Demichev M. Chaichian. Path Integral in Physics, Vol. I. Springer, 1998.

[4] F. Steiner C. Grosche. Handbook of Feynman Path Integrals. Institute of PhysicsPublishing, 2001.

[5] Sonia Stanciu. D-branes in kazama suzuki models. Nuclear Physics B, 526(1-3):295310, August 1998.

[6] Anton Kapustin and Dmitri Orlov. Remarks on a-branes, mirror symmetry, andthe fukaya category. Journal of Geometry and Physics, 48(1):8499, October2003.

[7] Wolfgang Tome. Path Integral on Group Manifolds. World Scientific, 1998.

[8] D. Peak and A. Inomata. Summation over feynman histories in polar coordinates.Journal of Mathematical Physics, 10(8):14221428, August 1969.

[9] M. Aldi and E. Zaslow. Coisotropic branes, noncommutativity, and the mirrorcorrespondence. Journal of High Energy Physics, 2005(06), June 2005.

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BIBLIOGRAPHY

[10] Y. Kazama and H. Suzuki. Characterization of n=2 superconformal modelsgenerated by the coset space method. Physics Letters B, 216(1-2):112116,January 1989.

[11] Y. Kazama and H. Suzuki. New n=2 superconformal field theories and superstringcompactification. Nuclear Physics B, 321:232268, 1989.

[12] Edward Witten. The n matrix models and gauged wzw models. Nuclear PhysicsB, 371:191245, 1992.

38

IntroductionPath Integral on Group ManifoldsPath IntegralPath Integral in Curved Spaces

Example SU(2) Lie GroupFinite times path integral

QuantizationFundamentalsMorse FunctionLagrangian and Coisotropic submanifolds

Quantization ProblemBrane QuantizationA new perspective on quantization

Path Integral Quantization of G/HBasic ConstructionAnalytic ContinuationSimplest Cycle of IntegrationAnother Cycle of Integration

Sigma ModelBoundary conditions of the A-modelExample SL(2,R)/U(1)

ConclusionsBibliography

Feynman Integral on Lie Groups

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