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The continuous spin representations of the Poincaré and Super-Poincaré groups and their construction by the Inönü-Wigner group contraction

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The continuous spin representations of the Poincaré and Super-Poincaré groups and their construction by the Inönü-Wigner group contraction
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THE CONTINUOUS SPIN REPRESENTATIONS OF THE POTNCARE ANT)
SiPER-POINCARE GROUPS AND THEIR CONSTRUCTION
BY THE INONU-WIGNER GROUP CONTRACTION


















By

ABU M.A.S. KHAN


















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

'UNIVERSITY OF FLORIDA


2004





































Copyright 2004 by

Abu M.A.S. Khan

















To my mother and my late father















ACKNOWLEDGMENTS

I am boundlessly grateful to my mother for all her support and sacrifices without which it would have been impossible for me to continue higher studies. My father, who died at my early age, has always been an inspiration to me, through the memories he left with me, for his enthusiasm, respect and encouragement for knowledge and learning. I am also indebted to my brother for taking care of our family during my stay abroad without which it would have been very difficult for me to continue research.

I am very grateful to Professor Pierre Ramond for his patience and support, and teaching me how to do research. I am highly indebted to my teachers Shubhash Chandra Datta at my high school and Professor Khorshed A. Kabir for their continuous encouragement and help. I cannot express how important their support was at times for me.

I would like to express my gratitude to the Bangladeshi community here at Gainesville who never let me feel that I was out of my home, particularly Jaha Hamida who constantly and tirelessly supplied the food and tea at the department. I seldom had to go out for lunch! Special thanks go to Professor Khandker A. Muttalib who helped me to get admitted here and for his support whenever I needed it, and of course the great volleyball team members. I will definitely miss them. I thank my friends for encouraging me all through my studies.


iv

















TABLE OF CONTENTS
Eagt
ACKNOW LEDGMENTS ...................................... iv

A B ST R A C T . . . . . . . . . . . . . . . . . . . . . . vii

CHAPTERS

1 INTRODUCTION .................................... .. .. 1

2 THE CONTINUOUS SPIN REPRESENTATIONS OF THE POINCARE GROUP ... 4

2.1 Light-Cone Form of the Poincar6 Algebra . . . . . . . . . . . 4
2.2 Continuous Spin Representations . . . . . . . . . . . . . 7
2.2.1 Four Dimensions . . . . . . . . . . . . . . . . 7
2.2.2 Five Dim ensions . . . . . . . . . . . . . . . . 8
2.2.3 Higher Dimensions . . . . . . . . . . . . . . . 9

3 THE CSR IN SUPER-POINCAREt ALGEBRA . . . . . . . . . . . 12

3.1 Super-Charges in Light-Cone Form . . . . . . . . . . . . . 12
3.2 Non-nilpotent Light-cone Vector and the SUSY CSR . . . . . . . . 14
3.2.1 Four Dimensions . . . . . . . . . . . . . . . . 14
3.2.2 Five Dim ensions . . . . . . . . . . . . . . . . 16
3.3 Nilpotent Light-Cone Translations and the CSR . . . . . . . . . 17
3.3.1 Four Dimensions . . . . . . . . . . . . . . . . 17
3.3.2 Ten Dim ensions . . . . . . . . . . . . . . . . 18

4 INoNU-WIGNER GROUP CONTRACTION . . . . . . . . . . . . 20

4.1 IW Group Contraction . . . . . . . . . . . . . . . . 20
4.2 Contraction of SO(3) group with periodic Boundary Condition . . . . . 22

5 CONTRACTION OF THE FIVE DIMENSIONAL MASSLESS REPRESENTATIONS 26

5.1 KK Reduction of the Poincar6 Algebra ........................ 26
5.2 Contraction of the Orbital Rotations ...... ......................... 27
5.3 Contraction of the Internal Part and the CSR ......................... 29

6 CONTRCATION OF THE MASSIVE REPRESENTATION IN FIVE DIMENSIONS 32

6.1 Non-linear Realization and the Majorana Theory . . . . . . . . . 32
6.1.1 SO(4) E(3) -- SO(3) -+ E(2) ...................... 34
6.1.2 SO(4) SO(3) -4 E(2) ................................. 36
6.2 Representation of the Wavefunction . . . . . 37

7 CONCLUSION ..... .................................... 42

APPENDICES

A CSR AND THE CONFORMAL GROUP .................. . . . 44


V









B ANGULAR MOMENTUM IN CYLINDRICAL COORDINATES WITH
PERIODIC BOUNDARY CONDITION ............................... 46

REFERENCES ......................................... ........ 48

BIOGRAPHICAL SKETCH .................................... 50


vi















Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE CONTINUOUS SPIN REPRESENTATIONS OF THE POINCARE AND
SUPER-POINCARE GROUPS AND THEIR CONSTRUCTION BY THE INNU-WIGNER GROUP CONTRACTION By
Abu M.A.S. Khan

December 2004

Chair: Pierre Ramond
Major Department: Physics

We study the continuous spin representation (CSR) of the Poincar6 group in arbitrary dimensions. In d dimensions, the CSRs aie characterized by the length of the light-cone vector and the Dynkin labels of the SO(d- -3) short little group which leaves the light-cone vector invariant. In addition to these, a solid angle d-3 which specifies the direction of the light-cone vector is also required to label the states. We also find supersymmetric generalizations of the CSRs. In four dimensions, the supermultiplet contains one bosonic and one fermionic CSRs which transform into each other under the action of the supercharges. In a five dimensional case, the supermultiplet contains two bosonic and two fermionic CSRs which is like N = 2 supersymmetry in four dimensions. When constructed using Grassmann parameters, the light-cone vector becomes nilpotent. This makes the representation finite dimensional, but at the expense of introducing central charges even though the representation is massless. This leads to zero or negative norm states. The nilpotent constructions are valid only for even dimensions.

We also show how the CSRs in four dimensions can be obtained from five dimensions by the combinations of Kaluza-Klein(KK) dimensional reduction and the Indnii-Wigner group contraction. The group contraction is a singular transformation. We show that the group contraction is equivalent to imposing periodic boundary condition along one direction and taking a double singular limit. In this form the contraction parameter is interpreted as the inverse KK radius. We apply this technique to both five dimensional regular massless and massive representations. For the regular massless case, we find that the contraction gives the CSR in four dimensions under a double singular limit and the representation wavefunction is the Bessel function. For the massive case, we use Majorana's


vii









infinite component theory as a model for the SO(4) little group. In this case, a triple singular limit is required to yield any CSR in four dimensions. The representation wavefunction is the Bessel function, as expected, but the scale factor is not the length of the light-cone vector. The amplitude and the scale factor are implicit functions of the parameter y which is a ratio of the internal and external coordinates. We also state under what conditions our solutions become identical to Wigner's solution.


viii















CHAPTER 1
INTRODUCTION
All elementary particles in nature are labelled by their masses and spins or helicities. The underlying mathematical principle for this characterization is the representations of the Poincar6 group. In 1939, E. P. Wigner first classified all representations of the four dimensional Poincar6 group in a classic paper [1]. These classifications are characterized by two Casimir eigenvalues: the mass-squared M2 and the square of the Pauli-Lubanski vector W2. The orthogonality between the Pauli-Lubanski vector and the four-momentum vector defines the little group which leaves the momentum vector invariant. The little group gives the spin (helicity) of the massive (massless) particles respectively. We summarize the classification in the following:


Representation First Casimir Second Casimir Little Group

Massive representation M2, M real W2 = M2j(j + 1) SO(3)

Regular massless representation M= 0 W2 0 SO(2)

Tachyonic representation M2, M imaginary W2 = M2j(j + 1) SO(3)

Continuous spin representation M = 0 W2 2 E(2)


Among these four, only the first two representations are realized in nature. The massive particles, like protons, neutrons, 7r-mesons, etc., belong to the massive representations of different spin, and the massless particles like photons are described by the regular massless representations. Tachyons have imaginary mass, and violate causality as they travel faster than the speed of light [2]. They are unphysical and do not occur in nature. In field theory, the appearance of the tachyonic mass term is removed by shifting the vacuum, a process known as the Higgs mechanism.

The continuous spin representation (CSR), which is the main subject in this dissertation, is characterized by the zero mass and non-zero finite length of the Pauli-Lubanski vector. In four dimensions, the Pauli-Lubanski vectors are related to the translation generators which together with the homogeneous rotation generator form the inhomogeneous little group. Group theoretically this is the E(2) algebra of the little group. In this representation, an action of a finite boost on any state produces an infinite tower of massless states with helicities ranging from minus to plus infinity. Even though the CSR is mathematically very sound, there are various arguments against


1








its existence in nature. Wigner [3] himself argued that since the CSR gives an infinite heat capacity of the vacuum, it is unphysical and does not exist in nature. L. Abbott [4] showed that the CSR violates causality and K. Hirata [5] further showed that in this representation the fields become non-local. Moreover, S. Weinberg and E. Witten [6] proved that in four dimensions there cannot exist any particle with helicity higher than two. Because of these strong arguments, the CSR has never attracted much attention.

Despite these facts, there are still reasons for further studying the CSR, in particular in the context of models for extended objects in higher dimensions, as for example, String theory and Mtheory. According to these models, the infinite slope limit of the Regge trajectories yields an infinite helicity tower of massless particles. Mathematically this is the representation of the Euclidean group which is the little group of the Poincare group.

This is our main motivation to study the CSR. We will not address how the problems associated with the CSR can be cured, but will concentrate on their mathematical properties. We hope that these studies may provide better insights for how to avoid or cure the problems associated with the CSR. Our aim is two-fold. First, we study how these representations and their states can be characterized in any dimensions by the Casimirs of the little groups and if these representations have any supersymmetric extension. Because of the supersymmetry, the light-cone vector can be constructed with or without using the Grassmann parameters. This means the light-cone can be made nilpotent and as a consequence the representation can become finite-dimensional.

Second, we study how these representations can be obtained from larger groups. Of particular interest in physics are the Conformal groups which contain the Poincar6 group by embedding and the larger Poincar6 group, (Anti) de Sitter groups which give the Poincar6 group by group contraction only. It has been known for sometime that the CSR has no conformal extension [7, 8]. We prove this in light-cone language in Appendix A. Therefore, we will look into group contractions.

Group contraction has been introduced by In6nii and Wigner [9] in 1953, and independently by Segal [10]. The group contraction is a singular transformation to obtain the Euclidean group from a homogeneous group. In6nii and Wigner (IW) showed explicitly how the contraction of the SO(3) group yields the E(2) group which is algebraically identical to the Poincar6 little group in four dimensions. We show that the IW group contraction can be realized in a different way which is more physical. Imposing a periodic boundary condition and taking a double singular limit yield the same conclusion as IW. We consider the periodic boundary condition as the Kaluza-Klein(KK) dimensional reduction process. One singular limit can be identified with the infinite KK radius






3


limit whose inverse is the contraction parameter and the other singular limit is the spin. We apply this technique to both regular massless and massive representations in five dimensions. We find the conditions for the CSR to exist in four dimensions and find the representation wavefunction. We also compare with Wigner's solution for the CSR.

The plan of this dissertation is the following: chapter 1 is the introduction, chapter 2 reviews the CSR and extends it in any dimensions, chapter 3 discusses the supersymmetric extension of the CSRs, chapter 4 reviews the group contraction, chapter 5 discusses the contraction of the five dimensional regular massless representation, chapter 6 is the contraction of the massive representation in five dimensions, and finally chapter 7 is the conclusion.















CHAPTER 2
THE CONTINUOUS SPIN REPRESENTATIONS OF THE POINCARE GROUP

In this chapter, we introduce the continuous spin representations(CSRs) of the Poincar6 group. For convenience, we express the generators in light-cone form [11, 12]. We start with the general form of the Poincar6 algebra in any dimensions, and review and construct the continuous spin representation in four and five dimensions respectively. We also present the Casimirs for arbitrary dimension. More details can be found in Wigner [1] and Brink et al. [13].

2.1 Light-Cone Form of the Poincard Algebra The Poincar6 group is the semi-direct product of the abelian translation group and homogeneous Lorentz group for rotations. The Poincar6 generators satisfy the commutation relations, [P, P)= 0 (2.1)

[4" P)= i(7"P" 7""P), (2.2)

[Mm', Mc"'] i(qgaM" + 7avMog + n"3MA + 770M"" (2.3)

where r7" = (-1,1, - 1) and A, V, 0 = 0, 1, - (d 1); and P4 and Mo'3 are the generators for translation and rotation respectively. All representations of the Poincar6 group have been classified by E. P. Wigner in a classic paper [1] in 1939. These are characterized by the Casimir operators which are the squared mass, M2, and the squares of the Pauli-Lubanski forms which in d space-time dimensions are defined as


W -e.- = (2.4)


where n = 1, 3,- (d -- 3) for d even and n = 0, 2,- (d 3) for d odd. Their squares are the Casimir operators of the light-cone little group. In the following, we rewrite the Poincar6 group generators in light-cone language, originally introduced by Dirac [11] for spinless particles and later extended for arbitrary spin by Bacry and Chang [12], and characterize the representations by the Casimirs of the light-cone little group.

The light-cone variables are defined as

X = 1 (Xo0 Xd.-I) =j l(PI Pd- 1)
v'2-.


4









and similarly for any other variables. In this formalism, the canonical commutation relations become [x: p 1 j= -i and [ x' pJ J = i V .


where i, j = 1, 2, (d 2) are the transverse directions. In light-cone form, the space-time translation generators become

P- = 2p Pi = p,


where P-, the light-cone Hamiltonian, is expressed in terms of transverse momenta by using the mass-shell condition. Since P-, the conjugate variable to x+, is constrained by the mass-shell condition, we set x+ = 0. The Lorentz generators can be grouped into two types: those which transform the transverse plane into itself are called "kinematic" and those which transform out of that plane are called "Hamiltonian" types respectively by Dirac. The kinematic generators are given by

M+i P+, M+- x-p+ Mij = i xipi + Si (2.5)

where S'i obey the SO(d 2) Lie algebra of the transverse little group [Si Ski] i(ik Sil + 51l Sik 6il Sik 6k Sil) (2.6)

The Hamiltonian-like boosts are


M i = zpi -{i, P~}+ (T pi S .i) (2.7)

The vector, T, is called the light-cone translation vector. It transforms as a vector under the SO(d 2) rotations,

[ S.j, T k] (sikTi 6jkTi) (2.8)

and satisfies the following commutation relation, [ T, T ] = i M2Sij (2.9)

Eqs.(2.8-2.9) form the algebra of the little group. There are four ways to satisfy the algebra:

M J 0: There are two types of representations in this case depending on whether M is real,

positive or imaginary.

1. M real and positive: This is known as the massive representation. In this case, TI/M

and S'i form the homogeneous little group SO(d 1). In four dimensions, it is the SO(3)






6


little group and all massive elementary particles obey this representation. The states are

labelled by the mass M and the spin j.

2. M imaginary: In this case, the little group is exactly the same as in the previous case. This

is known as the tachyonic representation. No tachyonic particles have been observed in Nature. The tachyons are unphysical particles because these travel faster than the speed

of light [2].

a M = 0: In this case, there are also two types depending on whether T's are zero or not.

1. T = 0: This corresponds to the familiar massless representations which describe particles

with a finite number of degrees of freedom. These are realized on states that satisfy T' p+,pi; {ak} > = 0 k = 1, - Rank of SO(d 2) .

where {ak} are the Dynkin labels of SO(d 2) representations. In four dimensions, the little group is SO(2) and its eigenvalue helicity along with the zero-mass uniquely

determine the representation. The Pauli-Lubanski vector is light-like.

2. T' > 0: In this case, Tt are the c-number components of a transverse vector. The vectors

T and the rotation generators SU' form the E(d 2) inhomogeneous little group. The states are labelled by the length of the light-cone vector and the subgroup SO(d 3) of

the SO(d 2) little group which leaves T' invariant,

T'l p+,pt; C', {ak} > = I p ,p'; ', {ak} > k = 1,..- .Rank of SO(d 3).

where {ak} are now the Dynkin labels of SO(d 3) subgroup which we call the "short little group." In four dimensions, there is no such group and the states are simply labelled by an space-like vector of constant magnitude ', i = 1,2. These span two distinct representations, called "continuous spin representations" by Wigner [1] in his original work. The Pauli-Lubanski vector is space-like. The CSRs describe a massless state with

an infinite number of integer-spaced helicities.

In the following sections, we review continuous spin representation in four dimensions, construct the state in detail in five dimensions, and finally end this chapter by introducing the general formula for the Casimirs of the little group in any dimensions.









2.2 Continuous Spin Representations These representations are characterized by the zero mass and the finite length of the commuting light-cone translations T' which together with Sj form the E(d 2) Euclidean little group in ddimension. In the following, we first briefly review the properties of the CSR in four dimensions following Wigner's work (in light-cone formulation) and show how the infinite tower of helicity states is generated, and extend it into the five dimensions and calculate the Casimirs for higher dimensions.

2.2.1 Four Dimensions

In four dimensions, the little group is formed by the two-vector T' and SO(2) rotation. Together these satisfy the E(2) algebra,

[T' Ti] = 0 and [Sn Tk ] i(jikTi 5ikT ) ; for i,j = 1, 2.

Since S12 and T' do not commute, helicity is no longer a good quantum number; however T2T' does commute. The two Casimirs that uniquely label the representation are the zero mass-squared and the square of the Pauli-Lubanski vector given by W2 = T2T = 2

where we used Eq.(2.4) for four dimensions and worked in a frame where p' = 0 for simplicity.

It follows that a finite boost creates an infinite number of integer-spaced helicities. To see the appearance of the infinite number of helicity states, we construct the light-cone raising and lowering operators,

Tz= T1 iT2

which satisfy the relations, [S12,T ] = kT and [T ,T] =0 .

It is easy to show that

s12 (T 1 > )=(m t 1)(T*| >), which clearly shows that the state T I > has helicity (m 1). Therefore, an action of an infinitesimal boost,
M = M-1 i M-2

changes the helicity by one unit because of the following relation


M Ip+,pi=0;EQa> = +p' =0;E, a>
p









Therefore an action of a finite boost produces states with possible helicities ranging from minus to plus infinity in integer steps. This is an infinite dimensional representation. There are two types of representations: those with all integer (single-valued) and those with all half-odd integer (double-valued) helicities, known as bosonic and fermionic CSR respectively.

As we have stated in the introduction, the CSRs have no obvious physical applications, explicitly in local field theories, but the appearance of an infinite number of states may indicate a connection with non-local theories of extended objects. This motivates their study in more general contexts.

2.2.2 Five Dimensions

In five dimensions, the light-cone little group generator S'i and T, for i, j = 1, 2, 3, satisfy the following E(3) algebra,

[T,Ti] = 0 and [StI, Tk] i (ikTi SikT )

Using Eq.(2.4), we get the following two Pauli-Lubanski forms,

0 form: W EijkTS (2.10)

2 form: Wi, = EijkTk,

where we used a frame pi = p = 0. The square of the two-form We, W% TI kk E 72 (2.11)

is one Casimir, identical to the four dimensional case. The other Casimir, the zero-form, can be thought of as the projection of the generator Sik along P. The vector Ti then acts as a "quantization axis," along which W assumes the values

W 1 3
0, i2 1, ,- (2.12)

It follows that there are two types of representations corresponding to each integer and half-odd integer values of W/vf E. For each value of W/v/2 2, there corresponds one infinite dimensional representation. Unlike four dimensions, there are infinite numbers of both bosonic and fermionic CSRs in higher dimensions. The states are no longer characterized by the light-cone little group but by its SO(2) short little group orthogonal to TI. Each CSR is labelled by the length of T' and the eigenvalue m of the short little group which is the helicity.

It is straightforward to find the CSRs in terms of the eigenstates of SO(3), the full little group. Let 1j, m > be the eigenstate of the little group. These states are required to be eigenstates of T, the SO(2) rotations about it, and of the Casimir EikTiS3k. Since T' is a vector under SO(3) rotation,






9


its action on each jj, m > yields a linear combination of j = j, j 1 states. Therefore, the CSR eigenstates are infinite linear combinations of eigenstates of the full little group SO(3). To construct a CSR state, let the light-cone vector be aligned along the third direction. This allows us to identify T3 with the tensor operator To which is the same as YO' in spherical basis. The action of T.1 on the state Ij, m > yields


T0J1j, m > = a j+1, m > +a md j, m > +a?"0|j -, (2.13)

where the a's are proportional to Clebsch-Gordan coefficients


O" = (+ m)(j-m+1), -j
a" = i s/(j'+m)(j' -m+1) ; j'=j l, -j'
Let us define a state,
00
F > = Z f jm >
j=l- im
This will be a CSR eigenstate if it is an eigenstate of both T3 and the zero form Casimir,

1 1
1 ik-% jk | F > =Em|F>, m=0, ,k1,. .
2 '2

We find that, after using Eq.(2.13) and little algebra, the coefficients must satisfy the following recursion relations

(1m1+p-) fl(-) P + a(Jm+p-2,m) fm I + II+P-lIm)- ) fm
a- mJ+p + MIm+p-2 + (a- flMI+P-1 -with p = 2, 3,. .. and

(II lf(m) 1+ (a~m~m) -fE-) f =0, mJ+1 0 m

since fl1 = 0. Clearly for each value of j, the SO(3) angular momentum, there are two CSRs: bosonic and fermionic types corresponding to whether m is integer or half-odd integer, and there are an infinite number of CSRs of each type. The other CSR states can be obtained by acting the SO(3) raising and/or lowering operator on IF >.
2.2.3 Higher Dimensions

In higher dimensions, we just list the Casimirs and explain how the states are labelled, for more detail see Brink et al. [13]. Following the previous section, it is clear that the Casimirs of the short little group and the length-squared of the highest form uniquely label the representation.






10


From Eq.(2.4), the Pauli-Lubanski n-form for the light-cone little group SO(d 2) can be written as
El...nt,, ,... id- T id-2Si +Ii.+2 ... Sld-4id-3
n! 2 ( 3
The Casimir operators are simplest to calculate in a frame where only Tid-2 # 0. The square of the highest form is the squared length of the vector T, a common feature for all dimensions. The Casimir operators of the short little groups for n > 1 are given by

For little group SO(4) and SO(5):

W... =- 21; 1
W .. i2 = 2 { (S)(d-n-3)/2 2(S-n~3) ; 1
For little group SO(8) and SO(9) :


2 (S2)(d-n-3)/2 2(Sd-n-3)) 1 S (S)3 + 2S2SI -2S ; d-n=9.

In general for the little group SO(d)
k
W2.2A2PS2 kP)+ B2(SI)k + C2(SI)PS
p=O

where k = (d n 3), and A2p, B2p and C2p's are numerical constants. In the above, Si

are the generators of SO(d 3) subgroup of SO(d 2) perpendicular to TId-2 and S St 12St .L2Z.. %IPS1


In odd dimensions only, the extra Casimir operator is provided by the Pauli-Lubanski zero-form W ijk...nT'Sj ... Sm.
2~ (v)!

The CSR states are labelled by E and the solid angle in (d 3) dimensions Qd-3 which give the length and direction of f respectively, as well as by (ai, - a,), the Dynkin labels of SO(d 3),


SP+ Pi ; E Qd-3 ; {ar} > ,






11


where {a,} is the Dynkin label of the SO(d 3) short little group with r being the rank. These differ from the usual massless representations in that they are characterized by a space-like vector, and contain an infinite number of states. In the next chapter, we extend the Poincar6 group to include supersymmetry and find the supersymmetric CSR(SCSR).















CHAPTER 3
THE CSR IN SUPER-POINCARE ALGEBRA The Super-Poincar6 algebra includes the Poincar6 algebra and the spinorial supercharge. In this chapter, we discuss how to construct the supersymmetric (SUSY) CSR. This chapter is based on Brink et al. [13]. A supermultiplet consists of an equal numbers of bosonic and fermionic states, and when a supercharge acts on these, they transform into each other. The supercharges are constructed using the Grassmann parameters. Therefore, to close the algebra, the lorentz generators must also be extended to include the Grassmann parameter dependent terms. Since the light-cone translations transform as a vector under the transverse rotations, it can be constructed without or with using the Grassmann parameters. The first one is of course the usual type of vector as in the previous chapter, whereas the second one is necessarily nilpotent because of the Grassmann parameter whose square is zero. In the following two sections, we investigate these two possibilities.

3.1 Super-Charges in Light-Cone Form The generators of the Super-Poincar6 algebra satisfy the commutation relations in Eqs.(2.1-2.3) and the following for the spinorial supercharge,


[QA ,p] = 0 (3.1)
1
[" QA] = --1 (r"'Q)A ,(3.2)
2
{QA Q'} = (F'Pp0)AB (3.3)

where A, B are spinor indices. In the above, rAV"= [ ru rv]
2

where the F matrices satisfy the anticommutation relation, {F", J2} = 2"; d .

The supercharges can be realized linearly using Grassmann variables and their derivatives [14] as QA = OA + !(F"'Pp F)ABO (3.4)
2


12






13


and their conjugates as

C c 2(rprO)DC9D (3.5)

where we used

rorstro = r,

and OA = 5, &A = ,and 6A is the complex conjugate of OA with A,.- D running over the

spinor indices. 6, 6, 8 and 6 are anticommuting Grassmann parameters. Given the supercharges, it is always possible to define the covariant derivatives, DA = aA (r pyr0ABO


and their hermitian conjugates. The covariant derivatives always anticommute with the supercharges and their conjugates,

{QA DB} = {Qt DB} = 0. (3.6)

Acting on the wavefunction <), the above anticommutativity is equivalent to the following constraint, D
which is a common constraint in any dimension. This indicates that the supercharges and hence all other generators are reducible. This constraint reduces the number of Grassmann parameters by a factor of two. There may be further reducibility depending on whether the Grassmann parameters are of Dirac type or Majorana type. These two constraints depend on the number of space-time dimensions. The irreducible form is obtained by imposing all constraints allowed. In d-dimension, a complex spinor have 2d/2(2(d4-)/2) components for d even(odd). If p is the number of constraints imposed, the number of independent complex spinor is 2 /2--p(2 (d--)/2-p) for d even(odd)1. Let a, b, c, ... run over the irreducible components. Imposing all constraints, the irreducible forms of the kinematic supercharges become

8 1 & 1
Qi =a + a ta = + p+oa, (3.8)

and the dynamic supercharges become


Qa = f(pa/p+)Qa A f*(Pa/p+)Qat (3.9)


1 The detailed analysis can be found in [15, 16].






14


where f (pa/p+) and its complex conjugate are functions of the ratio ,p, determined by the constraint DT = 0. The kinematic and dynamic components are determined by using the light-cone projector. The irreducible forms of the Lorentz generators can be written as M = X p2 -- x'p, + S'3 (3.10)

M+- --p+ + S+-- (3.11)

M-S = xp. -- {x IP} + (T' ) S+- (3.12)


where


So + 10(-Yij )b + cc) (3.13)

S+- 0(Y+--)bb + c.c.) (3.14)


with y'3 and y+- being the reduced Dirac submatrices consistent with the constraints. M+' remain the same as before. The indices i, j run over transverse space.

3.2 Non-nilpotent Light-cone Vector and the SUSY CSR

In this section, the light-cone vector is considered like an ordinary vector as in the Poincar6 group. Following the discussion in the previous section, we write down the irreducible generators below and check various commutator whether the Super-Poincar6 algebra closes or not.

3.2.1 Four Dimensions

In four dimensions, a complex spinor has four components. We can impose either chirality or Majorana condition and the covariant derivative condition, a common constraint for any dimensions, to reduce the number of independent component to one. Let 6 be the Grassmann parameter. In Weyl representation, following Eq.(3.4) and imposing the chirality condition and DD = 0, we can write the kinematic supercharge [13, 14] as a-h-t 0. (3.15)
Q+ = -+ -P+ + = + --p6.(.5

and the dynamic supercharges are given by


Q- = p+Q+ Q_ = (3.16)









where p(p) = L(p ip2). The generators M+-, M12 and M can be expressed in the following irreducible forms,

M+- -4P+ +6 (3.17)
2 ( 0 00

m12= x1 p2 X 2P1S12 + 1 ( -0 (3.18)
2 aO 00

M-a = -pa _Ia p-}+ I (T p a 0a + (3.19)
2 PT+ 2p+ 6ao ao

where a, b = 1, 2. The above generators and the supercharges have to satisfy the following conmnutation relations to close the algebra,


{M+- Q ] [M2, Q*] [M+i M-3 ] [M a, Q+] [Mi1 Q-,] [Im 2, Q]F


i Q*,
2 2
-i(6juM+- Mi3) =0,


1
=-72=Q ,


which are derived by applying the constraints to Eqs.(3.1-3.3) directly. After a long and tedious calculation, it can be shown that the irreducible generators satisfy these commutation relations. However, the last two relations, Eqs.(3.24-3.25), require that Q+ to commute with T'. An easier to see this is to choose a frame where p' = 0. In this frame, the dynamic supercharge vanishes and these commutators become


[T', Q+] = 0.


(3.26)


Since the light-cone vector commutes with the supercharge, we can implement supersymmetry on the continuous spin representations without having to change the supercharges. Let Im > be a state with helicity m. So its supersymmetric partner Q+jm > has felicity (m 1/2). So an action of a finite boost on the states Im > and m 1/2 > generates the CSR states Fm >c,, and Im -1/2 >CSR respectively. So the CSR supermultiplet is


)SCSR CmR- 1/2 >cSR


(3.27)


which is like ordinary (T' = 0) supermultiplet, but accompanied by an infinite tower of states.


(3.20) (3.21) (3.22) (3.23)

(3.24) (3.25)


I I')






16


3.2.2 Five Dimensions

In five dimensions, the spinors have four complex components which can be reduced to two by imposing the covariant derivative constraint, the only constraint allowed. In the representation, S= iol & i ri = U.3 0,7-i i=1,2,3 ; and Y4 = o.2 I, (3.28)

the light-cone projectors, P = --!r+Fr, split up Eq.(3.3) into the following,


Qia Q } = P 6.ab and {Qsa Qt = i(.p-)ab (3.29)

where a = 1, 2. The constraint, D

ao02+a p+___ a

and similarly for their complex conjugates. This allows us to set (03, 94) = 0. Substituting these, the supercharges in Eqs.(3.4-3.5) reduce to the irreducible forms a + P Q P (3.30)
0 v'_2 + a6.+ ;72

with a, b = 1, 2, and the dynamic supercharges are now given by


Q-a = -i + and hence Qt-a = i Q) (3.31)

The internal part of the Lorentz generators S+ and S'i become

o~~), s ~'k(a~+-. (3.32)


where now 0 = (01, 02) and = (0, 0). The remaining Lorentz generators remain unchanged. It is quite easy to see that the supercharges transform as SO(3) spinors,

1
[M"i Q+a] = 2I(a Q+) (3.33)
2

These charges can be made to act on the CSRs, for which the relevant group is that of the short little group SO(2) which leaves the light-cone translation vector invariant. Let us align T' along the third direction. This causes the supercharges to split up into two types according to their action on a helicity state,
1 1
[M12, Q+1 = 2 Q+1 [M2, Q+2] = 1Q+2 (3.34)






17


Clearly the supercharges Q+i and Q 2 lower the helicity and the remaining two Q+2 and Qti raise it. In terms of the helicity m, the supermultiplet consists of the following states,


m >CSR ~ m >CSR
1
Q+ I m >CSR and Q+2 m>CSR M >CSR

Q+&142 m >CSR C m SR

It contains two bosonic and two fermionic CSRs, with the same structure as the ordinary (T = 0) massless N = 2 supermultiplet in four dimensions. The important difference is that it contains not only the ordinary states but their copies under the boosts proportional to T. This yields as usual an infinite number of SO(3) polarization states. The action of supersymmetry is the same as in the normal case, but the CSR supermultiplets contain an infinite number of ordinary massless supermultiplets of ever-increasing spin. This construction has obvious generalization to any higher dimensions.

3.3 Nilpotent Light-Cone Translations and the CSR

In the previous section, we have shown that the CSR has supersymmetric extension. The light-cone translation vector was ordinary type, i.e., as in Poincard algebra. In this section, we construct the light-cone vectors T' using the Grassmann parameters. This construction will, of course, make the light-cone vector nilpotent, such that at least T2 is nonzero but the higher orders are. If T2 becomes zero this will not lead to any CSR. A finite boost will now terminate after the second order term resulting a finite number of helicity state. Since it is not possible to construct SO(d) vector using only one set of Grassmann parameter, we have to introduce at least two sets of Grassmann parameters which automatically make T2 nonzero, but the higher orders are zero. This can be thought of N = 2 supersymmetry. It should be noted that if we require higher degree of nilpotency, we have to introduce more Grassmann parameters and thereby will be able to consider higher N supersymmetry. In the following, we construct nilpotent light-cone translation in four and ten dimensions.

3.3.1 Four Dimensions

Since it is not possible to build a SO(2) vector with one Grassmann parameter, we introduce two complex Grassmann variables 01 and 62. We define the light-cone vectors as


T = -(Z1 2 + Z12) (3.35)

7 (ZO102 -- Z012) (3.36)






18


where Z is a complex c-number parameter, and p+ is used to ensure its proper commutation relation with M+-. The square of the Pauli-Lubanski vector is, W2 = 21Z12(p+)201 2 2. (3.37)


The kinematic supercharge and its conjugate are

9 1 1
Q' = + p + ; Q" = + -p+Oa, (3.38)
00a /2 60,, V/

where a = 1, 2, because of two Grassmann parameters. This is equivalent to N = 2 supersymmetry. However the light-cone translations no longer commute with these supercharges. To close the SuperPoincar6 algebra, the dynamic supercharges must be altered to the new form


Q i+ ab at a+t + iZ k9 (3.39)

which ensure the commutation relations of Qa with the boosts M-. These supercharges also transform correctly under transverse rotation S12 which now reads, 12 = S12 + -(Oa, Oaaa) (3.40)
2

Let us compute the anticommutator between the supercharges. We find that,


{ QS, Q_} I {Q0+, > Qbt = iEZ (3.41)

These non-zero anticommutator together with the Super-Poincar6 algebra (anti)commutators form the supersymmetry algebra with central charge, even though the representation is massless. It is well-known that the supersymmetry with central charge leads to negative or zero norm states for massless representation [16], it only makes physical sense for massive representation.

Even though by constructing the nilpotent light-cone translation vector, we remedied the infinite vacuum heat capacity problem which was Wigner's argument against the existence of the CSR, but at the expense of negative or zero norm states.

3.3.2 Ten Dimensions

Following previous section, the ten dimensional case is straightforward. The light-cone little group is SO(8) which has the magic triality property. In ten dimensions, we can impose both Majorana and chirality conditions, in addition to the D







19


Grassmann parameters to form SO(8) vector. In the representation, r0 = io. i ri = U3 O yi (3.42)


where i = 1,... 9 and -yis are 8 x 8 gamma matrices, the supercharges and the light-cone vectors can be written as


Q(a) = + 1p+O ; a = 1,2. and T' = ip+ZOlYi02 (3.43)


where Z is now real. These transform as Qf 8, Q(2) 8, and T 8,


under the SO(8) transverse rotations given by

-i = SO 017 a( + 0 (3.44)
2 ( 6 7, 2

where 7'j is the reduced tensor. The generator S+- and the dynamic boost reduce to the following forms,


+ \ l 2 ~ + 8 2 ) an d S, = +1_2
i+ = (01 +02 an S =i ZO1~Y% -P+ +
2 ao1 502 + +

To satisfy the algebra, the dynamic supercharges must change to the new form, Q(a) _ y Q(a) i ab. V2Z6, a, b = 1, 2. (3.45)

where e12 = 1 = _621. The anticommutators between the kinematic and dynamics supercharges are no longer zero,

{Qa) ,Q9b} = ijabV Zv (3.46)

indicating the N = 2 supersymmetry with central charge. Even though our construction leads to supersymmetry but the representations necessarily contain negative and zero-norm states. Although it is interesting to note that central charges occur naturally whenever the light-cone translations are built out of the Grassmann variables, the representations contain negative and zero-norm states; at best they could be used as ghost compensators of some unknown theory.

This construction does not seem to generalize to odd dimensions. To see it, consider eleven dimensions with T quadratic in the Grassmann numbers. There a quadratic product of a Grassmann spinor transforms as 2- and 3-forms, so that to make a vector we need some c-number tensors, either a one or two-form, but the commutation with the supercharge does not have the right form.















CHAPTER 4
INONU-WIGNER GROUP CONTRACTION In this chapter, we review the group contraction first introduced by Inonui and Wigner [9], and also independently by Segal [10]. Later Saletan [17] discussed it for arbitrary homogeneous Lie groups in more general context. Here we restrict to the IW type contraction. We start with a brief review of the IW's original approach to group contraction, and then we show that the same result can be reproduced by imposing periodic boundary condition to the homogeneous SO(3) group and taking two singular limits. The different between these two approaches is the interpretation of the contraction parameter.

4.1 IW Group Contraction

In6nii and Wigner [9] used homogeneous SO(3) group to introduce the group contraction. They showed explicitly how to obtain the Euclidean group E(2) by contracting the SO(3) group under singular limit. They applied the contraction to the algebra generators, identifying those which are well defined under contraction, and derived the two-dimensional wavefunction on which the Euclidean translation vectors act, the Bessel functions J,.

Let S' (i = 1, 2, 3) be the generator of the SO(3) group which satisfy the angular momentum algebra,

[Si S = Eijk Sk

The states are labelled by the eigenvalue of the Casimirs S2 and S3,


Si Si = s (s + 1) S3 = M .

Since -s < m < s, this spans a (2s+1)-dimensional representation. The representation wavefunction is of course the spherical harmonics.

Let us define another set of generators as

Si' = lim ES, ,
C-0



Si is related to the generators Si' by Poincar6 duality, i.e., S' = IEiikSik.


20






21


where E is an arbitrary parameter, called the contraction parameter. For fixed m, S3' = 0, and the remaining two generators and S3 satisfy the following algebra, [ S3, S]= 'Cab Sa Sb'] = 0, (4.1)

where a, b = 1, 2. These commutation relations form the Euclidean algebra only if the Casimir is also nonzero. To find the Casimir, we multiply the SO(3) Casimir by the contraction parameter, (eS3)2 + (cSa)2 = E2s(s + 1) ,

and taking the limit E -- 0, while keeping m fixed, yields the following,


(Sa')2 = lim E2s(s + 1) = ifs fixed. (4.2)
C-0=2 ;ifs cc.

where E = es is nonzero finite as E -- 0 and s ox. In the first case, we obtain homogeneous SO(2) group with S3 being the Casimir operator, whereas the second case yields the Euclidean group under the double limit. Since s sets the range of m, and s -+ oo, m can now have infinite range. Thus it forms an infinite dimensional representation. This is the most essential feature of the contraction process. The Casimir eigenvalue of the E(2) group now reads, sa'a 2 (4.3)

Since Sa's are like momenta, rather than deriving from the SO(3) generator, IW identified it with the differential operator for translation, Sa' = a= 1,2. (4.4)
Oxa

Therefore, Eq.(4.3) reduces to the Laplace wave equation in two dimensions. In cylindrical coordinates, the solution of the wave equation is the well-known Bessel function, < (p, ) = NmJm(Ep)emo,


where p = (x1)2 + (x2)2. This is the representation wavefunction of the E(2) algebra. In their paper, they also derived the following identity, /(1 in)! 1/2
lim (I + i)!) Pm(cos (=p/)) = Jm(Bp) I-- ( ( + m)!)






22


which justifies that the contracted algebra is well defined, in particular the identification of the contracted generator Sa' with the differential operator for translation in Eq.(4.4). In their original paper, In6nu and Wigner applied their method to the Poincard group, using the inverse speed of light as a contraction parameter, and find it to contract to the non-relativistic Galilei group only if the starting point is the tachyonic representation!
4.2 Contraction of SO(3) group with periodic Boundary Condition

In this section, we apply the periodic boundary condition to the SO(3) algebra by putting one direction on to a circle of radius R. The contraction parameter is now the inverse radius. We show that under contraction, the representation wavefunction is the same as the previous section. In the following, we retrace IW's steps, stressing the geometrical picture of the contraction procedure. The IW contraction of SO(3) to E(2) amounts to the study of a dynamical system with SO(3) symmetry restrained to a space whose boundary condition breaks that symmetry. We switch to cylindrical coordinates, and seek solutions which are periodic in z,

z z + 27rR (4.5)

4(p,0, z) = 4D (p, 0)ein/ (4.6)
n

where n is the mode number. Let j2 be the eigenvalue of the Casimir (SiJ)2 when it acts on this wavefunction. Note that in spherical basis the Casimir eigenvalue is j(j + 1). For very large spin j2 ;:: j(j + 1), and eventually we will take it to infinity. So we no longer distinguish between j2 and j(j + 1). Therefore, we write,

Sij S's = j2 ,(4.7) where i, j = 1, 2, 3. In the following, we use the inverse radius, 1/R, as the contraction parameter. The angular momentum operators are given by

S12 = -i (4.8)

S23 = -i -zsin0-+psin0 (4.9)
OP az p ao)

S31 = -i (zcos0 a-- pcos-0 zsin (4.10)
Op az p 890









Acting on the wave function, the factor j can be replaced by 9. Substituting this, the square of the generators become

(S12)2 = 2 (4.11)
-202
(S23)2 = sin2 82 cos20 8 2sin cos0 82 psin2o 8
__2_2-+-z si _[ p p Op p 88p z2 ap
2psin20 in 9 2sin~cos9 & cos29 82 sin cos0 8 z R9p p2 + p2 802 z2 09
2 sin0 cos0 in 0 1 in p2 sin2 n 2 1 z R 89 z R z2 R2, (4.12)
2 [282 sin20 8 2 sin0 cos 0 82 p cos20 a
( o p p 8908p z2 Op
2pcos20in 8 2sin0cos6 8 sin20 2 sin0cos0 8 z R2 5+p2 892+ z2 89
2sin~cos in 8 1 in p2cos2o n2 1 z R 89 z R z2 R2. (4.13)

When R is very large, all terms proportional to inverse R can be dropped2. Using Eqs.(4.11-4.13) into Eq.(4.7) and dropping the terms proportional to 1/R, we obtain the following differential equation,

+2 1i i a2
-Z +z2 892 =j2, (4.14)

which acts on the wavefunction of the form



Since there is no mode dependence, we have dropped the subscript n from the wavefunction. For the above form of the wavefunction, the angular part decouples from the radial part and can be written as
1 d 2E)
-- m(4.16)

where m, the azimuthal quantum number, is an integer or half-odd integer depending on whether the wave function is single-valued or double-valued. Substituting this and rearranging, the differential equation Eq.(4.14) reduces to

2 d2 d + j2 M2
p2 +p ) P2 m2 = 0 (4.17)



2 The complete solution with finite R is the Whittaker function. See Appendix B.






24


which is the Bessel differential equation. The regular solution is

4(p, 0) = N. J.(qp)emeO (4.18)
m

where the scale factor is given by q = This is the representation wavefunction for very

large radius R.

Let us now define,

Sal = lim Eab (4.19)
R-+oo R

while keeping m, eigenvalue of S3, fixed. Using Eqs.(4.9-4.10), we get,

8 cosO & 4.0
S" = ij(- sin ,- (4.20)
R 49p p 00

S2' = i(cos p p 8) (4.21)
R ap P io

It can easily be checked that the Sa's, a = 1,2, commute with each other and that they transform as a 2-vector under S12, forming a representation of the E(2) algebra as in Eq.(4.1).

Let =2 be the eigenvalue of the Casimir operator for infinite radius limit. Therefore, the Casimir operator


(Si')2 + (S2')2 2 (4.22)

reduces to the following differential equation, z2 / 2 18e 18 N 2
R 2 Y p 2 + a I a = 2 (4 .2 3 )

where we used limR..- ( = 0 and Eqs.(4.20-4.21). For the wavefunction of the form as in Eq.(4.15) and the angular dependence as in Eq.(4.16), the above differential equation becomes (after some rearrangement)
d2 d =2 2- 2 (4.24)
p 2 + p + (z2/R2) p

whose regular solution is the well-known Bessel function,

I(p,O) = NmJm(qp)emO (4.25)


where the scale factor is defined as = Since the solutions in Eq.(4.18) and in Eq.(4.25)

have to be identical for infinite radius limit, this implies that the scale factors in Eqs.(4.17,4.24) must also be identical. Therefore, equating these two scale factors, we obtain a relation between the






25


Casimir eigenvalues of the SO(3) and E(2) groups,
2
2 (4.26)


where now both j and R tend to infinity, but their ratio E remains fixed and nonzero. This solution is of course the same as that of In6nii and Wigner. The generalization of the above construction to any dimensions is also quite straightforward. In general, for d dimensions, compactifying one direction gives the following relation j2
=2 d (4.27)
d-1 R2

where j2 is the quadratic Casimir of the homogeneous group SO(d) and =2_1 is the length squared of the E(d 1) group with R being the radius of the circle.















CHAPTER 5
CONTRACTION OF THE FIVE DIMENSIONAL MASSLESS REPRESENTATIONS

In this chapter, we discuss how the contraction of the five dimensional regular massless representation yields the CSR in four dimensions. We apply the techniques developed in the previous chapter. Since the Poincar6 group has an orbital and an internal part, we need to address how the contraction process effects these two. For the orbital rotation part, we show explicitly how the five-momentum and the rotation generators involving the compactified direction combine to produce the four-momentum. For the internal part, the scenario is the same as the IW group contraction process. We end this chapter by constructing the CSR in four-dimension and the representation wavefunction of the Poincar6 group.

5.1 KK Reduction of the Poincar4 Algebra Kaluza and Klein [18, 19] reduced one dimension by putting it on to a circle of finite radius. We apply it to the five-dimensional Poincar6 algebra, by putting the third direction on to a circle of radius R,

x= x3 +27rR.

The generators act on functions of the form


(X= ,D(x4)ei,, (5.1)
n
where p = -,1,2 and n is the mode number. The momentum along the third direction is

quantized,

p3 = (5.2)
R
This changes the light-cone Hamiltonian and two transverse boots in the following way, papa +(3)2+ M2 PaP,,+ M'
P - -- =+ (5.3)
2p+ -- 2p+M-a X-pa Ixa,P-} + (T a ,- s3a -Pbsab (5.4)

where a, b = 1, 2 and Mn is the KK mass. The single state of mass M in five dimensions generates an equally spaced tower of states of mass MW. Looking at M-1 we identify the light-cone translation


26






2-


vector,

Ta Ta+ n3a (5.5)
R
which satisfy the algebra


[S ab fc] = (3 acb 6bcja) (5.6)

[?a ,f] iM,2aS (5.7)

The generators S'2 and 1! form the SO(3) light-cone little group in four dimensions. Analyzing the mass term, we find that M, increases with the mode number and gives the well-known infinite KK tower of masses starting at Mo = M. These are the general expression for the KK reduction of the five dimensional representations.

For the KK reduction of the regular massless representation in five dimensions, we simply substitute M = 0 and T = 0, i = 1, 2, 3. The resulting Light-cone Hamiltonian, the transverse boosts and the light-cone translations become papa + (p3 2 papa+_M2
P = 2p+ (5.8)

M a Xp a I{xap-l 1 (ns3a pbsab) (5.9)
2+ 2p+R

Ta = S3a (5.10)
R

where now M, = A. The little group is formed by the generators S12 and T' and the satisfy the algebra in Eqs.(5.6-5.7).

The remaining Lorentz generators of the four-dimensional group, M+-, M12 and M+a for a = 1, 2 are not changed. Note that when evaluated at x3 = 27rR, the generators that rotate into the third direction become (for large R) like the momenta M_-3 a ( a \
R = 2 -2rp P+ R,]

i= 27pin 3
R =R2 R
M +3 = 27rp
R

In the following two sections, we show in details how the contraction of the orbital and internal parts lead to the CSR in for dimensions.

5.2 Contraction of the Orbital Rotations At the group level, the contraction of the orbital rotations results in shifting the parameters for translations. This has been studied in detail by E. Wigner, Y. Kim and others in [20, 21]. We









consider their construction for five dimensions with periodic boundary condition and contract it to four dimensions.

In five dimensions, there are five translational and ten rotational parameters. We choose to use usual space-time coordinates rather than light-cone coordinates (only for this section),

-" = (t, X z, 2, X3, X4) (x+,1- ) i = 1. 2,3. We want to contract with respect to the subgroup which leaves the third direction invariant. Let us consider the boost between t and x3 axes. Using x3 = 27rR, the transformation equation can be written as

1 cosh# L

t = 27rR sinh 0 cosh / t0 t (5.11)

\1 0 0 1 7 1

where sinh # = v and cosh = v1i7v with v being the velocity along the third direction. Since there is no velocity along the compactified direction, we need to take the limit v -+ 0 as R -+ 00 such that Rv remains fixed. This gives


t' = t + at + to (5.12)


where at = 27rRv is a constant and to is an arbitrary initial parameter. That is, the parameter for translation along time is shifted by at as a result of contraction of the t x3 rotation matrix.

Let us now consider the rotation between x' and x' axes for i = 1,2, 4. The transformation equation for rotation can be written as

( cos 0 27rR sin 0' a, Xi
sinl i' a3
ll =Cos0i __ 1 (5.13)

0 0 1 5 1

where 0' is the rotation angle. Since after the compactification, there is no rotation on the i 3 plane, we need to take the limit 0' -+ 0 as R -+ oc such that Rp' remains constant. This gives = X +ax. +a, (5.14)

where a, = 27rRoi is a constant and ai is an arbitrary initial parameter. From the above, we see that the rotations between the invariant direction and other directions of the original group become the translations of the contracted group which is a generic feature. The generators of the contracted






29


group can be read off using new shifted parameters. As for example, consider the rotation on the 3 1 plane. The product of the group elements for translation along first direction and the rotation on 3 -- 1 plane is

ialp' iW31L 31 ialp' i27r (RO' ) L3 R- oo e-e as e ,-ae- 2 ,._n) R-+ i(al+agl)pl


which is again a translation along the first direction, but with a new parameter (a, +a,,). Therefore, using standard method we can write down the differential form for p1 with (a, + ai) being the translational parameter along the first direction and similarly for any other directions.

The light-cone forms of the above results are straightforward. We find,

X' = x +a +a and x' +ai+a, ; i=1,2 (5.15)

where a = (at aX4) and ao' = 1 (to i a4). Under boost along the forth direction, x

transform as

ai' = Xe*z (5.16)

where A is the rapidity parameter and is given by


e= 1v4
1 V4

with v4 being the velocity along the forth direction. The transverse directions remain unchanged. Therefore, in the boosted frame, the orbital rotation generators after contraction can be written as L+- = _x-p+ L+i = _Xip+ ,

12 = X1p2 x2 I, L = z-p1 1i'p-}

where i = 1, 2 and x+ = 0 is used because of mass-shell condition. This is exactly the same form of the Poincar6 algebra in four dimensions for spinless particles Dirac obtained while introducing the front form dynamics [11.

5.3 Contraction of the Internal Part and the CSR In this section, we apply contraction to the internal part. The generators Ta and S12 satisfy the algebra (for R -) oo),

[Sal Tc = i( acfb bcfa) and [fa, ] = 0, (5.17)

where Tf = JSaa and a, b, c = 1, 2. Let the internal rotations are parameterized by three pairs of canonical coordinates and their conjugate momenta which satisfy the following commutation






30


relations,

7' -] = iPo i, j = 1, 2, 3 = (a, 3) Unlike the orbital case, we do not put the third coordinate 973 on to a circle. Instead we assume r73 is very large and so its conjugate momentum 7r3 is very small. As R -- oc, 9/' also goes to infinity such that n is well defined. Therefore, in the infinite radius limit, the terms proportional to 7r3 can be
R
ignored, and the terms proportional to r7' becomes a function of y ~ Note that being a ratio of the internal and external coordinates, the parameter y bears the signature of the extra dimension. This is the difference between the contraction process introduced in Sec.4.2 and the internal part in this section. Therefore, in the contraction limit, we can write, a= y7r' ,(5.18)


where y = 7. The light-cone vector differs by a factor of y from the Euclidean vector Sa' defined by Wigner (see Sec.4.1). We now follow the same procedure as in Sec.4.1. The Casimir of the SO(3) little group is,

(S12)2 + = j(j+ 1) (5.19)
M11
Divide by R2 and taking R -- o3 while keeping m fixed, we find,

0 if n =0.
22
n2=2
T"Ta R2 0 ifn # j= fiedR -+oo.(5.20) E2:if n 7 0, j, R -+ oc, "y=E = fixed. The first two clearly belong to the regular massless representation, and the last one is the CSR in four dimensions. Notice that for zero mode, the representation is always regular massless. Clearly only the double limit yields the CSR as expected. Combining Eqs.(5.18,5.20), we find,


(fa)2 y2(7ra)2 = 2 (5.21)

which in cylindrical coordinates gives the differential equation,

2 d2 d + g2P2
p2d + Pd+( -m2) = 0. (5.22)

The regular solution is the well-known Bessel function,


m






.31


In their paper [9], IW obtained the following solution, rw( p,O) = ZAm e"Jm(Ep) (5.24)

At y = 1 these two solutions become identical. Therefore the complete Poincard wavefunction can be written as

,D(X-, p,, ,y) = 'Nm e -ixp-P)ei"' Jm.(-p/y) (5.25)

For a given y, there are only two types of CSR: one bosonic and one fermionic corresponding to whether m is integer or half-odd integers respectively.















CHAPTER 6
CONTRCATION OF THE MASSIVE REPRESENTATION IN FIVE DIMENSIONS

In this chapter, we consider the contraction of the five dimensional massive representation based on the paper by Khan and Ramond [22]. The little group is SO(4). Instead of realizing linearly, we choose to construct the generators using three pairs of canonical coordinates and the conjugate momenta, which is a particular non-linear realization of the SO(4) little group. This non-linear realization allows us to use the Majorana infinite component wave theory [23], introduced long ago. It is amusing that the SO(4) symmetry properties of the Majorana model are exactly same as those of the Bohr's non-relativistic hydrogen atom. In the following sections, we show how to construct the non-linear realization, identify the generators consistent with the five dimensional Poincar6 generators in Dirac-Bacry-Chang representation, explain the contraction procedure and finally obtain the representation wavefunction. We also compare our result with Wigner's form of the CSR in four dimensions.

6.1 Non-linear Realization and the Majorana Theory

The linear realization of the generators of the SO(4) little group in five dimensions requires four internal canonical coordinates and their conjugate momenta which satisfy the commutation relations,

[7i ri] = i i = 1, 2, 3. and [t74 7r4] = i (6.1)

To realize non-linearly, we express the fourth coordinate and its conjugate momentum in terms of the remaining variables as


17 = (77r' i) and 7r = (7r (6.2)

where M is the mass, p is an internal parameter with the dimension of mass and 7 =/r7p. Using these, the generator S' can be expressed as

r4 i = 7 4 77 i (.4




I ( itS ij Pi7 T ) __. (6.3)


32






33


Using S' =, EtikSk, we find,

MS4i = eijkltiSk -i (6.4)
7

which is exactly the same form as the Laplace-Runge-Lenz(LRL) vector and it satisfies the following commutation relations,

[MS MS4j] = iM2S3 (6.5)

[ si MS4k] i(ikMS4i SikMS4,) (6.6)

This allows us to relate the mass M with the Bohr Hamiltonian, M= --2p 1, (6.7)


where H, the Bohr Hamiltonian, is given by H (6.8)
2p 7

So the parameter M is the mass of the electron with the electric charge set to one.

On the other hand, the generators of the massive little group in five dimensions satisfy the commutation relations,


[ St Tk] i(sikTj 6ikTi) (6.9)

[Ti, Ti] = i M2Sj (6.10)

The commutators in Eqs.(6.5-6.6,6.9-6.10) suggest to identify the light-cone vector Ti with the LRL vector MS4,. Therefore we can use the mathematics of the hydrogen atom to find the spectrum of the light-cone little group. In this nonlinear realization, the light-cone Hamiltonian and the boost generators become

pp -+ m2 pP' 2p H
2p+P- 2p+ (6.11)

= p {x, P }+- (T' -pSi). (6.12)

Since SO(4) ~ SO,(3) x SO_(3), the commuting generators of SO (3) group can be identified with the combinations
1 T k
2 S r- & 1 .

Since


SiT' = T'Si = 0 ,






34


the two Casimir eigenvalues of the SO(4) are same which is given by


C, = Sk Sk = j(j + 1) (6.13)
-4 2p H

where j is the eigenvalue of the SO (3) angular momentum algebra. Therefore the eigenvalue of the Bohr Hamiltonian is
H p/2
(2j + 1)2
and so the mass M can be expressed in terms of j, m2 JU2 2
M2 .(6.14)
4C2+1 (2j+1)2

So at each mass level, the states assemble in a representation of SO(4) and generate the spectrum of the infinite component Majorana theory.

We now apply contraction to this model. Since SO(4) D SO(3) D SO(2), we need to contract at least twice to obtain the E(2) representation of the little group in four dimensions. There are two ways to proceed:

1. We contract. staying in five dimensions first, and then use KK reduction and contract to four

dimensions, i.e.,

SO(4) 1st E(3) KK -,SO(3) 2nd E(2)
contraction reduction contraction

2. We apply the KK reduction first and then contract, i.e.,

SO(4)- KK : SO(3) Double : E(2) .
reduction contraction

In the following two sections, we find the conditions to obtain the CSR in four dimensions for these two cases.

6.1.1 SO(4) -- E(3) SO(3) ---+ E(2)

The states are eigenstates of the diagonal subgroup which is the angular momentum, and given by the sum of SO+(3) and SO-(3) groups of SO(4),


S Si = s(s + 1), (6.15)

where s = 0,1, 2,-.- 2j. Therefore each mass level is (2s + 1)-fold degenerate. The contraction scenario is very similar to the IW contraction process SO(3) E(2) where m is kept fixed while j goes to infinity. In this case, the contraction process for SO(4) E(3) is to take j, the quantum number for SO(4), to infinity while the quantum number s for SO(3) remains finite. Since Mj and









j are inversely related, the j -- oc limit leads to zero mass and commuting light-cone vectors,


M =4 0, andso [Ti,Tk3 0, (6.16)
1 2j+1I

and thus the generator SU and T' form the E(3) algebra. Note that this contraction process do not change the space-time dimensions. This is the CSR in five dimensions.

To find the Casimir of the E(3) group, we multiply the SO(4) Casimir by a contraction parameter E,

S TT = 42j(j1) (6.17)

As E 0 and j oc, such that Ej remains fixed, we get T T = A2 (6.18)


That is, the length of the translation vector is fixed, and this is a Casimir of the E(3) group. The other Casimir is given by the zero form,


W = EijkT'Sik = T'S = 0 (6.19)

Comparing to Eq.(2.10) in Sec.2.2.2, we readily see that these are the Poincar6 Casimirs and the remaining Casimir is of course the mass squared which is zero. These Casimirs uniquely determine the representation in five dimensions.

The above analysis also indicates that we could have chosen P or 1/j as the contraction parameter, because both of these lead the mass operator to zero. However, if E = A, then following Eq.(6.18), we find null light-cone vector which yields the regular massless representation, not the CSR. The other choice 1/j does not change the above result at all. Therefore, 1/j is actually the contraction parameter.

After the first contraction, the light-cone hamiltonian reduces to P- = and the boost 2p+
generators do not change at all. The KK reduction changes these two generators to


p = p+ M (6.20)
2p+

M- 1 } Ix' + I4 (Ta_-pbSab) (6.21)

where a, b = 1, 2, and M, = g. The new light-cone vector is now


Ta = T a+ nS 30 (6.22)
R






36


The generators Ta/M, and S'2 form the massive SO(3) little group. Let j be the eigenvalue of the KK reduced SO(3) algebra. It is now quite clear by now that to obtain a CSR in four dimension, we must take the limits, j --> oo and R 0c such that the ratio remains finite and nonzero, TaT" = __ :2 = finite (6.23)

This is the Casimir of the little group in four dimensions which together with the zero mass-squared uniquely label the representation. We find the representation wavefunction in Sec.6.2.

6.1.2 SO(4) SO(3) E(2)

This sequence is a two-step process to obtain the light-cone little group E(2) in four dimensions from the SO(4) little group in five dimensions: first we compactify along the third space (orbital) direction by making it periodic. The light-cone Hamiltonian becomes

1
--= (papa+ m )2


for a = 1, 2, with
M? A2 n2
3n 4- = 2 (6.24)
(2j-+1)2 R2 (

where we used p3 = n/R (momentum along the third direction). For finite j and R, there are no massless states, and the massive little group is SO(3) in four dimensions, generated at each mass level by

s3 and 1 (T- + S3.) a =1,2.
Mn R
deduced by looking at M-'. The spectrum is made up of massive particles with spin j', which depends on j and n through the quadratic Casimir


(s3)2 + (Ta + 3-) (Ta + nSa) = j'(j'+ 1) (6.25)

Eq. (6.24) clearly shows that a necessary condition for the mass operator to vanish is to take the double singular limit where both j and R tend to infinity. But this is not enough because the following condition
+11~i22 < 0c,

must satisfy to yield the CSR. This requires, in addition to the double limit, the ratio L to remain
R
finite and nonzero. To see it explicitly, divide Eq.(6.25) by R2 and rearranging, Tafa = ( 2/R2 + 722) (j2 2)






37


where we used S3 = m. The double limit certainly does not give any nonzero value, but the following triple limit

R -4 oc such that remain finite (non-zero)

does give a finite value. Therefore, in this triple limit, we find,


(Ta)2 2 ) = '2 (6.26)
(2j/R)2 + R -(.6

which is the Casimir in four dimensions. Together with the mass squared (which is zero) it uniquely labels the representation. This is what we expected, because the generator S12 was not affected by any of the reduction and/or contraction processes. The contraction only affected the quadratic Casimir eigenvalue which sets the range of quadratic Casimir of its maximal subgroup. The above analysis suggest how to identify the contraction parameter and the required singular limits to obtain any CSR: look at the mass operator and find what limits provide zero mass and if these limits keep the light-cone vector non-zero and finite. Unlike the previous section, the contraction parameter has a physical meaning which is the inverse KK radius.

6.2 Representation of the Wavefunction

Since we have the form of the T's in terms of the internal variables, we can explicitly find the representation of the wave function for the light-cone vectors !f. First note that in t' the contraction parameter j is not explicitly present. Therefore if we apply the contraction it will only include the R -- oc limit, not the j -- oo. As a result we will not obtain commuting vectors which is required to obtain the CSR. Since the contraction parameter j appears through the Hamiltonian when it acts on the wavefunction, we rewrite Ti in terms of the Hamiltonian as in the following, T' = Cijk7F3Lki~

= 77 2p 7 ri)7ri


where the Hamiltonian is given in Eq.(6.8). So the light-cone vector, Ta, can be expressed as 1= 2 +aH + na7r3 H] + (y + 2i .-)7ra (6.27)

where y = n73/R. The first term is dependent on the contraction parameters whereas the second term is independent and also both terms are separately hermitian. Since H ~ 0(1/j2) and 1/7 ~-O(1/R), the first term vanishes as j, R -- c. It should be noted that if we evaluate the commutator in the first term in Eq.(6.27) and then consider the contraction, the resulting light-cone vector becomes non-hermitian and the whole analysis becomes physically meaningless. It is not well






38


understood why this happens, but to maintain the hermiticity, we have to consider the contraction without computing the commutator. Using

-=7r=77 y .
ay

and dropping the first term, the light-cone vector becomes ( y 2i + iyb ir7r a (6.28)
ay

It can be easily checked that the above form satisfies the E(2) algebra,


[fa |b] = 0 and [ Sab ,c ] = i (6ac pb b j c pa)

which is required to obtain any CSR in four dimensions. We now find the representation of the wavefunction for this E(2) algebra.

The square of the light-cone vector is,


(Ta)2 [(7b7b)2 b b --2iy + 5 4 2y (y2 (2iy2 6y) (y2 + 6iy 6) (7r2) On a wavefunction of the form, <1(77) = eY( 7 Y)

this gives the following form of the differential equation, (?7brl) __i77, Ib 2 Yd (7r) =
5 + 2yy2 +6 + 6) ( (6.29)
dy Y Y2 d

where E = E or =' corresponding to Cases 1 and 2 respectively. To solve the above equation, let (7ra)2 = 112(y) (6.30)

when it acts on 4(r7, y), that is the representation of the rf dependent part of the wave function is the Bessel function, namely, 0(7y) = Jm(I(y)p)eim0F(y) where F(y) is a real function. In polar coordinates (r7' = pcoso and 272 = psin9), we use, aad ad%22 d d
-i7rr=- dp and (r7a)2 2 + p


and rewrite Eq.(6.29) as

2 2 d d 2d2 d
P dp + 6+2 +y +6y +6 (2JmF) = -2JmF. (6.31)






39


Making use of the following identities, d .
Jm(fl(y)p) = g (Jm Jm+i) (6.32)
dp 2

dJm(II(y)p) = dy (Jm- Jm+i) (6.33)
dy 2 dy

and after straightforward algebra, the left-hand side of Eq.(6.31) can be expressed as a linear combination of Bessel function of different orders, p2_F+ p2yfl3f'F +p2 2 r2j,2F
4 2 4 ) 2 + Jm+2

+ (3pU3F +6py12 II'F + pyU13F' + 2py2flf2F


+py2112'F' + Py2n2I/F (p2H4F
+ -_ 2 p2yl3 fl'F + 2y211,2 F + 2y2II"F + 4y2flf'F'
2

+Y2112F" + 12yIfI'F + 6yfl2F' + 6fI2F Jm,
2

where the 'prime' denotes derivative with respect to the argument which are suppressed for simplicity. By matching the coefficients of Bessel functions of different order with the right-hand side of Eq.(6.31), we get three constraints. Equating the coefficient of Jm 2 to zero gives the first constraint, p 2r 2 dUl 2 d,\2
4 (2+2y + y2 ( ) F(y) = 0 ,

which is satisfied if

+2y +Y2 = 0. (6.34)
dy Y
The solution to this equation is given by


f(y) = Y (6.35)
y

where II0 is a constant to be determined by the boundary condition. The constraint obtained by equating the coefficients of Jm, provides no new result. Finally the remaining constraint is from the coefficients of Jm which gives a differential equation for F(y), d2F 2 dF 2
dy2 + y+ lF = 0, (6.36)






40


where we used II(y) = Io/y and its derivatives. The solution is


F (y) = -. A sin Ey + Bcos
y ( Ho\o /

where A and B are constants. Since F(y) is regular at y = 0, we set B =0. Therefore the wave function can be written as


S(p, 0, y) = Nm sin ")) Jm (6.37)

where Nm is the overall normalization constant.

In their paper [9], IW found the following wavefunction for the E(2) algebra,


DIw(p, 9) = Am e" Jm(=p) (6.38)

where =2, the square of the Pauli-Lubanski vector, is the second Casimir of the Poincar6 group, and labels the CSR in four-dimensions. The Euclidean vector is linear in 7ra and as a result its length appeared as a scale factor in the Bessel function. In four dimensions there are only two type of CSRs, fermionic and bosonic types corresponding to half-odd and integer values of m. The amplitude is also constant.

Although similar in form, there are differences between Eq.(6.37) and the IW form in Eq.(6.38). In our case, the internal momentum (Eq.(6.35)) and amplitude are not constant, but functions of y which is the ratio of the internal and external coordinates. It is due to the fact that the Euclidean light-cone vector T' is not linear in 7ra (in addition to the y dependence). Moreover, unlike IW case, the length of the light-cone vector does not appear as a scale factor in the Bessel function, even though it is the CSR. To find how our CSR is related to IW's CSR, we must find a relation between H and E. Let us assume H = E = Ilo/y. The only solution we get, following Eq.(6.34), is = = 0 which corresponds to the regular massless representation, not the CSR. Therefore II $ = is the only possibility and these are quite new. On the other hand, we may assume that, instead of II, HO = E. Substituting this into Eq.(6.37) and setting y = 1, the solution becomes exactly the same as that of Wigner's CSR apart from the overall phase factor which has no physical effect (the overall constant factor can be absorbed into the normalization factor). For any other values of y, we have a different kind of CSR because of the non-equality between the scale factor and length of the Euclidean vector. There is no physical reason for Ho and B to be equal and y = 1, but this is the only condition to obtain IW's result.






41


Finally, the raising and lowering operators are defined as f = j4 iT2


In polar coordinates, these become eO 2 (2y d++ ( d id (6.39)


= e(2 + d 4-P d ( d id (6.40)
y dp dPp pdO

which acts on 0(p, 0, y). It is quite obvious that the states t'I > and Tj > have helicities (m + 1) and (m 1) respectively. The remaining generator T3/R becomes

= (7r a)2


under contraction. The complete Poincar6 wavefunction is

b(x+, xa; p, 0, y) = ENm sin -- e- x emOJm ( -2i)

We do not understand what is the physical interpretation of the parameter y. But being a ratio of the internal and external coordinates, definitely it relates the internal structure to the external. This may help to understand better the spectrum of the higher dimensional theory.















CHAPTER 7
CONCLUSION

In this dissertation, We have studied two things: first, how the CSRs and the states are characterized in any dimensions with and without supersymmetry and, second, how these representations can be obtained from larger groups by group contraction.

We have shown that in any dimensions, the CSR is characterized by the length of the light-cone vector and the Dynkin label of the short little group which leaves the light-cone vector invariant. We have explicitly shown that, unlike in four dimensions, there are infinitely many CSRs of both bosonic and fermionic types in five dimensions. The states require one more label Qd-3, the solid angle in d dimensions, which specifies the orientation of the light-cone vector. In the supersymmetric case, we have considered the light-cone vector to be non-nilpotent and nilpotent. In the case of non-nilpotent light-cone vector, it commutes with supercharge. The supermultiplet is like ordinary (zero length light-cone vector) supermultiplet, except that each bosonic and fermionic partner is actually the CSR. In five dimensions, the supermultiplet is like N = 2 CSR supermultiplet in four dimensions. The spinorial generators do not have to change their forms. When the light-cone vector is nilpotent, however, we obtain a finite dimensional representation, because the higher order of T' except T2 is zero. To make it work we have to introduce more Grassmann parameters, and hence there is no N = 1 supersymmetry if T is nilpotent. To get both T' and T2 nonzero, we need two sets of Grassmann parameters. Introducing more Grassmann parameters will give rise to higher N supersymmetry. The nilpotent construction remedied the infinite heat capacity problem of Wigner, but at the expense of introducing the central charge even though the representation is massless. We have shown this in four and ten dimensions. The nilpotent constructions do not generalize to odd dimensions.

To obtain CSIts from higher dimensions, we need group contraction. First we have shown that the IW group contraction is equivalent to imposing the periodic boundary condition and taking the singular limits. The different between our and Wigner's formalism is in the interpretation of the contraction parameter. In our case, it is the inverse KK radius. We have applied this technique to both five dimensional regular massless and massive representations.


42






43


In the regular massless case, we have found that the double limit, j -+ oo and R oo, yields the CSR if the ratio 1 is nonzero finite and for non-zero mode. For zero mode, the representation
R
is always regular massless in four dimensions. The representation wavefunction is the Bessel function, and the scale factor now includes a parameter y which is a ratio of the internal and external coordinates. This bears the signature of the higher dimension.

In the massive case, we use the infinite component Majorana theory as our model which can be thought of the non-linear realization of the homogeneous SO(4) little group in internal space. We have found that a simple double limit, like in the regular massless case, does not yield any CSR even though the mass term vanishes. We need a triple limit to obtain any CSR in four dimensions. The representation wave function is the Bessel function, but the scale factor does not include the length of the light-cone vector. In fact, the amplitude and scale factor are implicit functions of the parameter y. This is the key difference with Wigner's solution. At y = 1 and if Ilo = E, our solution and that of Wigner become identical, even though there is no physical reason for these conditions. For any other value of y, the solutions are quite new.

Our analysis of the contraction of the five dimensional algebra suggests how to identify the contraction parameter correctly which will lead to the Euclidean algebra, essential for the existence of the CSR. We need to look at the mass term first and find the parameters which will make the mass term zero, and then look at the quadratic Casimir of the non-contracted group and find if this parameter leads to non-zero length; if not it is not the CSR. This procedure obviously generalizes to higher dimensions.















APPENDIX A
CSR AND THE CONFORMAL GROUP The conformal group is one of the most familiar group which contain the Poincar6 group by embedding. In d-dimension, the SO(d, 2) Conformal group include the Dilatation and Conformal translation generators, in addition to the Poincar6 generators. It has been known [7, 8] for long time that only the regular massless representation of the Poincar6 group has unique Conformal extension. But there is no elegant proof for that. In this appendix, we proof this in light-cone language.

We express the generators of the Conformal algebra using the same variables as for the Poincar6 algebra. We add to the Poincar6 generators, the dilatation generator

1
D = 1(X.P- p-x) (A.1)
2

for the scale transformation and the special conformal transformations generated by K" = 2xM"' + x2p4 = 2x"D x 2p4 + 2x"S"" (A.2)

where y = 0, 1, - (d 1), and we used, M = X4P" x p" + SA" (A.3)

In addition to the commutation relations of the Poincar6 algebra, these two generators satisfy the following commutation relations,


[MP" D] = 0 [D ,p") ip" [D K"I = -iKt [Mv K"] = i (7"K' -_ -v"Kt [ pu K"] = -2i (7""D M"')

To obtain the light-cone forms of the Dilatation and the Conformal translation generators, we first express the generators in light-cone coordinates and apply boost along the (d 1)-direction, D = 1 (-{a~,p+} {+, p} + {xp}) (A.4)

K+ = -2x+M+- + X2P+ + 2xiM+i (A.5)

K- = 2r-M+~ + 2x'M-' + X2p (A.6)

K' = 2x-M+ + 2x+M- -,+ 2x3 Mi' + x2p,


44






45


where i, j = 1, -. (d 2). Using the light-cone forms of the Poincar6 generators from Eqs.(2.5,2.7) and setting x+ = 0, K-- and K can be written as

K- = 2x- (-x-p + xip') -xxp + + (X'T' x'pjS"),
p
K' = 2xI (-X--P+ 4xi p) x xip- + 2xi S'.

To check if the above forms satisfy the algebra, we calculate the commutators [P-, K'] and [pi, K-]. Substituting the light-cone forms, we calculate

[P- K'] = 2i x--Pi- {xi P-}-- S), (A.7)

[K-, p'] = 2i xzPi 1{xi, P-}+ I(T' pii) (A.8)

On the other hand these two commutators are given by FP- K'] = [K- p'] = 2iM-, (A.9)

where M- is given in Eq.(2.7). So comparing Eqs.(A.7-A.9), it is clear that we must set T' = 0, to satisfy the Conformal algebra. Therefore, the light-cone forms of the conformal generators are K+ _-x'xp+ (A.10)

K' = 2xD -- xjx'p" + 2x3 SI' (A.11)

K-- = 2x~D x x p~ xpISi2 (A.12)

where now the dilatation generator is given by D = (- p+}+{X ,pi}) (A.13)

Since T' = 0, it implies that this is the regular massless representation, not the CSR. Therefore, to obtain the CSR, we must use group contraction which produce the Euclidean group, an essential ingredient for the CSR to exist.















APPENDIX B
ANGULAR MOMENTUM IN CYLINDRICAL COORDINATES WITH PERIODIC BOUNDARY CONDITION The cylindrical form of the angular momentum operator is given in Eqs.(4.8-4.10) which acts on the wavefunction of the form as in Eq.(4.6). When these operators act on the wavefunction, we can replace 1 by ". Substituting this, the squares of S3 and S-0 become
z .


(S23)2 = -z2 [sin2 + a2 cos2 0 C9
p + p


2sin cosO 02
p -0&p


2psin20in & 2sin~cosO 0 cos2 0a2 sin0cos0 0
z R p p2 p;2 2 z2 50
2sin cosO in 0 1 in p2 sin2o n2 1
z R 0O z R z2 R2


(S31) 2 2 [COS 2 02 sin2 9 0 2 sin0 cos 0 a2 pcos2 9 0
ap2 p (0p p 09p z2 Op
2pcos20 in a 2sin0cos0 0 sin2 02 sin0cos0 z R Op p209 p2 Q92+ z2 09
2 sin 0cos 0 in 0 1 in p2 cos2o n 2 + z R 0 z R z2 R2

and also (S12)2 = Therefore


(B.1)


(B.2)


$2 (Sij)2 =


1 p 2in> 0 P Z2 zR } op
1 9 2 2in p2 n2+Z2) 002 zR Z2R2 .


Let s2 be the eigenvalue of 2. The eigenvalue equation now becomes


p 2inp' 0 z2 zR / Op

+ 2in z 2 7R


1 1 9 2 +P2 Z2) 02 2RI 2D(PO) = 0.


Using separation of variables, we write


cDn(p, 0) = RnZ(p)E(0) ,


46


psin 9 0
z2 Op


a2 1 V8p + -p


(B.3) (B.4)


(B.5)


(B.6)


2 9 Z -






47


and substituting into the eigenvalue equation, the differential equation factors into a radial and an angular part,

_1 __1 27+ 1 p 2inp) 1 wz, (B.7)
-+ + (B.7)
p2 z2g2 p Z2 zR I, (9p
sl2 2in p2n2 1 02E
+z2 R -z2R2 + 2 gg = 0 (B.8)
I zR O ]

Let the solution of the angular part is of the form

E(0) = eimo (B.9)

up to a normalization constant and where m is the azimuthal quantum number. Substituting this, we obtain the differential equation of the radial coordinate only, 2d 2 2 P_2 inLp d ZP
z2+p2 R(p) + (- P R ) dp

+ 2m2(+ +s2- R)R(P) = 0.

The most general solution of the above differential equation is p2(RZ+2m v/(s2 -m2+1) m 4inz+ Rp2
R(p) = p e 42(R AM --,
244 inz+ R 2' 2z2 R

+BW(VR(s2-m2+1) m 4inz+Rp2
2 4inz +R 2' 2z2vKR

where M and W are Whittaker functions defined in terms of confluent hypergeometric functions as


Ma,b(X) = e i X Fi(b -a+ 2b + 1; x) (B.10)
2'
Wa,b(X) = e-~2 xu U(b- a+ 2b+ 1; x) (B.11)
2'

where 1F1 and U are the confluent Hypergeometric functions of first and second kind respectively. Their integral representations are given by

1F1(a, b; x) = F(b) fxetal(1 t)b-a-idt (B.12)
P(b a)F(a) Jo

U(a, b; x) = e-Xtta-l(l + t)b-a-ldt
r (a) o

As x goes to zero, the function U is divergent and 1F1 is finite. For physically relevant solution, we need R(p) to be finite and well-defined at the origin and also as R oo. Therefore, we set B = 0. So the solution is given by the Whittaker function M.















REFERENCES

[1] E. P. Wigner. On unitary representations of the inhomogeneous lorentz group. Annals of
Mathematics, 40:149-204, 1939.

[2] V. Bargmann. Irreducible unitary representations of the lorentz group. Ann. Math., 48:568-640,
1947.

[3] E. P. Wigner. International atomic energy agency. Theoretical Physics, 1963. [4] L. F. Abbott. Massless particles with continuous spin indices. Phys.Rev., D13:2291, 1976. [5] Kohji Hirata. Quantization of massless fields with continuous spin. Prog. Theor.Phys., 58:652,
1977.

[6] Edward Witten and Steven Weinberg. Limits on massless particles. Phys.Lett., B96:59, 1980. [7] E. Angelopoulos, M. Flato, C. Fronsdal, and D. Sternheimer. Massless particles, conformal
group and de sitter universe. Phys. Rev., D23:1278, 1981.

[8] Tsu Yao. in Lectures in Theoretical Physics: De Sitter and Conformal Group and Their Applications, Edited by A. 0. Barut and W. E. Brittin. Colorado Associated University Press,
Colorado, USA, 1971.

[9] E. InbnQi and E.P. Wigner. On the contraction of groups and their representations.
Proc.Nat.Acad.Sci., 39:510-524, 1953.

[10] I. E. Segal. A class of operator algebras which are determined by groups. Duke Math. J., 18:221,
1951.

[11] P. A. M. Dirac. Forms of relativistic dynamics. Rev.Mod.Phys., 21:392, 1949. [12] H. Bacry and N. P. Chang. Kinematics at infinite momentum. Annals of Physics, 47:407, 1968. [13] L. Brink, A. Khan, P. Ramond, and X. Xiong. Continuous spin representations in poinacr6 and
super-poincar6 groups. J.Math.Phys., 43:6279, 2002.

[14] Anders K. H. Bengtsson, Ingemar Bengtsson, and Lars Brink. Cubic interaction terms for
arbitrarily extended supermultiplets. Nucl. Phys., B227:41, 1983.

[15] J. Polchinski. String theory. vol. 2: Superstring theory and beyond. Cambridge, UK: Univ. Pr.
(1998) 531 p.

[16] M. F. Sohnius. Introducing supersymmetry. Phys. Rept., 128:39-204, 1985. [17] Eugene J. Saletan. Contraction of lie groups. J. Math. Phys, 2:1, 1961. [18] Th. Kaluza. On the problem of unity in physics. Sitzungsber. Preuss. Akad. Wiss. Berlin,
Math. Phys., K1:966-972, 1921.

[19] 0. Klein. Quantum theory and 5-dimensional theory of relativity. Z. Phys., 37:895-906, 1926. (20] Y. S. Kim and E. P. Wigner. Cylindrical group and massless particles. J. Math. Phys., 28:1175,
1987.


48






49


[21] D. Han, Y. S. Kim, Marilyn E. Noz, and D. Son. Internal space-time symmetries of massive
and massless particles. Am.J.Phys., 52:1037, 1984.

[22] Abu M. Khan and Pierre Ramond. Continuous spin represtations from group contraction.
Submitted to Journal of Mathematical Physics, hep-th/0410107, UFIFT-HEP-04-12, 2004.

[231 E. Majorana. Teoria relativistica di particelle con momento intrinseco arbitrario (relativistic
particle theory with arbitrary intrinsic moment). Nuovo Cimento, 9:335-344, 1932.















BIOGRAPHICAL SKETCH

I was born in Dhaka, Bangladesh. While in high school, I became more and more interested in science, especially in mathematics and physics. This motivated me to do my undergraduate studies in physics at the University of Dhaka, Bangladesh. In 1997, I went to the University of Cambridge. There I completed the Part III of the Mathematical Tripos in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge in 1998. At Cambridge, I became more interested in theoretical physics. In 1999, I enrolled in the Ph.D. program of the Department of Physics at the University of Florida, and since 2001 I have studied in the High Energy Theory Group.


50









I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.



Pierre Ramon Chair Distinguished Professor of Physics

I certify that 1 have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosr'phy.



Khandker A. Muttalib Professor of Physics

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.



Charles B. Thorn Professor of Physics

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.



John Yelton
Professor of Physics

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.



Alexander Turull Professor of Mathematics










This dissertation was submitted to the Graduate Faculty of the Department of Physics in the
College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy.

December 2004


Dean, Graduate School




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TIIE CONTINUOUS SPIN REPRESENTATIONS OF THE POTNCARE AND SUPER-POINCARE GROUPS AND THEIR CONSTRUCTION BY THE INONU-WlGNER GROUP CONTRACTION By ABU M.A.S. KHAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNfVTiRSITY OF FLORIDA IN PARTIAL FULFILLMENT OF TIIE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004

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Copyright 2004 by Abu M.A.S. Khan

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To my mother and my late father

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ACKNOWLEDGMENTS I am boundlessly grateful to my mother for all her support and sacrifices without which it would have been impossible for me to continue higher studies. My father, who died at my early age, has always been an inspiration to me, through the memories he left with me, for his enthusiasm, respect and encouragement for knowledge and learning. I am also indebted to my brother for taking care of our fanaily during my stay abroad without which it would have been very difficult for me to continue research. I am very grateful to Professor Pierre Ramond for his patience and support, and teaching me how to do research. I am highly indebted to my teachers Shubhash Chandra Datta at my high school and Professor Khorshed A. Kabir for their continuous encouragement and help. I carmot express how important their support was at times for me. I would like to express my gratitude to the Bangladeshi community here at Gainesville who never let me feel that I was out of my home, particularly Jaha Hamida who constantly and tirelessly supplied the food and tea at the department. I seldom had to go out for lunch! Special thanks go to Professor Khandker A. Muttalib who helped me to get admitted here and for his support whenever I needed it, and of course the great volleyball team members. I will definitely miss them. I thank my friends for encouraging me all through my studies. iv

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' t TABLE OF CONTENTS pag e ACKNOWLEDGMENTS iv ABSTRACT vii CHAPTERS 1 INTRODUCTION 1 2 THE CONTINUOUS SPIN REPRESENTATIONS OF THE POINCARE GROUP ... 4 2.1 Light-Cone Form of the Poincare Algebra 4 2.2 Continuous Spin Representations 7 2.2.1 Four Dimensions 7 2.2.2 Five Dimensions 8 2.2.3 Higher Dimensions 9 3 THE CSR IN SUPER-POINCARE ALGEBRA 12 3.1 Super-Charges in Light-Cone Form 12 3.2 Non-nilpotent Light-cone Vector and the SUSY CSR 14 3.2.1 Four Dimensions 14 3.2.2 Five Dimensions 16 3.3 Nilpotent Light-Cone Translations and the CSR 17 3.3.1 Four Dimensions 17 3.3.2 Ten Dimensions 18 4 INONU-WIGNER GROUP CONTRACTION 20 4.1 IW Group Contraction 20 4.2 Contraction of 50(3) group with periodic Boundary Condition 22 5 CONTRACTION OF THE FIVE DIMENSIONAL MASSLESS REPRESENTATIONS 26 5.1 KK Reduction of the Poincare Algebra 26 5.2 Contraction of the Orbital Rotations 27 5.3 Contraction of the Internal Part and the CSR 29 6 CONTRCATION OF THE MASSIVE REPRESENTATION IN FIVE DIMENSIONS .32 6.1 Non-linear Realization and the Majorana Theory 32 6.1.1 S0{4) --4 E{3) -> 50(3) £(2) 34 6.1.2 50(4) --4 50(3) E{2) 36 6.2 Representation of the Wavefunction 37 7 CONCLUSION 42 APPENDICES A CSR AND THE CONFORMAL GROUP 44 V

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B ANGULAR MOMENTUM IN CYLINDRICAL COORDINATES WITH PERIODIC BOUNDARY CONDITION 46 REFERENCES 48 BIOGRAPHICAL SKETCH 50 vi

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirenierits for the Degree of Doctor of Philosophy THE CONTINUOUS SPIN REPRESENTATIONS OF THE POINCARE AND SUPER-POINCARE GROUPS AND THEIR CONSTRUCTION BY THE INONtj-WIGNER GROUP CONTRACTION By Abu M.A.S. Khan December 2004 Chair: Pierre Ramond Major Department: Physics We study the continuous spin representation (CSR) of the Poincare group in arbitrary dimensions. In d dimensions, the CSRs aie characterized by the length of the light-cone vector and the Dynkin labels of the S0(d-3) short little group which leaves the Ught-cone vector invariant. In addition to these, a solid angle i1d-.3 which specifies the direction of the light-cone vector is also required to label the states. We also find supersymmetric generalizations of the CSRs. In four dimensions, the supermultiplet contaias one bosonic and one fermionic CSRs which transform into each other under the action of the supercharges. In a five dimensional case, the supermultiplet contains two bosonic and two fermionic CSRs which is like N = 2 supersymmetry in four dimensions. When constructed using Grassmann parameters, the light-cone vector becomes nilpotent. This makes the representation finite dimensional, but at the expense of introducing central charges even though the representation is massless. This leads to zero or negative norm states. The nilpotent constructions are valid only for even dimensions. We also show how the CSRs in four dimensions can be obtained from five dimensions by the combinations of Kaluza-Klein(KK) dimensional reduction and the Inonii-Wigner group contraction. The group contraction is a singular transformation. We show that the group contraction is equivalent to imposing periodic boundary condition along one direction and taking a double singular limit. In this form the contraction parameter is interpreted as the inverse KK radius. We apply this technique to both five dimensional regular massless and massive representations. For the regular massless case, we find that the contraction gives the CSR in four dimensions under a double singular limit and the representation wavefunction is the Bessel function. For the massive case, we use Majorana's vii

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infinite component theory as a model for the 50(4) little group. In this case, a triple singular limit is required to yield any CSR in four dimensions. The representation wavefunction is the Bessel function, as expected, but the scale factor is not the length of the Ught-cone vector. The amplitude and the scale factor are implicit fimctions of the parameter y which is a ratio of the internal and external coordinates. We also state imder what conditions our solutions become identical to Wigner's solution. viii

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CHAPTER 1 INTR013UCTI0N All elementary particles in nature are labelled by their masses and spins or helicities. The underlying mathematical principle for this characterization is the representations of the Poincare group. In 1939, E. P. Wigner first classified all representations of the four dimensional Poincare group in a classic paper [1]. These classifications are characterized by two Casimir eigenvalues: the mass-squared and the square of the Pauli-Lubanski vector W^. The orthogonality between the Pauli-Lubanski vector and the four-momentum vector defines the little group which leaves the momentum vector invariant. The little group gives the spin (helicity) of the massive (massless) pai-ticles respectively. We summarize the classification in the following: Representation First Casimir Second Casimir Little Group Massive representation M^, M real W'^ --= -M^j{j + 1) 50(3) Regular massless representation M -= 0 = 0 50(2) Tachyonic representation M^, M imaginary W'^ = -M'^j{j + 1) 50(3) Continuous spin representation M =-0 W"^ — £^(2) Among these four, only the first two representations are realized in nature. The massive particles, like protons, neutrons, 7r-mesons, etc., belong to the massive representations of diflterent spin, and the massless particles Uke photons are described by the regular massless representations. Tachyons have imaginary mass, and violate causality as they travel faster than the speed of light [2]. They are unphysical and do not occur in nature. In field theory, the appearance of the tachyonic mass term is removed by shifting the vacuum, a process known as the Higgs mechanism. The continuous spin representation (CSR), which is the main subject in this dissertation, is characterized by the zero mass and non-zero finite length of the Pauli-Lubanski vector. In four dimensions, the Pauli-Lubanski vectors are related to the translation generators which together with the homogeneous rotation generator form the inhomogeneous little group. Group theoretically this is the £'(2) algebra of the little group. In this representation, an action of a finite boost on any state produces an infinite tower of massless states with helicities ranging from minus to plus infinity. Even though the CSR is mathematically very sound, there are various arguments against 1

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its existence in nature. Wigner [3] himself argued that since the CSR gives an infinite heat capacity of the vacuum, it is unphysical and does not exist in nature. L. Abbott [4] showed that the CSR violates causaUty and K. Hirata [5] further showed that in this representation the fields become non-local. Moreover, S. Weinberg and E. Witten [6] proved that in four dimensioris there cannot exist any particle with helicity higher than two. Because of these strong arguments, the CSR has never attracted much attention. Despite these facts, there are still reasons for further studjdng the CSR, in particular in the context of models for extended objects in higher dimensions, as for example, String theory and Mitheory. According to these models, the infinite slope limit of the Regge trajectories yields an infinite helicity tower of massless particles. Mathematically this is the representation of the Euclidean group which is the little group of the Poincare group. This is our main motivation to study the CSR. We will not address how the problems associated with the CSR can be cured, but will concentrate on their mathematical properties. We hope that these studies may provide better insights for how to avoid or cure the problems associated with the CSR. Our aim is two-fold. First, we study how these representations and their states can be characterized in any dimensions by the Casimirs of the little groups and if these representations have any supersymmetric extension. Because of the supersymmetry, the light-cone vector can be constructed with or without using the Grassmann parameters. This means the light-cone can be made nilpotent and as a consequence the representation can become finite-dimensional. Second, we study how these representations can be obtained from larger groups. Of particular interest in physics are the Conformal groups which contain the Poincare group by embedding and the larger Poincare group, (Anti) de Sitter groups which give the Poincare group by group contraction only. It has been known for sometime that the CSR has no conformal extension [7, 8]. We prove this in light-cone language in Appendix A. Therefore, we will look into group contractions. Group contraction has been introduced by Inonu and Wigner [9] in 1953, and independently by Segal [10]. The group contraction is a singular transformation to obtain the Euclidean group from a homogeneous group. Inonii and Wigner (IW) showed explicitly how the contraction of the 50(3) group yields the E{2) group which is algebraically identical to the Poincare little group in four dimensions. We show that the IW group contraction can be realized in a different way which is more physical. Imposing a periodic boundary condition and taking a double singular limit yield the same conclusion as IW. We consider the periodic boundary condition as the Kaluza-Klein(KK) dimensional reduction process. One singular limit can be identified with the infinite KK radius

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if 3 limit whose inverse is the contraction parameter and the other singular limit is the spin. We apply tliis technique to both regular massless and massive representations in five dimensions. We find the conditions for the CSR to exist in four dimensions and find the representation wavefunction. We also compare with Wigner's solution for the CSR. The plan of this dissertation is the following: chapter 1 is the introduction, chapter 2 reviews the CSR and extends it in any dimensions, chapter 3 discusses the supersymmetric extension of the CSRs, chapter 4 reviews the group contraction, chapter 5 discusses the contraction of the five dimensional regular massless representation, chapter 6 is the contraction of the massive representation in five dimensions, and finally chapter 7 is the conclusion.

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CHAPTER 2 THE CONTINUOUS SPIN REPRESENTATIONS OF THE POINCARE GROUP In this chapter, we introduce the continuous spin representations(CSRs) of the Poincare group. For convenience, we express the generators in light-cone form [11, 12]. We start with the general form, of the Poincare algebra in any dimensions, and review and construct the continuous spin representation in four and five dimensions respectively. We also present the Casimirs for arbitrary dimension. More details can be found in Wigner [1] and Brink et al. [13]. 2.1 Light-Cone Form of the Poincare Algebra The Poincare group is the semi-direct product of the abelian translation group and homogeneous Lorentz group for rotations. The Poincare generators satisfy the conunutation relations, [p^P1 = 0, (2.1) [M^^P''] = iiri^^P" -T]''^P^) (2.2) [M*^", M"''] = iiri^^M"^ + ri'^'M'^'' + ri^^M^"" + rif^^M"") (2.3) where t?*^"^ = (-1, 1, • • • ,1) and /x, i/, a, /3 = 0, 1, • • • {d-l); and P'^ and M"^ are the generators for translation and rotation respectively. All representations of the Poincare group have been classified by E. P. Wigner in a classic paper [1] in 1939. These are characterized by the Casimir operators which are the squared mass, M^, and the squares of the Pauli-Lubanski forms which in d space-time dimensions are defined as ,ni 2^ (^=f^) (^^)! where n 1, 3, • • (d 3) for d even and n = 0, 2, • • • (d 3) for d odd. Their squares are the Caiiimir operators of the light-cone little group. In the following, we rewrite the Poincare group generators in light-cone language, originally introduced by Dirac [11] for spinless particles and later extended for arbitrary spin by Bacry and Chang [12], and characterize the representations by the Casimirs of the light-cone little group. The light-cone variables are defined as X 4

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5 and similarly for any other variables. In this formalism, the canonical commutation relations become [x^,p'^] = -i and [x' j^] = i5'^ where i,j = 1, 2, • • • (d 2) are the transverse directions. In light-cone form, the space-time translation generators become ^ ^ fg + Ml ^ p. ^ ^. ^ where P~ the light-cone Hamiltonian, is expressed in terms of transverse momenta by using the mass-shell condition. Since P~ the conjugate variable to x"*", is constrained by the mass-shell condition, we set x"*" = 0. The Lorentz generators can be grouped into two types: those which transform the transverse plane into itself are called "kinematic" and those which transform out of that plane are called "Hamiltonian" types respectively by Dirac. The kinematic generators are given by M+* = x'p+ M+= x-p+ M'^ = xV x^p' + S'^ (2.5) where 5'^ obey the SO{d 2) Lie algebra of the transverse little group [ S'^ 5*^' ] i [S''' 5^' + 5^^ S''' 5'^ S^'' 6^'' S'^) (2.6) The Hamiltonian-Uke boosts Eire = x-p' -l{x\p--} + -^{T -p^S'^) (2.7) 2 p+ The vector, T', is called the light-cone translation vector. It transforms £is a vector imder the SO{d — 2) rotations, [5'^ r'=] = i{S'''P S^''T) (2.8) and satisfies the following commutation relation, [T\T^] = iM'^S'^ (2.9) Eqs.(2.8-2.9) form the algebra of the Uttle group. There are four ways to satisfy the algebra: • M ^ 0: There are two types of representations in this case depending on whether M is real, positive or imaginary. 1. M real and positive: This is known as the massive representation. In this case, T'/M and S''^ form the homogeneous httle group S0(d-1). In four dimensions, it is the 50(3)

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fi 6 little group and all massive elementary particles obey this representation. The states are labelled by the mass M and the spin j. 2. M imaginary: In this case, the little group is exactly the same as in the previous case. This is known as the tachyonic representation. No tachyonic particles have been observed in Nature. The tachyons are unphysical particles because these travel faster than the speed of light [2]. • M = 0: In this case, there are also two types depending on whether J^s are zero or not. 1. T' ~ 0: This corresponds to the familiar massless representations whirJa describe particles with a finite nimiber of degrees of freedom. These are realized on states that satisfy T\ p+,p'; {ofc} > 0 fc = 1, • • • ,Rank of SO{d 2) where {ak} are the Dynkin labels of SO{d 2) representations. In four dimensions, the little group is 50(2) and its eigenvalue helicity along with the zero-mass uniquely determine the representation. The Pauli-Lubanski vector is light-like. 2. T' > 0: In this case, T' are the c-number components of a transverse vector. The vectors T' and the rotation generators 5*^ form the E{ = e \p-^,p';e,M > fe = l, Rank of 50(d3). where {ak} are now the Dynkin labels of SO{d — 3) subgroup which we call the "short little group." In four dimensions, there is no such group and the states are simply labelled by an space-like vector of constant magnitude = 1,2. These span two distinct representations, called "continuous spin representations" by Wigner [1] in his original work. The Pauli-Lubanski vector is space-like. The CSRs describe a massless state with an infinite number of integer-spaced helicities. In the following sections, we review continuous spin representation in four dimensions, construct the state in detail in five dimensioas, and finally end this chapter by introducing the general formula for the Casimirs of the little group in any dimensions.

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7 2.2 Continuous Spin Representations These representations are characterized by the zero mass and the finite length of the commuting light-cone translations T* which together with S'^ form the E{d 2) Euclidean little group in ddimension. In the following, we first briefly review the properties of the CSR in four dimensions following Wigner's work (in light-cone formulation) and show how the infinite tower of helicity states is generated, and extend it into the five dimensions and calculate the Casimirs for higher dimensions. 2.2.1 Four Dimensions In four dimensions, the little group is formed by the twovector T and 50(2) rotation. Together these satisfy the £^(2) algebra, [r,T^]=0, and = i(<5*'=TJ-(5^'=r) ; for2,j = l,2. Since S^^ and T' do not commute, helicity is no longer a good quantum number; however T'T'' does commute. The two Casimirs that uniquely label the representation are the zero mass-squared and the square of the Pauli-Lubanski vector given by where we used Eq.(2.4) for four dimensions and worked in a frame where = 0 for simplicity. It follows that a finite boost creates an infinite number of integer-spaced helicities. To see the appearance of the infinite number of hehcity states, we construct the light-cone raising and lowering operators, T = T^iT'^ which satisfy the relations, ^_5i2^j,j ^ ^rj. [T,rT] =0. It is easy to show that 5i2(r| >) = (ml)(T| >) wiiich clearly shows that the state T^\ > has helicity (mil). Therefore, an action of an infinitesimal boost, changes the heficity by one imit because of the following relation

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Therefore an action of a finite boost produces states with possible helicities ranging from minus to plus infinity in integer steps. This is an infinite dimensional representation. There are two types of representations: those with all uiteger (single-valued) and those with all half-odd integer (double-valued) helicities, known as bosonic and fermionic CSR respectively. As we have stated in the introduction, the CSRs have no obvious physical applications, explicitly in local field theories, but the appearance of an infinite number of states may indicate a connection with non-local theories of extended objects. This motivates their study in more general contexts. 2.2.2 Five Dimensions In five dimensions, the light-cone little group generator 5*-' and T\ for i,j 1,2,3, satisfy the following B(3) algebra, [r,TJ] = 0, and [S'^ ,T^] = i{5'^T^ 5^''T) Using Eq.(2.4), we get the following two Pauli-Lubanski forms, 0-form: W = e.j.T^S^" ^^^^^ 2 form : Wij = etjkT'' where we used a frame = p~ = 0. The square of the two-form WijW'^ r'=T'= = (2.11) is one Casimir, identical to the four dimensional case. The other Casimir, the zero-form, can be thought of as the projection of the generator S^'' along T\ The vector then acts as a "quantization axis," along which W assumes the values W 13 0,i 1,^,--. (2.12) V2E "' 2' 2 It follows that there are two types of representations corresponding to each integer and half-odd integer values of W/\/2E. For each value of W/^/2E, there corresponds one infinite dimensional representation. Unlike four dimensions, there are infinite numbers of both bosonic and fermionic CSRs in higher dimensions. The states are no longer characterized by the light-cone little group but by its 50(2) short little group orthogonal to T\ Each CSR is labelled by the length of T' £md the eigenvalue m of the short little group which is the helicity. It is straightforward to find the CSRs in terms of the eigenstates of 50(3), the fuU little group. Let \j, m > be the eigenstate of the Uttle group. These states are required to be eigenstates of T, the 50(2) rotations about it, and of the Casimir CijkT'S^'^. Since T* is a vector under 50(3) rotation.

PAGE 17

9 its action on each \j,m > yields a linear combination of j = j,j 1 states. Therefore, the CSR eigenstates are infinite linear combinations of eigenstates of the full little group 50(3). To construct a CSR state, let the hght-cone vector be aligned along the third direction. This allows us to identify with the tensor operator which is the same as in spherical basis. The action of on the state \j, m > yields T^\j m > = a'i'^^lj + 1 m > +a[,^"'^| j m > j 1 m > (2.13) where the a's are proportional to Clebsch-Gordan coefficients a^o^""^ = ^/{j + m){j -m + 1) -j =Em\F>, m = 0, i 1, . . We find that, after using Eq.(2.13) and little algebra, the coeflficients must satisfy the following recursion relations (\m\+p,m) Am) (\m\+p-2,m) ,(m) (\m\+p-l,m) -,n ,(m) p, J\m\+p'^"+ J\m\+p-2'^ ^-)J\m\+p-l with p = 2,3, . and (|m|+l,m) ,{m) / (|m|,m) ,(m) p, •'|m|+l + '."O -) J\m\ since /j^^^i = 0Clearly for each value of j, the 50(3) angular momentum, there are two CSRs: bosonic and fermionic types corresponding to whether m is integer or half-odd integer, and there are an infinite niunber of CSRs of each type. The other CSR states can be obtained by acting the 50(3) raising and/or lowering operator on |F >. 2.2.3 Higher Dimensions In higher dimensions, we just Ust the Casimirs and explain how the states are labelled, for more detail see Brink et al. [13]. Following the previous section, it is clear that the Casimirs of the short little group and the length-squared of the highest form uniquely label the representation.

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10 Prom Eq.(2.4), the Pauli-Lubanski n-form for the Ught-cone Httle group SO{d 2) can be written as c Ctn + llTi + 2 . C'trf-4d-3 ^l---Jn7.+ l---J 1 are given by • For little group 50(4) and 50(5): = E^Sl; l

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11 where {a^} is the Dynkin label of the SO{d 3) short little group with r being the rank. These differ from the usual massless representations in that they are characterized by a space-like vector, and contain an infinite number of states. In the next chapter, we extend the Poincare group to include supersymmetry and find the supersymmetric CSR(SCSR).

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CHAPTER 3 THE CSR IN SUPER-POINCARE ALGEBRA The Super-Poincare algebra includes the Poincare algebra and the spinorial supercharge. In this chapter, we discuss how to construct the supersymmetric (SUSY) CSR. This chapter is based on Brink et al. [13]. A supermultiplet consists of an equal numbers of bosonic and ferraionic states, and when a supercharge acts on these, they transform into each other. The supercharges are constructed using the Grassmann parameters. Therefore, to close the algebra, the lorentz generators must also be extended to include the Grassmann parameter dependent terms. Since the light-cone translations transform as a vector under the transverse rotations, it can be constructed without or with using the Grassmann parameters. The first one is of course the usual type of vector as in the previous chapter, whereas the second one is necessarily nilpotent because of the Grassmann parameter whose square is zero. In the following two sections, we investigate these two possibihties. 3.1 Super-Charges in Light-Cone Form The generators of the Super-Poincare algebra satisfy the conmiutation relations in Eqs.(2. 1-2.3) and the following for the spinorial supercharge, [Qa p1 = 0 (3.1) [M'^^ Qa] -^(r'^'^Q)^ (3.2) {Qa QL) = (rV)AB (3.3) where A, B are spinor indices. In the above, r'"' = -^[r r], where the T matrices satisfy the anticommutation relation, {r'', r} = 2r/^^ fx,u = 0,.-,d. The supercharges can be realized linearly using Grassmann variables and their derivatives [14] as QA = dA + \{T''p^TUBe^ (3.4) 12

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13 and their conjugates as QL = ^c + ^(r%r%c^^ (3.5) where we used and dA = Ba -= and 6 a is the complex conjugate of 6a with A, • • ,D running over the spinor indices. 0, 6, d and B are anticommuting Grassmann parameters. Given the supercharges, it is always possible to define the covariant derivatives, DA = dA-\{Ty^T)ABe^ and their hermitian conjugates. The covariant derivatives always anticommute with the supercharges and their conjugates, {Qa, Db} = {Q^A, Db} = 0. (3.6) Acting on the wavefunction the above anticommutativity is equivalent to the following constraint, = 0 (3.7) which is a common constraint in any dimension. This indicates that the supercharges and hence all other generators are reducible. This constraint reduces the number of Grassmann parameters by a factor of two. There may be further reducibility depending on whether the Grassmarm parameters are of Dirac type or Majorana type. These two constraints depend on the number of space-time dimensions. The irreducible form is obtained by imposing all constraints allowed. In d-dimension, a complex spinor have 2'*/^(2('^~-'^/^) components for d even(odd). If p is the niunber of constraints imposed, the niunber of independent complex spinor is 2''/^~p(2*'^~^'/^~p) for d even(odd)^. Let a, 6, c, ... run over the irreducible components. Imposing all constraints, the irreducible formis of the kinematic supercharges become and the dynamic supercharges become ' g = f(j>yp+)Ql Q'^J = r{p''/p^)Qt (3.9) ^ The detailed analysis can be found in [15, 16].

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14 where f{p'/p'^) and its complex conjugate are functions of the ratio ^ determined by the constraint D$ = 0. The kinematic and dynamic components are determined by using the light-cone projector. The irreducible forms of the Lorentz generators can be written as M'J = x^p' + S'^ (3.10) M+= -x-p+ + S+-, (3.11) = x-p'-l-{x\P-} + ^{T-p^S'^)-^S+, (3.12) 2 p+ p^ where SiJ = S'^ + (^^e%f^)abd'' + c.c?j (3.13) S+= Q^<'(7+-)a65^ + c.c.) (3.14) with 7*^ £md 7"*"" being the reduced Dirac submatrices consistent with the constraints. M"*"' remain the same as before. The indices i,j run over traiisverse space. 3.2 Non-nilpotent Light-cone Vector and the SUSY CSR In this section, the light-cone vector is considered like an ordinary vector as in the Poincare group. Following the discussion in the previous section, we write down the irreducible generators below and check various commutator whether the Super-Poincare algebra closes or not. 3.2.1 Four Dimensions In four dimensions, a complex spinor has four components. We can impose either chirality or Majorana condition and the covariant derivative condition, a common constraint for any dimensions, to reduce the number of independent component to one. Let 6 be the Grassmarm parameter. In Weyl representation, following Eq.(3.4) and imposing the chirality condition and = 0, we can write the kinematic supercharge [13, 14] as and the dynamic supercharges are given by Q_ = Ql = ^Ql (3.16)

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15 where p{p) = -^{p^ ip^). The generators M+~, M^^ and M""* can be expressed in the following irreducible forms, = (3.17) = .V_.V + S+ i(|-?|), (3.18) M= x-p"4{.-,p-} + ip(r-/S"') + i^(4+j|) (3.19) where a, 6 = 1, 2. The above generators and the supercharges have to satisfy the following commutation relations to close the algebra, (3.20) (3.21) (3.22) (3.23) (3.24) (3.25) which are derived by applying the constraints to Eqs.(3. 1-3.3) directly. After a long and tedious calculation, it can be shown that the irreducible generators satisfy these commutation relations. However, the last two relations, Eqs. (3.24-3.25), require that Q+ to commute with T-. An easier to see this is to choose a frame where = 0. In this frame, the dynamic supercharge vanishes and these commutators become [T\Q+\ = 0 (3.26) Since the light-cone vector commutes with the supercharge, we can implement supersynunetry on the continuous spin representations without having to change the supercharges. Let |m > be a state with hehcity m. So its supersymmetric partner Q+\m > has hehcity (m 1/2). So an action of a finite boost on the states |m > and |m1/2 > generates the CSR states \m >csk and |m — 1/2 >csTi respectively. So the CSR supermultiplet is [M+, Q] Q] = t\q, Q] = 0, > Q^] [M2 1"^ >CSR l^-l/2>csf which is Uke ordinary (T' = 0) supermultiplet, but accompanied by an infinite tower of states. (3.27)

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16 3.2.2 Five Dimensions In five dimensions, the spinors have four complex components which can be reduced to two by imposing the covariant derivative constraint, the only constraint allowed. In the representation, r = ia^/, r = a^{8)(7' i = 1,2,3; and 7" = 7 (3.28) the light-cone projectors, V = -^r^fT^^, split up Eq.(3.3) into the following, {Qa, Qit} = ^P'^^ab and {Qa QU} = i(^ (3-29) where a = 1, 2. The constraint, = 0, gives the following relation, d f a -p and similarly for their complex conjugates. This allows us to set {6^,d^) = 0. Substituting these, the supercharges in Eqs.(3.4-3.5) reduce to the irreducible forms with a,b= 1,2, and the dynamic supercharges are now given by Q_a = -i (|-^Q+) and hence ^-"^^(qI^^) (3-31) The internal part of the Lorentz generators S'^~ and 5'^ become where now 6 = {6^,0^) and ^ = The remaining Lorentz generators remain unchanged. It is quite easy to see that the supercharges transform as 50(3) spinors, \M'^ Q+a] = \e'^H
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17 Clearly the supercharges Q+i and Q+2 ^o^er the helicity and the remaming two Q+2 and Qj^^ raise it. In terms of the helicity m, the supermultiplet consists of the following states, + ,1 Q+i|m>csa and QV2I'^>csr ~ I'ti-2>csr. Q+iQ+2\m>csK ~ |m-l> OSR It contains two bosonic and two fermionic CSRs, with the same structure as the ordmary (T = 0) massless N = 2 supermultiplet in four dimensions. The important difference is that it contains not only the ordinary states but their copies under the boosts proportional to T. This yields as usual an infinite number of 50(3) polarization states. The action of supersymmetry is the same as ui the normal case, but the CSR supermultiplets contain an infinite number of ordinary massless supermultiplets of ever-increasing spin. This construction has obvious generalization to any higher dimensions. 3.3 Nilpotent Light-Cone Translations and the CSR In the previous section, we have shown that the CSR has supersymmetric extension. The light-cone translation vector was ordinary type, i.e., as in Poincare algebra. In this section, we construct the Ught-cone vectors T' using the Grassmann parameters. This construction will, of course, make the light-cone vector nilpotent, such that at least is nonzero but the higher orders are. If becomes zero this will not lead to any CSR. A finite boost will now terminate after the second order term resulting a finite number of helicity state. Since it is not possible to construct SO{d) vector using only one set of Grassmann parameter, we have to introduce at least two sets of Grassmann parameters which automatically make nonzero, but the higher orders are zero. This can be thought of = 2 supersymmetry. It should be noted that if we require higher degree of nilpotency, we have to introduce more Grassmann parameters and thereby will be able to consider higher N supersymmetry. In the following, we construct nilpotent light-cone translation in four and ten dimensions. 3.3.1 Four Dimensions Since it is not possible to build a 50(2) vector with one Grassmann parameter, we introduce two complex Grassmann variables 9i and 82We define the light-cone vectors as = ^(zeih + 26162) (3.35) r2 = l^^Z6,62 26162) (3.36)

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18 where Z is a complex c-number parameter, and p+ is used to ensure its proper commutation relation with M+~. The square of the Pauli-Lubanski vector is, W2 = 2|Z|2(p+)2eiMi^2 (3.37) The kinematic supercharge and its conjugate are where a = 1, 2, because of two Grassmann parameters. This is equivalent to iV = 2 supersymmetry. However the light-cone translations no longer commute with these supercharges. To close the SuperPoincare algebra, the dynamic supercharges must be altered to the new form Q = 4 ^^^"'^f Q-^ = ^QV+ iZe''% (3.39) which ensure the commutation relations of with the boosts M~. These supercharges also transform correctly imder transverse rotation S^"^ which now reads, ^12 ^ _5l2 ^ 1 (^ag^ gaQ^-^ (3 40) Let us compute the anticommutator between the supercharges. We find that, {Ql,Q'L} = ie"" Z {Ql\ Q'J } = ie'^ Z (3.41) These non-zero anticommutator together with the Super-Poincare algebra (anti)commutators form the supersymmetry algebra with central charge, even though the representation is massless. It is well-known that the supersymmetry with, central charge leads to negative or zero norm states for massless representation [16], it only makes physical sense for massive representation. Even though by constructing the nilpotent light-cone translation vector, we remedied the infinite vacuum heat capacity problem which was Wigner's argument against the existence of the CSR, but at the expense of negative or zero norm states. 3.3.2 Ten Dimensions Following previous section, the ten dimensional case is straightforward. The light-cone little group is 50(8) which has the magic triality property. In ten dimensions, we can impose both Majorana and chirality conditions, in addition to the = 0 constraint. These reduce the number of independent real spinor components to eight. As in four dimensions, we use two sets of real

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19 Grassmann parameters to form 50(8) vector. In the representation, where i = 1, • • • 9 and 7*s are 8 x 8 gamma matrices, the supercharges and the hght-cone vectors can be written as = ^ + ; a = 1,2. and r = ip+Z^n^ (3-43) Ooa v 2 where Z is now real. These transform as Q^l^ ~ 8 ~ 8e and T ~ 8„ mider the SO (8) transverse rotations given by where 7*^ is the reduced tensor. The generator 5+~ and the dynamic boost reduce to the following forms, To satisfy the algebra, the dynamic supercharges must change to the new form, g(a) ^ __JPLyq(:) -ie'^b^ze a,6-l,2. (3.45) V2p+ where = 1 = -e^^. The anticommutators between the kinematic and dynamics supercharges are no longer zero, {Q^+^ = ie"'\/2Z, (3.46) indicating the N = 2 supersjarunetry with central charge. Even though our construction leads to supersymmetry but the representations necessarily contain negative and zero-norm states. Although it is interesting to note that central charges occur naturally whenever the light-cone translations are built out of the Grassmarm variables, the representations contain negative and zero-norm states; at best they could be used as ghost compensators of some unknown theory. This construction does not seem to generalize to odd dimensions. To see it, consider eleven dimensions with T quadratic in the Grassmann numbers. There a quadratic product of a Grassmann spinor transforms as 2and 3-forms, so that to make a vector we need some c-number tensors, either a one or two-form, but the commutation with the supercharge does not have the right form.

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CHAPTER 4 INONti-WIGNER GROUP CONTRACTION In this chapter, we review the group contraction first introduced by Inonii and Wigner [9], and also independently by Segal [10]. Later Saletan [17] discussed it for arbitrary homogeneous Lie groups in more general context. Here we restrict to the IW type contraction. We start with a brief review of the IW's original approach to group contraction, and then we show that the same result can be reproduced by imposing periodic boundary condition to the homogeneous 50(3) group and taking two singular hmits. The different between these two approaches is the interpretation of the contraction parameter. 4.1 IW Group Contraction Inonii and Wigner [9] used homogeneous 50(3) group to introduce the group contraction. They showed explicitly how to obtain the Euclidean group E{2) by contracting the 50(3) group under singular limit. They applied the contraction to the algebra generators, identifying those which are well defined under contraction, and derived the two-dimensional wavefunction on which the Euclidean translation vectors act, the Bessel functions J„. Let 5' (i = 1,2,3) be the generator of the 50(3) group which satisfy the angular momentum algebra^. The states are labelled by the eigenvalue of the Casimirs 5^ and 53, 5^5^ = s(s + l) 53 m. Since — s < m < s, this spans a (2s-|-l)-dimensional representation. The representation wavefunction is of course the spherical harmonics. Let MS define another set of generators as 5'' = Um e5' ^ 5' is related to the generators 5'^ by Poincare duality, i.e., 5' = \e^^^S^'^. 20

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21 where e is an arbitrary parameter, called the contraction parameter. For fixed m, S^' = 0, and the remaining two generators and satisfy the following algebra, [53,5'] = ie,,S'' [5', 5*'] = 0, (4.1) where a, 6 = 1,2. These commutation relations form the Euclidean algebra only if the Casimir is also nonzero. To find the Casimir, we multiply the 50(3) Casimir by the contraction parameter, [eS^f + (e5)2 = e^sis + 1) and taking the limit e ^ 0, while keeping m fixed, yields the following, 0 ; if s = fixed. (4.2) H'^ ; if s — > 00. (S^y = lime2s(s + l) = { ^ £—0 ^ ' where H = es is nonzero finite as e 0 and s — > oo. In the first case, we obtain homogeneous 50(2) group with 5^ being the Casimir operator, whereas the second case yields the Euclidean group under the double limit. Since s sets the range of m, and s — > oo, m can now have infinite range. Thus it forms an infinite dimensional representation. This is the most essential feature of the contraction process. The Casimir eigenvalue of the .E(2) group now reads, 5'"5"' = =2 (4.3) Since 5"'s are like momenta, rather than deriving from the 50(3) generator, IW identified it with the differential operator for translation, 5"' = a =1,2. (4.4) Therefore, Eq.(4.3) reduces to the Laplace wave equation in two dimensions. In cylindrical coordinates, the solution of the wave equation is the well-known Bessel function, $(p,^) = Y.NM~py^' where p = \J[x^)'^ -I[x^YThis is the representation wavefunction of the £(2) algebra. In their paper, they also derived the following identity. ([]mV\ .^^drrdi) ^ncos(H,/o) = j.(H,),

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22 which justifies that the contracted algebra is well defined, in particular the identification of the contracted generator 5' with the differential operator for translation in Eq.(4.4). In their original paper, Inonii and Wigner applied their method to the Poincare group, using the inverse speed of light as a contraction parameter, and find it to contract to the non-relativistic GaUlei group only if the starting point is the tachyonic representationl 4.2 Contraction of S0(3) group with periodic Boundary Condition In this section, we apply the periodic boundary condition to the 50(3) algebra by putting one direction on to a circle of radius R. The contraction parameter is now the inverse radius. We show that under contraction, the representation wavefunction is the same as the previous section. In the following, we retrace IW's steps, stressing the geometrical picture of the contraction procedure. The rW contraction of 50(3) to E{2) amounts to the study of a dynamical system with 50(3) symmetry restrained to a space whose boundary condition breaks that symmetry We switch to cyhndrical coordinates, and seek solutions which are periodic in z, z ^ z + 2tiR (4.5) ^{p,9,z) := ^$„(p,5)e'-/, (4.6) n where n is the mode number. Let be the eigenvalue of the Casimir (5'-')2 when it acts on this wavefunction. Note that in spherical basis the Casimir eigenvalue is j(ji' + 1). For very large spin i(j + 1)) S'lid eventually we will take it to infinity. So we no longer distinguish between and j{j + !) Therefore, we write, S'^S'^ = f (4.7) where i,j = 1,2, 3. In the following, we use the inverse radius, 1/i?, as the contraction parameter. The angular momentimi operators are given by (4.8) „ d zcosO d =: d = -i ^-2 sin = —i 1 z cos 6 dz p de d zsraB d 'dz (4.9) (4.10)

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23 Acting on the wave function, the factor ^ can be replaced by ^. Substituting this, the square of the generators become (5^2)2 d 2sin6'cos(9 d'^ psin^O d dddp dp Q ^ c cos sin^^— + — + op^ p op 2psm^9in d 2sin9cos6l d cos^d dP_ singcos^ d z R dp 2 sin 6 cos 6 in d 36 p^ 1 in p2 sin^ Q X? ROe z R J?2 {S 31\2 = -Z o.d'^ sin cos^ ^— + 6 d 2sin6icos^ d"^ pcos^B d dp^ P dp dOdp dp 2pcos^6in d 2sinecos^ d sin^f d'^ sin^cosS d + o ^ + —^^ + Rdp 86 de^ d6 + 2 sin 6 cos 6 in d 1 in p^ cos^ 6 n 2 1 (4.11) (4.12) (4.13) z Rd6 zR 22 When R is very large, all terms proportional to inverse R can be dropped2. Using Eqs.(4.11-4.13) into Eq.(4.7) and dropping the terms proportional to l/R, we obtain the following differential equation, —z d^ 1 d pdp which acts on the wavefunction of the form 52 (4.14) ^{p,6) = 1l{p)Q{6) (4.15) Since there is no mode dependence, we have dropped the subscript n from the wavefunction. For the above form of the wavefimction, the angular part decouples from the radial part and can be written as (4.16) e dd^ = — m where m, the azimuthal quantum number, is an integer or half-odd mteger depending on whether the wave function is smgle-valued or double-valued. Substituting this and rearranging, the differential equation Eq.(4.14) reduces to 2 (f d = 0 (4.17) ^ The complete solution with finite R is the Whittaker function. See Appendix B.

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24 which is the Bessel differential equation. The regular solution is ^p,e) = Y.^Mqpy"'', (4.18) m where the scale factor is given by 9 = {^^-^^^This is the representation wavefunction for very large radius R. Let us now define, 5' = lim eab^ (4.19) while keeping m, eigenvalue of S^, fixed. Using Eqs.(4.9-4.10), we get, 0I' .zfdsm0d\ It can easily be checked that the S's, a = 1,2, commute with each other and that they transform as a 2-vector under 5^^, forming a representation of the E{2) algebra as in Eq.(4.1). Let be the eigenvalue of the Casimir operator for infinite radius Umit. Therefore, the Casimir operator (51? + (52')2 = =2, (4.22) reduces to the following differential equation. where we used limi?_oo x ~ ^ Eqs.(4. 20-4.21). For the wavefunction of the form as in Eq.(4.15) and the angular dependence as in Eq.(4.16), the above differential equation becomes (after some rearrangement) whose regular solution is the well-known Bessel function, ^p,e) = Y.NmJmiqp)e'""> (4.25) m where the scale factor is defined &s q = (p^^Since the solutions in Eq.(4.18) and in Eq.(4.25) have to be identical for infinite radius limit, this implies that the scale factors in Eqs. (4.17,4.24) must also be identical. Therefore, equating these two scale factors, we obtain a relation between the

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25 Casimir eigenvalues of the 50(3) and E{2) groups, =2 = 4, ^4.26) where now both j and R tend to infinity, but their ratio H remains fixed and nonzero. This solution is of course the same as that of Inonii and Wigner. The generalization of the above construction to any dimensions is also quite straightforward. In general, for d dimensions, compactifying one direction gives the following relation H2_, = I (4.27) where is the quadratic Casimir of the homogeneous group SO{d) and E^_i is the length squared of the E{d 1) group with R being the radius of the circle.

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CHAPTER 5 CONTRACTION OF THE FIVE DIMENSIONAL MASSLESS REPRESENTATIONS In this chapter, we discuss how the contraction of the five dimensional regular massless representation yields the CSR in four dimensions. We apply the techniques developed in the previous chapter. Since the Poincare group has an orbital and an internal part, we need to address how the contraction process effects these two. For the orbital rotation part, we show explicitly how the five-momentum and the rotation generators involving the compactified direction combine to produce the four-momentimi. For the internal part, the scenario is the same as the IW group contraction process. We end this chapter by constructing the CSR in four-dimension and the representation wavefunction of the Poincare group. 5.1 KK Reduction of the Poincare Algebra Kaluza and Klein [18, 19] reduced one dimension by putting it on to a circle of finite radius. We apply it to the five-dimensional Poincare algebra, by putting the third direction on to a circle of radius R, = + 2ttR The generators act on functions of the form ^x) = ^$„(j;'^)e''"^'/' (5.1) n where /x = -|-,-,l,2 and n is the mode number. The momentum along the third direction is quantized, = 1(5.2) This changes the light-cone Hamiltonian and two transverse boots in the following way, ^ 2^ = (^-^^ M- a;-p'^-l{a;^p-} + -L (T'^ + ^53a_p6^a6^ ^ (5 4) where a, 6 = 1,2 and M„ is the KK mass. The single state of mass M in five dimensions generates an equally spaced tower of states of mass M„. Looking at M"" we identify the light-cone translation 26

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27 vector, which satisfy the algebra f r + ^5^" (5.5) ^^ab -fej ^ i(<5"=f''-<5'"=f") (5.6) [f" = iMlS"^ (5.7) The generators 5^^ and form the 50(3) Hght-cone little group in four dimensions. Analyzing the mass term, we find that Mn increases with the mode number and gives the well-known infinite KK tower of masses starting at Mo = M. These are the general expression for the KK reduction of the five dimensional representations. For the KK reduction of the regular massless representation in five dimensions, we simply substitute M -0 and T' = 0, i = 1,2,3. The resulting Light-cone Hamiltonian, the transverse boosts and the light-cone translations become P = — 2^T = (5-8) = x'p''-'^{x\p-} + ~{^'^S^''-p^S''^) (5.9) f = ^53 (5.10) where now M„ = The little group is formed by the generators and and the satisfy the algebra in Eqs.(5.6-5.7). The remaining Lorentz generators of the four-dimensionsil group, M+~, M^^ and M"*"" for a = 1,2 are not changed. Note that when evaluated at = 2nR, the generators that rotate into the third direction become (for large R) like the momenta — = -27VP+ In the following two sections, we show in details how the contraction of the orbital and internal parts lead to the CSR in for dimensions. 5.2 Contraction of the Orbital Rotations At the group level, the contraction of the orbital rotations results in shifting the parameters for translations. This has been studied in detail by E. Wigner, Y. Kim and others in [20, 21]. We

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28 consider their construction for five dimensions with periodic boundary condition and contract it to four dimensions. In five dimensions, there are five translational and ten rotational parameters. We choose to use usual spacetime coordinates rather than light-cone coordinates (only for this section) We want to contract with respect to the subgroup which leaves the third direction invariant. Let us consider the boost between t and axes. Using = 2i{R, the transformation equation can be written as I A t' ~ 27ri?sinh/3 cosh/3 to 0 0 1 ^ cosh/5 ^ l\ \ J (5.11) where sinh p = v and cosh /? — \/T+ v^, with v being the velocity along the third direction. Since there is no velocity along the compactified direction, we need to take the limit u — 0 as i? — > oo such that Rv remains fixed. This gives (5.12) where Ot = 2'nRv is a constant and to is an arbitrary initial parameter. That is, the parameter for translation along time is shifted by at as a result of contraction of the t — x^ rotation matrix. Let us now consider the rotation between and axes for i = 1,2,4. The transformation equation for rotation can be written as ( x^'\ \ 1 / v coscp' 27ri2sin(/>' at \ 0 cos ' 0 a3 2irR X1 (5.13) V 1 I where is the rotation angle. Since after the compactification, there is no rotation on the i 3 plane, we need to take the limit 0' -> 0 as -+ oo such that remains constant. This gives x' = a;' -IGj;. 4Oi (5.14) where ax, = 27ri?0' is a constant and ttj is an arbitrary initial parameter. Erom the above, we see that the rotations between the invariant direction and other directions of the original group become the translations of the contracted group which is a generic feature. The generators of the contracted

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29 group can be read off using new shifted parameters. As for example, consider the rotation on the 3-1 plane. The product of the group elements for translation along first direction and the rotation on 3 — 1 plane is which is again a translation along the first direction, but with a new parameter (ai +aa;i). Therefore, using standard method we can write down the difi'erential form for with {a\ + a^i) being the translational parameter along the first direction and similarly for any other directions. The Ught-cone forms of the above results are straightforward. We find, x'^ — x^ + a^+a^ and a;'' = + Oj + a^. ; i = l,2 (5.15) where = •^(ataj;4) and — -^{to 04). Under boost along the forth direction, transform as x^' e^:c (5.16) where A is the rapidity parameter and is given by with Vi being the velocity along the forth direction. The transverse directions remain unchanged. Therefore, in the boosted frame, the orbital rotation generators after contraction can be written as L"*"" = —x'p'^ = — = x'p^-xY, = x-p^ \{x\p-} where i — 1, 2 and x"*" = 0 is used because of mass-shell condition. This is exactly the same form of the Poincare algebra in four dimensions for spinless particles Dirac obtained while introducing the front form dynamics [11]. 5.3 Contraction of the Internal Part and the CSR In this section, we apply contraction to the internal part. The generators f and satisfy the algebra (for R-* oo), [5"* f^^] = i(^=f*-5*<^2^) and [f" f*] = 0 (5.17) where T= ^5^ and a,6,c = 1,2. Let the internal rotations are parameterized by three pairs of canonical coordinates and their conjugate momenta which satisfy the following commutation

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30 relations, [ri\ T^ ] i,i = 1,2,3 = (a,3) Unlike the orbital case, we do not put the third coordinate 77^ on to a circle. Instead we assume 77^ is very large and so its conjugate momentum tt^ is very small. As i? — oc, r?^ also goes to infinity such that ^ is well defined. Therefore, in the infinite radius limit, the terms proportional to tt^ can be 3 ignored, and the terms proportional to rj^ becomes a function of y ~ ^ Note that being a ratio of the internal and external coordinates, the parameter y bears the signatin-e of the extra dimension. This is the difference between the contraction process introduced in Sec.4.2 and the internal part in this section. Therefore, in the contraction limit, we can write, f yn^ (5.18) where y = ^ The light-cone vector differs by a factor of y from the Euclidean vector 5' defined by Wigner (see Sec.4.1). We now follow the same procedure as in Sec.4.1. The Casimir of the 50(3) little group is, Divide by and taking R —^ (x while keeping m fixed, we find. 0 ; if n = 0. 0 ; if n 0, j = fixed, 00. (5.20) 52 ; if n 7^ 0, j, 00, ^ = = = fixed. The first two clearly belong to the regular massless representation, and the last one is the CSR in four dimensions. Notice that for zero mode, the reprasentation is always regular massless. Clearly only the double limit yields the CSR as expected. Combining Eqs.(5. 18,5.20), we find, (f ")2 = 2/2(^a-)2 ^ ^2 ^ (5 21) which in cylindrical coordinates gives the differential equation, The regular solution is the well-known Bessel function, $(p,^,2/) = 53iV„J„(H/>/j/)e'"^r (5.23)

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31 In their paper [9], IW obtained the following solution, ^IW{P,0) Y.^rnS'"'' Jm{Sp) (5.24) m At 2/ = 1 these two solutions become identical. Therefore the complete Poincare wavefunction can be written as ^x-,p\p,e,y) = 5];iV„e-'(-"P^-=^'')e'-''j^(Hp/y). (5.25) m For a given y, there are only two types of CSR: one bosonic and one fermionic corresponding to whether m is integer or half-odd integers respectively.

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CHAPTER 6 CONTRCATION OF THE MASSIVE REPRESENTATION IN FIVE DIMENSIONS In this chapter, we consider the contraction of the five dimensional massive representation based on the paper by Khan and Ramond [22]. The little group is S0{4). Instead of realizing linearly, we choose to construct the generators using three pairs of canonical coordinates and the conjugate momenta, which is a particular non-linear realization of the 50(4) httle group. This non-linear realization allows us to use the Majorana infinite component wave theory [23], introduced long ago. It is amusing that the 50(4) symmetry properties of the Majorana model are exactly same as those of the Bohr's non-relativistic hydrogen atom. In the following sections, we show how to construct the non-linear reahzation, identify the generators consistent with the five dimensional Poincare generators in Dirac-Bacry-Chang representation, explain the contraction procedure and finally obtain the representation wavefunction. We also compare our result with Wigner's form of the GSR in four dimensions. The linear realization of the generators of the 50(4) httle group in five dimensions requires four internal canonical coordinates and their conjugate momenta which satisfy the commutation relations. To realize non-Unearly, we express the fourth coordinate and its conjugate momentum in terms of the remaining variables as 6.1 Non-linear Realization and the Majorana Theory [ri' TT^] = i5'^ 2 = 1,2,3. and [t?"* tt'*] = i (6.1) (6.2) where M is the mass, /.t is an internal parameter with the dimension of mass and r) — \Jrfrf. Using these, the generator 5^' can be expressed as 5' '4t (6.3) 32

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33 Using e'^'^S'', we find, MS^' = eyV5'=--^-^-i7r' (6.4) V which is exactly the same form as the Laplace-Runge-Lenz(LRL) vector and it satisfies the following commutation relations, [MS^' MS*^] = iM^S'^ (6.5) [5'^,M5'"=] i{5'^MS^^ -5^''MS*') (6.6) This allows us to relate the mass M with the Bohr Hamiltonian, M = v/^^, (6.7) where H, the Bohr Hamiltonian, is given by (6.8) So the parameter is the mass of the electron with the electric charge set to one. On the other hand, the generators of the massive little group in five dimensions satisfy the commutation relations, [S'^ ,T^] = i(S'''T^ S^^T') (6.9) [T\P] iM^S'^ (6.10) The commutators in Eqs.(6. 5-6.6,6. 9-6. 10) suggest to identify the light-cone vector T' with the LRL vector M5^'. Therefore we can use the mathematics of the hydrogen atom to find the spectrum of the light-cone Uttle group. In this nonlinear realization, the light-cone Hamiltonian and the boost generators become = x-p'l{x' P} + (r p^S'^) (6.12) Since 50(4) ~ 50+ (3) x 50_(3), the commuting generators of 50(3) group can be identified with the combinations 2 V v^^2jni Since S'T = T'S' = 0

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34 the two Casimir eigenvalues of the 50(4) are same which is given by 1 ^5fc5fe_^ j ^ ^.(^+ 1) ^ (6.13) 4 V 2^// where j is the eigenvalue of the 50(3) angular momentum algebra. Therefore the eigenvalue of the Bohr Hamiltonian is (2j + l)2 and so the mass M can be expressed in terms of j, So at each mass level, the states assemble in a representation of S0(4) and generate the spectrum of the infinite component Majorana theory. We now apply contraction to this model. Since -50(4) D 50(3) D 50(2), we need to contract at least twice to obtain the E{2) representation of the little group in four dimensions. There are two ways to proceed: 1. We contract, staying in five dimensions first, and then use KK reduction and contract to four dimensions, i.e., 50(4) ls^_^ ^(3) 50(3) — ^ E{2) contraction reduction contraction 2. We apply the KK reduction first and then contract, i.e., 50(4) 50(3) E{2) reduction contraction In the following two sections, we find the conditions to obtain the CSR in four dimensions for these two cases. 6.1.1 50(4) £(3) ^ 50(3) - E{2) The states are eigenstates of the diagonal subgroup which is the angular momentimi, and given by the sum of 50+ (3) and 50_(3) groups of 50(4), S'S' = s{s + 1) (6.15) where s ^ 0, 1, 2, • • • 2j. Therefore each mass level is (2s + l)-fold degenerate. The contraction scenario is very similar to the IW contraction process 50(3) E{2) where m is kept fixed while j goes to mfinity. In this case, the contraction process for 50(4) E{3) is to take j, the quantum number for 50(4), to infinity while the quantum number s for 50(3) remains finite. Since M, and

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35 j are inversely related, the j -> oo limit leads to zero mass and commuting light-cone vectors, Mi = ^ 0, and so fr',r'=] 0, (6.16) and thus the generator 5'^ and T' form the E(3) algebra. Note that this contraction process do not change the space-time dimensions. This is the CSR in five dimensions. To find the Casimir of the £(3) group, we multiply the 50(4) Casimir by a contraction parameter e, (65%5*) + ^rr = Ae^j{j + l). (6.17) As e ^ 0 and j — oo, such that ej remains fixed, we get rr = (6.18) That is, the length of the translation vector is fixed, and this is a Casimir of the JS(3) group. The other Casimir is given by the zero form, W = tijkTS^^ = TS' = 0 (6.19) Comparing to Eq.(2.10) in Sec.2.2.2, we readily see that these are the Poincare Casimirs and the remaining Casimir is of course the mass squared which is zero. These Casimirs uniquely determine the representation in five dimensions. The above analysis also indicates that we could have chosen /U or l/j" as the contraction parameter, because both of these lead the mass operator to zero. However, if e = /i, then following Eq.(6.18), we find null light-cone vector which yields the regular massless representation, not the CSR. The other choice 1/j does not change the above result at all. Therefore, 1/j is actually the contraction parameter. After the first contraction, the light-cone hamiltonian reduces to P~ = |^ and the boost generators do not change at all. The KK reduction changes these two generators to ^ 2p+ (6-20) M- = x-p-i{x",p-} + ^ (f"-p''5"'') (6.21) where a, 6 = 1, 2, and M„ = ^. The new light-cone vector is now r + -^53 (6.22)

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36 The generators f "/M„ and 5^^ form the massive 50(3) httle group. Let j be the eigenvalue of the KK reduced SO (3) algebra. It is now quite clear by now that to obtain a CSR in four dimension, we must take the limits, j — > oo and oo such that the ratio ^ remains finite and uonzero, fafa ^ ^ 62 ^ finite (6.23) This is the Casimir of the Uttle group in four dimensions which together with the zero mass-squared uniquely label the representation. We find the representation wavefunction in Sec.6.2. 6.1.2 50(4) ^ 50(3) ^ E{2) This sequence is a two-step process to obtain the light-cone little group £'(2) in four dimensions from the 50(4) little group in five dimensions: first we compactify along the third space (orbital) direction by making it periodic. The light-cone Hamiltonian becomes P= ^(pV + m^), for a = 1 2, with where we used = n/i? (momentum along the third direction). For finite j and R, there are no massless states, and the massive little group is 50(3) in four dimensions, generated at each mass level by 5^ and A_iT-^^S^-), a = 1,2. deduced by looking at M"". The spectrum is made up of massive pairticles with spin which depends on j and n through the quadratic Casimir Eq.(6.24) clearly shows that a necessary condition for the mass operator to vanish is to take the double singular limit where both j and R tend to infinity. But this is not enough because the following condition must satisfy to yield the CSR. This requires, m addition to the double hmit, the ratio ^ to remain finite and nonzero. To see it expUcitly, divide Eq.(6.25) by R^ and rearranging, lv(2i-M)Vi?2+" ; U2 ^2 I'

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37 where we used — m. The double limit certainly does not give any nonzero value, but the following triple limit j j' j,j',R~^ oo such that — — remain finite (non-zero) R R does give a finite value. Therefore, in this triple limit, we find, (^"'^ = [ww*"')^^' ^ • which is the Casirair in foiu dimensions. Together with the mass squared (which is zero) it uniquely labels the representation. This is what we expected, because the generator 5^^ was not aflPected by any of the reduction and/or contraction processes. The contraction only affected the quadratic Casimir eigenvalue which sets the range of quadratic Casimir of its maximal subgroup. The above analysis suggest how to identify the contraction parameter and the required singular limits to obtain any CSR: look at the mass operator and find what limits provide zero mass and if these limits keep the light-cone vector non-zero and finite. Unlike the previous section, the contraction parameter has a physical meaning which is the inverse KK radius. 6.2 Representation of the Wavefunction Since we have the form of the T's in terms of the internal variables, we can explicitly find the representation of the wave function for the light-cone vectors T. First note that in T" the contraction parameter j is not exphcitly present. Therefore if we apply the contraction it will only include the R ^ oo limit, not the j oo. As a result we will not obtain commuting vectors which is required to obtain the CSR. Since the contraction parameter j appears through the Hamiltonian when it acts on the wavefunction, we rewrite T* in terms of the Hamiltonian as in the following, = rf (2fxH + (7, TT i)7r' where the Hamiltonian is given in Eq.(6.8). So the fight-cone vector, T", can be expressed as f = (2fiv''H +—-^'y'^TT^ ^ [ ^^ H + (2/ + 2i r? • 7r)7r'' (6.27) where y nrj^/R. The first term is dependent on the contraction parameters whereas the second term is independent and also both terms are separately hermitian. Since H ~ 0(1/ j'^) and 1/77 ~ 0{1/R), the first term vanishes as j,R 00. It should be noted that if we evaluate the commutator in the first term in Eq.(6.27) and then consider the contraction, the resulting light-cone vector becomes non-hermitian and the whole analysis becomes physically meaningless. It is not well

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38 understood why this happens, but to maintain the hermiticity, we have to consider the contraction without computing the commutator. Using d T) T! — rfix'^ — ly dy and dropping the first term, the light-cone vector becomes rpa 42i + iy^ rf'TT*' ] tt" dy (6.28) It can be easily checked that the above form satisfies the £(2) algebra, [f\ = 0 and f"] i(5"^f''-5''"f") which is required to obtain any CSR in four dimensions. We now find the representation of the wavefimction for this £^(2) algebra. The square of the light-cone vector is, d {ri^Try irf-K^ ~1iy + 5 42y On a wavefunction of the form, 'dyj (2iy'-6y)-f-{y' + 6iy-6) dy this gives the following form of the differential equation, (r,V)^ (5 + 2,^) (y^^ + 6, A + 6))] (.^) = (6.29) where H = E or H' corresponding to Cases 1 and 2 respectively. To solve the above equation, let (6.30) when it acts on (i>{T]'^, y), that is the representation of the ry dependent part of the wave function is the Bessel function, namely, Hv'.y) = Jmmy)p)e'"''F{y) where F{y) is a real function. In polar coordinates {rj^ — pcosO and 77^ = psinO), we use, and rewrite Eq.(6.29) as dp2 d2 62/3+6 dy J dp \ dy'^ dy (6.31)

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39 Making use of the following identities, j-Jmmp) = 5 j^^i) (6.32) dp L fjmimp) = ^^(Jm-l-Jm+l) (6.33) dy 2 dy and after stredghtforward algebra, the left-hand side of Eq.(6.31) can be expressed as a linear combination of Bessel function of different orders, \-r+ — 2 — + — 4 ) r'"-' + +py^n^Il!F' + ^^^-^ 1 (Jm-l Jm+l) + p2yn3n'F + 2y2n' V + iy'^mi'T + 4t/2nn'p' where the 'prime' denotes derivative with respect to the argument which are suppressed for simplicity. By matching the coefficients of Bessel functions of different order with the right-hand side of Eq.(6.31), we get three constraints. Equating the coefficient of Jm2 to zero gives the first constraint, which is satisfied if The solution to this equation is given by n(2/) = 5 (6.35) y where IIo is a constant to be determined by the boundary condition. The constraint obtained by equating the coefficients of Jmi provides no new result. Finally the remaining constraint is from the coefficients of Jm which gives a differential equation for F{y], cPF 2dF ^ ^ dy^ y dy ^

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where we used Ii{y) = Ho/y and its derivatives. The solution is FM = l(Asi„(|)+Bcos(|)), where A and B are constants. Since F{y) is regular at y = 0, we set B = 0. Therefore the wave function can be written as ^ipAy) = (y-(1^)) ^'""'J(^^) (6.37) where Nm is the overall normalization constant. In their paper [9], IW found the following wavefunction for the E{2) algebra, ^iw{p,e) Y^Ame'^^'JmiEp) (6.38) m where E^, the square of the PauU-Lubanski vector, is the second Casimir of the Poincjire group, and labels the CSR in four-dimensions. The Euclidean vector is linear in tt" and as a result its length appeared as a scale factor in the Bessel function. In four dimensions there are only two type of CSRs, fermionic and bosonic types corresponding to haJf-odd and integer values of m. The amplitude is also constant. Although similar in form, there are differences between Eq.(6.37) and the IW form in Eq.(6.38). In our case, the internal momentum (Eq.(6.35)) and amplitude are not constant, but functions of y which is the ratio of the internal and external coordinates. It is due to the fact that the Euclidean hght-cone vector T is not linear in tt" (in addition to the y dependence). Moreover, unlike IW case, the length of the light-cone vector does not appear as a scale factor in the Bessel fimction, even though it is the CSR. To find how our CSR is related to IW's CSR, we must find a relation between II and E. Let us assume U = E — Uo/y. The only solution we get, following Eq.(6.34), is E = 0 which corresponds to the regular massless representation, not the CSR. Therefore 11 ^ E is the only possibility and these are quite new. On the other hand, we may assume that, instead of n, IIo = E. Substituting this into Eq.(6.37) and settmg y = 1, the solution becomes exactly the same as that of Wigner's CSR apart from the overall phase factor which has no physical effect (the overall constant factor can be absorbed into the normalization factor). For any other values of y, we have a different kind of CSR because of the non-equality between the scale factor and length of the EucUdean vector. There is no physical reason for IIo and S to be equal and y = 1, but this is the only condition to obtain IW's result.

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41 Finally, the raising and lowering operators are defined as f = f 1 if 2 In polar coordinates, these become which acts on (l>{p,6,y). It is quite obvious that the states T'^\ > and T"\ > have helicities (m + 1) and (m — 1) respectively. The remaining generator T^/R becomes T = ('"'^ under contraction. The complete Poincare wavefunction is We do not understand what is the physical interpretation of the parameter y. But being a ratio of the internal and external coordinates, definitely it relates the internal structure to the external. This may help to understand better the spectrum of the higher dimensional theory.

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CHAPTER 7 CONCLUSION In this dissertation, We have studied two things: first, how the CSRs and the states are characterized in any dimensions with and without supersymmetry and, second, how these representations can be obtained from larger groups by group contraction. We have shown that in any dimensions, the CSR is characterized by the length of the light-cone vector and the Dynkin label of the short little group which leaves the light-cone vector invariant. We have explicitly shown that, unlike in four dimensions, there are infinitely many CSRs of both bosonic and fermionic types in five dimensions. The states require one more label Cld-3, the sohd angle in d dimensions, which specifies the orientation of the light-cone vector. In the supersymmetric case, we have considered the Ught-cone vector to be non-nilpotent and nilpotent. In the case of non-nilpotent light-cone vector, it commutes with supercharge. The supermultiplet is like ordinary (zero length light-cone vector) supermultiplet, except that each bosonic and fermionic partner is actually the CSR. In five dimensions, the supermultiplet is like N = 2 CSR supermultiplet in four dimensions. The spinorial generators do not have to change their forms. When the Ught-cone vector is nilpotent, however, we obtain a finite dimensional representation, because the higher order of T* except is zero. To make it work we have to introduce more Grassmann parameters, and hence there is no A'^ — 1 supersymmetry if T' is nilpotent. To get both T' and nonzero, we need two sets of Grassmann parameters. Introducing more Grassmann parameters will give rise to higher N supersjTumetry. The nilpotent construction remedied the infinite heat capacity problem of Wigner, but at the expense of introducing the central charge even though the representation is massless. We have shown this in four and ten dimensions. The nilpotent constructions do not generalize to odd dimensions. To obtain CSRs from higher dimensions, we need group contraction. First we have shown that the IW group contraction is equivalent to imposing the periodic boundary condition and taking the singulax limits. The different between our and Wigner s formalism is in the interpretation of the contraction parameter. In our case, it is the inverse KK radius. We have applied this technique to both five dimensional regular massless and massive representations. 42

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43 In the regular massless case, we have found that the double limit, j — > oo and R—>^oo, yields the CSR if the ratio ^ is nonzero finite and for non-zero mode. For zero mode, the representation is always regular massless in four dimensions. The representation wavefunction is the Bessel function, and the scale factor now includes a parameter y which is a ratio of the internal and external coordinates. This bears the signature of the higher dimension. In the massive case, we use the infinite component Majorana theory as our model which can be thought of the non-linear realization of the homogeneous 50(4) little group in internal space. We have found that a simple double limit, like in the regular massless case, does not saeld any CSR even though the mass term vanishes. We need a triple limit to obtain any CSR in four dimensions. The representation wave fimction is the Bessel function, but the scale factor does not include the length of the light-cone vector. In fact, the amplitude and scale factor are implicit functions of the parameter y. This is the key difference with Wigner's solution. At 1/ = 1 and if Hq = E, our solution and that of Wigner become identical, even though there is no physical reason for these conditions. For any other value of y, the solutions axe quite new. Our analysis of the contraction of the five dimensional algebra suggests how to identify the contraction parameter correctly which will lead to the Euclidean algebra, essential for the existence of the CSR. We need to look at the mass term first and find the parameters which will make the mass term zero, and then look at the quadratic Casimir of the non-contracted group and find if this parameter leads to non-zero length; if not it is not the CSR. This procedure obviously generalizes to higher dimensions.

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APPENDIX A CSR AND THE CONFORMAL GROUP The conformal group is one of the most familiar group which contain the Poincare group by embedding. In d-dimension, the SO{d, 2) Conformal group include the Dilatation and Conformal translation generators, in addition to the Poincare generators. It has been known [7, 8] for long time that only the regular massless representation of the Poincare group has imique Conformal extension. But there is no elegant proof for that. In this appendix, we proof this in light-cone language. We express the generators of the Conformal algebra using the same variables as for the Poincare algebra. We add to the Poincare generators, the dilatation generator D = ^(x-p + p-x) (A.l) for the scale transformation and the special conformal transformations generated by = 2x^M'"' + xV "^x^D x^f^ + Ix^S'"'' (A.2) where = 0, 1, • • • (d 1), and we used, M*"" = x^p" x'^p" + S>"' (A.3) In addition to the commutation relations of the Poincare algebra, these two generators satisfy the following commutation relations, [M^" D]=0, [D,p^] = ip^ [D K''] = -iK'^ [M"^ K'']^i (ry'^^/r-^ ~ ri'^'^K'^) , K"] = -2i {ri''''D M^"") To obtain the light-cone forms of the Dilatation and the Conformal translation generators, we first express the generators in light-cone coordinates and apply boost along the {d l)-direction, D = ^(-{^^P+}-{:^+,p-}-^{x^p'}) (A.4) K-^ = -2x+M+-f xV + 2x'M+' (A.5) K= 2x-M+' + 2x'M-' + zV (A.6) K' = 2x-M+' + 2x+M-'+ 2x^M'^ + xV 44

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45 where i, j = 1, • • (d 2). Using the Ught-cone forms of the Poincare generators from Eqs.(2.5,2.7) and setting x+ = 0, K and K' can be written as = 2x+ x'f) x'x'p+ ^ {x'T x'p>S'^) K' = 2x' (-a;"p+ + x'p') x^x^p' + 2x^S'^ To check if the above forms satisfy the algebra, we calculate the commutators [P~ K^] and \p\ K~]. Substituting the light-cone forms, we calculate [P-,K'] = 2i(^x-p'-^{x\ P-}-^S'^^ (A.7) (A.8) [K-,p'] = 2i(^x-p'-\{x\P-} + ^{r-piS'^)Y On the other hand these two commutators are given by [P, K'] [K, p'] = 2iM-\ (A.9) where M"* is given in Eq.(2.7). So comparing Eqs.(A.7-A.9), it is clear that we must set — 0, to satisfy the Conformal algebra. Therefore, the light-cone forms of the conformal generators are K+ = -x'x'p^ (A.10) K' = 2x'D-x^x^p' + 2x^S'^ (A.11) K =r. 2x-D-x'x'p-^xyS^\ (A.12) p+ where now the dilatation generator is given by D = ^-{-{x-,p+] + {x\p'}) (A.13) Since 0, it imphes that this is the regular massless representation, not the CSR. Therefore, to obtain the CSR, we must use group contraction which produce the EucUdean group, an essential ingredient for the CSR to exist.

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APPENDIX B ANGULAR MOMENTUM IN CYLINDRICAL COORDINATES WITH PERIODIC BOUNDARY CONDITION The cylindrical form of the angular momentum operator is given in Eqs.(4.8-4.10) which acts on the wavefunction of the form as in Eq.(4.6). When these operators act on the wavefunction, we can replace ^ by ySubstituting this, the squares of 5^^ and 5'^ become iS 23\2 sm 52 cos'^e d 2 sin 61 cos ^ psiri^d dp' + p dp + dddp 2p sm^e in d 2 sin (9 cos 6* 5 cos^ 6 8'^ + — z Rdp 2 sin 6 cos 6in d z 'Rde p1 in ~z'r 89 p^ sin^ ( 9^2 R? z^ 8p sin ^ cos 0 8 ^2 -QQ (B.l) and also (5^2)2 52 sin^e 8 2sin6>cos6' 8"^ p cos Q d P dp de8p 2pcos^e in 8 2sin6>cos6' 6 sin^ 6 8"^ ^ T — 7^ + z R 8p 2 sin 6 cos 0 in 8 ~z 'RdO 86 p^ 89^ + 2 8p sin ^ cos ^ 8 +z "Se^ Therefore 52 = (5*^)2 = 1 in p2 (.Qg2 0 ^2 ? R? 8^ n dp^ \p p 2inp\ 8 7^ 7r) Yp 2in 89'^ p^n^ zR z^R'^ (B.2) (B.3) (B.4) Let s2 be the eigenvalue of 5^. The eigenvalue equation now becomes 0^ (I + dp p 2inp\ 8 7 ~1r) dp "2 2m 1 i. 8^ 89^ s +-7; zR z'^R^ (B.5) Using separation of variables, we write ^n{p,9) = 7ln{p)e{9) (B.6) 46

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47 and substituting into the eigenvalue equation, the differential equation factors into a radial and an angular part, p2 2^ -1 1 d'^Tln fl P 2inp\ 1 dUn 7^,^ + ^2 zR ) Tin dp s2in p^ri^ 1 d^Q = 0 Let the solution of the angular peirt is of the form Q[6) = e im6 (B.7) (B.8) (B.9) up to a normalization constant and where m is the azimuthal quantum number. Substituting this, we obtain the differential equation of the radial coordinate only, dp2 np) + 2inzp\ d R J dp -nip) + -mM 1 + -T7 2inz ^, 1 n{p) p^/ i?2 R The most general solution of the above differential equation is 's/R{^^ 0 n{p) AM s' + l) m V4 inz \Rp^ BW 2yjAinz^R 2' 2z'^s/R y/Ris^ -vn? + l) m •.jAinz + Rp^\ 2y/Ainz + R 2 2z'^VR where M and W are Whittaker functions defined in terms of confluent hypergeometric functions as Ma,6(x) = e-i x'>+^ iFi{b-a+'^,2b + l;x) Wa,b{x) e-^^x^+iU{b-a+^,2b+l;x) 1 (B.IO) (B.ll) where iFi and U are the confluent Hypergeometric functions of first and second kind respectively. Their integral representations are given by T{b a)r(a) Jq U{a,b;x) = — / e-^'i"-Hl + <)''--idf W Jo (B.12) As a; goes to zero, the function U is divergent and iFi is finite. For physically rele\'ant solution, we need 7l{p) to be finite and well-defined at the origin and also as R —> oo. Therefore, we set B = 0. So the solution is given by the Whittaker function M.

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RI;^;FERENCES [I] E. P. Wigner. On unitary representations of the inhomogeneous lorentz group. Annals of Mathematics, 40:149-204, 1939. [2] V. Bargmann. Irreducible unitary representations of the lorentz group. Ann.Math., 48:568-640, 1947. [3] E. P. Wigner. International atomic energy agency. Theoretical Physics, 1963. [4] L. F. Abbott. Massless particles with continuous spin indices. Phys.Rev., D13:2291, 1976. [5] Kohji Hirata. Quantization of massless fields with continuous spin. Prog.Theor.Phys., 58:652, 1977. [6] Edward Witten and Steven Weinberg. Limits on massless particles. Phys.Lett, B96:59, 1980. [7] E. Angelopoulos, M. Flato, C. Pronsdal, and D. Sternheimer. Massless particles, conformal group and de sitter universe. Phys. Rev., D23:1278, 1981. [8] Tsu Yao. in Lectures in Theoretical Physics: De Sitter and Conformal Group and Their Applications, Edited by A. 0. Barut and W. E. Brittin. Colorado Associated University Press, Colorado, USA, 1971. [9] E. Inonii and E.P. Wigner. On the contraction of groups and their representations. Proc.Nat.Acad.Sci., 39:510-524, 1953. [10] I. E. Segal. A class of operator algebras which are determined by groups. Duke Math. J., 18:221, 1951. [II] P. A. M. Dirac. Forms of relativistic dynamics. Rev.Mod.Phys., 21:392, 1949. [12] H. Bacry and N. P. Chang. Kinematics at infinite momentum. Annals of Physics, 47:407, 1968. [13] L. Brink, A. Khan, P. Ramond, and X. Xiong. Continuous spin representations in poinacre and super-poincare groups. J.Math.Phys., 43:6279,2002. [14] Anders K. H. Bengtsson, Ingemar Bengtsson, and Laxs Brink. Cubic interaction terms for arbitrarily extended supermultiplets. Nucl. Phys., B227:41, 1983. [15] J. Polchinski. String theory, vol. 2: Superstring theory and beyond. Cambridge, UK: Univ. Pr. (1998) 531 p. [16] M. F. Sohnius. Introducing supersymmetry. Phys. Rept, 128:39-204, 1985. [17] Eugene J. Saletan. Contraction of lie groups. J. Math. Phys, 2:1, 1961. [18] Th. Kaluza. On the problem of unity in physics. Sitzungsber. Preuss. Akad. Wiss. Berlin, Math. Phys., Kl:966-972, 1921. [19] 0. Klein. Quantum theory and 5-dimeiisional theory of relativity. Z. Phys., 37:895-906, 1926. [20] Y. S. Kim and E. P. Wigner. Cylindrical group and massless particles. J. Math. Phys., 28:1175, 1987. 48

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49 [21] D. Han, Y. S. Kim, Marilyn E. Noz, and D. Son. Internal space-time symmetries of massive and massless particles. Am.J.Phys., 52:1037, 1984. [22] Abu M. Khan and Pierre Ramond. Continuous spin represtations from group contraction. Submitted to Journal of Mathematical Physics, hep-th/0410107, UFIFT-HEP-04-12, 2004. [23] E. Majorana. Teoria relativistica di particelle con momento intrinseco arbitrario (relativistic particle theory with arbitrary intrinsic moment). Nuovo Cimento, 9:335-344, 1932.

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BIOGRAPHICAL SKETCH I was born in Dhaka, Bangladesh. While in high school, I became more and more interested in science, especially in mathematics and physics. This motivated me to do my undergraduate studies in physics at the University of Dhaka, Bangladesh. In 1997, I went to the University of Cambridge. There I completed the Part III of the Mathematical Tripos in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge in 1998. At Cambridge, I became more interested in theoretical physics. In 1999, I emolled in the Ph.D. program of the Department of Physics at the University of Florida, and since 2001 I have studied in the High Energy Theory Group. 50

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I certify that I have read this study and that in my opinion it conforms to eicceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pierre Ramoiicff Chair Distinguished Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Khandker A. Muttalib Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. > I Charles B. Thorn Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. John Yelton Professor of Physics T certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Alexander Turully Professor of Mathematics

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This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 2004 Dean, Graduate School