COHERENT STATES OF THE POINCARE GROUP, RELATED FRAMES AND TRANSFORMS

Mohammed Rezaul Karim

A Thesis in

The Special Individualized Programme

Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

Concordia University Montreal, Quebec, Canada

October, 1996

@Mohammed Rezaul Karim, 1996

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ABSTRACT

Coherent States of the Poincar6 Group, Related Ekames and 'Itansforms

Mohammed Rezaul Karim, P h. D.

Concordia University, 1996

We construct here families of coherent states for the full Poincari group, for repre-

sentations correspouding to mass nt > 0 aud arbitrary integral or half-integral spin.

Each family of coherent states is defined by an affine section in the group and con-

stitutes a frame. The sections, in their turn, are determined by particular velocity

vector fidds, the latter always appearing iu dual pairs.

We discretize the coherent states of Poincar6 group in 1 -space and 1 -time dimensions

and show that they form a discrete frame, develop a transform, similar to a windowed

Fourier transform, which we call the relativistic windowed Fourier transform. We also

obtain a reco~~struction fornlula.

Finally, we perform numerical computations. We evaluate the discrete frame oper-

ator numerically and present it graphically for different sections and windows. We

also reconstruct some functious, compare reconstructed functioxis with the original

ones gapliically. We compare the reconstruction scheme of the relativistic windowed

Fourier transform with that of the standard windowed Fourier transform.

ACKNOWLEDGEMENTS

I would like to thank and express my gratitude to those who assisted me in the

successful completion of this thesis.

I am indebted to my principal supervisor, Prof. S. T. AIi, for his encouragement, con-

stant guidance a11 through the progress of the research and in the final preparation

of this thesis and also for partial support through his NSERC grants. Dr. Aii was

excellent supervisor and I am fortunate in having had his guidance.

I am also grateful to the two other members of my supervising committee, Prof. J.

Harnad and Prof. M. Frank. I took several courses with Prof. Harnad and learned

many things from him in Differential Ckometry, Hamiltonian Systems and Complex

Analysis which were helpful iu writing this thesis. I would like to thank Prof. Frank,

who taught me Classical Dyua~nics which was crucial for the understanding of the

subject matter of this thesis.

I would also like to thank Prof. hl . A. Malik for a fruitful djscussion on the periodiza-

tion of compactly supported functious and for his exicouragemeu t .

I ackuowledge the assistance of Prof. Y. P. Chaubey in programming in FORTRAN-87

and plotting using S-plus.

I would like to thank Prof. P. Gora and Prof. J. Seldin for ins ta lhg FORTRAN47

and Maple-V, and for their help during the computational work of this thesis.

I gratefully remember the valued advice and encouragement of Prof. T.N. Srivastava

and Prof, M. Ahmad (of UQAM).

I would like to thank all the secretaries of Department of Mathematics k Statistics,

Concordia University. In particular, I am very much thankful to Nancy Gravenor for

her cooperation during all these years.

Finally, 1 wish to express my special thanks to my wife and daughter, for their encour-

agement, moral support and the sacrifices they made during my Ph.D. programme.

Contents

.............................................................. List of Figures VUI

................................................................. Introduction - 1

1 Coherent States. Generalized Wavelet Tkansforms. Frames 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Coherent States 9

. . . . . . . . . . . . . . . . . 1.1 .1 Square IntegrableRepresentatious 12

1.2 Generalized Wavelet Transforxns . . . . . . . . . . . . . . . . . . . . . . 16

1.2.1 The Weyl-Heisenberg Group . . . . . . . . . . . . . . . . . . . . 18

. . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 The Affine Group 20

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Franles 22

2 The ,8 .Duality And Spin Coherent States 25

2.1 Notational Conve~ltions . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

. . . . . . . . . . . . . . . . 2.2 Phase Space For Massive. Spin-s Particles 35

2.3 The 8- Duality and Space-Like Sections . . . . . . . . . . . . . . . . . 40

2.4 Relativistic Frames and Coherent States . . . . . . . . . . . . . . . 54

3 A Relativistic Windowed Fourier Transform

. . . . . . . . . . . . . . . 3.1 The Mathematical Formalism and Notation 69

. . . . . . . . . . . . 3.2 Periodizatiou of Compactly Supported Functions 74

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Frame Operator 77

4 Numerical Computations Using The Relativistic Windowed Fourier

Transform 84

. . . . . . . . . . . . . . . . . . . . . . . . 4.1 Numerical Evaluatiolls of f 85

. . . . . . . . . . . . . . . . . . . . . . . . 4.2 Reconstruction of Functions 92

. . . . . . . . . . . 4.3 Comparison with the Windowed Fourier Transform 98

................................................................. Conclusion 102

.............................................*... .............. Bibliography 106

vii

List of Figures

Frame operator for Galilean section and triangular window. . . . . . . .

Frame operator for the Lorentziag section and triangular window. . . .

Frame operator for the sylnmetric section and triangular window. . . .

Frame operator for the Galilean section and the smooth window. . . . .

Frame operator for the Lorentzian section and the smooth window. . .

Frame operator for the symmetric section and the smooth window. . .

Reconstruction of Ct2 for the Galilean section and the triangular window.

Reconstructiou of e-~' for the Loreutzian section and the triangular

window. . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . .

Reconstruction of Ct2 for the symmetric section and the triangular

window, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Reconstruction of em'' for the Galilean section and the smooth window.

Reconstructiou of a discoutinuous function for the Galilean section and

the sniooth window. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Recoustruction of c d ( t ) e-"1Lh(t)2 using RWFT, WFT and smooth win-

dow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -101

Introduction

This thesis is based on two related notions- 'coherent states' and 'wavelet trans-

forms' and in the f is t chapter we describe them in order to set up the background.

'Coherent states' were first introduced by SchrGdinger [74] in 1926 as a system of

non-orthogonal wave functions to demoustrate the transition Lorn quantum mechan-

ics to classical mechanics. These states are just displaced forms of the ground state

of the harmonic oscillator and they minimize the uncertainty relation of Heisenberg,

(Aq Ap = f ). The completeness of these states was first noted by voo Neumam [60].

Siuce its discovery by Schriidinger, this idea did not get much attention until 1963,

when Glauber [39, 40) and later Klauder [SO, 511 rediscovered them. Glauber intro-

duced them in the context of quantum optics during the study of the coherence of

light beams and coined the term 'coherent states' (CS), because of their property of

maintaining the same pattern over long distances and time. Glauber also surveyed the

properties of CS, specially the expausion of arbitrary states iu terms of CS. The CS

introduced by Schr6dinger and Glauber are customarily known as the canonical co-

herent states. Klauder set forth the group theoretical foundation of CS and he stated

the resolution of the identity, a central property of coherent states, iu the present

form [52]. He also paranletrized the CS by pphke space points.

A connectiou betweeu the canonical CS and certain types of unitary irreducible group

representations was uoted by Klauder [SO, 511, Gilmore [37, 381 and Perelomov [62].

They observed iudepeudeilt Iy that the co~lstruction of the caxmnical CS is actually

associated to the unitary representatiou U of the Weyl-Heisenberg group GWH and

that the CS could be obtained by acting on the harmonic oscillator ground state with

U. Unlike Kiauder, Gillnore and Perelomov did not use the whole group GWH but the

coset space X = Gww /Z, wbere Z is the center of GWH (i.e., the set of all elements

commuting with every element of Gws) and the CS are indexed by the points of the

coset space X. In addition, the representation U is 'square integrable' in the sense

that for 1) E 'H = L2(R, dx),

wbere t/ is the invariaat measure on x.

Taking any locally compact goup G, rather than the Weyl-Heisenberg group GwH,

the followiug geueralization call be made: Let U be a unitary, irreducible representa-

tion of G in an abstract Hilbert space 'H, r ) E 'H and H, the isotropy group of r). (Let

G be a group acting 011 a set X and let x E X. The isotropy group of x, denoted by

G,, is the subgroup of all ele~tieuts of G leaving x fixed, G, = {g E G 1 g;c = x ) ) . If

X = GIH,, is the correspoudiug coset space the CS system associated with U is the

set of vectors

where 00 : X -, G is a Bore1 function defining a section in the group.

Although the method of Gilmore and Perelomov was very suitable for the extraction

of CS of most of the groups, it was soon discovered that their method was not capable

of handling at least some goups, for example, the Galilei and Poincark groups (9, 10,

1 1 , 12, 66, 671. A possible way to overcome the drawbacks of the Gilmore-Perelomov

method was suggested by Ali [2] and later developed in a series of papers by Ali,

Antoine and Gazeau [3,4,5,6, 7, 8, 141. Unlike in the Gilmore-Perelomov framework,

in the new method, CS are colltructed using a homogeneous space X = G I H , where

H is a suitable closed subgroup of G and H does not necessarily coincide with H, and

one needs to fiud a scctiotr, a : A' -, G. Then the CS are defined to be the vectors

It ought to be noted here that the iutroduction of sections in the above sense was

done implicitly by Prugovetki [GG, 121, and in the Gilmore-Perelomov frame-work the

sect ion ~ ( x ) appears as well, often in1 plici tly.

The history of wavelet anlysis is only about 20 years old, and yet in this short period

of time it has become very popular among mathematicians, physicists and engineers

alike. The reason behind this popularity is its synthetic nature, i.e., it is a synthesis of

a variety of fields, for instance, the Li ttlewood-Paley decomposition in mathematics,

coherent states expausion in quantum mechanics, sub-band coding in signal processing

etc. 'Wavelets' were first introduced by Morlet [59] iu 1982, as a convenient tool to an-

alyze seismic da t a Wavdets are actually coherent states of the a f h e group, satisfying

some admissibilty conditions. After successful applications in seismic study, Morlet

and Grossmanu [44] studied them extensively and developed a mathematical founda-

tion for wavelet theory. Then Meyer observed a connection between the method of

signal analysis and the techniques used in the study of singular integral operators. Af-

ter that, Daubechies, Grossmaun and Meyer [27] constructed a special type of frames,

generalizing the concept of a basis iu a Hilbert space. The next major breakthrough

was due to Mallat [55, 561 and Meyer [58] through their introduction of multiresolu-

tion a7talysis, and Daubechies' constructiou of families of orthouormal wavelets with

compact support [29, 30, 261 can be treated as a giant leap in the progress of wavelet

analysis. At the heart of wavelet analysis is the wavelet transform (WT) which is

defined in terms of wavelets. Because of the presence of a translation and a dila-

tion parameter, the WT is extrexliely efficient in reconstructing images/signals even

at points of discoutinuities. Nowadays wavelet analysis is a complete subject in its

own right and has applications in many different fields, for instance, signal analy-

sis [53], nunlerical analysis r21) and physics [20], neural uetworks [47], fractal image

and texture [49], teleco~nxllunication [75], etc., and the literature keeps growing .

Many of the techniques used in wavelet analysis have been around for quite a long

time but there the term wavelet was not used explicitly. Fourier analysis is a kind of

wavelet analysis in the broader sense of the term. The limitation of Fourier analysis

is that it offers either an all-time or au all-frequency description of a signal, nothing

in between. But for practical purposes, we need both. To overcome this situation,

Gabor [34] introduced the 'windowed Fourier transform' (WFT) (also knwon as 'short-

time Fourier transform' ). In the WFT a window function is chosen to localize the

signal and then the window is shifted repeatedly. The WFT and the wavelet transform

are two equivalent transforms, although they are also different. It is possible to switch

from one to another without losing any information [33].

Coherent states for the Poincar6 group e(l, 1)' in 1-space and 1-time dimesions

have beeu studied extensively in (6, 81 for unitary irreducible representations (UIR's)

corresponding to nlass trt > 0. These states are indexed by the points of the ho-

mogeneous space I' = p : ( ~ , 1)/T, where T is the subgroup of time translation. An

afine scction a : r -+ <(I , 1) has beeu defined iu order to construct these coherent

states. Several types of special sections are meutioned and for each section its dual

is also defined aud there exists a section which is dual to itself. The resultiug co-

herent states coustitute a fra.me and under certain specific situations t his frame can

be made a tight frame. Consequenly the: CS generate analogs of a, resolution of the

identity. The full PoiucarC group ?$(I, 3) was studied earlier and the corresponding

CS were coustructed in [12, 671 for UIR's correspouding to mass m > 0 and spin

s = 0,1,2,. . .. No idea of sectio~is was used there and in the context of ~ : ( l , 1) CS,

they were related to a particular section, cr = 00, the Galilean section. Construction

of CS of ~:(1,3) corresponding to 772 > O and spin-; representation was reported

iu [65]. These CS did not form a frame and thus led to no resolution of the identity.

Coherent states of the De Sitter and Poincarii groups were also studied for particular

sections in [35].

In chapter 2 we extend the results iu [6, 81 for ~ i ( l , l ) CS to any UIR of ~ : ( 1 , 3 ) , for

1 m > 0 and s = 0, 5, 1, $, 2,. . . . Jus t like in the case of ~ i ( l , I ) , each family of CS

is defined in terms of a particular affiue section, and it constitutes a frame. Under

certain specific conditions this frame can be made tight, thus leading to a resolution

of the identity. Siuce the base space of T$(1,3) is much larger than that of F$(l, I ) ,

the duality of the sections in the case of ?$(I, 3) is better understood. The duality of

sectious can be interpreted iu terms of Loreutz invariant fibratious of Minkowski space.

Each affine section can be characterized by a Cvector field u(p) = (uo(p) , u(p)), where

p is a relativistic 4-momento~n on the mass hyperboloid V ; and the Cvector field

u(p) maps V$ onto itself. Then the dual sectioti u*(p) which is also a 4-vector field

can be obtained by applying the Lorentz boost /\ to ~ ( p ) = ( u o ( p ) , -u(p)), pointwise

for all p. It should be mentioned here that the CS obtained in this thesis are similar

to the vector CS studied iu [69]. Vector coherent states are obtained from a set of

linearly indepeudent vectors of a representation space of an isotropy subgroup of a

goup G, and each state is written as a linear combination of these vectors. But the

situatiou we consider here is ~ ~ ~ u c h more general than that in (691 in the sense that the

subspace generated by the 'fiducial vector' in our case is not stable under the action

of any non-trivial subgroup of 7$(1,3).

In chapter3, we discretize the CS of e(l, 1) obtained in [6, 81 by periodizing a corn-

pactly supported window fullctiou and show that this discretized family of CS forms

a discrete frame. We calculate the frame operator for different sections and for var-

ious conditions on the window functiou. Under some specific situations, the operator

f' becomes a multiple of the identity operator and consequently the frame becomes

tight. We develop a transform, similar to the windowed Fourier transform, which

we call the relativistic windou~cd Fourier tnmnsfonn. We also obtain a reconstruction

fornula using the frame operator F and the relativistic windowed Fourier transform.

In chapter 4, we perform some numerical computations. We evaluate the operator T

obtained in chapter3 for different sactioxls and window functions. Using the recon-

struction formula of chapter 3 , we recoustruct a fuilction under various assumptions on

the sections and the window functiou, and compare the reconstructed function with

the original one both i~u~~lerically and graphically. I t is observed that for a smooth

window the reconstructioxl scheme does a better job than that for a non-smooth

wiudow. However different sections play more or less the same role under identical

conditions. For comparison, discretizing the CS of the Weyl-Heisenberg group, we

obtain the correspoudiug frame operator and the recollstructiou formula. We finish

this chapter by reconstructing a function using t h e reconstructiou formula obtained

in chapter 3 and that obtained using the Weyl-Heisenberg CS. Comparing these two

schemes we see that the recoustructed values are in close agreement.

In the Couclusiou, we briefly describe the results we obtained earlier and suggest

possible applications. We also indicate possible extensions of some of the results.

Chapter 1

Coherent States, Generalized Wavelet ~ransforms, Frames

In this chapter, we give an overview of 'coherent states', 'generalized wavelet trans-

forms' and 'frames' - 11otio11s central to this thesis. In the description of coherent

states, we place emphasis on the group-theoretical background, i.e. how coherent

states are related to group representations. By the term 'generalized wavelet' (GW)

transform, we mean here a specific type of a tra~~sform originating from the coherent

states of an arbitrary group. Here we shall give a brief description of: i) a Gabor

transform and ii) a urauclct trarrsfonn, both of which are examples of generalized

wavelet transforms; the Gabor transform originating from the coherent states of the

Weyl-Heiseuberg goup and the u~avclrt transform related to the affine group. Finally,

a t the end of this chapter, we give a short description of frames and some of their

properties.

1.1 Coherent States

We begin with the canonical coherent states. The canonical CS are an overcomplete

(i.e., the set remains complete upon removal of at least one vector) and non-ort hogonal

system of Hilbert space vectors. Starting with

the canouical CS f,,, are defined by

with 11 / I t 2 = 1 . The ca~otiical coherent states have many remarkable properties and

here we focus on some of them. Let us define the fonnal operator

where I ) ( f,, I is the projection operator on the state I I,,,); by ( f,, I we mean

Dirac's 'bra' vector and I f,,,) the 'ket' vector. The scalar product of a bra vector

( f,, I and a ket vector I h,,p ) will be written ( J q , I h,,p ). Then for $ E 31 = L2(R, dx) ,

and almost for all x we have

1 roo roo

(Using Fubini's theorem)

From (1.3) we can write

where I is the ideutity operator 011 7-1. This property is known as the 'resolution of

the identity'. Froxu (1.4) we observe immediately that the canonical coherent states

are: linearly dependent, i.e., ally canonical coherent state can be written as a linear

combinatiou of the rest. That is,

Define the kernel Ii : R2 x W2 -, @ by

Then

and K satisfies the properties

where for z € @, 5 indicates its complex conjugate. To check ( i i i ) , we note that

(Using Fubini's theorem)

A kernel with the properties (i) ,(ii) and (iii) is known as a reproducing kernel [15].

As we shall see later (Section 2.1) in a group theoretical context, for an arbitrary

f E 3-1 = L2(P, dz), f9,Jz) = e - y e'pZ f(s - p), Vz E I, are the coherent states of

Weyl-Heisen berg group.

1.1.1 Square Integrable Representations

Let G be a locally compact group, 'H a Hilbert space (over @) and g -, U ( g ) a strongly

continuous unitary irreducible representation (UIR) of C; in 'H. Let H c G be a closed

subgroup and

be the left coset space. The elenle~lts of X are deuoted by x , which are cosets gH,

g E G. Assulne that there exists an invariant lneasure 11 on X. Let a : X -, G

be a (global) measurable (Borel) sectiotr (a map which associates to each x E X , a

~ ( x ) E G such that the coset o ( x ) H is exactly x), F a positive operator on 'H with

finite rank n. Suppose that F has the diagoual representatiou

where

and denote by P, NL the projection operator and the subspace of 3-1 :

Using the operator F and the section a, define the positive operator valued function

F, : X -, t ( ~ ) + :

Definition 1.1 The representation U is said to be squarc intcgrablc m o d ( H , a) , if

there exists a positive operator F, of finite rank, and a bounded positive operator A,

on 3-1 with the bounded inverse, such that {X, F,, A,) is a reproducing triple, that is,

one bas

in the sense of weak couvergence. In this case we call F a rcsolt~tion generator and

the vectors

= F ~ U , u E N I (1.17)

admissible vectors mod(H, cr). We also say that the section a is admissible for the

representation I / . (We say a sequence of vectors jk E 7f (k = 1,2,3, . . .) converges

1 3

weakly to the vector / if limk,, ( fk I h ) = ( / I h ), for all h E 'H.)

For each u;, i = 1,2, . . . ,n , let 11' = F'$ui, and define

If (1.16) holds we call the set (1 -18) a family of covariant coherent states. Then the

'modified resolution of identity' is giveu by

We call the relation (1.19) a modified resolution of identity, because the operator A,

on the right hand side of it is not r~ecessarily au identity operator as in (1.4), but

rather a positive bounded operator with a bouxided inverse.

At this point, let us show how the carionical CS, Perelomov-CS and the vector CS

fit iuto the above definition of covariant CS (CCS). In the definition of CCS, if we

replace the izwariant subgroup H by the center of G, then CCS reduces to Perelomov

type of CS. Since the family of canonical CS is a special case of Perelomov CS,

the canonical CS also fits into the definition of CCS. Before going on to show how

the vector coherent states (VCS) are iu couformity with the definition of CCS we

briefly recall the co~istructioo of VCS. Let Go be a semisi~nple Lie group which has a

faithful representation and go be the correspoudiug Lie algebra. Let g be the complex

extension of go axid G be the corresponding Lie group. Let I(o be a compact semisimple

14

subgroup of Go with Lie algebra ko and k the corresponding extension of ko with I(

the corresponding Lie group. Then g can be decomposed as g = n+ + k + n-, where

n+ and n- are respectively the spaces of positive and negative roots. Let P be a

parabolic subgroup of G with Lie algebra p = n+ + k. Let U be a unitary irreducible

representation (UIR) of Go acting on the Hilbert space 'H and u be a UIR of KO acting

on H, C H. Let T be the extension of [I to G. Let e = { e i ) be a basis for n-. Then

an arbitrary vector in TL can be writtten as a e = Ci ziei, where zi are complex

numbers. Treatiilg z e as a representative of a coset space P exp(z e ) E GfP ,

z = { z i ] become t h e coordinates of GIP. If for 1 a ) E 71, and for any non-zero

x E n+, T ( x ) l a) is not in Xu, I b can be treated as an isotropy group that leavs

'H, invariant. If {ai) is an orthox~ormal basis for 'H,, the vector coherent states are

defined by

4(z) = C ( ai I T ( e x p ( = e)) 4 ) 10; ), for$ E 7-10 t

Clearly, 4 : GIK -, 'H, is a holomorpl~ic function. The group action is given by

What we observe here is that the constructio~i of vector coherent states is a kind of geu-

eralization of Perelomov's method, in the seuse that in both methods an isotropy sub-

group of the group in consideration is used. In Perelomov's case a 1 -dimensional pro-

jection operator F = la ) ( a I is used, whereas in the case of VCS an n-dimensional

projection operator F = C;=, la;) (ai I is used. Thus, if the closed subgroup H in

the CCS is an isotropy group that leaves Nu invariant, the CCS are exactly vector

CS.

1.2 Generalized Wavelet Transforms

In the previous section we gave a brief description of coherent states and their prop-

erties. Here we use cohereut states to define generalized wavelet transforms. For

example, we use the coherent states of the Weyl-Heisenberg group to define the Ca-

bor trausfortn and those of the afFmrt group for the wavelet transform. A signal ( e g .

music, speech etc.) evolves with time and its frequency changes with time. When

a signal is represented by its Fourier transfortn, i t gives information only regarding

the frequeucy of the signal a i d no information coucer~ling its time evolution. But

in practice we need both time- aud frequency-localization. In that sense the Fourier

transform is uot a useful way to represent a signal. Fortunately, there are at least two

types of transforms which have this desired property. They are i) Gabor Transform

(also kuowu as the windowed Fourier transforln or short-time Fourier transfor~n) and

ii) wauclct transform. Botl~ of these transforms are related to square integrable group

representations in the sense defined above.

The Gabor transform [34] of a signal f E L2(R) is defined by

where g E L2(B) is a fixed function, known a s a window-junction or mother wavelet.

In the literature (1.20) is known as a continuous Gabor transform. Signal analysts use

its discrete version, where t and w are discretized and written as t = do, w = mwo,

where m and n are integers, and wo, to > 0 are fixed. The discrete version of (1.20)

is given by

On the other hand, the continuous wavelet transform of a signal f E L2(R), for the

mother wavelet II, is defined by

where n (> 0), I E W. A possible discretizatiou of a and 1 is written as a = aom,

b = nbonom, where tn and tz are iutegers and a0 > 1 and bo > 0 are fixed. Then the

discretized version of wavelet transfor111 is given by

In both cases the mother wavelet $J must satisfy the admissibility condition:

The functions

and

define coherent states of the Weyl-Heisenberg group and of the afbe group respec-

tively, as we will now demonstrate.

1.2.1 The Weyl-Heisenberg Group

Let GWH be the Weyl-Heisenberg group, GwH = T x W x R, where T is the set of

complex numbers of modulus one. The group multiplication is given by

with identity elemex~t ( 1 ; 0,O) and inverse ( t ; q , P)-' = ( t - ' ; -q, - p ) .

A unitary irreducible representation ( I of GWH , actiug on the Hilbert space

is given by

where g = ( t ; q, p ) E GWH. The center of GWH is Z = (t; 0,O) and the left coset space

X is defined by

X = G w H / Z B2 (1.30)

An arbitrary element g E GWH has, according to (1.30), the following coset decom-

position:

g = ( t ;+1y) = ( l ; x , ~ ) ( t ; O l o )

If ( q , p ) E B2 are the global coordiuates of X, then since

( t ; 2, y) (I; (I? p ) = (1; I + (I, y + p ) (e'(qy-p) t o 1 , 0 0) 9

the action of GWH on XI is given by

and the invariant measure 0x1 S, under this action, is dq dp.

The sectiou 00 : X -+ GwH is defined by

Then the coherent states f,, of the Weyl-Heisenberg group are given by

1.2.2 The Affine Group

The affine g o u p is the set

and has the natural action z u a x + b on R. The group multiplication law is given

B

( a , b)(nr, I ' ) = (an', I + nb') (1.37)

with identity element (1, 0) and the illverse ( a , 6)-I = (d, -a-'B). This group

is not unimodular and the left Haar xneasure is a-2 dn db, the right Haar measure

a-' dcl db [43]. The unitary irreducible representation ( I of Gan, acting on the Hilbert

space 3C = L Z ( R , d x ) , is given by [l6, 17)

The square integrability of the represeutation [I and the admissibility of a vector

I/J E LZ(W, dx) in this case reduces to the condition:

It is counected to our definition of square integrability of a representation and ad-

missibility of a vector in the sense that here F = I 4 ) ( $ I (n = 1 ) is a multiple of

I-dimensional projection operator on 'H = L2(B, dx), and the trivial section a ( g ) = g

may be used, because the representation is square integrable with respect to the whole

group. The condition (1 .XI) cau also be written as

where 6 is the Fourier transform of +. From (1.40) we observe that if I&) # 0, the

integral does not converge, so it is necessary to assume $(0) = 0. That is, L

11 is admissible if &(0) = 0

i.e., the mean of 7,b is zero. Now we see that the coherent states of the affine group

G,/ are the vectors :

t,!l&) = (&z, ~ ) ? / J ) ( . Z )

that is,

with 11, an admissible vector.

1.3 Frames

The concept of a frame was first introduced by DuEn and Schaeffer [31] in 1952,

in the context of non-harmonic Fourier analysis and has also been reviewed in [77].

Here we will define it and give a brief description of some its properties, for later use,

specially in chapter 3. Let 'H be a separable Hilbert space (over @) and X be a locally

compact space, Y a regular Bore1 measure on X with support equal to X.

Definition 1.2 A set of vectors {rl$}z, in 7f is a framc if for all x E X the vectors

{ q i ) , i = 1, 2, . . . N, are linearly iudependent, and there exist two numbers A, B > 0

such that for all 4 E 'H one has

It is appropriate to uote here that the definition of frame given above is much more

general than the one given in [3l, 28, 48, 45, 25, 461. If N = 1 and X is a discrete

space with v a counting nleasura i.e., v(z) = Cz="=_, 6(2 - n) , where n is an integer, it

reduces to the usual definition of frame used in the literature and we call it a discrete

frame, otherwise, it will be called a contitiuous frame. The numbers A, B are called

the frame bounds. The fra~ne: is tight if A = B. The frame is ezact if it ceases to be

a frame whenever any single elemeut is deleted from the sequence. In general, the

set of vectors { v ~ ) ~ l is uot an orthogonal basis , even a tight frame may not be an

orthonomal basis but an ortltonomal basis is a tight frame with A = B = 1.

Let {4;, i = 1, 2,. . . m} be a discrete frame and q5 be any vector in 'H. Then an

operator S defined by

is known as the frame operator. It can be shown [31] that

i) S is a bounded linear operator with

A I 5 S 5 B I , (I is the identity operator)

ii) S is invertible with

iii) {S-14i} i s a frame, called the dun1 jramr or rrciprocal frame of { # i } -

which we call the rcco7rstructioa fotmula.

v) The decomposition of 4 E 7f is, in general, not unique in the sense that if we write

4 = Cgl ai4i, where ai = (dIS-*4;), then it is possible to f i ~ d a set of complex

numbers pi such that 4 = xz, Bi4i, where

Example 1.1 Let {e,L)r=l be ail orthononnal basis for 'H. Then

i) {el, el, ez, ez, e3, e ~ , . . .) is a tight inexact frame with bounds A = B = 2 (since

the eigenvalues of the frame operator are always 2), but is not an orthonormal basis;

a1 though it contains one.

ii) {el, e2/2, e3/3, . . .} is a complete orthogonal sequence; but not a frame (since,

in this case, the eigenvalues of the frame operator lie between 0 and 1 and are not

bounded away from 0 and consequently, the inverse of the frame operator is not

bounded).

iii) {2el, ez, e ~ , . . .) is a non-tight exact frame with bounds A = 1, B = 2 (since, the

miniinurn sigeuvalue of the frame operator is 1 and the maximum eigenvalue is 2).

h practice, the reconstructiou of a signal / image using the recoustructiou formula,

could be highly efficient, in the sense that the sun1 converges very rapidly and can be

truncated after a few terms, if IB/A - 1 I<< 1. It is a matter of fact that the frames used

in wavelet analysis (we mean both Gabor wavelets and affine wavelets) are obtained by

discretiziug a set of coherent states of the respective group. We have already shown in

the previous sectioii of this chapter that Gabor wavelets are associated to the coherent

states of the Weyl-Heisenberg group and affine wavelets or siinply wavelets with those

of the affine group.

Chapter 2

The P-Duality And Spin Coherent States

In the previous chapter, we gave a brief description of coherent states, along with

some examples. Here our aim is to cotlstruct the coliemut states of the Poincarg group

'P4(1,3), in 3-space and I-time dimensions, for a unitary irreducible representation

U&, correspouding to a positive lnass m and arbitrary spin s, by adopting the same

technique as used in extracting the coherent states of the Weyl-Heisenberg group in

the previous chapter. Then we will show that the resulting coherent states form a

frame, which we call a relativistic frame. In sectiou 2.1, we give a brief description

of Minkowski space and sotne xlotational conveutions, we collstruct cohereut states of

the full Poiucark group and discuss &duality respectively in section 2.2 and section

2.3. Finally, in section 2.4, we show that the coherent states of the full Poincare group

form frames aud analyse tba frames for various known sectious.

2.1 Notational Conventions

We denote the coordinates of a point of the four-dimensional space-time continuum

Rlq3, known as Minkowski space, by x p = (so, x), with

z0 = d , the time coordinate (2- 1)

2 x = (zl = z , t = y, z3 = z), the spatial coordinates (2-2)

In what follows we sliall assume f i = c = 1 . We use the Greek indices to denote

the compouents of four-vectors taking values 0, 1, 2, 3, and Roman indices to denote

the components of orditlary space vectors taking values 1, 2, 3 . We write covariant

vectors with subscripts and contravariant vectors with superscripts. Thus np is a

coutravariant vector and tlie corresponding covariaut vector a, is obt aiued by

where

is the space-time mctn'c tcnsor. (2.3) gives no = no n k = -ak, k = I, 2, 3. Here

the convention of suimniug over repeated indices is to be understood.

The Minkowski scalar product of two four-vectors up and V is defiued by

and the Miukowski norm of crp is

Because of the uegative signs in the metric tensor (2.4), the scalar product npn,

is no longer positive definite, it can be positive, uegative or zero. The vectors in

Minkowski space are classified into three categories, depeuding on whether the norm

of n@ is positive, negative or zero. The vector np is said to be spacelike if its norm

is negative; it is called light-likr if the I I O ~ I ~ I is zero; and it is called time-like, if the

uorm is positive.

An affine transforn~atioo in Mikowski space is defined by

with the A: satisfying

Here A: is the matrix of the Lorentz trausfor~natiou aud the vector a" represents a

simple translation of the space-time axes. (2.7) is known as an ir~hornogetreous Lorentz

transformation and if a' = 0, it is called a homogrncous Lorentr trrmfomation. The

homogeueous Lorentz transformation leaves the scalar product invariant

and forms a group kuown as the Lorentz group (we denote it by L).

Topologically the matrices of the Lorentz group L consist of four disconnected pieces.

Cr V From (2.8) we have gI4,A, A. = 900 = 1 wllicl~ implies

Then we have either

A: 2 1 07* A:< -1

Takiug the determinaut of both sides of (2.8) we get

det ( l \ ) = f 1 (2.12)

t

Thus we have the following deco~nposition of the Lorentz group into four pieces [72] :

L: : d e t ( A ) = + 1 , sign A: = +l . This one itself forms a goup, known as the

proper, orthochronous Lorentz group.

1 t- : det( l \ ) = - 1, sign /\: = + 1 . This piece has an element I, defined by

and is called space inversion operation.

1 L- : det (A) = -1, sign /\: = -1. This piece has an element I,, known as a time

reversal operation, is defined by

0 1 2 3 rtz = (-x ,x ,x ,x )

L: : det (A) = + I , sign A: = - 1. This one has an element ISt = I, I t .

From these four pieces we call build the following three subgroups of the Lorentz

group:

L? = L: u L! , the orthochronous Loreutz group (direction of time unchanged).

t+ = t: u t i , the proper Loreutz group.

to = L:UL~, the ort hochronous Lorentz group (spatial directions unchanged).

The inhomoge~eous Lorexltz tramfonnatious form a group, known as the Poincard

group. W e denote this group by ~ L ( 1 , 3 ) (in 1-time and 3-space dimensions) which

can be identified as T4 @ ~: (1 ,3 ) , where T4 cz W1,J is the group of space-time trans-

lations and @ denotes the semi-direct product. Since t: and SL(2, @) are physically

equivalent (as they are 2 - 1 homomorphic [70]) here we use SL(2, C), the universal

covering group of ti instead to include the half integral spins [76]. Here by 'universal

covering group' we mean: Let G be an admissible topological group and (G, r ) a

covering of the topological space G. By introducing a multiplication into G, it is pos-

sibletoshow that bitselfis a topological group and n : e -+ G is acontinuous

homomorphism with discrete kernel

which is a discrete iuvariant subgroup and G = G / N . In this case G' is called a

couering p u p of G. Given any admissible group G, there exists a simply connected

covering group of G. It is detsrmiued up to isomorphism and is known as the universal

covcring group of G [64]. We write [13]

Elements of ~ 4 ( 1 , 3 ) will be denoted by

(a, A ) , = (no, a) E W I ~ , A E SL(2, @)

The multiplication law is

(a, A) (d , A') = ( a + Aa', AA') (2.15)

where A E .~:(1,3) (the proper, orthochronous Lorentz group) is the Lorentz tran-

formation corresponding to A:

L

where: Tr[M] deuotes tlie trace of the matrix M,

and a (ol, 02, u3) are tlie Pauli matrices. We take the following specific realization

for the Pauli matrices:

Let

be the forward mass hyperboloid. It is noted that the invariant measure under

the action of I\ E 3) on V$ is and A acts on V$ in the following manner:

where a - k = apk,, = ko12 - k - u

If p4 is the dual group of T4, then the orbits of an arbitrary ii = (fro, &, f i2 , T i3 ) E p4

under the action of SL(2, C) are of the form,

here we identify m with mass. In total there are. six different orbits (2 for m2 > 0,

1 for t n Z < 0, and 3 for 772 = 0) [IS]. Correspondingly, there are six different

classes of unitary irreducible representations of 7': (1,s) [I 9, 541. For our purposes

we need the represelltations correspolldiug to the orbit of ti = (771 , 0, 0, 0), rn > 0.

The corresponding representatiolls denote particles of mass rn > 0 and spin s =

0, i, 1, i, 2, . . .. Tliese represe~~tations are

WfY = Qs+l(& ~2

of Cs+'-valued functions c$ on V:, which are

carried by the Hilbert spaces [XI.

square integrable in the sense:

The representation is defined by

where PO is the (2s+1)-dimensional irreducible spinor representation of SU(2) (carried

by Pi' ) and

is the image in SL(2, @) of the Lorentz boost AC, which brings the four-vector (m, 0 )

to the 4-vector k in V::

The matrix form of the Loreritz boost is

which could be written a s

where Vk is the 3 x 3 symmetric matrix

It is noted that Ar. and V , have followiug useful properties which can be easily verified:

and

1 k * - I , (Akp) = - k n + K p , ( A P ~ ) o = -(kopo + k P) = y - (2.30)

t l l nz

the ullderliue denoting the spatial part of a 4-vector, while

ko(Aa) - k(Akp)o = k O v ~ ' ~ ~

from whicli it follows that

Also, since

we have,

2.2 Phase Space For Massive, Spin-s Particles

In general, a phasc spacc of a systenl with n degrees of freedom is a 2n-dimensional

space. The notion of phase space goes back to the Hamiltoniau formulation of classical

mechanics [41], where a dynaniical state of a system at a given instant is completely

determined by its n postiou coordinates q l , gz, - qn, and the n corresponding conju-

gate momenta p, , h, - py1. Then the 272-dimensional space r whose points have the

coordinates ( q l , q-2, - q,&; PI, yz, - - - pa ) is known as phase space. The phase space

r for a massive relativistic particle with arbitrary spiu s corresponding to a Wigner

representation call be written a s [I]

where T denotes t h e subgroup of time translations. For A E SL(2 , C), let

where h ( k ) is defined in (%.24), be its Cartan decomposition. An arbitrary element

(a, A) E ~ : ( 1 , 3 ) has the left coset decomposition,

according to (2.36). Thus , elements in l? have the global coordinatization, i-e., any

point on l? has the coordinates (q, p) E W6: with

In terms of these variables, the action 0 : ~ ! ( 1 , 3 ) x r + r of ~ 1 ( 1 , 3 ) on r is,

according to (2.37), given by

Then using (2.58) aud writit~g the new coordiuates as (q', p'), we get

which can be written as

where A E ~ : ( 1 , 3 ) is related to A by (2.16) and m' = (Ak P ) ~ .

Now we want to show that the invariant measure under this action is dqdp: The

Jacobian matrix J of the transfonnatiou (2.40) is given by

r3q' A = V k - - = (p' @ kt) x nrp0 ' a~

J = ( ~ ;)=( 0 3 B = f i + & ( k @pt) (2.42)

where Vk is defined in (2.28) and o3 is a 3 x 3 zero matrix. Since J in (2.42) is an

upper triangular block matrix, the determinant det(J) = det(A) - det(B) = 1 which

implies

dq' clp' = d q dp (2.43)

Hence the invariant measure v on r, in the variables (q, p), is dqdp.

Next, in terms of these variables let us define the basic section,

which we call the the Galilean section, if h(p) = I . Let II : ~:(1,3) + I' defined

by H(g) = g (T x SU(2)) E I' such that ll(oo(q, p)) = (q, p), then any other section

a = oo o Il : r + P:(I, 3) is defined by:

where j : W6 --+ R and 7Z : R -, SU(2) are smooth functions. We work with a f ine

sections, for which the function f is taken to be of the form

where, 9 : R3 + R, and 19 : R3 -+ W3 are smooth functions of p alone. In calculating

the frame operator, we need to evaluate au integral, (see (2.120)) where it is necessary

for f to Lave the above form. We will later see that as far as the construction of CS is

concerned, y only iutroduces an inessential phase and it does not make any difference

even if we set y = 0 (see(2.120)). Moreover, we also impose the restriction that

R(q,p) = R(p) be a function of p alone (we need this for the evaluation of the

integral (2.120) below). Thus writing,

we see that

where M(p, 6) is the 3 x 3 real matrix

We shall analyze (2.51) extensively later. Let us simply note here that

so that if det[M(p,g)] # 0,

Also, assunling M(p, 19) to be coi~ti~~uous in p and 6, and uon-singular for all (q, 19)

(i.e., q = 0 $ = 0 in (2.50)) and s i ~ ~ c e det[M(O, a)] = 1 , it follows that

2.3 The P- Duality and Space-Like Sections

Since the matrix M ( p , 8) in (2.50) has an inverse, we easily obtain from there,

where P(p) is the 3-vector field

Solviug for 9(p) , this gives

Let us also introduce the d u d vector fields P' and a',

where V, is the: matrix defimd in (2.28). The significance of these dual quantities

will shortly become clear. First note that

and

Let us try to get a better understanding of the P -P* duality [13] as defined in (2.56)

and (2.58). Since in order to satisfy the positivity condition of the Jacobian Jx(k)

(see (2.124)) we will need IIP(p)ll < 1 (see again (2. IN)), let us define the relativistic

4-veloci ty n(p) by

Theu, by (2.55), the point i = (h,q) E T4 satisfies

that is, c j lies on the spacelike hyperplaw with normal vector r2(p), determined by

P(p). Let us denote this hyperplane by S$ 111 particular, for the Galzlmn section:

while for the Lorcntr section,

Note that these two sections are related by the duality of (2.58), i.e., PoS(p) = Pc(p).

From (2.62) and (2.63), we see that, with llPll < 1, Vp, we can associate to P(p) the

time-like 4-vector field u(p) = mn(p) which is normal to the space-like hyperplane Ct

in T4. More generally, if p + P(p) is a p-dependent 3-vector field, we can associate

to it a ray [u(p)] of pdependent relativistic Cvector fields u(p) in the following way:

The 4-vector u(p), and llence the ray [u(p)] = R+u(p) is time-lika, light-like or space-

like according as IIP(p)II < 1 , IIP(p)II = 1 , or IIP(p)II > 1, respectively. Under a

Lorentz trausformation A , tc@) -, Au(p) and

P(P) =

Of course, such a trausforznatiou preserves the property of u being time-like, Light-like

or space-like, and hence of the equivalent properties of IIP(p)ll being <, =, or > 1.

Now let Ak be a Loreutz boost. To any u(p) E [u(p)], let us associate its k-conjugate

4-vector field under Ak:

Clearly,

u(p) = l \ p k ( ( p ) * ( u + ( ~ ) ) * ' ~ =

Let be the k-conjugate 3-velocity associated to [u'*'(~)], i.e.,

Theu, using (2.80) and (2.31),

In particular, the Cvector field

depeuds on p only, (like u(p) itself). We call uS(p) the dual of the. 4-vector field u ( p ) .

Thus, for arbitrary u(p) E [u(p)] ,

and P(p), P*(p) satisfy the duality relationship in (2.58). Taking the dot product

of P(p) with p on both sides of (2.58), using the explicit form for V, in (2.28), and

rearranging, we get the relation

Similarly, one may verify the matrix relation

Note also, that u*(p) is time-like if u(p) is time-like and viceversa. Hence,

Physically, to each ordinary 3-vector velocity P(p), u(p) associates a relativistic 4-

velocity 4 p ) (= u(p)/[u(p)-u(~)l+, if u(p)-u(p) Z 0 and = u(p)/uo(p) if u(p)-u(p) = 0),

while P * * ~ ( ~ ) is the velocity obtained by mlativistically adding the 3-velocity +(p)

to the 3-velocity associated to the boost A,.

Below are details of some particular space-like affiue sections and a light-like limiting

section, all of which have interesting physical iu terpretations.

44

I. The Galilean section Q:

As noted in (2.64), for this sectiou

2 . The Lorfr i t z section u?:

This time (see (2.65))

iu other words, the Galilean aud Lorentz sectious are duals to each other.

3. Thc syntmrlric scctiotc o,:

This section is self-dual, being given by

Again, IIPc(p)ll < I , Vp. Note that iu a stlose: us lies half-way between 00 and

ot-

4. The limiting sect ions o*:

These sections are duals of each other and are both light-like, being given by

In this liniitiug situation, IIP+(p)ll = IIP-(p)II = 1 , Vp.

A t this poiut we state and prow the following proposition (for later usages).

Proposition 2.1 Thc follou~ing conditions arc c q u i v a h t :

I . The $-vector = (h, 4) is spacc-likr, i - c . ,

for all p E V:.

9. For all unit vectors 6 E W3 and for all p E W3,

4. For a l l p E Vm+,

~ o l l f i ( ~ ) l l < 77t + P - PI = lm + P $(p)l.

5. For all p E W3, thc 3-vectorficld P obeys

IIP(p)II < I *

6. For all p E W3, the 3-vector field P* obeys

IIP*(P)II < 1-

Note that (2.88) shoua, i7t particular, that

P Po tj space-like (16(p) - -11 < m, m

for all p E V;.

Proof:

We start with 1. The condition rewritten as

implies by (2.49) and (2.50) that

The expression withiu the square brackets is easily seen to be the matrix S. In other

words,

q S(P, q q > 0, 'jq E R ~ , q # 0.

which is equivalent to 2. Next note that S is a matrix of the type

Such a matrix has eigeuvectors

b x a A

e = e+ and e,, e+ -e - = U , Ilb all '

where e+ and e- are linear combinations of a and b. Moreover e has eigenvalue 1 ,

while e+ have eigenvalues 1 + a bf Ilall llblly respectively. Since [la11 llbll 2 a by the

matrix A is positive definite if and only if 1 + a b - Ilallll bll > 0. Applying this result

to S yields (2.87) and hence 2. Going back to (2.93) take q = i, an arbitrary unit

vector in W3, and rearrange to obtain

which can be factorized to yield

Since (6 p) + [(e P)2 + n?]i > 0, and (ii p) - [(6 p)* + m2]f < 0, the above

inequality holds if and only if

and this is the same as 3. To see that 1 . and 4. are equivalent, use (2.53) to rewrite

(2.49) as

Then,

In this equation, for fixed Ilqll, if we choose $ to lie along 9(p), we get

Thus, for any 11611, we can find a 4 for which

Now, is spacelike ++ lio12 < llqllZ H lio12/114112 < 1 , and in general, from (2.95)

Hence, if

then tj is space-like. Conversely, if i is space-like, then lio12/1G112 < 1, and since for

fixed 11q1(2 there is a always a q for which (2.97) holds, it follows that space-like

+ (2.98). Since by (2.52) and (2.54), tn + p 9 (p) > 0, we see that 1 . 4. The

equivalence of 4. and 5. follows from the definition of /3 iu (2.56). Finally (2.78)

shows that 5. e 6.

We end this section by proving the stability of the class of affine sections under the

action of P:(I, 3). If a : r -+ P:(I, 3) is any section then, as shown in [6], for

arbitrary (a, A) E ~:(1,3), c(o,~) is again a section where,

(a, A)-'(q, p) being the translate of (q, p) by (a , A)-' under the action (2.40) and

Let 24 denote the class of all affine space-like sections, defixled by (2.47) - (2.50) and

satisfying the conditiot~s of Proposition 2.1, but with cp not necessarily assumed to be

zero. Then we have the result:

Proposition 2.2 I f o E U, titen q a , ~ , E U, for all (a , A ) E ~:(1,3).

Proof:

If 9 is included in the definition of the sectiou, the relations (2.49) and (2.50) generalize

From this it follows that

4 ~ ) - Pp(p)7 414 - i =

with n(p) given by (2.62). Next, write I\ = Akp, where p is a rotation. Then

so that writing

in (2.99), we get (see (2.40))

with Vk as in (2.28). Thus, if o is the affine section corresponding to the quantities

P and 9, and (ti', p') = o(ql, pi), then

Let

and nt(p) = I \ n ( p ) . Clearly, P' = - An/(l\t& satisfies IIP'II 5 1 if llPll 5 1. Further-

more,

Thus, qatAt (q, p) is again an affine sectiou corresponding to the quantities P' and p',

with

q.e.d.

In view of this result, startiug with any family of coherent states R = {q&),

we may generate an entire class of covariantly translated families Gn(a,A) of other

coherent states, using the natural action (2.99) of ~ : ( 1 , 3 ) on the space of sections.

If o is characterized by P and y, then g(a,A) is characterized by P' and p', the

relationship between 9 and 9' being given by (2.1 11) above.

2.4 Relativistic Frames and Coherent States

We now take an arbitrary affine section o, and going back to the Hilbert space Wf&

in (2.21), choose a set of vectors i = 1, 2 , . . . ,2s + 1, in it to define the formal

operator (see (1.16) and (2.23)):

From the general defixlitioxl (1. IS), i n order for the set of vectors

to constitute a family of cohereut states for the representation (J&, the integral in

(2.1 12) must converge weakly, and define A, as a bounded operator with a bounded

inverse.

To study the couvergeuce properties of the operator integral in (2.112) we have to

determine the convergence of the ordinary integral

for arbitrary 4, * E lib. In (2.48) set

where and p(p) are t h e matrices in the Lorentz group ~: (1 ,3 ) which correspond

to A ( ~ ) and R(p), respectively. Then.

'h.~) ( k ) = exp(-iX(k) q)vs(v(k, p ) ) q i ( i \ ( p ) - l k ) , (2.1 16)

where

BPO X(k) = --9(p) n~ + M(p$)'k,

is a one-to-one fullction of k haviug tlle property that:

X(k) = X(kt) implies k = kt and ko = kto

and

Substituting into (2.1 14) yields

In order to perform the k, kt integrations in (2.120), we need to change variables:

k 4 X(k). At this point w e compute the Jacobian Jx(k) of this transformation from

(2.1 17). To this end, using (2.51) for M (p, r P ) we write the components of X(k) as

where &(p) , i = 1, 2, 3; are the coxnponellts of 6(p). Then

which has the detertui~lat~t

1 det[Jr(k)] = 1 + -8(p) = [kop - kpo] rn ko

Since at k = p = 0, det[,&(k)] = 1, and since in order to change variables we need

det[&(k)] # O, we must impose the condition that

We shall show that the conditiou(2.124) holds for all k, p E R3 if and only if the

Pvector t j = (&, 4) is space-like. To this end we first rewrite (2.124) in terms of P'.

Let p(k -t p) be the rotation matrix defined by

( it is a rotation, siuce the deterininant of the matrix is 1 and for any Cvector

a = (1, O), (A;' A~I\,A;' = 1 ) which could be written as

and acting on the vector (m,O) with both sides of the equation we obtain,

Then using (2.61) we can write,

and plugging in the explicit form of V,, from (2.28) and simplifying we get,

by virtue of (2.30). So

which implies

after making the use of (2.127).

Thus, it would appear that the positivity of det[&(k)], as required by (2.124), would

be ensured if the second tern1 within the square brackets in (2.131) did not exceed 1

in magnitude, in other words, (2.124) would seem to require that IIPU(p) 11 < 1, V p.

Proposition 2.3 Thc condition (!?.I24), det[&(k)] > 0, hold9 for all k , p E IK3

if and only i f thr 4-vector = (h, q) is span-like, i.e., if and only i f any one of the

equivalent conditions in Pro positiorz 2.1 is sat isf id.

58

Suppose that ij is space-like. Then by Proposition 2.1, IIP'JJ < 1. Hence, since

p(k + f i ) is a rotation matrix,

so that

i-e., det[Jx(k)] > 0 (2.131)).

Conversely, assume that det[&(k)] > 0. Tlien by (2.122),

Taking k in the direction of d(p) and letting Ilk11 -t ca, the above inequality implies

that

which, in view of condition 4. of Proposition 2.1, implies that 4 is space-like.

q.e.d.

From now on, unless otherwise stated, we shall work with space-like affine sections o;

actually the only exception will be the two limiting sections Q in (2.83) and (2.84).

Thus we shall assume that the conditions of Proposition 2.1 hold.

59

Going back to the computation of I+ * + in(2.120), we note that dq integration yields

a Cmeasure in X, and hence making the change of variables k + X, integrating and

rearranging (using (2.131)) we obtain,

where A,(k, p ) is the (2s + 1) x (2s + 1)-matrix kernel

where p ( p ) , p(k -+ p) and v(k, p) are given by ((2.1 Is), (2.125) and (2.119), respec-

tively. Assumiug the integral (2.1 14) to exist for all t$, + E Xb, let us write

Then the operator A, iu (2.1 12) is a matrix -valued multiplication operator:

At this point we make two simplifying assumptions on the the nature of the vectors

qi EX+, i = 1,2 ,..., 2s+1.

1. Assumption of rotational invariance of the operator x:!:' Iv')(r)'l, i.e., V R E

S W ,

This implies that

where 7 E L2(V;, e), and thus we may take for 77' E I f s the vectors

tbe ei being the canonical unit vectors in a?"+' (i-e., ii = (h i j ) , j = 1,2, . . + ,2s +

1)-

2. Assumption of rotational invariance of Irl(k)12 in (2.137):

We shall generally refer to these two assumptions as the assumption of rotational

invariance. With this assumption, the kernel A,(k, p ) in (2.133) simplifies to

On 'HfY define the operators (Po, P),

We shall also denote the analogous operators on L2(V:, $) by the same symbols.

Note that P;' is a bounded operator with spectrum [0, $1. Then, with the above

simplifications (2.1 34) becomes

where (-), denotes the LZ(V,l;, e) expectation value with respect to the vector in

(2.137). Hence for the operator A, (see (2.135))

provided this supremum exists. 0x1 the other hand, since I I@' ( -~k '~) l l < 1 and

Ilp(k -, l\,lP)tII = I, from (2.140) we get,

The two extreme values in the above inequality are reached for the limiting sections

o* (see (2.82)). Thus, we have the result:

Lemma 2.1 If IIP(p)ll 5 1, Vp, then a@(k,p) is a bounded function satisfying

' . Suppose now that q lies in the domain of P;, l.e.,

and set

Theu (2. LE), (2.140 and (2.145) toget l~er imply

Lemma 2.2 If thr assumption of rotational irtvan'ance on qi , i = 1,2,. . . ,2s + 1 is

t

satisfied, and if q E Dont(Po5), tkrn for all P such that IIP(p) 11 5 1, Vp,

As a consequeuce of this lenuna we see tha t both the operator A, in (2.112) and its

inverse, A;', are bounded, with

Indeed, collecting all these results we obtain,

Proposition 2.4 Let rli , i = 1,2, . . . , %+ 1, satisfy the condition of rotational inuari-

arzce. Then for each /3 satisfying IIP(p)II 5 1, Vp, the set of vectors Gm in (2.113) is a

family of spin-s cohcntct states, forming a rank-(2s + 1 ) frame 3{7&,,,, A,, 2s + 1 ),

if and only i f r ) E DO~(P$) . The operator A, acts via multiplication by u bounded

invertible function AJk) givcrc by (2.142) and A,' via multiplication by the function

Ai1(k) . Moreover,

Note that since we are assuming rotational invariance, we could just as well have

dona without the restriction, R ( q , p ) = R(p) , iu defining the sections a in (2.48).

The following coustructiou xiow enlerges for building spin-s coherent states for the

representations [I&, (see (2.23)) of lmss 7 n > 0 and s = 0, $, 1, $, 2, . . ., of ~ : ( 1 , 3 ) :

1. Choose: a function P : R3 + R3 such that l Ip(p) 11 < 1 , Vp, or equivalently,

a map u : V: -+ V$ as iu (2.66); choose an arbitrary measurable fullctiou

R : R3 + SIJ(2) and construct the corresponding aftine space-like (or in the

limit, the affine light-like) section a, using (2.57), (2.48), (2.49) and (2.50).

2. Choose au r) E L2(V$ &/ko) , satisfying (2.139) and (2.146) and form the vec-

tors q' i = 1,2, . . . , 2s + 1, usiug (2.138).

64

3. Construct the family, Ge, of coherent states i]ip,p) using (2.112).

While this procedure provides us with a large class of CS and frames, the latter are

generally not tight, i.e., A, is not, in general, a multiple of the identity. A few special

cases worked out below will make this statement clearer. For computational purposes,

the following expressions prove useful (assuming rotational invariance):

and,

1 . The Galilcan section :

From (2.79), (2.80) and (2.152),

and using the rotational iuvariance of l v (k ) 12,

Hence,

so that the frame is tight.

2. The Lorentz section :

Rom (2.81) and (2.140),

so that

A, (k) = A@) = (2~)~nz(~~'),,I~.+~.

Thus,

and once again the frame is tight.

3. The symmetric s c c t i o ~ ~ a, :

From (2.82) and (2.153),

Thus,

and

The operator A, = A, is given by

To determine the spectrum of A,, note that the functio~i f : [m, oo) + R+,

defined by

is uniformly bounded for all po E [m, c] , for any finite c > m. Also f'(k0) # 0,

for all ko, po > 778 (here denotes the derivative of f) and

Thus,

which, by virtue of (2.162), implies that

Hence, in this case, the frame is never tight.

Chapter 3

A Relativistic Windowed Fourier Transform

In the previous chapter we constructed the coherent states of the full Poincar4 group

T$(1,3) and showed that they fornl coutinuous frames. Here we discretize the coher-

ent states of ~ i ( l , 1) and show that discretized CS of e(l, 1) can also be made into

discrete frames and develop a transform, analogous to the windowed Fourier trans-

form, which we call the relativistic uhrdouwd Fotmricr transform (RWFT). Finally, we

obtain a recot~struction formula for any function ou the Hilbert space ?l= L2(v;, $)

using the RWFT.

3.1 The Mat hemat ical Formalism and Not at ion

Udike the previous chapter (which dealt with ?:(I, 3)), here we restrict ourselves to

pl(1,l)) i.e., instead of dealing with a 3-component space vector, we work with a

single-component space vector. Most of the relations and statements which were true

for ~ : ( 1 , 3 ) in Chapter 2, are also true for ~ : (1 ,1 ) . Let G = ~ ; ( l , l ) be the Poincark

group in one space and one time dimensions. It acts on the Minkowski space BlV1

whose points we denote by z = (lo, x), xo = t , x E R (we assume that ti = c = 1) .

The metric is g = diag(1, -1). The elements of e ( 1 , l ) are denoted by (a; I\), where

a = (a0, a) E W2 is a space-time translation and A is a Lorentz boost. Let V,+ denote

the forward mass hyperbola,

for some mass m. The matrix A may be parametrized by a vector p = (PO, p) E V;:

The group n~ultiplication is defined by the semi-direct product:

and the inverse and the identity elen~ents are respectively (a; A)-' = (+-*a; A-')

and (0, I), where I is the 2 x 2 identity matrix. The elements Ap of the Lorentz group

act on V$ in the natural manner,

This action is transitive and by a straightforward calculation, it can be shown that the

invariant measure on )I: is dk/ko. We work with the unitary irreducible representation

Uw of P:(I, I), on the HiIbert space

given, for g = (a;/\,) E P$, I ) , by

where k n = 4 n o - k a. We call (Iw the Wigncr rcprcscntation of ~ : ( l , 1) for the

mass m. Since for ally t$ E acv,

mc mc' dp x 4(&' k') - - da -

ko 6 Po

we see that the Wigner representation is not square integrable with respect to the

whole group e(l, 1) . However, as in the e ( 1 , 3 ) case, it is square iutegrable with

respect to the homogeneous space r = e(l, 1)/T, where T is the subgroup of time

translations. Then r has global coordinatizatiou (q, p) E W2, and the left invariant

measure is dqdp (61. The action of P:(I, 1 ) on r corresponding to (2.40) is :

where

In case of e(l, 1 ) the particular section : r 4 e(l, 1) is defined by

Then any other measurable section reduces to

where f : R2 --+ R is a smooth function. The relations (2.48), (2.49) and (2.50)

reduce respectively to

The function f (q, p) is defined in (2.47) and in this case tp : R -+ R and 9 : W 4

IP are continuous functions of p alone. As we saw in Chapter 2. that 9 played an

inessential role for our purposes, here we set cp = 0. It is to be noted here that all

the relations from (2.55) to (2.61) also hold true for the case of ~ : ( 1 , 1 ) where P(p)

has to be treated as a I-dimensional vector field and V, = E. Consequently (2.61)

reduces to

Then the coherent states of pL(1, l ) , for an arbitrary section a(q,p), are the set of

vectors

for an q~ in the domain defined in (2.141)). The coherent states in (3.16) satisfy

the frame condi tion

(3.17)

(see (1.19)) where both A, aud A,-' are bounded operators.

3.2 Periodization of Compactly Supported Functions

Let Y : W -t @ be a compactly supported function, with support [a, b] which means

Y is zero outside the interval [a, b] and the length L of the support is finite, i.e.,

L = b - a < m. We want to periodize the function Y in the following manner:

Let F : W -, @ be a periodic function with period L such that

F has the Fourier series decomposition

and,

where

Theu

where b(x - x') is the Dirac delta function defined, in the sense of distributions, by

It has the property that for a sufficiently smooth function f ( x )

For an arbitrary section u(q, p) (see (3.1 1)) and 4, j E Z, where by Z we denote the

set of integers, we write the discretized form of coherent states iu (3.16) as

where, Aj denotes the: discretized version of A,, for time being we take 4% > 0 and

will fix it later and

Let a = (no , a), b = (bo , b), pj = (pi*, pi), and k = (ko, k) E V$ satisfying

2 ao2 -a2= m , a0 > 0, a E R etc.

Let ~ ( k ) = O if k 6 [b, a], i.e., the length of the support of ~ ( k ) is b - a. Then

and the length of the support of ~ ( ~ 7 ' k ) is

Let ~ ( A Y ' k) = fj(Xj(k)), where Xj(k) is given by (3.27) and

Then the length of the s c ~ p o r t Lj of ij(Xj(k)) is given by

Then, for any E 7frv, we write

where 9 indicates complex conjugate of r).

We define the formal operator T

in order to calculate the discretized frame operator. We will study the convergence

properties of this operator in the next section.

3.3 The Frame Operator

Now we want to study the convergence properties of the operator defined in (3.32).

To do this, for arbitrary 4, 1C, E 'Hw, we consider the formal sum

using (3.31).

Now let r ) ( ~ y l k ) = ij(Xj(k)) a d setting Aqj = a, where Lj is the length of the L,

support of @(Xj(k)) and defined in (3.30), we cau write

Since,

changing variables k' -, Xj(kt) yields

Using (3.23) we write,

Then using J f ( x ) b ( x - xo)dx = f (XO) we get,

which implies

a = m sinh(&), a0 = nt cosh(8.); b = rn sinh(&,), b0 = n cosh(Ob) (3.41)

Also,

I @ ( ~ j ) = - t ~ 1 [pj - ~joP*(pj)] (see (3-15))

Now substituting (3.40), (3.41), (3.42) in (3.39) and writing

we have,

where we have written,

B R ( O j ) = t d l (a* (ej)).

Let Bj = jOo , where Bo > 0 (fixed) (we call Bo the step size), then f takes the form:

For any compactly supported function 6, with finite length of the support L, the sum

in (3.45) contains at best LIBo + 1 te rns only, i.e., it is a finite sum. We observe that

each term in the sum is positive and bounded for any t E W and j E Z, consequently,

the operator f is strictly positive and bounded. Hence the discretized version q t j

of the cohereut states in (3.16) form a frame, more appropriately, a discrete frame.

Depending upon different situations, this frame could be tight or non-tight. Here we

now show it by a~~alysing some particular sections and imposing restrictions on i j .

Note that the contributiox~s of the sectious come from the function @' in (3.45).

I. Thc Galikan scction 00:

For this section 8(pj) = 0, that implies (P* ( jOo) = j$ (using (3.42) and

(3.44)) and

For different values of t, F is different, so in this case the frame is never tight.

By contrast, in the corresponding continuous case the frame was tight [6].

2. The Lorentz section ct:

For this section 6(pj) = % that implies B'(jOo) = 0 and

If l i ( t - j90)12 = c d ( t - jOo) in the support of +, i.e.,

1 o otherwise

where

t - ~i~d@.)] , , [ t - sinb ( e b ) ] , ) -

80

if n is au integer

( integral part of n + 1 otherwise ,

if n is an integer

( iutegral part of t z otherwise ,

is a constant operator slid therefore, the frame is tight.

3. The symmetric scctio~r 0.:

For this section 79(pj) = ,,,g,,. which implies @'(jBo) = $ 3 ~ 2 and

In this case, F is a non-constant operator and so the frame is never tight.

Note that if @* is a constaut fu~iction, say for example, OS(jOo) = n (a constant),

theu

Then if we have l i ( t - j O 0 ) I 2 = cosh(t - j O a + a) in the the support of ij(t - jOo),

P(t) becomes a constant operator a11d hence the frame is tight.

Again, substituting (3.40) aud (3.41) iu (3.27) and (3.30); using (3.42) and (3.44);

writing i ( t ) = q(cosh(t), siuh(t)) and Bj = jOo, the discretized version of coherent

states in (3.16) takes the form,

Also, assuming &(t ) = 4(cosh(t), sinh(t)), using (3.54), we can write the scalar prod-

uct in (3.31) as

which we call the discretized relativistic windowed Fourier transform of the

function &t) for the window-function + ( t ) ) .

Now

- -1 where I is the identity operator on lirv and [T] = $. Then we can write,

We call (3.58) the reconstruction formula. Substituting (3.45), (3.54), and (3.55) in

(3.58), we can reconstruct any fuxlctio~l 4 E L2(Vi , e). In the next chapter, we will

analyse the reconstructiou formula and reconstruct some functions in Xw numerically.

Chapter 4

Numerical Computations Using The Relativistic Windowed Fourier Transform

We devote this chapter to liuxnerically

rccor~dructio tr fo m u l a obtained i x i t I1 e

computing the operator F and applying the

previous chapter in various situations. First

we evaluate the operator f ( t ) for different values of t, various window functions and

sections. Then we reconstruct several functions using the reconstruction formula,

for different window functions, sections and the step sizes. We also evaluate the

original functions at the same values of t as t he reconstructed ones and compare them

graphically. Finally, we show the effectiveness of the relativistic windowed Fourier

transform in comparison with the general wiudowed Fourier transform (also known

as the Gabor transform).

4.1 Numerical Evaluations of T

Here we rewrite the operator F :

For the Galilean sections (4.1) turns into (see (3.46))

Unless otherwise stated, from now 011 we assume:

which implies

f ( t ) = C cosh(0.025j) l+(t - 0.025j)I2. cash ( t ) j=-,

Let the window functiou i j be colnpactly supported with the support [-I, 11 and

defined by

which we call the triangular u~irtdow.

Figure 4.2: Frame operator for the Lorentziau sectiou and triangular window.

For the Lorentzian section, asssunling everything is same as in the case of Galilean

section, the operator ?(t ) takes the form (see (3.47)):

From (4.7) we have the above Figure.

Figure 4.3: Frame operator for the symmetric section and triangular window.

For the symmetric section

Using (4.8) we produce the above Figure:

Figure 4.4: Frame operator for the Galilean section and the smooth window.

We now want to calculate F(t ) for the smooth window

(1- t2 )10 if -1 s t 5 1 *4 = [

0 otherwise

For the smooth window (4.9) and Galilean section the operator f ( t ) becomes

froin which we get the above Figure.

Figure 4.5: Frame operator for the Lorentzian section and the smooth window.

Corresponding to the smooth wiudow and the Lorentzian section f ( t ) takes the form:

which gives the above Figure;

Figure 4.6: Frame operator for the sylllnletric section and the smooth window.

Finally, for the symmetric section the expression of P(t) corresponding to the smooth

window becomes

From (4.11) we: produce the above Figure:

4.2 Reconstruction of Functions

In this section, we reconstuct some functions using (3.58). Denoting 4, [F(t)]- I and

ijtG respectively by 4, [ ~ ( t ) ] - ' and qc we can write (3.58) as

where

I sinh(t - jOO + W ( j $ ) ) vr j ( t ) = exp -ire s i n h ( 9 ) c o s h ( 9 + (PS(j&)) ] sit - j 00 )

(see (3.54)) and T ( t ) is given in (4.1). Let

First we construct 4 ( t ) for the triangular wixldow (non-smooth window) (see (4.5))

and different sections, then we do the same for the sn~ooth window (4.9). For the

Galilean section with the assumption (4.4) and (4.15); (4.14) and (3.55) change re-

spectively to:

and

#

Figure 4.7: Reconstruction of eot2 for the Galilean section and the triangular window.

Then substituting (4.16), (4.17) and (4.5) iu (411~) we get

where T ( t ) is given by (4.6). Usiug (4.18) and (4.15); we have the above Figure:

Figure 4.8: Reconstructiou of e-L' for the Lorentzian section and the triangular win- dow.

For the Lorentzian section, the reconstruction fornlula becomes:

exp [id sinh(z - 0.025 j ) ]

+ [T jr)l-l 2 L(F)J /0.025j exp [id sinh(x - 0.025 j ) ]

k - 0 0 j= [(&)I 0.0% j-1 0 025

where T ( t ) is given by ((4.7). For (4.19) and (4.15) we have the above Figure.

Figure 4.9: Reconstructiou of for the symmetric section and the triangular win- dow.

The reconstruction fonnula correspouding to the symmetric section is given by

L(&)J o.o2sj+l

* ( t ) = [ ~ ( t ) ] - ' 1 exp sink(x - 0.0125j)

0.025 j cosh(0.0125 j ) t=-m i= r(,&)i I

where T ( t ) is given by (4.8). For (4.20) aud (4.15) we have the above Figure.

95

Figure 4.10: Reconstruction of Ct2 for the Galilean section and the smooth window.

Observation: We see from the Figure 4.7, Figure 4.8 and Figure 4.9 that the dif-

ferent sections play more or less the same role in the reconstruction scheme and the

accuracy of approximation of a function by its reconstructed counterpart is not very

high when the triangular (non-smooth) wiudow is used. From now on we use only the

Galilean section and some snlooth window.

For the smooth wiudow function q defined in (4.9), the reconstruction formula for the

Galilean section becomes

where T ( t ) is give11 by (4.10). Using (1.21) and (4.15) we: have the above Figure:

Figure 4.1 1: Recoustructiox~ of a discoutinuous function for the Galilean section and the smooth witldow.

0 bservation: From the Figure 4.7, Figure 4.8, Figure 4.9 and Figure 4-10, we see

that for a better qproximation of a function by the reconstruction formula we should

use a smooth window.

So far we observed that the reconstruction scheme goes well for smooth functions and

smooth windows and at this point we want to see how does it work with discontinuous

functions. To this end we take the following discontinuous function:

( 0 otherwise.

Substituting (4.22) in (4.21) we have the above Figure.

The Figure 4.1 1 tells us that in the vicinity of xo where a function f (x) is discontinu-

ous, the reconstructed functiou starts ringing and the accuracy of approximation is not

high. A situation like this in Fourier analysis is known as the Gibbs ' phetromerion [61].

It also tells us that f(xo) converges to

where f (xi) and /(I;) are respectively the right- and left-hand limit of f (x) at xo.

It is worthwhile meutioxiiug here that the inverse Fourier transform of a function f at

a point xo couverges to (4.23) [Gl ] .

4.3 Comparison with the Windowed Fourier Transform

In this section, we discretize the coherent states of Weyl-Heisenberg group aud fol-

lowing the techniques of chapter-:3 we come up with the corresponding frame operator

and t he reconstruction fonnula. Finally we reconstruct a functiou using both t h e rel-

ativistic windowed Fourier transforn~ aud the usual windowed Fourier transform and

compare them.

The discretized version of the collerent states of the Weyl-Heisenberg g o u p is given

by:

where po, go > 0, m, n are integers and po, go must satisfy the condition

The condition (4.25) is necessary for &,,(I) to be complete and to form a frame [18,

62,271. Let 4(2) be a compactly supported function with support L = z. The frame

operator T : L2(W, d x ) P ( Z 2 ) is defined by

For 11 E 'H = L2(W, dx), we c o l d e r the formal sum

which implies

and the corresponding reconstruction formula is :

Let p, = r and

Then we can write

(1 - t2)l0 if -1 5 t 5 1

0 otherwise.

and

Now writing cosh(t) e-si1111(t)2 for 4 in (4.21) and + in (4.32) and setting Bo = 0.01

(instead of 0.025) in (4.11), qo = 0.01 in (4.32), we have the: Figure 4.12 (in the next

page)

From the Figure 4.12, we observe that the recoustruction scheme goes well both in

relativistic windowed Fourier and windowed Fourier techniques and the respective

reco~istructed values are virtually the same.

In chapter 2, we constructed

that they formed frames. We

coherent states of the full Poincare group and we saw

believe that these coherent states will have applications

to spin dependent problems in atomic and nuclear physics, as well as to image recon-

struction problems, using the discretized versions of these frames. In chapter 3, we

discretized the coherent states of the PoincarC group ?:(I, 1) and obtained discrete

frames and a recoustruction formula. We observed in chapter4 that the reconstruc-

tion of a signal is much more accurate if we use a smooth window function, instead of

a nou-smooth window functiori and the Galileau - , Lorentzian - and symmetric sec-

tions all play similar roles in the recoilstructioxi programme. As we also saw that the

reconstruction of a signal by relativistic wiudowed Fourier transform (RWFT) was as

good as the reconstructiou by the Gabor transform (GT), the RWFT can be used as

a substitute of the CiT. The RWFT has applications in many fields, including pattern

recognition, signal and image reconstruction, detection and extraction of unknown

signals etc.

Some Specific Applications:

States and observables in quantunl mechanics can be viewed in a very special way

in terms of coherent states. For example, if we consider the hydrogen atom, we can

describe its continuuln and bound state wave functions, dipole operators, etc., in

terms of an overcomplete basis cousistiug of Galilean coherent states as:

where the Sn,c,(po, r) are the Sturmian functions [68]. The Sturmian functions are

solutions of one of the Sturm-Liouville problems and form a complete discrete basis

set in the Hilbert space. For computational purposes, for example, matrix elements

of multi-photon process in the non-relativistic case [57, 711, the states ( r I q, p ) are

suitable. This is also true in the intermediate relativistic case, where the Dirac or the

Feymau-Gell-Mam equation is able to describe the interaction of a charged spin-f

particle with the electromagnetic field. Some work iu atomic physics in this direction

has been done recently iu [73]. The spin-Sturmiw functions for the Feynman-Gell-

Maun equations were obtained in [22]. Using these wave functions one can construct

relativistic coherent states in the light of the present work:

In constructing these collerent states one bas the freedom iu the choice of available

sections, in addition to the already existiug freedom in the choice of the Sturmian

functions. The various sections o also have applications to relativistic statistical

mechanics in the con~putatiou of distribu tiou functioils [36].

We now show, as a specific exa~nple, how the RWFT can be used in detection and

extraction of unknown signals: In detecting a signal by a radar a 'threshold' is to

be set. A threshold is an electronic device that produces an output signal when the

receiver output, averaged over the several repitition periods, exceeds a predetermined

level. If the threshold voltage level is adjusted to lie well above the rms 'noise' output

of the receiver, false alarms caused by noise may be kept to any desired low rate. The

use of high threshold setting will also result in failure to note the presence of actual

target when their signals are relatively weak. Hence the probability of detection will

be a function both of signal-to-noise ratio and the threshold setting. Here by 'noise'

we mean very small random fluctuating voltages that unaviodably are present in the

input circuit of the receiver. An observed wavefornl may be a signal mixed with

noise or a noise alone. h their work, Chen and Qian [24] used the following steps in

detecting an unknown signal:

1) representing the uoisy signal in the joint timefrequency domain by using Gabor

transforms;

2) determiniug the mean of the Rayleigh distribution of the background noise in the

joint time-frequency domain;

3) thresholdiug the time-frequeucy coefficients;

4) definiug the time-frequency coefficients which are above the threshold;

5) measuriug the time-frequency location, time duration and frequency range from

the coefficients above the threshoId.

In the step 2) , by the Rayleigh distribution we mean,

where p is related to the mean value by, mean = fip. After the detection of the signal, it is extracted by using the inverse Gabor transform.

The localization of the time and frequency by the Gabor transform makes it possible

for de-noising, signal detection, and signal extraction in the time-frequency domain.

Finally the extracted signal is reconstructed by a Gabor expansion. We can use the

RWFT instead of the Gabor transfor111 in the above procedure to detect, extract and

reconstruct an ullknown signal.

Some Possible Extensions of the Present Work:

The immediate extension one can do is to discretize the cohereut states of the full

Poincar6 group to obtain discrete frames and a recoustruction formula, as in the

case of ~:(1,1). These could be applied in the reconstruction of signals and images.

Since each transform (e-g., wavelet transform, RWFT, GT etc.) arose from the CS of

different groups a d the discretizatio~l of CS in each case is done iu different ways, one

transform may be more suitable for a certain class of functions than for another class.

So one cau also explore the possibility of finding the classes of functions suitable, for

the reconstructiou formula obtained in chapter& so that the reconstruction scheme

does a better job, at least for that kind of functions, than other available schemes. It

is our intention to proceed in this direction.

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