harmonic analysis on the poincaré group

Commun. math. Phys. 12, 331--350 (1969)

Harmonic Analysis on the Poincar6 Group* I. Generalized Matrix Elements

NGIu~M XVAN HA1

Laboratoire de Physique Th6orique et I-Iautes Energies** Orsay, ~rance

Received February 5, 1969

Abstract. The generalized matrix elements of the unitary representations of the Poincar6 group are computed as eigendistributions of a complete commuting set of infinitesimal operators on the group. The unitary representations of the Poincar6 group can be reconstructed from these matrix elements in a Hilbert space of square integrab]e functions over the space of eigenvaines. Some properties of these distributions (which are measures) are given and some characters are computed from the explicit formulae for the matrix elements.

Introduction

I t was suggested by Lv~gAT [1] to construct the quantum fields on the Poincar6 group instead of Minkowskian space (which is a homogeneous space of Poinear6 group). Numerous a t tempts were made to enlarge the homogeneous space since then [2, 3]. In the a t t empt to construct this field theory, i t was necessary to obtain the "exponentials" of the Poinear6 group, tha t is to say to begin to s tudy the Fourier transform on the group.

The harmonic analysis on the universal covering group of the proper orthochronons Poincar6 group (in this paper this covering group is named improperly Poincar6 group) is led trough with the simple idea of using the "exponentials" of the group. These "exponentials" must be the eigenfunctions of the infinitesimal operators and the method is therefore straightforward. In this paper, we first compute the eigenfunctions and in the following one, we shall s tudy the Fourier transform. Chapter 0 is devoted to notations and to some useful definitions.

In Chapter I , we derive the infinitesimal operators (in more mathema- tical terms: the one-sided invariant vector fields on the group manifold) by using the Lie algebra of the group. We get ten left generators (which are ten independent right invariant vector fields) and ten right generators. In the enveloping algebra we choose a complete subset of commuting operators, which include the two Caslmir operators (namely p2 and W2), four left operators, and four right operators.

* In partial fulfilment, of the requirements for the degree of Doeteur d'Etat ~s-Sciences Physiques, Facult6 d'Orsay, Universit6 de Paris 1969.

** Laboratoire associ6 au C.N.R.S.

332 N.X. HAI:

In Chapter II , we compute the eigendistributions of this Abelian subset. The eigendistributions, which depend on the set of eigenvMues, are in ~act measures on the Poincar6 group and when properly normalized can be handled as matrix elements of irreducible representations of the group, expressed on a continuous basis. By this way, we compute the matrix elements for all ~,he unitary irreducible representations, and we get a fairly simple expression for them.

In Chapter III , we reconstruct the unitary ix'reducible representations from its matrix elements and we give some properties of the matrix elements. As an application, we compute the characters for non-zero mass representations.

Chapter O. Notation and Symbols

A point in space time is denoted by a latin letter or by its components:

x = (x0, x ) = (xo, x~, x2, x3) .

The metrics is positive on time like vectors, ~he Minkowskian product is

x y = xoy o - x y .

C and R are the complex and real number fields, 2V the set of integers. The S t ( 2 , C) group is the group el two by two complex uuimodular

matrices, its elements are denoted by capitals: X, Y, Z, and the matrix elements of X are X ~ with i , ~ = 1, 2.

We need some subgroups of S L (2, C): S U (2): unitary unimodular subgroup, its eIements are U, V, W . . . S U ( 1 , 1): an dement of this group is called 0 (or P, Q . . ) and

satisfies the relations 011 = 0~2; 012 = 0~1. ST(2) : triangular subgroup, its elements are called R(or S, T . . . )

and satisfy the relations R21 = 0, RI~ = R~2. a a

Wese t O i l - OX~ and 0 " - a x * •

We define the differentiation matrices ~x, ax*, 0x¢ by

X = ~x~I x ~ ] ' ax=\a~l a~/,

and we use the notation

L j=1,2

A B . . . C ' O x = ( A B . . . C ) ' ~ x .

In the same way we define ° °)

a~= (a~o,a,~)= ~ o ' a~ , a ~ ' a:~ •

Harmonic Analysis on the Poincar~ Group 333

Some particular four vectors are useful: 0 3 03

1 = (1,0,0,0); 1 = (0,0,0,1); 1 = (1,0,0,1);

we shall use the same notat ion with an M replacing the l ' s in these formulae:

0

M = ( M , 0 , 0 , 0 ) , etc.

Using the standard Pauli matrices ~0, ~1, ~2, ~a, we define

H (x) = X°ao + xlax + x~a~ + xaa3 ,

and ~ny X belonging to SL(2 , C) realizes a Lorentz transformation when we give its result X x on any x by the formula:

I t ( z x ) = x t t ( x ) x t .

The stabilizator of a time-like (a light-like or a space-like) four 0

vector is isomorphic to S U (2) (to S T (2) or ¢~) S U (1, 1 )) which stabilizes 1

1 or . The homogeneous spaces i.e. the spaces of left cosets of these subgroups, can be identified to the corresponding orbits, tha t is:

1. the positive energy shell of the hyperboloid k ~ = 1 ; 2. the positive light cone; 3. the hyperboloid k ~ =~ - 1.

For example, in the first case, the one.to-one correspondence is given by 0

k = X - 1 1 ,

and the group acts on its homogeneous space by the canonical trails- formation

k-+ X k .

A canonical decomposition of SL(2 , C) is obtained once a section in the classes is given, i.e. when one has chosen an element in each class. The simplest choice is to take the unique positive Hermit ian element in each class, which is named HA. We then have the relations

0 0 H~ -I I = k -= X - I i

X = UHk ( ~ t h U ~ S U (2)).

I n the other cases, the results are the same and the formulae too are similar: i) S U ( 2 ) : ( U C S V(2))

x = UH~

B~ =[1 + ko - k a ] [2 + 2,~o]-~/~ (1) 0

k = X - 1 1 . 23 Commun.ma~h. Phys.,Vol. 12

334 N.X. H~a:

H~ is a C °~ function of k.

fi) S T ( 2 ) : (S E S T ( 2 ) )

X = S T ,

03 k = X - 1 1 ,

Te is a C ~° funct ion of k for any k on the light cone.

iU)

~' = (kl -}- ik~)]2:¢

NU(1, 1 ) : ( 0 E S U ( 1 , 1))

X = 0 F ~

F~ = {[ko + (1 - l k l ) ~ ] [21kl - 2]-1/~}

{ r lk l + h,:, - it,~,o-., + ,;ho"~] [2 lk l ([kl + k~)]-~,'~} 3

k = X - 1 1 .

(1')

(I")

da0 _

d 8 U -

if S = ( ; '~t~ $19 g-~epl2]

1 d~a

I dSa 2 ~ a o

We define the no rm

llXP ; !_ffTr X X ¢ = o.

This funct ion is left and right invariant by S U (2) translations and its reciprocal hs~perbolie cosine can be used to define a distance between the

the normalizat ion of the other measures:

1 _ d dSS = a 2 ~ ~odSI~dS%

/~k is a C °~ function of/c o and ~. The invar iant measure of S L ( 2 , C) splits into an invaxiant measure on the subgroup and an invar iant measure on the homogeneous space. We take it to be

d 6 X ~- d 3 U d3k/]co = d a S dak/[co -~- d sO dak//~0 (2)

where d 3 U is the normalized H a a r measure of S U(2) and the Eq. (2) fix

Harmonic Analysis on the Poincar4 Group 335

classes of S U (2). We quote some relations:

IIH~Hr~II ~ = p .

ItHz -~ UH~II ~ = k . V k

IIEr ~ 0 F ~ V = k~(tl0II ~ - 1) + 1 (3)

ItT~ -~ S Z k i l ~ = ~( l iS~ u - 1) ÷ 1

lIHkli ~=/Co; IIFkti ~ = Iki; iIT~lI ~-~ (ko2+ 1)/2k 0.

The Poincar6 group. ~¥e denote its elements by Greek letters ~, fl, 7 - . . or by a couple

(x, X ) or (y, Y ) . . . where x CR 4 and X C S L ( 2 , C). The product is denoted multiplicatively and is defined by

(x, X ) (y, Y ) = (x + X x , X Y).

The two-sided invariant measure is taken to be

dlO o~ = d4x do X .

Chapter I. The Infinitesimal Operators

The Poincar6 group ~ being endowed with its invariant measure, one can introduce the Hi]bert space of square integrable functions on ~ , which is called ~f2(~). The group acts on this Hilbert space by the formulae :

( u (.)/) (7) = I( ~-~ 7)

( v (.)/) (7) = / ( r . ) .

These are the left and right regular representations of the group. These representations are uni tary and one can easily check the relations

u( . ) u ( ~ ) = u ( ~ ) ; Vt(.) = u ( . -1)

v(~) v(fi) = v(~fl); v*(~)= v(~-l) .

The involution I of ~ z ( ~ ) , defined by

(I/) (~) = 1(.-i) ,

real~es a uni tary equivalence between these two regular representations. When a one-parameter subgroup is given by its generator Q, we can

derive an infinitesimal operator on the group manifold. Giving the one- parameter subgroup in its exponential ~orm, let us define

i0 ia Qal = - ~ U (ei*°)[ Qa/ = - ~ V (e~*Q)I ;

whenever / is a smooth function, for examp]e C ~, these expressions make sense and we get the infinitesimal operators associated to any Q. Each Qg (resp. Qa) is a right invariant (resp. left invariant) vector field on the 23*

336 N . X . H)~:

group manifold. They have the same commutat ion relations as the generators of the group itself and any Q~ commutes with any Q~.

In this way, one can exhibit a complete basis of right (resp. left) invariant vector fields by taking the Q~ 's (resp. Qa's ) associated with the four components P~ of the energy momentum and the six generators of the homogeneous Lorentz group. These last generators can be written as two vectors M and N (corresponding respectively to rotations and to boosts: see for example ref. [4]). We give here the explicit expression of the infinitesimal operators:

Pg = i0~

2 M g = - 2 i x A a~ - ~ X • a x + X C a • ~x* (4)

2 N ~ = - 2 i x ax. - 2 i x o ax - i a X • a x - i X ¢ a • ~ x t

and P a = - X - 1 P g

2 M ~ = X a . O x - a X t "Ox¢ (5)

2 N ~ = i X a . O x + i a X * . a x , •

The symbol A stands for the vector product in R ~. We can then easily extend the domain of the infinitesimal operators

to include the distributions on the Poinear6 group. First, let us define convolution on the group: if S and T are dis-

tributions let us set

< S , T, ~> = <S(~) ® T(fi), ~(~fl)>

whenever this expression makes sense, for any ~ in 9 (~) (which is the classical Schwartz space of functions on ~ ) and with S(~)® T(fl) as the product distribution on the product space ~ × ~ .

Let then ~ be the Dirac measure at the point ~ E ~ . We define

u(~) T

V(~)T This definition is consistent with a function. Finally, we define

Q g T = -

Q ~ T = -

(this derivative is to be taken in

~ s a t T

---- T $ 8~_1 .

the previous one when T is given by

i ~ U (ei~Q) T

a iS[ V(e~)T distribution space [5]) then, one gets

the following relations (which are expected !):

<Q~T, ~> = - <T, Qg~>

and the Q's appear as first order differential operators.

~rmonic Analysis on the Poincar6 Group 337

Let us now consider the algebra generated by these infinitesimal operators. We can easily construct a set. of commuting operators in a standard way. First, among those generated by left operators, as the last ones have the commutat ion rules of the Lie algebra of Poinear6 group, we can choose the six operators:

Pg, Wo~, W~

where, as usual, the W,'s are the Pauli-Lubanski operators

and Moo is related to the M's and N's in the standard way. As the left operators commute with the righ~ ones we can also take the six opera~ors

Pa, Waa, W~

but in fact, we get the relations

and

and we have only ten algebraically independent commuting infinitesimal operators.

We can compute explicitly the Pauh-Lubanski operators:

2 w . o = - ox + x * ( P . a ) . a x ,

2Wo = - P g o a X • Ox + PgoX¢a • Ox¢+ iPg A a X . Ox+ i P , A (X 'a ) . Ox¢

2 Wa o = X (Pa a). Ox - (Pa a) X ¢. Ox¢ (6)

2 Wa = PaoXa" O x - PaoaX ¢" Ox¢ - iPa A (Xa) "Ox- iPa A (aX¢). Ox,.

Whenever P~ = M 2 > 0, let us define (h stands for d or g)

S~, = s M -1 Brat (e is the sign of the eigenvalue of Pao)

Sh + = Shl + iSh~

S~ = Shl - iSh~.

On the eigenspaee defined by Po = - M, we get the simpler explicit form:

S~o=O

2 S t = - a X . O x ÷ X t a . O x t ~g+ * * * $ Xa1021 + = XlzO~2 - X2xOa~ - X220a 2

S ~ = X210115 * ~- X22012* * - Xl1021 - X 1202z (7)

2s. * * * * X21 021 -- X22 022

- Xl1011 - Xa~012 + X210~1 + X~O2z

338 N.X. I-IAz:

o and when Pa = M

S~o = 0

2Sd = X a . a x - a X * . a x , ~ 2 $ * * $

= - X ~ . a x l - Xe2a21 ÷ Xx~O~2 ÷ X2~0~2

S~- * * * * = _ XllOz2 - X21a22 ÷ X120n + X2~21

Xl l ~11 -- X21 ~21 ÷ X12 012 ÷ X22 022

÷ XnO, 1 + X~I 0~, - X ~ O ~ - X 2 ~ 2 .

I n fac t these opera tors are first order differential opera tors s tab i l iza tor group, and in th is case, on the S U (2) subgroup.

(8)

on the

Chapter II. The Matrix Elements as Eigendistributions

The solut ion of the equat ions

P ~ , ( x , X ) = - k' ~p(x, x )

can be wr i t t en in t he form

(x, X) = e~ ~' ~ ~' (X) .

The equat ions Pa YJ (x, X ) = k y~ (x, X ) can then be wr i t t en as

( x - l k ' - k) V ( x ) = 0 .

These equat ions have a non-zero solut ion only when k and k' belong to the same orbi t and the i r solut ion is a d i s t r ibu t ion the suppor t of which is the submani fo ld defined b y the equa t ion X - ~ k ' - k = O.

1. The Non.Zero Physical Mass Case

The c o m p a t i b i l i t y condi t ions are

k ~ = k' 2 = M ~ > 0

0 One can solve the pa r t i cu l a r case when k = k' = e M (s is the sign

of energy) b y se t t ing ~ ' ( x ) = ~o(X) ~ 6 3 ( x )

where co Oa (X) is the measure defined b y

f ¢ o ~ a ( X ) ~ ( X ) d 6 X = f g ~ ( u ) a a u Z~(2, G) S ~](2)

( this is t he i n v a r i a n t measure of t he submani fo ld of S U (2)). I f one defines (with p2 = if2 and P0q0 > 0) the Dirae measure on the

mass shell a t the po in t q:

a)68(P, q) =[Plt ~a(p _ q ) O(poqo) ,


then one can write

In fact when one parametrizes X by setting X = Utt~, X becomes a function of U E S U(2) and of p which belongs to the positive mass shell of the hyperboloid p2 = 1, and one gets

0

X-11 = p

d6 X = ds U d~P Po

0

When the integration over dap is performed, one gets p = t so tha t X = U and

S U(2)

The remaining equations are

( W ~ + M2s(s + 1)) eo88(X) ~oo(X ) --- 0

(Wg~ - M~') ~o~(X) ~0(X) = 0

(Wd3 - M~) o~3(X) ~po(X) = 0 , 0 0

With the conditions Pg = - M and Pa = M, the W'~s, up to the constant factor M, are the infizfitesimal operators of the S U(2) subgroup, and these equations lead to the matrix elements of many representations of the SU(2) group. We can exhibit them easily by finding a generating function, namely the function

~(X) = (XI~ + X12 + X21 + X~2) ~'~

which is an eigenfunction of W 2 with - M2s(s + 1) as eigenvalue. The matrix elements are obtained by expanding ~5 into eigenfunctions

of Wga and Waa; this can be easily done by performing a Fourier integration over the two one-parameter subgroups generated by Waa and Wg a. The normalization is then computed in such a way as to satisfy the correct muJtipIicative property for the matrix elements.

The results are C 2n 2~

(9)

C = [ ( s - ~')! (s + 2')! ( s - ~)! (s ÷ ~)!]1/2/(2s)[

(throughout this paper a! = F(a ÷ 1) is the factorial function).

340 N.X. HAI:

These functions are matr ix elements of irreducible representations of SU(2) . W h e n 2s E N, the representations are un i ta ry and the matr ix elements have the usual properties:

+ s

L~,~(X*) = 1%,~(X)

/ s r a (~ ) =/~za,(X) (~ is the transposed matr ix o~ X)

and when X belongs to S U (2), the two last relations give

l ~ , (x*) = t*~,~ (x ) .

I n fact, these functions are defined over the whole group SL(2, C). Now we can write the solution of our problem in the part icular case:

0 ~(X, X ) - - e i eMxo) (~a(X) / s ) , a ( X ) ,

then the general solution can be obtained by translat ing this part icular solution :

v(sk 'k2 '2; (x, X)) = #I¢'~a)(~8(Hk, XH~-I)/~rz(H~,XH~ -1) (10)

and we write it in a shor t -hand nota t ion

= e~k' ~co~a/~r~(Hk, XH~-l) .

The ranges of the eigenvalues are restricted by the conditions: 1. k and k' belong to the same sheet of the mass hyperboloid

k ~ = k ' 2 = M 2 > 0

2 . 2 s belongs to iV and defined by

2 , ~ ' = - s , - s + l , . . . , + s , and H~ is

o H~-I M = ek .

2. The Zero-Mass Case

The compatibi l i ty conditions are now

k' ~ = k s = 0 and k~k o > 0 .

03 I n the particular case where we have k o = k~ = e 1 (s is the sign of the energy) the solution is

v,' ( x ) = ~1 ( x ) ~ ~3 ( x )

where v~a (X) is an invariant measure carried by S T(2) and defined by

f ~,Sa(X) q~(X) d6X = f ~(S) daS. SL(2, C) s T(~)


I n the same wa y as for non-zero mass, we define the Dirae measure on the zero-mass shell a t the point q:

v~a(P, q) = ]Po] ~(P - q) O(poqo) and we can write

Now, the W~'s are operators of the Eucl idian group of the plane and the remaining equations give rise to the mat r ix elements of this group, which can be found for example in V ] ~ ] ~ K ~ ' S book [6]. For us, it is ve ry easy to get t hem by the same method, with a generating function. We take

~p(i) = e x P 2 i (rX12 + rX~2 + 2sq)}

w h e n X = ( o ~ xl~ ] e - ¢ C / z ] •

This funct ion is an eigenfunetion of W~ with - r ~" as eigenvalue and we expand it (in the same way as before) into eigenfunctions of Waa and Was; after normalization, the result is

1 f d ~ e x p ~ /~,~,~(X) = ~ - o 01)

• i (rX12e -in + rX~2e in + 2 ( t ' - ~)~ + ( t ' + ~)T)

and when r = 0, this matr ix element is zero unless t = ~'; r mus t be a real positive number ; the eigenvahie of W 2 is - r 2, t ha t of Saa (= Wda ) is 1 and t h a t of Wga(= - Sg3) is 2 ' ; d indicates the par i ty of 21 and 2) / ( i f ~' = 0, t and i ' are integers and if ~' ~- 1, t and ~' are half integers). I n the case where r = 0 (finite spill), one has t = ~'.

W i t h these expressions, one can compute the character of S T(2) (r :~ 0)

2u

0

( ~ ' means a summat ion over all )~ such t h a t 2~ has the par i ty of e) and

also the or thogonal i ty relations:

S T(2)

These functions are mat r ix elements and we have the relations

~" ]r~ r a (X) ]r~ ~.a ( Y) = / ~ , a ( X Y ) (for r 4 0)

/~,~ (x*) =/*,~, (X).

342 N.X. ttAI:

We can then write the particular solution and obtain the general one by translat ing it :

v(re k'k)J ),; (x, X)) = e*k" ~v63(T~,X T [ 1) ],~z,z(Tk, X TF 1) (12)

= #k'~v3a/~r~(T~:,XTF. I) in shorGhand notat ion, and the result is similar for the case r = 0.

The ranges of the parameters are given by : - k and k' belong to the same sheet of the light cone. -- if r > O, s = O, ~ and ~' are integers

= 1, ~ and ~' are half integers - if r = O, the conditions are the same, but moreover Z = ~'.

3. The Unphysical Mass Case The compatibi l i ty conditions reduce to

k ~ = / ~ , 2 = _ M 2 =< O .

3 In the particular case where one has k = k' = M, the solution is

~' (X) = ~1 (X) ~ (X)

where $~3(X) is an invariant measure of S U(1, l) defined by

f ~ ( X ) q~(X)d6X= f ~o(O) d~O. SL(2, C) S U(1,1)

As for the other eases, one can define the Dh'ae measure on the imaginary mass shell (defined by q) at the point q:

~3(p, q)= ]Pol 53(p _q) O(poqo)

and then one can write

Now, the I~,'~ are operators of the S U (1, 1) subgroup and the remaining equations lead to its mat r ix elements. Matrix elements of the S U(1, 1) group are given by BARGMA~ [7] or BARUT and FRO~-SDAL [8]. F r o m our standpoint , they are easily obtained by using a generating function, the same as for the S U(2) subgroup;

~(X) = (Xn + XI~ + X~I + X22) ~

and the corresponding eigenvalue for W 2 is now M l s (s ÷ 1). One te rm of its formal expansion (with ce ÷ fi = 28) wri t ten as

(Xn + X~2) ~ (Xs~ + X22)~ is an eigenfunction of S~o(Sgo is defined to be - M -1 . Was ) with I

2 - ( f i - c¢) as eigenvalues. Now, we expand it into eigenfunctions of

Harmonic Analysis on the Poincur6 Group 343

Sao(Sao = M -1 " Wao) by using an integrat ion over the one-parameter rota t ion subgroup generated by Sao, and we normalize it by a factor N~(2', 2). The result is

N,(~', ~) 2~ y~ra(X) -- 2~ f dcf(Xlle*~ + Xl~e-~)s+a"

o (13)

• ( / 2 1 e i g A- i 2 2 e - i c p ) s - ) . ' e - 2 i ) . e p "

When the un i t a r i ty relation and the composition relations

yJ~z,~(X -1) = ~ r ( X ) * (for X C S U ( 1 , 1))

are required, the ranges of the parameters are restr icted and the nor- realization factor is sett led up to a phase fac tor :

1 l. s = - ~ - + io (Q > 0); 22 and 2 2' are integers having same par i ty

and any sign ~v~ (2 ' , 2) = 1

2. - 1 __< 2 s < 0; 2, 2' are integers of any sign

N8(2', 2) = [(s + 2)! (2' - s - 1)l]1/~ [(,s + 2')l ( 2 - s - 1)1]-1/2

3. 2s is an integer >-_ - 1, 22 and 22 ' are integers of the same par i ty as 2s and

a) 2 a n d 2 > s

N ~ ( 2 ' , 2) = [(s + 2)! (2 ' - s + 1)!]~/~ [(s + 2 ' ) ! ( 2 - s - 1 ) ! ] - ~ / ~

b) ~ a n d 2 ' < s

_~'~(2', 2) = [ (s - 2 ) ! ( - 2 ' - s - 1)!]~/2 [(8 - i t ' ) ! ( - 2 - s - 1 ) ! ] -~ /~ .

These mat r ix elements have the same normalizat ion as those of Bargmaml in the discrete case (3a and 3b).

To distinguish between them, we introduce a parameter s which can be s = 0 : cont inuous series with 2 and 2' integer (case 1 or 2) s = 1 : cont inuous series with 2 and 2" hal~ integer (case 1) s = + : discrete series with positive 2, 2' (case 3a)

= - : discrete series with negative 2, 2' (ease 3b). Then the values of a and e completely fix the representat ion and we

can denote the matr ix elements by the symbol

L~,~(X) •

F r o m BA~GMA~'S work (or directly f rom the expressions of the matr ix elements given here in the case of the discrete series) we get the orthogo- nal i ty relations :

344 N.X. HA~:

1. Discrete series with s ~ 0

f ]*sr~(O)/~181~;~l(O)d30= 2 ( 2 s + I) ~s81~,~ld~.~,~; • (14) s g (1,1)

I 2. Continuous series with s = - ~- + i# (# > 0 and s = 0,1)

f /~x,~(O)/~,~,( )d30 - 4e ~(q - Co) ( ~ ' ~ . ~ . (143

B y summing up the diagonM matr ix demen t s given in their explicit form, we can compute the characters Z~s ~or:

1. the continuous series - I f T r X > 2, we set T r X = 2~ch~/2 with ~ real a n d ~ = ± I a n d w e get

Z~,(X) = ~%h(2s ÷ 1 ) - ~ . s h - ~ -1 . (15)

- I f T r X < 2, we get z,~(x) = o

2. The discrete series - I f T r X > 2, we set T r X = 2~eh~/2 and we get

Z~s(X) = 12 ~'r2s~--~ (2s + 1)In/2[ [sh~/2[-x . (15')

-- I f T r X < 2, we set ~.~X = 2cos ~v/2 (~ is taken to be the angle of the rota t ion conjugate to X), and we get

Z~(X) _- 1_2 ~°~'~e~(2~ + ~)~/~ (sin ~/2) -~

Now, we define F~ ~or k ~ = - M ~ by setting

3 F i - I M = k .

Then by t ranslat ing the part icular solution by F~ and .F~,, we get the general one

~p(~s~'I¢~')~; (x, X)) = e ~ ' ~ ( F ~ . X F i -~) £ ~ , A F ~ , X F i -~) (16)

= e " ~ ' ~ , ~ ( F ~ , X F i - ~ ) .

Chapter IH. Properties of the Eigendistribufions

Let us define a double index (~, a) which, as we shall see, characterizes the representat ions:

a) Non-zero physical masses M > 0

v = M ; ~ = ( s , s ) ( w i t h e = ± 1 and 2 s E N )

s is the sign of the energy, s is the spin of the representat ion and then 2 , ~ , ' = - s , - s ÷ l . . . . . + s .


b) Zero mass v = 0; a = (e, e',/~) e is the sign of energy; e' d is t inguishes be tween

the r ep resen ta t ions : - s ' = + or - : f inite spin represen ta t ion , 2# ~ N and the he l ic i ty is e'/x - e ' = 0, 1: infini te spin representa t ion , A and 2' are integers if

e' = 0, half in tegers if e' = 1, the e igenvalue of W 2 is - #s. e) I m a g i n a r y masses i M

= i M , a = ( s , e ) .

I n the same way, e dis t inguishes be tween the represen ta t ions : - if s is an in teger >= O, s = ± gives t he sign of $ and X' which are

in tegers a n d s2, sA' > s 1

- if s is a half in teger > - ~ - , s = ~ gives t he sign of 2, A' which

are half integers and cA, s2 ' > s 1

- if s is cont inuous and - ~- < s < 0, e = 0, 2 and 2' are integers

I - i f s = - ~ + i @ w i t h @ > 0 ; e = 0 , 1 a n d 2 A a n d 2 X ' a r e o f t h e

same p a r i t y as e. A n d in a n y ease, k and k' belong to the same mass shell defined b y

k s = k'S = v~. (and in t he eases a and b, t h e y mus t have the same sign of energy).

Now we recons t ruc t t he u n i t a r y represen ta t ions f rom the m a t r i x elements . F i r s t , we define t he space 3 - ~ of t he pa r ame te r s (k, 2) :

a) 3 - 8 _~,~ = 5 ~ M X Is

b) ~-o**',----- 3¢r~ × I8,, (17)

C) ~"iJlse ~ ~ i M X ISe.

I n these formulae, ~ / i s the mass shell of mass M a n d sign of energy e, 38"~ is the e-energy l ight cone, ~fiM is t he i m a g i n a r y mass shell (k s = - M s) and discre te spaces for the p a r a m e t e r A are given b y

I ~ = { - 8 , - s + 1 . . . . , s }

I~,~. = {e'r} if e = ~ (2r t hen belongs to N)

= { z + e ' / 2 } i f 8' = o , 1 ( 1 ,) Z is t~hen t h e add i t i ve group of in tegers and Z + ~- i ts t r an s l a t ed b y ~-

I ~ = { e ( s + l ) , e ( 8 + 2 ) . . . . } if e = 4-

= {z + 8 / 2 } i f 8 = 0 , a .

W e define the measure on J '~o b y t a k i n g the p roduc t of the i n v a r i a n t measure over t he mass shell (we normal ize i t ~o be dSk/MSlko[ in non- zero mass case and dSk/[ko] in t he zero-mass case) b y the n a t u r a l measure

346 N.X. Ha~:

over the discrete space Ia for the index 2, which is denoted ~ :

v, E J , a ~ v = (k, 2)

d,k Z (ff v 2 4 0)

d 3 k y~ tk01 ~ ~ ~ = 0

Y ~ is a locally compact space and we can define the matr ix elements as measures over Y-~ × J ~ . Let us denote them by the symbol (v and v' belong to Y-,,~) :

~Z~,(~)

and we realize the representations in the space L ~ ( Y ~ , d # (T)):

( ~ * ( ~ ) W) (*) = f ~ Z ' (~) 9 (~') d/~ (~').

In fact, it is easily seen tha t this integral exists for almost all v' and defines an element of L2(~-~,, d#) which can be written as

+ s

{qlM~s(X, X)~} (k, 2) = e ~ ~v, ]s~r(HkXHyl, k)q~(X-~k, 2') ~ ' = - - S

{~'°'"~(x, X)q~} (k, 2) = e ~ S f,'~z'(T~XT~l-~})cP( X-~k, 2') (18) ;/ E I g r

{q,, ,z-(z, x)q~} (~ 2) = e , ~ S /-~,(F~XF:~I-~)~( X -~ , ,~').

In these formulae, k' and k belong to the right mass shell and the discrete space I , has been defined just before. With all the properties we established for the matr ix elements of the respective subgroup, we can check tha t ~hese formulae define all the unitary representations of the Poincar~ group (of non-null energy momentum). They are Wigner's well- known formulae and we established them as a subsidary result f rom our computation of the "exponentials" of the group. These exponentials are needed for establishing the explicit formulae for Fourier transforms on the Poincar~ group.

We might also note tha t this gives a method of determination of irreducible representations of a Lie group; one computes ~he eigendistributions and adjust the normalization to an adequate measure on t, he index spaces. This works for the Poincar~ group and might work for other Lie groups.

The matr ix elements, computed as eigendistributions are, in fact, fairly simple measures on the group, whose carriers are algebraic sub- manifolds. We shall s tudy their properties from a measure-theoretic point of view.


1. Unitarity One extends to distributions the involution I defined in Chapter I by

setting ( I T, 9) : (T, 1 9 )

where T is a distribution, ~ a test function and

( I q~) (a) = ~ (a-~). When the right-hand side is computed, the following relations are obtained :

Z~(~k'k;. '~; (x, X)) = : ( ~ k k ' 2 ; . ' ; (x, X))

I y~(rek' k ~' ~; (x, Z)) = ~o*(rekk'~$'; (x, X)) (19)

I ~ ( ~ k ' k Z ' 2 ; (x, X)) = W*(~s~';A'; (x, X)) .

This is the unitarity of the generalized matrix elements expressed over a continuous basis indexed by (k).).

2. Composition Relations As we have established tha t the eigendistributions are matrix

elements of unitary irreducible representations of the Poincar6 group, the relation

f ~ , (a) ~ 5 " (8)d ~ (~') = ~ , (a/5)

mnst be interpreted as a weak integral of a measure:

f ~ 9 ( a ) {f~$5"(fl) 9(z") d#(z")} d/~(z') = f ~g, ,(af i) qo(z") d/x(z")

and has a meaning for ~ny 9 continuous with compact support. We can also have another interpretation as a weak integral of measure over the group itself, that is

f q/~,(a) {f ¢Z~%,,(fl) ~(fl) d~0fl} d/,(~') = l(~0).

The left-hand side is easily computed by applying the Fubini theorem on the product space 3-,~× ~ and when the integration over d#(v') is performed first, the expected result is obtained:

f + s d 8 k' y~(skk' 2A' ; a) F(sk' k" 2' 2" ; 8) [M~I~A I -- ~o(skk" AA" ; aft)

. d ~ k " f __r v,(re'kk';~Z'; a) ~,(r~'k'k"~'Z"; yJ(re'kk",~Z" ; afl)

¢/-~ "~" E l d r

d~k' = y)(eskk" ]~." ; aft). f z yJ(eskk'2~.'; a) ~(esk'k"3, '2"; 8) M*]kg]

(20)

348 N.X. H ~ :

3. Orthogonatity Relations

They are s imilar to t he re la t ion

+co

f eik~e-ik'~dx = 27r (5(k-- k') - - o o

which m u s t be unde r s tood as

+ c o

f e ~:k~ (e -~k'~, 9a(k'))dx = 2or (~(k - k'),g0(/~')) = 2~q~(k) . - - o o

This re la t ion is t rue when ~ is smooth enough (for example when q0 is integTable and differentiable). This can be seen as a weak in tegra l of d i s t r ibu t ions on the index spaces, a n d th is me thod gives a jus t i f icat ion for t he formal computa t ions , a t leas t in t he phys ica l non-zero mass case.

Ano the r w a y to u n d e r s t a n d such a fo rmula is to t ake i t as a Bessel- Pa r seva l fo rmula for t he Four i e r t rans form, then the o r thogona l i ty re la t ions hold for al l the represen ta t ions presen t in t he P lanchere l measure (see P a r t I I ) .

W e jus t l is t now the resul t which are easi ly compu ted fo rma l ly :

(~asl = (2~)~ ~4(]~, _ k~)M~o~8(k, kl) 2-5~4-Y ~ : ~ 4

f yv(re'k'k)/)~; ce) * . . . . (rlel kJzl ).121 , 5) d1°o~ (21)

(2 j r )~4(k , , 3 ~(r - r~) = - kl)~,~ (k, kl) ~7~ ~ a ' ~ . ~ ' ~

(2~) ~

w h e n 2 s ~ N a n d e = ~ : , a n d

= (2~r)~6*(k ' - k~) M ~ ( k , k~) ~ , ~ 8 ~ , ~ C 7 ~ ( 0 ) ~(q - 0o)

1 when s . . . . ~- + i o and e = O, 1 ; C~(O) = 4~ ( t a n h z ~ ) ( ~ - ~ .

d. The Character o] hTon-Zero Mass Representations

A m a t r i x e lement is a measure on the Poincar6 group, so we can define, for a n y tes t funct ion ~ the q u a n t i t y


When this function of v is integrable, it defines a functional of ~ which is in fact the trace of the operator

~,(~) = f ~,(~:) q~(~) d~O~

defined in the representation space. This computation has been ah'eady done by Joos and SCHRADER [9],

but we were able to prove that, with an adequate limiting process to define the in te~al

5-va

it makes sense for any ~p continuous with compact support in the physical mass case; the character is a measure on the group.

In the non physical mass case, we need someting more, namely that is continuously differentiable on the group, then the integral makes

sense and defines a measure. The characters can be written pointwisely, they are invariant by

conjugation and we give here their expressions for the diagonal element of each conjugation class:

a) Physical mass:

~M(~0(X, X) -- ~(n)X,(~) f #~(ko~,-~,~) dk3 (22) 4sin2 ~/2 d k0

?,,+,~,~ 0] where X == \0 e-(,7+~)/~]

sin(28+ I)~/2 X~ (~) = sin ~/2

ko=l/ + b) Non physical mass

~(~)

+ Z~(X) ~(~) + ~(~ + 2=)

where X has the same definition as before, k a = ~ + M ~ and Z~s(X) are the characters of the representations of S U (1, 1) we computed in Chapter I I , For more details one can see ref. [9, 10].

References

1. LURgAT, F. : Quantum field theory and the dynamical role of spin. Physics 1, 95 (1964).

2. Kr~BE~G, A.: Internal coordinates and explicit representations of the Poinear6 group. Nuovo Cimento, Ser. I0, 53 A, 592 (1968).

3. Fucks, G.: Fields on homogeneous spaces of the Poincar6 group. Preprint Ecole Polyteehnique, France.

24 Commun. ma~h. Phys.,VoL 12

350 N.X. H ~ : Harmonic Analysis on the Poincar~ Group

4. SCEW~]~ER, S.S.: Introduction to relativistic quantum theory. Evanston- Elmsford: Row and Peterson 1961.

5. SCaW~TZ, L. : Th6orie des Distributions. Tomes I, IL Paris: Hermann 1957/59. 6. V~mx~KI~, N. ¥ . : Special functions and group representations. ~Ioseow:

Naul~ 1965, (In Russian). 7. BARGIVIA~-~, V. : Irreducible unitary representations of the Lorentz group.

Ann. Math. 48, 568--640 (1947). 8. B ~ u % A. O., and C. FRONDSDAL: On non-compact groups: Representations

of She 2 + 1 Lorentz group. Trieste preprint 1965. 9. Joos, H., and R. S. SOm~ADE~: On the primitive characters of the Poincar6

group, Commun. Math. Phys. 7, 21--50 (1968). 10. FucHS, G., and P. RENO~D : The characters of the Poinc~r6 group, Preprint

Ecole Polytechnique. France (to appear in J.~I.P.).

NG]~EM X ~ N H~_t Laboratoire de Physique Th~orique et Hautes Energies FacuR~ des Sciences Batiment 211 91 Orsay, France

harmonic analysis on the poincaré group

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