Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations
1969 A new approach to representations of the Lorentz group William Henry Greiman Iowa State University
Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Physics Commons
Recommended Citation Greiman, William Henry, "A new approach to representations of the Lorentz group " (1969). Retrospective Theses and Dissertations. 4655. https://lib.dr.iastate.edu/rtd/4655
This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. This dissertation has been microiihned exactly as received 69-15,613
GREIMAN, William Hemr, 1942- A NEW APPROACH TO REPRESENTATIONS OF THE LORENTZ GROUP. Iowa State University, P!h.D., 1969 Physics, general
University Microfilms, Inc., Ann Arbor, Michigan A NEW APPROACH TO REPRESENTATIONS OF THE LORENTZ GROUP
by
William Henry Greiman
A Dissertation Submitted to the
Graduate Faculty in Partial Fulfillment of
The Requirements for the Degree of
DOCTOR OF PHILOSOPHY
Major Subject: Physics
Approved:
Signature was redacted for privacy.
Signature was redacted for privacy.
Head
Signature was redacted for privacy.
Iowa State University Of Science and Technology Ames, Iowa
1969 i î
TABLE OF CONTENTS
Page
I. INTRODUCTION 1
A. Previous Work 1
B. Work Done in This Thesis 3
II. LIE GROUPS 5
A. Introduction 5
B. Lie Algebras 8
C. invariant Integration 10
D. Representations of Lie Algebras and Lie Groups 12
E. Decompositions of Lie Groups 13
III. REPRESENTATIONS OF THE LORENTZ GROUP AND POINCARE GROUP 16
A. The Lie Algebra 16
B. Representations of the Lorentz Group 19
C. Representations of the Pol nearé Group 21
IV. AN IWASAWA DECOMPOSITION OF THE LORENTZ GROUP 25
A. The Lie Algebra 25
B. Representations of the Conformai Group 26
C. Representations of the Lorentz Group in a Conformai Group Basis 28
D- Expansion Coefficients 29
v. MATRIX ELEMENTS OF THE LORENTZ GROUP IN AN ANGULAR MOMENTUM BASIS 33
A. Matrix Elements of Pure Lorentz Transformations 33
B. An Integral Representation 39
C. Asymptotic Behavior of Matrix Elements 40
D. Orthogonality Relations 42 i i i
VI. HARMONIC FUNCTIONS FOR THE POINCARE GROUP 4$
A. An Angular Momentum Basis 45
B. A Conformai Group Basis 48
VII. APPENDIX 53
A. The General Expansion Coefficient 53
B. An Integral Representation 54
C. A Special Case of the Iwasawa Decomposition 56
VIII. LITERATURE CITED 59
IX. ACKNOWLEDGEMENT 61 I
1. INTRODUCTION
A. Previous Work
Symmetries have long been used to make predictions about the behavior of a system of particles, even when the form of the interaction between particles is not known. If the symmetry can be expressed as
invariance of physical quantities under a group of transformations, the behavior of the system will be greatly restricted.
The most widely applied group of transformations has been ^he group of rotations and reflections in three dimensions. The usual procedure is
to express physical amplitudes as a sum of irreducible representations of this group. Products of amplitudes may be expressed in terms of irre ducible representations with the aid of the Clebsch-Gordan series and recoupling coefficients. This procedure is known as harmonic analysis and provides a systematic approach to symmetry groups.
While the importance of the Lorentz group In physics is well estab lished, the unitary representations of the Lorentz group have been used infrequently in the solution of physical problems. One reason is that the various functions necessary for harmonic analysis on the Lorentz group have not been studied as thoroughly as in the case of the rotation group.
The finite dimensional representations of the Lorentz group have long been known. These representations are obtained from the representa tions of the rotation group in four dimensions by the "unitary trick."
Since the Lorentz group Is noncompact any unitary representation must be infinite dimensional. In 19^5 Dlrac (1) pointed out several unitary representations and posed the problem of classifying all Irreducible unitary representations. In 19^7 Harish Chandra (2) proved that In unitary 2
representations the two invariants of the Lorentz group have the values
2 2 2 2 J«K= -tv 3 J-K=^-v-l
where ^ is half integral and v is real. In the case ^ = 0 v may also
be pure imaginary with [v[ Gel'fand and Naimark (3) (4) constructed representations of the Lorentz group on the space of complex valued functions f (z), z = x + iy for which Jj* |f (z) 1^ dx dy < œ . While the space of complex valued functions is convenient from a mathematical standpoint^ it has not been used in the solution of physical problems. The main disadvantage is that the complex variable z has no immediate physical interpretation. In physical problems one encounters functions of one or more four vectors which transform according to the Lorentz group. For example scalar fields satisfy the transformation law X) (AX) » Dolginov et al. (5)(6)(7)j derived unitary representations of the Lorentz group in a form which is more useful to physicists. The resultSj which were obtained by analytic continuation of unitary representations of the four dimensional rotation group^ are not in the most convenient form. In 1965 Strom (8) obtained certain matrix elements by integrating differential relations. The general matrix element was then to be found by ladder operators. In 196/ Strom (9) found an expression for the general matrix element by a different method. In the above work states in a representation are labeled by the ro tation group. However, the Lorentz group also contains the conformai 3 group in a plane and the three dimensional Lorentz group as subgroups. In certain problems it may be convenient to take a basis in which states are labeled by one of the latter subgroups. For example, it would be more convenient to expand a function of a lighflike four vector in terms of the conformai group basis since the conformai group in a plane is the little group of a light-like vector. A number of advantages of the conformai group basis are discussed by Chang and 0'Raifeartaigh (10). In addition they derive the transformation coefficient relating angular momentum states and conformai group states for the case j = m. In this thesis the angular momentum and conformai group bases are studied. The relationship between these bases is expressed by the Iwasawa decomposition of the Lorentz group. This decomposition implies that every Lorentz transformation may be written as the product of a rotation and a conformai group element. This relationship provides a simple derivation for many of the functions needed for harmonic analysis on the Lorentz group. B. Work Done in This Thesis Chapter 2 contains a number of well known results concerning Lie groups which will be required in later chapters. Chapter 3 is a summary of well known results concerning the Lorentz group and Poincaré group. In chapter 4 representations of the Lorentz group are constructed in a conformai group basis. The matrix elements of a general Lorentz transformation are derived in this basis. Finally the expansion coef ficient connecting a conformai group state with an angular momentum state is derived. These results are original with the exception of the expansion coefficient f-' for the case j = m which has also been calculated by Chang 4 and 0'Raîfeartaigh (10). Matrix elements of the Lorentz group in an angular momentum basis are considered in chapter 5. Expressions for these matrix elements have been derived by several authors (7)(10). However^ the method of section A is original and results in several new expressions for the matrix elements. In addition the symmetries of matrix elements are very easy to derive by this method. In section B a new integral representation for the matrix elements is derived and is used in section C to derive the asympto tic form of the matrix elements. The orthogonality relations for the matrix elements have been listed by Dolginov and Moskalev (7)= An alter native proof of the orthogonality relations is presented in section D. In chapter 6 harmonic functions for the Poi ncare group are considered» In section A harmonic functions in an angular momentum basis are derived. Any wave function may be expanded in terms of these functions. In section B a corresponding set of Lorentz harmonics is derived in a conformai group basiso The results of chapter 6 are original with the exception of the angular momentum basis for the case of spin zero which has been derived by (11 )(12)(13). This work suggests a number of possible directions for future work. In order to expand functions of space-like four vectors it would be useful to have matrix elements in a basis labeled by the 3 dimensional Lorentz groupj 0(2^1). The expansion coefficients relating the 0(2^1) basis to the conformai group and rotation group bases would be useful. Finally the Clebsch-Gordan coefficients for the Lorentz group have been studied very little. Most of the available results are in terms of the space of functions of a complex variable. 5 11. LIE GROUPS A. Introduction An abstract group is a set of elements G which possess a law of compo sition that associates with every ordered pair of elements x,y a unique element z in G. The following conditions must be satisfied. 1. Associative law: x(yz) = (xy)z (2.1) 2. Unit element: There exists in G a left unit e such that for any x in G ex = X (2.2) 3. Inverse: Every element x in G possess a left inverse -1 X such that X ^x = e (2.3) -1 The group postulates imply that e is also a right unit and x is a right inverse. Groups do not arise in physics as abstract groups but as transforma tions on vector spaces. They may be discrete groups such as the rotations and translations which are symmetries of crystal fields or they may be continuous groups. The continuous groups which have occurred in physics have been Lie groups for example^ the rotation group, the Lorentz group, and SU(3)- Lie groups have an analytic structure which simplifies the study of their propert ies. In this chapter we consider properties of Lie groups which will be required in later chapters. We begin with the definition of a Lie group. A group G is a Lie group if: — 6 1. Elements of G In a neighborhood of the unit element can be put in a one to one correspondence with points x'(i = 1, . . . , r) in a region of a r dimensional Euclidian space with the unit element corresponding to the origin. 2. The coordinates z' of the product of elements x,y are analytic functions of x'j y'. i i, 1 r 1 z = cpcp' (x(x% J . . . ,x ; y , 3 y*") = cp'(x,y) . (2.4) The group postulates impose several conditions on the functions tp S inee cp'(Ojy) = y' and cp'(x,0) = x' (2.5) the functions cp' have the Taylor series cp i(x,y), \ = X i+. y i+ . i x-'yJ k + r where we have assumed summation on repeated indices and the remainder r is of the order four in x and y. The commutator q = x y X V ' (2.7) of two elements x,y will play a special role in the development of Lie groups. In order to derive the Taylor series for the commutator, we first - ] find a second degree approximation for x in terms of x. The result, which follows from 2.6, is 7 (x ') ' = -x' + ai. x-^x^ + 0(3) . (2.8) J K Relations 2.6, 2.7, and 2.8 imply that the commutator has the Taylor ser ies q' = (ajk " + 0(3) • (2.9) The numbers S'k " ' ^kj are called the structure constants of the group G. The structure constants clearly satisfy the relation Cjk = - Ckj . (2.11) A further condition on the structure constants arises from the associative law cp'[x,cp(y,z)] = cp'[cp(x,y),z] . (2.12) If we substitute the Taylor series for cp' in 2.12 we have the condi tion k^tm ^-Lj^mk ^/Cmk '^•Lkm ^-Lmk ^m^k ' (2.13) If we anti symmetr ize 2.13 with respect to the indicies k^-L^m and use definition 2.10, we find It has been proven (14) that Lie groups with the same structure constants are isomorphic in some neighborhood of the unit element and that there exists a Lie group for every set of numbers cj satisfying 2.11 and 2. 14. B. Lie Algebras We now consider the relationship between the structure constants and a Lie algebra. In order to define the Lie algebra of a group we consider all one dimensional submanifolds x'(s) (curves) of the group containing the unit element at s = 0. The tangents to these curves at s = 0 (2.15) S=0 form a r dimensional vector space. This vector space becomes the Lie algebra for the group when we define a suitable product of two vectors. If x'(s) and y'(s) are curves in G with tangent vectors rj' and f ' we define the Lie product c' = (2.16) to be the tangent of the commutator z'(s') = [x(s)y(s)x ^ (s) y ^ (s) ]' where s ' = s^ . (2.17) The Lie product is therefore given by c' = • (218) In order to simplify the notation we introduce a basis E. for the Lie algebra and make the following definitions. 77= E.77' = cJi^E. . (2.19) Conditions 2.11 and 2.14 on the structure constants imply two condi tions on the Lie product. 1. (2.20) 2. J??] + [[C.,77] == 0 9 Having defined a Lie algebra^ we list several important facts about Lie algebras and Lie groups. In order to state the following definitions more concisely we define the Lie product of sets R.,G^ to be the set = {the set of all Ç = where Definitions: A Lie algebra is said to be Abe 1 ian if [G,G] = 0 A subset ^ of a Lie algebra is said to be a subalgebra if [R,R] cR . A suba 1 gebra _R of a Lie a 1 gebra jG is ca 1 1 ed an idea 1 if [G,R]CR . A Lie algebra is called semi s imp le if it has no nonzero AbeIi an ideals. We note several properties of Lie groups which are reflected in their Lie algebras. 1. Abelian groups have Abe 1ian Lie algebras. 2. If ^ is the Lie algebra of a subgroup R and G_ is the Lie algebra of the group G then R^ is a subalgebra of G^. 3. If R is an invariant subgroup of G then the Lie algebra ^ of R is an ideal in Car tan's criterion for a Lie algebra to be semi s impie will be impor tant in the following work. To state it we introduce the symmetric bilinear form (Killing form) on the Lie algebra JG as follows: If E. is a basis for G^ with 1 0 [E.,Ej] = f = f'E. , V=ri'E. , (2.21) we define the Killing form to be B(f.7?) = f'77^ • (2.22) Car tan's criterion: G is semi s impie if and only if the form B( j ) is nondegenerate, (i.e. the determinant of is nonzero) . C. Invariant Integration The topological properties of a Lie group are related to the Killing form on the Lie algebra of the group. To see this relationship we consider integration on a group. We say a Lie group G is compact if for every real continuous function f defined on G we can define an integral I = J'f(x) p(x)dx (2.23) where p is independent of f and I has the following properties: 1. j'p(x)dx = 1 (2^ 24) 2. If f(x) > 0 but is not identically zero Jf(x) p(x)dx > 0 . (2.25) 3. If a is an element of G J'f (ax) p(x)dx = j f (x) p(x)dx . (2.26) The integral I is colled the invariant integral on G„ If z' - T'(x^y) is the law of composition for the Lie group, the funct ion f,(x) =• k (2.27) Idet^ 1 ?)y"^ y=0 satisfies 2.25 and 2.26. To see this we define w = ax and change the 11 variable of integration. j*f (w)p (a 'w)dx = j'f(w)p[cp(a %w)] jdet ^ |dw (2.28) S inee [co(a ',w),y]| = [a 'jcpCw^y)]! = (a %w) (w;,y)| (2.29) r*w ?)y y=0 we have p[(o(a ',w)] = (2.30) Idet (a ^ jw) I which completes the proof- If the integral of 2,27 exists the group is compact since 2.24 may be satisfied by normalizing 2.27» If G is compact I has the following addi tional properties (14). J'f (xa) p (x)dx = Jf (x)p(x) dx (2.31) Jf(x ')p(x)dx = J'f (x)p (x)dx The relationship of the Killing form on the Lie algebra to the topology of the group is seen in following theorems (15). I f G. is the Lie algebra of a compact group^ the Killing form on G is negative semi def in i te, i.e. 6(77,77) ^ 0 for all 77c G. (Weyl) If G is semi s impie then G is compact if and only if the Killing form on G^ is negative definite, i.e. B(77,7^) < 0 for all 77€G. If 2.24 does not converge the group is said to be noncompact. 12 D. Representations of Lie Algebras and Lie Groups Suppose that to each element f of a Lie algebra there corresponds a linear transformation A(^) of a vector space U, which in general is of different dimension than the Lie algebra, with the properties: A(f + 77) == A(f) + A (77) A (eg) = cA(g) 3 c a scalar (2.32) A([fjT?!) = Ai^)Ain) - A(T7)A(^) Such a correspondence of operators is called a representation of the Lie a Igebra G_. In the cases we consider, a representation of a Lie algebra will determine uniquely a representation of the Lie group with the following properties: 1. For every element s of the Lie group there corres ponds a linear transformation D(s) of the vector space U. 2. D(st) = D(s)D(t) 1 , (2.33) 3. D(s"^) = D '(s) . Iff is an element of the Lie algebra, a corresponding element of the Lie group will be represented by D = exp[A(f)] = E ^ ° (2,3^) n We will not prove this result but note that if D(s) is a representa tion of a Lie group and if s'(T) is a one parameter curve in G with «' = I ' (Z-35) T=0 then the operators 13 A(f) = D[S(T)]1 (2.36) ' T=0 represent the corresponding Lie algebra. We call a representation reducible if there exists a subspace R of U which is invariant under the transformations D(s). In other words if ja'^cR then (a' ^ = D (s) for all seG. (2.37) If there are no invariant subspaces in U we say the representation is irreducib1e. if the vector space U has a Hermitian inner product (ccjp) and if where lo:' > = D (s) (2.38) 1P'> = D(s)lP> for all jo:), seG then the representation is said to be unitary. E- Decompositions of Lie Groups We list two decompositions of semi s imp le Lie groups which will be used in later chapters. We omit the proofs (16) which are long^ since the detailed structure of all semi s impie Lie algebras must be considered. Cartan's theorem: If G is a noncompact semi simple Lie algebra G^ has a direct sum decom position G = J€K satisfying: 1. ^ is a subalgebra ^ [K,£l C-K [K,K] C 2. The Killing form restricted to ^ is negative definite. The Killing form restricted to J<^ is positive definite- ]k 3. If G îs a connected Lie group with Lie algebra £ then G has the unique decomposition G = RA (2.39) where R is the image of and A is the image of K, under the exponential map. If _K' is a maximum Abe 1ian subalgebra of Kj A can be decomposed as A = RjA'R^^ (2.40) where A' is the image of J<'. Therefore G has the decomposition G = RgA'R^ . (2.41) The Iwasawa decomposition: I f is a Car tan decomposition of a noncompact semi s imp le Lie algebra G^ and K' is a maximum Abe 1 ian subalgebra of_|^ then there is a basis E. for G^with the properties: 1. K' is diagonal and has real eigenvalues. ra,E.l = X. (a)E. (2.42) CceK' A.. real — I 2. The set of vectors E. which do not have identically zero eigenvalues, X j (c^), are called root vectors. This set of root vectors may be decomposed into two ni 1potent subalgebras as follows: Let OC^ be a regular element of J<', i.e. X . ^ 0» Then N . = f E ; \ ;(a ) > 0} —T I I O (2.43) N_ = [E.; X;(%) < 0} 15 are ni 1potent subalgebras of G_. A Lie algebra jL is ni 1potent if [L'L] *i [L.CL.L]! i- [L,L-| [Lf C J • » • » • • -13 = 0 K'@N ^ is a solvable subalgebra. A Lie algebra ^ is solvable if 1.] = ^ k I2 " ^ ill L = 0 • —n ^ is the direct sum of the subalgebras If RjAjN are the connected subgroups correspondin respectively to the subalgebras JLjJi'then G = RAN and this decomposition is unique. 16 III. REPRESENTATIONS OF THE LORENTZ GROUP AND POINCARE GROUP A. The Lie Algebra The homogeneous Lorentz group is the group of real linear transforma tions C on the four dimensional space X^j X^ ) which leave the form )(% = (x°)^ - (x^)^ - (x^)^ - (x^)^ (3.1) invariant. The metric tensor for this space is Ij u = V = 0 g""^' = 9,^^ = ^-l,Ai=v=l;,2j3 (3.2) Oj V . Invariance of 3.1 implies that a Lorentz transformation ty where X'"" = x\ must satisfy the equation . (3.3) The Poincare group contains, in addition to Lorentz transformations, translations a. X"'' = x^ + (3.4) We denote elements of the Poi ncare group by (a, -t) where a is a 4 vector and is a 4 x 4 matrix satisfying 3.3 The transformati on law X'^' = x"" (3.5) implies a multiplication law (a., V) = (a + -La ) (3.6) for the Poincare group. We next derive the Lie algebra for the proper Lorentz group to illus trate the results of chapter 2. A general Lorentz transformation may be written in the form 17 C = 9", + u"^ . (3.7) Only six of the numbers are independent, since 3-3 implies the following ten relations among the u'"^. juv ^ ..Vju ^ jj.a yy u"":" g =0 (3.8) ^07 We choose 1 , /(V vu \ '"'Atv ^ (u - U ) U to be the six independent parameters of the Lorentz group. The parameters of the product element iM = t(u) -tCv) follow from the multip!ication law 3-6. „«6 . u'^P 4. vW + g , u"'' (3.9) 7 A. In order to find the structure constants of the Lorentz groupj we ~ ojs construct the Taylor series for the parameters w We antisymmetrize 3-9 on 3 and use 3-8 to obtain a second order ^ ae . ap _ . approximation for u and v - The result is -ap -ap ^~ap W =u +V ~ jU V ~ O n I We may anti symmetr ize the coefficient of u v on Uj v and 03 o and define ° - 3",, 9"p' 9VB where A(\j T) anti symmetri zes the indices A.j T. The structure constants are then (f»P . - a"P . (3.12) uv, ap (Tp ooj Mv 18 We introduce a basis -i = î for the Lie algebra. The factor -i insures that will be Hermi tian in a unitary representation of the group. In this basis we have the commutation relations "ap' = ' (Sva "yp + 9»p "va - %% • 9«a"vp > • (3.13) If we introduce a basis -iP for trans 1 at ions, we find the following jU commutation relations for the Poincare group. '"«V' fpl = 'Kp''» - pj = 0 (3.14) It follows from the commutation relations that the Lorentz group is a noncompact semi s impie group. However the Poi ncare Lie Algebra has an Abelian ideal, P^j and the group is therefore not semi simple. In the usual treatment of the Lorentz group operators (3-15) k' = M°' are defined. Physically J' is the angular momentum (generator of rotations and k' is the generator of pure Lorentz transformations. The commutation relations for J and K are [j', jj] = i j" rj% KJ] = ! c'j'' K" rK% K-j] = -i ( 'j"" . (3.16) We see from 3.16 that 3.15 Is a Cartan decomposition of the Lie algebr Car tan's theorem implies that any Lorentz transformation may be written as 19 the product of a pure Lorentz transformation and a rotation. L = e~"^*^ = R((o)A(Q:) (3-17) If we choose as a maximum Abe 11 an subalgebra of J< we may further decompose 3-17- L = R e"'"'^3 R' = e-iej2 î YJ3 ^-'*7^3 (3.18) B. Representations of the Lorentz Group We now consider unitary representations of the Lorentz group. The Lorentz group has two independent invariants. These may be chosen as 2 2 —f —* J - K and J • K . In an irreducible representation these operators are scalar multiples of the unit operator. Nai mark (17) has shown that in unitary representations they have the values - 1 J ' K = (3.19) where v is a real number and is a half integer. If ^ = 0^, v may also be a pure imaginary number. The first case is called the principal ser ies_, and the second is called the supplementary series. The vectors in a representation are labeled by the rotation subgroup- We then have states j/tj v, j^ m). In this basis the generators have the following representation (17)° J+k, j, m> = \l, vj m+n J_1j, m> = ^ m-1) I Lj, V, j;, m> = m|t, V;, j, m\ K_^ I Vj 7(j+ l; v) j-L, v, j+ 1^ m+l> 20 TU^T) OH-I 7(j;^,v)l^,v,j-l, m+1 > K_l-tj Vj j, m) = -io:jj^^j^_^ 7 (j+ 1 v) v^ j+ I^, m-l> j(j + O J%m-1> - '0=:!^^ ^i-1 ^ 0 ^ j" ' ^ ^ > UjV,j,m> = - 7(j+ l; -tj v) 1^^ v^ j+ 1, m> \ + iajj^ 7 (j v) 1^^ v^ j-l^m> (3.20) j Cj +1 ) •' where J+ = J] ± '^2 ^ ^+ = 1^1 ± '"^2 ?(j;i,v)=. y ' "i,n = f(j-m)0+ n)]^ • We assume that states are normalized (U V, m\^, V, j % m'^ = 6jj , 0^^, (3.21) and the relative phase of states in a representation is fixed by 3.20. From 3.20 we see that -t is the minimum value of angular momentum occurring in a representation. The problem of finding matrix elements of an arbitrary Lorentz trans formation is reduced to finding the matrix elements of pure Lorentz transformations in the z direction by the decomposition 3.18. A general matrix element then has the form (t, V, j, m|L 1^;, V, jm"> = ,^,(L) = min(j^j') : . :; , ;i Z DJ^(4),e, Y)AJJ (a, v) dJ , (0, P, 7) (3.22) • / • • I \ UK I ' Il I II11 n=- mi n (jJ J ) where _ A^-' (a. t- v) = (Ij V. j, n |e ' 3 j n>. (3-23) 21 ; I; ' The explicit form of may be found in (18), will be calculated in chapter 5« C. Representations of the Poincare Group The Poincaré group is not semisimple, since it is a semidirect product of the invariant Abeli an subgroup T of translations and the Lorentz group. Representations of the Poincare group may be constructed by using little group techniques (19). We begin by partially labeling states of the Poincare group by irreducible representations of the translation subgroup. Therefore we have a basis in which P^ is diagonal i.e. p'"^ 1 p > = p^ 1 p > (3.24) We next consider the effect of a Lorentz transformation on a fixed representation, p^ , of the translation group. Since the translation subgroup is invariant, we obtain either the same representation p^ or a new irreducible representation p Those Lorentz transformations which leave p^ unchanged are called the little group of p^. Since P^P^ commutes with all generators of the Lorentz group, its value is invariant and characterizes the irreducible representation of the Poincare group. Physically P^P^ = m^ is the square of the mass, and we consider the case of positive mass. We choose a fixed p^^ in each representation and a fixed transformation L(p, p) for each p^ in the representation which transforms p^ into p^. We then have u . , / ~\ I |p" > = L(p, p) 1 p" > p^p^ = . (3.25) 22 The conventional choice for is P^ = (m, 0,0,0). (3.26) The labeling is completed by specifying which irreducible representa tion of the little group is carried by (p). The little group of p is the rotation group in three dimensions. Therefore the states are labeled by 2 the values of J and acting on jpV We define a general state by s, = L(p, p) ( p, s, (3c27) where 1 Pj s, = s(s+l) 1 p, s, X) J3 I s, X) = X I p, s, X> . (3.28) We consider two cases for L(p, p). In the first case we choose a pure Lorentz transformation, i.e. L(p, p) = L(p) = e (3.29) where p^ = m (cosh CK, p sinh Q!) . Th is basis will be called the canonical basis for the Poincare group. In the second case we choose L(p, p) to be the product of a rotation and an acceleration in the z direction. H(p) = ^-iKja (3.3 0) where p^ = m (cosh a, sinh a sin G cos (jl , sinh a sin G sin (j), sinho: cos G) . This basis is called the helicity representation, since IT • is diagonal and has the value 23 The Irreducible representations of the Poincare group are therefore characterized by two numbers m_, the mass, and s, the angular momentum of a state of zero three momentum. Physically s is the spin of a particle having this representation. We assume that states in an irreducible representation are invarlantly normali zed — I f (jl is a wave function defined by A. 4\(P) = (P;, Sj \1 (j) > (3.32) then the inner product of two wave functions is (t *') = s (p) (p) • (3.33) X o The transformation properties of the basis states under an arbitrary element of the Poincare group may be derived by using the Cartan decomposi tion of the Lorentz group. If L is an arbitrary Lorentz transform we have L L(p) = L(p') L H(p) = H(p') (3.34) where L(p) and H(p) are defined by 3-29 and 3-30. Therefore the basis states transform as follows e'^*^ L(p, s, = e'P ^ D®,^(R^) )p', s, e'^ ^ L|p, S;, X>^ = e'P (R^) | p', s, ^ (3»35) X k Finally we consider the spin operator S . The spin operator is defined to be = L(p) L(p) ' (3.36) 24 when acting on a basis state of momentum p. It follows from 3.29 and 3.30 that is diagonal in the canonical —* —* basis and P - S is diagonal in the helicity basis. IPj = Xjp, P • S|p, s, X = 1p1 X |p/ s^ (3.3 7) With the aid of the operator identity e^Be ^=B+[AjB]+-^, [AJ[AJB]]+. .• • (3.38) we may express 3*36 as mS = P° J - P X K - • (3.3 9) o The following commutation relations for follow at once from 3-36. [s/, gj] = iE^Jk gk [s'j p^g =0 (3.40) 25 IV. AN IWASAWA DECOMPOSITION OF THE LORENTZ GROUP Ao The Lie Algebra The sets {J.} and [K.} defined by 3•15^ form a Cartan decomposition of the Lorentz group. If we define P = + Kg J G = , (4. I ) we see that the sets {J.} [K., } and [Pj Q.} form an Iwasawa decomposition I J of the Lie algebra. Therefore every proper Lorentz transformation may be decomposed uniquely in the form L = RAN (4.2) where R is a rotation, A is an acceleration in the z direction, and H _ 2-1(aP + bS). (4.3) In the new basis the commutation relations are [P. Q] = 0 rjj, P] = iS [K^ P] = iP [J], K3] = 0 Q] = -iP [K^, Q] = 'Q. [J,, P] = ÎK3 [Jg, P] = -ÎJ3 [J,, K3] = i(J,- P) [J,, 0] = ÎJ3 [Jg, Qj = 'Kg [Jg, Kg] = '(Jg - %) [Jj, Jj] = iC;jkJk ' From 4.4 we see that [P, K^ } generate a subgroup. This sub group is isomorphic with the conformai group in a plane. P,Q. generate translations, generates rotations in the plane and K^ generates scale changes- In this chapter we construct representations of the Lorentz group in a basis labeled by this subgroup. 26 B. Representations of the Conformai Group In this section we construct representations of the group generated by {Pj Qj K^}" It follows from the commutation relations that exp(i2TiJ^) is an invariant of the group. Only single and double valued representations occur in the Lorentz group, so we discuss the cases = e = + 1 . (4.5) 2 2 We take a basis in which and P + Q. are diagonal. In this basis we have Jgl X, — X I r, Xj e) (4.6) where X = 0, + 1, + 2 . . . for e = 1 X = ± Y f ' ' ' fo'' E = -1 and (p^ + Q^) I r, X, e) = r^l r, X, 0 The commutation relations 4.4 and the identity = B + [A, b] + -j, [A, [A, B]] + . . . (4.8) impl y i i [J], P+] = ± P+ (4.9) where P+ = P ± 'G . We make a choice of phase for states in a representation by taking jl, Oj 1) and 11, -I) as standard states in the single and double valued representations and defining a general state by 27 le^j X3 e > = e'^'^3 |]^ Xj e) P+ I Tj Xj c ) = r)r, X + 1, c) . (4.10) We assume states are invariantly normalized. (r, 6 ( r', e > = r Ô (r - r ' ) (4.11) For wave functions ^i^^r) = (r^ X | (j)'> the inner product is then We now calculate matrix elements of finite transformations. A general transformation may be written in the form T(e, cc, a, b) = e-'GJ; ^-i{aP + bd) _ (4.13) The result of exp[~i(aP + bQ.)] acting on a state is found by expanding the exponential. ^-i(=P + bS),r, jr, X> where 7 = We use the fact that commutes with P to obtain + — s |r, x>. (4. >4) n^ m n. We use 4.10 and define a new index of summation, X' = X + m -n in order to write 4.14 in the form CO (TO . z |r. X') . (4.15) \'=-,T3 n=o HI ni (n+\'-X)i The final result follows from the definition of Bessel's function (20) 28 ^-î(aP + = ^ ^ ^ J%,_% (v/a2 + b2 r) |r, X '> (4.16) X' Is/aZ + b2/ ^ The matrix element of a general transformation follows from 4.6, 4.10 and 4.16. (r'j X' I 7(9, a, a, b) |r, X ^ = r'6(r' - e^r) e-"''® ^ (-^a^ + b^" r) (4.17) C. Representations of the Lorentz Group in a Conformai Group Basis We now construct representations of the Lorentz group in a basis of the form I'to "^3 '"j where r, X label states in a representation. The representations are characterized by the two invariants /tv = J • K and - 1 = . (4.18) In terms of the basis [J., P, Q.} these are U+ = (iK^ ± J3 - I)^ - - 0.2 + 2J^ P_ (4.19) where J+ = Jl ± ÎJ2 The effect of J_^ acting on a state can be related to by using 4.10 and 4.19. J+ It, V, r, X> = 29 2~ [a+îv)^ + - (ÎK^ ± X)2] r, X± 1> (4.20) Iwasawa's decomposition, 4.2, implies that any Lorentz transformation may be written in the form L = T(e, a, a, b) (4.21) where T is defined by 4.13 and its matrix elements are given by 4.17. We calculate the matrix elements of the rotation about the y axis by expanding in an angular momentum basis. The result is (•Ij V, r X' je '4^2 \ t, v, r, \> = CO s r J X j't/j Vj j, X ^ d , (^) Vj jj X j'Lj j=max(x,X') (4.22) The explicit form of the expansion coefficient v, r, xj-L, v, j, X / is given in the next section. Since the left side of 4.22 is defined to be 6 , rfi (r-r') when w ^ = Oj the right side must be considered to be a distribution. In other words if Y(r) is a square integrable function, we give the right side of 4.22 meaning by making the definition j' (4, V, r % X' je '4^2 v, r, X> Y (r) = o CO y. (I, V, r x' 1 tj, V, j, X'> dJ, ((|)) j=max(x, X' ) / ^ (tj V, J, Xj-t, V, r, X) %(r) , (4.23) o D. Expansion Coefficients In this section we derive and solve differential equations for the expansion coefficients relating the conventional angular momentum basis to the conformai group basis. We introduce the notation 30 (t, v_, j, mj-Lj Vj r, m > = (r, v) (4.24) for the expansion coefficient. It follows from 4.10 that -j- jj mj^j Vj r, m^ = Vj mlK^j-C/j Vj r^ . (1^.^25) Two differential relations for f^ may be derived by multiplying 4.20 on the left by an angular momentum basis state. ( (tj, Vj jy "1 I 27 [6± + r^ - (i ± X)^] |-L, v, r^ X ± l!> (4.26) With the aid of 4.24 and 4.25 we obtain 2rJ(J ± m + 1) (j - m) fjJ^j [ il — iv)^ + - (r — m - 1)^1 f^ _ (4.27) In the cases m = j and m = -j 4.27 becomes a second order differential equation since fj+i = f-j-i = ° • The solutions are J + 1 fj (O I, v) = Cj rJ Z^_.^(r) (r, t, v) = Cj rj*' (4-28) where Z is a modified Bessel function. Since the angular momentum states Ai are normalized, we have < C, V, j, m I V, J m') = ÔJJ , ônim' = no , - T IT" m I x| U j \ m'> X o 3! = X f fi (r) . (4.29) o -1/2 i Therefore r f^ must be square integrable and is a modified Bessel function of the third kind (20)» The genera] expansion coefficient is found by using h,2f as ladder operators on the functions fj and f^j. The result, which is proved by induction in appendix A, is u v) = ^ n?o (1, n)(-2j. n) Viv-j^n^n - Ba, V, J, (,, n)(-2j. n) Vlv-J-m-.n where (a, n) = a(a+l) .... (a+n-I) B (tj, V;, }, m) = 1 • 2JI (2j -f I).' ]2 .r(j+ îv+l)r(j-îv+])(j+m)!(j-m)!(j+^)](j-^)i ' (^.30) We list several properties of the functions f^. From 4.3 0 we note that v) = -v) = f^(r, m, v) = im) . (4.31) The functions fj^ have the following integral representation which is derived in appendix B. 32 v) = 2 where (_1)j-t2m-t-:Vr(j_;v+i) c{u \>, J:, m) = U + m)l r(-t-iv+l ) r (i+m)l (i+>fji (2i+0 1^ (4.32) [2r(j-iv+l )r(j+ Iv+l ) (j-m)] (j-^)ij ^ and 2^2 'S a hypergeometric function (21). According to 4.29 the expansion coefficients satisfy the orthogonality relation ^ (r, & v) fjjj {r, ^ v) = 6.ji • (4.33) o . Finally we list an additional orthogonality relation which is proven in chapter 6. ^ . J C (•"' t, v) f^ir, V, v') m=-j o r 33 V. MATRIX ELEMENTS OF THE LORENTZ GROUP IN AN ANGULAR MOMENTUM BASIS A. Matrix Elements of Pure Lorentz Transformations The results of chapter 3 reduce the problem of finding matrix elements of an arbitrary Lorentz transformation to that of finding matrix elements of an acceleration in the z direction- In this section we calculate matrix elements of pure Lorentz trans formations in the z direction by two methods. The first method is simple and the resulting expression exhibits the index symmetries of the matrix element explicitly. The second method yields an expression which is useful when a state of low angular momentum is involved in the matrix e 1 ement. We calculate the matrix element /J Vj j m> (5.1) m by transforming to a conformai group basis. The result, which follows from 4.10 is (a, v) = p f^/r) f^ "(yr) o (5.2) a where y = e . It is clear from 5.2 that the index symmetries v) = f|(r^ m, v) = t, -v) (5.3) imply corresponding symmetries for A^-* . (a, v) = A^-" (a, m, v) = A^^ (a, -v) = aJJ ' (a, I, vV' (5.4) 34 ii ' Two additional symmetries may be derived for - If we make the change of variable r' = yr in 5-2 we find (a, v) = (-CKj v) . (5.5) If we substitute the identity e'^^2 e"'"'^3 = e'^'^3 (5.6) in 5«I we obtain ' (a, v) = (-l)j"j' A^J' (-c, Z, v). (5.7) We next substitute the explicit form of the expansion coefficient f-' in 5.2 and evaluate the integral. Since two expressions for f-* are m m given in chapter 4, the integral may be evaluated in four ways. However the symmetry («, -L, v) = A^j (a, -v) reduces this to two cases. We choose these to be (a) f-^ and f-^ are both of the first form in m m 4.30 and (b) is of the first form and f-^ is of the second form. We m m evaluate the first case below. The second case is similar and we list the result. We substitute 4.30 i n 5.2 to obtain A^-^' (CKj t.j v) = Vj j, m)B(-L^ j m) 2 S n=o p=o n+r, (m-J, n) Cc-j, n) (-i v-j, n) (m-j p) (^-j % p) (i v-j p) (-2)"+P (l,n) (-2j,n) (l,p) (-2j', p) y' r • (5.8) The integral in 5» 8 is evaluated with, the aid of the formula (20) 35 f t ^ K (at) K (bt) dt = •J U V (|)X-i p(±wri) r('T""^) ) /I -V°M-À) _ 2 r /1+v+^-X 1+v-Ai-X .1^.1 b_ \ src-x) 2 1 -ReX- 1 ReM |- \ Rev |> 0 . (5.9) The final result Is min 4 ^ min 4, ^ ;; I j-m J -m AjJ, (pCj -tj, v) = N(-t,j Vj j'j m) % E n=o p=o (-j-j '-I. n+p) (m-j, n) (^-j, n) (m-j % p) (t-j p) (-j-j '+/(,+m, n+p) (1,n) (-2j, n) (l,p) (-2j p) ^-L+i v+rrrf-1 ^ (/L+m+1, j+ iv+l-n; j+j'+2-n-p; 1-y^) valid for ^ + m > 0 (5»10) where (-L+m)! (j+j '--t- m)l N('Lj V, jy j'y m) = (j+j'+i)! r(j-iv+i) r(j'+iv+i) 1 2 2j:(2j+i)! r (j+iv+i) r (j-iv+i) 2j • j (2j'+i )'. r(j'+iv+i )rO'-'v+1) (j+m): (j-m)i (j+t): (j-t)L (j'+m)i(j'-m)i (j'-^)I (j'+t)i The second case is evaluated in the same way. The result is , j-m j'+m («J -tjf v) = N' (t, V, j, j'y m) S 2 n=o p=o 36 (-j-j n+p) (l-jj n) (-i v-jj n) (<,-j \p) (-i v-j ',p) (-J-J Vj n+p) (1,n) (-2j, n) (1,p) (-2j % p) Ti (I+^riv, j+ l-m-n; j+j'+2-n-p; 1-y^) (5.11) where r(1+t-i v) r (J+J '+ ]-^+iv) N' (t, Vj j'y m) = (j + j ' + l)i '(j-m)i 2ji (2j+ l)l (j'+m)i 2j (2j'+ l)i (j+m)i (j+t)! (j-'t)ir(j+ iv+l )r(j-îv+l ) (j '-m)i (j '+Z)i (j '-t)!r(j '+ ÎV+1 )r(j '-ÎV+1 We note that while the above results were derived for the principal series they are also valid for the supplementary series. It is interesting to note that if the factorials are expressed as gamma functions, equation 5.11 may be obtained from 5.10 by the substitution v) = (a^ im) . Other forms may be obtained with the aid of symmetries expressed in 5.4, 5-5 and 5*7" We now derive an alternative expression for Aj^-^ . This derivation, which is not as straight forward as the previous one, depends on the resu1t sinh a o (5-12) The proof of 5»12 follows from 4.4 and 4.8. The matrix element A"^ ^ (CK, v) may be written as O-.jlur |S,-S> (5.13) 37 by using the properties of and J_ At this point we use 5-12 to move the exponential in 5-13 to the left of the ladder operators. The matrix element in S»13 then becomes =-'="3 Is. -s> = O.j'le"'" 3 (e"j^ - sinha)-'"'" |s, -s> . (5.14) Since commutes with 5.14 may be expressed as 'T (J':") (J,j|e-'OK3 pj -s> (5.15) q=o where is a binomial coefficient. We note that if j > s. the sum begins at q = j - s since |s. -s> = 0 for n > 2s. We change the Index of summation to p=q+s-j and obtain Tril" ) (-s;nh 5.16 is also valid for s > j since the binomial coefficient is zero unless j - s + p :> 0. We use the result 1 2 (2s-p)l 2sl Is, -s> = Is, s-p> and 5.13 to obtain the expression 1 2 (j+m)i (s-m)J s-m ly v) = _(s+m) i (j -m ) '. 2jI _ p=o 1 j-(2s^j2 a(s-m-p)(.^,„„^)J-s+p q^j]g-!«K3 p^J-s+p (5., 7) 38 We calculate the matrix element in 5.17 by inserting a complete set of conformai group states. The result is ^Cl(j-s+p) J fj rJ-s+P-l 16 Ji s-p where y = , (5.18) We evaluate the integral in 5.18 by substituting the explicit form of f. The final result is m v) = M(4., Sj (1-y^)-• ^ s-m p ({rs,n)(iv-s,n)(j+ l-iv, p-n)(j-&+l, p-n) S S (. ) p=o n=o j (l,n)(-2s,n)(l,p-n)(2j+2, p-n) , 2 p — ) 2^] (J 1^ j+ iv+1; 2j+2+p-n; 1-y ) (5.19) 2 where V, j, Sj m) = (j+m)I(s-m)! 2sl (2s+])i (j+-L) ^ (j-^l) ' F(j+ iv+1) r(j-iv+l) (s+m)J (j-m)i 2ji (2j+ ])i (s+^)I (s--t) ' r(s+iv+l ) r(s-iv+l )_ Several remarks concerning 5.19 are in order- We note that any matrix element can be related to the case j >s>m->-t^Oby using 5.4, 5-5 and 5.7" This results in the minimum number of terms in 5.19° For example if s = m we have ^ (nJ v) = M(/Cj Vj j, Sj s) yS+^^'v+1 s 2F, (j j+ iv+1; 2j +2; l-yZ) . (5.20) 39 It is clear that 5*19 correctly satisfies (0, I,, v) = 6jg B. An Integral Representation An integral representation for Aj|^-^" may be derived by substituting the integral representation of fj^ in 5*2 and changing the order of integration. This integral representation will be useful in discovering ii the asymptotic behavior of Aj^-^ . Upon substitution of 4.32 in 5.2 we obtain ^3 -tj v) = Citj Vj j, m) vj j S m) y^ 00 œCO œoo r 2 V 2 r rt2 / dr J dt J* ds e-s-t-St - \ï ^2t-2w+] c:: •> -I • «•") y>0 The order of integration may be changed and the r integration performed. —^ The result is the integral representation 00 CO (oCj I3 v) = Q.itj Vj j'j m) y^ ' ^ J dt J ds b o .-s-t^-iv -t+iv j+^+1, Irj; J '+/C+1, t-j'l ^ nF-C t) ,F,( s) (5.22) ^g^y2^^^-m+l ^+m+lj {,-iv+1; ^+m+l, ^+iv+I ; where Q.il'3 V, jj J m) = 40 r(j-iv+l) r(j'+ i V+I ) (-I)J J r(-t-îv+l) r(-L+îv+l) [U+m)î]^ (2j+ l ) (2j '+ 1 ) (j+m)i (j+t)i (j '+m)i (j '+&)! (j-m)l (j'-m)l (j'-.L)i r(j-iv+1) r(j+ iv+l) r(j'-iv+l) r(j'+ iv+l) Valid for v real, y > 0, ^ > |m| . The case ^ < |mj follows from the index symmetries 5-4. C- Asymptotic Behavior of Matrix Elements The asymptotic form of may be derived from 5*22 by noting that interchange of integration is also justifiable in the case ^ > m > 0, y = 0 and y"^'^ A"^"' is continuous at y = 0. The result is m lim y"^' ^ ^ ^ (a, v) = y- 0 " -.f f-iv, t-j I V, j, j S m) e t 2'^2^^,+m+l, ^-i v+1 ; 1-- The integrals in 5-23 are Laplace transforms and may be evaluated (22). We obtai n V, jj jm) r(^-iv+n r(m+iv) 2^] ; 1) j t-j'j m+i v : ') - (5.24) The hypergeometric functions may be expressed as f functions (21). 41 The result Î s limit y"^' ^ ^ ^ (a, v) = y- 0 m J_ (j-m)c' 0+^)1 (j'-m).' (j'+^)i (2j+ l )(2j'+I) 2 (j-&)' (j+m)l 0(J'+m)! r(m+iv) r(j-iv+l) r(j '~ iv+1 ) (-1)"^ ^ (5.25) r(-L-iv+l) |r(j-iv+l ) r(j'-ÎV+1) 1 ^>m>0 Equation 5»25 may be extended with the aid of 5«4 and 5»5 to obtain I^-m iv+1 (a, t, v) -tj m > 0 y « 1 ^ j-t+m {+i v+l ^ > 0, m < 0 y « 1 - j-C-m j-i v-1 tn > 0 y » 1 - l^+m j+i V-1 ^ > Oj m < 0 y » 1 . (5.26) i î If m = 0,Ag is real and from 5.10 we find ± 2 (J+t);(j'+4):(2J+ 1)(2j'+1) v). (j-t)! (j'-t)l r(iv) 2y^^^ cos (\jQ:+(J)) (5.27) r(t+iv+1) y « 1 where _.A r(iv) r(&+iv+i) r(j-iv+i) r(j'-i^+t) ^ ' T _ —— |r(iv) r(t+iv+i) r(j-iv+i) r(j'-iv+i)| The cases ^ = 0 and y » 1 follow from 5-4 and 5»5» 42 D. Orthogonality Relations We consider the integral TT 2at °° 2 it 2jt I = J* d(|) p sin 6 dQ J* dY p si nh a doc J dp J dy where D. , is defined by 3•22. J mj kn ' The integrations over rotation matrices may be performed (18) to obtain (4n)3 ' ~ S;;| 6.. I 6__l 6 jj' kk' mm' nn' (2j+ l)(2k+l) S r sinh^adD! A:^''^"(q:, v) (a, v'). (5.29) \ o ^ ^ • k in view of the asymptotic behavior of A:" we may write 5»29 as A. (4%)3 ' ~ Ô;jj' ; I 6..kk' I mm', ànn' (2j+l)(2k+l) jt(2j+ 1)(2k+l) 2—2 5.,, (5(v-v')+ 6-0 6(v+v')) I 2U +^ ) ^ +GU,tj 'C'V, ' J "^3V, ^ J Jj, J k)k)^^ . (5.3 0) For fixed v'j jj and k_, G is a continuous function of v on any closed interval containing v' since it has a uniformly convergent integral representation (23) CO G(v) = J* g(v, a) dtx . (5.31) o 2 2 However 5*28 must be zero unless &= and v' = v since for fixed 43 k and n the functions (L) = Dj; k,(L) (5.32) carry a representation of the Lorentz group and 5*28 may be considered to be the inner product. If we take matrix elements of the differential 2 2 operators (8) representing J • K and J - K we find (-Lv - = 0 (5.33) - v^)l = 0 It follows that G is identically zero and 25*4 *Jj' ^kk' ^mm' ^nn' [ô(v-v') + 6(v+v')] (5«34) t > Oy V real - In the derivation of 5-34 ^ was assumed to be nonnegative. In certain expansions it will be convenient to use the symmetry "KknC-) - "S'kn (L) and assume v>0, t = 0^ + -^ ^ + . . . . In this case 5.34 becomes zS,'* ' ' V' 5(v-v') Sjj, Gkk, 6^. (5-35) V •> 0, -T = OJ + + 1, .... A sum over all irreducible representations in the principal series will then have the form 44 S ^ dv . ^=-00 o This form has the advantage of including each representation once while the form CO 00 2 J" dv J( = 0 -co contains the ^=0 representation twice. 45 VIo HARMONIC FUNCTIONS FOR THE POINCARÉ GROUP A. An Angular Momentum Basis We consider a basis for the Poincare group in which states are labeled by irreducible representations of the Lorentz group. States in this basis have the form (mj s ; Vj jJ mx V > 0_, ^ = 0^ + Y , + ? j .». . (60 i ) where m is the mass, s is the spin, and the numbers Vj m specify a representation of the Lorentz group in an angular momentum basis. We assume states have the normalization / M, s; Vj j, m I M, s; C, v';, j', m'> = ô(v-v') 6.y 6^, . (6.2) If L is a Lorentz transformation, the basis 6.1 has the transforma tion law LjM, s ; I, V, j, m> = S ' im(L)lM, s ; v, jm'> j ',m' J ^ 'J" (6.3) where the transformation matrices are defined by 3.27. In terms of the basis 6.1, a wave function ^^P, \) = (Pj s, \|Y> (6.4) h has an expansion Y(p, \) = Z / dv aj^ (p, s, (6.5) o where jp, s, .\> is a helicity state, h Yj'^(p, s,x) = p, s, \jM, s; t, V, j, m > (6.6) 46 and ^jm ^ (M, s; V, m ) Y) , (6.7) The explicit form of follows from the definition of the jm hellcity basis, 3.30, and the relation ( Pj s, \(M, s; V, j, m > = (6.8) N(H, s, t, V) 6J3 6^^ The result Is Yj|!^(pj, s,\) = p, s, Xle"^'^3 e'9J2e:^J3 |m, s ; v_, j, m > = N(M, s; v) (-a, v) (0, - ({)) (6.9) where D;^ Isa rotation matrix (18) and Af-* Is defined by 5°1. A.ni A. The magnitude of N is fixed by 6.2. To see this we expand 6.2 In a hell ci ty basi s. -3 » .I .1 z s, < (p. s, X) (6.10) \ "^O Upon the change of variable sinh^cc ctx sin 6 d0 dd) Po 6.10 becomes 2 œ ir Zjx — S f sinhadarj sin6d0|* d(j) \ o o b N(M, s, v) A^-^ (3, v) -% -0) N(M, s, v') A®j'(a, v') 0^^, (0, -e, -({)) (6.11) 47 Sj v) 1^ 2 = ^2 ^2 (2s+l)^ 6^, 6(v-v') 6jj, 6^, _ The last step follows from the orthogonality relation 5°34c The final result is then vj-^Cp. s, x) = 1 (-a, v) -e, -(j)) (6.12) jr 2s+l where a choice of phase has been made. A representation (Mj s) of the Poincare group contains only values 2 2 of t satisfying •< s . Therefore the Lorentz harmonics for low spin may be expressed in a simple form with the aid of 5»19" For example the cases s = 0 and s = 1/2 may be expressed in the form (p- = QXM, 5. J, X) DJ^(0, -e,-*) (i-y2)j-s F, (j-\+l, j+ iv+1, 2J+2, 1-y ) 2 1 a where y = e and Q.(Mj Sj Jj \) = (j+s)î 2s] (j+\)i r(j+ iv+l) r(j-iv+l) (6.13) 31 M (j-s)i 2ji (2j+l)! (s+\)i(s-\)i r(s+iv) r(s-ivj In the case of spin zero (6.13) becomes 1 2r(j+iv+l) r(j-iv+l) 2 ^jm M(2j+1) r(iv) r(-iv) 48 -1/2 -1/2-j (sinh a) ^-1/2 -iv (cosh a) (9, (6.14) where is a spherical harmonic and Pisa Legendre function. The Legendre functions with upper index -j-1/2 may be expressed as a finite number of terms (21). For example 9 1/2 sin P (cosh a) = (—) Y7Ô -1/2-iv v(sinh cc) The asymptotic behavior of the functions implies that the \ -5 -1 functions Y?^ are square integrable with the measure Pq dp unless % = ± Finally we note that the expansion coefficient 6.7 is given by ajm = ^j'm (P' (6.15) B. A Conformai Group Basis We now consider harmonic functions of the form (P, s, x) = ^(P, s, X I M, s; y, r^ m> (6,16) where v, r, m) is a conformai group basis state for the Lorentz group. This set of functions may be obtained from the previous section with the aid of the expansion coefficients 4.24. The result is xjrjjj (P;, s, \) = s fj (r, v) YfjJJ (p, s, \). (6.17) j=m However a much simpler expression may be obtained with the aid of the Iwasawa decomposition. According to the Iwasawa decomposition, the Lorentz transformation h"\p) e"'4U3 = e'^'^3 e'GJ2 (6.18) 49 may be expressed in the form ^IQ:K3 giejz ^ . (6.19) The explicit form of RAN, which is derived in appendix C, is RAN = e'S'JZ e/a'KS where sin 8 ^ ^ cosh oc - cos G sinh (3 ,0 ~cc ' e = cosh a - cos 6 sinh a si nh C% sin 6 ^ cosh cc ~ cos 6 sinh q: • (6.20) Therefore 6.16 may be written as ''rm (P' = H The final result, which follows from 4.17 and 6.3 is (P. s, X) = ^ ^ rm s (- Q') f^ (^'r, v) à^_Jbr) (6.22) n=-s where J is a Bessel function df is a rotation matrix and f^ is defined Xn n by 4.24. We note that the simplest set of harmonic functions is obtained in the basis \Pj, s, = L(p, p ) I p, s, \> where 50 L{p, ?) = g-IJst . (6.23) in this basis the harmonic functions are (P;, s, \) = x 4îr ^Èr (6-^'') a ' id) bj e and e ~ may be expressed in terms of p. The result is - "/Px + Py b = Pq • P-z -a ' ^o e M (6.25) ^Px + Py • The orthogonality relations z r jfp xfv (p, s, I)* )f:^;(p, s, \) = fi^i 6(v-v') r6(r-r') 6^i (6.26) follow at once from 6.17 and 6.2. The orthogonality relations may also be calculated from 6.22. If we make the change of variable = M -y bdb dd) (6.27) Po y , a ' where e = Vj 51 we find ' r X- (P,fr, S,« .)••\ XJ-.-.CP, S. X) = o [({.W)a'^+v'^)]^ 2:1^ (2s+1 ) S .f (ry, v) f^ (r'y, V, v') \ o y- X Vx"""' Jm'-x(br') J" e"'d* [ a^+v^)a'^+v'^)]^ rô(r-r') 6__i n(2s+l) """" s f ^ f^ *(y, t, V) f= (y, V, V). (6.28) Therefore the expansion coefficients f^ satisfy the orthogonality \ re 1 ati on S r -4 ff " (Va v) f%(y, v') = \=-s o ^ ^ 5,., 6(v-v') . (6.2%) As in the previous section, any norma]izable function may be expanded in terms of the harmonic functions rm CO CO dr wLv 't'Cp, s, k) = S Jdv f — a^^ (p, s, \) m o o 52 where _ / a - s (p. S. xf Y(p, s, X) . (6.30) rm ^ «JVn_J2po *"1 Finally we list the harmonic functions for the cases s = o and 1 s - 2 • ± re"' K. (re"') . IV 1 ~l/2v / _L JL\ _ 1 r ] 12 im(|) rm 2' 2' ~ jtM ^r(l/2+i v)r(1/2-i v) Jm-1/2 (br)[e r] ^ ~l/2v / 1 ]\ r 1 i2 im(|) ^ rm 2' ~2' ~ «M 't(1/2+i v)r(1/2-i v) Jm+l/2(br) [e r] ) Here J and K are Bessel functions. 53 VII. APPENDIX A. The General Expansion Coefficient In this appendix we prove 4.3 0. We consider the case of lowering operators acting on fj• The proof for raising operators is similar. We prove 4.30 by induction on m. For m = j 4.30 reduces to 4.28 and r r^J*^K^_.^(r) K^^j^(r) dr = [ B v, j)] = (7.1) Next we prove that the step down operator acting on fjj^ yields fj|^ ^. For convenience we write 4.3 0 as fi(r, t, v) = B(m) S a^{m) rJ*'"" , (7-2) We obtain an expression for f^^^ from 1.2. by operating with the step down operator 4.27- t, v) = 2r(J^mî + r" - (e+m-,)2] S a„(m) rJ*'-" (7-3) where e = r d dr We use the relations e K^(r) = -w K,(r) - rKy_,(r) = M K^(r) - (r) (7-4) to obtain r(.t+iv)^+r^-(e+m-l)^] rJ+ ^ " Kt-iv+m+n-j 54 t4te-j+n)(iv+J-n) K^.j^+m+n-J +2(m+J-n)r ] . (7-5) A recurrence relation for a^(m) follows from 7*2^ 7-3^ and 7-5° The result is [(j+m-n) a^ (m) +2 (^-j+n) (i v+j-n) a^_^(m)] (7-6) Upon substitution of the explicit form of a^(m) in 7-6 and use of the properties of (x, n), (x^n)(x+n) =(Xj n+1), (x, n) = x(x+l, n-1 ) (7.7) we find (-2)"(m-l-jj n) it~jj n) (-iv-j,n) a_(m-l) = (7.8) (l^n) (-2j, n) which completes the proof. B. An Integral Representation In this appendix we drive the integral representation 4.32. We begin with 4.30. f;i,(r, v) = B(&, V, J, m) S n (-2)"(-m-j, n)(t-J% n)(iv-J, n) r^*^ " ÔTÔT^jT^Ô Kj+m-n-^-iv B(-C, V, j, m) = 55 1 2 21 • (2, + nj 2^^ ' (j+m)i (j-m)i (J+&)1 (j--L)i r(j+ iv+l) r(j-iv+l) We assume + m ^ 0 and define a new index of summation n' = j - ^ - n (7.9) The symbols (a,n) appearing in 4.3 0 may be written in the form n'). The result is U-j, n) = r O-i+1) (I, "•) (-m-j, n) = (-1)" rO-l-iH-l) r(m+^+l ) (nH-t+l, n' ) (iv-j,n) = r(J-ivH-l) v+1 ) (-t-i v+lj n ' ) (-2J, „) = r(2J+') rU+t+i ) (t+J+l, n') (1, n) = ±±^ ru-t+n (7.10) (t-j J, n ' ) With the aid of 7"1P, 4.30 may be written in the form . r(j+nH-i) r(j-iv+i) r(j+t+i ) fl/r, t, v) = BUj V, j, m) r(^-iv+i) r(t+m+i) r(2j+i) " (1, n')(m+'t+lj n')(-t-iv+l, n ') (n +m iv) . (7.11) Finally we substitute the integral representation (20) 56 K^(z) = Y (|)" r e't" 4? t"»"' (7.12) ° larg z| < 5 for K to obtain fi(r. t. v) = CU, V, i, m) m 2 e-t-TE (m-'v-l /J+t+L -t-j ; ) y (7.13) 2*^2 W^+1, -t-iv+i; where r(j-iv+i) C(t, V, jj m) = r(&-iv+l) (-t+m).' 1 (j+m)i (j+t)! (2j+ l) 2 _2r(j+iv+I) r(j-iv+?) (j-m)] C. A Special Case of the Iwasawa Decomposition We prove 6.19 by choosing the following representation for the Lie a 1gebra. 1 ,01 1 ,0-i J = Iflo ) 1 2^10^ — 2M?(: n)0' •^3 2^0-T' K, = •<2 =l(-Vo) S = î<ô-i ) f = o 1 = (o'o' (7.14) The corresponding representations of one parameter subgroups are CO? G/2 sin e/2 :4J2 _ •^^3 . if V -sin -0/2 COS •0/2 57 = (: :i ^ e'-. :: 0 = 1- ^a/lj (7.15) Therefore a general Lorentz transformation has the following representation in the Iwasawa decomposition. RAN = e i(tlJ3 ,iYJ3 ia-Ks ,!aP ^ibCl o: ' . dM-Y a' . a ' , . (b+Y ^" 2 g, r 2- + ' -2- -2- + ' [e cos le sin ^ + (b+ia) e cos —] a ' . Y-A a ' . Ô+Y a ' ^ . Y-d) ~ô~ ' •J o ' ~'ô~ •*" ' —- Q' 2 gI ^ 2.6' [-e sin Y"] [® cos ^ - (b+ia) e sin —] > (7.16) The left side of 6.19 has the representation a a / "2 Q ~2 . e\ cos Y sin 2 ^iaKg = a a (7.17) 2 . G G -e sin Y cos 2 Upon equating 7-16 and 7»17 we find a = (j) = Y = 0 I e = cosh cx - cos € sinh a sin 0' = ^ ^ 0 < Q' < Ti cosh a - cos e sinh a 58 b- - sin « sinh a (7.,8) cosh CÙ - cos € sînh a which completes the proof. 59 VIII. LITERATURE CITED 1. Dirac, P. A. M.j Roy. Soc. Proc. A183, 284 (1945). 2. Chandra, H., Roy. Soc. Proc. AT89, 372 (19^7). 3. Gelfandj I. M. and Naimark, M. A., Izv. Akad. Nauk S.S.S.R. Ser. Mat. 11, 411 (1947). 4. Naimark, M. A., Usp. Mat. Nauk 19 (1954). 5. Dolginov, A. Z., Soviet Phys. JETP X, 589 (1956). 6. Dolginov, A. Z. and Toptygin, I. N., Soviet Phys. JE TP TO, 1022 (I96O). 7. DoTginov, A. Z. and MoskaTev, A. N., Soviet Phys. JE TP 10, T202 (I960). 8. Strom, S., Ark, Fys. (Sweden) 29, 467 (1965). 9. Strom, S., Ark. Fys. (Sweden) 34, 2T5 (T967). TO. Chang, S. and 0'Raifeartaigh, L., Unitary Representations of SL (2, C) in an E(2) Basis, Preprint, Institute for Advanced Study, Princeton, New Jersey, T 968. TT. Hussain, M., Lorentz Harmonics, Unpublished Ph.D. thesis, Glasgow, Scotland, Library, University of Glasgow^ 1965» 12. Fischer, J., Niederle, J. and Raczka, R., J. Math. Phys. Jj 816 (1966). 13. Zmuidzinas, J. S., J. Math. Phys. 764 (I966). 14. Pontryagin, L. S., Topological Groups, Gordon and Breach, New York, 1 966. 15. Hermann, R., Lie Groups for Physicists, W. A. Benjamin, Inc., New York, 1 966. 16. Iwasawa, K., Japan, J. Math. 19, 405 (1948). 17* Naimark, M. A., Linear Representations of the Lorentz Group, Pergamon Press, New York, 1964. 18. Rose, M. E., Elementary Theory of Angular Momentum, John Wiley and Sons, Inc., New York, 1957- 19. V/igner, E., Ann. Math. 40, 149 (1939)» 20. Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G., Higher Transcendental Functions, Vol. 2, McGraw-Hill Book Company, New York, 1953- 60 21. Erdélyij A., Magnus, V/., Oberhettînger, F., and Tricomi, F. G., Higher Transcendental Functions, Vol. 1, McGraw-Hill Book Company, New York, 1953. 22. Erdélyî, A., Magnus, Vf., Oberhettî nger, F., and Tricomi, F. G., Tables of Integral Transforms, Vol. 1, McGraw-Hill Book Company, New York, 1954. 23. Titchmarsh, E. C., The Theory of Functions, 2nd éd., Oxford Press, London, 193 9- 61 IX. ACKNOWLEDGEMENT The author wishes to thank Dr. D. L. Pursey for his advice and encouragement during the course of this work.