Fortschritte der Physik / Progress of Physics: Band 25, Heft 2 1977 [Reprint 2021 ed.] 9783112519387, 9783112519370

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM A U F T R A G E D E R P H Y S I K A L I S C H E N

GESELLSCHAFT

DER DEUTSCHEN DEMOKRATISCHEN

REPUBLIK

VON F. K A S C H L U H N , A. LÖSCHE, R. R I T S C I I L U N D R. R O M P E

HEFT2

• 1977 • B A N D 25

A K A D E M I E - V E R L A G EVP 1 0 , - M 31728

•

B E R L I N

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Zeitschrift „Fortschritte der Physik" Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur Lösche, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 1 0 8 Berlin, Leipziger Straße 3 - 4 ; F e m r u f : 2200441; Telex-Nr. 114420; Postscheckkonto: Berlin 35021; B a n k : Staatsbank der D D R , Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R -104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik.' Gesamtherstellung: V E B Druckhaus „Maxim Gorki", D D R - 7 4 Altenburg, Carl-von-Ossietzky-StraOe 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der P h y s i k " erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je B a n d : 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je Heft IS,— M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/25/2. © 1977 by Akademie-Verlag Berlin. Printed in the German Democratio Republic. AN (EDV) 57618

Fortschritte der Physik 25, 8 3 - 9 9 (.1977)

The Transformation Behaviour of Fields in Conformally Covariant Quantum Field Theories W . R ü h l a n d B . C. Y u n n

Fachbereich Physik der Universität Kaiserslautern,

BRD

Abstract We study the transformation behaviour of any covariant component obtained from a spinorial field operator by Fourier decomposition on the centre of the universal covering group of the conformal group for four dimensional Minkowski space.

1. Introduction

According to a proposition by S c h r o e r and S w i e c a [1] an operator field 0 A g(x) in a conformally covariant quantum field theory is Fourier analyzed on the centre of the universal covering group of the conformal group 6 + 00 0 ab( x )

= L z"®AB(x) n = — oo

(1)

where Z represents the generator of the centre of Q. Obviously £ is connected with the eigenvalue of Z Z&Ai>(x) Zr* =

(2)

and thus is a real number restricted to 0 ^ f < 1.

(3)

Z is assumed to be unitary in (1), (2). In this article we study the transformation behaviour of the J-component (1) for arbitrary ji,h^ A < +j1 (4) The transformation behaviour of this ^-component is in general that of a non-unitary representation of G. We call it "degenerate" if jl'j2 = 0 and "most degenerate" if jl = j2 = 0. Under certain conditions the representation is reducible. The invariant subspaces may carry unitary irreducible representations of the discrete series. These irreducible representations were studied in detail in Pvefs. [2, 3, 4]. 7

Zeitschrift „Fortschritte der Physik", Hett 2

84

W . ROTO, and B . C. YUNN

The most elegant approach seems to consist in formulating the representations first for spaces of functions defined on the universal covering space M of compactified Minkowski space M. The results can then easily be projected on the one-sheeted space M. The realization on M allows us to exploit the advantages of expansions in terms of the (discrete) canonical basis, whereas the realization on M admits a Fourier expansion into the (continuous) plane wave basis. All topological notions are easier to describe in the realization on M. We proceed as follows. In Section 2 we describe the representations realized on M. Dual and conjugate representations are introduced by means of bilinear and sesquilinear invariant forms in Section 3. The extension of the carrier spaces from test function spaces to distribution spaces is discussed in this context. The single-sheeted space M is studied in Section 4. In this Section 4 we give the elements of the canonical basis in either realization. The invariant subspaces carrying the discrete series representations are characterized by the elements of the canonical basis by which they are spanned. In Section 5 we introduce the intertwining operators in either realization first as certain convolution operators. Their extension to a maximal domain in the parameters is performed by a Fourier transformation and analytic continuation. They turn into multiplication operators this way. A new type of invariant subspaces is discovered by this procedure. In Section 6 we extend the group 0 by an element of order two, the "R-inversion". Representations of this extended group are obtained from those of G by a standard procedure [6']. Finally we give some preliminary results on trilinear invariant forms in Section 7. 2. Representations realized on spaces of functions over M We study functions h)>-

fe/i1)

of equivalent representations as follows. B y (±) and the image space S ) ^ 1 contains invariant subspaces ^"„WiiT). Provided the normalizing factors in (109) are well defined and finite, the image is J 5 ",)« 1 '^'. 8 annihilates a subspace ^0M*(±> of This subspace is necessarily invariant and consists of functions with support consisting of (i)-timelike and spacelike momenta. Thus we have the invariant subspaces JroM„(±)

c

^0w* c

95

Transformation Behaviour of Fields

I t is obvious from (107) that whenever a is not an integer we can define an inverse of the operator Ka,b{p)• If neither alternative (108) holds in addition, S is a one-to-one continuous map of onto and yf are equivalent representations. If (108) holds for one signature then Ka-*h-*h-» J

J

Dh(e-i)

8f(x)

Dh(s)}

T r {g(p)^

d*p

Dh(b)

f(p) & . $ ) }

x

(110)

KaM(p)

with 9 as defined by (104) and (33), still acting solely on Ka,b(p)In the cases (108), the form (110) reduces to a positive definite invariant form on which is equivalent to a scalar product on This scalar product on JF0w'i(T> defines a unitary representation of the discrete series.

6. The ii-inversion The .¿¡¡-inversion is usually defined by CC—•£*;' = -

4 ,

+ 0

(111)

or X - ^ X '

=

(112)

- ( X ) ~ \

On S, X Ri it can be defined by U

u' u

—«1 —u

=

U'

=

- U .

(113)

{ M )

(114)

R induces an automorphism AR on G: M

€ G,

A

r

M - ^ A

{ M ) = R ~

1

r

(115)

M R .

Here M is defined by its action on S3 X Ri (15a, b) and (18). Projecting we have M-

\U

/ M M )

G

on

SU(2,2)

V j

I) T

_i,'T\ ¿V

(116)

96

W.

R i t a and B .

C. YUNN

correspondingly. Inserting the submatrices of AR(M) into (15a, b) yields the action of AR(M) on S 3 X Note that det {-BW

+ i t ) = det ( - C T L + D)

(117)

defines the left hand side on (S3 X Rj) X G. Given a pair of functions

Hs)

y>T g @zhh. Here u',

are as in (113). Then TR2 = 1 necessitates ^ 2 = 1.

(119)

The ansatz (118) solves the constraint TR^TMTR. Thus we end up with the following alternative due to

£) CLIFFORD

(120)

[5]:

1. If jx — j2 both spaces 3>XW' and are identified and = ¡u2. We obtain a pair of inequivalent irreducible representations of the extended group belonging to the signs fix = ¿ 1 . Restricting the group to G gives one irreducible representation. 2. If =j= j2 different choices of fit and fi2 lead to equivalent representations. Thus choose fix = ,«2 = + 1 - We obtain the irreducible representation of the extended group that reduces into two inequivalent irreducible representations of G, in the space 0 SfJ^. We mention that E leaves T invariant. Thus we have either a single = j2) or a pair (ji =1= ji) °f discrete series representations of the same branch (holomorphic or antiholomorphic) that build up the representations of the extended group. Let f{x) (g{x)) transform as («, f, jx, j2) ((

exp i e i arg d

X | det (E -

U,-1

• U2)\°>

exp

X

ft.a.O, 0, 0).

Covariance of the function ip (122) is guaranteed if and only if «1 + 02 + 03 + 4 = 0 and

«2 + Cl +

(123)

+ 4 = 0

01 + 02 — Ss = 0 02 + ei +

= 0-

(124)

Then the respresentation comes out as

Xt:

(«„ 03, 0 , 0 , 0 )

«3 =

implying

a

i + 02

03 = —6l — 02 = 01 + 02 £s A C i + f 2 m o d i .

(125) (126)

Among the parameters g2> 03 there exist only two relations (112) leaving a linear combination of them free. We choose the following two kernels K1:o1=Q1

(127)

K2:c2=Q2.

(128)

We expect that any kernel K entering (122) is a linear combination of the kernels K1 and K2. Let us make this precise in the following two proposals: Proposal 1 : Any bilinear map K of ^ 0 0 0 and into 3>000 that is covariant in the sense K{T*

x T*>) = T*>K

(129)

is of the convolution type (122) with a distribution kernel operating on the product space (except for a set of "singular" combinations Xi and %2 for which K can be obtained as a "limit" of the expression (122)). Proposal 2 : Any convolution kernel of type (122) is a linear combination of the kernels Kx and K 2 . The nuclearity of the spaces 3>000 enters crucially into Proposal 1 [7]. By (129) the form of the kernel can be restricted to be of the type (122). Proposal 2 can be proved for smaller groups by explicit computation using an expansion into the canonical basis [5]. For the group G we do not know any direct way of proving Proposal 2. An analytic continuation from a similar statement for principal series representations may be the most elegant proof.

98

W . RÜHL a n d B . C . Y U N N

We mention finally that the inverse problem of a linear map Kd of .@000 into a tensor product % 0 0 (x) S>a00 that obeys the covariance law K*T» = (T* ® T*') Kd

(130)

is solved by an ansatz analogous to (122).

8. Conclusion

The field component 0A^{x) of a quantum field (oi, f, ju j2) with * = - d - j 1 - j From (66) we have

has a transformation behaviour i

.

(131)

Um4>ÌA(®) UM-i = |det (TX + Q)\* exp »(-2Ç - j, + j2) arg det+ (TX + Q) X {CHQT + XTt) 0((X>) DH(Q + TX)}AÈ .

For a special conformai transformation

that is usually considered this simplifies to

8 = 0, R = Q = E, T = b°E + bo = T* X' = X(E + We define

(132)

x; = (x, +

a(b, x) = det (E + TX) = 1+2bx

a(b, x)-1 + bV.

(133) • (134) (135)

a+(b, x)1 = lim a(b, wf, ). € C a—• o

(136)

o-{b, x)1 = a+(b, x)1.

(137)

W€T

and Then (132) can be rewritten as UM0(Ai(x) U^r1 = c+(b, X