Fortschritte der Physik / Progress of Physics: Band 16, Heft 9 1968 [Reprint 2021 ed.] 9783112500569, 9783112500552

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE D E R PHYSIKALISCHEN GESELLSCHAFT IN D E R DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

B A N D 16 • I I E F T 9 • 1968

A K A D E M I E

- V E R L A G

•

B E R L I N

Informationstheorie Einführung in die statistische Theorie der elektrischen Nachrichtenübertragung Von Prof. Dr.-Ing. PETER FEY (Elektronisches Rechnen und Regeln, Band 3) 3., berichtigte Auflage 1968. VIII, 217 Seiten - 96 Abbildungen - 16 Tabellen - gr. 8° — Lederin 27,— M Die in den ersten Bänden behandelte Theorie und Praxis der Verarbeitung digitaler und analoger Information in Rechenmaschinen wird im III. Band ergänzt durch die Theorie der Übertragung diskreter und kontinuierlicher Informationen. Die Information wird hierbei, im Gegensatz zu den determinierten Prozessen bei der Verarbeitung nach Shannon als statistische Kategorie aufgefaßt. In der deutschen Literatur ist die Einbeziehung wahrscheinlichkeitstheoretischer Methoden in die Analyse von informationsverarbeitenden und übertragenden Systemen erst in letzter Zeit erfolgt. Im vorhegenden Buch wird der gesamte Komplex der Nachrichtenübertragung über gestörte Übertragungskanäle, beginnend bei der Nachrichtenquelle, statistisch beschrieben und die damit in Verbindung stehenden Probleme der störungsfreien Übertragung durch geeignete Kodierungen behandelt. Durch zahlreiche Beispiele wird dem Leser das Verständnis für die statistische Betrachtungsweise erleichtert und die Verbindung zur praktischen Anwendung hergestellt. Bestellungen

durch eine Buchhandlung

erbeten

Verzeichnis der in der Buchreihe „Elektronisches Rechnen und Regeln" bisher erschienenen Bände auf Wunsch direkt vom Verlag. AKADEMIE-VERLAG

•

BERLIN

BEZUGSMÖGLICHKEITEN Sämtliche Veröffentlichungen unseres Verlages sind durch jede Buchhandlung im In- und Ausland zu beziehen. Falls keine Bezugsmöglichkeit vorhanden ist, wende man sich in der Deutschen Demokratischen Republik an den AKADEMIE-VERLAG, GmbH, DDR-108 Berlin, Leipziger Straße 3—4 in der Deutschen Bundesrepublik an KUNST UND WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 in Österreich an den GLOBUS-Buchvertrieb, Wien I, Salzgries 16 in Nord- und Südamerika an Gordon and Breach Science Publishers, Inc., 150 Fifth Avenue, New York, N. Y. 100 11 U.S.A. im sozialistischen Ausland an die Buchhandlungen für fremdsprachige Literatur bzw. den zuständigen Postzeitungsvertrieb bei Wohnsitz im übrigen Ausland und -Import GmbH, DDR-701 Leipzig, Leninstraße 16. an den Deutschen Buch-Export

Fortschritte der Physik 16, 491—543 (1968)

Coupling of Space-Time and Internal Symmetry G. C. HEGEREELDT a n d J .

Institut für Theoretische

Physik,

HENNIG

Universität Marburg,

Marburg

Contents Introduction

491

I.

Embeddings and Extensions

494

LI. 1.2. 1.3. 1.4. 1.5. 1.6.

The McGlinn theorem and some generalisations General embeddings and redefinitions Semidirect couplings Unifications Miscellaneous results Extensions

494 498 502 506 517 519

II.

Mass Splitting

522

I I . 1.

T h e o r e m s b y O'RAIFEABTAIGH a n d JOST

522

II.2.

Mass Splitting from infinite-dimensional groups and algebras

526

III.

Field Theoretic Approaches

528

111.1. A Lagrangian approach

528

1 1 1 . 2 . MICHEL'S e x t e n s i o n p r o p o s a l

529

111.3. An iS-Matrix approach

531

Appendix

537

Literature

541

Introduction Since the success of isospin and SU(3), internal symmetry groups have acquired an established place in the phenomenological description of elementary particles. One of the most important features of present high energy physics is the experimental fact that elementary particles and resonances can be grouped into multiplets which correspond to irreducible representations of internal symmetry groups such as 8U(2) and ${7(3). The members of a multiplet possess equal Spin and parity, and their masses are not too far apart, as for example the baryon SU(3)octet which consists of the isospin multiplets N, S , 2 , and A. Thus internal sym35

Zeitschrift „Fortschritte der Physik", Heft 9

492

G . C. H E G E R F E L D T a n d J . H E N N I G

metry groups yield as a first result an ordering of elementary particles into multiple ts of particles of similar properties, and representation theory tells one which multiplets may be realized. Incomplete multiplets then give hints to the possible existence of further particles with corresponding properties. This situation is similar to the early days of the periodic system of chemical elements. But the use of internal symmetry groups does not end here, as the term 'symmetry group' already suggests. For further applications one usually assumes t h a t the interaction between particles is invariant not only under space-time transformations, but also under the internal symmetry group. Mathematically, this can be described as follows. The physical states correspond to vectors of a Hilbert space 3€, or more precisely, to unit rays in 36. Let the interaction be characterized by some operator, in scattering theory for instance by an tS*-matrix. Lorentz invariance is then guaranteed by the existence of a representation of the Poincaré group by unitary operators in 3€ which commute with the S-matrix. In variance under the internal symmetry group is expressed in an analogous manner; there are unitary operators in 3€ which commute with the interaction operator and which form a group isomorphic to the internal symmetry group. This assumed invariance implies physical consequences which can be tested experimentally. The particles of a multiplet differ in mass to some extent — for isospin the mass difference is very small, but for other proposed symmetries like 8U(3) the mass splitting is considerable. This fact is usually explained by the assumption that the symmetry is more or less 'broken' and t h a t the interaction is only 'approximately' invariant under the internal symmetry. The noninvariant part of the interaction should then change only the mass, but not the discrete quantum numbers of spin and parity. In the limit of exact symmetry, all particles of a multiplet should have the same space-time quantum numbers, in particular the same mass. Then one can assume that, for exact invariance, space-time symmetry and internal symmetry are completely 'independent' and unrelated to each other, and expressed mathematically, that the operators of the Poincaré group $ commute with the operators of the internal symmetry group (3. The total symmetry then simply corresponds to the direct product (x) © of the corresponding groups. For a long time, this has been the accepted picture. However, there are certain aspects of it which are rather unsatisfactory. First of all, one knows very little about the nature and mechanism of symmetry breaking, except for isospin which is broken by electromagnetic interactions. Furthermore, one does not know how the breaking affects theoretical predictions based on exact invariance. And then a principal difficulty ; the transition from broken to exact invariance, t h a t is the 'switching off' of the symmetry breaking part of the interaction, is certainly a nontrivial mathematical procedure. This becomes immediately evident by the following remark. By switching off a part of the interaction, one changes the masses. But the mass spectrum, together with spin, characterizes the representation of the Poincaré group in 3t. Hence one changes this representation, i.e., the operators corresponding to Poincaré transformations. One obtains another, inequivalent representation of For broken and exact symmetry one thus deals with two different mathematical structures. Because of these difficulties, another approach, advocated in particular at Coral Gables 1964 [2, 56], attracted considerable interest and initiated a large number of investigations. As then formulated and treated by M C G L I N N [ 6 3 ] the approach wants to avoid symmetry breaking and tries in particular to explain the mass splitting within multiplets by purely group theoretical means.

Coupling of Space-Time and Internal Symmetry

493

The starting point is physically very convincing. Let P„v = 0, 1,2, 3, denote the components of the four-momentum operator. The mass of a (stable) particle is an eigenvalue of the mass operator M =

1/P,P'.

(1)

The P, are also generators for infinitesimal translations in space-time, they belong to the Lie algebra of the Poincaré group $ (cf. appendix). In case of the direct product ^ ® (3, all elements of (3 commute with the P „ hence also with M2. I t then follows immediately that all particles of an S-multiplet belong to the same mass. For let \m) belong to the mass m. Then for any s £ ) = s (M21m))

= s(m 2 1m)) = m 2 (s |m>).

(2)

If, however, 5(5 and (3 are coupled in Such a way that 6 (ref. [22], p. 237). As for (4.7.a), D& is the Lie algebra of the complex orthogonal group SO (10, 0). All real forms of D5 are defined by certain bilinear invariants [55]. These are also invariant under $0(3,1), except for the compact case S = SO(10). Bs is isomorphic to $0(11, C). Its real forms are S0(p, q) (p + q = 11) of which obviously only $0(10, 1) contains $0(10) and L. Thus G = $0(10,1), $ = $0(10). However, by inspection of the basic representation of SO (10,1), one can see that SO (10,1) contains no subalgebra isomorphic to P. Hence case x) is not possible. /?) Assume G = Gx © G'x, Gxf*a Q'x and simple. For $ there are two possibilities: = § 1 © ,q) with j> + q = 5. Only the compact $0(5) = SO (5,0) does not contain L, hence S — SO (5). The smallest nontrivial representation of SO (5) is 4-dimensional, but it is not real (B2 & C2 is the complex extension of Sp (4) and hence SO(5) is mapped onto the unitary Sp(4:)). Therefore the only nontrivial coupling of R and SO (5) and hence also of E and -F is by a 5-dimensional representation, which is irreducible for SO (5). But A3, the complex extension of SU (4), has no irreducible 5-dimensional representation, hence F =)= Aa. Case (c) is also impossible because the only simple real form of S = Ax © A t is L so that S would contain L. As for (d) and (e), B3 pa SO (7, C) and G2 have no nontrivial representation of dimension less than 7, and dim R = 4 in this case. Thus F 4= B3, G2. P) Assume F to be nonsimple. Then F =F1@F'-i with Fx en F[ and Fx simple. As in case bx • -^•QiiVb- Similarly, one can introcude a basis tap in T, «, /? = 1, 2, with t^ -> ga, r e f- [«> vi>] = t„p,

0,

[ y p , xa] =

-tMp.

(4.25)

T:\

P2 =

1 2" ('12

+

ya,