Linear representations of the lorentz group

LINEAR REPRESENTATIONS OF THE LORENTZ GROUP M A NAIMARK Translated by ANN SWINFEN and O J M A R S T R A N D Translation edited by H K F A R A H A T DEPARTMENT OF MATHEMATICS THE UNIVERSITY OF SHEFFIELD P E R G A M O N PRESS OXFORD · L O N D O N · E D I N B U R G H · NEW YORK PARIS F R A N K F U R T 1964 PERGAMON PRESS LTD Headington Hill Hall, Oxford & Fitzroy Square, London W.l PERGAMON PRESS (SCOTLAND) LTD & Teviot Place, Edinburgh PERGAMON PRESS INC 122 East 55th Street, New York 22, NY GAUTHIER-VILLARS ED 55 Quai des Grands-August ins, Paris PERGAMON PRESS G.m.b.H Kaiserstrasse 75t Frankfurt am Main Distributed in the Western Hemisphere by THE MACMILLAN COMPANY · NEW YORK pursuant to a special arrangement with Pergamon Press Limited Copyright © 1964 Pergamon Press Ltd This translation has been made from M A Naimark9s book entitled JIHHEHHHE IIPEflCTABJIEHHX rpynnbl JIOPEH1JA (Lineinyye predstavleniya gruppy Lorentsa) published in 1958 by Fizmatgiz, Moscow Library of Congress Catalog Card Number 63-10025 PRINTED IN POLAND PWN-DRP PREFACE THE study of the linear representations of a given group is one of the most important problems of the theory of groups; it has a great number of applications in various branches of mathematics and of theoretical physics A specially important part in theoretical physics is played by the representations of the three-dimensional rotation group and of the Lorentz group The importance of the former is bound up with the fact that a knowledge of the repre­ sentations of the three-dimensional rotation group enables us to describe in invariant form the physical magnitudes and equations of non-relativistic mechanics This invariance is the mathematical expression of the independence of the laws of non-relativistic mechanics of the choice of the coordinate system The represen­ tations of the Lorentz group play an analogous part in relativistic mechanics; in this case the invariance is with respect to the trans­ formations of the Lorentz group, and this is the mathematical expression of the independence of the laws of relativistic mechanics of the choice of inertial system of reference (see §§ 2, 17 and 18) The finite-dimensional irreducible representations of the com­ plete Lorentz group and proper Lorentz group are well known (see, e.g Ref [7a]) and are widely used in quantum mechanics; it is known that none of these representations is unitary, with the exception of the trivial unity representation The study of the infinite-dimensional representations of the Lorentz group may prove useful for the further development of quantum theory Apart from this, the study of the infinite-dimensional repre­ sentations of the proper Lorentz group is a good introduction to the general theory of infinite-dimensional representations of semi-simple Lie groups Indeed, it was this simplest example of the proper Lorentz group, which elucidated the characteristic properties of the representations of complex semi-simple Lie groups xi XI1 PREFACE For example the representations of the so-called complementary series are not included in the decomposition of the regular repre­ sentation into irreducible components; while for commutative and compact groups, the decomposition of the regular representation yields all the irreducible representations of the group Moreover, many methods of the general theory of infinite-dimensional repre­ sentations of semi-simple Lie groups were originally worked out for the simplest case of the proper Lorentz group A complete description of all the irreducible unitary representa­ tions of the Lorentz group, to within equivalence, has been given in papers by I M Gelfand and the author(see Refs [12a] and [12b]) It was found that the formulae obtained for the operators of the representations retain a definite meaning also for those values of the parameters for which these representations cease to be unitary (see Ref [12b]) The question thus arises as to whether the resulting formulae determine, in some sense, all irreducible representations and not only the unitary ones This problem was solved by the author in Ref [28c], where all the completely irre­ ducible representations of the proper Lorentz group were found, to within equivalence The definition of equivalence of two repre­ sentations of the Lorentz group given there seems to us to be the most natural one in the theory of representations ; from the point of view of this definition the spaces of two equivalent representations need not be isometric, so that it is the formulae which are essential for the representations and not the norm of the space The method of investigation is a development of the method, previously applied by the author in Ref [28b] for the description of all irreducible unitary representations of the complex classical groups It is to be noted that this method can also be applied to the description of all completely irreducible linear representations of semi-simple Lie groups (not only the unitary ones"*") The author recently used his method in Ref [28f] for the exact formulation and solution of the problem of describing all completely t F A Berezin recently developed another method of describing the irre­ ducible representations of complex semi-simple Lie groups Ref [5b] PREFACE Xlll irreducible representations of the complete Lorentz group to within equivalence In connection with these results, the problem of finding all the representations (not only the completely irreducible ones) of the proper Lorentz group and the complete Lorentz group, remains unsolved.* The solution of this problem has important applications especially to the theory of relativistically invariant equations, which still lacks a conclusive treatment The present book is devoted to a systematic exposition of the theory of linear representations of the proper Lorentz group and the complete Lorentz group Having in mind physicist readers, the author has endeavoured to make the exposition as elementary as possible, so that the reader is not expected to have any special mathematical knowledge beyond what is acquired in university courses on analysis and analytical geometry The necessary supple­ mentary information is given in the text itself or in the appen­ dices The book consists of four chapters The first two chapters are of an introductory nature; they contain an exposition of the basic material on the three-dimensional rotation group, on the complete Lorentz group and the proper Lorentz group, as well as the theory of representations of the three-dimensional rotation group in the form in which it will be needed in the chapters that follow In addition, the second chapter contains an exposition of the necessary basic information from the general theory of group representations Chapter III is devoted to the representations of the proper Lorentz group and the complete Lorentz group The first three sections are of a more elementary nature and contain a description of the completely irreducible representations of the proper Lorentz group in infinitesimal form and of spinor representations In order to make the exposition simple the author has imposed on the representations some supplementary conditions; later on (in §§13 and 15) it is proved that these conditions in fact hold for any t Added in proof This problem was recently discussed by Zhelobenko XIV PREFACE completely irreducible representation of the proper Lorentz group Then, in §§ 10-14, we give the theory of infinite-dimensional repre­ sentations of the proper Lorentz group in integral form, the theory of characters (traces) and Plancherel's formula for the proper Lorentz group Finally, §§15 and 16 contain the formulation and solution of the problem of describing all completely irreducible representa­ tions of the proper Lorentz group and complete Lorentz group to within equivalence; the detailed exposition of these matters is here given for the first time The last chapter (Chapter IV) deals with the theory of invariant equations As mentioned above, this theory cannot yet be considered complete; nevertheless, in view of the important applications of the theory, the author considered the inclusion of this chapter to be justified The author expresses his sincere thanks to I M Gelfand, M I Grayev, D P Zhelobenko and S V Fomin who have read through the manuscript of the book and made many valuable suggestions M A NAIMARK CHAPTER It THE THREE-DIMENSIONAL ROTATION GROUP AND THE LORENTZ GROUP § The Three-dimensional Rotation Group General definition of a group An aggregate G of elements g9h, is called a group, if (1) in G there is defined the product g h of any two elements g, he G in such a way that the product of these two elements g, he G also belongs to G; (2) (#i g2)93 = 9i( and |(f;, n n η'η)\ > 1, but this contradicts the continuity of the bilinear form Thus, |(f, i/)| < C when |f| = 1, \η\ = 1, where C is some constant Now letting ξ' = f, ?/ei?, we see that ξ,η' = -— η for arbitrary vectors III \(ξ',η')\ toi < C, and so |(f.ii)| ξ = ξ On the other hand x ' e l for xeX; therefore equation (3) follows from remark in § 13 Sub­ section REFERENCES A I AKHIYEZER and V B BERESTETSKH, Quantum electrodynamics (Kvantovaya elektrodinamika), Moscow (1953) N I AKHIYEZER and I M GLAZMAN, The theory of linear operators in Hilbert space (Teoriya lineinykh operatorov v GiFbertovom prostranstve), Moscow (1950) N J BHABHA, Relativistic wave equations for the elementary particles, Rev Mod Physics, 17, 200-216 (1945) V BARGMANN, Irreducible unitary representations of the Lorentz group, Ann of Math (2) 48, 568-640 (1947) F A BEREZIN, (a) Laplace operators on semi-simple Lie groups, Dokl Akad Nauk SSSR, 107, 9-12 (1956) (b) Laplace operators on semi-simple Lie groups, Trudy Mosk Matem Ob-va 6, 371-463 (1957) F A BEREZIN and I M GELFAND, Some remarks on the theory of functions on symmetric Riemann manifolds, Trudy Mosk Matem Ob-va, (1956) B L VAN DER WAERDEN, (a) Die gruppentheoretische Methode in der Quantenmechanik, Springer, Berlin (1932) (b) Modern algebra, Ungar, New York (1940) A WEIL, L'integration dans les groupes topologiques et ces applications Actualités Sci et Indust 869, Paris (1938) N Ya VILENKIN, Bessel functions and representations of the group of Euclidean motions, Uspekhi Matem Nauk, 9, 69-112 (1956) 10 I M GELFAND, (a) On one-parameter groups of operators in a normed space, Dokl Akad Nauk SSSR, 25, 711-716 (1939) (b) Lectures on linear algebra (Lektsii pò lineinoi algebre), Moscow (1948), English translation, Interscience tract (c) The centre of an infinitesimal group ring, Matem Sbornik, 26(68), 103-112 (1950) (d) Spherical functions on symmetric Riemann spaces, Dokl Akad Nauk SSSR, 70, 5-8 (1950) 11 I M GELFAND and M I GRAYEV, (a) On a general method of decompo­ sition of the regular representation of a Lie group into irreducible represent­ ations, Dokl Akad Nauk SSSR, 92:2, 221-224 (1953) (b) Analogue to PlanchereFs formula for classical groups, Trudy Mosk Matem Ob-va, 4, 375-408 (1955) 441 442 REFERENCES 12 I M GELFAND and M A NAIMARK, (a) Unitary representations of the Lorentz group, Journ of Physics X, 93-94 (1946) (b) Unitary representations of the Lorentz group, Izv Akad Nauk SSSR, ser matem., 11, 411-504 (1947) (c) Normed rings with involutions and their representations, Izv Akad Nauk SSSR, ser matem., 12, 445-480 (1948) (d) The connection between unitary representations of a complex unimodular group and its unitary subgroup, Izv Akad Nauk SSSR, ser matem., 14, 239-260 (1950) (e) Unitary representations of the classical groups, Trudy Matem In-ta im V.A STEKLOVA, 36, 1-288 (1950) 13 I M GELFAND and Z Ya SHAPIRO, Representations of the group of rotations in three-dimensional space and their applications, Uspekhi Matem Nauk, 7, 3-117 (1952); English translation Amer Math Soc Translations, (2) 2, 207-316 (1956) 14 I M GELFAND, and A M YAGLOM, (a) General relativistic-invariant 15 16 17 18 19 20 21 22 23 24 equations and infinite-dimensional representations of the Lorentz group, Zhur Eksper i Teoret Fiz., 18, 703-733 (1948) (b) Pauli's theorem for general relativistic-invariant equations, Zhur Eksper i Teoret Fiz., 18, 1096-1104 (1948.) V L GINZBURG and I Ye TAMM, On the theory of spin, Zhur Eksper i Teoret Fiz 17, 227-237, (1947) R GODEMENT, A theory of spherical functions I, Trans Amer Math Soc, 73, 496-556 (1952) M I GRAYEV, On a general method of computing traces of infinitedimensional unitary representations of real simple Lie groups, Dokl Akad Nauk SSSR, 103: 3, 357-360 (1955) P A M DIRAC, Unitary representations of the Lorentz group, Proc Roy Soc, A 183, 284-295 (1945) R J DUFFIN, On the characteristic matrices of covariant systems, Phys Rev 54, 1114 (1938) N V YEFIMOV, A short course in analytical geometry, Moscow (1954), English translation in preparation by Pergamon Press S IZMAILOV, On the quantum theory of particles possessing internal ro­ tational degrees of freedom, Zh Eksper i Teoret Fiz., 17, 629-647 (1947) R COURANT and D HILBERT, Methods of Mathematical Physics, Vol I, Interscience (1953) A G KUROSH, Theory of groups (Teoriya grupp), Moscow (1948), English translation, Chelsea (1955) L D LANDAU and E M LIFSHITZ, (a) The theory of fields (Teoriya polya), Moscow, (1948), English translation published as The Classical Theory of Fields, Pergamon Press (1959) REFERENCES 25 26 27 28 29 30 31 32 33 443 (b) Quantum mechanics (Kvantovaya mechanika), English translation published by Pergamon Press, (1960) LE COUTEUR, The structure of linear relativistic wave equations, I, II, Proc Roy Soc A, 202, 288-300, 394-407 (1950) L A LYUSTERNIK and V I SOBOLEV, Elements of functional analysis, (Elementy funktsionaPnogo analiza), Moscow (1951), English translation, Constable L Loo MIS, An introduction to abstract harmonic analysis, Van Nostrand, New York (1953) M A NAIMARK, (a) Rings with involution, Uspekhi Matem Nauk 3, 52-145 (1948), English translation, Amer Math Soc Translation No 25 (1950) (b) On the description of all unitary representations of the complex classical groups, I, II, Matem Sbornik 35 (77), 317-356 (1954); 37 (79), 121-140 (1955) (c) On linear representations of the proper Lorentz group, Dokl Akad Nauk SSSR, 97, 969-972 (1954) (d) Linear representations of the Lorentz group, Uspekhi Matem Nauk, 9, 19-93 (1954); English translation, Amer Math Soc Translation, Series 2, (1957) (e) Normed rings, Normirovannye koFtsa, Moscow (1956), English translation, Noordhoff, Groningen (f) On irreducible linear representations of the complete Lorentz group, Dokl Akad Nauk SSSR, 112, 583-586 (1957) V NEMYTSKn, M SLUDSKAYA and A CHERKASOV, A course in mathematical analysis (Kurs matematicheskogo analiza), Vol I, Moscow (1944) V PAULI, Relativistic theory of elementary particles (Relyativistskaya teoria elementarnykh chastits), Moscow, (1947) I G PETROVSKU, Lectures on the theory of integral equations (Lektsii po teorii integraFnykh uravnenii), Moscow (1951), English translation, Graylock (1957) L S PONTRYAGIN, Continuous groups (Nepreryvnye gruppy), Moscow, (1954), English translation, Princeton (1946) R POTTER, (a) Sur les systèmes d'équation aux dérivées partielles linéaires et du premier ordre, quatre variables invariantes dans toute transform­ ation de Lorentz, C R Acad Sci Paris 222, 638-640 (1946) (b) Sur la définition du vecteur-courant en théorie des corpuscules Cas du spin demi entier, C R Acad Sci Paris, 222, 855-857 (1946) (c) Sur la définition et les propriétés du vecteur-courant associé un corpuscule de spin quelconque, C R Acad Sci Paris, 222, 10761079 (1946) 444 REFERENCES 34 I M RYZHIK and I S GRADSHTEIN, Tables of integrals, sums, series and products (Tablitsy integralov summ ryadov i proizvedenii), Moscow, (1951) 35 V.l SMIRNOV, A course of higher mathematics, Vol Ill, Moscow, (1958), Vol IV, Moscow (1958), Vol V, Moscow, (1960), English translations of all three volumes in preparation by Pergamon Press 36 E WILD, On first order wave equations for elementary particles without subsidiary conditions, Proc Roy Soc, A, 191, 253-268 (1947) 37 M FIERZ and W PAULI, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc Royal Soc, A, 173, 211-232, (1939) 38 HARISH-CHANDRA, (a) Infinite irreducible representations of the Lorentz group, Proc Roy Soc A, 189, 372-401 (1947) (b) On relativistic wave equations, Phys Rev., 71, 793-805, (1947) (c) Plancherel formula for complex semi-simple Lie groups, Proc Nat Acad Sci., 37, 813-818, (1951) (d) The Plancherel formula for complex semi-simple Lie groups, Trans Amer Math Soc, 76, 485-528 (1954) INDEX Abelian group Absolutely convergent series 69 Adjoint operator 41, 85 Admissible operator 242 Decomposable invariant equations 345-346 Dense set 69, 239 Density of energy 404 definite 404 Differentiable 31, 32 Differential 31, 191 Dimension 26, 28 Dirac equation 415 Direct sum 79, 82 Dotted indices 129 Duffin's equation 416 Banach space 69 Basic infinitesimal operators 37 Basis of linear space 26 Bilinear form 382 Bounded linear functional 71 set of matrices 197 Burnside's Theorem 424 Eigenvalue 44 Eigenvector 44 Elementary spherical function 164 Energy momentum 404 Equation(s) Dirac 415 Duffin 416 invariant 329, 357, 378, 402 indecomposable 345 Pauli-Fierz 420 Equivalent representations 28, 239 Essentially bounded function 153 Euclidean space 30 Eulerian angles Canonical basis 48 Cauchy-Buniakowsky inequality 83 Charge 403 Closed direct sum 79 set 69 Closure of operator 433 of set 69 Commutative group Commutator 35 Complementary series 171, 180 Complete Lorentz group 24 ring of operators 429 series of representations 293 space 68 Completely irreducible 234, 242 Completion 179 Conjugate representation 235 space 71 Continuation by continuity 179 Continued direct sum 212 Continuous 29, 32, 72, 73, 382 Convergent in norm 68 Function elementary spherical 164 Lagrangian 396 positive definite 206 Functional, linear 70 Fundamental sequence 68 Galilean transformation 19 General Lorentz group 22 General Lorentz transformation Graph of an operator 432 Group ring 197, 300 445 20 446 INDEX Hermitian adjoint 204 form 382 operator 41, 86 Hilbert -Schmidt kernel 206 space 83 isometric 84 Identity representation 137 Indecomposable invariant equation 345 Induced representation 251, 301 Inertial system of coordinates 19 Infinitesimal 34, 98 Integral 74, 75 operator 205 Invariant bilinear form 382 equation 328, 329, 357, 378, 402 integral 13, 189 subspace 28 Inverse element operator 204 240 Involution Irreducible representations 29, 287 Isometric operator 84 space 84 Isomorphic group 89 Non -degenerate bilinear form 382 -singular class 141 Norm 68, 69, 71 Normed linear space 68 One-parameter group 36, 97 Operator 27, 239 Orthogonal complement of set 63 sets 63 sum of linear operators 64 of subspaces 63, 87 vectors 63 Orthogonality relations 62 Orthonormal basis 64 Pauli -Fierz equation 420 theorem 409 PlanchereFs formula 210 Planck's constant 401 Positive definite function 206 time-like vector 23 Principal series of representations 138 Projection operator 86 Projective spin 402 Proper Lorentz group 24 Quantity of a representation 328 Lagrangian function 396 Linear combination 26 dependence 26, 79 functional 70 multiplicative 258 operator 27 space 25 Linked representations 373 Lorentz transformation 22 Reflexive space 72 Regular representation 211,232 Representation (see also specific types, properties, etc.) 28, 199 Residue class 139 Restriction of a representation 238 Ring with involution 204 Rotation group Mass of particle 401 Matrix of an operator 434 Multiple of a representation 81 Scalar product 30 Singular class 141 Space of a representation 28 INDEX Spinor mapping 123 representation 52, 127 State 401, 402 Stereographic projection Subspace 25 Subgroup 89 Sum of series 69 Symmetric ring with involution 204 Tensor 327 Time-like vector 23 Two-valued representation 51 Unimodular group 120 Unitary 11, 12, 30, 85, 168 Unit element representation 137 Vector of current 404 Volume element 16 Weight 44, 48, 105 447 VOLUMES PUBLISHED IN THE SERIES IN PURE AND APPLIED MATHEMATICS Vol Vol Vol Vol Vol Vol Vol Vol Vol 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