A. J. Macfarlane

On the Restricted Lorentz Group and Groups Homomorphically Related to It

A study is made of the real restricted Lorentz group, L, and of its relationship(a) to the group, SL(2C), of complex unimodular two‐dimensional matrices, and(b) to the group, O3, of orthogonal transformations in a complex space of three dimensions.The discussion of case (a) is an improved version of the treatment by Wightman. Its notable features are, firstly, that it gives important formulas in new concise forms and their proofs in an elegant and economical manner, and, secondly, that it deals with the nontrivial matter of proving the internal consistency of the formalism. To illustrate the practical utility of the theory, the product of two nonparallel pure Lorentz transformations is studied. In the discussion of case (b), explicit formulas realizing the isomorphism of O3 and L are obtained. These formulas are new and have been applied, for illustrative purposes, to the derivation of the transformation properties under L of the electromagnetic field vectors, regarded as a complex three‐vector (E + iH). ...

Algebraic Tabulation of Clebsch‐Gordan Coefficients of SU3 for the Product (λ, μ)⊗ (1, 1) of Representations of SU3

An algebraic tabulation is made of the Clebsch‐Gordan (CG) coefficients of SU3 which occur in the reduction into irreducible representations of the direct product (λ, μ)⊗ (1, 1) of irreducible representations of SU3. Full explanation is made of the method of handling the complications associated with the possible double occurrence of the representation (λ, μ) itself in the direct product. The phase convention employed is an explicitly stated generalization of the well‐known Condon and Shortley phase convention for SU2. The relationship of the CG coefficients associated with the direct product (1, 1)⊗ (λ, μ) to those coefficients already mentioned is also exhibited.

Matrix Elements of the Octet Operator of SU3

All the nonvanishing matrix elements of all the components of the tensor operator which belongs to the regular representation (the octet) of SU3 have been evaluated. Of special interest is the component Y, for it is usual in the broken unitary symmetry theory of strong interactions to assume that the interactions which break exact SU3 invariance have the same transformation properties as Y. Previously, matrix elements of Y connecting states of the same irreducible representation of SU3 have been given by Okubo in the form of the mass formula. Knowledge of all the matrix elements of Y is essential however if one is to do more than evaluate one‐particle matrix elements in the broken unitary symmetry theory. Our method provides such knowledge for all components of the octet tensor operator with little more effort than is needed to treat Y alone.

Weyl reflections in the unitary symmetry theory of strong interactions

SummaryRecent work has clearly demonstrated the fact that useful predictions in theNe’eman, Gell-Mann unitary symmetry theory of strong interactions follow from consideration of invariance under the Weyl reflections (generalized charge symmetry operations) ofSU3. Here we describe a fairly rapid and general algebraic method for obtaining the effect of the Weyl reflections on the basis vectors of an arbitrary irreducible representation (IR) ofSU3. The important feature of the method is that it applies to those basis vectors of the IR, which belong to the nonsimple weights of the IR and which can therefore not be treated by inspection of the weight diagram of the IR. Results are given for certain IR’s ofSU3 relevant to the Ne’eman-Gell-Mann theory — the 8, 27 and 35 component IRs (1.1) (2.2) and (4.1) of SU3.RiassuntoLavori recenti hanno chiaramente dimostrato che dallo studio dell’invarianza rispetto alle riflessioni di Weyl (operazioni di simmetria di carica generalizzata) diSU3 seguono utili predizioni nella teoria della simmetria unitaria diNe’eman, Gell-Mann delle interazioni forti. Qui si descrive un metodo algebrico generale abbastanza rapido per ottenere l’effetto delle riflessioni di Weyl sui vettori basilari di una rappresentazione irriducibile (IE) arbitraria di SU3. Una caratteristica importante del metodo è che esso si applica a quei vettori basilari dell’IR, ehe appartengono ai pesi non semplici dell’IR e che quindi non possono essere trattati con l’ispezione del diagramma dei pesi dell’IR. Si dànno i risultati per alcune IR di SU3 importanti nella teoria di Ne’eman e Gell-Mann-gli IR a 8, 27 e 35 componenti (1.1), (2.2) e (4.1) diSU3.

Electromagnetic properties of stable particles and resonances according to the unitary symmetry theory

SummaryWe show that a theory whoseSU3-invariant strong interactions are perturbed by electromagnetic interactions alone may be obtained formally by a certain unitary transformation of a theory whoseSU3-invariant strong interactions are perturbed by merely charge-independent interactions. By exploiting readily available information on the latter theory, we can give a rapid derivation of relationships between the electromagnetic properties of various particles and resonances. While many new results are presented, principal emphasis is on the power of the method and on the recognition of the precise nature of the assumptions necessary for the derivation of results. For example, of the seven familiar relations between the magnetic moments of baryons only two require use of the assumption that magnetic moments depend linearly on the electromagnetic charge-current density; the others exemplify relations which hold in identical form for all electromagnetic properties of baryons — electric and magnetic form factors, electromagnetic self-energies, Compton scattering amplitudes etc.RiassuntoDimostriamo che si può ottenere formalmente una teoria in cui le interazioni forti invarianti rispetto adSU3 sono perturbate solo da interazioni elettromagnetiche con una trasformazione unitaria di una teoria in cui le interazioni forti invarianti rispetto adSU3 sono perturbate da interazioni indipendenti soltanto dalla carica. Sfruttando informazioni facilmente ottenibili su quest’ultima teoria, possiamo dare una rapida deduzione delle relazioni fra le proprietà elettromagnetiche di varie particelle e risonanze. Mentre si riferiscono molti nuovi risultati, si mette in particolare rilievo l’efficacia del metodo ed il riconoscimento della natura esatta delle ipotesi necessarie per dedurre i risultati. Per esempio, delle sette familiari relazioni fra i momenti magnetici dei barioni solo due richiedono l’uso dell’ipotesi che ci sia una dipendenza lineare fra i momenti magnetici e la densità della corrente di carica elettromagnetica; le altre sono esempi di relazioni valide in forma identica per tutte le proprietà elettromagnetiche dei barioni — fattori di forma elettrici e magnetici, autoenergie elettromagnetiche, ampiezze dello scattering di Compton, etc.

Generalized Shmushkevich Method: Proof of Basic Results

We here derive certain orthogonality properties of the Clebsch‐Gordan (CG) coefficients of an arbitrary compact group G. Our discussion recognizes the fact that the irreducible representations (IRs) of G need not be equivalent to their complex conjugates and that the same IR can appear more than once in the reduction of the direct product of two IRs of G. The properties obtained allow the development of a generalized Shmushkevich method for directly writing down consequences of the invariance of particle interactions under G. The discussion given is sufficiently general to apply to the currently interesting cases of SU3 and G2.

Triality Type and its Generalization in Unitary Symmetry Theories

Within the context of an extension of the SU3‐symmetry theory recently suggested by Gell‐Mann and further developed by the authors, certain aspects of the theory of the special unitary groups are examined. The plurality type of a given representation is introduced as the generalization of the triality concept to SUn+1 and is shown to be associated with a multiplicative conservation law. Theorems for the reduction of representations of SUn+1 with respect to SUn⊗ U1(n) are derived which are subsequently used to relate plurality type to the existence of fractional eigenvalues for the generator Y1(n) of U1(n).

Representation Mixing forSU3andG2

A theory whose exact invariance with respect to SU 3 is broken by interactions transforming like the hypercharge generator of SU 3 is studied in perturbation theory. Physical particle states corresponding to unperturbed SU 3 multiplets are obtained by mixing into these multiplets possible impurities corresponding to other SU 3 multiplets, these impurities being as general as the perturbation allows. The effect of symmetry breaking on various physical quantities like masses and magnetic moments can then be obtained in terms of mixing parameters, and interesting formulas follow various restrictions of their values. Similar considerations applied to G 2 show that many of the conclusions which made exact G 2 invariance unattractive remain for approximate G 2 invariance.

REPRESENTATION MIXING FOR SU

A theory whose exact invariance with respect to SU 3 is broken by interactions transforming like the hypercharge generator of SU 3 is studied in perturbation theory. Physical particle states corresponding to unperturbed SU 3 multiplets are obtained by mixing into these multiplets possible impurities corresponding to other SU 3 multiplets, these impurities being as general as the perturbation allows. The effect of symmetry breaking on various physical quantities like masses and magnetic moments can then be obtained in terms of mixing parameters, and interesting formulas follow various restrictions of their values. Similar considerations applied to G 2 show that many of the conclusions which made exact G 2 invariance unattractive remain for approximate G 2 invariance.