N. Mukunda

Tensor Methods and a Unified Representation Theory of SU3

Starting with irreducible tensors, we develop an explicit construction of orthonormal basic states for an arbitrary unitary irreducible representation (λ, μ) of the group SU3. A knowledge of the simple properties of the irreducible tensors can then be exploited to obtain a variety of results, which ordinarily require more abstract algebraic methods for their derivation. As illustrative applications, we (i) derive Biedenharns expressions for the matrix elements of the generators of SU3, (ii) compute the matrix elements of octet‐type operators for the case (λ, μ) → (λ, μ), and (iii) develop an explicit unitary transformation connecting the isospin and the U‐spin states in any arbitrary irreducible representation.

Irreducible Representations of Generalized Oscillator Operators

All of the irreducible representations are found for a single pair of creation and annihilation operators which together with the symmetric or antisymmetric number operator satisfy the generalized commutation relation characteristic of para‐Bose or para‐Fermi field quantization. The procedure is simply to identify certain combinations of these three operators with the three generators of the three‐dimensional rotation group in the para‐Fermi case, and with the three generators of the three‐dimensional Lorentz group in the para‐Bose case. The irreducible representations are then easily obtained by the usual raising and lowering operator techniques. The applicability of these techniques is demonstrated by a simple argument which shows that the commutation relations require that the generator to be diagonalized have a discrete spectrum.

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.

Algebraic Tabulation of the Clebsch‐Gordan Coefficients for Reduction of the Product (λ, μ) ⊕ (3, 0) of Irreducible Representations of SU(3)

Tables of Clebsch‐Gordan coefficients of SU (3) for the reduction of the product (λ, μ)⊗ (3, 0) of representations of SU(3) are constructed by use of the tensor method. Derivations of some crossing and symmetry relations for the SU3 Clebsch‐Gordan coefficients are given, and the Clebsch‐Gordan coefficients for the product (μ, λ)⊗ (0, 3) are related to those mentioned above. The phase convention used in compiling the tables is stated and explained.

Relativistic-particle interactions-a third world view

SummaryA new form for describing interacting relativistic classical particles with invariant world-lines is presented. In contrast to the use of independent particle variables or of «centre of mass» and relative co-ordinates previously considered in the literature, we introduce a Lorentz matrix and an associated conjugate four-dimensional angular momentum as collective variables; these are supplemented with a suitable number of internal variables. The methods of constrained Hamiltonian dynamics are used to ensure that the number of independent degrees of freedom corresponds precisely to that for a system of a given number of particles. Independent particle variables can be defined in the formalism, and interactions included. The eleventh-generator form for relativistic Hamiltonian dynamics is essential to make relativistic invariance, invariant world-lines and interaction compatible with one another.RiassuntoSi considera un nuovo modo per descrivere particelle classiche relativistiche in interazione con linee duniverso invarianti. Invece di usare variabili indipendenti per ogni particella o le variabili del «centro di massa» e le coordinate relative già usate nella letteratura precedente, si introducono variabili collettive usando una matrice di Lorentz e un momento angolare quadridimensionale corrispondente; a queste si aggiunge un adeguato numero di variabili interne. Si utilizzano i metodi di dianamica hamiltoniana con vincolo per ridurre opportunamente il numero di variabili in modo che il numero di gradi di libertà corrisponda al numero di particelle. Nel formalismo si possono definire le variabili indipendenti per ogni particella e includere le interazioni. Il formalismo dellundicesimo generatore per la dinamica relativistica è essenziale perchè invarianza relativistica, linee di universo invarianti e interazione siano mutuanmente compatibili.РезюмеПредлагается новая форма для описания взаимодействующих релятивистских классическнх частиц с инвариантными мировыми линиями. В противоположность использованию независимых переменных частиц или «центра масс» и относительных координат, ранее рассмотренных в литературе, мы вводим матрицу Лоренца и сопряженный четырехмерный угловой момент, как коллективные переменные; к этим переменным добавляется соответствующее число внутренних переменных. Методы для ограниченной гамильтоновой динамики используются для доказательства, что число независимых степеней свободы точно соответствует чуслу степеней свободы для системы с данным числом частиц. Независимые переменные частиц могут быть определены в этом формализме и включены взаимодействия. Форма генераторов для релятивистской гамольтоновой динамики обладает, по существу, релятивистской инвариантностью, инавариантными мировыми линиями и взаимодействиями, совместимыми друг с другом.

Classical models for Regge trajectories

Two classical models for particles with internal structure and which describe Regge trajectories are developed. The remarkable geometric and other properties of the two internal spaces are highlighted. It is shown that the conditions of positive time-like four-velocity and energy momentum for the classical system imply strong and physically reasonable conditions on the Regge mass-spin relationship.

Null phase curves and manifolds in geometric phase theory

Bargmann invariants and null phase curves are known to be important ingredients in understanding the essential nature of the geometric phase in quantum mechanics. Null phase manifolds in quantum-mechanical ray spaces are submanifolds made up entirely of null phase curves, and so are equally important for geometric phase considerations. It is shown that the complete characterization of null phase manifolds involves both the Riemannian metric structure and the symplectic structure of ray space in equal measure, which thus brings together these two aspects in a natural manner.

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.