The Branching Rules for SU(2N)⊃(SU(N)⊃O(N)⊃O(3))⊗SU(2) and the Highest Weight States for SU(N) and O(N)

Simple analytical recurrent formulae of the branching rules for group chain SU(2N)⊃(SU(N)⊃O(N)⊃O(3))⊗SU(2) are obtained. The highest weight states for the irreducible representations [2^al^b] or {nS} of group SU(N) and for the irreducible representations (2^α1^β) or (υS) of group O(N) are constructed respectively     

The Representation of Bosonic Oscillator

The bosonic oscillator is discussed in quite a different method compared with the conventional one. We first strictly deduce an algebraic identity. After introducing an important,boundary condition, we deduce the characteristic equation of number operator N. From this equation,we discuss some properties of bosonic oscillator including its Hilbert space.     

Hamiltonization of the Singular Systems with Hessian Variable Rank

The difficulties in Dirac's algorithm of Hamiltonization are discussed. By comparison of Sudarshan and Mukunda's approach dealing with the singular systems with Hessian variable rank, we study in detail the validity, application and presupposition of our extended Dirac's consistency conditions. It is given and shown that the Hamiltonization of thegeneral singular systems with the varying rank Hessian matrix can be carried out by the algorithm of the extended Dirac's consistency conditions.     

From Dirac's Contact Transformation to Dirac's Conjecture

The finite Dirac's contact transformation according to Dirac's physical idea is obtained. It is shown that its generators belong to the smallest evolution closed subalgebra of the first class constraints but do not exhaust all the first class constraints in general. Then the extended Dirac's conjecture is proposed and is applied to Cawley's counter example of Dirac's conjecture successfully.     

THE GAUGE OPERATORS FOR PSEUDOSCALAR FIELDS AND THE BAKER-HAUSDORFF FORMULA

In this paper, we present the explicit expressions of the gauge operators for pseudoscslar fields in a gauge theory coupled vector and axial-vector fields with the aid of the method of operator algebras.The gauge operators of the pure gauge field theory under the chiral group SU(N)×SU(N) are also presented. P9oreover, the explicit expression of the Baker-Hausdorff formula is obtained for a special case and the general situation is discussed.     

QUANTUM FIELD THEORY IN EXTERNAL SOURCE:COHERENT STATE AND EIGENSTATES OF FIELDS

The real scalar field and the spinor field with external sources present are studied. The relationship between external sources and coherent states is given. In the case of real scalar field the eigenstates of field and its canonically conjugate field are derived in terms of coherent state and functional integration techniques. For a fermion field, it is demonstrated that Schwinger's external sources technique can lead to the spinor coherent states, with which we can also derive the eigenstates of fie3d operators and their conjugate field operators.The Yang-Feldman equation with the Grassmann sources is also obtained.     

NEW OPERATOR IDENTITIES GAINED VIA COHERENT STATE FORMULATION

By the properties of the coherent state and the technique of performing integrations within ordered products as in Ref. [1], we derive some new useful operator ordering identities. Their applications, especially in studying the Boglyubov transformations both in one and two dimensional cases, are presented. The explicit non-unitary operators which turn the coordinate or momentum eigenstate to the corresponding coherent state are also given. Moreover,by using t h e Schwinger angular momentum theory and the technique mentioned above we f!nd the normal product form of the rotation operator, with which the coefficient of the rotation D-matrix can be easily obtained.     

Space-Like Form of Bethe-Salpeter Equation: Many Particles

The Gell-Mann and Low's technology and Bethe-Salpeter equation for manyparticles are first discussed. Then the space-like form of Bethe-Salpeter equation involving both the bound state and scatter state are extended to many-particle case. Consequently we build the formalism of the space-like equation with explicit Lorentz covariant form and without the difficulty of ghost states automatically.     

On the Dromion Interaction of the JMKP Equation: Equipotential Plots

The dromion interaction of the (3+1)-dimensional conditional integrable model, Jimbo-Miwa-Kadomtsev-Petviashvilli (JMKP) system, is re-studied by means of the equipotential surface plots. The result shows that the interaction among dromions of the JMKP system is elastic which corrects the conclusion of our previous paper (J. Math. Phys. 38 (1997) 3123). firtherrnore, the interaction is also irrotational. That is to say, there are not exchanges of any physical quantities like energy, momentum and angular momentum.     

The Branching Rules for U(7)⊃0(7)⊃G2⊃0(3)

The branching rules for U(7)⊃0(7)⊃G₂ and G₂⊃0(3) for at most fourcolumned IRs of U(7)) and O(7) are given respectively by using the method of decomposition of Kronecker products of IRs of groups. The needed formulae for decomposition of Kronecker products of IRs of O(7) and G₂ are given also.