Conformal invariance of relativistic equations for arbitrary spin particles

We show that any Poincaré-invariant equation for particles of zero mass and of discrete spin provide a unitary representation of the conformal group, and find an explicit expression of the conformal group generators in terms of Poincaré group generators.     

KAC-MOODY ALGEBRA FOR TWO DIMENSIONAL PRINCIPAL CHIRPL MODELS

A Darbou transformation depending on single continuous parameter t is constructed for a principal chiral field. The transformation forms a nonlinear representation of the group for any fixed value of t. Part of the kernel in the Riemann- Hilbert transform is shown to be-related to the Darboux trans- formation with its generators forming a Kac-Moody algebra. Conserved currents associated with the Kac-Moody algebra of the linearized equations and the Noether current for the group transformations with fixed value of t are obtained.     

Gauge formulation of a vector-spinor superalgebra

We present a gauge formulation of the Poincaré algebra extended to include fermionic generators belonging to the vector-spinor representation of the Lorentz group. The gauge action of the theory, quadratic in curvature tensors, is derived.     

Analyticity and Lie generators

We consider and answer in the negative the question whether, given a Lie group representation, analyticity of a vector for the representatives, in the differentiated representation, of a set of Lie generators of the Lie algebra implies analyticity for the group representation.     

Inhomogeneous inverse differential realization of multimodeSU(2) group

The generators and irreducible representation coherent state of the multimodeSU(2) group are constructed by using the inverse operators of the multimode bosonic harmonic oscillator, and the inhomogeneous inverse differential realization of the multimodeSU(2) group are derived.     

Non-Standard Schwinger Fermionic Representation of Unitary Group

The non-standard Schwinger fermionic representation of the unitary group is studied by using n-fermion operators. One finds that the Schwinger fermionic representation of the U(n) group is not unique when n≥3. In general, based on n-fermion operators, the non-standard Schwinger fermionic representation of the U(n) group can be established in a uniform approach, where all the generators commute with the total number operators. The Schwinger fermionic representation of U(C _n ^m ) group is also discussed.     

On Equivalence of Two Realizations for a Nonlinear Lie Algebra

Recently,there are two independent approaches related to a class of nonlinear Lie algebras of three generators;and the realizations of these generators are achieved respectively in Schwinger-boson and position representation.However,by use of the representation transformation between these two representations,the equivalence of the two realizations is therefore proved.     

Another solution of 2D Ising model

The partition function of the Ising model on a two-dimensional regular lattice is calculated by using the matrix representation of a Clifford algebra (the Dirac algebra), with number of generators equal to the number of lattice sites. It is shown that the partition function over all loops in a 2D lattice including self-intersecting ones is the trace of a polynomial in terms of Dirac matrices. The polynomial is an element of the rotation group in the spinor representation. Thus, the partition function is a function of a character on an orthogonal group of a high degree in the spinor representation.     

Spiders for rank 2 Lie algebras

A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namelyA ₂,B ₂, andG ₂. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger.The author was supported by an NSF Postdoctoral Fellowship, grant #DMS-9107908.     

SU(2)群的非齐次逆微分实现

利用玻色振子的逆算符构造了SU(2)群的生成元和不可约表示的相干态,进而导出了SU(2)群的非齐次逆微分实现.     

Superselection rule against the existence of real colour excited states

In a three triplet model the generator of the baryonic charge is supposed to receive contribution from the two hermitian commutative generators of the colour group SU(3)^c. Consequently, the group SU(3)^c cannot be used as a conventional symmetry group, its singlet representation being the single one suitable for the classification of the real hadron multiplets.     

SU(1,1)群的非齐次逆微分实现

利用玻色振子的逆算符构造了SU(1,1)群的生成元和不可约表示的相干态,导出了SU(1,1)群的非齐次逆微分实现.     

An elementaryU(2) theory of radiation fields with spinJ

An elementary theory for a radiation field with any spinJ is presented. This is a natural extension of Maxwell's equations for the electromagnetic field. The idea is to use the generators for theU(2) group in a multidimensional representation. These generators are a linear combination of the ones for infinitesimal Lorentz transformations. The constants of the motion in this formalism are discussed. As an example, angular distributions of the Poynting vector are given.Supported in part by grants from the Research Corporation and the Mitsubishi Fund.Parts of this work were done in partial fulfilment of the requirements for the M.A. Degree at Western Michigan University.     

Towards an explicit construction of the sine-Gordon field theory

We sketch a program for the explicit construction of the sine-Gordon and the massive Thirring-model fields. This construction only works in the phase of the model in which the infinite set of “soliton conservation laws” are valid. The procedure entails two steps of which we only indicate explicitly the first, namely the determination of the S-matrix leading to the sine-Gordon spectrum.     

<Emphasis Type="Italic">SU</Emphasis>(5) symmetry of<Emphasis Type="Italic">spdfg</Emphasis> interacting boson model

The extended interacting boson model withs-, p-, d-, f- andg-bosons included (spdfg IBM) is investigated. The algebraic structure including the generators, the Casimir operators of the groups at theSU(5) dynamical symmetry and the branching rules of the irreducible representation reductions along the group chain are obtained. The typical energy spectrum of the symmetry is given.     

Hidden algebra of the N-body Calogero problem

A certain generalization of the algebra gl(N, ) of first-order differential operators acting on a space of inhomogeneous polynomials in ^N−1 is constructed. The generators of this (non-) Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the N-body Calogero model can be represented as a second-order polynomial in the generators of this algebra. The representation given implies that the Calogero Hamiltonian possesses infinitely-many finite-dimensional invariant subspaces with explicit bases, which are closely related to the finite-dimensional representations of the above algebra. This representation is an alternative to the standard representation of the Bargmann-Fock type in terms of creation and annihilation operators.     

A Possible Way for Constructing Generators of the Poincaré Group in Quantum Field Theory

Starting from the instant form of relativistic quantum dynamics for a system of interacting fields, where amongst the ten generators of the Poincaré group only the Hamiltonian and the boost operators carry interactions, we offer an algebraic method to satisfy the Poincaré commutators.We do not need to employ the Lagrangian formalism for local fields with the N?ether representation of the generators. Our approach is based on an opportunity to separate in the primary interaction density a part which is the Lorentz scalar. It makes possible apply the recursive relations obtained in this work to construct the boosts in case of both local field models (for instance with derivative couplings and spins ≥ 1) and their nonlocal extensions. Such models are typical of the meson theory of nuclear forces, where one has to take into account vector meson exchanges and introduce meson-nucleon vertices with cutoffs in momentum space. Considerable attention is paid to finding analytic expressions for the generators in the clothed-particle representation, in which the so-called bad terms are simultaneously removed from the Hamiltonian and the boosts. Moreover, the mass renormalization terms introduced in the Hamiltonian at the very beginning turn out to be related to certain covariant integrals that are convergent in the field models with appropriate cutoff factors.     

Fundamentals of nonassociative classical field theory

A nonassociative classical field theory is constructed. Octonion algebra is studied. The octonion is represented as the sum of a quaternion and an associator. The octonion algebra is expanded and Lorentz group generators are specified in terms of octonion bases in one of the subalgebras. Lorentz vectors and spinors are constructed in the nonassociative algebra. The representation of the Lorentz group in terms of spin and the associator is obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 22–27, November, 1986.     

The metaplectic semigroup and related topics

There exists an extension of the projective representation of the real symplectic group associated with the CCR's to a projective representation of a semigroup S such that $\text{Sp(2n,}\text{R}\text{) ? S ? Sp(2n, C)}$. It is proven that S is connected, and the universal covering semigroup S? of S is then constructed. It is also shown that the projective representation V? of S can be lifted to a strongly continuous representation V of S. The representation operators are studied; it is shown that they are injective contractions, and that they leave a subspace isomorphic to the Schwartz space invariant. Finally, the one- parameter contraction subsemigroups of the representation are discussed and their generators are computed.