Ultralocal quantum field theory in terms of currents

The paper considers the possibility of constructing ultralocal theories, whose Hamiltonians contain no gradient terms and are therefore diagonal in position space, entirely in terms of currents with an equal time current algebra replacing the canonical commutation relations. It is shown that the free current theory can be defined in terms of a certain representation of the current algebra related to the group,S L(2,R). This representation is then constructed by using certain results of Araki and in the process a new infinitely divisible state on the universal covering group ofSL(2,R) is displayed. An ultralocal free theory can also be constructed for the canonical fields, and its relation to the free current theory is shown to involve a certain renormalization procedure reminiscent of the thermodynamic limit.Research sponsored by the Air Force Office of Scientific Research under Contract No. F 44620-71-C-0108 and Contract No. AF 49(638) 1545.     

Q operators for the simple quantum relativistic Toda Chain

We investigate the simple quantum relativistic Toda chain. The ultralocal simple Weyl algebra pair is associated with each site of the chain. Weyl’s q is considered to be inside a unit circle. Both independent Baxter operators Q are constructed explicitly as series in local Weyl generators. The operator-valued Wronskian of Q-s is also calculated.     

Covariant representations of the causal logic

An algebra is found whose representations are in a one-to-one correspondence with covariant, current generated, representations of the causal logic.     

Representations of the Sugawara SU(2) current algebra

The Sugawara SU(2) current algebra is exponentiated in order to obtain the group. Further, the representations of the current algebra are recovered from the representations of the group.Generators of the three-dimensional Euclidean group E, (space translations and space relations) are constructed. An example showing the relation with Yang-Mills theory is presented.     

String propagation in gravitational wave backgrounds

The Conformal Field Theory of the current algebra of the centrally extended 2-d Euclidean group is analyzed. Its representations can be written in terms of four free fields (without background charge) with signature (-+++). We construct all irreducible representations of the current algebra with unitary base out of the free fields and their orbifolds. This is used to investigate the spectrum and scattering of strings moving in the background of a gravitational wave. We find that all the dynamics happens in the transverse space or the longitudinal one but not both.     

Superalgebras,symplectic bosons and the Sugawara construction

New representations of affine Lie algebras are constructed using symplectic bosons of the sort that occur naturally in the BRST treatment of fermionic string theories. These representations are shown to have analogous properties to the current algebra representations in terms of free fermion fields, though they do not act in a positive space. In particular, the condition for the Sugawara construction of the Virasoro algebra to equal the free one is the existence of a superalgebra with a quadratic Casimir operator, paralleling the symmetric space theorem for fermionic field constructions. Both results are seen to be particular cases of a more general super-symmetric space theorem, which arises from considering an affinisation of the superalgebras. These algebras are realised in terms of free fermions and symplectic bosons and lead to a super-Sugawara construction of the Virasoro algebra. The conditions for this to equal a Virasoro algebra obtained from the free fields are provided by the super-symmetric space theorem.     

The Lie-Poisson structure of integrable classical non-linear sigma models

The canonical structure of classical non-linear sigma models on Riemannian symmetric spaces, which constitute the most general class of classical non-linear sigma models known to be integrable, is shown to be governed by a fundamental Poisson bracket relation that fits into ther-s-matrix formalism for non-ultralocal integrable models first discussed by Maillet. The matricesr ands are computed explicitly and, being field dependent, satisfy fundamental Poisson bracket relations of their own, which can be expressed in terms of a new numerical matrixc. It is proposed that all these Poisson brackets taken together are, representation conditions for a new kind of algebra which, for this class of models, replaces the classical Yang-Baxter algebra governing the canonical structure of ultralocal models. The Poisson brackets for the transition matrices are also computed, and the notorious regularization problem associated with the definition of the Poisson brackets for the monodromy matrices is discussed.Suported by the Deutsche Forschungsgemeinschaft, Contract No. Ro 864/1-1Supported by the Studienstiftung des Deutschen Volkes     

Possible representations of local current algebra

Mass operators which lead to possible representations of local (vector) current algebra are discussed.     

多项式角动量代数的代数表示及实现

本文利用三参数李群求代数表示的方法求出多项式角动量代数的代数表示及其酉表示，找到一个能同时承载李代数及相对应的多项式角动量代数的基底，并在该基底下求出两种代数之间的联系，利用该联系则也可求出多项式角动量代数的代数表示.最后求出多项式角动量代数的单玻色实现及其在有限维多项式函数空间的微分实现. 关键词： 多项式角动量代数 Higgs代数 su(2)代数     

Realisations of the representations of para-Fermi algebras—Part IV: Fock spaces of para-Bose and para-Fermi operators

The representations of the para-Fermi algebra in the Fock spaces of para-Bose and para-Fermi operators are constructed. The unitary equivalence of different representations is proved. The Bardeen-Cooper-Schrieffer pair creation and annihilation operators and the four fermion interaction appear as particular realisations of the para-Fermi algebra. The para-Fermi algebra representations in quantum mechanics are discussed.     

Polynomial associative algebras of quantum superintegrable systems

The integrals of motion of classical two-dimensional superintegrable systems, with polynomial integrals of motion, close in a restrained polynomial Poisson algebra; the general form of the quadratic case is investigated. The polynomial Poisson algebra of the classical system is deformed into a quantum associative algebra of the corresponding quantum system, and the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. The finite-dimensional representations of the algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the roots of algebraic equations in the quadratic case.     

Lie algebras and representations of continuous groups

It is shown that every finitely generated continuous group has a subgroup generated by its infinitesimal transformations. This subgroup has a group algebra which is the Lie algebra of the group. By obtaining complete systems in the Lie algebra and complete rectangular arrays, it is shown that these can yield matrix representations of the continuous group. Illustrative examples are given for the rotation groups and for the full linear groups. It would seem that all the finite motion representations can be obtained by these methods, including spin representations of rotation groups. But the completeness of the method is not here demonstrated.     

Irreducible representations of Virasoro-toroidal Lie algebras

Toroidal Lie algebras and their vertex operator representations were introduced in [MEY] and a class of indecomposable modules were investigated. In this work, we extend the toroidal algebra by the Virasoro algebra thus constructing a semi-direct product algebra containing the toroidal algebra as an ideal and the Virasoro algebra as a subalgebra. With the use of vertex operators and certain oscillator representations of the Virasoro algebra it is proved that the corresponding Fock space gives rise to a class of irreducible modules for the Virasoro-toroidal algebra.To A. John Coleman on the occasion of his 75th birthday     

Representation Theory of Lattice Current Algebras

Lattice current algebras were introduced as a regularization of the left- and right moving degrees of freedom in the WZNW model. They provide examples of lattice theories with a local quantum symmetry . Their representation theory is studied in detail. In particular, we construct all irreducible representations along with a lattice analogue of the fusion product for representations of the lattice current algebra. It is shown that for an arbitrary number of lattice sites, the representation categories of the lattice current algebras agree with their continuum counterparts. Received: 25 April 1996 / Accepted: 14 April 1997     

Solutions of the Yang-Baxter Equation and Temperley-Lieb Algebraic Structure

The necessary and sufficient condition for the representations of Temperley-Lieb algebra arising from representations of Heck algebra is presented. Using the condition, the representations of braid group with a quadratic reduction relation are divided into two kinds. One has a representation of Temperley-Lieb algebra behind it and another has not. Some concrete examples are discussed with the condition and their Baxterizations are also mentioned.     

Superfields as Representations

We study irreducible and reducible representations of the generalized Lie algebra of Wess and Zumino. The algebra is integrated to a group with the help of Grassmann algebras and the representations of the algebra are made into representations of the group. We construct invariant sesquilinear forms that are positive definite for the Wess-Zumino algebra over the complex field. We define irreducible superfields for any spin J as specific realizations of such representations. All superfields appearing in the literature are either equivalent to one of these or built up out of these superfields.     

Local Currents for a Deformed Heisenberg–Poincaré Lie Algebra of Quantum Mechanics,and Anyon Statistics

We set out to construct a Lie algebra of local currents whose space integrals, or “charges”, form a subalgebra of the deformed Heisenberg–Poincaré algebra of quantum mechanics discussed by Vilela Mendes, parameterized by a fundamental length scale ℓ. One possible technique is to localize with respect to an abstract single-particle configuration space having one dimension more than the original physical space. Then in the limit ℓ→0, the extra dimension becomes an unobservable, internal degree of freedom. The deformed (1+1)-dimensional theory entails self-adjoint representations of an infinite-dimensional Lie algebra of nonrelativistic, local currents modeled on (2+1)-dimensional space-time. This suggests a new possible interpretation of such representations of the local current algebra, not as describing conventional particles satisfying bosonic, fermionic, or anyonic statistics in two-space, but as describing systems obeying these statistics in a deformed one-dimensional quantum mechanics. In this context, we have an interesting comparison with earlier results of Hansson, Leinaas, and Myrheim on the dimensional reduction of anyon systems. Thus motivated, we introduce irreducible, anyonic representations of the deformed global symmetry algebra. We also compare with the technique of localizing currents with respect to the discrete positional spectrum.     

Unitary representations of the Virasoro and super-Virasoro algebras

It is shown that a method previously given for constructing representations of the Virasoro algebra out of representations of affine Kac-Moody algebras yields the full discrete series of highest weight irreducible representations of the Virasoro algebra. The corresponding method for the super-Virasoro algebras (i.e. the Neveu-Schwarz and Ramond algebras) is described in detail and shown to yield the full discrete series of irreducible highest weight representations.     

Cohomology of some nilpotent subalgebras of the Virasoro and Kac-Moody Lie algebras

The homology of the Lie algebra of algebraic vector fields in the complex line with trivial 3-jet at the point 0 with the coefficients in irreducible highest weight representations of the Virasoro Lie algebra is calculated. The same is done for vector fields with trivial 1-jets at two distinguished points. The class of quasi- finite representations of the Virasoro Lie algebra naturally arises which is the substitute for the class of finite-dimensional representations. The similar results for Kac-Moody Lie algebras are given as well as some conjectures and announcements.