quantum field theory as group algebra

A group theoretical approach to dynamical quantization in general, and quantum field theory in particular, is developed. This approach opens possibilities of new quantization schemes. Some of these schemes are discussed in detail. They offer certain advantages such as relaxation of the conventional principles of unitarity and causality on the one hand and the possibility to attach some meaning to the formal solutions of the equations of unitarity and causality in terms of functional integrals on the other.     

On the quantum group and quantum algebra approach toq-special functions

Two interpretations ofq-special functions based on quantum groups and algebras have been presented in the literature. The connection between these approaches is explained using as an example the case whereU _q (sl(2)) is the basic structure.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.     

On the integrability of the Lie algebra of the conformal group in quantum field theory

It is shown that the infinitesimal conformal symmetry implies (in any quantum field theory which satisfies the Wightman axioms without invoking locality and global Poincaré symmetry) that there exists a uniquely defined unitary representation of the universal (-sheeted) covering group of the Minkowskian conformal groupSO _e (4,2)/ ₂. Proof was obtained using sufficient conditions for the integrability of a representation of a Lie algebra given by [8].     

On the constraint algebra of quantum gravity in the loop representation

Although an important issue in canonical quantization, the problem of representing the constraint algebra in the loop representation of quantum gravity has received little attention. The only explicit computation was performed by Gambini, Garat, and Pullin for a formal point-splitting regularization of the diffeomorphism and Hamiltonian constraints. It is shown that the calculation of the algebra simplifies considerably when the constraints are expressed not in terms of generic area derivatives but rather as the specific shift operators that reflect the geometric meaning of the constraints.     

Unbounded functionals on a group algebra and applications to quantum theory

A certain class of positive functionals on a group algebra is examined that is pertinent to the induced representations of Frobenius and Mackey. Though these functionals are not bounded in theL ¹ norm, continuity still persists to an extent that secures the existence of a continuous group representation obtained from Gelfand's construction. The theory thus developed provides a new aspect of both the improper states in quantum theory and the induced representations of groups. The method is applied to the Poincaré group and it is shown that the representations, in which particles can be accommodated, are determined up to unitary equivalence by unbounded functionals of a simple structure. It is stressed that representations describing an infinitely degenerate vacuum emerge from mass nonzero representations as the mass tends to zero.     

The spectrum of the algebra of skew differential operators and the irreducible representations of the quantum Heisenberg algebra

The left spectrum of a wide class of the algebras of skew differential operators is described. As a sequence, we determine and classify all the algebraically irreducible representations of the quantum Heisenberg algebra over an arbitrary field.     

低维量子磁性和量子海森伯反铁磁体之有效场论

文章对低维量子磁性的基本问题和相关研究进展作了简单评述，强调了量子非线性Sigma模型在研究量子海森伯反铁磁体的低能物理方面所起的作用以及理论本身存在的疑难问题，并简单介绍了作者最近提出的克服这些疑难问题的一个新建议．     

低维量子磁性和量子海森伯反铁磁体之有效场论

文章对低维量子磁性的基本问题和相关研究进展作了简单评述,强调了量子非线性Sigma模型在研究量子海森伯反铁磁体的低能物理方面所起的作用以及理论本身存在的疑难问题,并简单介绍了作者最近提出的克服这些疑难问题的一个新建议.     

差错基、量子码与群代数

本文找到了一种研究优质差错基和量子纠错码的新方法,即群代数方法, 它为差错基和量子码提供了一种代数表示. 利用这种代数表示, 建立了一系列关于最一般量子纠错码的线性规划限. 关键词： 群代数 差错基 量子纠错码 量子信息     

The quantum harmonic oscillator on a circle and a deformed Heisenberg algebra

We show that the Heisenberg-type algebra describing the first levels of the quantum harmonic oscillator on a circle of large length L is a deformed Heisenberg algebra. The successive energy levels of this quantum harmonic oscillator on a circle of large length L are interpreted, similarly to the standard quantum one-dimensional harmonic oscillator on an infinite line, as being obtained by the creation of a quantum particle of frequency w at very high energies. Received: 29 March 2001 / Revised version: 17 July 2001 / Published online: 31 August 2001     

On the quantum super Virasoro algebra

The quantum super-algebra structure on the deformed super Virasoro algebra is investigated. More specifically we established the possibility of defining a nontrivial Hopf super-algebra on both one and two-parameters deformed super Virasoro algebras.     

q-bosons and the Lie-deformed Heisenberg algebra

It is shown that the non-Hermitian realization of a Lie-deformed Heisenberg algebra given by Jannussis et al. is closely related with the q-Heisenberg-Weyl algebra of Biedenharn and Macfarlane with q being a phase (q = e^iθ, with θ real). The physical implications of this result are stressed.     

Scalar potential for the gauged Heisenberg algebra and a non-polynomial antisymmetric tensor theory

We study some issues related to the effective theory of Calabi–Yau compactifications with fluxes in type II theories. At first the scalar potential for a generic electric Abelian gauging of the Heisenberg algebra, underlying all possible gaugings of R–R isometries, is presented and shown to exhibit, in some circumstances, a “dual” no-scale structure under the interchange of hypermultiplets and vector multiplets. Subsequently a new setting of such theories, when all R–R scalars are dualized into antisymmetric tensors, is discussed. This formulation falls in the class of non-polynomial tensor theories considered long ago by Freedman and Townsend and it may be relevant for the introduction of both electric and magnetic charges.     

A bicovariant differential algebra of a quantum group

A bicovariant differential algebra of four basic objects (coordinate functions, differential 1-forms, Lie derivatives and inner derivations) within a differential calculus on a quantum group is shown to be produced by a direct application of the cross-product construction to the Woronowicz differential complex, whose Hopf algebra properties account for the bicovariance of the algebra. A correspondence with classical differential calculus, including Cartan identity, and some other useful relations are considered. An explicit construction of a bicovariant differential algebra on GL_q(N) is given and its (co)module properties are discussed.     

On the representations of quantum oscillator algebra

It is shown that for q<1, the quantum oscillator algebra has a supplementary family of representations inequivalent to the usual q-Fock representation, with no counterpart at the limit q=1. They are used to build representations of SU _q (1,1) and E(2) in Schwinger's way.     

The Heisenberg ferromagnet as a quantum field theory

We consider a lattice of spin 1/2 ions, described by the discrete form of the current commutation relationsJ _i J _(i) =1/2, [J _i ,J _i ]=i _ij J _i where =1, 2, 3 andi label the lattice sites. The algebra is realized as the Clifford algebra over a Hilbert space. The equations of motion are specified by a formal Hamiltonian of the Heisenberg form: , wheref _ij 0 and only a finite numberQ of ions are linked to any given lattice site. We prove that the Hamiltonian is non-negative in a representation of , and has a ground state exhibiting ferromagnetism. The time displacement group acts continuously on , inducing automorphisms. is asymptotically abelian with respect to the space translations of the lattice.The model is an example of an algebraic quantum field theory and possesses a broken symmetry, the rotation group 0(3). The consequent Goldstone theorem is proved, namely, there is no energy gap in the spectrum ofH.     

An analytic representation of quantum field theory

The connection between a space of quadratically integrable functions of real variablesq and a Hilbert space of analytic functions of complex variablesz established byBargmann is used to introduce quantised field operators for which the -functions of the commutation relations inq-space are replaced by analytic kernel functions inz-space, and a reference to distributions can be avoided.Bargmann's representation is first somewhat modified, so that the derivative terms in the field equations retain their form in the new representation. Local interaction terms inq-space obtain a non-local appearance inz-space. The transition to a 4-dimensional formulation inz-space has to resort to a Euclidean metric. The equations can be derived directly by starting from an action integral inz-space, and applying a variational calculus in which variations are restricted to analytic functions. Explicit analytic expressions are given for free field propagators.     

Hopf algebra primitives in perturbation quantum field theory

The analysis of the combinatorics resulting from the perturbative expansion of the transition amplitude in quantum field theories, and the relation of this expansion to the Hausdorff series leads naturally to consider an infinite dimensional Lie subalgebra and the corresponding enveloping Hopf algebra, to which the elements of this series are associated. We show that in the context of these structures the power sum symmetric functionals of the perturbative expansion are Hopf primitives and that they are given by linear combinations of Hall polynomials, or diagrammatically by Hall trees. We show that each Hall tree corresponds to sums of Feynman diagrams each with the same number of vertices, external legs and loops. In addition, since the Lie subalgebra admits a derivation endomorphism, we also show that with respect to it these primitives are cyclic vectors generated by the free propagator, and thus provide a recursion relation by means of which the (n+1)-vertex connected Green functions can be derived systematically from the n-vertex ones.     

Representations of aq-deformed Heisenberg algebra

We extend the symmetric operators of theq-deformed Heisenberg algebra to essentially self-adjoint operators. On the extended domains the product of the operators is not defined. To represent the algebra we had to enlarge the representation and we find a Hilbert space representation of the deformed Heisenberg algebra in terms of essentially self-adjoint operators. The respective diagonalization can be achieved by aq-deformed Fourier transformation.