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.     

Two-dimensional quantum electrodynamics as a model in the constructive quantum field theory

We investigate a transformation of two-dimensional quantum electrodynamics ((QED)₂)-type models into sine-Gordon models in the constructive method.     

Thermo field dynamics of a quantum algebra

Thermo field dynamics (a real-time finite-temperature quantum field theory) is used to construct a reduction formula for the causal multipoint Green function. The method is applicable to the case in which the field operators form some algebra. The application to current problems in physics is suggested.     

Jordan algebra field theory

It is shown that the set of observable functionals associated with a constrained field theory satisfying two given assumptions is a Jordan algebra under the symmetric Dirac bracket composition law.     

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].     

Two-dimensional quantum gravity and current algebra on a torus

The euclidean generating functional of the correlation functions of the energy-momentum tensor in conformal field theory on a torus is derived. The quantization of the effective action for induced two-dimensional gravity on the torus is still related to the SU(2)×SU(2) current algebra.     

Weil algebra structure and geometrical meaning of BRST transformation in topological quantum field theory

The Weil algebra structure of the BRST transformation of topological quantum field theory is investigated. This structure appears in the gauge and ghost fields sector and is common to both topological quantum field theory and BRS gauge fixed non-abelian gauge theory. By the Weil algebra structure, we can derive the descent equations of topological quantum field theory which generate the Donaldson polynomials. The algebraic structure also reveals the geometrical meaning of the ghost fields ψ and ? in topological quantum field theory as the components of the total curvature.     

Local quantum field theories involving the U(1) current algebra on the circle

The OPE algebra Q=Q(g ² ) generated by a pair of oppositely charged currents (z,±g)(|z|=1) of spin is specified by the leading terms in the small distance expansions of (z ₁,g)(z ₂, -g) and (z ₁,g)(z ₂,g). The current (z,g) splits into a product of a U(1)-Thirring field and a Zamolodchikov-Fattev parafermionic current. The quasilocal(i.e.single-or double-valued) representations of Q are classified. The level k states involve 2(k+1) (ks–k+1) lowest weights (dimensions). The results can be viewed as an extension of the (known) representation theory of the SU(2) current algebra in the bosonic case corresponding to even values of g ² and of the N=2 extended superconformal algebra in the fermionic case corresponding to odd g ².     

Renormalization of a quantum field theory model in a curved space-time with torsion

A quantum field theory model that contains interacting non-Abelian gauge fields, scalar fields, and spinor fields is considered in a curved space-time with torsion. The cone-loop counterterms are found. It is shown that the multiplicative renormalization condition requires a nonminimal coupling of the matter with the gravitational field.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 94–100, August, 1985.     

A model with superconducting solution in quantum field theory

By considering a theory based on a simple model the mathematical possibilities are analysed when the same energy operator may have in quantum theory two (approximatively) orthogonal ground states and eigenfunction systems. The consequences for the elementary particle physics are discussed.

     

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.     

Toward a rigorous quantum field theory

This paper outlines a framework that may provide a mathematically rigorous quantum field theory. The framework relies upon the methods of nonstandard analysis. A theory of nonstandard inner product spaces and operators on these spaces is first developed. This theory is then applied to construct nonstandard Fock spaces which extend the standard Fock spaces. Then a rigorous framework for the field operators of quantum field theory is presented. The results are illustrated for the case of Klein-Gordon fields.     

Renormalization of a non-renormalizable quantum field theory

We present a rigorous construction of the continuum limit for a perturbatively non-renormalizable euclidean quantum field theory: the Gross-Neveu model in two dimensions with bare propagator $\textcolor{red}{}\text{/|p|}{}^{\mathrm{2??}}$ for ?>0 small. The construction proceeds via the control of the flow of Wilson's renormalization group around the non-gaussian ultraviolet stable fixed point in the space of effective actions.     

Representations of BRS algebra

In the canonical formulation of gauge theories the BRS transformation plays a fundamental rôle. The generator of this transformation along with the ghost number forms an algebra called the BRS algebra. Certain properties of this algebra are essential to the proof of unitarity of the S matrix in the physical sector and also to the discussion of color confinement.In the present paper we present all the possible representations of the BRS algebra in the light of indefinite metric.     

Duality in quantum field theory

We postulate a convergent version of operator product expansions on the vacuum and explore some of their consequences. They lead to structures much reminiscent of dual resonance models.     

Hyperfunction quantum field theory

The quantum field theory in terms of Fourier hyperfunctions is constructed. The test function space for hyperfunctions does not containC functions with compact support. In spite of this defect the support concept ofH-valued Fourier hyperfunctions allows to formulate the locality axiom for hyperfunction quantum field theory.     

Quaternionic quantum field theory

We show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal and the fermions, as well as the bosons, obey a second order wave equation. The theory takes the form of either a functional integral with quaternion-imaginary Lagrangian, or a Schrödinger equation and transformation theory for quaternion-valued wave functions, with a quaternion-imaginary Hamiltonian. The connection between the two formulations is developed in detail, and many related issues, including the breakdown of the correspondence principle and the Hilbert space structure, are discussed.