Comments on lightlike translations and applications in relativistic quantum field theory

In the algebraic framework of quantum field theory we consider one parameter subgroups of lightlike translations. After establishing a few preliminary properties we prove a certain cluster property and then exhibit the close connection between such subgroups and a class of type III factors. A few applications of this connection are also discussed.     

Conformally invariant equations of the quantum field theory

The equations invariant under the transformation of the conformal algebra are obtained using the Casimir operators. The connections among the components of the field are explicitly derived in the case of indecomposable representations of the conformal algebra which give rise to e.g. the Maxwell equations with currents. The free field equations are also incorporated in the conformally covariant scheme.     

Construction and Borel summability of planar 4-dimensional euclidean field theory

We use the methods of [1] to show that the planar part of the renormalized perturbation theory for ₄ ⁴ -euclidean field theory is Borel-summable on the asymptotically free side of the theory. The Borel sum can therefore be taken as a rigorous definition of theN limit of a massiveN×N matrix model with a +trg ⁴ interaction, hence with wrong sign ofg. Our construction is relevant for a solution of the ultra-violet problem for planar QCD. We also propose a program for studying the structure of the renormalons singularities within the planar world.     

Canonical transformations in quantum field theory and solitons

In a sine-Gordon model we discuss a transformation which can be viewed as a quantum image of the Bäcklund transformation. The soliton is described in terms of condensation of quanta; their confinement is ascribed to the non-implementability of unitary transformations among unitarily inequivalent representations of canonical commutators and to the related topological conservation law.     

On the decomposition of wightman functionals in the euclidean framework

We consider Wightman functionals having the property that the Schwinger functions can be represented by a Euclidean invariant measure on the space of tempered distributions. The strong form of the Osterwalder-Schrader positivity condition is shown to imply that the measure is positive and some restrictions on the Schwinger functions are discussed which guarantee that this condition holds. The ergodic decomposition of the measure leads to a decomposition of the Wightman functional into such with cluster property. We discuss also the role of the positivity condition in connection with a general criterion for the existence of a decomposition.     

On the spectrum of internal symmetries in the algebraic quantum field theory

It is shown that the point spectrum of internal symmetries is always symmetric. It is a group provided the intersection of all local subalgebras is trivial.     

Crystal field theory and the Shubnikov point groups

Two pieces of theory which have so far remained unconnected, crystal field theory and the theory of corepresentations of non-unitary groups, are brought together here for the study of the splitting of atomic energy levels in a crystalline field with the symmetry of one of the magnetic (Shubnikov) point groups. The cases of the various possible relative strengths of the crystalline field and of spin-orbit coupling are considered.

Tables are presented which enable the splitting of any atomic energy level to be obtained very easily in a crystalline field with the symmetry of any one of the 58 magnetic point groups. Examples of the use of these tables are given.

A discussion is given of the relevance of Kramers' theorem to the energy levels of electrons in surroundings with the symmetry of any one of the 58 magnetic point groups.     

A variational approach to bound states in quantum field theory

We consider here in a toy model an approach to bound state problem in a nonperturbative manner using equal time algebra for the interacting field operators. The potential is replaced by offshell bosonic quanta inside the bound state of nonrelativistic particles. The bosonic dressing is determined through energy minimisation, and mass renormalisation is carried out in a nonperturbative manner. Since the interaction is through a scalar field, it does not include spin effects. The model however nicely incorporates an intuitive picture of hadronic bound states in which the gluon fields dress the quarks providing the binding between them and also simulate the gluonic content of hadrons in deep inelastic collisions.     

Bound states and scattering from bound states in the light-front field theory dynamics

Starting from the Weinberg rules we derive a covariant form of the relativistic Schrödinger equation and formulate the bound state problem in the light-front field theory dynamics. We present an explicit rule for embodying the two-body subsystem in the three-body space and demonstrate that the cluster decomposition property is explicitly preserved in the light front field theory dynamics. As an application of these results we write amplitudes forπd→nN ^*, πd→πpn, andπd→πd, in the impulse approximation, in terms of the internal bound state wave functions and two-body reducedt-matrix elements.     

The G-central decomposition of states of statistical systems in the algebraic and in the operational description

We consider a group of automorphisms of a compact convex set X. Using simplical measure techniques we find a decomposition of points of X into “G-primary” points. The decomposition of Guichardet and Kastler for quasi-invariant states of a separable C¹- algebra is generalized.     

Ultraviolet asymptotics of the Green and vertex functions in supergauge invariant field theory

Ultraviolet asymptotics of the Green and vertex functions in supergauge invariant field theory for the SU(N) group is established. Renormalization of the gauge field coupling constant is calculated and the condition for asymtotic freedom is found.     

Some applications of thermal field theory to quark-gluon plasma

We briefly introduce the thermal field theory within imaginary time formalism, the hard thermal loop perturbation theory and some of its applications to the physics of the quark-gluon plasma, possibly created in relativistic heavy-ion collisions     

Remarks on degenerate theories of the conformal invariant quantum field theory

It is remarked that conformally invariant degenerate theories can be found along the current algebra approach to quantum field theory. The only input is the definition of the product of single-valued fields with other fields.     

Poincaré gauge invariant spinor theory and the gravitational field. II

The nonlinear equation for an abstract noncanonical 2-component Weyl spinor field — as used with the inclusion of internal symmetries in Heisenberg's nonlinear spinor theory of elementary particles — which is invariant under scale, phase, and Poincaré transformations is modified in such a way as to become invariant under spacetime dependent phase gauge and Poincaré gauge transformations. In such an equation a phase gauge field B _m , six Lorentz gauge fields A[]m and four translation gauge fields g_m have to be introduced. It is demonstrated that all these fields can be identified as certain combinations of the Weyl spinor field, and hence should be considered in a rough sense as bound states of this spinor field. In particular the electromagnetic field B_m and the gravitational field g m appear as S-states and P-states of a spinor-antispinor system. The noncanonical property and the operator character of the spinor field is essential for this result. The relation between the translation gauge field and the spinor field involves a fundamental length. In a classical geometrical interpretation this relation leads to Einstein's equation of gravitation without cosmological term in a Riemannian space without torsion if the fundamental length is identified with Planck's length. It is shown that this equation is covariant under the larger symmetry group of phase gauge and Poincaré gauge transformations. The modified nonlinear equation constructed solely from a single 2-component Weyl field hence seems to incorporate in an extremely compact way electromagnetic and gravitational interaction in addition to non-mass-zero interactions. In this equation no arbitrary dimensionless constants enter. The considerations can be generalized to Dirac spinor fields and to spinor fields involving additional interior degress of freedom.An abridged version of this paper was presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971.     

Theory of fourier hyperfunctions and its applications to quantum field theory

A new type of Fourier hyperfunctions is introduced. The axiomatic quantum field theory in terms of Fourier hyperfunctions is shown to include Wightman's formulation of tempered fields and its generalizations. The complete equivalence is established between the axioms for Wightman Fourier hyperfunctions and those for Green's functions by eliminating from the latter the linear growth condition of Osterwalder and Schrader.     

Bose condensation,solitons and canonical transformations in quantum field theory

In an quantum sine-Gordon model in 1+1 dimensions, the condition that the vacuum expectation value of the Heisenberg scalar field reproduces static soliton solution to the classical equation is satisfied when the LSZ asymptotic quantum field undergoes a canonical transformation. This can be viewed as a quantum image of the Bäcklund transformation and the soliton is described in terms of condensation of quanta. Their confinement is ascribed to the non-implementability of unitary transformations among unitary inequivalent representations of the canonical commutation relations.Invited talk at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–21, 1981.It is a pleasure to thank Professor J. Niederle and the Organizing Committee for the opportunity of participating in this Symposium and for the kind hospitality.     

Contribution to the theory of transformations in metals

The premise is developed that the elementary act of lattice-restructuring in first-order phase transitions constitutes a transition of a cluster of ions from one phase into another. Based on a microscopic analysis, a procedure is proposed for determining the cluster parameters from the properties of the material and from the thermodynamic parameters of the phase transformation. The concept of lattice-restructuring rate is introduced. Starting from the cluster model of the lattice-restructuring, expressions are obtained for the activation energy of the growth of the new phase. The results are used to calculate numerically - transformations in rapidly heated iron. The results are in satisfactory agreement with experiment.Translation of Preprint No. 194, Lebedev Institute of Physics, Academy of Sciences of the USSR, Moscow, 1990.