The sigma model (dual) representation for a two-parameter family of integrable quantum field theories

We introduce and study two-parameter families of integrable field theories. The perturbative and non-perturbative methods are used to justify their factorized scattering theory in the form of the direct products of two S-matrices of the sine-Gordon model. The Bethe ansatz technique is applied for the calculation of the observables in the strong coupling region. The results are in the exact agreement with ones following from the sigma model action which is a two-parameter U(1) ⊗ (1) symmetrical deformation of the O(4) non-finear sigma model. The application of the sigma model representation to related perturbed conformal field theories is discussed.     

Integrability of coupled conformal field theories

The massive phase of two-layer integrable systems is studied by means of RSOS restrictions of affine Toda theories. A general classification of all possible integrable perturbations of coupled minimal models is pursued by an analysis of the (extended) Dynkin diagrams. The models considered in most detail are coupled minimal models which interpolate between magnetically coupled Ising models and Heisenberg spin ladders along the c < 1 discrete series.     

Operator approach to conformal invariant quantum field theories and related problems

Conformal quantum field theory is analysed from a global point of view. The use of the covering transformations leads to a global decomposition theory in which the basic building blocks are nonlocal quantum fields which live on each light cone separately. As an application of the new formalism arbitrary order-disorder mixed n-point functions of the Ising field theory are explicitly calculated.     

Integrable quantum field theories and conformal field theories from lattice models in the light-cone approach

The light-cone lattice approach to two-dimensional quantum field theories is generalized to a large class of vertex models with any number of possible states per link and any simple Lie group of symmetry. Starting from a given lattice model, different scaling limits are defined leading to conformal field theories or to massive integrable quantum field theories, for which the lattice hamiltonian, momentum and currents are constructed. For a large set of models, the complete mass spectrum is also exhibited. Our approach applies equally well to chiral fermionic theories (like the chiral Gross-Neveu) and to bosonic models like the principal chiral model.     

Non-integrable quantum field theories as perturbations of certain integrable models

We approach the study of non-integrable models of two-dimensional quantum field theory as perturbations of the integrable ones. By exploiting the knowledge of the exact S-matrix and form factors of the integrable field theories we obtain the first-order corrections to the mass ratios, the vacuum energy density and the S-matrix of the non-integrable theories. As interesting applications of the formalism, we study the scaling region of the Ising model in an external magnetic field at T ∼ T_c and the scaling region around the minimal model M_2,7. For these models, a remarkable agreement is observed between the theoretical predictions and the data extracted by a numerical diagonalisation of their Hamiltonian.     

Regular quantum field theories

Mathematically regular and more precise versions of quantum field theories are discussed. A new class of representations called minimal wave-packet representations is introduced. Several possibilities of constructing nonconventional, bounded interaction operators (nonpolynomial, nonlinear, explicitly or implicitly nonlocal) corresponding to the traditional ⁴ or ³ interactions are reviewed. The problem of macrocausality is discussed. A procedure of renormalization of regular theories is indicated.     

Nonequilibrium quantum field theories

Combining the Feynman-Vernon influence functional formalism with the real-time formulation of finite-temperature quantum field theories we present a general approach to relativistic quantum field theories out of thermal equilibrium. We clarify the physical meaning of the additional fields encountered in the real-time formulation of quantum statistics and outline diagrammatic rules for perturbative nonequilibrium computations. We derive a generalization of Boltzmann's equation which gives a complete characterization of relativistic nonequilibrium phenomena.     

Two-dimensional quantum field theories containing a massless scalar field

The following new findings are briefly reported:

1. A consistent quantum theory can be formulated for a free massless scalar field in two-dimensional spacetime.
2. Satisfactory operator solutions in terms of asymptotic fields can be constructed in the Thirring and Schwinger models.
3. Gauge invariance is spontaneously broken in the Thirring model as well as in the Schwinger model.

     

Toward finite quantum field theories

The properties that make theN=4 super Yang-Mills theory free from ultraviolet divergences are (i) a universal coupling for gauge and matter interactions, (ii) anomaly-free representations, (iii) no charge renormalization, and (iv) if masses are explicitly introduced into the theory, then these are required to satisfy the mass-squared supertrace sum rule _s=0,1/2(–1)^2s+1(2s+1)M _s ² =0. FiniteN=2 theories are found to satisfy the above criteria. The missing member in this class of field theories are finite field theories consisting ofN=1 superfields. These theories are discussed in the light of the above finiteness properties. In particular, the representations of all simple classical groups satisfying the anomaly-free and no-charge renormalization conditions for finiteN=1 field theories are discussed. A consequence of these restrictions on the allowed representations is that anN=1 finiteSU (5)-based model of strong and electroweak interactions can contain at most five conventional families of quarks and leptons, a constraint almost compatible with the one deduced from cosmological arguments.     

A discrete approach to topological quantum field theories

First, we describe a rather general scheme for constructing three-dimensional euclidean topological quantum field theories, whose basic building blocks are provided by the representation theory of a certain class of (bi-)algebras. Secondly, we discuss in some detail examples, where the algebra is either the function algebra of a finite group, the group algebra of a finite group or a deformation of the enveloping algebra of a classical simple Lie group.     

Boundary conditions in quantum field theories

Several interesting and important quantum field theories must contain the coupling constant in the boundary conditions. The theories considered include quantum electrodynamics of spin-1/2 fermions and gauge field theories.     

Galilean quantum field theories and a ghostless Lee model

Galilean quantum field theories, i.e. kinematically consistent non-relativistic quantum theories with an infinite number of degrees of freedom, are considered. These theories transcend the frame of ordinary quantum mechanics by allowing genuine particle production processes to be described. The general structure of such theories is discussed and contrasted with the typical structure of relativistic quantum field theories which they may serve to illustrate a contrario. Despite the mass superselection rule, and due to the weakening of local commutativity conditions, galilean quantum field theories are much less constrained than relativistic ones. The CPT and spin-and-statistics theorems do not hold here, neither does Haag's theorem.Second-quantized quantum mechanics, some many-body theories (such as the polaron model) and static models are briefly examined, giving simple examples and counterexamples of the general properties asserted.A Lee model with complete nonrelativistic kinematics is studied and shown to give a consistent non-trivial example of a galilean quantum field theory. In this GaliLee model, while all the desirable features of the usual Lee model remain, the ghost problem disappears and the local coupling limit gives meaningful expressions for the physical quantities. The (V N ) sector is solved for the physical V-particle whose renormalization constant is finite for local coupling, and for the N- scattering amplitude, which obeys an exact effective range formula in the same local limit. The elementarity of the V-particle is discussed in relation with theZ=0 rule and Levinson's theorem which is found wanting. The case of an unstable V-particle is also considered, and leads, for local coupling, to an exact Breit-Wigner formula for the N- scattering cross-section.Revised Version. September 1966.     

Topological quantum field theories on manifolds with a boundary

We consider a class of exactly soluble topological quantum field theories on manifolds with a boundary that are invariant on-shell under diffeomorphisms which preserve the boundary. After showing that the functional integral of the two-point function with boundary conditions yields precisely the linking number, we use it to derive topological properties of the linking number. Considering gauge fixing, we obtain exact results of the partition function (Ray-Singer torsion of manifolds with a boundary) and theN-point functions in closed expressions.