Braided groups and algebraic quantum field theories

We introduce the notion of a braided group. This is analogous to a supergroup with Bose-Fermi statistics ±1 replaced by braid statistics. We show that every algebraic quantum field theory in two dimensions leads to a braided group of internal symmetries. Every quantum group can be viewed as a braided group.     

Homotopy theory of algebraic quantum field theories

Letters in Mathematical Physics - Motivated by gauge theory, we develop a general framework for chain complex-valued algebraic quantum field theories. Building upon our recent operadic approach to...     

Quantum simulation of quantum field theories in trapped ions

We propose the quantum simulation of fermion and antifermion field modes interacting via a bosonic field mode, and present a possible implementation with two trapped ions. This quantum platform allows for the scalable add up of bosonic and fermionic modes, and represents an avenue towards quantum simulations of quantum field theories in perturbative and nonperturbative regimes.     

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.     

Quantum group structure of quantum Toda conformal field theories (I)

We consider the quantization of non-affine Toda field theories in the light-cone and lattice formalisms. The vertex operators are constructed and their braiding is found to be a consequence of the fundamental commutation relations satisfied by the monodromy matrix. For certain values of the coupling, which correspond to the minimal models, the truncation of the operator algebra is closely tied to the quantum group structure.     

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.     

Quantum logics seen as quantum testability theories

We confute logical relativism and forward an alternative epistemological thesis according to which nonstandard truth-theories are considered theories of some metalinguistic concepts which do not coincide with truth, this latter concept being exhaustively described by Tarski's truth theory. We illustrate our viewpoint by showing that quantum logics can be interpreted as quantum physical theories of the metalinguistic concept of testability in the framework of a suitable classical language (with Tarskian semantics).     

Quantum field theories of vortices and anyons

We develop the quantization of topological solitons (vortices) in three-dimensional quantum field theory, in terms of the Euclidean region functional integral. We analyze in some detail the vortices of the abelian Higgs model. If a Chern-Simons term is added to the action, the vortices turn out to be anyons, i.e. particles with arbitrary real spin and intermediate () statistics. Localization properties of the interpolating field, scattering theory and spin-statistics connection of anyons are discussed. Such analysis might be relevant in connection with the fractional quantum Hall effect and two-dimensional models of HighT _csuperconductors.     

Quantum field theories in all dimensions

In this paper we exhibit a large class of hermitian scalar field theories satisfying the Wightman axioms. For eachd>0, and each polynomialP, we exhibit a collection of theories which are loosely but legitimately based on aP() interaction ind space dimensions. One of the features of the construction is that the Wightmann-point function of each theory is a sum of finitely many integrals associated with Feynman-like graphs. Thus, it is in closed form.     

Quantum equivalence of dual field theories

Motivated by the study of ultraviolet properties of different versions of supergravities duality transformations at the quantum level are discussed. Using the background field method it is proven on shell quantum equivalence for several pairs of dual field theories known to be classically equivalent. The examples considered include duality in chiral model, duality of scalars and second rank antisymmetric gauge tensors, vector duality and duality of the Einstein theory with cosmological term and the Eddington-Schrödinger theory.     

Finiteness of quantum field theories and supersymmetry

The consequences of finiteness are studied for a general renormalizable quantum field theory by analysing the finiteness conditions resulting from the requirement of absence of divergent contributions to the renormalizations of the parameters of an arbitrary gauge theory. In all cases considered, the well-known two-loop finite supersymmetric theories prove to be the unique solution of the finiteness criterion.     

Integrable quantum field theories and Bogoliubov transformations

The Federbush, massless Thirring and continuum Ising models and related integrable relativistic quantum field theories are studied. It is shown that local and covariant classical field operators exist that generate Bogoliubov transformations of the annihilation and creation operators on the Fock spaces of the respective models. The quantum fields of these models are closely related or equal to quadratic forms implementing these transformations, and hence formally inherit the covariance and locality of the underlying classical field operators. It is proved that the Federbush and massless Thirring fields on the physical sector do not satisfy the equation of motion. Closely related fields are defined that do satisfy it, and which lead to the same S-matrix, but these fields are presumably non-local. Bethe transforms are constructed for the various models, and on the unphysical sector the relation with the field theory approach is established.     

General quantum field theories and strict locality

A class of general quantum field theories without explicit use of fieldsA(x) is defined by a set of postulates for the von Neumann algebras ? _C of local observables. The vacuum state is cyclic with respect to any ? _C , and the Borchers tube theorem is shown to hold. Some ? _C are factors and not of finite type. A property of local observables called strict locality is formulated and expressed by means of a necessary and sufficient condition for the algebras ? _C . It is proved for finite regions.     

Solving Toda field theories and related algebraic and differential properties

Toda field theories are important integrable systems. They can be regarded as constrained WZNW models, and this viewpoint helps to give their explicit general solutions, especially when a Drinfeld–Sokolov gauge is used. The main objective of this paper is to carry out this approach of solving the Toda field theories for the classical Lie algebras, following Balog et al. (1990) [5]. In this process, we discover and prove some algebraic identities for principal minors of special matrices. The known elegant solutions of Leznov (1980) [10] fit in our scheme in the sense that they are the general solutions to our conditions discovered in this solving process. To prove this, we find and prove some differential identities for iterated integrals. It can be said that altogether our paper gives complete mathematical proofs for Leznov’s solutions.