Ivan T. Todorov

Majorana representations of the Lorentz group and infinite component fields

A self‐contained exposition is given of the theory of infinite‐component fields with special emphasis on fields transforming under the Majorana representations of the Lorentz group, for which the scalar vertex function is written down explicitly for particles with arbitrary momenta and spins. The problem of spin and statistics for such fields is analyzed. A class of coupled representations of SL(2, C) is studied, containing unitary as well as nonunitary representations (including the Dirac 4‐component spinors), for which invariant first‐order equations can be written down for the free field. Most of the results are known (either from old or from recent publications), but are presented here in a unified way, their derivation sometimes being simplified. Among the few new points we mention: (1) The location of singularities of the matrix elements of some infinite‐dimensional representations of SL(2, C) for complex values of the group parameters. (2) The construction of an infinite‐component local Fermi field...

Rationality of conformally invariant local correlation functions on compactified Minkowski space

Abstract: Rationality of the Wightman functions is proven to follow from energy positivity, locality and a natural condition of global conformal invariance (GCI) in any number D of space-time dimensions. The GCI condition allows to treat correlation functions as generalized sections of a vector bundle over the compactification of Minkowski space M and yields a strong form of locality valid for all non-isotropic intervals if assumed true for space-like separations.

Canonical Quantization of the Boundary Wess-Zumino-Witten Model

We present an analysis of the canonical structure of the Wess-Zumino-Witten theory with untwisted conformal boundary conditions. The phase space of the boundary theory on a strip is shown to coincide with the phase space of the Chern-Simons theory on a solid cylinder (a disc times a line) with two Wilson lines. This reveals a new aspect of the relation between two-dimensional boundary conformal field theories and three-dimensional topological theories. A decomposition of the Chern-Simons phase space on a punctured disc in terms of the one on a punctured sphere and of coadjoint orbits of the loop group easily lends itself to quantization. It results in a description of the quantum boundary degrees of freedom in the WZW model by invariant tensors in a triple product of quantum group representations. In the action on the space of states of the boundary theory, the bulk primary fields of the WZW model are shown to combine the usual vertex operators of the current algebra with monodromy acting on the quantum group invariant tensors. We present the details of this construction for the spin 1/2 fields in the SU(2) WZW theory, establishing their locality and computing their 1-point functions.

Renormalization of massless Feynman amplitudes in configuration space

A systematic study of recursive renormalization of Feynman amplitudes is carried out both in Euclidean and in Minkowski configuration spaces. For a massless quantum field theory (QFT), we use the technique of extending associate homogeneous distributions to complete the renormalization recursion. A homogeneous (Poincare covariant) amplitude is said to be convergent if it admits a (unique covariant) extension as a homogeneous distribution. For any amplitude without subdivergences — i.e. for a Feynman distribution that is homogeneous off the full (small) diagonal — we define a renormalization invariant residue. Its vanishing is a necessary and sufficient condition for the convergence of such an amplitude. It extends to arbitrary — not necessarily primitively divergent — Feynman amplitudes. This notion of convergence is finer than the usual power counting criterion and includes cancellation of divergences.

Gauge dependence of world lines and invariance of theS-matrix in relativistic classical mechanics

The notion of world lines is studied in the constraint Hamiltonian formulation of relativistic point particle dynamics. The particle world lines are shown to depend in general (in the presence of interaction) on the choice of the equal-time hyperplane (the only exception being the elastic scattering of rigid balls). However, the relative motion of a two-particle system and the (classical)S-matrix are independent of this choice.

Partial wave expansion and Wightman positivity in conformal field theory

Abstract A new method for computing exact conformal partial wave expansions is developed and applied to approach the problem of Hilbert space (Wightman) positivity in a nonperturbative four-dimensional quantum field theory model. The model is based on the assumption of global conformal invariance on compactified Minkowski space (GCI). Bilocal fields arising in the harmonic decomposition of the operator product expansion (OPE) prove to be a powerful instrument in exploring the field content. In particular, in the theory of a field L of dimension 4 which has the properties of a (gauge invariant) Lagrangian, the scalar field contribution to the 6-point function of the twist 2 bilocal field is analyzed with the aim to separate the free field part from the nontrivial part.

Coherent state operators and n-point invariants for U q ((sl(2))

We write down a complete set of n-point Uq(sl(2)) invariants (using a polynomial basis for the irreducible finite dimensional Uq-modules) that are regular for all nonzero values of the deformation parameter q.

Infinite dimensional Lie algebras in 4D conformal quantum field theory

The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of two-dimensional chiral conformal field theory, to a higher (even) dimensional spacetime. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, VM(x, y), where the M span a finite dimensional real matrix algebra closed under transposition. The associative algebra is irreducible iff its commutant coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite-dimensional Lie algebra: a central extension of corresponding to the field of reals, of u(∞, ∞) associated with the field of complex numbers, and of so*(4∞) related to the algebra of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N) and , respectively.

CONFORMAL INVARIANCE AND RATIONALITY IN AN EVEN DIMENSIONAL QUANTUM FIELD THEORY

Invariance under finite conformal transformations in Minkowski space and the Wightman axioms imply strong locality (Huygens principle) and rationality of correlation functions, thus providing an extension of the concept of vertex algebra to higher dimensions. Gibbs (finite temperature) expectation values appear as elliptic functions in the conformal time. We survey and further pursue our program of constructing a globally conformal invariant model of a hermitean scalar field L of scale dimension four in Minkowski space-time which can be interpreted as the Lagrangian density of a gauge field theory.

Classical and Quantum Two-body Problem in General Relativity

The two-body problem in general relativity is reduced to the problem of an effective particle (with an energy-dependent relativistic reduced mass) in an external field. The effective potential is evaluated from the Born diagram of the linearized quantum theory of gravity. It reduces to a Schwarzschild-like potential with two different ‘Schwarzschild radii’. The results derived in a weak field approximation are expected to be relevant for relativistic velocities.