Nikolay M. Nikolov

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.

Vertex Algebras in Higher Dimensions and Globally Conformal Invariant Quantum Field Theory

We propose an extension of the definition of vertex algebras in arbitrary space–time dimensions together with their basic structure theory. A one–to–one correspondence between these vertex algebras and axiomatic quantum field theory (QFT) with global conformal invariance (GCI) is constructed.

Four-dimensional conformal field theory models with rational correlation functions

Recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying Wightman axioms is used to construct a family of soluble models in four-dimensional Minkowski spacetime. We consider in detail a model of a neutral scalar field of dimension two. It depends on a positive real parameter c, an analogue of the Virasoro central charge, and admits for all (finite) c an infinite number of conserved symmetric tensor currents. The operator product algebra of is shown to coincide with a simpler one, generated by a bilocal scalar field V(x1,x2) of dimension (1,1). The modes of V together with the unit operator span an infinite-dimensional Lie algebra V whose vacuum (i.e. zero-energy lowest-weight) representations only depend on the central charge c. Wightman positivity (i.e. unitarity of the representations of V) is proven to be equivalent to c.

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.

Globally conformal invariant gauge field theory with rational correlation functions

Operator product expansions (OPE) for the product of a scalar field with its conjugate are presented as infinite sums of bilocal fields Vκ(x1,x2) of dimension (κ,κ). For a globally conformal invariant (GCI) theory we write down the OPE of Vκ into a series of twist (dimension minus rank) 2κ symmetric traceless tensor fields with coefficients computed from the (rational) 4-point function of the scalar field. We argue that the theory of a GCI hermitian scalar field L(x) of dimension 4 in D=4 Minkowski space such that the 3-point functions of a pair of Ls and a scalar field of dimension 2 or 4 vanish can be interpreted as the theory of local observables of a conformally invariant fixed point in a gauge theory with Lagrangian density L(x).

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.

Jacobi identity for vertex algebras in higher dimensions

Vertex algebras in higher dimensions, introduced previously by Nikolov, provide an algebraic framework for investigating axiomatic quantum field theory with global conformal invariance. We develop further the theory of such vertex algebras by introducing formal calculus techniques and investigating the notion of polylocal fields. We derive a Jacobi identity which together with the vacuum axiom can be taken as an equivalent definition of vertex algebra.

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.

Euclidean Configuration Space Renormalization, Residues and Dilation Anomaly

Configuration (x-)space renormalization of Euclidean Feynman amplitudes in a massless quantum field theory is reduced to the study of local extensions of associate homogeneous distributions. Primitively divergent graphs are renormalized, in particular, by subtracting the residue of an analytically regularized expression. Examples are given of computing residues that involve zeta values. The renormalized Green functions are again associate homogeneous distributions of the same degree that transform under indecomposable representations of the dilation group.