Particle versus Field Structure in Conformal Quantum Field Theories

TL;DR

This paper explores the incompatibility of particle structures with interactions in conformal quantum field theories, proposing particle-like excitations with anomalous dimensions and connecting the framework to AdS formulations for broader perturbative access.

Contribution

It introduces a novel perspective on particle-like excitations with anomalous dimensions in conformal QFT and links the spectra to the Lorentz invariant quadratic invariant, expanding the perturbative approach via AdS.

Findings

Particle structure is incompatible with interactions in conformal QFT.

Spectra of anomalous dimensions are discrete and related to a Lorentz invariant quadratic invariant.

AdS formulation enables perturbative access to a wider class of conformal theories.

Abstract

I show that a particle structure in conformal field theory is incompatible with interactions. As a substitute one has particle-like exitations whose interpolating fields have in addition to their canonical dimension an anomalous contribution. The spectra of anomalous dimension is given in terms of the Lorentz invariant quadratic invariant (compact mass operator) of a conformal generator $R_{\mu}$ with pure discrete spectrum. The perturbative reading of $R_{0\text{}}$as a Hamiltonian in its own right i.e. associated with an action in a functional integral setting naturally leads to the AdS formulation. The formal service role of AdS in order to access CQFT by a standard perturbative formalism (without being forced to understand first massive theories and then taking their scale-invariant limit) vastly increases the realm of conventionally accessible 4-dim. CQFT beyond those for which one had to use Lagrangians with supersymmetry in order to have a vanishing Beta-function.