Strings from interacting quantum fields

TL;DR

This paper extends the derivation of holographic duals from free quantum fields to interacting theories, showing how boundary correlators relate to bulk propagators and revealing the emergence of the holographic dimension from field theory amplitudes.

Contribution

It introduces a perturbative approach to derive holographic duals for interacting quantum field theories, generalizing previous free-field results and exploring the structure of correlators and the holographic dimension.

Findings

Two-point functions expressed as sums over boundary-to-boundary propagators in AdS and other geometries.

Three-point functions represented as sums over products of bulk-to-boundary propagators.

Identification of the holographic dimension with a continuum limit of Schwinger parameters.

Abstract

We generalize Gopakumar's microscopic derivation of Witten diagrams in large N free quantum field theory [1] to interacting theories in perturbative expansion. For simplicity we consider a matrix scalar field with $\Phi^h$ interaction in d dimensions. Using Schwinger's proper time formulation and organizing the sum over Feynman diagrams by the number of loops $\ell$, we show that the two-point function in the massless case can be expressed as a sum over boundary-to-boundary propagators of massive bulk scalars in $AdS_{d+1}$ with masses determined by $\ell$. The two-point function of the massive theory has the same structure given by a sum over boundary-to-boundary propagators but on a geometry different than AdS. The coefficients in the sum contain information on the putative string geometry dual to the interacting QFT. We also consider the three-point function in the field theory and show that it can again be given as an infinite sum, this time over the products of three bulk-to-boundary propagators. The issue of divergences and renormalization is discussed in detail. We also notice an intriguing similarity between field theory and string amplitudes. In particular we observe that, in the large-N limit, embedding function of string in the holographic direction corresponds to a continuum limit of Schwinger parameters of Feynman diagrams in the limit where $\ell$ diverges. This provides an interpretation of the holographic dimension emerging directly from field theory amplitudes.