Relativistic causality and position space renormalization

TL;DR

This paper reviews the development of position space renormalization in quantum field theory, highlighting its mathematical structure, historical context, and recent connections to number theory, with specific results in massless 4 theory.

Contribution

It provides a historical survey, clarifies the role of residues as quantum periods, and demonstrates integration techniques in massless 4 theory, linking physics and number theory.

Findings

Residues identified with quantum periods

Successful integration over internal vertices in massless 4 theory

Dilation and conformal anomalies demonstrated

Abstract

The paper gives a historical survey of the causal position space renormalization with a special attention to the role of Raymond Stora in the development of this subject. Renormalization is reduced to subtracting the pole term in analytically regularized primitively divergent Feynman amplitudes. The identification of residues with "quantum periods" and their relation to recent developments in number theory are emphasized. We demonstrate the possibility of integration over internal vertices (that requires control over the infrared behavior) in the case of the massless $\varphi^4$ theory and display the dilation and the conformal anomaly.