# Causality and Unitarity via the Tree-Loop Duality Relation

## TL;DR

This paper derives causality and unitarity constraints directly from the tree-loop duality relation in quantum field theory, providing a momentum space approach that simplifies handling contact terms.

## Contribution

It introduces a novel derivation of causality and unitarity conditions at the graph level using the tree-loop duality relation in momentum space.

## Key findings

- Derivation of Bogoliubov causality condition at the graph level
- Representation of graphs in terms of lower order cut graphs
- General unitarity relation (Cutkosky rule) obtained from absorptive parts

## Abstract

The tree-loop duality relation is used as a starting point to derive the constraints of causality and unitarity. Specifically, the Bogoliubov causality condition is ab initio derived at the individual graph level. It leads to a representation of a graph in terms of lower order cut graphs. Extracting the absorptive part gives then the general unitarity relation (Cutkosky rule). The derivation, being carried out directly in momentum space, holds for any local (polynomial) hermitian interaction vertices. This is in contrast to the technical difficulties arising from contact terms in the spacetime approach based on the largest time equation.