Time Ordering, Energy Ordering, and Factorization

TL;DR

This paper explores the relationship between time-ordered and energy-ordered operator integrals, demonstrating how irreducible time-ordered expressions can replace energy ordering for infrared separation in gauge theories, with added factorization benefits.

Contribution

It introduces the use of irreducible time-ordered expressions as an alternative to energy ordering for infrared separation, highlighting their factorization advantages.

Findings

Irreducible time-ordered expressions can replace energy-ordered ones for infrared separation.

Both time-ordered and energy-ordered integrals can be decomposed into irreducible factors.

Energy ordering was originally designed for infrared separation in gauge theories.

Abstract

Relations between integrals of time-ordered product of operators, and their representation in terms of energy-ordered products are studied. Both can be decomposed into irreducible factors and these relations are discussed as well. The energy-ordered representation was invented to separate various infrared contributions in gauge theories. It is shown that the irreducible time-ordered expressions can be used to accomplish the same purpose. Besides, it has the added advantage of being factorizable.