Deriving Z[J] from the time evolution operator

TL;DR

This paper demonstrates a method to derive the vacuum-to-vacuum transition amplitude Z[J] in quantum field theory directly from the time evolution operator, bypassing the traditional path integral approach.

Contribution

It introduces a novel derivation of Z[J] from the time evolution operator, providing an alternative to the standard path integral method.

Findings

Z[J] can be derived directly from the time evolution operator

The approach offers a new perspective on calculating Greens functions

Simplifies the derivation process in quantum field theory

Abstract

An important quantity in quantum field theory is the vacuum-to-vacuum transition amplitude in the presence of an external source. This quantity is often designated by Z[J] and is the generator for the n-point Greens functions. In textbooks Z[J] is often derived using a path integral formulation, however it will be shown here that it is possible to derive Z[J] directly from the time evolution operator without using the path integral formulation.