# From Green Function to Quantum Field

## TL;DR

This paper introduces a pedagogical approach to quantum scalar fields, emphasizing the role of the Wightman function derived from retarded Green functions, with applications to curved spacetimes and causal sets.

## Contribution

It demonstrates how to construct the Wightman function from retarded Green functions, facilitating analysis in curved spacetimes and causal set frameworks.

## Key findings

- Shows the Wightman function as a tensor on field configurations
- Provides a method to derive $W(x,y)$ from retarded Green functions
- Identifies conditions for zero-entropy or 'purity' in the theory

## Abstract

A pedagogical introduction to the theory of a gaussian scalar field which shows firstly, how the whole theory is encapsulated in the Wightman function $W(x,y)=\langle\phi(x)\phi(y)\rangle$ regarded abstractly as a two-index tensor on the vector space of (spacetime) field configurations, and secondly how one can arrive at $W(x,y)$ starting from nothing but the retarded Green function $G(x,y)$. Conceiving the theory in this manner seems well suited to curved spacetimes and to causal sets. It makes it possible to provide a general spacetime region with a distinguished "vacuum" or "ground state", and to recognize some interesting formal relationships, including a general condition on $W(x,y)$ expressing zero-entropy or "purity".