From Green Function to Quantum Field

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 $W(x,y)=⟨\ddot{I}•(x)\ddot{I}•(y)⟩$ 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".

Version 2 contains minor improvements and additions, including the observation that the "inverse" of the commutator-form Δ^{xy} provides the Klein-Gordon inner product. plainTeX, 29 pages, 7 figures. Most current version will be available at http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/157.G2f.pdf (or wherever my home-page may be)

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