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 $W(x,y)=⟨\varphi (x)\varphi (y)⟩$ 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)

References in corpus (9)

• The Necessity of the Hadamard Condition
• A Distinguished Vacuum State for a Quantum Field in a Curved Spacetime: Formalism, Features, and Cosmology
• Scalar Field Theory on a Causal Set in Histories Form
• On a Recent Construction of "Vacuum-like" Quantum Field States in Curved Spacetime
• "Vacuum-like" Hadamard states for quantum fields on curved spacetimes
• Entanglement Entropy in Causal Set Theory
• A Ground State for the Causal Diamond in 2 Dimensions
• Spacetime Entanglement Entropy in 1+1 Dimensions
• How to Measure the Quantum Measure

Cited by in corpus (21)

• Interpreting Theories without a Spacetime
• Towards Spacetime Entanglement Entropy for Interacting Theories
• Studies on the SJ Vacuum in de Sitter Spacetime
• Entanglement structure of quantum fields through local probes
• Algebraic Classical and Quantum Field Theory on Causal Sets
• The art of the state
• Entanglement Entropy of Causal Set de Sitter Horizons
• On the Entanglement Entropy of Quantum Fields in Causal Sets
• Insights on Entanglement Entropy in $1+1$ Dimensional Causal Sets
• Covariant Jacobi Brackets for Test Particles
• Sorkin-Johnston vacuum for a massive scalar field in the 2D causal diamond
• Entanglement Entropy of Disjoint Spacetime Intervals in Causal Set Theory
• In-in correlators and scattering amplitudes on a causal set
• A Spacetime Calculation of the Calabrese-Cardy Entanglement Entropy
• Everything You Always Wanted to Know About How Causal Set Theory Can Help with Open Questions in Cosmology, but Were Afraid to Ask
• Quantization, dequantization, and distinguished states
• Aspects of Quantum Fields on Causal Sets
• On the (Non)Hadamard Property of the SJ State in a $1+1$D Causal Diamond
• Entanglement Entropy and Causal Set Theory
• Localizing quantum fields with time-dependent potentials
• Geodesic structure of spacetime near singularities