2012/04/20 by Stan Gudder, Gudder, Stan · 1 citation
Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Noncommutative and Quantum Gravity Theories #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1204.4506
openalex publication_date 2012/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The basic framework for this article is the causal set approach to discrete quantum gravity (DQG). Let Qn be the collection of causal sets with cardinality not greater than n and let Kn be the standard Hilbert space of complex-valued functions on Qn. The formalism of DQG presents us with a decoherence matrix Dn(x,y), x,y∈ Qn. There is a growth order in Qn and a path in Qn is a maximal chain relative to this order. We denote the set of paths in Qn by Ωn. For ω, ω'∈Ωn we define a bidifference operator \varbigtriangledownω, ω'n on Kn⊗ Kn that is covariant in the sense that \varbigtriangledownω, ω'n leaves Dn stationary. We then define the curvature operator \rscriptω, ω'n=\varbigtriangledownω, ω'n-\varbigtriangledownω', ωn. It turns out that \rscriptω, ω'n naturally decomposes into two parts \rscriptω, ω'n=\dscriptω, ω'n+\tscriptω, ω'n where \dscriptω, ω'n is closely associated with Dn and is called the metric operator while \tscriptω, ω'n is called the mass-energy operator. This decomposition is a discrete analogue of Einstein's equation of general relativity. Our analogue may be useful in determining whether general relativity theory is a close approximation to DQG.