An Einstein Equation for Discrete Quantum Gravity


S. Gudder

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Causal set approach, Discrete quantum gravity


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, ω0 ∈ Ωn we define a bidifference operator 5n ω,ω0 on Kn ⊗ Kn that is covariant in the sense that 5n ω,ω0 leaves Dn stationary. We then de- fine the curvature operator Rn ω,ω0 = 5n ω,ω0 − 5n ω0 ,ω. It turns out that Rn ω,ω0 naturally decomposes into two parts Rn ω,ω0 = Dn ω,ω0 +T n ω,ω0 where Dn ω,ω0 is closely associated with Dn and is called the metric operator while T n ω,ω0 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.

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