Evidence for the continuum in 2D causal set quantum gravity

Abstract

We present evidence for a phase transition in a theory of 2D causal set quantum gravity which contains a dimensionless non-locality parameter ǫ ∈ (0, 1]. The transition is between a continuum phase and a crystalline phase, characterised by a set of covariant observables. For a fixed size of the causal set the transition temperature β Causal set theory(CST) is a discrete approach to quantum gravity which combines local Lorentz invariance with a fundamental discreteness [1]. The spacetime continuum is replaced by a locally finite poset or causal set, with the order relation ≺ being the analog of the spacetime causal order. The continuum quantum gravity path integral is thus replaced by a discrete sum over causal setswhere Ω is a sample space of causal sets and S[C] is an appropriately chosen action. The combination of local Lorentz invariance with fundamental discreteness gives rise to a non-locality in the continuum approximation, making the extraction of local geometric data highly non-trivial. The recent construction of a discrete Einstein-Hilbert action, the Benincasa-Dowker action for causal sets [2], thus allows us for the first time to begin a serious study of the causal set partition function Z CST . Apart from a choice of action, Z CST also depends crucially on the sample space Ω. A natural starting choice for Ω is the collection of countable 1

A Model for the Universe that Begins to Resemble a Quantum Computer

“…as the quantum propensity of the event A ⊆ Ω n [1,3,6]. In general, µ ρ is not additive on 2 Ωn , so µ ρ is not a measure.…”

“…In general, µ ρ is not additive on 2 Ωn , so µ ρ is not a measure. However, µ ρ is Grade-2 additive [1,3,6] in the sense that if A, B, C ∈ 2 Ωn are mutually disjoint, then:…”

“…Precluded events have been extensively studied in [2][3][4]7,8], and they are considered to be events that never occur. We shall give simple examples later that show that if A is precluded and B ⊆ A, then B need not be precluded.…”

“…In P 4 , the antipodal pairs are: (1, 7), (1, 11), (1, 13), (1,14), (2, 7), (2, 11), (2, 13), (2,14) (4, 7), (4, 11), (4, 13), (4,14), (8.7), (8,11), (8,13), (8,14) According to the coevent formulation [3,5,7,8], precluded events do not occur, so we can remove them from consideration. What is left can occur in some a homomorphic realization of possible universes [3,7,8]. We can remove a precluded event from Ω n (or P n ), which is as large as possible, but there is no unique way of doing this, in general.…”

“…We first present a theory of discrete quantum gravity in terms of causal sets (causets) [1][2][3]. Unlike previous sequential growth models, the basic elements of this theory are a special type of causet called a covariant causet (c-causet).…”