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Register a new userContinuous measurement on a causal set with and without a boundary
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Authors
Roman Sverdlov
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The purpose of this paper is two-fold. First, we would like to get rid of common assumption that causal set is bounded and attempt to model its scalar field action under the assumption that it isnt. Secondly, we would like to propose continuous measurement model in this context.
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The goal of this paper is to introduce one of t
A Universe with finite age also has a finite causal scale $chi_S$, so the metric can not be homogeneous for $chi>chi_S$, as it is usually assumed. To account for this, we propose a new causal boundary condition, that can be fulfil by fixing the cosmological constant $Lambda$ (a free parameter for gravity). The resulting Universe is inhomogeneous, with possible variation of cosmological parameters on scales $chi simeq chi_S$. The size of $chi_S$ depends on the details of inflation, but regardless of its size, the boundary condition forces $Lambda/8pi G $ to cancel the contribution of a constant vacuum energy $rho_{vac}$ to the measured $rho_Lambda equiv Lambda/8pi G + rho_{vac}$. To reproduce the observed $rho_{Lambda} simeq 2 rho_m$ today with $chi_S rightarrow infty$ we then need a universe filled with evolving dark energy (DE) with pressure $p_{DE}> - rho_{DE}$ and a fine tuned value of $rho_{DE} simeq 2 rho_m$ today. This seems very odd, but there is another solution to this puzzle. We can have a finite value of $chi_S simeq 3 c/H_0$ without the need of DE. This scale corresponds to half the sky at $z sim 1$ and 60deg at $z sim 1000$, which is consistent with the anomalous lack of correlations observed in the CMB.
The goal of this paper is to propose an approach to the formulation of dynamics for causal sets and coupled matter fields. We start from the continuum version of the action for a Klein-Gordon field coupled to gravity, and rewrite it first using quantities that have a direct correspondent in the case of a causal set, namely volumes, causal relations, and timelike lengths, as variables to describe the geometry. In this step, the local Lagrangian density $L(f;x)$ for a set of fields $f$ is recast into a quasilocal expression $L_0(f;p,q)$ that depends on pairs of causally related points $p prec q$ and is a function of the values of $f$ in the Alexandrov set defined by those points, and whose limit as $p$ and $q$ approach a common point $x$ is $L(f;x)$. We then describe how to discretize $L_0(f;p,q)$, and use it to define a discrete action.
In this paper we address the non-locality issue of quantum field theory on a causal set by rewriting it in such a way that avoids the use of dAlembertian. We do that by replacing scalar field over points with scalar field over edges, where the edges are taken to be very long rather than very short. In particular, they are much longer than the size of the laboratory. Due to their large length, we can single out the edges that are almost parallel to each other, and then use directional derivatives in the direction of those edges (as opposed to dAlembertian) along with a constraint that the derivatives are small in the direction perpendicular to those edges, in order to come up with a plane wave. The scalar field is thought to reside at the future end of those edges, which renders the seemingly nonlocal effects of their large length as physically irrelevant. After that we add by hand the interaction of those plane waves that would amount to 4-vertex coupling of plane waves.
An important probe of quantum geometry is its spectral dimension, defined via a spatial diffusion process. In this work we study the spectral dimension of a ``spatial hypersurface in a manifoldlike causal set using the induced spatial distance function. In previous work, the diffusion was taken on the full causal set, where the nearest neighbours are unbounded in number. The resulting super-diffusion leads to an increase in the spectral dimension at short diffusion times, in contrast to other approaches to quantum gravity. In the current work, by using a temporal localisation in the causal set, the number of nearest spatial neighbours is rendered finite. Using numerical simulations of causal sets obtained from $d=3$ Minkowski spacetime, we find that for a flat spatial hypersurface, the spectral dimension agrees with the Hausdorff dimension at intermediate scales, but shows clear indications of dimensional reduction at small scales, i.e., in the ultraviolet. The latter is a direct consequence of ``discrete asymptotic silence at small scales in causal sets.
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