Investigation of the sprinkling process and local structures for the discretization of fields on causal sets
Sprinkling is a process to create causal sets from a given spacetime manifold. Using the Poisson distribution, a random, uniformly distributed set of points (called a sprinkle) is selected by this process. Each sprinkle is a causal set (a locally finite partially ordered set) where the partial ordering is inherited from the causal structure of the underlying spacetime manifold. We reviewed this probability process in the second part of (Fewster et al., 2021).
In standard classical and quantum field theory, we describe elementary particle with classical and quantum fields that are defined everywhere on a spacetime manifold and the dynamics follow field equations (like the Klein-Gordon equation for a scalar boson). Similarly, we define fields on the analogue discrete structures of causal sets. The first main difference is that the field equations have to be discretized, in particular the differential operators that appear in the field equations. There are standard approaches to do this operator discretization such that the results agree with those obtained on a spacetime manifold in the limit the sprinkling density becomes arbitrarily large. While such discretizations are typically non-local, we investigated a more recent discretization technique, which is local.
Causal set sprinkling
The sprinkling process can be broken down into two main steps. First, we need to find a way to select points from a region of a spacetime in a random and uniform way. Second, we have to determine the partial ordering by testing all pairs of points if they are causally related.
An implementation for differently shaped regions in Minkowski spacetime is included in my repositories using different programming languages. I started my source code developments on causal sets sprinkling in MATLAB. Later on, I continued developments of causal set sprinkling with Python.
Field equation discretization with local structures
In the beginning of my doctoral research, a new discretization method for the differential operator of the scalar field equations was proposed using a preferred past structure (Dable-Heath et al., 2020). We analysed this local structure in sprinkled during as part of my projects and published the results in the first part of (Fewster et al., 2021). In that publication, we showed that there are ways to find unique preferred pasts that also fulfils a list of desired properties for elements in a sprinkle intrinsically with a high probability.
We describe numerical and analytical investigations of causal sets sprinkled into spacetime manifolds. The first part of the paper is a numerical study of finite causal sets sprinkled into Alexandrov subsets of Minkowski spacetime of dimensions 1+1, 1+2 and 1+3. In particular we consider the rank 2 past of sprinkled causet events, which is the set of events that are two links to the past. Assigning one of the rank 2 past events as ’preferred past’ for each event yields a ’preferred past structure,’ which was recently proposed as the basis for a causal set d’Alembertian. We test six criteria for selecting rank 2 past subsets. One criterion performs particularly well at uniquely selecting — with very high probability — a preferred past satisfying desirable properties. The second part of the paper concerns (infinite) sprinkled causal sets for general spacetime manifolds. After reviewing the construction of the sprinkling process with the Poisson measure, we consider various specific applications. Among other things, we compute the probability of obtaining a sprinkled causal set of a given isomorphism class by combinatorial means, using a correspondence between causal sets in Alexandrov subsets of 1+1 dimensional Minkowski spacetime and 2D-orders. These methods are also used to compute the expected size of the past infinity as a proportion of the total size of a sprinkled causal set.
@article{FewsterHawkinsMinzRejzner:2021,author={Fewster, Christopher J. and Hawkins, Eli and Minz, Christoph and Rejzner, Kasia},title={{Local structure of sprinkled causal sets}},eprint={2011.02965},archiveprefix={arXiv},primaryclass={gr-qc},doi={10.1103/PhysRevD.103.086020},journal={Phys. Rev. D},volume={103},number={8},pages={086020},month=aug,year={2021},}
The framework of perturbative algebraic quantum field theory (pAQFT) is used to construct QFT models on causal sets. We discuss various discretised wave operators, including a new proposal based on the idea of a ’preferred past,’ which we also introduce, and show how they may be used to construct classical free and interacting field theory models on a fixed causal set; additionally, we describe how the sensitivity of observables to changes in the background causal set may be encapsulated in a relative Cauchy evolution. These structures are used as the basis of a deformation quantization, using the methods of pAQFT. The SJ state is defined and discussed as a particular quantum state on the free quantum theory. Finally, using the framework of pAQFT, we construct interacting models for arbitrary interactions that are smooth functions of the field configurations. This is the first construction of such a wide class of models achieved in QFT on causal sets.
@article{DableHeathFewsterRejznerWoods:2020,author={Dable-Heath, Edmund and Fewster, Christopher J. and Rejzner, Kasia and Woods, Nick},title={{Algebraic Classical and Quantum Field Theory on Causal Sets}},eprint={1908.01973},archiveprefix={arXiv},primaryclass={math-ph},doi={10.1103/PhysRevD.101.065013},journal={Phys. Rev. D},volume={101},number={6},pages={065013},month=mar,year={2020},}