Paul Balduf4d ago
As is well known, the Einstein-Hilbert action of general #relativity, i.e. Einstein #Gravity, can not be "simply" turned into a #quantumFieldTheory in the usual perturbative way. Over the decades, this has given rise to dozens of proposed solutions, either by modifying the Einstein-Hilbert action (e.g. introducing a fourth derivative term), or by modifying quantum field theory (e.g. replace it by #stringTheory ), or by examining the possibility for fully non-perturbative quantum field theory (e.g. #asymptoticSafety studies via functional renormalization).
One of the approaches that somehow combines the first and second viewpoint is #causalSetTheory . There, one assumes that spacetime is fundamentally discrete, not like a lattice, but more like a random #graph and the edges in this graph represent causal relations (i.e. being in the future or past light cone). Viewed from afar, this might look like a 4-dimensional spacetime, but looked closely, there is only discrete structure. The edges (causal connections) do not only join nearest neighbours, but can reach far along the lightcone, this gives rise to non-trivial effects, and it makes it surprisingly hard to simulate such a causal set (I've tried that once, but without much success).
The causal sets community has produced a 1 hour video to explain these concepts to a general audience interested in #physics
https://www.youtube.com/watch?v=5EoZE1rOHAo
Causal Sets : The quantum theory that predicted dark energy

YouTube
Show threadCharo del Genio3d ago
@paulbalduf that's interesting... what your describe is what in graph theory we would call a directed small-world graph. So, what's the difficulty in simulating them, if you don't mind me asking?
Show threadPaul Balduf3d ago
@paraw These graphs represent posets which are locally finite, but still, they can have very many edges adjacent to any given vertex. The difficulty is that the edges must represent consistent order relations in the poset. Say I want to generate a random partially ordered set by adding one edge at a time at random. In a conventional random graph I can just do that, but here, I need to check whether I already have an order relation between the two vertices in question. Typically one takes the graph to consist only of the immediate successors in the order, that is, I need to traverse these edges to see if they eventually lead to the vertex in question. One can store the full order relation, i.e. an array that contains for each pair of vertices whether one is in the future of the other, but even then it is quite costly to do even a single update.
Since the structure is so "non-local", it takes many updates for a Monte Carlo algorithm to reach an equilibrium state. In 2015, the authors of https://arxiv.org/abs/1504.05902 reached 85 elements at most, and they restricted themselves to naturally labeled posets (which makes it easier because it is clear from the labeling whether an element can be in the future or in the past). For the actual physical application, one typically wants to generate them weighted by some "action" function, which might by itself be costly to evaluate. It could be that by now there is a more efficient Monte Carlo algorithm, but I am not aware of one.
Onset of the Asymptotic Regime for Finite Orders

We describe a Markov-Chain-Monte-Carlo algorithm which can be used to generate naturally labeled n-element posets at random with a probability distribution of one's choice. Implementing this algorithm for the uniform distribution, we explore the approach to the asymptotic regime in which almost every poset takes on the three-layer structure described by Kleitman and Rothschild (KR). By tracking the n-dependence of several order-invariants, among them the height of the poset, we observe an oscillatory behavior which is very unlike a monotonic approach to the KR regime. Only around n=40 or so does this "finite size dance" appear to give way to a gradual crossover to asymptopia which lasts until n=85, the largest n we have simulated.

arXiv.org
Show threadCharo del Genio3d ago

@paulbalduf I see! This is an interesting problem! I wonder... is there a (polynomial) graphicality theorem for this kind of structure? If so, it may be possible to create an efficient sampling algorithm for these graphs.

Question: given any two vertices, is the relationship between them necessarily strict? That is, can there be any two vertices that are equivalent under the imposed relation of order?

Also, is the order on the vertices imposed a priori? If so, then natural labelling should be algorithmically equivalent to any other labelling.

Show threadPaul Balduf2d ago
@paraw I've only been following #causalSetTheory from the sidelines, so don't quote me on this. However, my understanding is that
1. This is a partially ordered set, so there are pairs of vertices without either being in the future of the other (in physics, they would be "spacelike separated", i.e. not causally connected). Also I think it is allowed that pairs of distinct vertices are equivalent in terms of relations, i.e. 2 vertices have all the same future and past relations, but being distinct. I'm not sure if a situation like that would perhaps been penalized from imposing a physically motivated weighting function (maybe, this situation would correspond to some piece of spacetime "existing twice", which, however, might be fine on very small quantum scales)
2. The simulation paper I shared used labelled vertices, which makes the simulation simpler. In my understanding, in the original physical theory the vertices are not necessarily labelled and the order emerges dynamically, and can change during the simulation.
Under these conditions, a large uniform random partially ordered set most likely consists of 3 layers only. This is why one wants to impose some non-uniform weighting function in order to produce "longer" partially ordered sets which can sensibly be interpreted as a space time.
Show threadChristoph Minz2d ago
@paulbalduf @paraw
Indeed, the space of all causal sets (#causet) is vast and not even the enumeration of (unlabelled) finite partial orders is known in general (OEIS: A000112).
I have worked with causets and created some tools (all shared via my website and GitHub), including a LaTeX package and an online tool (https://c-minz.github.io/assets/html/proset-editor.html). There you can select "predefined type", "random 2-order with n elements" and set some value for n to generate examples (#HasseDiagram).
PrOSET Editor

Editor for diagrams of partially ordered sets

Show threadChristoph Minz2d ago
@paulbalduf @paraw
To your points:
1. Causet models of spacetimes typically do not admit such identical vertices ("local symmetries"). Short summary: http://arxiv.org/abs/2410.02862
2. On a computer, you cannot avoid having a labelling. But such labelling can be changed arbitrarily, while people usually consider those that are order-preserving ("linear extensions" in maths and "natural labellings" in physics).
Do causal sets have symmetries?

Causal sets are locally finite, partially ordered sets (posets), which are considered as discrete models of spacetimes. On the one hand, causal sets corresponding to a spacetime manifold are commonly generated with a random process called sprinkling. This process keeps only a discrete set of points of the manifold and their causal relations (loosing the spacetime symmetries in each sprinkle). On the other hand, the main conjecture of causal set theory is that given an ensemble of causal sets there is a corresponding spacetime manifold and the continuum symmetries of it are like all manifold properties "reconstructable" from the partial orders of all the causal sets in the ensemble. But most generic finite posets have very few layers ("instances of time") in contrast to sprinkles with many layers in a sufficiently large spacetime region. In a recent project, I investigated the automorphism groups of (finite) posets in order to identify and classify their symmetries systematically. The comparison of local symmetries of generic posets (including Kleitmann-Rothschild orders) with sprinkled causal sets may help us to find those posets that can serve as discrete spacetime models in causal set theory.

arXiv.org
Show threadChristoph Minz2d ago
@paulbalduf @paraw
A somewhat physical intuitive explanation for the large class of 3-layer posets is the fact that causal sets may also simulate spacetimes of arbitrary high dimension (in the sense of order dimension or Minkowski space embedding dimension).
Show threadCharo del Genio2d ago
@cminz @paulbalduf thanks for the explanation and the resources. As I mentioned, I have no experience in this, but I worked (and still do) in graph algorithms. One set of questions I answered is how to generate and sample graphs with given sets of constraints. Also in all these cases, the general enumerations are unknown, but it is still possible to create efficient (polynomial) algorithms for construction and sampling. This is what sparked my interest after the first post. Thus, from my point of view, the question is: can the problem be rigorously expressed in terms of existence and construction of graphs with given structural constraints?
Show threadYasaman Yazdi2d ago

@paraw @cminz @paulbalduf

For the partial order, there exists both a convention where it is a strict causal precedence, as well as one where it is precedes or equal to. As mentioned, we can (though it is uncommon) get unrelated elements (sometimes called non-Hegelian) which have the exact same causal relations to every other element and are therefore indistinguishable based on their set of relations.

Show threadCharo del Genio
@ykyazdi @cminz @paulbalduf thank you all for your answers. This is really cool stuff. I should seriously think about doing something in the topic.
Apr 10 at 4:00pmWeb