arXiv subjects

Stav Zalel

Publications and source records attributed to Stav Zalel.

12 recordsLinked to original sources

arXiv2026

The causal set reduction formula

We derive a reduction formula for matrix elements on a causal set background. We derive an infinite tower of relations between correlators, akin to the Schwinger-Dyson equations of the continuum. Combining these two results we are able to express matrix elements in three different forms: as a path integral and as two distinct sums of correlators. We sketch the form that our method - which circumvents explicit use of differential equation of motion operators - takes in flat continuum spacetime where it provides an alternative expression for the standard LSZ result.

hep-th↗

arXiv2026

Sizes of witnesses in Covtree

Given a set $\Gamma$ of $k$ unlabelled posets, each of size $n$, we say that a poset $Q$ is a \emph{witness} to $\Gamma$ if $\Gamma$ is the set of downsets of size $n$ of $Q$. We say that $Q$ is a \emph{minimal witness} if it does not contain a proper downset that is itself a witness to $\Gamma$. Motivated by the causal set approach to quantum gravity, we study the upper bound on the size of minimal witnesses as a function of $n$ and $k$. We show that there is no linear upper bound of the form $n+k+c$ for any constant $c$. We introduce the \emph{exchange graph of downsets} as a new tool to study this scenario, and use it to show that all minimal witnesses $Q$ satisfy the bound $|Q|\leq nk-n$, and that when $k=3$ there is at least one minimal witness $Q$ that satisfies the bound $|Q|\leq \frac{3}{2}(n+1)$.

math.CO↗

arXiv2024

In-in correlators and scattering amplitudes on a causal set

Causal set theory is an approach to quantum gravity in which spacetime is fundamentally discrete at the Planck scale and takes the form of a Lorentzian lattice, or "causal set", from which continuum spacetime emerges in a large-scale (low-energy) approximation. In this work, we present new developments in the framework of interacting quantum field theory on causal sets. We derive a diagrammatic expansion for in-in correlators in local scalar field theories with finite polynomial interactions. We outline how these same correlators can be computed using the double-path integral which acts as a generating functional for the in-in correlators. We modify the in-in generating functional to obtain a generating functional for in-out correlators. We define a notion of scattering amplitudes on causal sets with non-interacting past and future regions and verify that they are given by S-matrix elements (matrix elements of the time-evolution operator). We describe how these formal developments can be implemented to compute early universe observables under the assumption that spacetime is fundamentally discrete.

hep-th↗

arXiv2023

Hopf algebras from poset growth models

We give a framework for growth models on posets which simultaneously generalizes the Classical Sequential Growth models for posets from causal set theory and the tree growth models of natural growth and simple tree classes, the latter of which also appear as solutions of combinatorial Dyson-Schwinger equations in quantum field theory. We prove which cases of the Classical Sequential Growth models give subHopf algebras of the Hopf algebra of posets, in analogy to a characterization due to Foissy in the Dyson-Schwinger case. We find a family of generating sets for the Connes-Moscovici Hopf algebra.

math.CO↗

arXiv2023

Observables for cyclic causal set cosmologies

In causal set theory, cycles of cosmic expansion and collapse are modelled by causal sets with "breaks" and "posts" and a special role is played by cyclic dynamics in which the universe goes through perpetual cycles. We identify and characterise two algebras of observables for cyclic dynamics in which the causal set universe has infinitely many breaks. The first algebra is constructed from the cylinder sets associated with finite causal sets that have a single maximal element and offers a new framework for defining cyclic dynamics as random walks on a novel tree. The second algebra is generated by a collection of stem-sets and offers a physical interpretation of the observables in these models as statements about unlabeled stems with a single maximal element. There are analogous theorems for cyclic dynamics in which the causal set universe has infinitely many posts.

gr-qc↗

arXiv2023

Covariant Growth Dynamics

The spacetime discreteness of causal set theory has enabled the formulation of novel spacetime dynamics. In these so-called "growth" dynamics, a causal set spacetime is generated probabilistically by means of a random walk on certain tree structures. The first growth dynamics - the Classical Sequential Growth models - were proposed more than two decades ago and their study has furthered our understanding of general covariance and covariant observables within causal set theory. In this setting, labels take the place of spacetime coordinates so that general covariance takes the form of label-invariance and covariant observables are those order-theoretic properties of the causal set which are label-independent. In recent years, these insights have led to a new formulation of growth dynamics which makes no reference to labels. Here we present an overview of these (manifestly) covariant growth dynamics.

gr-qc↗

arXiv2021

If time had no beginning

General Relativity traces the evolution of our Universe back to a Big Bang singularity. To probe physics before the singularity -- if indeed there is a ``before'' -- we must turn to quantum gravity. The Causal Set approach to quantum gravity provides us with a causal structure in the absence of the continuum, thus allowing us to go beyond the Big Bang and consider cosmologies in which time has no beginning. But is a time with no beginning in contradiction with a passage of time? In the Causal Set approach, the passage of time is captured by a process of spacetime growth. We describe how to adapt this process for causal sets in which time has no beginning and discuss the consequences for the nature of time.

gr-qc↗

arXiv2021

If time had no beginning: growth dynamics for past-infinite causal sets

We explore whether the growth dynamics paradigm of Causal Set Theory is compatible with past-infinite causal sets. We modify the Classical Sequential Growth dynamics of Rideout and Sorkin to accommodate growth "into the past" and discuss what form physical constraints such as causality could take in this new framework. We propose convex-suborders as the "observables" or "physical properties" in a theory in which causal sets can be past-infinite and use this proposal to construct a manifestly covariant framework for dynamical models of growth for past-infinite causal sets.

gr-qc↗

arXiv2020

The structure of covtree: searching for manifestly covariant causal set dynamics

Covtree - a partial order on certain sets of finite, unlabeled causal sets - is a manifestly covariant framework for causal set dynamics. Here, as a first step in picking out a class of physically well-motivated covtree dynamics, we study the structure of covtree and the relationship between its paths and their corresponding infinite unlabeled causal sets. We identify the paths which correspond to posts and breaks, prove that covtree has a self-similar structure, and write down a transformation between covtree dynamics akin to the cosmic renormalisation of Rideout and Sorkin's Classical Sequential Growth models. We identify the paths which correspond to causal sets which have a unique natural labeling, thereby solving for the class of dynamics which give rise to these causal sets with unit probability.

gr-qc↗

arXiv2020

A Criterion for Covariance in Complex Sequential Growth Models

The classical sequential growth model for causal sets provides a template for the dynamics in the deep quantum regime. This growth dynamics is intrinsically temporal and causal, with each new element being added to the existing causal set without disturbing its past. In the quantum version, the probability measure on the event algebra is replaced by a quantum measure, which is Hilbert space valued. Because of the temporality of the growth process, in this approach, covariant observables (or beables) are measurable only if the quantum measure extends to the associated sigma algebra of events. This is not always guaranteed. In this work we find a criterion for extension (and thence covariance) in complex sequential growth models for causal sets. We find a large family of models in which the measure extends, so that all covariant observables are measurable.

math-ph↗

arXiv2020

A manifestly covariant framework for causal set dynamics

We propose a manifestly covariant framework for causal set dynamics. The framework is based on a structure, dubbed covtree, which is a partial order on certain sets of finite, unlabeled causal sets. We show that every infinite path in covtree corresponds to at least one infinite, unlabeled causal set. We show that transition probabilities for a classical random walk on covtree induce a classical measure on the $σ$-algebra generated by the stem sets.

gr-qc↗

arXiv2017

Evolution of Universes in Causal Set Cosmology

The causal set approach to the problem of quantum gravity is based on the hypothesis that spacetime is fundamentally discrete. Spacetime discreteness opens the door to novel types of dynamical law for cosmology and the Classical Sequential Growth (CSG) models of Rideout and Sorkin form an interesting class of such laws. It has been shown that a renormalisation of the dynamical parameters of a CSG model occurs whenever the universe undergoes a Big Crunch-Big Bang bounce. In this paper we propose a way to model the creation of a new universe after the singularity of a black hole. We show that renormalisation of dynamical parameters occurs in a CSG model after such a creation event. We speculate that this could realise aspects of Smolin's Cosmological Natural Selection proposal.

gr-qc↗