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Causal sets is a model for quantum gravity that postulates a fundamental spacetime discreteness. Mathematically, a causal set is a partially ordered set. The elements of the causal set are spacetime events and the order relation between 2 elements is defined to reflect that the corresponding events are causally related.

The causal set hypothesis is based on a series of results proved by Malament, Hawking, King, McCarthy and Levichev that, in essence, show that in Lorentzian manifolds causality information is sufficient to reconstruct other geometric and topological structure up to a conformal factor. This conformal factor is related to the volume of the manifold considered. In causal sets the volume can be related to the number of elements.

Causal sets as a concrete model for quantum gravity was conceived by Rafael Sorkin who formulated the above ideas in precise terms and coined the causal sets slogan Order + Number = Geometry.

History and Motivation
Put in stuff about conception of causality as fundamental in Lorentian manifolds (Riemann, finkelstein, penrose etc.), describe the causality theorems and the haupvermutung

Definition
A causal set (or causet) is a set $$C$$ with a partial order relation $$\prec$$ that is
 * Irreflexive: For all $$x \in C$$, we have $$ x \nprec x $$.
 * Antisymmetric: For all $$x, y \in C$$, we have $$ x \prec y$$ and $$y \prec x$$ implies $$x = y$$.
 * Transitive: For all $$x, y, z \in C$$, we have $$ x \prec y$$ and $$y \prec z $$ implies $$ x \prec z $$.
 * Locally finite: For all $$x, z \in C$$, we have card$$ (\{y \in C | x \prec y \prec z\}) < \aleph_0$$.

Here card($$A$$) denotes the cardinality of a set $$A$$.

In the context of causal sets the set $$C$$ represents the set of spacetime events and the order relation $$x\prec y$$ implies that event $$x$$ causally precedes event $$y$$.

It is a choice of convention to use the irreflexive relation, the reflexive convention can also be used. Causal relations in a Lorentzian manifold (without closed causal curves) satisfy the first three conditions. It is the local finiteness condition that introduces spacetime discreteness.

Kinematics
''write about what kind of objects can be used (chains, antichains etc, these are already defined under partial order so I don't think we need to define them again) and what kind of geometric quantities we expect to be able to reproduce in brief. Also mention that since in dealing with kinematics we don't worry about the causet itself doing anything, we are working with a fixed causet.''

Causal set theory aims to reconstruct continuum geometric and topological information like spacetime dimension, geodesics, curvature, homology etc starting from the causal set. The standard objects available in a causet are chains, antichains, intervals, extremal elements etc. These must be used for the reconstruction. Also, at the level of kinematics a fixed causal set can be considered.

Sprinkling
''why this is needed to get the fixed causet and its use. Mention the KR poset and why we can't use a typical poset and how it might not be a manifold. Local lorentz invariance.''

Scalar Field Theory
A real scalar field on a causet can be defined as a map $$\phi:C\rightarrow\mathbb{R}$$. From the continuum perspective this can be thought of as a scalar field on a manifold $$\mathcal{M}$$ restricted to points that make up the causet. In order to define a free field theory one only needs the Wightman function $$W(x,y)=\langle0|\phi(x)\phi(y)|0\rangle$$ where $$x$$ and $$y\in\mathcal{M}$$. In usual quantum field theory this is provided through the specification of the Feynman propagator, however in causet theory, it is natural to instead start with the retarded propagator.

Entanglement Entropy
mention the causal set BH

Action
boundary terms, phase transitions

Non-Locality
Poincare' violations

Philosophical Implications
''I added this in because I've found so much literature around this! I'd say its worth a mention''

still not sure where to mention quantum measure and the double path integral