The concept of causal set – or causet, for short – is a concept with an attitude: In itself it is just a partially ordered set (or poset, for short), but meant to be understood as a set of spacetime events subject to the relation of causality.
As such, causal sets play a role in:
computer science, where causal sets model concurrent computations: there exists a morphism $a \to b$ in the poset if $a$ and $b$ are states of some machine and if operating the machine can take it from state $a$ to state $b$. A simple example would be type of computation which can be done in a single step $in \to out$, but which needs to be done on two inputs of the same kind. The causal set modelling this situation is
relativistic field theory, where every globally hyperbolic Lorentzian structure equips the underlying set of points in a manifold naturally with the structure of a causal set: there is a morphism from the point $x$ to the point $y$ in the manifold precisely if $y$ is in the future of $x$, i.e. precisely if there exists a smooth path from $x$ to $y$ whose tangent vector is everywhere non-spacelike with respect to the Lorentzian metric. Moreover, the Lorentzian metric on a manifold can essentially (need to dig out the details here, see discussion at smooth Lorentzian space) be reconstructed from this poset structure and from a measure. This has led to some attempts to use posets as a foundational concept for relativistic physics.
For more see smooth Lorentzian space
The only technical distinction between the notion of posets and that of causal sets is that for a causal set the under-over category $x\darr X\darr y$ for all objects $x$ and $y$ in the poset (the category of “two-step time evolution paths” from $y$ to $x$) is required to be finite. This means these are required to have a finite set of objects (and hence necessarily, being a poset, a finite set of morphisms).
A causal set, or causet is a poset $X$ such that for all objects $x,y$ the interval $[x,y] = \{z \mid x \le z \le y\}$ is finite.
Discussion in relativistic field theory:
Nomaan X, Quantum Field Theory On Causal Sets, in Handbook of Quantum Gravity, Springer (2023) [arXiv:2306.04800]
Stav Zalel, Covariant Growth Dynamics, in Handbook of Quantum Gravity, Springer (2023) [arXiv:2302.10582]
See also:
Last revised on June 9, 2023 at 08:56:16. See the history of this page for a list of all contributions to it.