nLab causet

Idea

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

  $$\left\lbrace \array{ (in_1, in_2) &\to& (out_1, in_2) \\ \downarrow && \downarrow \\ (in_1, out_2) &\to& (out_1,out_2) } \right\rbrace$$

• 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).

Definition

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.

Related concepts

• causal structure

• causal order

• causal complement, causal closure

• causal locality, causal additivity

• causal perturbation theory

• time-ordered product

• S-matrix

References

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:

• Wikipeida, Causal set

• Tim Porter, Enriched categories and models for spaces of dipaths