causet in nLab

Contents

Contents

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→ba \to b in the poset if aa and bb are states of some machine and if operating the machine can take it from state aa to state bb. A simple example would be type of computation which can be done in a single step in→outin \to out, but which needs to be done on two inputs of the same kind. The causal set modelling this situation is

  {(in 1,in 2) → (out 1,in 2) ↓ ↓ (in 1,out 2) → (out 1,out 2)} \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 xx to the point yy in the manifold precisely if yy is in the future of xx, i.e. precisely if there exists a smooth path from xx to yy 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↓X↓yx\darr X\darr y for all objects xx and yy in the poset (the category of “two-step time evolution paths” from yy to xx) 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 XX such that for all objects x,yx,y the interval [x,y]={z∣x≤z≤y}[x,y] = \{z \mid x \le z \le y\} is finite.

• 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 and quantum gravity:

• 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]

• Steven Carlip, Causal Sets and an Emerging Continuum [arXiv:2405.14059]

• Heidar Moradi, Yasaman K. Yazdi, Miguel Zilhão: Fluctuations and Correlations in Causal Set Theory [arXiv:2407.03395]

See also:

• Wikipedia, Causal set

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