Causal sets
This partial order has the physical meaning of the causality relations between spacetime events.
For some decades after the formulation of general relativity, the attitude towards Lorentzian geometry was mostly dedicated to understanding its physical implications and not concerned with theoretical issues.
[1] However, early attempts to use causality as a starting point have been provided by Hermann Weyl and Hendrik Lorentz.
[2] Alfred Robb in two books in 1914 and 1936 suggested an axiomatic framework where the causal precedence played a critical role.
[1] The first explicit proposal of quantising the causal structure of spacetime is attributed to Sumati Surya[1] to E. H. Kronheimer and Roger Penrose,[3] who invented causal spaces in order to "admit structures which can be very different from a manifold".
The program of causal sets is based on a theorem[4] by David Malament, extending former results by Christopher Zeeman,[5] and by Stephen Hawking, A. R. King and P. J.
The conformal factor that is left undetermined is related to the volume of regions in the spacetime.
Causal sets was initiated by Rafael Sorkin who continues to be the main proponent of the program.
He has coined the slogan "Order + Number = Geometry" to characterize the above argument.
The program provides a theory in which space time is fundamentally discrete while retaining local Lorentz invariance.
The difficulty of determining whether a causal set can be embedded into a manifold can be approached from the other direction.
We can create a causal set by sprinkling points into a Lorentzian manifold.
By sprinkling points in proportion to the volume of the spacetime regions and using the causal order relations in the manifold to induce order relations between the sprinkled points, we can produce a causal set that (by construction) can be faithfully embedded into the manifold.
To maintain Lorentz invariance this sprinkling of points must be done randomly using a Poisson process.
When defining these we must remember to rely only on the causal set itself, not on any background spacetime into which it might be embedded.
is a chain consisting only of links such that In general there can be more than one geodesic between two comparable elements.
Tests of this conjecture have been made using causal sets generated from sprinklings into flat spacetimes.
This involves algorithms using the causal set aiming to give the dimension of the manifold into which it can be faithfully embedded.
The algorithms developed so far are based on finding the dimension of a Minkowski spacetime into which the causal set can be faithfully embedded.
By computing the maximal chain length (to estimate the proper time) between two points
These estimators should give the correct dimension for causal sets generated by high-density sprinklings into
An ongoing task is to develop the correct dynamics for causal sets.
The most popular approach to developing causal set dynamics is based on the sum-over-histories version of quantum mechanics.
Elements would be added according to quantum mechanical rules and interference would ensure a large manifold-like spacetime would dominate the contributions.
This model, due to David Rideout and Rafael Sorkin, is known as classical sequential growth (CSG) dynamics.
[10] The classical sequential growth model is a way to generate causal sets by adding new elements one after another.
Rules for how new elements are added are specified and, depending on the parameters in the model, different causal sets result.
In analogy to the path integral formulation of quantum mechanics, one approach to developing a quantum dynamics for causal sets has been to apply an action principle in the sum-over-causal sets approach.
Sorkin has proposed a discrete analogue for the d'Alembertian, which can in turn be used to define the Ricci curvature scalar and thereby the Benincasa–Dowker action on a causal set.
[11][12] Monte-Carlo simulations have provided evidence for a continuum phase in 2D using the Benincasa–Dowker action.
