Over the past few years I have worked on a range of questions in Causal Set Theory and Lorentzian geometry which roughly fall into the following categories:

- Quantum Dynamics
- Geometric Reconstruction
- Scalar field theory on causal sets

I describe some of this work below.

### Quantum Dynamics

Constructing a feasible dynamics is perhaps the most challenging aspects of any approach to quantum gravity. In CST there are two routes to take, sidestepping the very important issue of quantum interpretation for the time being .

**Quantum Sequential Growth**

The first is a growth dynamics where a causal set is “grown” element by element starting from a single element. A classical dynamics of this stochastic process has been well studied, but there have been challenges in constructing a full quantum dynamics, including finding the right quantum version of Bell causality. In this approach the dynamics is determined by the choice of quantum measure or decoherence functional.

The simplest quantum dynamics to study is the complexification of the classical transitive percolation growth, characterised by a single parameter. In the classical case, this parameter is a probability which takes values in [0,1] while in the quantum case, it can take values in the complex plane. Several years ago, we showed that this simple model fails to “extend” to the full set of covariant observables. Recently with Stav Zalel we have found other classes of dynamics for which this extension is possible, and hence are candidate models for quantum gravity.

**Continuum Inspired Dynamics **

The second approach is to use the continuum inspired partition function using the discrete Einstein-Hilbert (or Benincasa-Dowker) action, to calculate the expectation value of observers. Read here about recent work which shows that the Link action can suppress the entropy of the dominant class of Kleitmann-Rothschild causal sets.

Instead of a Wick rotation, the path integral can be analytically continued to a statistical partition function by introducing a new parameter akin to an inverse temperature. This gives us a theory of statistical Lorentzian geometries which can be evaluated using standard MCMC methods. Read here about recent work that extends our earlier 2d work to 3d and non-trivial topologies.

*The two phases of 2d causal set quantum gravity *

##### Scalar field theory on causal sets

An important question both fundamentally and phenomenologically is coupling quantum matter to CST. Because of the inherent non-locality of the continnum-like causal set, one cannot appeal to Hamiltonian approaches to quantisation. The Sorkin-Johnston proposal, which has been formulated for a free scalar field theory, uses the Peierls bracket spacetime formulation of QFT to define a unique ground state, starting purely from the Green’s functions. The discrete Green’s functions have been defined for causal sets that are approximated by 2d and 4d Minkowski spacetime. We have recently generalised this to include small causal diamonds in generic curved spacetimes as well as to all of de Sitter spacetime. We have studied the SJ vacuum in various cases — the massive case in the 2d diamond as well as in de Sitter spacetime. More recently we have studied Sorkin’s Spacetime Entanglement Formula in the discrete and continuum setting using the SJ vacuum.