The prescription of a Hamiltonian to a causal set is complicated by the lack of a continuous space or time variables in the granulated universe. The dynamics of a causal set come from parental actions that occur between causal subsets of particles that lead to progeny and from those family histories then emerges what we call space and time. Thus, by its very definition, a causal set is always changing and today, the universe of discrete matter decays just as it's discrete actions grow.

The dimensions of discrete matter and action form step operators of both anti-commuting and non-commuting conjugate Hilbert spaces and therefore form a density matrix with a quadratic Hamiltonian. The Hamiltonian factors into an anti-commuting spin = 2 gravity and a non-commuting spin = 1/2 charge operators and so the granulated universe unifies charge and gravity.

Although the notion of a granulated universe has been around since 1991 (Sorkin 1991), it has not really received a very wide acceptance by many in mainstream science. Although causal set granulation does formulate a quantum density matrix for gravity, the granulated quantum gravity has not yet resulted into any further insights into the conundrums of gravity. Just as the quantization of electromagnetic fields resolved the singularities of quantum charge, granular quantum gravity does remove the singularities of gravity, i.e., black holes.

The gravity of a black hole deflects and blue-shifts light as the figure shows. The skimming light exchanges momentum with the black hole but if the light skims closer than about a wavelength to the event horizon during its journey, that light phase entangles the black hole phase at the two points shown and the photon gains energy but is not absorbed. This phase entanglement means that the states and entropy of the black hole are now in a superposition of light states with those in spacetime.

The black-hole surface represents a phase boundary between light, which facilitates the action of matter, and the black-hole matter and actions that exist as transitive, non-circular, and finite granules. The notion of a granulated black hole is also consistent with the holographic principle, which supposes that black holes preserve information as a hologram on their surfaces.

The entanglement of light's quantum phase stores quantum information in space and time and the entanglement of action's quantum phase stores information for black-hole granules.

Causal Sets: Discrete Gravity, Sorkin

Causal sets and the deep structure of spacetime, Dowker 2008

A Classical Sequential Growth Dynamics for Causal Sets, Rideout and Sorkin, 2004

Entanglement Entropy in Causal Set Theory, Sorkin and Yazdi, 2018

Discrete Aether and Action as Fundamental

FQXi Discrete Aether and Action as Fundamental