Axiomatic Quantum Field Theory in Discrete Spacetime via Multiway Causal Structure: The Case of Entanglement Entropies

The causal set and Wolfram model approaches to discrete quantum gravity both permit the formulation of a manifestly covariant notion of entanglement entropy for quantum fields. In the causal set case, this is given by a construction (due to Sorkin and Johnston) of a 2-point correlation function for a Gaussian scalar field from causal set Feynman propagators and Pauli-Jordan functions, from which an eigendecomposition, and hence an entanglement entropy, can be computed. In the Wolfram model case, it is given instead in terms of the Fubini-Study metric on branchial graphs, whose tensor product structure is inherited functorially from that of finite-dimensional Hilbert spaces. In both cases, the entanglement entropies in question are most naturally defined over an extended spacetime region (hence the manifest covariance), in contrast to the generically non-covariant definitions over single spacelike hypersurfaces common to most continuum quantum field theories. In this article, we show how an axiomatic field theory for a free, massless scalar field (obeying the appropriate bosonic commutation relations) may be rigorously constructed over multiway causal graphs: a combinatorial structure sufficiently general as to encompass both causal sets and Wolfram model evolutions as special cases. We proceed to show numerically that the entanglement entropies computed using both the Sorkin-Johnston approach and the branchial graph approach are monotonically related for a large class of Wolfram model evolution rules. We also prove a special case of this monotonic relationship using a recent geometrical entanglement monotone proposed by Cocchiarella et al.