Scalar field Green functions on causal sets

Abstract

We examine the validity and scope of Johnston's models for scalar field retarded Green functions on causal sets in 2 and 4 dimensions. As in the continuum, the massive Green function can be obtained from the massless one, and hence the key task in causal set theory is to first identify the massless Green function. We propose that the 2-d model provides a Green function for the massive scalar field on causal sets approximated by any topologically trivial 2 dimensional spacetime. We explicitly demonstrate that this is indeed the case in a Riemann normal neighbourhood. In 4-d the model can again be used to provide a Green function for the massive scalar field in a Riemann normal neighbourhood which we compare to Bunch and Parker's continuum Green function. We find that the same prescription can also be used for deSitter spacetime and the conformally flat patch of anti deSitter spacetime. Our analysis then allows us to suggest a generalisation of Johnston's model for the Green function for a causal set approximated by 3 dimensional flat spacetime.

Non-Vacuum Solutions, Gravitational Collapse and Discrete Singularity Theorems in Wolfram Model Systems

“…Our rationale for choosing a massive scalar field as the underlying stress-energy model (rather than the dust or perfect fluid models commonly used in relativistic astrophysics) is to enable more direct comparison with pure Wolfram model evolution: in recent work [28], we showed how a massless scalar field theory (obeying the discrete Klein-Gordon equation) could be defined over an arbitrary Wolfram model system, building upon the previous work of Dowker and Glaser [29], Sorkin [30] and Johnston [31] in the context of causal set theory. Following an ansatz proposed by Dowker et al [32], as well as a more direct approach outlined by Johnston [33][34], the discrete massless Green's functions can naturally be extended to the massive case. To the best of our knowledge, no comparable proposal has yet been made for equipping arbitrary causal sets/Wolfram model evolutions with matter fields consistent with either the dust or perfect fluid forms of the continuum stress-energy tensor.…”

“…These general forms were proposed by Dowker et al [32] as a generic ansatz for obtaining massive causal set Green's functions in arbitrary (integer) numbers of dimensions, based on the formal expansion of massive scalar field Green's functions G…”

High performance algorithms for quantum gravity and cosmology

“…One attempt is the Classical Sequential Growth model[75], which provides a stochastic growth model for causal sets. Subsequent work has examined the dynamics of scalar fields propagating across causal sets[76][77][78][79], including those with variable topology[80]. Other recent work uses a top-down approach with Monte Carlo dynamics to evaluate the gravitational partition function furnished by the Einstein-Hilbert action S EH ,…”