Suppression of non-manifold-like sets in the causal set path integral

Abstract

While it is possible to build causal sets that approximate spacetime manifolds, most causal sets are not at all manifold-like. We show that a Lorentzian path integral with the Einstein-Hilbert action has a phase in which one large class of non-manifold-like causal sets is strongly suppressed, and suggest a direction for generalization to other classes. While we cannot yet show our argument holds for all non-manifold-like sets, our results make it plausible that the path integral might lead to emergent manifold-like behavior with no need for further conditions. *

The causal set approach to quantum gravity

“…This can be viewed as an effective, continuum-like dynamics, arising from the more fundamental dynamics described above. A recent analytic calculation in Loomis and Carlip (2018) showed that a sub-dominant class of non-manifold-like causal sets, the bilayer posets, are suppressed in the path integral when using the BD action, under certain dimension dependent conditions satisfied by the parameter space. This gives hope that an effective dynamics might be able to overcome the entropy of the non-manifold-like causal sets.…”

“…Indeed, there is a hierarchy of sub-dominant causal sets which are non manifold-like (Dhar 1978(Dhar , 1980Kleitman and Rothschild 1975;Promel et al 2001), with the set of bilayer posets B being the next subdominant class. A recent calculation by Loomis and Carlip (2018) shows that B is suppressed by the BD action when the mesoscale and dimension satisfy certain conditions. The only relations in a bilayer poset are links.…”

“…Moreover, the action itself reduces to a simple sum over n and N 0 . In the limit of large n, Loomis and Carlip (2018) consider p to be a continuous variable using which the partition function Z B can be expressed as an integral over p Z B = dp|B p,n |e iS(p)/ = e iµn dp|B p,n |e 1 2 iµλ0pn 2 +o(n 2 )…”

Boundary contributions in the causal set action

“…One is to construct a quantal version of the classical sequential growth models for causal sets [1][2][3][4] in which causal sets grow in a process of accretion of new elements. In contrast to this process approach, the other strategy is to construct what might be called state sum models where a sum over causal sets -usually of a fixed cardinality -is defined using a weight for each causal set given by the exponential of (i times) an action for the causal set [5][6][7][8].…”

If time had no beginning: growth dynamics for past-infinite causal sets

“…We say that an order C is a convex-rogue if there exists another order D that is not isomorphic to C and that has the same convex-suborders as C. In that case we say that C and D are a convex-rogue pair. 4 Note that the convex-subcausets (convex-suborders) are ordered by inclusion.…”