Causal Set Theory

Causal set theory is an approach to quantum gravity which assumes, as a starting point, that spacetime is fundamentally discrete. A causal set is essentially a partial order with a discreteness axiom. It is this primitive structure that should replace the continuum spacetime of General Relativity at short distances/high-energies. See here 1,2,3 for some nice introductory review.

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Non-local d’Alembertians

In the seminal work [Sorkin, ArXiv:0703099] R. Sorkin introduced the discrete causal set version of the wave operator in two dimensions. This operator, once averaged over causal sets well approximating flat spacetime, reduces to the standard d’Alembertian in the continuum limit and in this sense represents the causal set analogue of the continuum operator. However, the fluctuations of the discrete operator were divergent in the continuum limit. This led the author to introduce a new intermediate scale, the non-locality scale, which serves to eliminate this problem. The new scale is a free parameter of the model and as such its value can be in principle constraint by looking at the phenomenology of the model. Since then several works have extended the 2-dimensional operator to generic dimensions and curved spacetimes, resulting also in the discovery of the discrete version of Einstein’s action.

Once averaged over causal sets well approximating a given spacetime, the discrete d’Alembertians give rise to effective non-analytic, non-local wave operators with infinitely many derivatives. In [Belenchia, Benincasa, Dowker, Class.Quant.Grav. 33 (2016) no.24, 245018] we have rigorously proven that the minimal 4-dimensional non-local operator reduces to the standard d’Alembertian and a term dependent on the Ricci scalar. In [Belenchia, Class.Quant.Grav. 33 (2016) no.13] I have proved that the same local limit appears for every generalized non-local d’Alembertian (which are an infinite family of operators) in curved spacetime, a relevant fact in connection to Einstein’s equivalence principle. See also my contribution to the XIV Marcel Grossmann Meeting.

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Non-local Quantum Field Theory

Once the dynamical equation for a scalar field is given, the temptation to quantize the theory is always strong. In [Belenchia, Benincasa, Liberati, JHEP 1503 (2015) 036] we follow this temptation and quantize the theory. While the theory is a bit hostile to quantization, we succeed and find a theory that shows some very nice peculiarities.

Pole structure of the two-point function in the non-local theory in 4D.

As an example, in [Belenchia et al., Phys. Rev. D 93, 044017 (2016)] we investigate the spectral dimension obtained from these non-local continuum d’Alembertians. The spectral dimension is a dimension indicator which is particularly useful when considering models of quantum gravity. It coincides with the usual notion of dimension in a classical flat spacetime, but it is in general different when the microscopic structure of spacetime is considered. What is particularly interesting is that in many QG models the spectral dimension presents a dimensional reduction to 2 dimensions at high-energies (i.e. short distances). This reduction seems to be so ubiquitous that it could be considered as a general feature of QG.

For the non-local theory, we also find a universal dimensional reduction to 2 dimensions, in all dimensions. 

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Entanglement Entropy in Causal Set Theory

Entanglement entropy is a direct consequence of the correlation structure of states in a local relativistic quantum field theory. It turns out that the entanglement entropy of the vacuum state of a quantum field is not only nonzero, but it is also infinite. This divergence can be traced back to the fact that the field lives on a continuum, thus allowing for correlations in the state over arbitrarily small scales. It is not surprising then, that this divergence can be removed by introducing an ultraviolet cutoff that effectively restricts the correlations to be on length scales greater than a cutoff scale.

However, the ad-hoc introduction of a UV cutoff is usually not a covariant procedure and it is in any case not fully justified by the theory itself. Causal set theory offers a structure that, by being fundamentally discrete, does not need any cutoff and it is fully covariant. Thus the hope is that entanglement entropy for fields living on a casual set is always finite.

In [Belenchia et al., Class.Quant.Grav. 35 (2018) no.7, 074002 ] we study this problem. While the correlations are clearly not divergent at short distances, the entanglement entropy still shows some divergence. We trace down these divergences to the non-trivial center of the algebra of observables of the field theory in a subregion and show how and why these divergences can be tamed and that the area-law, characteristic of the entanglement entropy in the continuum, can be reobtained in the appropriate limit.