9.3 Entropy in non-commutative theories and in models with minimal length

One might have hoped that the UV divergence of the entanglement entropy could be cured in a natural way were the structure of spacetime modified on some fundamental level. For example, if spacetime becomes non-commutative at short distances. This idea was tested in the case of simple fuzzy spaces in [68, 67]. Although the area law has been verified, the entanglement entropy appears to be sensitive to the size of the ignored region, a phenomenon, which may be understood as a UV-IR mixing typical for the non-commutative models.

A holographic calculation of the entanglement entropy in non-commutative Yang–Mills theory was considered in [13, 14]. This calculation for a strip of width l shows that for large values of l ≫ lc compared to some characteristic length l ∼ 𝜃λ1∕2∕𝜖 c, where 𝜃 is the parameter of non-commutativity and 2 λ = gYM N is the ’t Hooft coupling, then the short-distance contribution to the entanglement entropy shows an area law of the form

1∕2 S ∼ Neff A(Σ-), Neff = N 2(𝜃λ--- ), (335 ) 𝜖2 𝜖
while for smaller values l ∼ lc the entropy is proportional to the volume. As seen from Eq. (335View Equation) the non-commutativity does not improve the UV behavior of the entropy but leads to the renormalization of the effective number of degrees of freedom that may be interpreted as a manifestation of non-locality of the model.

The other related idea is to consider models in which the Heisenberg uncertainty relation is modified as Δx Δp ≥ ¯h(1 + λ2(Δp )2) 2, which shows that there exists a minimal length Δx ≥ ¯hλ (for a review on the models of this type see [113]). In a brick-wall calculation the presence of this minimal length will regularize the entropy as discussed in [27, 225, 210, 154, 156].

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