### 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 shows that for large values of
compared to some characteristic length , where is the parameter of non-commutativity
and is the ’t Hooft coupling, then the short-distance contribution to the entanglement entropy
shows an area law of the form

while for smaller values the entropy is proportional to the volume. As seen from Eq. (335) 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
, which shows that there exists a minimal length (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].