### 9.4 Transplanckian physics and entanglement entropy

One way to check whether the entanglement entropy is sensitive to the way the conventional theory is
completed in the UV regime is to study the possible modifications of the standard Lorentz invariant
dispersion relation at large values of momentum (or at short distances). A typical
modification is to break the Lorentz invariance as follows . This issue was studied
in [147] and [40] in the context of the brick-wall model. However, the conclusions made in
these papers are opposite. According to [147] the entropy is still UV divergent, although the
degree of divergence is modified in a way, which depends on the form of function . On
the other hand, [40] claims that the entropy can be made completely UV finite. In a similar
claim [186] suggests that the short-distance finiteness of the 2-point correlation function should imply
the UV finiteness of the entanglement entropy. The entanglement entropy in a wide class of
theories characterized by modified (Lorentz invariant or not) field operators (so that the UV
behavior of the modified propagator is improved compared to the standard one) was calculated
in [184]. The conclusion reached in [184] (see also discussion in Sections 2.12 and 3.13 of this
review) agrees with that of [147]: no matter how good the UV behavior of the propagator is, the
entanglement entropy remains UV divergent. That the short-distance regularity of correlation
functions does not necessarily imply that the entanglement entropy is UV finite was pointed out
in [183].