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 ω2 = k2 at large values of momentum k (or at short distances). A typical modification is to break the Lorentz invariance as follows 2 2 2 ω = k + f (k ). This issue was studied in [147Jump To The Next Citation Point] and [40Jump To The Next Citation Point] in the context of the brick-wall model. However, the conclusions made in these papers are opposite. According to [147Jump To The Next Citation Point] the entropy is still UV divergent, although the degree of divergence is modified in a way, which depends on the form of function f(k2). 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 [184Jump To The Next Citation Point]. 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].
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