## 3 Non-Locality in Higher Dimensional Spaces  
The list of studies connected to quantum non-locality of the previous sections, albeit uncomplete and a little schematic, demonstrates as this is not only a fundamental point for understanding the very foundations of quantum mechanics, but it is also becoming an important resource for emerging quantum technologies. This prompts the need of a clear understanding of quantum non-locality and, in particular, of its compatibility with special relativity. Even if one can rigorously demonstrate that it can not lead to any superluminal transmission (signalling) [130], peaceful coexistence between special relativity and QM would require more, i.e., it would be necessary understanding a coherent description among different observers [131]. For instance, how to reconcile two observers that observe a different temporal order in the collapse of two entangled particles? The answers to this question span from being an unsolved problem [132; 133; 134; 135; 136; 137; 138; 139], requiring a preferred foliation to relativistic space-time, to being only apparent, since accounts of entangled systems undergoing collapse yielded by different reference frames can be considered as no more than differing accounts of the same process and events [140]; there is a form of holism associated to QM description of composite systems, the factorizable state after collapse on a certain hypersurface is merely one way of slicing together local parts, entangled states are superposition of such splicings. Be that as it may, rebus six stantis, an undoubted difficulty in understanding the collapse in composite systems persists. Our suggestion is that one should explore the possibility that this problem can be solved in higher dimensional spaces. Let us consider a \(n>4\) space-time with \(n-4\) more spatial dimensions and let us suppose that, while the usual fields only \"live\" in \(3+1\) dimensions, the collapse is mediated by \"some field\" propagating also in the other dimensions. Since every n-dimensional space can be \"contracted\" around a single point of a higher dimensional space (for example a plane can be wrapped as tied as one wishes around a single point in three dimensional space), two far points in the \"ordinary\" space can be as close as wanted considering extra-dimensions. Incidentally, this \"contraction\" is an isometry and leaves invariant Gaussian curvature. Thus, if this higher dimensions are extremely small, every event of the 3+1 dimensional space could be connected with any another event in an \"extremely short\" effective distance through the compactified dimensions. Therefore, the collapse of one of two remote entangled particles could cause the collapse of the other with the propagation of a subluminal signal through the compactified dimensions. Quantum non-locality would reduce to the fact that only wave function collapse would be affected by these extra dimensions. This kind of phenomenon could either be included in a more general theory going beyond quantum mechanics (for example, some Planck scale theory already predicts compactified dimensions) or find a use in building relativistic collapse models, for example a simple generalization of [PERSON]'s one [141] should be able to incorporate it. For instance, one could think to some general non-unitary evolution operator  
\[U(t)=Exp\{-i\int d^{4}xd^{n}yO_{I}(x,y,t)\} \tag{10}\]  
where \(O_{I}(x,y,t)=H_{I}(x,t)+C_{I}(x,y,t)\), \(H_{I}(x,t)\) being the usual Hermitian-Hamiltonian density of quantum fields leaving in a 3+1 dimensional space (x), while \(C_{I}(x,y,t)\) is not Hermitian and eventually induces the collapse. When one traces over the degrees of freedom corresponding to the not Hermitian component, \(C_{I}(x,y,t)\), the usual unitary evolution is eventually restored, as in 't Hooft finite degrees of freedom model [142]. Lastly, it would be worth investigating if this extra component could contribute to solve other issues of modern physics, as dark matter and dark energy.