Bell's Theory with no Locality assumption: putting Free Will at work

We prove a version of the Bell's Theorem that does not assume Locality but only the Effect After Cause Principle (EACP) according to which for any Lorentz observer the value of an observable cannot change because of an event that happens after the observable is measured. Since the EACP is compatible both with Locality and with Non-Locality, Locality cannot be considered as the common cause of the contradictions obtained in all versions of Bell's Theory. By definition, all versions of Bell's Theorem assume Weak Realism according to which the value of an observable needed in the discussion of Bell's Theorem is well defined whenever the measurement could be made and some measurement is made. As a consequence of our results, Weak Realism becomes the only hypothesis common to the contradictions obtained in all versions of Bell's Theory. This work indicates that it is Weak Realism, not Locality, that needs to be negated to avoid the contradictions in microscopic Physics associated to Bell's Theory, at least if one refuses as false the de Broglie-Bohm Hidden Variable theory because of its essential violation of Lorentz invariance. This paper completes with much more details the genuine Bell Theorem part of a previous paper. That paper also offered a treatment of the GHZ entanglement, a treatment which did not suffer from the lack of clarity of the definition of the EACP.Comment: 13 pages, 0 figures: Conference in honor of Pierre Coullet's 60th Birthday; Vina del Mar, Chile, December 2009. This paper completes the original Bell Theorem part of Charles Tresser:"Bell's theory with no locality assumption", Eur.Phys.J. D 58, 385-396 (2010); (DOI: 10.1140/ep jd/e2010-00122-8

A geometric study of Wasserstein spaces: Hadamard spaces

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space W(X). In this paper we investigate the geometry of W(X) when X is a Hadamard space, by which we mean that $X$ has globally non-positive sectional curvature and is locally compact. Although it is known that -except in the case of the line- W(X) is not non-positively curved, our results show that W(X) have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for W(X) that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in W(X).Comment: This second version contains only the first part of the preceeding one. The visibility properties of W(X) and the isometric rigidity have been split off to other articles after a referee's commen

Random Sorting Networks

A sorting network is a shortest path from 12...n to n...21 in the Cayley graph of S_n generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n->infinity the space-time process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at half-time converges to the projected surface measure of the 2-sphere. We prove that, in the limit, the trajectories are Holder-1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux.Comment: 38 pages, 12 figure