Convexity and the Euclidean Metric of Space-Time

Abstract

Abstract:We address the reasons why the "Wick-rotated", positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point out the potential physical significance that functional analytical dualities play in this framework. Following the spirit of the variational principles employed in classical and quantum Physics, such Hilbert spaces dominate in a generalized functional integral approach. The metric of space-time is induced by the inner product of such Hilbert spaces.

The Legendre Transform in Non-Additive Thermodynamics and Complexity

“…As such they are always sensitive to the details of their environment since they can probe its entirety. This is evident in the form of the modified Legendre transform applicable to them: it has a power law form (39). Naturally as s → ∞ someone recovers the conventional Legendre transform, which mirrors the fact that when q → 1, the Tsallis entropy "reduces" to the BGS entropy.…”

“…Hence, we are led to the following fact: it is not natural form the viewpoint of the Legendre transform to use (33) for a Tsallis entropy-based thermodynamics. Instead, if one wants to remain faithful to the "spirit", namely the features of the Legendre transform that practically uniquely characterize it, then for all calculations in a Tsallis entropy-based Thermodynamics, one should use (39) instead. This is the main point of the present work.…”

“…One can generalize (46) to the case of s-concave functions, by using the modified Legendre transform (39) instead of (33). Using the notation of ( 38), (39) we conjecture that for nonnegative, s-concave functions f ∈ Conc s (R n ) we should have ˆRn e q (− f )…”

An Entropy for Groups of Intermediate Growth

“…Moreover bcd = I (32) and the group generated by b, c, d is Z 2 2 . Furthermore (ad) 4 = (ac) 8 = (ab) 16 = I…”

“…There is no possibility for such a group to have intermediate growth as G does. Hence, even if one may initially suspect that due to(30),(31),(32) the generators b, c, d may act in a analogous way to real generators of an N = 4 supersymmetry algebra, or as the set of almost complex…”