Convexity and the Euclidean metric of space-time

TL;DR

The paper explores why the Euclidean space-time metric obeys the Pythagorean theorem, linking it to the convexity properties of functional spaces, especially Hilbert spaces, in quantum gravity frameworks.

Contribution

It proposes that the Euclidean metric's Pythagorean form arises from the convexity and smoothness properties of Hilbert spaces used in quantum gravity models.

Findings

Hilbert spaces extremize convexity and smoothness moduli

Functional dualities have potential physical significance

Space-time metric is induced by Hilbert space inner products

Abstract

We address the question about 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.