Application of Dvoretzky’s Theorem of Measure Concentration in Physics and Cosmology

Abstract

Using Dvoretzky’s theorem in conjunction with Bohm’s picture of a quantum particle inside a guiding quantum wave akin to De Broglie-Bohm pilot wave we derive Einstein’s famous formula E = mc 2 as the sum of two parts E(O) = mc 2 /22 of the quantum particle and E(D) = m c 2 (21/22) of the quantum wave where m is the mass, c is the speed of light and E is the energy. In addition we look at the problem of black holes information in the presence of extra dimensions where it seems initially that extra dimensions would logically lead to a hyper-surface for a black hole and consequently a reduction of the corresponding information density due to the dilution effect of these additional dimensions. The present paper argues that the counterintuitive opposite of the above is what should be expected. Again this surprising result is a consequence of the same well known theorem on measure concentration due to I. Dvoretzky. We conclude that there are only two real applications of the theorem and we expect that many more applications in physics and cosmology will be found in due course.