**References**:

**RE**: Your ArXiv posting: “Space-Time Quantization (

**STQ**), Elementary Particles and Dark Matter

**Dear Prof. Dr. Auguste Messen**:

I am pondering this line from your paper, “Planck’s length L was initially considered to show that c and h, combined with G, define universal units for length, time and mass measurements. Later on, it appeared that the metric of general relativity breaks down inside a spherical mass M when its radius is of the order of r_{o}= GM/c^{2} (page 14).

Confession: Our work originated from a high school geometry class; http://bblu.org tells that story. It continues as a base-2 chart of 200+ notations from the Planck base units to the Age of the Universe: https://81018.com/chart It is a large, horizontally-scrolled file that simulates the big bang but *without the bang*. It is an “all natural” approach to inflation.

Even though STQ uses the quantum of length __ a__ which is smaller than Planck’s length L, wouldn’t it still be subject to all those studies that fall well within pure mathematics and below the measurable space-time threshold currently at notation 67? Wouldn’t aspects of pointfree geometries, bifurcation theory, Langlands programs, cellular automaton, etc. apply? Wouldn’t it still be subject to a base-2 progression that would look a lot like our progression today?

Thank you.

Most sincerely,

Bruce

Bruce Camber

http://bblu.org has the history/work within secondary schools.

http://81018.com is a site for ongoing work for the general public.

http://quietexpansion.com will become a site for personal reflections.

04 November 2016

Dear Bruce,

I am glad to see your curiosity and that a teacher could arouse your interest in mathematical power laws. Concerning **Space-Time Quantization** (STQ) and the** Planck length** (PL), I can tell you the following:

- To find out if space and time are continuous or not, it was essential to develop a theory of STQ, where the value of the quantum of length a was treated as an unknown parameter. To impose
*a priori*some particular value would not be better than the usual assumption (a = 0). - In regard to “states of motion”, it appeared that the highest possible energy would be hc/2a. This value is finite when a ≠ 0 and leads to changes of the usual theory when the energy of material particles tends towards hc/2a. Nevertheless, there are no logical inconsistencies.
- STQ is not only possible, but real, since it allows us to define “particle states” in agreement with a great amount of highly remarkable data, accumulated during more than half a century. This results from possible modulations of functions at the smallest possible scale in space and time when a ≠ 0.
- The actual value of a is irrelevant for elementary particle physics. It is only necessary that a ≠ 0. The smallest measurable distance could thus be a function a(t) of the cosmic time variable t, while c, h and G are constants. However, the highest possible energy was always hc/2a. This defines the total positive energy content of our universe, which could be in a “zero-energy state”, since bound states of material particles correspond to negative energies.
- According to General Relativity, our universe can correspond to the surface of a hypersphere in a 4-D space. Its radius R(t) could then change in the course of time, but Einstein introduced a cosmological constant ∧ to compensate the effects of the gravitational constant G. Lemaître (who was my teacher during 4 years at the UCL) introduced and justified the idea that our universe should have started in the simplest possible state and thus as a “single quantum”. Assuming that R(0) = 0, he found that our universe is expanding. He predicted even that this expansion would be accelerated when > 0. This has been confirmed by observations in 1998. I can explain why > 0, because of STQ.
- I will also show (in an article, to be published next year) that the initial “single quantum” was an elementary particle of special type, allowed by STQ. It was transformed according to the rules of this theory and produced an increasing number of elementary particles. Since they got bound to one another, this increased the negative and positive energy content of our universe.
- The initial value of R(t) was not zero, but determined by the condition that the available positive energy was hc/2a and that it was concentrated as much as possible by gravitational forces. This yields a relation between a o = a(0) and G, as well as h and c. It turns out that a o was the PL.
- Planck deduced its value from h, c and G, to get a natural system of units that is identical in the whole universe (even for ET civilizations). Later, it was assumed that the PL is related to quantum gravity. We might assume, for instance, that the PL is the Compton wavelength h/Mc of a body of mass M and also the Schwarzschild radius of this mass. Some proponents of quantum gravity assume that there are flexible “atoms of space” with dimensions of the order of the PL, but this would also be quite arbitrary. Space and time should be defined by possible results of measurement and not by hypothetical pieces of space.

Cordial greetings, Auguste Meessen, 04 November 2016

This email has been approved by Auguste Messen to be posted here.