Saturday, December 9, 2017

Minkowski Space, Natural Numbers, arithmetics and quantum mequanics, pure coincidence?

Bored as I was, I started to check what was the smaller tuple of natural numbers (n1, n2, n3,...) that could match next equation:

 N = n1^2 + n2^2 + n3^2 +.... where N is *any* natural number.

First I tried with just tuples of two numbers (n1, n2) and it's pretty easy to check that many natural numbers have no representation as the sum of squares of (n1, n2).

The second try with (n1, n2, n3). A much bigger subset of N can be adjusted to sums of squares of (n1, n2, n3), but still a few numbers escape to the rule.


What about 4-length tuples (n1, n2, n3, n4). This time everything works!. There is no single natural number that can NOT be represented as sum of squares of, at most, four other natural numbers.

Next step was to Google Qwant around to see if somebody else had arrived to the same result, or it was just that I have a divine inspiration that no other human being is able to reach. ... To my dis-fortune a few (hundreds) of results in Qwant pointed out to Lagrange four-square-theorem, developed a few centuries ago. Not only that, but Diophantus was already aware of the problem more than two millennium ago. Ummm, .... time to try something new:

If any natural number can be represented as

 N = n1^2 + n2^2 + n3^3 + n4^2

what about trying something like ?:
 

  N = n1^2 + n2^2 + n3^3 -  n4^2

And voila!. Again, it's always possible to always  find  a 4-tuple (n1,n2,n3,n4) that for any natural number N  match the previous equation. In matrix terminology it means that 

                         | 1 0 0 0 |   n1
 N =  (n1, n2, n3, n4) x | 0 1 0 0 | x n2
                         | 0 0 1 0 |   n3
                         | 0 0 0 -1|   n4

Where the diagonal matrix can be interpreted as the non-euclidian Minkowski matrix of Relativity.
Using a different terminolgy, let's call |Ψ> = (n1, n2, n3, n4) the vertical matrix and <
Ψ| the transposed one. Now let's suppose that Minkowski matrix is the "distance" observable using quantum mechanics. What we get is:

N =
<Ψ| M | Ψ>

Where N, any order-able distance in the "continuous" space-time, is now the macroscopic result of a quantum measurement in the internal 4th dimensional non-order-able quantum one.

Basically put, the four dimensional Minkosky space is one of the two smallest multidimensional spaces compliant with the rule of being able to generate an arbitrary measure of distances (any natural number N).