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The Courant conjecture
(CoR) p. 763: “Families of spherical waves for arbitrarily time-like lines exist only in the case of two and four variables, and then only if the differential equation is equivalent to the wave equation“.
The Hamiltonian dynamical (energy) operator in the proposed UFT supports the Berry-Keating conjecture (which is in line with the Hilbert-Polya conjecture). The related dynamical energy Hilbert space provides the appropriate framework for well-posed wave/radiation hyperbolic PDE models. An operator is only well-defined in combination with a defined domain (!). In case of the proposed domain (the compact (closed, connected) "unit quaternions") there are no mechanical "cause-action" (initial and boundary value) conditions required. Regarding the Courant conjecture the following related facts may support an appropriate answer:
- the S(1) and S(3) are the only spheres with a "continuous" group structure, (EbH) 7.2
- the spheres S(0), S(1), S(3), and S(7) are the only parallelizable spheres
- „Thurston‘s geometrization theorem stating that every closed connected 3-manifold can be decomposed in a canonical way into eight pieces that each have one of eight types of geometric structure. It is an anlogue of the uniformization theorem for two-dimensional surfaces, which states that every simple connected Riemann surface can be given one of the three geometries, Euclidian, spherical, or hyperbolic", Wikipedia
The Kummer conjecture
"Every of the three prime residue classes p=1 mod 3 contains an infinite number of primes and the three classes have the densities 1/2, 1/3, and 1/6", (HaH) S. 453
H. Hasse referred to it as "it would be probably more productive for number theory to work on it than working on Fermat's Last Theorem", (HaH), S. 453.
The Kummer conjecture might be answered by new arithmetic concepts developed in combination with
- the proposed two-semicircle method for binary number theory
- the proposed Krein space based UFT
- the definition of frames for Krein spaces (extending the notion of J-orthogonal bases of Krein spaces), which are the sum of three orthonormal bases of a Krein space, (GiJ), (KaS).
Derbysire's question
All in all the combination of the above thoughs may allow a first answer to the question raised by J. Derbysire, (DeJ) p. 295:
“What on earth does the distribution of prime numbers have to do with the behavior of subatomic particles?“
Related papers
 Braun, K., The Kummer conjecture and the two-semicircle method |
(CoR) Courant R., Hilbert D., Methods of Mathematical Physics, Volume II, Wiley Classics Edition, 1989
(DeJ) Derbysire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press, Washington, D. C., 2003
(GiJ)
Giribet J. et al., On frames on Krein spaces.pdf |
(HaH) Hasse H., Vorlesungen über Zahlentheorie, Springer-Verlag, Belin, Göttingen, Heidelberg, 1950
Hasse H., Die Kummersche Vermutung fuer kubische Charaktere |
(KaS)
Karmakar S. et al., Frames on Krein Spaces.pdf |
