Quaternion Integers

The number theory of quaternion integers has connections to many beautiful parts of mathematics. The quaternions themselves are selected out from all the structures in mathematics as the only associative real non-commutative division algebra and so play a rather unique role in the entire landscape of mathematics. Its unit sphere is the only Euclidean sphere that is a non-Abelian Lie group meaning that it is a sphere on which we can calculate! Again it is this uniqueness as well as the fact that we see this sphere SU(2) in particle physics as well as the only Euclidean sphere that is commutative U(1) which is the unit sphere in the unique commutative and associative complete division algebra. It makes you think, like why “chicken tastes like anything”. Quaternions were once very popular but then came the fall. Three dimensional mafia has taken over. A popular textbook of Gibbs killed the quaternions around 1900 as the Dot product and cross product replaced things ). Anyhow, quaternions are cool because they are singled out in a nice way and because they appear to be relevant in the stuff we are made off. SU(2) after all is related to the weak force.

As for the integers, Hurwitz nailed it. The only text which has been in par so far on presenting the material is the book of Conway and Smith. But it is not only the presentation which counts. It is rather easy to present something which has been developed already. Much harder is to develop something new and present it nicely in the same time. The lectures of Hurwitz are of this type. His lecture notes are highly original and can still be read today. What makes the number theory of quaternion integers special is that it deals in a non-commutative ring. It is therefore rather different than the number theory in rings of integers within algebraic number fields, finite degree extensions of the rational numbers within the complex numbers. The topic is not only part of ring theory but but belongs into the area of algebra type structures, as the quaternions are an associative algebra over the reals. Because the complex numbers are not in the center of the algebra , one does not consider the quaternions to be an algebra over the complex numbers. As for the number theory however, things are a bit different than in number theory in commutative rings, but it is close enough. There is an essentially unique prime factorization. We have a convenient way to identify and count primes. Hurwitz did that: over the only even rational prime p=2, there are exactly 24 quaternion primes; over each odd rational prime there are exactly 24(p+1) quaternion primes.

Here is something interesting about the symmetries: Hurwitz showed in his lectures that any permutation of the quaternions preserving the arithmetic structure is given by where is an other quaternion that is invertible Obviously scaling with a real non-zero number does not change this so that we can assume . In other words, the symmetries are given by with . The rotation group is the symmetry group. Naively, we might initially had guessed that it is because these are the rotations in the quaternions. But no, we have also to worry about the algebraic structure. Hurwitz then investigates the symmetries of the integers. One can take any of the 24 units and form . Similarly as we got SO(3) rather than SU(2), we get now the symmetry group A4 and not the binary tetrahedral group The alternating group A4 has 12 members while the binary tetrahedral group has 24 members.

I had my office printer do last night a larger red copy of the 24-cells. I took some code written 15 years ago for this video in order to 3D print the 24 cell. I turned it first in 4D first so that 6 vertices of one of the octahedrons are on a plane. Less supporting material was so needed. Back to the point which Hurwitz makes and which is a bit surprising. There are more symmetries than just the “gauge bosons” in the units: every quaternion prime of norm p=2 (there are 24 of them and they form an other 24 cell of the form (1+i) U, if U is the 24 cell) defines a symmetry of the quaternion integers. Proof: . It is a bit surprising because 1/(1+i)=(1-i)/2 is not an integer (also not a Hurwitz integer). But we see that the primes belonging to p=2 are also contribute to some of symmetry. The prime p=2 is special also in the Gaussian integer case as it is the only ramified case. The primes of the form 4k+1 split and the primes 4k+3 do not. Thinking in terms of primes in the context of particles, we see that all 24 even primes can also serve as an other type of symmetry. All the odd primes produce quarks belonging to Hadrons (either mesons if two of them can be moved into each other by a unit or Baryons if three of them can be turned into each other using units). We have also associated all primes in the Gaussian integers as Leptons (either electrons if they are not on an axis or neutrini if they are on an axis). But also there, there had been the prime p=1+i which was ramified and special. But in the case of Gaussian integers, there was not a second “heavier symmetry” because the algebra is commutative. In any case, one could think that both the units U (massless) and the scaled units (1+i) U (massive) can serve as gauge bosons and so force carriers.

An other thing which is a bit puzzling in the particle-prime allegory is that in the Gaussian integers, we can associate primes of the form a+ib like 1+2i as electrons. But as quaternion integers, it counts as a quark entangled with an other Hurwitz prime as a meson. Baryons triples of quarks never have a quark member of the form (a,b,0,0). Maybe that is why they are so stable; we need some other ingredient to get via metacombination and different factorization to a meson. For larger primes, the chance to get to such a resolution is much bigger providing some sort of “explanation” why larger (and so heavier) primes decay so fast. If we have an electron and Baryon together and they move randomly (not really randomly but tossed around by many other virtual particles and anti-particles surrounding them), then we some chance that they recombine in a different way and fly apart, explaining how a neutron can decay into a proton and electron. It would be nice to study this experimentally and to see how if neutrons and protons are close together there is more stability. We after all see that the nucleus of the hydrogen atom is stable.

Even if one does not like hese particle allegories it motivates to study number theoretical functions on the quaternion primes that do not depend on the order of the prime factorization. This was one of the reasons to take the number of primes at a simplex as determining the interaction rule in the standard model cellular automaton.