Calculus without limits # Quaternions and Particles

The standard model of particle physics is not so pretty, but it is successful. Many lose ends and major big questions remain: is there a grand unified gauge group? Why are there three generations of particles? Why do neutrini oscillate? How is general relativity included? (See for example page 540 in Woit’s online monograph).

When experimenting with quaternion primes, especially in connection with Goldbach, I recently stumbled over a combinatorial structure within quaternion integers which resembles some structures seen in the standard model. This structure is purely combinatorial and could suggest that some relations seen in elementary particles appear unavoidable. Maybe there is more behind it, maybe it is naive, maybe it is just amusing like this story. For now, I see it as a caricature showing that the structures seen in the Standard model are not so arbitrary. Nice would be to show that the structure is unavoidable. Here is a recent new write-up. It might score a bit high both on the crackpot index for primes as well as the crackpot index for physics. I leave it to you to judge. For me it has just been fun and will also in future be a bit motivation to learn more about the physics of particles.

The starting point is the mathematical structure of division algebras. The requirement of being a division algebra restricts the structure severely: according to a theorem of Hurwitz, (Frobenius 1878, Hurwitz 1922, Mazur 1938), there are only 4 normed division algebras and 3 associative normed division algebras. In a physical frame work with quantum evolution and representability of observables as operators, where we want to have both associativity and completeness, the list of normed division algebras which are associative and complete drops down to two: they are the complex numbers C and the quaternions H. It is rare in mathematics that a categorical structure has so few constituents. There are other ways to distinguish C and H: these are the only linear spaces for which the unit sphere is a continuous Lie group. As symmetries are of enormous important in physics as stressed by Emmy Noether, this is a relevant feature also distinguishing the complex numbers and the quaternions.

The higher arithmetic in C and H are both quite well established: one has prime factorization and fundamental theorem of arithmetic in each case. While the arithmetic in C is quite old as it has been done by Gauss, the prime factorization in H is more complex: The work from Hurwitz to Conway and Smith have established the fundamental theorem of arithmetic in H: the factorization is unique there only up to unit migration, recombination and meta commutation. (The word higher arithmetic is due to Davenport and more adequate than “elementary number theory” as we know that elementary number theory is one of the most difficult topics in mathematics overall.)

Lets start with C, the associative commutative normed division algebra. The number theory in C deals with the structure of the Gaussian integers. There are three type of primes, the inert ones, the split ones and the ramified ones. The inert ones are the rational primes of the form 4k+3 together with the negative ones. So, the inert Gaussian primes are { …-11,-7,-3,3,7,11…,-11i,-7i,-3i,3i,7i,11i,…}. Then there are the split primes which come in groups of 8. They are of the form a+i b with p = a2 + b2 being prime. The 8 primes belonging to p=5 are 1+2i,1-2i,-1+2i,-1-2i, 2+i,2-i, -2+i,-2-i. Then there are the ramified primes: they belong to the rational prime p and there are exactly 4 elements: {1+i,1-i,-1+i,-1-i}. Now there are two symmetries we can let work on these primes. The first is multiplication with the units U = {1,i,-1,-i}. The second is the dihedral group generated by U and conjugation a+ib -> a-ib. The equivalence classes modulo V correspond one to one to the rational primes. For the equivalence classes modulo U, it depends on the type. For every rational prime p=4k+1 there are two equivalence classes a+ib, b+ia for which the arithmetic norm a2 = b2=p. For ramified primes, there is only one equivalence class 1+i with arithmetic norm 2. For inert primes p=4k+3, there is also only one U-equivalence class, but now, not the norm but the square root of the norm is equal to the prime p.Now, what does this have to do with particle physics? The starting point is quadratic reciprocity. Given two primes p,q, there is a number (p|q) called the Jacobi symbol. Now, primes of the form 4k+1 behave like Bosons and primes of the form 4k+3 behave like Fermions: the reason is the quadratic reciprocity theorem of Gauss: (p|q) = (q|p) if and only if one of the primes p,q is a Boson and (p|q) = -(q|p) if both are Fermions. The fact that primes of the form 4k+1 behave like Bosons is not surprising as they are actually the product of two Gaussian primes a+ib, a-ib by the Fermat two square theorem. We can now look at these two primes as a Fermion pair. An other input comes from the double cover V of U, which allows to define a quantity called charge. Double covers are everywhere in physics like Spin(n) double covering SO(n). A fancy way to describe this through a short exact sequence 1->Z/(2Z) -> V -> U -> 1 telling that dihedral groups covers the cyclic group. Since a+ib and a-ib are not U-equivalent if N(a+ib) is an odd prime, we look at them as Fermions of different charge. The most natural choice is to call them electron and positron. The neutral 4k+3 Fermions are then the neutrini. We still have to place the primes with arithmetic norm 2. Since their V and U equivalence classes are the same, they are neutral. They are also light. Why not associate them with the Higgs particle?

Lets look at the primes in the quaternion algebra H. The Lagrange four square theorem prevents the existence of neutrini type primes on the coordinate axes. All primes (a,b,c,d) have at least 2 entries which are nonzero. Now, primes in H come in two types, there are the Lipschitz primes (a,b,c,d), where all a,b,c,d are integers or then the Hurwitz primes (a+1/2,b+1/2,c+1/2,d+1/2), where a,b,c,d are integers. In both cases, a quaternion integer is a prime if N(a,b,c,d) = a2 + b2 + c2 + d2 is prime. Lets again look at symmetries. The first one is the group U of units in the algebra. It is a finite group, the binary tetrahedral group. The first group \$V\$ is the group generated by the permutations of coordinates and the conjugation (a,b,c,d) -> (a,-b,-c,-d). We call U the weak group of symmetries because it is generated by a discrete subgroup of SU(2). V is the strong group of symmetries because it is generated by a discrete subgroup of U(3). There is a third group W which is a subgroup of V and which only take even permutations. It is a subgroup of SU(3). As V is a double cover of W, giving an equivalence class of V and a sign called charge determines an equivalence class in W. The equivalence classes in V are easy to describe. Every integer is modulo V equivalent to an integer (a,b,c,d) with 0 ≤ a &leq b ≤ c ≤ d. Such an integer together with a charge determines an equivalence class in W. Here comes the interesting part: what are the weak equivalence classes within the strong ones? It is a bit surprising that the answer is very simple. Except for the integers with arithmetic norm 2, the equivalence classes consist either of two or three elements. When seeing this there is almost no other reflex possible as seeing the odd primes as “hadrons” and the equivalence classes either as baryons or mesons. The individual elements in the equivalence classes are then the quarks.

Here is the situation from a number theory perspective: the fundamental theorem of algebra for quaternions (see the book of Conway and Smith) can be restated that any quaternion integer z of arithmetic norm p can be written for any factorization p=p1 …pn be written modulo V as a product [z1] … [zn], where [zi] is an equivalence class of primes with arithmetic norm pi. In the case when the primes are all odd and adjacent primes have different norm, the factorization is given as an ordered product of mesons or baryons. We hope to be able to assign charge to quarks in a natural way by looking at all possible factorizations and assign charge in a simple combinatorial way. Its clear that this leads to charges of individual members which are multiple of 1/6 and as we believe actually to a multiple of 1/3. We have not done this combinatorics yet but instead assigned charge ad hoc to individual elements of the equivalence classes. Our assignment is simple and unique but it has the disadvantage yet that it rules out baryons of charge 2 which have been observed in nature. I hope that looking at all possible factorization of a quaternion integer modulo the symmetry groups U,V,W allows to define charge in a more natural way. As for now, this is a rather concrete combinatorial problem.

Here is a picture of all the hadrons in the case of the prime p=107. There are two baryons and two mesons. The first baryon has charge 1, the second has charge 0. Then there are two mesons of charge 1. 