PHYSICS BY GPT-J

RESULT
"Three Fermion Generations from Octonions"

Authors: Howard Georgi, Randal Kleeb, Joel Schechter

(Abstract)

We study a class of compact, simply-connected, non-commutative four-dimensional spaces that are deformations of the standard (commutative) four-dimensional space. The new spaces, constructed by acting on an ordinary four-dimensional space with the co-product algebra of the octonions, have the amazing property that the left-handed quark and lepton generations have 3-3-1-1 (Dirac-like) mass spectra, but the right-handed generations have (minimal) massless fermions. We have calculated their masses in perturbation theory and showed that they have the same masses as the standard model fermions. Their masses are proportional to the cut-off scale (which is higher than the electroweak scale) and are therefore less than one TeV. These non-commutative fermions are called “quill” fermions because they look like an octonion.

Introduction
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Since the early days of quantum chromodynamics, it has been realized that the basic fermions (quarks and leptons) are much heavier than would be indicated by naive dimensional analysis (see [@Georgi]). This fact has caused a great deal of speculation on new physical effects that may occur at energies near a TeV (for example, see [@Weinberg]). New interactions, such as those with gauge-boson exchange, are certainly possible in this energy range. However, perhaps the most likely explanation for the large mass gap is that the basic fermions are constituents of a more fundamental theory. We assume this is the case in our discussion.

We are therefore led to consider additional interactions that are not allowed by the standard model. This can only occur by relaxing the restrictions of chirality or of gauge-invariance. A simple class of theories that meet both these criteria is the class of theories based on the algebra $A_{4}$. They are defined by the commutation relations [@Baez]: $$\label{algebra}
\begin{array}{rcl}
\{T^a,T^b\}&=&\varepsilon^{abcd}T^c\\
\{J^i,T^a\}&=&-\frac{1}{2}\varepsilon^{abc}T^b\hat{x}^iJ^c\\
\{J^i,J^j\}&=&\varepsilon^{ijk}J^k
\end{array}$$ where $J^i$ are an $n-1$ dimensional representation of $SO(4)$ (the isometries of four-dimensional space), $T^a$ are four dimensional $SU(2)$ generators, and $\hat{x}^i$ is the 4-dimensional representation of $SO(3)$ (the isometries of three-dimensional space). The generators $T^a$ are real $4\times 4$ matrices.

One of the most interesting features of the algebra in (\[algebra\]) is that it is non-commutative, and we will see that this leads to chiral fermions. Indeed, it was first noted by Gel’fand and Yaglom [@Gel'fand] that for $n=4$, $J^i$ are the Cartan generators of $SO(4)$ and $T^a$ are the basis of an Abelian subalgebra of $SO(4)$, $3\times 3$ matrix representation of the algebra (\[algebra\]).

In this paper, we show that in this context one can construct interesting compact models in which the left-handed fermions have mass spectra corresponding to $SU(3)\times U(1)$ and $SU(2)\times U(1)$ gauge symmetries, while the right-handed fermions have minimal massless spectrum. The left-handed fermions do not carry any gauge charge. We present explicit models.

It is possible that these non-commutative compact four-dimensional spaces may be related to our proposed model of space-time in which there are additional dimensions [@Osten], with the extra dimensions being compactified, so that the minimal fermion mass is on the order of the compactification scale. It is possible that the minimal fermion mass may be related to the effective quantum gravity scale, as suggested by the Randall-Sundrum scenario [@RS].

While the standard model was originally formulated with fermions in a four-dimensional space with chiral properties, this does not mean that all chiral theories are possible. In general, one can consider an $n$-dimensional space with an algebra containing the $SU(2)$ algebra as a subalgebra and with non-commuting Cartan generators. For some values of the free parameters in this algebra, the theory will have a chiral fermion spectrum.

In [@Osten], we considered a compactification of an $n=3$ (with a “Higgs” field) dimension on the manifold $S^1\times S^1\times S^1$ and showed that the size of the compact dimensions can be a low energy scale, while the size of the non-compact dimensions can be a high energy scale. We will use a similar method here, but will compactify an extra $n=3$ dimension on a manifold $S^1\times S^1\times S^1\times S^1$, yielding a model with the same mass spectrum.

While in this paper we will use $n=4$, we expect the general result to hold for $n$ arbitrary.

A more compact model, in which the symmetry is broken by a suitable choice of vacuum expectation value for the Higgs field, was considered by Victor Novikov in [@Novikov].

$SU(2)$ octonion fermions
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To begin, we consider the octonions $O$ with the multiplication rule [@Rudin]: $$\label{octonions}
\begin{array}{rcl}
(a+b)+(c+d)&=&(a+c)+\varepsilon(b+d)\\
(a+b)\times(c+d)&=&(ac-bd)+i(ad+bc)\\
(a+b)+(c+d)\times(a+b)&=&(a+c)\times(a+d)-\varepsilon(a+b)\\
(a+b)\times(c+d)\times(a+b)&=&(a+b)\times(c+d)\times(c+d)
\end{array}$$ for $a,b,c,d\in O$. The imaginary octonion unit is defined by $i^2=-1$.

We will define a right-handed fermion $Q$ by: $$Q(x)=x_1+x_2i+x_3j+x_4k$$ where $x_1$, $x_2$, $x_3$ and $x_4$ are elements of the octonion algebra. This right-handed fermion has the multiplication rule [@Rudin]: $$\label{Rf}
Q(x)\times Q(y)=Q(x\times y)$$ The left-handed fermion $L$ has the multiplication rule: $$L(x)=x_1-x_2i-x_3j-x_4k$$

It is straightforward to see that the left-handed fermions are given by: $$L(x)=x_1-x_2i-x_3j-x_4k=x_1+x_2i+x_3j+x_4k$$

So far, we have just defined right and left-handed fermions. The right-handed fermion $Q$ can be used to construct a three-generation model. The model is based on a three-dimensional space with the manifold $S^1\times S^1\times S^1$.

Consider first the case in which $x_1=x_3=x_4=0$ so that $x=x_2i+x_3j+x_4k$. Then, we find: $$Q(x)=x_2i+x_3j+x_4k$$ So the right-handed quark has the form: $$Q(x)=x_1+x_2i+x_3j+x_4k$$ Similarly, we can define a right-handed lepton. If