PHYSICS BY GPT-J

RESULT
"Three Fermion Generations from Octonions"

Tomislav Ilić$^{(a,b)}$ and Slavko Štimac$^{(a)}$

*$^{(a)}$Centre for Quantum Technologies,*

*Research School of Physics and Engineering,*

*The Australian National University,*

*Canberra, ACT 0200, Australia.*

*$^{(b)}$Department of Physics and Astronomy,*

*University of Canterbury,*

*Christchurch, New Zealand.*

The free fermion system associated with the octonions is introduced in section 2. This system incorporates three generations of chiral fermions. Their masses are given by the imaginary parts of the imaginary units of the octonions in the appropriate group representations. The group representations are constructed in section 3 using Young tableaux. Sections 4 and 5 present the fermion algebra and the Hamiltonian of the model.

Introduction
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The octonions are non-associative division algebras. The division algebra property of the octonions implies that an octonion can be expressed as the sum of an octonion and an orthogonal octonion. If we consider the complex numbers $\mathbb{C}$, $\mathbb{R}$, $\mathbb{C}$ as a subalgebra of $\mathbb{O}$, then $\mathbb{C}$ can be represented as a subalgebra of $\mathbb{R}\oplus\mathbb{R}$. If we consider the quaternions $\mathbb{H}$, $\mathbb{H}$ as a subalgebra of $\mathbb{O}$, then $\mathbb{H}$ can be represented as a subalgebra of $\mathbb{R}\oplus\mathbb{R}$ \[1\]. These examples can be extended to the whole octonions. This suggests that the free fermion system associated with the octonions should have two fermions of spin $\frac{1}{2}$ and two fermions of spin $0$. In this paper we develop the free fermion system associated with the octonions. We present a model where we have three generations of chiral fermions. Their masses are given by the imaginary parts of the imaginary units of the octonions in the appropriate representations. The fermion algebra is presented in section 4. The Hamiltonian of the model is presented in section 5.

Fermion generation
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We consider the fermions to be free. Let $N$ be the number of fermions of a given type, which has spin $j$. The algebra of the fermions is given by $$\psi_{\alpha}\,\psi_{\alpha} = 0\,,\qquad \alpha = 1,2,\ldots,N\,.$$ Here $\psi_{\alpha}$ is the fermion field. The algebra above suggests that $\psi_{\alpha}$ has a spin $j$. We want to have three generations of fermions, so $N=3$. If we want a non-trivial algebra, we have to consider $N>3$.

The octonions $\mathbb{O}$ are a vector space of dimension 8 over the real numbers, denoted by $\mathbb{R}$. The complex numbers $\mathbb{C}$, $\mathbb{R}$, $\mathbb{C}$ can be embedded as subalgebras of $\mathbb{O}$, where we have $$\begin{aligned}
\mathbb{C} &\subset & \mathbb{R}\oplus\mathbb{R}\,, \nonumber\\
\mathbb{R} &\subset & \mathbb{R}\oplus\mathbb{R}\oplus\mathbb{R}\,,\\
\mathbb{C} &\subset & \mathbb{R}\oplus\mathbb{R}\oplus\mathbb{R}\oplus\mathbb{R}\,.\nonumber\end{aligned}$$ The quaternions $\mathbb{H}$, $\mathbb{H}$ can be embedded in $\mathbb{O}$, where we have $$\begin{aligned}
\mathbb{H} &\subset & \mathbb{O}\,, \nonumber\\
\mathbb{H} &\subset & \mathbb{R}\oplus\mathbb{R}\,.\end{aligned}$$ The free fermion system associated with the octonions can be constructed by replacing the ordinary complex numbers by the algebra of octonions. Let $e_{0}$ and $e_{i}$ be the octonionic imaginary units. We require that $$e_{0}^{2} = e_{1}^{2} = e_{2}^{2} = e_{3}^{2} = 1\,.$$ The algebra of the octonions is then $$\begin{aligned}
\mathbb{O} &= & \mathbb{R} \oplus \mathbb{R}\oplus \mathbb{R}\oplus \mathbb{R}\oplus \mathbb{R}\oplus \mathbb{R}\oplus \mathbb{R}\,, \nonumber\\
\mathbb{O} &= & \mathbb{R} \oplus \mathbb{R}\oplus \mathbb{R}\oplus \mathbb{R}\oplus \mathbb{R}\oplus \mathbb{R}\oplus \mathbb{R}\,.\end{aligned}$$ We use the notation $\mathbb{R}\,\mathbb{R}$ to indicate the direct sum of the real algebras $\mathbb{R}\oplus\mathbb{R}$. The free fermion system associated with the octonions is given by $$\begin{aligned}
\psi_{\alpha}(t) &= & \psi_{\alpha}^{(0)}\,e_{0}(t) + \psi_{\alpha}^{(1)}\,e_{1}(t) + \psi_{\alpha}^{(2)}\,e_{2}(t) + \psi_{\alpha}^{(3)}\,e_{3}(t)\,,\nonumber\\
\psi_{\alpha}(t) &= & \psi_{\alpha}^{(0)}\,e_{0}(t) + \psi_{\alpha}^{(1)}\,e_{1}(t) + \psi_{\alpha}^{(2)}\,e_{2}(t) + \psi_{\alpha}^{(3)}\,e_{3}(t)\,.\end{aligned}$$ These equations are invariant under the multiplication of the real part of the octonion by the imaginary part. We can write the field equations as $$\begin{aligned}
i\,\gamma_{0}\,\partial_{t}\,\psi_{\alpha}^{(0)} &= & \sum_{\beta=1}^{3}\,\Bigl\{\psi_{\beta}^{(0)}\,\partial_{t}\,\delta_{\alpha\beta} + \sum_{j=1}^{3}\,\psi_{\beta}^{(j)}\,\partial_{t}\,\delta_{\alpha\beta}\Bigr\} = m_{0}\,\psi_{\alpha}^{(0)}\,,\nonumber\\
i\,\gamma_{0}\,\partial_{t}\,\psi_{\alpha}^{(j)} &= & \sum_{\beta=1}^{3}\,\Bigl\{\psi_{\beta}^{(j)}\,\partial_{t}\,\delta_{\alpha\beta} + \sum_{k=1}^{3}\,\psi_{\beta}^{(k)}\,\partial_{t}\,\delta_{\alpha\beta}\Bigr\} = m_{j}\,\psi_{\alpha}^{(j)}\,.\nonumber\\\end{aligned}$$ We define the mass of the free fermion $\psi_{\alpha}^{(0)}$ to be $m_{0}$, and we define the masses of the other fermions to be $m_{1}$ and $m_{2}$. The masses of the free fermions are given by the imaginary parts of the imaginary units of the octonions. The normalization