J. G. Moxness | Searcharxiv

arXiv2023

The Isomorphism of 3-Qubit Hadamards and $E_8$

This paper presents several notable properties of the matrix $\mathbb{U}$ shown to be related to the isomorphism between $H_4$ and $E_8$. The most significant of these properties is that $\mathbb{U}$.$\mathbb{U}$ is to rank 8 matrices what the golden ratio is to numbers. That is to say, the difference between it and its inverse is the identity element, albeit with a twist. Specifically, $\mathbb{U}$.$\mathbb{U}$-$ (\mathbb{U}$.$\mathbb{U})^{-1}$ is the reverse identity matrix or standard involutory permutation matrix of rank 8. It has the same palindromic characteristic polynomial coefficients as the normalized 3-qubit Hadamard matrix with 8-bit binary basis states, which is known to be isomorphic to E8 through its (8,4) Hamming code.

quant-ph↗

arXiv2023

The Isomorphism of $H_4$ and $E_8$

This paper gives an explicit isomorphic mapping from the 240 real $\mathbb{R}^{8}$ roots of the $E_8$ Gosset $4_{21}$ 8-polytope to two golden ratio scaled copies of the 120 root $H_4$ 600-cell quaternion 4-polytope using a traceless 8$\times$8 rotation matrix $\mathbb{U}$ with palindromic characteristic polynomial coefficients and a unitary form $e^{\text {i$\mathbb{U}$}}$. It also shows the inverse map from a single $H_4$ 600-cell to $E_8$ using a 4D$\hookrightarrow$8D chiral left$\leftrightarrow$right mapping function, $ φ$ scaling, and $\mathbb{U}^{-1}$. This approach shows that there are actually four copies of each 600-cell living within $E_8$ in the form of chiral $H_{4L}$$\oplus$$φH_{4L}$$\oplus$$H_{4R}$$\oplus$$φH_{4R}$ roots. In addition, it demonstrates a quaternion Weyl orbit construction of $H_4$-based 4-polytopes that provides an explicit mapping between $E_8$ and four copies of the tri-rectified Coxeter-Dynkin diagram of $H_4$, namely the 120-cell of order 600. Taking advantage of this property promises to open the door to as yet unexplored $E_8$-based Grand Unified Theories or GUTs.

math.GR↗