We rewrite the Born-Infeld Lagrangian, which is originally given by the determinant of a 4 × 4 matrix composed of the metric tensor g and the field strength tensor F , using the determinant of a (4·2 n )×(4·2 n ) matrix H4·2n . If the elements of H4·2n are given by the linear combination of g and F , it is found, based on the representation matrix for the multiplication operator of the Cayley-Dickson algebras, that H4·2n is distinguished by a single parameter, where distinguished matrices are not similar matrices. We also give a reasonable condition to fix the parameter.
We calculate the eigenvalue ρ of the multiplication mapping R on the Cayley-Dickson algebra An. If the element in An is composed of a pair of alternative elements in An−1, half the eigenvectors of R in An are still eigenvectors in the subspace which is isomorphic to An−1. The invariant under the reciprocal transformation An × An ∋ (x, y) → (−y, x) plays a fundamental role in simplifying the functional form of ρ. If some physical field can be identified with the eigenspace of R, with an injective map from the field to a scalar quantity (such as a mass) m, then there is a one-to-one map π : m → ρ. As an example, the electro-weak gauge field can be regarded as the eigenspace of R, where π implies that the W-boson mass is less than the Z-boson mass, as in the standard model.
We examine the possibility of high-Tc superconductivity mediated by the elementary excitation of an electron stripe which is realized by neutralizing local polarization mainly caused by the c-displacement of ions on the CuO 2 plane. Under the translation invariance of the electronic state of the neutralization charges, it is found that both itinerant and localized electrons and/or holes can coexist in the stripe phase. If the frequency of the localized charge oscillation tends to soften along the stripe direction, superconductivity with Tc around 100 K can be realized, due to the enhancement of the coupling constant.
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