The Quantum Mechanics based on Tessarines with Nonlinear Conjugation

This paper investigates the generalization of the quantum mechanical formalism based on the algebra of tessarines. We circumvent the rigid constraints of Hurwitz's theorem on normed division algebras by transitioning from linear operators to a fundamentally nonlinear, state-dependent component-wise conjugation (a semi-involution structure). It is demonstrated that the resulting nonlinear quantum field theory possesses a strictly positive-definite scalar probability density and fully preserves the property of homogeneity, which is critical for $U(1)$ gauge invariance and conservation laws. In the diagonal sector, the model seamlessly reduces to the standard linear Schrödinger equation.