Bicomplex quantum mechanics: I. The generalized Schr\"odinger equation

TL;DR

This paper introduces bicomplex numbers and extends the Schr"odinger equation into this framework, analyzing its properties, symmetries, and the Born rule within bicomplex quantum mechanics.

Contribution

It develops the bicomplex number system, formulates the generalized Schr"odinger equation, and explores its symmetries and Born formulas, extending quantum mechanics into a new algebraic setting.

Findings

Bicomplex numbers form a commutative ring with zero divisors.

Derived the bicomplex Schr"odinger equation and its continuity equations.

Established the form of the Born rule for bicomplex wave functions.

Abstract

We introduce the set of bicomplex numbers $\mathbb{T}$ which is a commutative ring with zero divisors defined by $\mathbb{T}=\{w_0+w_1 \bold{i_1}+w_2\bold{i_2}+w_3 \bold{j}| w_0,w_1,w_2,w_3 \in \mathbb{R}\}$ where $\bold{i^{\text 2}_1}=-1, \bold{i^{\text 2}_2}=-1, \bold{j}^2=1,\ \bold{i_1}\bold{i_2}=\bold{j}=\bold{i_2}\bold{i_1}$. We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schr\"odinger equation and found the continuity equations. The discrete symmetries of the system of equations describing the bicomplex Schr\"odinger equation are obtained. Finally, we study the bicomplex Born formulas under the discrete symetries. We obtain the standard Born's formula for the class of bicomplex wave functions having a null hyperbolic angle.