Infinite-Dimensional Bicomplex Hilbert Spaces

TL;DR

This paper introduces and studies infinite-dimensional bicomplex Hilbert spaces, extending finite-dimensional results and establishing foundational properties with applications to quantum mechanics.

Contribution

It defines bicomplex Hilbert spaces, proves key theorems like Riesz representation and Schwarz inequality, and explores applications in quantum mechanics.

Findings

Established properties of bicomplex Hilbert spaces

Proved Riesz representation theorem for bicomplex functionals

Derived bicomplex Schwarz inequality

Abstract

This paper begins the study of infinite-dimensional modules defined on bicomplex numbers. It generalizes a number of results obtained with finite-dimensional bicomplex modules. The central concept introduced is the one of a bicomplex Hilbert space. Properties of such spaces are obtained through properties of several of their subsets which have the structure of genuine Hilbert spaces. In particular, we derive the Riesz representation theorem for bicomplex continuous linear functionals and a general version of the bicomplex Schwarz inequality. Applications to concepts relevant to quantum mechanics, specifically the bicomplex analogue of the quantum harmonic oscillator, are pointed out.