Bicomplex Quantum Mechanics: II. The Hilbert Space

Using the bicomplex numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{T} \cong {\hbox{Cl}}_{\mathbb{C}} (1,0) \cong {\hbox{Cl}}_{\mathbb{C}} (0,1) $$\end{document} which is a commutative ring with zero divisors defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{T} = \{w_0 +w_1 {\bf{i}}_{\bf 1} +w_2 {\bf{i}}_{\bf 2} + w_3 {\bf{j}} \vert w_0, w_1, w_2, w_3 \in \mathbb{R}\}$$\end{document} where i12 =  − 1, i22 =  − 1, j2 = 1 and i1i2 = j = i2i1, we construct hyperbolic and bicomplex Hilbert spaces. Linear functionals and dual spaces are considered on these spaces and properties of linear operators are obtained; in particular it is established that the eigenvalues of a bicomplex self-adjoint operator are in the set of hyperbolic numbers.