Bicomplex Quantum Mechanics: II. The Hilbert Space

Abstract.

Using the bicomplex numbers

$$ \mathbb{T} \cong {\hbox{Cl}}_{\mathbb{C}} (1,0) \cong {\hbox{Cl}}_{\mathbb{C}} (0,1) $$

which is a commutative ring with zero divisors defined by

$$ \mathbb{T} = \{w_0 +w_1 {\bf{i}}_{\bf 1} +w_2 {\bf{i}}_{\bf 2} + w_3 {\bf{j}} \vert w_0, w_1, w_2, w_3 \in \mathbb{R}\}$$

where i ² ₁  =  − 1, i ² ₂  =  − 1, j² = 1 and i ₁ i ₂  = j = i ₂ i ₁ , 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.