Bicomplex Quantum Mechanics: II. The Hilbert Space

Abstract

Using the bicomplex numbers $\mathbb{T}$ which is a commutative ring with zero divisors defined by $\mathbb{T}=\{w_0+w_1{i}_1+w_2{i}_2+w_{3}j|w_0,w_1,w_2,w_3\in \mathbb{R}\}$ where $i_1^2=-1,i_2^2=-1,j^2=1,i_1{i}_2=j=i_2{i}_1$, we construct hyperbolic and bicomplex Hilbert spaces. Linear functionals and dual spaces are considered 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.

25 pages, no figure

References in corpus (2)

• Bicomplex quantum mechanics: I. The generalized Schrödinger equation
• Representation of the contextual statistical model by hyperbolic amplitudes

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• Bicomplex Riesz-Fischer Theorem
• Determinant, Characteristic Polynomial, and Inverse in Commutative Analogues of Clifford Algebras
• Bicomplex k-Fibonacci quaternions
• Bicomplex Linear Operators on Bicomplex Hilbert Spaces and Littlewood's Subordination Theorem
• On Commutative Analogues of Clifford Algebras and Their Decompositions