Basics Of Functional Analysis With Bicomplex Sc... Link

(with respect to (i)): (z = z_1 + z_2 j), where (z_1, z_2 \in \mathbbC_i) (complex numbers with unit (i)).

is a set equipped with addition and scalar multiplication by C2the complex numbers sub 2 Basics of Functional Analysis with Bicomplex Sc...

The introduction of inner products leads to the study of bicomplex Hilbert spaces. These spaces are essential for applications in physics, particularly in bicomplex quantum mechanics. In this framework, wave functions take values in the bicomplex ring, providing a more flexible language for describing physical phenomena that involve multiple phases or dimensions. The orthogonality of vectors in these spaces is defined relative to a bicomplex-valued inner product, which must be managed to account for the zero divisors mentioned previously. (with respect to (i)): (z = z_1 +

(most important for analysis): Define (\mathbfe_1 = \frac1 + k2) and (\mathbfe_2 = \frac1 - k2). These satisfy (\mathbfe_1 + \mathbfe_2 = 1), (\mathbfe_1^2 = \mathbfe_1), (\mathbfe_2^2 = \mathbfe_2), and (\mathbfe_1 \mathbfe_2 = 0). Then every bicomplex number (z) decomposes uniquely as: [ z = \alpha \mathbfe_1 + \beta \mathbfe_2 ] where (\alpha, \beta \in \mathbbC_i) (or (\mathbbC_j)). This representation converts multiplication to component-wise operations: (z \cdot w = (\alpha \gamma) \mathbfe_1 + (\beta \delta) \mathbfe_2). In this framework, wave functions take values in