Bicomplex Hermitian Clifford analysis

Abstract

Complex Hermitian Clifford analysis emerged recently as a refinement of the theory of several complex variables, while at the same time, the theory of bicomplex numbers motivated by the bicomplex version of quantum mechanics is also under full development. This stimulates us to combine the Hermitian Clifford analysis with the theory of bicomplex number so as to set up the theory of bicomplex Hermitian Clifford analysis. In parallel with the Euclidean Clifford analysis, the bicomplex Hermitian Clifford analysis is centered around the bicomplex Hermitian Dirac operator Open image in new window: \(C^\infty (\mathbb{R}^{4n} ,\mathbb{W}_{4n} ) \to C^\infty (\mathbb{R}^{4n} ,\mathbb{W}_{4n} )\), where \(\mathbb{W}_{4n}\) is the tensor product of three algebras, i.e., the hyperbolic quaternion Open image in new window, the bicomplex number \(\mathbb{B}\), and the Clifford algebra ℝ0,4n. The operator Open image in new window is a square root of the Laplacian in ℝ4n, introduced by the formula Open image in new window with Kj being the basis of Open image in new window, and \(\partial _{Z_j }\) denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra \(\mathbb{B} \otimes \mathbb{R}_{0,4n}\) whose definition involves a delicate construction of the bicomplex Witt basis. The introduction of the operator Open image in new window can also overturn the prevailing opinion in the Hermitian Clifford analysis in the complex or quaternionic setting that the complex or quaternionic Hermitiean monogenic functions are described by a system of equations instead of by a single equation like classical monogenic functions which are null solutions of Dirac operator. In contrast to the Hermitian Clifford analysis in quaternionic setting, the Poisson brackets of the twisted real Clifford vectors do not vanish in general in the bicomplex setting. For the operator Open image in new window, we establish the Cauchy integral formula, which generalizes the Martinelli-Bochner formula in the theory of several complex variables.