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 |D:C^{\infty }({\mathbb{R}}^{\mathrm{4}n},{\mathbb{W}}_{\mathrm{4}n})\to C^{\infty }({\mathbb{R}}^{\mathrm{4}n},{\mathbb{W}}_{\mathrm{4}n}), where {\mathbb{W}}_{\mathrm{4}n} is the tensor product of three algebras, i.e., the hyperbolic quaternion \widehat{\mathbb{B}}, the bicomplex number \mathbb{B}, and the Clifford algebra {\mathbb{R}}_{\mathit{n}}. The operator D is a square root of the Laplacian in {\mathbb{R}}^{\mathrm{4}n}, introduced by the formula D|={\displaystyle {\sum }_{j=\mathrm{0}}^{\mathrm{3}}{K}_j{\partial }_{Z_j}} with K_jbeing the basis of \widehat{\mathbb{B}}, and {\partial }_{Z_j} denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra \mathbb{B}\otimes {\mathbb{R}}_{0,4\mathit{n}} whose definition involves a delicate construction of the bicomplexWitt basis. The introduction of the operator D 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 D, we establish the Cauchy integral formula, which generalizes the Martinelli-Bochner formula in the theory of several complex variables.

Key words： Bicomplex numbers; Hermitian Clifford analysis; Witt basis; Cauchy integral formula

References

1 Abreu Blaya R, Bory-Reyes J, Brackx F, De Schepper H, Sommen F. Cauchy integral formulas in quaternionic Hermitean Clifford analysis. Complex Anal Oper Theory, 2012, 6: 971-985

2 Brackx F, Bureš J, De Schepper H, Eelbode D, Sommen F, Souček V. Fundaments of Hermitean Clifford analysis. Part I: Complex structure. Complex Anal Oper Theory, 2007, 1: 341-365

3 Brackx F, Bureš J, De Schepper H, Eelbode D, Sommen F, Souček V. Fundaments of Hermitean Clifford analysis. Part II: Splitting of h-monogenic equations. Complex Var Elliptic Equ, 2007, 52: 1063-1079

4 Brackx F, De Knock B, De Schepper H, Sommen F. On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis. Bull Braz Math Soc, 2009, 40: 395-416

5 Brackx F, Delanghe R, Sommen F. Clifford Analysis. Boston: Pitman, 1982

6 Brackx F, Delanghe R, Sommen F. The Hermitian Clifford analysis toolbox. Adv Appl Clifford Algebr, 2008, 18: 451-487

7 Catoni F, Cannata R, Zampetti P. An introduction to commutative quaternions. Adv Appl Clifford Algebr, 2006, 16(1): 1-28

8 Damiano A, Eelbode D, Sabadini I. Algebraic analysis of Hermitian monogenic functions. C R Acad Sci Paris Ser I, 2008, 346: 139-142

9 Damiano A, Eelbode D, Sabadini I. Quaternionic Hermitian spinor systems and compatibility conditions. Adv Geom, 2011, 11: 169-189

10 Delanghe R, Sommen F, Souček V. Clifford Algebra and Spinor-Valued Functions. Dordrecht: Kluwer, 1992

11 Eelbode D. A Clifford algebraic framework for sp(m)-invariant differential operators. Adv Appl Clifford Algebr, 2007, 17: 635-649

12 Gilbert J, Murray M. Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge: Cambridge University Press, 1991

13 Gürlebeck K, Habetha K, Sprössig W. Holomorphic Functions in the Plane and n-dimensional Space. Basel: Birkhäuser Verlag, 2008

14 Gürlebeck K, Sprössig W. Quaternionic and Clifford Calculus for Physicists and Engineers. Chichester: Wiley, 1998

15 Kytmanov A. The Bochner-Martinelli Integral and Its Applications. Basel-Boston-Berlin: Birkhäuser, 1995

16 Lavoie R G, Marchildon L, Rochon D. Finite-dimensional bicomplex Hilbert spaces. Adv Appl Clifford Algebr, 2011, 21: 561-581

17 Mathieu J, Marchildon L, Rochon D. The bicomplex quantum Coulomb potential problem. Canad J Phys, 2013, 91(12): 1093-1100

18 Peña-Peña D, Sabadini I, Sommen F. Quaternionic Clifford analysis: the Hermitian setting. Complex Anal Oper Theory, 2007, 1: 97-113

19 Price G B. An Introduction to Multicomplex Spaces and Functions. New York: Marcel Dekker, 1991

20 Rocha-Chávez R, Shapiro M, Sommen F. Integral theorems for functions and differential forms in Cm. In: Research Notes in Mathematics, Vol 428. Boca Raton: Chapman & Hall/CRC, 2002

21 Rönn S. Bicomplex algebra and function theory. arXiv math 0101200v1.[math.CV], 2001

22 Ryan J. Complexified Clifford analysis. Complex Var Theory Appl, 1982, 1: 119-149

23 Sabadini I, Sommen F. Hermitian Clifford analysis and resolutions. Math Meth Appl Sci, 2002, 25: 1395-1413