Spin actions in Euclidean and Hermitian Clifford analysis in superspace

Abstract

In [4] we studied the group invariance of the inner product of supervectors as introduced in the framework of Clifford analysis in superspace. The fundamental group SO 0 leaving invariant such an inner product turns out to be an extension of SO(m) × Sp(2n) and gives rise to the definition of the spin group in superspace through the exponential of the so-called extended superbivectors, where the spin group can be seen as a double covering of SO 0 by means of the representation h(s)[x] = sxs. In the present paper, we study the invariance of the Dirac operator in superspace under the classical H and L actions of the spin group on superfunctions. In addition, we consider the Hermitian Clifford setting in superspace, where we study the group invariance of the Hermitian inner product of supervectors introduced in [3]. The group of complex supermatrices leaving this inner product invariant constitutes an extension of U(m) × U(n) and is isomorphic to the subset SO J 0 of SO 0 of elements that commute with the complex structure J. The realization of SO J 0 within the spin group is studied together with the invariance under its actions of the super Hermitian Dirac system. It is interesting to note that the spin element leading to the complex structure can be expressed in terms of the n-dimensional Fourier transform.

Hypermonogenic Plane Wave Solutions of the Dirac Equation in Superspace

“…a linear operator M satisfying t`M (t`) + M (t`) t`= 0. For an introduction to superrotations and their use in Clifford analysis we refer the reader to [12] and [14].…”

Hermitian Clifford Analysis on Superspace

“…In forthcoming work we will further develop this theory. This will include a deep study of the group realization of rotations in superspace and the invariance of the super Dirac operators under the action of these groups ( [9,10]). Also a Bochner-Martinelli formula in this setting will be established.…”

Bochner‐Martinelli formula in superspace

“…When doubling the bosonic dimension, it is possible to define a so-called complex structure J on the algebra P ⊗ C 2m,2n , see e.g. [24,25]. In general, J is an algebra automorphism over P ⊗ C 2m,2n defined by…”

“…More recently, harmonic and Clifford analysis has been extended to superspace by introducing some important differential operators (such as Dirac and Laplace operators) and by studying special functions and orthogonal polynomials related to these operators, see eg, previous studies . The basics of Hermitian Clifford analysis in superspace were introduced in De Schepper et al following the notion of an abstract complex structure in the Hermitian radial algebra, developed in Sabadini and Sommen and De Schepper et al Some particular aspects related to the invariance properties with respect to underlying Lie groups and Lie algebras in this setting have been already studied, see eg, De Schepper et al…”

“…The basics of Hermitian Clifford analysis in superspace were introduced in [24] following the notion of an abstract complex structure in the Hermitian radial algebra, developed in [23,35]. Some particular aspects related to the invariance properties with respect to underlying Lie groups and Lie algebras in this setting has been already studied, see e.g [25].…”