The fundamental solutions of the super Dirac and Laplace operators and their
natural powers are determined within the framework of Clifford analysis.Comment: 12 pages, accepted for publication in J. Math. Anal. App

Aronszajn

Bartocci

Berezin

Brackx

De Bie

Delanghe

F. Sommen

Gilbert

H. De Bie

Kostant

Leĭtes

Sommen

English

arXiv

Crossref

Fundamental solutions for the super Laplace and Dirac operators and all their natural powers

AbstractThe fundamental solutions of the super Dirac and Laplace operators and their natural powers are determined within the framework of Clifford analysis

De Bie, H.

Sommen, F.

Elsevier - Publisher Connector 

J. Math. Anal. Appl. 338 (2008) 1320–1328
www.elsevier.com/locate/jmaa
Fundamental solutions for the super Laplace
and Dirac operators and all their natural powers
H. De Bie ∗, F. Sommen
Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Ghent, Belgium
Received 26 April 2007
Available online 20 June 2007
Submitted by Steven G. Krantz
Abstract
The fundamental solutions of the super Dirac and Laplace operators and their natural powers are determined within the frame-
work of Clifford analysis.
© 2007 Elsevier Inc. All rights reserved.
Keywords: Clifford analysis; Dirac operator; Superspace; Fundamental solution
1. Introduction
In a previous set of papers (see [1–4]) we have developed the basic framework for Clifford analysis in superspace.
Clifford analysis in standard Euclidean space is a function theory of the so-called Dirac operator and as such a gen-
eralization of the theory of holomorphic functions of one complex variable. Basic references are [5–7]. A superspace
on the other hand is a generalization of the classical concept of space, where not only commuting variables are con-
sidered, but also a set of anti-commuting ones. These superspaces find their use in theoretical physics. We refer the
reader e.g. to [8–12].
We have first established the basic algebraic framework necessary for developing a theory of Clifford analysis
in superspace (see [1] and [2]). We have constructed the fundamental differential operators such as the Dirac and
Laplace operators, the Euler and Gamma operators, etc. Next we have constructed a theory of spherical monogenics
in superspace (see [3]), which was then used to introduce an integration over the supersphere and over superspace
(see [4]), inspired by similarities with the classical theory of harmonic analysis. Moreover it turned out that this
integration is equivalent with the one introduced in the work of Berezin (see [8,9]), although our approach offers
better insight. Indeed, the definition given by Berezin is not motivated by any connection with classical types of
integration, whereas our approach connects the integral over superspace with the well-known theory of integration in
Euclidean space (see the discussion in [4]).
* Corresponding author.
E-mail addresses: Hendrik.DeBie@UGent.be (H. De Bie), fs@cage.ugent.be (F. Sommen).0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2007.06.015
H. De Bie, F. Sommen / J. Math. Anal. Appl. 338 (2008) 1320–1328 1321All previous work is situated on the level of polynomial functions. A next step is to consider larger algebras and to
start the study of the function-theoretical properties of our differential operators. Thus is the aim of the present paper.
More precisely we will determine the fundamental solutions for all natural powers of the super Dirac operator.
The paper is organized as follows. We start with a short introduction to Clifford analysis on superspace. Then
we recapitulate the results on polyharmonic functions needed later on. In the next section we derive the fundamental
solutions for the super Laplace and Dirac operators and compare them with an ad hoc approach, inspired by the theory
of radial algebra. In the following section we extend this technique to construct fundamental solutions for all natural
powers. Finally we discuss how the present technique is valid for a still larger class of differential operators.
2. Clifford analysis in superspace
We first consider the real algebra P = Alg(xi, ei; x`j , e`j ), i = 1, . . . ,m, j = 1, . . . ,2n, generated by
• m commuting variables xi and m orthogonal Clifford generators ei ,
• 2n anti-commuting variables x`i and 2n symplectic Clifford generators e`i
subject to the multiplication relations
⎧⎪⎨
⎪⎩
xixj = xjxi,
x`i x`j = −x`j x`i ,
xi x`j = x`j xi
and
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
ej ek + ekej = −2δjk,
e`2j e`2k − e`2ke`2j = 0,
e`2j−1e`2k−1 − e`2k−1e`2j−1 = 0,
e`2j−1e`2k − e`2ke`2j−1 = δjk,
ej e`k + e`kej = 0
and where moreover all elements ei , e`j commute with all elements xi , x`j .
If we denote by Λ2n the Grassmann algebra generated by the anti-commuting variables x`j and by C the algebra
generated by all the Clifford numbers ei , e`j , then we clearly have that
P = R[x1, . . . , xm] ⊗ Λ2n ⊗ C.
The most important element of the algebra P is the vector variable x = x + x` with
x =
m∑
i=1
xiei, x` =
2n∑
j=1
x`j e`j .
One calculates that
x2 = x` 2 + x2 =
n∑
j=1
x`2j−1x`2j −
m∑
j=1
x2j .
The super Dirac operator is defined as
∂x = ∂x` − ∂x = 2
n∑
j=1
(e`2j ∂x`2j−1 − e`2j−1∂x`2j ) −
m∑
j=1
ej ∂xj .
Its square is the super Laplace operator:
 = ∂2x = 4
n∑
j=1
∂x`2j−1∂x`2j −
m∑
j=1
∂2xj .
The bosonic part of this operator is b = −∑mj=1 ∂2xj , which is the classical Laplace operator. The fermionic part is
f = 4∑nj=1 ∂x`2j−1∂x`2j . For the other important operators in super Clifford analysis and their commutation relations
we refer the reader to [1]. If we let ∂x act on x we find that
∂xx = m − 2n = M
1322 H. De Bie, F. Sommen / J. Math. Anal. Appl. 338 (2008) 1320–1328where M is the so-called super-dimension. This numerical parameter gives a global characterization of our superspace
and will be used in Remark 1. We will also need the following basic formulae (see [1])
∂x
(
x2s
)= 2sx2s−1, (1)
∂x
(
x2s+1
)= (M + 2s)x2s . (2)
In the case where m = 1, n = 0 this reduces to the familiar formula d
dx
xk = kxk−1.
Now we can consider several generalizations of the algebra P . This leads to the introduction of the function-spaces:
F(Ω)m|2n =F(Ω) ⊗ Λ2n ⊗ C
where F(Ω) stands for D(Ω), Ck(Ω), Lp(Ω), Lloc1 (Ω), . . . with Ω an open domain in Rm. Finally the space of
harmonic functions, i.e. null-solutions of the super Laplace operator, will be denoted by H(Ω)m|2n ⊆ C2(Ω) ⊗ Λ2n.
Similarly we denote by M(Ω)m|2n ⊆ C2(Ω)m|2n the space of monogenic functions, i.e. null-solutions of the super
Dirac operator. We have thatM(Ω)m|2n ⊆H(Ω)m|2n ⊗ C.
Now we have the following theorem, which generalizes a classical result in harmonic analysis.
Theorem 1. Null-solutions of the super Laplace and the super Dirac operator are C∞-functions, i.e.
H(Ω)m|2n ⊗ C ⊆ C∞(Ω)m|2n, M(Ω)m|2n ⊆ C∞(Ω)m|2n.
Proof. It suffices to give the proof for harmonic functions, as monogenic functions are also harmonic. So we consider
a function f ∈H(Ω)m|2n. Such a function can be written as
f =
∑
(α)
f(α)x`
α1
1 . . . x`
α2n
2n
with (α) = (α1, . . . , α2n), αi ∈ {0,1} and f(α) ∈ C2(Ω).
Expressing that f = bf + f f = 0 leads to a set of equations of the following type:
bf(α) =
∑
(β)
c(β)f(β),
∣∣(β)∣∣= ∣∣(α)∣∣+ 2,
with c(β) ∈ R. We conclude that for every (α) there exists a k ∈ N (k  n + 1) such that kbf(α) = 0. Hence f(α) is
polyharmonic and thus an element of C∞(Ω). 
We end this section with a few words on integration in superspace. The proper integral to consider is the so-called
Berezin integral
∫
B
(see [8,9] and [4]) which has the following formal definition:
∫
B
=
∫
Rm
dV (x )∂x`2n . . . ∂x`1
with dV (x ) the Lebesgue measure in Rm.
One can also define a super Dirac distribution as
δ(x) = δ( x )x`1 . . . x`2n = δ( x ) x`
2n
n!
with δ( x ) the classical Dirac distribution in Rm. We clearly have that
〈
δ(x − y), f (x)〉=
∫
B
δ(x − y)(x`1 − y`1) · · · (x`2n − y`2n)f (x) = f (y)
with f ∈D(Ω)m|2n.
H. De Bie, F. Sommen / J. Math. Anal. Appl. 338 (2008) 1320–1328 13233. Fundamental solutions in Rm
The fundamental solutions for the natural powers of the classical Laplace operator b are very well-known, e.g.
see [13].
We denote by νm|02l , l = 1,2, . . . , a sequence of such fundamental solutions, satisfying

j
bν
m|0
2l = νm|02l−2j , j < l,
lbν
m|0
2l = δ( x ).
Their explicit form depends both on the dimension m and on l. More specifically, in the case where m is odd we have
that
ν
m|0
2l =
r2l−m
γl−1
, γl = (−1)l+1(2 − m)4l l!(l + 2 − m/2)
(2 − m/2)
2πm/2
(m/2)
(3)
with r =√−x2. The formulae for m even are more complicated and can be found in [13].
Concerning the refinement to Clifford analysis, we clearly have that νm|02l+1 = ∂xνm|02l+2 is a fundamental solution
of lb∂x .
4. Fundamental solution of  and ∂x
From now on we restrict ourselves to the case where m = 0. The purely fermionic case will be discussed briefly in
Section 7.
Our aim is to construct a function ρ such that in distributional sense
ρ = δ(x).
We propose the following form for the fundamental solution:
ρ =
n∑
k=0
ak
(
n−kb φ
)
x`
2n−2k
,
with φ and ak ∈ R to be determined.
Now let us calculate ρ
ρ = (b + f )ρ
=
n∑
k=0
ak
(
n−k+1b φ
)
x`
2n−2k +
n∑
k=1
ak(2n − 2k)(2n − 2k − 2 − 2n)
(
n−kb φ
)
x`
2n−2k−2
= a0
(
n+1b φ
)
x`
2n +
n∑
k=1
[
ak − 2k(2n − 2k + 2)ak−1
](
n−k+1b φ
)
x`
2n−2k
.
So ρ is a fundamental solution if and only if
a0
(
n+1b φ
)= δ( x ) 1
n!
and ak satisfies the recurrence relation
ak = 4k(n − k + 1)ak−1.
The first equation leads to
φ = νm|02n+2, a0 =
1
n! .
1324 H. De Bie, F. Sommen / J. Math. Anal. Appl. 338 (2008) 1320–1328We then immediately find the following expression for the ak
ak = 4
kk!
(n − k)! , k = 0, . . . , n.
Summarizing we obtain the following theorem.
Theorem 2. The function νm|2n2 defined by
ν
m|2n
2 =
n∑
k=0
4kk!
(n − k)!ν
m|0
2k+2x`
2n−2k
,
with νm|02k+2 as in Section 3, is a fundamental solution for the operator .
Proof. It is clear that νm|2n2 ∈ Lloc1 (Rm)m|2n. Moreover, we have that νm|2n2 ∈H(Rm−{0})m|2n and that νm|2n2 = δ(x)
in distributional sense. 
Now suppose that m is odd, then the previous formula simplifies to
ν
m|2n
2 =
(m/2)
2(2 − m)πm/2
n∑
k=0
(−1)k+1
(n − k)!
(2 − m/2)
(k + 2 − m/2) r
2k+2−mx` 2n−2k, (4)
where we have used formula (3).
As we have that νm|2n2 = δ(x), a fundamental solution for the Dirac operator ∂x is given by ∂xνm|2n2 . This leads
to the following
Theorem 3. The function νm|2n1 defined by
ν
m|2n
1 =
n−1∑
k=0
2
4kk!
(n − k − 1)!ν
m|0
2k+2x`
2n−2k−1 −
n∑
k=0
4kk!
(n − k)!ν
m|0
2k+1x`
2n−2k
is a fundamental solution for the operator ∂x .
Remark 1. We could propose the following form:
g = 1
(x2)
M−2
2
for the fundamental solution of , where we have replaced m by the super-dimension M in the classical expression.
This technique is inspired by radial algebra (see [14]), which gives a very general framework for constructing theories
of Clifford analysis, based on the introduction of an abstract dimension parameter (in this case the super-dimension).
This leads partially to the correct result (see also [15]). Indeed, formally we can expand this as
g = 1
(x2)
M−2
2
= 1
(x2 + x` 2)M−22
= 1
(x2)
M−2
2
(
1 + x`
2
x2
)1−M2 =
n∑
k=0
(
1 − M2
k
)
x`
2k
(x2)
M
2 −1+k
.
The coefficients in this development are proportional to the ones obtained in Theorem 2 (see also formula (4)), so this
yields the correct result. This expansion is however only valid if m is odd.
The fundamental solution can of course be used to determine solutions of the inhomogeneous equation f = ρ.
We have for example the following
H. De Bie, F. Sommen / J. Math. Anal. Appl. 338 (2008) 1320–1328 1325Proposition 1. Let ρ ∈D(Ω)m|2n, then a solution of f = ρ is given by
f (x) = νm|2n2 ∗ ρ =
∫
B
ν
m|2n
2 (x − y)ρ(y).
5. Fundamental solution of k and k∂x
A similar technique as in Section 4 can be used for the polyharmonic case. First we expand k as
k =
k∑
j=0
(
k
j
)

k−j
b 
j
f .
This expansion is valid as b commutes with f .
Now we propose the following form for its fundamental solution:
ρ =
n∑
l=0
al
(
n−lb φ
)
x`
2n−2l
with φ and al ∈ R still to be determined. We calculate that
kρ =
n∑
l=0
al
k∑
j=0
(
k
j
)(

n−l+k−j
b φ
)

j
f x`
2n−2l
.
As we have that, using formulae (1) and (2) in the case where m = 0, M = −2n,

j
f x`
2n−2l = 4j (−1)j (n − l)!
(n − l − j)!
(l + j)!
l! x`
2n−2l−2j
, j  n − l,
this yields
kρ =
n∑
l=0
al
k∑
j=0
(
k
j
)
4j (−1)j (n − l)!
(n − l − j)!
(l + j)!
l!
(

n−l+k−j
b φ
)
x`
2n−2l−2j
.
Putting kρ = δ(x) leads to the following set of equations:
a0
n+k
b φ = δ( x )
1
n! , (5)
k∑
j=0
al−j
(
k
j
)
4j (−1)j (n − l + j)!
(n − l)!
l!
(l − j)! = 0, (6)
a−1 = a−2 = a−3 = · · · = 0. (7)
We immediately have that a0 = 1/n! and that φ = νm|02n+2k .
Equation (6) can be simplified by the substitution
al = 4l l!
(n − l)!bl
to
k∑
j=0
bl−j
(
k
j
)
(−1)j = 0, b0 = 1,
which has the solution (see the subsequent Lemma 1)
bl =
(
l + k − 1)
.
l
1326 H. De Bie, F. Sommen / J. Math. Anal. Appl. 338 (2008) 1320–1328We conclude that
al = 4l (l + k − 1)!
(n − l)!(k − 1)! , l = 0, . . . , n.
We can summarize the previous results in the following theorem.
Theorem 4. The function νm|2n2k defined by
ν
m|2n
2k =
n∑
l=0
4l
(l + k − 1)!
(n − l)!(k − 1)!ν
m|0
2l+2kx`
2n−2l
,
is a fundamental solution for the operator k .
In a similar vein we obtain the fundamental solution νm|2n2k+1 for the operator k∂x = ∂2k+1x by calculating ∂xνm|2n2k+2.
This leads to
Theorem 5. The function νm|2n2k+1 defined by
ν
m|2n
2k+1 =
n−1∑
l=0
2
4l(l + k)!
(n − l − 1)!k!ν
m|0
2l+2k+2x`
2n−2l−1 −
n∑
l=0
4l(l + k)!
(n − l)!k!ν
m|0
2l+2k+1x`
2n−2l
,
is a fundamental solution for the operator k∂x .
We still have to prove the technical lemma we used in the derivation of Theorem 4.
Lemma 1. The sequence (bl), l = 0,1, . . . , recursively defined by
min(k,l)∑
j=0
bl−j
(
k
j
)
(−1)j = 0, b0 = 1,
is given explicitly by
bl =
(
l + k − 1
l
)
.
Proof. Define the polynomial Rl(x) by
Rl(x) =
min(k,l)∑
j=0
(−1)j
(
k
j
)
(x + k − j − 1)!
(x − j)!
=
min(k,l)∑
j=0
(−1)j
(
k
j
)
(x + k − 1 − j) · · · (x + 1 − j).
We then have to prove that Rl(l) = 0.
We distinguish between three cases.
(1) l  k − 2.
We claim that for all t  k − 2
Rt(x) = (−1)t
(
k − 1
t
)
(x + k − t − 1) · · · (x + 1)(x − 1) · · · (x − t).
This can be proven using induction. The case where t = 1 is easily checked. So we suppose the formula holds for
t − 1, then we calculate
H. De Bie, F. Sommen / J. Math. Anal. Appl. 338 (2008) 1320–1328 1327Rt−1(x) + (−1)t
(
k
t
)
(x + k − 1 − t) · · · (x + 1 − t)
= (−1)t (x + k − t − 1) · · · (x + 1)(x − 1) · · · (x − t + 1)
((
k
t
)
x −
(
k − 1
t − 1
)
(x + k − t)
)
= (−1)t
(
k − 1
t
)
(x + k − t − 1) · · · (x + 1)(x − 1) · · · (x − t)
= Rt(x)
which proves the hypothesis. Now clearly Rl(l) = 0.
(2) l = k − 1.
Using the previous results, it is shown that in this case Rk−1(x) equals
Rk−1(x) = −(−1)k(x − 1) · · · (x + 1 − k)
so Rk−1(k − 1) = 0.
(3) l  k.
Now we have that
Rl(x) = Rk(x) = 0
so this case is also proven. 
6. A larger class of differential operators
The technique used above can be extended to a larger class of differential operators. Suppose we consider an
operator of the following form:
P = L(x, ∂x) + f
with L(x, ∂x) an elliptic operator in Rm and f the fermionic Laplace operator. Note that L(x, ∂x) and f clearly
commute. An interesting operator in this class is the super Helmholtz operator
 − λ2, λ ∈ R,
with L(x, ∂x) = b − λ2.
Denoting by μm|02k (k = 1,2, . . .) a set of fundamental solutions for the operators L(x, ∂x)k such that
L(x, ∂x)
jμ
m|0
2k = μm|02k−2j , j < k,
L(x, ∂x)
kμ
m|0
2k = δ( x )
we can now use the same technique as in Section 5 to obtain a fundamental solution μm|2n2k for the operator P k . This
leads to
μ
m|2n
2k =
n∑
l=0
4l
(l + k − 1)!
(n − l)!(k − 1)!μ
m|0
2l+2kx`
2n−2l
.
7. The purely fermionic case
In this case there is no fundamental solution. Indeed, determining the fundamental solution of f requires solving
the algebraic equation
f ν
0|2n
2 = x`1 . . . x`2n
which clearly has no solution, since there are no polynomials of degree higher than 2n.
1328 H. De Bie, F. Sommen / J. Math. Anal. Appl. 338 (2008) 1320–13288. Conclusions
In this paper we have developed a technique to construct fundamental solutions for certain differential operators
in superspace. In particular we have constructed the fundamental solutions of the natural powers of the super Dirac
operator ∂kx .
We envisage to use these fundamental solutions in a further development of the function theory of Clifford analysis
in superspace. There is no doubt that they will play an important role in the generalization of the Cauchy and Hilbert
transform to superspace.
Acknowledgment
The first author is a research assistant supported by the Fund for Scientific Research Flanders (F.W.O.-Vlaanderen). He would like to thank
Liesbet Van de Voorde for a discussion concerning Section 5.
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Fundamental solutions for the super Laplace and Dirac operators and all their natural powers 

De Bie, Hendrik

Sommen, Franciscus

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Fundamental solutions for the super Laplace and Dirac operators and all
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