Table Of ContentThe higher spin Laplace operator
Hendrik De Bie∗, David Eelbode†, Matthias Roels‡
Abstract
5
1 Thispaperdealswithacertainclassofsecond-orderconformallyinvariantoperators
0 actingonfunctionstakingvaluesinparticular(finite-dimensional)irreduciblerepre-
2
sentations of the orthogonalgroup. These operatorscan be seen as a generalisation
n of the Laplace operator to higher spin as well as a second order analogue of the
a
Rarita-Schwingeroperator. Toconstructtheseoperators,wewillusetheframework
J
of Clifford analysis, a multivariate function theory in which arbitrary irreducible
4
representations for the orthogonal group can be realised in terms of polynomials
2
satisfying a system of differential equations. As a consequence, the functions on
] which this particular class of operators act are functions taking values in the space
h
of harmonics homogeneous of degree k. We prove the ellipticity of these operators
p
and use this to investigate their kernel, focusing on both polynomial solutions and
-
h the fundamental solution.
t
a
m
1 Introduction
[
3
The aim of this paper is to introduce a framework to study a certain class of second-
v
4 orderconformallyinvariantoperatorswhichcanbeseenasgeneralisationsoftheclassical
7 Laplace operator.
9 Traditionally, these operators are mostly studied on either the conformal sphere Sm,
3
0 which is a flatmodel of conformal geometry, or curved analogues thereof which are mod-
. elled on aprinciplefibrebundle[7, 8,9]. Ausefulmethodto studytheseoperators is the
1
0 ambient space method developed by Fefferman and Graham [19], which requires embed-
5 ding a m-dimensional manifold into a space of dimension m+2. The main idea behind
1
their approach is that the ‘flat’ equations on the ambient space (the ones presented in
:
v
this paper) are in a suitable sense restricted to the embedded space-time, giving rise to
i
X the equations on the curved background.
r The advantage of using the Clifford analysis model for spin fields (see below) lies in the
a
fact that it leads to an encompassing framework in which both the dimension m as the
spinnumbercanbetreatedasaparameter. From thispointofview, theresultsobtained
in this paper form the scalar version of the function theory for the Rarita-Schwinger op-
erator on Rm which was developed in [6].
∗[email protected]
†[email protected]
‡[email protected]
1
Theaforementioned language of Cliffordanalysis refersto amultivariate functiontheory
which is often described as a generalisation of complex analysis to arbitrary dimension
m ∈ N. At the very heart of this theory lies the Dirac operator ∂ on Rm, a conformally
x
invariant first order elliptic differential operator generalising both the Cauchy-Riemann
operator ∂ and the operator introduced by P.A.M. Dirac in 1928 [12]. This operator
z
moreover satisfies ∆ = −∂2 (with ∆ the Laplace operator on Rm), which means that
x x x
Clifford analysis is a refinementof classical harmonicanalysis in m dimensions. We refer
the reader to the standard references [4, 11, 21] for more information. While classical
Clifford analysis is centred around the study of functions on Rm taking values in the
spinor space S, on which the Dirac operator is canonically defined, several authors have
been studying generalisations of the developed techniques to the so-called higher spin
theory [6, 15, 16, 17, 18]. This concerns the study of higher spin Dirac operators, acting
on functions on Rm taking values in arbitrary irreduciblerepresentations of Spin(m). So
far, the theory has been focusingon first-orderconformally invariant operators, reflected
in the fact that the functions under consideration take their values in irreducible half-
integer representations for thespingroup (the spinorspace beingtheeasiest case of such
a representation). This then leads to function theories refining (poly-)harmonic analysis
on Rm, see [17]. As mentioned earlier, we aim at extending these results to a certain
second-order conformally invariant operator acting on functions taking their values in
the simplest integer highest-weight representation. This will lead to analogues of the
Rarita-Schwinger function theory (see e.g. [6]). The existence of the operators we are
introducing follows from general arguments, see e.g. [5, 20]. However, our focus is quite
different: after developing explicit expressions for the higher spin Laplace operator, we
will study in depth its polynomial null-solutions as well as the fundamental solution.
The paper is organised as follows: after a brief introduction to Clifford analysis in
section 2, we will construct the higher spin Laplace operator in section 3. In section 4
we will construct all polynomial solutions for this operator, whereas section 5 will bede-
voted to the problem of constructing the fundamental solution for our operator. Finally,
in the last section we investigate the connection with the Rarita-Schwinger operator,
a conformally invariant operator acting on functions taking values in the irreducible
Spin(m)-module with highest weight λ = k+ 1,1,..., 1,±1 . We have also included
2 2 2 2
an appendix to prove some results of a more technical nature from section 3 (related to
(cid:0) (cid:1)
the conformal invariance and ellipticity).
Acknowledgement
This research was supported by the Fund for Scientific Research-Flanders (FWO-V),
project “Construction of algebra realisations using Dirac-operators”, grant G.0116.13N.
Theresultsinsection3and4wereoriginallyobtainedinthethirdauthor’s(unpublished)
master thesis [30].
2
2 Preliminaries on Clifford analysis
Letusfirstintroducethe(real)universalCliffordalgebraR asthealgebragenerated by
m
an orthonormal basis {e ,...,e } for the vector space Rm endowed with the Euclidean
1 m
inner product hu,xi = x u using the multiplication rules
j j j
P e e +e e = −2he ,e i =−2δ
a b b a a b ab
with 1 ≤ a,b ≤ m. The complex Clifford algebra C is then defined as the algebra
m
C = R ⊗ C. This algebra is Z -graded, and the even subalgebra (respectively the
m m 2
odd subspace) is denoted by means of C+ (resp. C−). We will not consider functions
m m
taking their values in C , but restrict the values to a suitable subspace, which is known
m
as the spinor space. This space can be realised as a matrix space, as is often done by
physicists, or as a subspace of the Clifford algebra (see [11, 21]):
Definition 2.1. In case m = 2n, the Witt basis {f ,f† : 1 ≤ j ≤ n} for C is defined
j j m
by means of
1 1
f = e −ie and f† = − e +ie .
j 2 2j−1 2j j 2 2j−1 2j
(cid:0) (cid:1) (cid:0) (cid:1)
The element I := (f f†)(f f†)...(f f†) ∈ C defines a primitive idempotent (I2 = I),
1 1 2 2 n n m
and in terms of this element one then introduces the complex vector spaces S± := C± I.
2n 2n
The main reason why it is better to consider spinor-valued functions is the following:
these spaces carry the irreducible spinor representations for the spin group or its Lie
algebra so(m). They are both realised inside the Clifford algebra:
Definition 2.2. The (real, compact) spin group Spin(m) can be defined as
2k
Spin(m) = ω :ω ∈Sm−1 ⊂ R ,
j j m
jY=1
with Sm−1 ⊂ Rm the unit sphere (viz. ω2 = −1). ThisLie group defines a double cover
j
for the orthogonal group, see e.g. [28].
Definition 2.3. The orthogonal Lie algebra so(m) can be realised as the space C(2) of
m
bivectors, defined as the linear hull of the elements e := e e with 1≤ a 6= b ≤ m. This
ab a b
vector space becomes a Lie algebra under the commutator [B ,B ] := B B −B B .
1 2 1 2 2 1
In case m = 2n, the spaces S± define inequivalent irreducible representations for the
2n
spin group and its Lie algebra (both actions given by multiplication in the Clifford
algebra C ). In case m = 2n+1 however, there is (up to isomorphism) a unique spinor
2n
space (the so-called space of Dirac spinors) which will be denoted as S and can be
2n+1
explicitly realised as S ∼= S± (whereby one can choose either the plus sign or the
2n+1 2n+2
minus sign, because as representation spaces for so(2n+1) they are equivalent).
3
Remark 2.1. From now on we will omit the subscript attached to the spinor spaces.
This subscript refers to the dimension of the underlying vector space, equal to m ∈ N
from now on. The superscript only matters in even dimensions (cfr. supra). In order
to avoid having to drag this parity sign along in what follows, we will work with Dirac
spinors in both even and odd dimensions (see next definition).
Definition 2.4. The Dirac spinor space S stands for S in case m = 2n+1 is odd,
2n+1
and for the direct sum S+ ⊕ S− in case m = 2n is even. Note that the latter is not
2n 2n
irreducible as a module for the Lie algebra so(2n).
The classical Dirac operator in Rm is given by ∂ = m e ∂ . It is the unique elliptic
x j=1 j xj
first-order conformally invariant differential operator acting on spinor valued functions
P
f(x) on Rm. It factorises the Laplace operator ∆ = −∂2 on Rm. A spinor-valued
x x
function f is monogenic in an open region Ω ⊂ Rm if and only if ∂ f = 0 in Ω. For a
x
detailed study of the (classical) theory of monogenic functions, see [4, 11, 21]. In [10, 21]
itwasshownthatevery (finite-dimensional)irreduciblerepresentation forthespingroup
(or its Lie algebra) with integer (half-integer) highest weight can be realised in terms
of scalar-valued harmonic (spinor-valued monogenic) polynomials of several (dummy)
vector variables, a convenient alternative for the spaces of traceless tensors often used
in physics. Our cases of interest are presented in the definition below, where from now
on we write ker(D ,...,D ) := kerD ∩...∩kerD .
1 n 1 n
Definition2.5. Fork ≥ ℓ, thevectorspace ofsimplicial harmonic polynomials isdefined
as
H R2m,C := P R2m,C ∩ker(∆ ,∆ ,h∂ ,∂ i,hx,∂ i).
k,ℓ k,ℓ x u u x u
Herethespace Pk(cid:0),ℓ R2m,(cid:1)C isthe(cid:0)space of(cid:1)C-valuedpolynomials depending ontwovector
variables (x,u) ∈ R2m, sometimes referred to as a matrix variable. The integers k and
(cid:0) (cid:1)
ℓ hereby refer to the degree of homogeneity in the variable x and u respectively. Recall
that h·,·i denotes the Euclidean inner product.
Note that this definition is notsymmetric with respect to x ↔ u which is reflected in the
dominant weight condition k ≥ ℓ. Also note that for ℓ = 0, the definition reduces to the
classical harmonicpolynomialsH (Rm,C),seee.g. [21]. Thevector spaceH R2m,C
k k,ℓ
is an irreducible Spin(m)-representation if m > 4. If ℓ = 0, this condition can even be
(cid:0) (cid:1)
relaxed to m ≥ 3. The regular action of the spin group on f ∈ H R2m,C is for
k,ℓ
all s ∈ Spin(m) given by H(s)[f](x,u) = f (s¯xs,s¯us), and the corresponding (derived)
(cid:0) (cid:1)
action of the orthogonal Lie algebra is given by:
dH(e )[f](x,u) = Lx +Lu f(x,u)
ij ij ij
:= x ∂ −x ∂ +u ∂ −u ∂ f(x,u).
(cid:0) i xj (cid:1)j xi i uj j ui
(cid:0) (cid:1)
Theoperators L arecalled theangular momentumoperators. Withoutproof(see [10]),
ij
we also mention that the highest weight is given by
λ = (k,ℓ,0,...,0) =: (k,ℓ),
4
where the length of this vector is equal to n for m ∈ {2n,2n+1}, and that the highest
weight vector is given by
w := (x −ix )k−ℓ (x −ix )(u −iu )−(x −ix )(u −iu ) ℓ .
k,ℓ 1 2 1 2 3 4 3 4 1 2
Wealsomentionthefollowingd(cid:0)efinition,underlyingtheclassicalHowedual(cid:1)pairSO(m)×
sl(2) for which we refer e.g. to [22, 24]:
Proposition 2.1. The Laplace operator, together with its symbol, span a Lie algebra:
1 1 m
sl(2) = Alg(X,Y,H) ∼= Alg − ∆ , |x|2,− E + .
x x
2 2 2
(cid:18) (cid:19)
(cid:16) (cid:17)
Here, E = x ∂ = r∂ (with r = |x| the norm of x ∈ Rm) denotes the Euler
x j j xj r
operator, measuring the degree of homogeneity in the variables (x ,...,x ).
1 m
P
In the last section, we will need the half-integer version:
Definition 2.6. For k ≥ ℓ, the vector space of simplicial monogenic polynomials is
defined as
S R2m,S := P R2m,S ∩ker(∂ ,∂ ,hx,∂ i).
k,ℓ k,ℓ x u u
Here the space Pk,ℓ R2(cid:0)m,S is(cid:1)the spac(cid:0)e of spi(cid:1)nor-valued polynomials depending on two
vector variables (x,u) ∈ R2m. The integers k and ℓ hereby refer to the degree of homo-
(cid:0) (cid:1)
geneity in the variable x and u respectively.
This definition is again not symmetric with respect to x ↔ u, which also here reflects
the dominant weight condition k ≥ ℓ. Also note that if ℓ = 0, the definition reduces to
the classical monogenic polynomials M (Rm,S), i.e. polynomial null solutions for ∂ .
k x
The vector space S R2m,S is an irreducible Spin(m)-representation if m is odd and
k,ℓ
splits into two inequivalent irreducible representations if m is even cfr. remark 2.1 (see
(cid:0) (cid:1)
also [10, 21]). The action of the spin group on simplicial monogenics is given by
L(s)[f](x,u) = sf(s¯xs,s¯us) ∀s∈ Spin(m) .
The corresponding action of the orthogonal Lie alg(cid:0)ebra is given b(cid:1)y:
1 1
dL(e )[f](x,u) = Lx +Lu − e f(x,u) = dH(e )− e [f](x,u).
ij ij ij 2 ij ij 2 ij
(cid:18) (cid:19) (cid:18) (cid:19)
Without proof, we also mention the highest weight
1 1 1 1
λ = k+ ,ℓ+ , ,..., =: (k,ℓ)′
2 2 2 2
(cid:18) (cid:19)
and the corresponding highest weight vector (see [10])
v := (x −ix )k−ℓ (x −ix )(u −iu )−(x −ix )(u −iu ) ℓI = w I ,
k,ℓ 1 2 1 2 3 4 3 4 1 2 k,ℓ
where I ∈ C is a primit(cid:0)ive idempotent realising the spinor space a(cid:1)s a left ideal (see
m
e.g. [11] for more details).
5
3 Construction of the higher spin Laplace operator
Itwas already mentioned that theLaplace operator (acting on C-valued fields)is related
to the Dirac operator (acting on spinor-valued fields). The operator which generalises
the Dirac operator to higher spin is the so-called Rarita-Schwinger operator (see [6, 29]
and also the last section), and in this section we will construct the generalisation of the
Laplace operator to the case of higher spin:
D :C∞(Rm,H )−→ C∞(Rm,H ).
k k k
Note that the space H (Rm,C), hereby plays the role of target space. This means that
k
elements of e.g. C∞(Rm,H ) are functions of the form f(x,u), where x ∈ Rm is the
k
variable on which the operator D is meant to act, satisfying f(x,u) ∈ H (Rm,C) for
k k
every x ∈ Rm fixed. Note that one might be tempted to think that D = ∆ , as
k x
the relation [∆ ,∆ ] = 0 clearly shows that ∆ is a rotationally invariant second-order
x u x
operator preservingthevaluesH ,buttheoperator∆ isnotconformally invariant with
k x
respect to the action of the inversion on H -valued functions (see below). This means
k
that we will have to add extra terms to ensure conformal invariance. For example, also
the second-order operator hu,∂ ih∂ ,∂ i is rotationally invariant and is well-defined on
x u x
H -valued functions, provided we can apply a projection onto the space C∞(Rm,H )
k k
because
hu,∂ ih∂ ,∂ i: C∞(Rm,H ) −→ C∞ Rm,H ⊕|u|2H ,
x u x k k k−2
(cid:16) (cid:17)
due to the classical Fischer decomposition, see e.g. [2] for more information. In other
words, for f(x,u) ∈ C∞(Rm,H ) we have:
k
hu,∂ ih∂ ,∂ if(x,u) = ϕ (x,u)+|u|2ϕ (x,u),
x u x k k−2
with ϕ ∈ C∞(Rm,H ). The projection of hu,∂ ih∂ ,∂ if onto C∞(Rm,H ) can be
i i x u x k
found as follows:
∆ hu,∂ ih∂ ,∂ if(x,u) = 2(2k+m−4)ϕ .
u x u u k−2
This implies that:
1 1
ϕ = ∆ hu,∂ ih∂ ,∂ if = h∂ ,∂ i2f.
k−2 u x u x u x
2(2k+m−4) 2k+m−4
We finally have that
|u|2
ϕ = hu,∂ i− h∂ ,∂ i h∂ ,∂ if.
k x u x u x
2k+m−4
!
We could try to define D as a linear combination of this operator with the Laplace
k
operator ∆ . Choosing the appropriate constant, this indeed works (see the appendix
x
for more details):
6
Theorem 3.1. The higher spin Laplace operator on Rm is the unique (up to a multi-
plicative constant) conformally invariant second-order differential operator
D :C∞(Rm,H )−→ C∞(Rm,H ),
k k k
defined by means of
4 |u|2
D f(x,u) = ∆ − hu,∂ i− h∂ ,∂ i h∂ ,∂ i f(x,u).
k x x u x u x
2k+m−2 2k+m−4
! !
Remark 3.1. For k = 1, we obtain the generalised Maxwell operator from [14], i.e.
4
D = ∆ − hu,∂ ih∂ ,∂ i :C∞(Rm,H )−→ C∞(Rm,H ).
1 x x u x 1 1
m
For the remainder of this section, we will give a sketch of the proof for the conformal
invariance of D , which justifies the presence of the constant − 4 in theorem 3.1.
k 2k+m−2
For a detailed proof, we again refer the reader to the appendix. In order to explain what
conformal invariance means, we need the following concept (see [13]):
Definition 3.1. An operator δ is a generalised symmetry for a differential operator D
1
if and only if there exists another operator δ so that Dδ = δ D. Note that for δ = δ ,
2 1 2 1 2
this reduces to the definition of a (proper) symmetry, in the sense that [D,δ ] = 0.
1
One then typically tries to describe the first-order generalised symmetries, as these span
a Lie algebra, see e.g. [27]. In this particular case, the first order symmetries will
span a Lie algebra isomorphic to the conformal Lie algebra so(1,m + 1). The higher
spin Laplace operator is clearly so(m)-invariant because it is the composition of so(m)-
invariant operators. This means that the angular momentum operators are symmetries
of the higher spin Laplace operator, in the sense that [D ,Lx + Lu] = 0. It is also
k ij ij
easy to see that the Euler operator and the infinitesimal translations are (generalised)
symmetries of D , in view of thefact thatD E = (E +2)D and [D ,∂ ]= 0. Finally,
k k x x k k xj
there is also a special class of generalised symmetries for the operator D which can be
k
definedin terms of theharmonicinversion for H -valued functions. Thisis an involution
k
mapping solutions for D to solutions for D , see the crucial relation (2) below.
k k
Definition 3.2. The harmonic inversion is a conformal transformation defined as
J : C∞(Rm,H ) −→ C∞(Rm,H )
R k 0 k
x xux
f(x,u)7→ J [f](x,u) := |x|2−mf , .
R |x|2 |x|2
(cid:18) (cid:19)
Note that this inversionconsists of the classical KelvininversionJ on Rm inthe variable
x composed with a reflection u7→ ωuω acting on the dummy variable u (where x = |x|ω),
and satisfies J2 = 1.
R
7
The special conformal transformations are then for all 1 ≤ j ≤ m defined as J ∂ J ,
R xj R
with J the harmonic inversion from above. More explicitly, we have:
R
J ∂ J = 2hu,xi∂ −2u hx,∂ i+|x|2∂ −x (2E +m−2), (1)
R xj R uj j u xj j x
whichfollows fromcalculations onan arbitraryH -valued functionf(x,u), seeappendix
k
proposition A.1. This special conformal transformation and ∂ , the generator of trans-
xj
lations in the e -direction, then define a model for a Lie subalgebra which is isomorphic
j
to sl(2):
Proposition 3.1. One has
sl(2) ∼= Alg J ∂ J ,∂ ,2E +m−2 .
R xj R xj x
(cid:0) (cid:1)
Proof. The classical commutator relations are verified after some straightforward com-
putations.
Proposition 3.2. The special conformal transformations J ∂ J from equation (1),
R xj R
with 1≤ j ≤ m, are generalised symmetries of the higher spin Laplace operator.
Proof. Thisresultcanbeprovedbycalculatingthecommutator[D ,J ∂ J ],forwhich
k R xj R
we refer to proposition A.2 in the appendix. In particular, it was shown that
J D J = |x|4D , (2)
R k R k
whichgeneralisestheclassicalresultJ∆ J = |x|4∆ fortheLaplaceoperator∆ acting
x x x
on C-valued functions (with J the classical Kelvin inversion), see e.g. [2].
The conformal invariance can be summarized in the following theorem:
Theorem 3.2. The first-order generalised symmetries of the higher spin Laplace oper-
ator D are given by:
k
(i) The infinitesimal rotations Lx +Lu, with 1 ≤ i< j ≤ m.
ij ij
(ii) The shifted Euler operator (2E +m−2).
x
(iii) The infinitesimal translations ∂ , with 1 ≤ j ≤ m.
xj
(iv) The special conformal transformations J ∂ J , with 1 ≤ j ≤m.
R xj R
These operators span a Lie algebra which is isomorphic to the conformal Lie algebra
so(1,m+1), whereby the Lie bracket is the ordinary commutator.
Besides conformal invariance, we also have another crucial property of the higher spin
Laplace operator. For more details, we refer to theorem A.1 in the appendix. We here
only mention the main conclusion:
8
Theorem 3.3. The higher spin Laplace operator D is an elliptic operator if m > 4,
k
which implies surjectivity of the map
D :C∞(Rm,H )−→ C∞(Rm,H ).
k k k
To conclude this section, we will introduce two other conformally invariant operators
which are called (dual) twistor operators. These operators are used in the next section
to describe the structure of the space of polynomial null solutions of the higher spin
Laplace operator. The twistor operators are defined as follows:
Definition 3.3. The twistor operator is the unique (up to a multiplicative constant)
conformally invariant operator defined as
π hu,∂ i :C∞(Rm,H ) −→ C∞(Rm,H ),
k x k−1 k
where π is a projection on ker∆ . The dual twistor operator is the unique (up to a
k u
multiplicative constant) conformally invariant operator given by
h∂ ,∂ i: C∞(Rm,H ) −→ C∞(Rm,H ).
u x k k−1
Since h∂ ,∂ if ∈ C∞(Rm,H ) for a function f ∈ C∞(Rm,H ), the calculation at the
u x k−1 k
beginning of this section shows that the twistor operator is explicitly given by
|u|2
π hu,∂ i := hu,∂ i− h∂ ,∂ i.
k x x u x
2k+m−4
This means that the higher spin Laplace operator is in fact a combination of theLaplace
operator and the composition of the twistor operator with the dual twistor operator, i.e.
4
D = ∆ − π hu,∂ ih∂ ,∂ i.
k x k x u x
2k+m−2
4 Polynomial null solutions
In this section we study ℓ-homogeneous polynomial solutions for D , i.e. polynomials
k
f(x,u) in two vector variables satisfying D f(x,u) = 0 and E f(x,u) = ℓf(x,u). The
k x
vector space of null solutions will from now on be denoted by ker D and this space is
ℓ k
not irreducible as a module for so(m), in contrast to the kernel of the Laplace operator.
Note that ker D does define an irreducible module for the real form so(1,m+1),
ℓ ℓ k
but this is beyond the scope of the present paper.
L
First, we use theorem 3.3, to compute the dimension of ker D as follows (ℓ ≥ k):
ℓ k
dim(ker D )= dim(P (Rm,H ))−dim(P (Rm,H )) .
ℓ k ℓ k ℓ−2 k
It thus follows for all ℓ ∈N\{0,1} that
m+ℓ−1 m+ℓ−3
dim(ker D ) = − dim(H ) .
ℓ k m−1 m−1 k
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
9
As this number coincides with dim(H ⊗H ), this suggests that the space ker D can
ℓ k ℓ k
be decomposed as follows for arbitrary ℓ ≥ k (hereby referring to [25] for the tensor
product decomposition rules):
k k−i
ker D ∼= H . (3)
ℓ k ℓ−i+j,k−i−j
i=0 j=0
MM
A straightforward subspace of ker D , is the space of harmonics in x and u intersected
ℓ k
with the kernel of the dualtwistor operator h∂ ,∂ i. This space is known as the space of
u x
Howe harmonics, see [22, 24], and it allows for the following decomposition into so(m)-
irreducible summands (where we use our explicit model involving simplicial harmonics):
k
A R2m,C := P R2m,C ∩ker(∆ ,∆ ,h∂ ,∂ i) = hu,∂ ijH R2m,C
ℓ,k ℓ,k u x u x x ℓ+j,k−j
j=0
(cid:0) (cid:1) (cid:0) (cid:1) M (cid:0) (cid:1)
It has the same structure as the space of double monogenics, and can thus be seen as
the analogue of the type A solutions for the Rarita-Schwinger operator discussed in [6].
As these solutions are also killed by the operator h∂ ,∂ i, which then implies that the
u x
operator D reduces to the ordinary Laplace operator ∆ , they can also be described
k x
as the solutions for D in the Lorentz gauge, in analogy with what is usually done in
k
physics.
We can prove by induction that decomposition (3) is in fact the correct one. To do so,
we need the following lemma:
Lemma 4.1. The following operator identity holds:
2k+m−6
h∂ ,∂ iD = D h∂ ,∂ i.
u x k k−1 u x
2k+m−2
In particular, this means that the dual twistor operator h∂ ,∂ i maps solutions of D to
u x k
solutions of D .
k−1
Proof. This can be proved by direct computations.
As a consequence of this lemma, the irreducible components of ker D also appear
ℓ−1 k−1
in decomposition (3). In particular, we obtain the following result:
Lemma 4.2. The vector space ker D has the following decomposition:
ℓ k
ker D ∼= A ⊕ker D .
ℓ k ℓ,k ℓ−1 k−1
Proof. Since A ⊂ kerh∂ ,∂ i, we have that A ∩ker D = {0}. The result then
ℓ,k u x ℓ,k ℓ−1 k−1
follows from the previous lemma and some basic linear algebra.
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