THE CAUCHY KERNELS OF ULTRAHYPERBOLIC CLIFFORD ANALYSIS

9

^th

International Conference on Clifford Algebras and their Applications in Mathematical Physics K. G¨urlebeck (ed.) Weimar, Germany, 15–20 July 2011

THE CAUCHY KERNELS OF ULTRAHYPERBOLIC CLIFFORD ANALYSIS

Ghislain R. Franssens

Belgian Institute for Space Aeronomy, Ringlaan 3, B-1080 Brussels, Belgium, E-mail: ghislain.franssens@aeronomy.be

Keywords: Cauchy Kernel, Ultrahyperbolic Clifford Analysis, Distribution.

Abstract. In this paper we consider the Cauchy kernels of Ultrahyperbolic Clifford Analysis

(UCA) for arbitrary signature (p, q). In contrast to Elliptic Clifford Analysis (ECA), the Cauchy

kernels of UCA are non-trivial distributions. We comment on a number of distributional tech-

nicalities, which arise in the derivation of the Cauchy kernels of UCA. We pay attention to a

necessary regularization process of the Cauchy kernel distributions and resolve the matter of

uniqueness of this process. In particular, we give a more detailed treatment of the subset of

signatures for which the Cauchy kernels satisfy Huygens’ principle. For the Huygens cases

with proper ultrahyperbolic signature, explicit functional value expressions are presented for

the occurring distributions.

1 INTRODUCTION

Let Ω be a domain in R ⁿ and P the canonical quadratic form of signature (p, q),

P (x) = | x t | ² − | x s | ² , (1)

=

∑ p i=1

(x _i ) ² −

∑ q i=1

(x _p+i ) ² , (2)

and Cl p,q the universal Clifford algebra generated by the inner product space R ^p,q , (R ⁿ , P ), n , p + q. Clifford Analysis over R ^p,q , called Ultrahyperbolic Clifford Analysis (UCA), is the study of a subset of functions taking values in Cl _p,q together with a Cl _p,q -valued first order vector differential Dirac operator ∂. In particular, we single out a subset of functions F ∈ C ^∞ (Ω, Cl _p,q ) satisfying a generalized Cauchy-Riemann equation of the form

∂F = − S, (3)

for given smooth, compact support S ∈ C _c ^∞ (R ⁿ , Cl p,q ) (possibly together with a boundary condition for F at infinity and integrability conditions on S). Functions satisfying ∂F = 0 (F ∂ = 0) are called left (right) Cl _p,q -holomorphic.

UCA is a non-trivial extension of the more familiar Elliptic Clifford Analysis (ECA) (p = 0 or q = 0), [1], [2], [7], and also includes Hyperbolic Clifford Analysis (HCA) (p = 1 or q = 1) as special cases. In the former case, the mathematics greatly simplifies, because ECA can be founded on ordinary function theory. HCA an proper UCA on the other hand, as we will see, requires a technically much more demanding approach based on distribution theory.

UCA is the proper mathematical setting for studying generalized physics in (flat pseudo- Riemannian) spaces with an arbitrary number p of time dimensions and an arbitrary number q of space dimensions. HCA, corresponding to p = 1 and q > 1, has direct relevance to physics, with in particular the case p = 1, q = 3 providing a function theory over Minkowski space that is directly applicable to electromagnetism and quantum physics, [8], [9].

2 CAUCHY KERNELS OF UCA

In order to exhibit the potential of HCA (and UCA), to serve as a mathematical tool to solve physics (and generalized physics) problems in our universe (and other conceivable universes with an arbitrary number of dimensions), the ultimate goal is to derive the solution of Boundary Value Problems and Riemann-Hilbert Problems in these function theories. Essential for this is the construction of integral representation theorems in HCA (and UCA), similar to the funda- mental theorem in ECA, e.g., [1, p. 52, Proposition 9.2], [4], and theorems derived from it. Such representation theorems are typically obtained from an integrated reciprocity relation holding for two functions, one being the Cauchy kernel C _x

₀

of R ^p,q . Important for the development of UCA is thus the determination of all its Cauchy kernels, ∀ p, q ∈ Z + , and the derivation of their properties.

Let x ₀ ∈ R ⁿ be a parameter point, playing the role of calculation point, D , C _c ^∞ (R ⁿ , R ) the set of smooth real-valued functions with compact support in R ⁿ and D ^′ the continuous dual of D , i.e., the linear space of distributions with support in R ⁿ . Denote by Cl ^k _p,q the grade k ∈ Z [0,n]

linear subspace of Cl _p,q . By definition, the Cl _p,q -valued vector distribution C _x

₀

∈ D ^′ ⊗ Cl ¹ _p,q satisfying

∂C _x

₀

= δ _x

₀

= C _x

₀

∂, (4)

with δ _x

₀

the delta distribution with support { x ₀ } , is called the Cauchy kernel relative to x ₀ . We do not need to specify any boundary conditions for C _x

₀

, any fundamental solution of (4) will do. The significance of C x

0

, as defined by (4), lies in the fact that the (left and right) convolution operator C _x

₀

∗ is a (left and right) Cl _p,q -valued and functional inverse of the Dirac operator ∂.

In those Clifford Analyses where associativity holds, this can be used to readily solve (3) by applying this operator to the left of it. Associativity holds of course in any Clifford algebra, but not in every UCA, due to the fact that triple convolution products are in general not associative.

Only if p = 1 or q = 1 and we are looking for causal solutions can we invoke the associativity of the occurring triple convolution products. Nevertheless, also in UCA with signatures (p, q) where triple convolution products are not associative, is C _x

₀

useful to solve (3), but then requires a more general method.

Since δ x

0

is scalar, the grade 2 component of (4) is ∂ ∧ C x

0

= 0. From Poincar´e’s lemma follows that there exists a scalar distribution g _x

₀

∈ D ^′ ⊗ Cl _p,q ⁰ such that any Cauchy kernel in UCA can be obtained as

C x

0

= ∂g x

0

. (5)

Substituting (5) in (4), and invoking associativity which holds since ∂ is a convolution operator with compact support distributional kernel, shows that g _x

₀

must be a fundamental distribution of the ultrahyperbolic equation

p,q g _x

₀

= δ _x

₀

, (6) since ∂ ² = p,q , the canonical d’Alembertian of signature (p, q).

Let P _x

₀

, P (x − x ₀ ) and A _{n − 1} , _Γ(n/2) ^2π

^n/2

. A (real) fundamental solution g _x

₀

of (6) is, [6, p.

280], [10, p. 137], (i) for n > 2,

g _x

₀

= − 1 (n − 2) A _{n − 1}

1 2

( e ^iq

^π2

(P _x

₀

+ i0) ^{1 −}

ⁿ²

+ e ^{− iq}

^π2

(P _x

₀

− i0) ^{1 −}

ⁿ²

)

, (7)

(ii) for n = 2,

g _x

₀

= 1 4π

1 2

( e ^iq

^π2

ln(P _x

₀

+ i0) + e ^{− iq}

^π2

ln(P _x

₀

− i0) )

, (8)

= 1 4

(

cos(qπ/2) 1

π ln | P | − sin(qπ/2)1 ₋ (P ) )

. (9)

Any Cauchy kernel C _x

₀

then follows formally from (5) as (i) for n > 2,

C x

0

= (x − x ₀ ) ^S A _{n − 1}

1 2

( e ^iq

^π2

(P x

0

+ i0) ⁻

ⁿ²

+ e ^{− iq}

^π2

(P x

0

− i0) ⁻

ⁿ²

)

, (10)

(ii) for n = 2,

C _x

₀

= (x − x ₀ ) ^S 1 2

(

cos(qπ/2) η ( ^P

^x0

) + sin( qπ/2)δ _P

_x

0

=0

)

. (11)

The superscript S in (10)–(11) denotes spatial conjugation, i.e., S : R ^p × R ^q → R ^p × R ^q such that x = (x _t , x _s ) 7→ x ^S = (x _t , − x _s ). In (11), η _(x) , ¹ _π x ^{− 1} is a normalized Cauchy’s principal value Pv _x ¹ and δ _P

_x

0

=0 is the delta distribution with support P _x

₀

= 0.

A justified calculation of C _x

₀

however, involves a number of distributional technicalities. The

most important one is related to the singularities of the distributions (P _x

₀

± i0) ^z at z = − n/2,

requiring a “regularization”.

3 REGULARIZATION OF CAUCHY KERNELS

The complex functions f _± (z) , ⟨ (P _x

₀

± i0) ^z , φ ⟩ , ∀ φ ∈ D , are complex analytic in z, except at the points z = − n/2 − k, ∀ k ∈ N , where they have simple poles. Indeed, the distributions (P _x

₀

± i0) ^z have the following Laurent series about the points z = − n/2 − k, holding in 0 < | z + n/2 + k | < 1, [6, p. 274–276],

(P _x

₀

± i0) ^z = c ^n/2+k _{− 1, ±}

z + n/2 + k + (P _x

₀

± i0) ⁻ ₀ ^(n/2+k) + O (z + n/2 + k) , (12) c ^n/2+k _{− 1, ±} = e ^{∓ iqπ/2} π ^n/2

4 ^k k!Γ(k + n/2) ^k p,q δ x

0

. (13)

From (7) and (5) follows that the Cauchy kernel, for n > 2, will contain the expressions (P _x

₀

± i0) ^{− n/2} , which (12) shows are non-existing distributions!

Remark however that (P _x

₀

± i0) ^z , as given by (12)–(13), are still defined at z = − n/2 on test functions vanishing in a neighborhood of x 0 . The quantities (P x

0

± i0) ^{− n/2} thus exist as partial distributions, i.e., as sequentially continuous linear functionals defined on a proper subspace D p

of D . As guaranteed by the Hahn-Banach theorem, any partial distribution f _p can be extended to a distribution f ε ∈ D ^′ such that ⟨ f ε , ψ ⟩ = ⟨ f p , ψ ⟩ , ∀ ψ ∈ D p (R ⁿ ). The distribution f ε is called an extension of f _p from D p to D . An extension is in general non-unique. Two extensions differ by a distribution that maps D p to zero.

This suggests that we define regularization as a functional extension process. Our chosen extension procedure uses a projection operator Π : D → D p , to replace an integral, convergent

∀ ψ ∈ D p but divergent for test functions φ ∈ D\D p , (e.g., in one dimension)

∫ + ∞

−∞

f (x) φ (x) dx, (14)

by the integral ∫ + ∞

−∞

f (x) (Πφ) (x) dx, (15)

which now by construction is convergent ∀ φ ∈ D .

The integral (15) is called a regularization of the integral (14). Equivalently, we could say that (15) defines a distribution which is an extension of the partial distribution defined by (14).

It is important to notice that this method is by no means unique, since infinitely many projection operators Π can be constructed, mapping D to D p .

A regularization of (P _x

₀

± i0) ^{− n/2} is then defined as any extension (P _x

₀

± i0) ⁻ _ε ^n/2 ∈ D ^′ . There is no natural reason to select out one particular extension, so regularization as a functional extension process is not canonical. The non-uniqueness of the extension process however can be reduced by demanding that regularization preserves homogeneity. We denote any homogeneity preserving extension with a subscript e. In our case,

(P _x

₀

± i0) ⁻ _e ^n/2 = (P _x

₀

± i0) ⁻ ₀ ^n/2 + c _± δ _x

₀

, (16) wherein (P _x

₀

± i0) ⁻ ₀ ^n/2 is the analytic finite part from (12) and c _± ∈ R is arbitrary.

We now define the Cauchy kernel C _x

₀

, for n > 2, as the distribution C x

0

, (x − x ₀ ) ^S

A _{n − 1} 1 2

(

e ^iq

^π2

(P x

0

+ i0) ⁻

n

e

2

+ e ^{− iq}

^π2

(P x

0

− i0) ⁻

n

e

2

)

. (17)

The arbitrariness in (16) then raises the question of the uniqueness of the Cauchy kernel (mod- ulo pseudo-harmonic functions). Fortunately, due to the identity (x − x ₀ ) ^S δ _x

₀

= 0 is (17) independent of the arbitrary constants c _± in (16).

The particular solution F of (3), caused by a given source function S, is then obtained as F (x ₀ ) = ⟨ C _x

₀

, S ⟩ _S , (18) wherein ⟨ , ⟩ _S denotes Schwartz pairing.

4 CAUCHY KERNELS SATISFYING HUYGENS’ PRINCIPLE

The complexity of UCA is due to the fact that g x

0

(and hence C x

0

) is a rather complicated distribution, whose form also profoundly depends on the parity of p and q. We define the Huygens cases as the subset of signatures

H , { (p, q) ∈ Z o,+ × Z o,+ : (p, q) ̸ = (1, 1) } . (19) Iff (p, q) ∈ H, then g _x

₀

has as support the null space of R ^p,q relative to x ₀ , and g _x

₀

is then said to satisfy Huygens’ principle. Hence from now on is 4 ≤ n. A (relative) simplification arises for the Cauchy kernels in the Huygens cases, which reflects the physical fact that in a universe with an odd number of time dimensions (p) and an odd number of space dimensions (q), distortion free communication is possible. This means that an emitted sharp pulse remains sharp under propagation. Equivalently stated, these are universes where “light” only exists on the null-space, relative to its source point. This fact is reflected in the mathematics by the occurrence of a multiplet delta distribution with support the null space of the observation point x ₀ (by reciprocity).

4.1 Distributions

In the ultrahyperbolic case, i.e., if (p, q) ∈ H : p > 1 and q > 1, g _x

₀

becomes g _x

₀

= δ ⁽⁽ⁿ _P

_x

^{− 2)/2 − 1)}

0

=0

( − 1) ^{(n − 2)/2} ((p − 2) /2) _{((n − 2)/2)} + ((q − 2) /2) _{((n − 2)/2)} , (20) with δ ⁽⁽ⁿ _P ^{− 2)/2 − 1)}

x0

=0 a multiplet delta distribution having as support the null space of R ^p,q relative to x ₀ , [3], and in the denominator the falling factorial polynomials

((q − 2) /2) _{((n − 2)/2)} (21)

= 2 ^{− (n − 2)/2} (q − 2) (q − 4) ... (q − (n − 4)) (q − (n − 2)) , (22)

= ( − 1) ^{(n − 2)/2} 2 ^{− (n − 2)/2} (p − (n − 2)) (p − (n − 4)) ... (p − 4) (p − 2) , (23)

= ( − 1) ^{(n − 2)/2} ((p − 2) /2) _{((n − 2)/2)} . (24)

Result (20) can be derived, either from (7), by taking into account the special form to which the distributions (P (x − x ₀ ) ± i0) ^z reduce if (p, q) ∈ H, or directly by showing that (20), using the explicit expression for

⟨

δ _P ^{((n − 2)/2 − 1)}

x0

=0 , φ

⟩

as given in [3], is a fundamental solution of the

generalized d’Alembertian.

The multiplet delta distribution δ _P ^(k)

x0

=0 can be expressed as the pullback, [5, p. 80], [10, p.

133], of the one-dimensional k-th derivative delta distribution δ ^(k) along P _x

₀

, i.e., δ ^(k) _P

x0

=0 = P _x ^∗

₀

δ ₀ ^(k) , [3]. However, the definition of pullback requires that the function P _x

₀

is a submer- sion, which in turn requires that we exclude the point x 0 from its domain. Hence, δ ^(k) _P

_x

0

=0 , as a pullback, is only defined on test functions that are vanishing in a neighborhood of x ₀ . We nevertheless want to define δ _P ^(k)

_x

0

=0 for all test functions in D , because we eventually also want to calculate F in the region supp S where F is not holomorphic. In physical terms: we want to be able to calculate the field F , generated by a given extended source function S, also in the source region supp S itself. Fortunately, for k < (n − 2) /2 the defining integrals for δ _P ^(k)

_x

0

=0

happen to exist for all test functions, [3]. Hence, g _x

₀

given by (20) is a distribution.

But for (n − 2) /2 ≤ k, the defining integrals for δ ^(k) _P

x0

=0 do not exist for test functions that are not vanishing in a neighborhood of x ₀ . Hence, the Cauchy kernel, which will involve a multiplet delta of the form δ _P ^{((n − 2)/2)}

x0

=0 , does not exist as a distribution. Fortunately, the quantities δ ^(k) _P

x0

=0 , for (n − 2) /2 ≤ k and n even, exist as partial distributions, defined on test functions vanishing in a neighborhood of x ₀ . We can again give them a meaning as a distribution by defining them as any extension.

It turns out however that there exists two natural sets of (homogeneity preserving) extensions of the partial distribution δ _P ^(k)

x0

=0 , which we denote by (

δ _P ^(k)

x0

=0

)

t,e

or (

δ _P ^(k)

x0

=0

)

s,e

. For the details how this arises, see [3]. For k = (n − 2) /2, these have the form

(

δ _P ^{((n − 2)/2)}

x0

=0

)

t,e

= (

δ ⁽⁽ⁿ _P ^{− 2)/2)}

x0

=0

)

0

+ c _t δ _x

₀

, (25)

(

δ _P ⁽⁽ⁿ

_x

^{− 2)/2)}

0

=0

)

s,e = (

δ ⁽⁽ⁿ _P

_x

^{− 2)/2)}

0

=0

)

0 + c s δ x

0

, (26)

with c _t , c _s ∈ R arbitrary. We are fortunate again that both sets coincide in our case k = (n − 2) /2 (this is no longer true for k > (n − 2) /2).

To calculate the Cauchy kernels in the Huygens cases we could use that δ _P ^(k)

x0

=0 = P _x ^∗

₀

δ ₀ ^(k) and apply the generalized chain rule. However, this rule requires that P x

0

is a submersion, which it is not at x ₀ . Since we do not want to exclude x ₀ from the domain of our test functions, we calculate ∂δ _P ^{((n − 2)/2 − 1)}

x0

=0 by applying the definition of generalized derivation. Then, the action of the Dirac operator ∂ on g x

0

yields (after extensive calculations) the two Cauchy kernels

C _t,x

₀

=

(x − x ₀ ) ^S (

δ _P ^{((n − 2)/2)}

x0

=0

)

t,e

((q − 2) /2) _{((n − 2)/2)} , (27)

C s,x

0

=

(x − x ₀ ) ^S (

δ _P ⁽⁽ⁿ

_x

^{− 2)/2)}

0

=0

)

s,e

((q − 2) /2) _{((n − 2)/2)} . (28)

Due to the presence of the factor (x − x ₀ ) ^S and since (x − x ₀ ) ^S δ _x

₀

= 0 we reproduce that the Cauchy kernel C _x

₀

is unique also in the Huygens case. Hence,

C _x

₀

=

(x − x 0 ) ^S (

δ ⁽⁽ⁿ _P

_x

^{− 2)/2)}

0

=0

)

0

((q − 2) /2) _{((n − 2)/2)} . (29)

4.2 Explicit expressions

For applications it is necessary that we have an explicit expression stating how any scalar distribution g x

0

and any Cauchy distribution C x

0

act on test functions, such as the source func- tion S in (3). As example of the complexities involved, we give the following expressions for the functional values, valid for the ultrahyperbolic Huygens cases (p, q) ∈ H : p > 1 and q > 1.

A. The multiplet delta distribution δ ^(k) _P

x0

=0 is given, ∀ k ∈ N and ∀ φ ∈ D , (i) for k < (n − 2) /2, by the equivalent expressions

⟨ δ _P ^(k)

_x

0

=0 , φ

⟩

= ( − 1) ^k

∫ + ∞ 0

ρ ^{(n−2−2k)−1} _t Ψ ^k _t,+ (ρ t ; q − 2) dρ t , (30)

= ( − 1) ^k

∫ + ∞ 0

ρ ⁽ⁿ _s ^{− 2 − 2k) − 1} Ψ ^k _s,+ (ρ s ; p − 2) dρ s ; (31) (ii) for (n − 2) /2 ≤ k and n even, by the in general non-equivalent extensions

⟨(

δ _P ^(k)

_x

0

=0

)

t,e , φ

⟩

= ( − 1) ^k

∫ + ∞ 0

ρ ⁻ _t ^{(2k − (n − 2)) − 1} (

Π _{2k − (n − 2)} Ψ ^k _t,+ )

(ρ _t ; q − 2) dρ _t (32) +c _t ( − 1) ^k lim

ρ

t

→ 0 d ^2k _ρ

_t

^{− (n − 2)} Ψ ^k _t,+ (ρ _t ; q − 2) , (33)

⟨(

δ _P ^(k)

x0

=0

)

s,e

, φ

⟩

= ( − 1) ^k

∫ _{+ ∞}

0

ρ ⁻ _s ^{(2k − (n − 2)) − 1} (

Π _{2k − (n − 2)} Ψ ^k _s,+ )

(ρ _s ; p − 2) dρ _s (34) +c s ( − 1) ^k lim

ρ

s

→ 0 d ^2k _ρ

_s

^{− (n − 2)} Ψ ^k _s,+ (ρ s ; p − 2) , (35) with c _t , c _s ∈ R arbitrary. The distribution

(

δ _P ^{((n − 2)/2)}

x0

=0

)

0

in (29) is given by either (33) or (35) with c _t = 0 = c _s .

B. The Cauchy kernel is given by, ∀ φ ∈ D ,

⟨ C x

0

, φ ⟩ = 1/2

(q − 2) (q − 4) ... (q − (n − 2))

(n ∑ − 2)/2 r=0

∫ + ∞ 0

ρ ^r (36)



 b

n−2

r

2

(q − 2) ( ψ ^S,S [

ω _t (

φ ◦ T _x ⁻¹

₀

)])

0,r (ρ, ρ)

− ( − 1)

ⁿ⁻²²

b

n−2

r

2

(p − 2) ( ψ ^S,S [

ω _s (

φ ◦ T _x ⁻¹

₀

)])

r,0 (ρ, ρ)



 dρ. (37)

The notation () _r,s denotes partial derivatives with respect to the first and second argument of orders r, s respectively. In (30)–(37) is further

Ψ ^k _{t, ±} (t; q − 2) , ( − 1) ^k 2

1 2 ^k

∑ k r=0

b ^k _r (q − 2) ( ± t) ^r ( ψ ^S,S [

φ ◦ T _x ^{− 1}

0

])

0,r (t, ± t) , (38) Ψ ^k _{s, ±} (s; p − 2) , 1

2 1 2 ^k

∑ k r=0

b ^k _r (p − 2) ( ± s) ^r ( ψ ^S,S [

φ ◦ T _x ^{− 1}

0

])

r,0 ( ± s, s) , (39) with b ^k _r (m) known polynomials in m ([3, Lemma 19]), T x

0

is translation over x 0 = (x t,0 , x s,0 ) and (

ψ ^S,S [

φ ◦ T _x ⁻

₀

¹ ])

(t, s) , ⟨

S _t ^{p − 1} × S _s ^{q − 1} , φ(x t,0 + tω t , x s,0 + sω s ) (

ω _S

^p−1

t

∧ ω _S

^q−1 s

)⟩

dR ,

(40)

is the bi-spherical mean function of the test function φ, with ⟨ , ⟩ _dR in (40) denoting integration (de Rham pairing), S _t ^{p − 1} (S _s ^{q − 1} ) the temporal p − 1 (q − 1) dimensional (spatial) unit sphere and ω t = x t / | x t | , ω s = x s / | x s | for x = (x t , x s ).

The projection operator Π _m used in the functional extension process in (33)–(35) is defined by

(Π _m [φ]) (x) , φ(x) −

∑ m l=0

φ ^(l) (0) x ^l l!

( 1 _l<m + 1 _m≤l 1 ₊ (1 − x ² ) )

, (41)

wherein 1 _p , 1 if p is true and 1 _p , 0 if p is false.

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