Generalised Maxwell Equations in Higher Dimensions

Abstract

This paper deals with the generalisation of the classical Maxwell equations to arbitrary dimension $m$ and their connections with the Rarita–Schwinger equation. This is done using the framework of Clifford analysis, a multivariate function theory in which arbitrary irreducible representations for the spin group can be realised in terms of polynomials satisfying a system of differential equations. This allows the construction of generalised wave equations in terms of the unique conformally invariant second-order operator acting on harmonic-valued functions. We prove the ellipticity of this operator and use this to investigate the kernel, focusing on both polynomial solutions and the fundamental solution.

Appendix A

In this appendix, we first consider the conformal invariance in more detail. The following technical result was already mentioned in expression (2):

Proposition 7.1

The following property holds for all $1 \le j \le m$:

$$\begin{aligned} {\mathcal {J}}_R\partial _{x_j}{\mathcal {J}}_R = 2\langle \underline{u},\underline{x}\rangle \partial _{u_j}-2u_j\langle \underline{x},\underline{\partial }_{u}\rangle +\left| \underline{x} \right| ^2\partial _{x_j}-x_j\left( 2{\mathbb {E}}_{x}+m-2 \right) \end{aligned}$$

Proof

Suppose $P_q(\underline{x};\underline{u})\in \mathcal {P}_{q}\left( {\mathbb {R}}^{m},{\mathcal {H}}_1 \right)$, i.e. ${\mathbb {E}}_{x}P_q=qP_q$. The action of ${\mathcal {J}}_R$ on $P_q$ is given by:

$$\begin{aligned} {\mathcal {J}}_R P_q=\left| \underline{x} \right| ^{2-m}P_q\left( \frac{\underline{x}}{\left| \underline{x} \right| ^2},\frac{\underline{x}\;\underline{u}\;\underline{x}}{\left| \underline{x} \right| ^2} \right) :=\left| \underline{x} \right| ^{2-m-2q}P_q\left( \underline{x};v(\underline{x}) \right) \!, \end{aligned}$$

where $v(\underline{x})$ is defined through the second equality. We thus get:

$$\begin{aligned} \begin{aligned} \partial _{x_j}{\mathcal {J}}_R P_q=&\left| \underline{x} \right| ^{2-m-2q}\left( \dot{\partial _{x_j}}P_q\left( \dot{\underline{x}},v(\underline{x}) \right) +\dot{\partial _{x_j}}P_q\left( \underline{x},\dot{v}(\underline{x}) \right) \right) \\&+ (2-m-2q)x_j\left| \underline{x} \right| ^{-m-2q}P_q \end{aligned} \end{aligned}$$

The first and the last term are the same as for the ${\mathbb {C}}$-valued case (using the inversion ${\mathcal {J}}$), which leads to the generalised symmetry

$$\begin{aligned} {\mathcal {J}}\partial _{x_j} {\mathcal {J}}= |\underline{x}|^2\partial _{x_j} - x_j(2{\mathbb {E}}_x+ m - 2). \end{aligned}$$

The remaining term gives:

$$\begin{aligned}&{\mathcal {J}}_R\left( \left| \underline{x} \right| ^{2-m-2q} \dot{\partial _{x_j}}P_q\left( \underline{x},\dot{v}(\underline{x}) \right) \right) \\&={\mathcal {J}}_R\left( \left| \underline{x} \right| ^{2-m-2q} \sum _{i=1}^m\frac{\partial v_i}{\partial _{x_j}} \dot{\partial _{v_i}}P_q\left( \underline{x},\dot{v}(\underline{x}) \right) \right) \\&={\mathcal {J}}_R\left( \left| \underline{x} \right| ^{2-m-2q}\sum _{i=1}^m\frac{\partial }{\partial _{x_j}}\left( u_i-2\frac{\langle \underline{x},\underline{u}\rangle }{\left| \underline{x} \right| ^2}x_i \right) \dot{\partial _{v_i}}P_q\left( \underline{x},\dot{v}(\underline{x}) \right) \right) \\&=-2{\mathcal {J}}_R\left( \left| \underline{x} \right| ^{2-m-2q}\sum _{i=1}^m\left( \frac{u_j x_i}{\left| \underline{x} \right| ^2} +\frac{\langle \underline{x},\underline{u}\rangle \delta _{ij}}{\left| \underline{x} \right| ^2}- 2\frac{\langle \underline{x},\underline{u}\rangle x_ix_j}{\left| \underline{x} \right| ^4} \right) \dot{\partial _{v_i}}P_q\left( \underline{x},\dot{v}(\underline{x}) \right) \right) \\&=2\left( -\left( u_j-2\frac{\langle \underline{x},\underline{u}\rangle x_j}{\left| \underline{x} \right| ^2} \right) \langle \underline{x},\underline{\partial }_{u}\rangle +\langle \underline{x},\underline{u}\rangle \partial _{u_j}-2\frac{\langle \underline{x},\underline{u}\rangle \langle \underline{x},\underline{\partial }_{u}\rangle }{\left| \underline{x} \right| ^2}\right) \end{aligned}$$

Simplifying the last term completes the proof. $\square$

We then arrive at the main proposition, stating that the special conformal transformations are generalised symmetries of the Maxwell operator:

Proposition 7.2

The special conformal transformations

$$\begin{aligned} {\mathcal {J}}_R\partial _{x_j}{\mathcal {J}}_R := 2\langle \underline{u},\underline{x}\rangle \partial _{u_j}-2u_j\langle \underline{x},\underline{\partial }_{u}\rangle +\left| \underline{x} \right| ^2\partial _{x_j}-x_j\left( 2{\mathbb {E}}_{x}+m-2 \right) , \end{aligned}$$

with $j\in \left\{ 1,\ldots ,m \right\}$ are generalised symmetries of the higher spin Laplace operator.

Using the fact that

$$\begin{aligned}{}[AB,CD]=A\, [B, C] D+AC\, [B, D] + [A, C]\, DB + C\, [A, D]\, B, \end{aligned}$$

we can prove the following technical lemmas:

Lemma 7.3

For all $1\le j\le m$, we have:

$$\begin{aligned} \![\Delta _x,{\mathcal {J}}_R\partial _{x_j}{\mathcal {J}}_R \!]=-4x_j\Delta _x+4\langle \underline{u},\underline{\partial }_{x}\rangle \partial _{u_j}-4u_j\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle . \end{aligned}$$

Proof

Follows from straightforward calculations. $\square$

Lemma 7.4

For all $1\le j\le m$, we have:

$$\begin{aligned} \![\langle \underline{u},\underline{\partial }_{x}\rangle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle , {\mathcal {J}}_R\partial _{x_j}{\mathcal {J}}_R \!]&=2\left| \underline{u} \right| ^2\partial _{u_j}\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle -4x_j\langle \underline{u}, \underline{\partial }_{x}\rangle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \\&\quad +\left( \langle \underline{u},\underline{\partial }_{x}\rangle \partial _{u_j}-u_j\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \right) \left( 2{\mathbb {E}}_{u}+m-2 \right) \!. \end{aligned}$$

Proof

Temporarily denoting $C_j := {\mathcal {J}}_R\partial _{x_j}{\mathcal {J}}_R$, we get:

$$\begin{aligned} \![\langle \underline{u},\underline{\partial }_{x}\rangle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle ,C_j \!]&=2\langle \underline{u},\underline{\partial }_{x}\rangle \langle \underline{x},\underline{\partial }_{u}\rangle \partial _{x_j}+2\langle \underline{u},\underline{x}\rangle \partial _{x_j}\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \\&\quad -\langle \underline{u},\underline{\partial }_{x}\rangle \partial _{u_j}\left( 2{\mathbb {E}}_{x}+m-2 \right) -2\langle \underline{u},\underline{\partial }_{x}\rangle x_j\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \\&\quad +u_j\left( 2{\mathbb {E}}_{x}+m-2 \right) \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle -2x_j\langle \underline{u},\underline{\partial }_{x}\rangle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \\&\quad +2\langle \underline{u},\underline{\partial }_{x}\rangle \left( {\mathbb {E}}_{u}+{\mathbb {E}}_{x}+m \right) \partial _{u_j}+2\left| \underline{u} \right| ^2\partial _{u_j}\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \\&\quad -2\langle \underline{u},\underline{x}\rangle \partial _{x_j}\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle -2\langle \underline{u},\underline{\partial }_{x}\rangle \partial _{x_j}\langle \underline{x},\underline{\partial }_{u}\rangle \\&\quad -2u_j\left( {\mathbb {E}}_{u}-{\mathbb {E}}_{x} \right) \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \\&=2\langle \underline{u},\underline{\partial }_{x}\rangle \langle \underline{x},\underline{\partial }_{u}\rangle \partial _{x_j}-\langle \underline{u},\underline{\partial }_{x}\rangle \partial _{u_j}\left( m-2 \right) \\&\quad -2\left( x_j\langle \underline{u},\underline{\partial }_{x}\rangle -u_j \right) \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle +u_j\left( m-2 \right) \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \\&\quad -2x_j\langle \underline{u},\underline{\partial }_{x}\rangle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle +2\langle \underline{u},\underline{\partial }_{x}\rangle \left( {\mathbb {E}}_{u}+m \right) \partial _{u_j} \\&\quad +2\left| \underline{u} \right| ^2\partial _{u_j}\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle -2u_j\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \left( {\mathbb {E}}_{u}-1 \right) \\&\quad -2\langle \underline{u},\underline{\partial }_{x}\rangle \left( \langle \underline{x},\underline{\partial }_{u}\rangle \partial _{x_j}+\partial _{u_j} \right) \\ \end{aligned}$$

Simplifying the last expression completes the proof. $\square$

Now we can put everything together, hereby again using the notation $C_j$:

$$\begin{aligned} {\mathcal {D}}_1C_j=&C_j{\mathcal {D}}_1+\![{\mathcal {D}}_1,C_j \!] \\ =&\left( C_j-4x_j \right) {\mathcal {D}}_1+4\left( \langle \underline{u},\underline{\partial }_{x}\rangle \partial _{u_j}-u_j\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \right) \\&-\frac{4}{m}\left( \langle \underline{u},\underline{\partial }_{x}\rangle \partial _{u_j}-u_j\langle \underline{u},\underline{\partial }_{x}\rangle \right) \left( 2{\mathbb {E}}_{u}+m-2 \right) -\frac{8}{m}\left| \underline{u} \right| ^2\partial _{u_j} \langle \underline{u},\underline{\partial }_{x}\rangle \\ =&\left( C_j-4x_j \right) {\mathcal {D}}_1. \end{aligned}$$

Finally, we also prove that the Maxwell operator is elliptic. We first start with the definition of ellipicity:

Definition 7.1

A linear homogeneous differential operator of second order ${\mathcal {D}}:{\mathcal {C}}^{\infty }\left( {\mathbb {R}}^m,\mathbb {V}_{\lambda } \right) \longrightarrow {\mathcal {C}}^{\infty }\left( {\mathbb {R}}^m,\mathbb {V}_{\mu } \right)$ is elliptic if for every non-zero vector $\underline{x}\in {\mathbb {R}}^m$ its principle symbol, which is a linear map $\sigma _x({\mathcal {D}}):\mathbb {V}_{\lambda }\longrightarrow \mathbb {V}_{\mu }$ obtained by replacing its partial derivatives $\partial _{x_j}$ with the corresponding variables $x_j$, is a linear isomorphism.

Theorem 7.5

The Maxwell operator ${\mathcal {D}}_1:=\Delta _x-\frac{4}{m}\langle \underline{u},\underline{\partial }_{x}\rangle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle$ is an elliptic operator.

Proof

To prove the theorem, we will show that for fixed $x\in {\mathbb {R}}^m$ the symbol of the Maxwell operator, which is given by

$$\begin{aligned} \sigma _x({\mathcal {D}}_1)=\left| \underline{x} \right| ^2-\frac{4}{m}\langle \underline{u},\underline{x}\rangle \langle \underline{x},\underline{\partial }_{u}\rangle : {\mathcal {H}}_1\longrightarrow {\mathcal {H}}_1, \end{aligned}$$

is a linear isomorphism. As the symbol is clearly a linear map, it remains to be proven that the map is injective. Since an arbitrary element of ${\mathcal {H}}_1$ can be written as $\sum _{j=1}^m \alpha _j u_j$ with $\alpha _j \in {\mathbb {C}}$ for all $1\le j \le m$, we have to show that the following system of equations has a unique solution:

$$\begin{aligned} \left( \left| \underline{x} \right| ^2-\frac{4}{m}\langle \underline{u},\underline{x}\rangle \langle \underline{x},\underline{\partial }_{u}\rangle \right) \left( \sum _j \alpha _j u_j \right) =0. \end{aligned}$$

With $c=\frac{4}{m}$, this can be written in matrix notation as follows:

$$\begin{aligned} \begin{pmatrix} \left| \underline{x} \right| ^2-cx_1^2&{}\quad -cx_1x_2&{}\quad \cdots &{}\quad -cx_1x_m \\ -cx_1x_2 &{}\quad \left| \underline{x} \right| ^2-cx_2^2&{}\quad \cdots &{}\quad -cx_2x_m \\ \vdots &{}\quad &{} &{}\quad \vdots \\ -cx_1x_m &{}\quad \cdots &{}\quad -cx_m x_{m-1}&{}\quad \left| \underline{x} \right| ^2-cx_m^2 \end{pmatrix} \begin{pmatrix} \alpha _1 \\ \alpha _2 \\ \vdots \\ \alpha _m \end{pmatrix} =0, \end{aligned}$$

In order to show that this system has a unique solution, it suffices to prove that

$$\begin{aligned} \begin{vmatrix} \left| \underline{x} \right| ^2-cx_1^2&\quad -cx_1x_2&\quad \cdots&\quad -cx_1x_m \\ -cx_1x_2&\quad \left| \underline{x} \right| ^2-cx_2^2&\quad \cdots&\quad -cx_2x_m \\ \vdots&&\quad \vdots \\ -cx_1x_m&\quad \cdots&\quad -cx_m x_{m-1}&\quad \left| \underline{x} \right| ^2-cx_m^2 \end{vmatrix} \ne 0. \end{aligned}$$

Using the notation $x=\begin{pmatrix}x_1&\cdots&x_m \end{pmatrix}^T$, the determinant can be written more compactly as

$$\begin{aligned} P(\underline{x}):=\mathrm {det}\left( \left| \underline{x} \right| ^2\mathbb {I}_m - cxx^T \right) \ne 0. \end{aligned}$$

As a function depending on the variable $\underline{x}\in {\mathbb {R}}^m$, it is clear that $P(\underline{x})$ is invariant under rotations, since

$$\begin{aligned} R_{\theta }\cdot P(\underline{x})&:=P(R_{\theta }^{-1}\underline{x}) \\&=\mathrm {det}\left( \left| \underline{x} \right| ^2\mathbb {I}_m - cR_{\theta }^{-1}xx^TR_{\theta } \right) \\&=\mathrm {det}\left( \left| \underline{x} \right| ^2\mathbb {I}_m - cxx^T \right) , \end{aligned}$$

with $R_{\theta }$ a matrix in $\mathrm{SO }(m)$. It thus follows that $P(\underline{x})$ is a constant multiple of $\left| \underline{x} \right| ^{2m}$. The constant can then be fixed by calculating $P(x_1,0,\ldots ,0)$. One finally finds that

$$\begin{aligned} \mathrm {det}\left( \left| \underline{x} \right| ^2\mathbb {I}_m - \frac{4 }{m}xx^T \right) =\left( 1-\frac{4}{m} \right) \left| \underline{x} \right| ^{2m}. \end{aligned}$$

Since this is non-zero if $\underline{x}$ is non-zero, this completes the proof. $\square$

Remark 4

In the special case $m = 4$, one has that ${\mathcal {D}}_1$ is not elliptic since the determinant $\mathrm {det}\left( \left| \underline{x} \right| ^2\mathbb {I}_m - \frac{4}{m}xx^T \right)$ is zero.

For the sake of completeness, we also mention that a classification of second order elliptic operators was obtained by e.g. Branson and Pilca in [4, 26]. In order to explain their results, we need the notion of a generalised gradient, also called a Stein-Weiss operator [28], which is a generalisation of the Cauchy-Riemann operator to arbitrary dimension $m$. Let $\mathbb {V}_{\lambda }$ be a finite-dimensional irreducible representation of $\mathfrak {so}(m)$ and consider smooth functions in ${\mathcal {C}}^{\infty }\left( {\mathbb {R}}^m,\mathbb {V}_{\lambda } \right)$. We call $V_1$ the $m$-dimensional vector space (indentified with ${\mathbb {C}}^m$). The gradient $\nabla$ of a function in ${\mathcal {C}}^{\infty }\left( {\mathbb {R}}^m,\mathbb {V}_{\lambda } \right)$ takes values in $\mathbb {V}_{\lambda }\otimes V_1$, i.e. it is a function in ${\mathcal {C}}^{\infty }\left( {\mathbb {R}}^m,\mathbb {V}_{\lambda }\otimes V_1 \right)$. Of course $V_1$ is the vector representation of $\mathfrak {so}(m)$, so $\mathbb {V}_{\lambda }\otimes V_1$ is a reducible representation where the irreducible summands occur with multiplicity one in the decomposition [14], i.e.

$$\begin{aligned} \mathbb {V}_{\lambda }\otimes V_1=\bigoplus _{\epsilon \subset \lambda }\mathbb {V}_{\lambda + \epsilon }. \end{aligned}$$

By $\epsilon \subset \lambda$ we mean the weights $\epsilon$ of $V_1$ such that $\mathbb {V}_{\lambda + \epsilon }$ occurs in the decomposition and we call them relevant weights. We denote $\Pi _{\epsilon }:\mathbb {V}_{\lambda }\otimes V_1\longrightarrow \mathbb {V}_{\lambda + \epsilon }$ the projection onto the associated irreducible summand.

Definition 7.2

For each relevant weight $\epsilon$ of $\mathbb {V}_{\lambda }$, i.e. for each irreducible component in the decomposition of $\mathbb {V}_{\lambda }\otimes V_1$, there is a generalised gradient $P_{\epsilon }$ defined by the composition:

$$\begin{aligned} {\mathcal {C}}^{\infty }\left( {\mathbb {R}}^m,\mathbb {V}_{\lambda } \right) \overset{\nabla }{\longrightarrow }{\mathcal {C}}^{\infty }\left( {\mathbb {R}}^m,\mathbb {\mathbb {V}_{\lambda }}\otimes V_1 \right) \overset{\Pi _{\epsilon }}{\longrightarrow }{\mathcal {C}}^{\infty }\left( {\mathbb {R}}^m,\mathbb {V}_{\lambda +\epsilon } \right) \end{aligned}$$

In our case $\lambda =(1)$, the representation $\mathbb {V}_{1}\otimes V_1$ decomposes as the direct sum

$$\begin{aligned} \mathbb {V}_{1}\otimes V_1=\mathbb {V}_{(2)}\oplus \mathbb {V}_{(0)}\oplus \mathbb {V}_{(1,1)}. \end{aligned}$$

This means that the relevant weights are given by

$$\begin{aligned} \epsilon \in \left\{ \epsilon _1=(1,0,\ldots ,0),\epsilon _2=(0,1,0,\ldots ,0),-\epsilon _1=(-1,0,\ldots ,0) \right\} . \end{aligned}$$

With a slight abuse of notation, we will use the symbol $\lambda$ for the target spaces in the arrow diagram below. In Question 1.5 (on page 338) in [4], the question was raised whether any combination of the form $\alpha _1 P_{\epsilon _1}^{*}P_{\epsilon _1}^{}+\alpha _2 P_{\epsilon _2}^{*}P_{\epsilon _2}^{}+\alpha _{-1} P_{-\epsilon _1}^{*}P_{-\epsilon _1}^{}$, is elliptic. Here $P_{\epsilon _i}^{}$ are generalised gradients, $P_{\epsilon _i}^{*}$ their duals and $\alpha _i$ are non-zero constants. For $\alpha _1=\alpha _2=\alpha _{-1}=1$, the question was answered in Theorem 1.4 (on page 336 in [4]), but unfortunately this is not the case for the Maxwell operator. To see that, let us first construct the following operators:

$$\begin{aligned}&(1)\xrightarrow {P_{\epsilon _1}^{}}(2)\xrightarrow {P_{\epsilon _1}^{*}}(1) \\&(1)\xrightarrow {P_{\epsilon _2}^{}}(1,1)\xrightarrow {P_{\epsilon _2}^{*}}(1) \\&(1)\xrightarrow {P_{-\epsilon _1}^{}}(0)\xrightarrow {P_{-\epsilon _1}^{*}}(1). \end{aligned}$$

Theorem 7.6

The generalised gradients mentioned above are explicitly given by:

$$\begin{aligned} P_{-\epsilon _{1}}&=\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \\ P_{\epsilon _1}&=\langle \underline{u},\underline{\partial }_{x}\rangle -\frac{1}{m}|\underline{u}|^2\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \\ P_{\epsilon _2}&= \langle \underline{v},\underline{\partial }_{x}\rangle -\frac{1}{2}\langle \underline{v},\underline{\partial }_{u}\rangle \left( \langle \underline{u},\underline{\partial }_{x}\rangle -\frac{\left| \underline{u} \right| ^{2}}{m}\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \right) \\&\quad - \frac{1}{m}\left( \langle \underline{u},\underline{v}\rangle -\frac{\left| \underline{u} \right| ^{2}}{m-2}\langle \underline{v},\underline{\partial }_{u}\rangle \right) \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle . \end{aligned}$$

Proof

It is obvious that the operator $P_{-\epsilon _1} = \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle$ maps ${\mathcal {H}}_1$-values to scalars. We will construct the operator $P_{\epsilon _2}$ as the construction of $P_{\epsilon _1}$ is completely similar. To do so, we need the decomposition of the space of double harmonic polynomials:

$$\begin{aligned} {\mathcal {H}}_1^u\otimes {\mathcal {H}}_1^v = \langle \underline{v},\underline{\partial }_{u}\rangle {\mathcal {H}}_{2}\oplus \langle \underline{u},\underline{v}\rangle {\mathcal {H}}_{0} \oplus {\mathcal {H}}_{1,1}\!, \end{aligned}$$

Consider then again $f\in {\mathcal {C}}^{\infty }\left( {\mathbb {R}}^m,{\mathcal {H}}_1 \right)$ and note that $\langle \underline{v},\underline{\partial }_{x}\rangle f$ is harmonic in both $\underline{u}$ and $\underline{v}$. This means that

$$\begin{aligned} \langle \underline{v},\underline{\partial }_{x}\rangle f=\langle \underline{v},\underline{\partial }_{u}\rangle \varphi _{2}+\varphi _{1,1}+\langle \underline{u},\underline{v}\rangle \varphi _{0}, \end{aligned}$$

(10)

with $\varphi _i\in {\mathcal {H}}_i$. Acting with $\langle \underline{\partial }_{u},\underline{\partial }_{v}\rangle$ on this expression easily gives

$$\begin{aligned} \langle \underline{\partial }_{u},\underline{\partial }_{v}\rangle \langle \underline{v},\underline{\partial }_{x}\rangle f = \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle f =\langle \underline{\partial }_{u},\underline{\partial }_{v}\rangle \langle \underline{u},\underline{v}\rangle \varphi _{0} = ({\mathbb {E}}_u + {\mathbb {E}}_v + m)\varphi _0 = m\varphi _0\,, \end{aligned}$$

and the substitution in Eq. (10) leads to

$$\begin{aligned} \langle \underline{v},\underline{\partial }_{x}\rangle f=\langle \underline{v},\underline{\partial }_{u}\rangle \varphi _{2}+\varphi _{1,1}+\frac{1}{m}\langle \underline{u},\underline{v}\rangle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle f\!. \end{aligned}$$

Next we determine $\varphi _{2}$, which can be done using the action of $\langle \underline{u},\underline{\partial }_{v}\rangle$:

$$\begin{aligned} \langle \underline{u},\underline{\partial }_{v}\rangle \langle \underline{v},\underline{\partial }_{x}\rangle f&=\langle \underline{u},\underline{\partial }_{v}\rangle \langle \underline{v},\underline{\partial }_{u}\rangle \varphi _{2}+\frac{1}{m}\langle \underline{u},\underline{\partial }_{v}\rangle \langle \underline{u},\underline{v}\rangle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle f \\ \implies \langle \underline{u},\underline{\partial }_{x}\rangle f&={\mathbb {E}}_{u}\varphi _{2}+\frac{1}{m}\left| \underline{u} \right| ^{2}\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle f\!. \end{aligned}$$

The latter expression can then be written as

$$\begin{aligned} \varphi _{2}=\frac{1}{2}\left( \langle \underline{u},\underline{\partial }_{x}\rangle f-\frac{1}{m}\left| \underline{u} \right| ^{2}\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle f \right) . \end{aligned}$$

Putting everything together, we find an expression for $\varphi _{1,1}$:

$$\begin{aligned} \varphi _{1,1} = \langle \underline{v},\underline{\partial }_{x}\rangle f-\frac{1}{2}\langle \underline{v},\underline{\partial }_{u}\rangle \left( \langle \underline{u},\underline{\partial }_{x}\rangle -\frac{\left| \underline{u} \right| ^{2}}{m}\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \right) f -\frac{1}{m}\langle \underline{u},\underline{v}\rangle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle f, \end{aligned}$$

which is the generalised gradient that we where looking for. $\square$

To calculate the dual operators $P^{*}_{\epsilon _i}$ of the generalised gradients, we will first give a definition of a formal adjoint of such a gradient.

Definition 7.3

An arbitrary function $f\in L^2({\mathbb {R}}^m,{\mathcal {H}}_k)$ can be written as $f=\sum _A f_A(\underline{x}) H_A(\underline{u})$, where $\left\{ H_A(\underline{u}) \right\} _A$ is a basis of ${\mathcal {H}}_k$ and $f_A(\underline{x})$ are scalar-valued smooth functions. The inner product on $L^2({\mathbb {R}}^m,{\mathcal {H}}_k)$ can be defined as follows:

$$\begin{aligned} \langle f,g\rangle =\sum _{A,B} \left( \int _{{\mathbb {R}}^m} \overline{f_A(\underline{x})} g_B(\underline{x}) \mathrm {d}\underline{x} \right) \![H_A,H_B \!]_F, \end{aligned}$$

where $[\cdot ,\cdot ]_F$ is the Fischer inner product on harmonics given by

$$\begin{aligned} \![H_A,H_B \!]_F:=\left. \left( \overline{H_A(\underline{\partial }_{u})}H_B(\underline{u}) \right) \right| _{\underline{u}=0} \end{aligned}$$

Theorem 7.7

The formal adjoints of the generalised gradients considered in theorem 7.6 are given by:

$$\begin{aligned} P^{*}_{\epsilon _1}&=-\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \\ P^{*}_{\epsilon _2}&=- \langle \underline{\partial }_{v},\underline{\partial }_{x}\rangle \\ P^{*}_{-\epsilon _1}&=-\langle \underline{u},\underline{\partial }_{x}\rangle . \end{aligned}$$

Proof

We will prove that $\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle$ is the dual of $-P_{\epsilon _1}$. Take $f\in L^2({\mathbb {R}}^m,{\mathcal {H}}_2)$ and $g\in L^2({\mathbb {R}}^m,{\mathcal {H}}_1)$ and consider

$$\begin{aligned} \langle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle f,g\rangle&=\sum _{A,B}\sum _{j=1}^m\left( \int _{{\mathbb {R}}^m} g_B(\underline{x}) \overline{\partial _{x_j}f_A(\underline{x})} \mathrm {d}\underline{x} \right) \![\partial _{u_j}H_A,H_B \!]_F \\&=-\sum _{A,B}\sum _{j=1}^m\left( \int _{{\mathbb {R}}^m} \overline{f_A(\underline{x})}\partial _{x_j}g_B(\underline{x}) \mathrm {d}\underline{x} \right) \![\partial _{u_j}H_A,H_B \!]_F, \end{aligned}$$

where the second equality follows from integration by parts. Remains to calculate the dual of $\partial _{u_j}$ with respect to the Fischer inner product on harmonic polynomials. Since $u_jH_B$ is not harmonic anymore, we need to do a projection on its harmonic part. Using the classical Fischer decomposition, this is given by $\left( u_j - \frac{1}{m}\left| \underline{u} \right| \partial _{u_j} \right) H_B$. Putting everything together, we have

$$\begin{aligned} \langle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle f,g\rangle&=-\!\sum _{A,B}\sum _{j=1}^m\left( \int _{{\mathbb {R}}^m} \overline{f_A(\underline{x})}\partial _{x_j}g_B(\underline{x}) \mathrm {d}\underline{x} \right) \![H_A,\left( u_j - \frac{1}{m}\left| \underline{u} \right| ^2\partial _{u_j} \right) H_B \!]_F \\&=\langle f,-\left( \langle \underline{u},\underline{\partial }_{x}\rangle -\frac{1}{m}\left| \underline{u} \right| ^2\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \right) g\rangle , \end{aligned}$$

which proves that $\langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle$ is indeed the dual of $-P_{\epsilon _1}$. The proof of the other statements is completely similar. $\square$

Calculating the compositions of the generalised gradients with their duals, we arrive at the following operators:

$$\begin{aligned} P_{\epsilon _1}^{*}P_{\epsilon _1}^{}&=-\Delta _x-\frac{m-2}{m}\langle \underline{u},\underline{\partial }_{x}\rangle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \nonumber \\ P_{\epsilon _2}^{*}P_{\epsilon _2}^{}&=-\frac{1}{2}\left( \Delta _x-\langle \underline{u},\underline{\partial }_{x}\rangle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle \right) \\ P_{-\epsilon _1}^{*}P_{-\epsilon _1}^{}&=-\langle \underline{u},\underline{\partial }_{x}\rangle \langle \underline{\partial }_{u},\underline{\partial }_{x}\rangle . \nonumber \end{aligned}$$

(11)

It is clear that the Maxwell operator can be written as a linear combination (but with non-zero coefficients different from $1$) of these second order elliptic operators:

$$\begin{aligned} {\mathcal {D}}_1 \ =\ -\frac{m}{2(m-1)}P_{\epsilon _1}^{*}P_{\epsilon _1}^{}-\frac{m-2}{m-1}P_{\epsilon _2}^{*}P_{\epsilon _2}^{}+\frac{4}{m} P_{-\epsilon _1}^{*}P_{-\epsilon _1}^{}, \end{aligned}$$

Remark 5

It is worth mentioning that in the special case $m = 4$, one has that ${\mathcal {D}}_1 \sim P^*_{\epsilon _2}P_{\epsilon _2}$. This is not clear from the expression above, but can be observed from formula (11). This was of course to be expected, as the classical Maxwell equations are expressed in terms of a $2$-form.