# A Gelfand Transform for Spinor Fields on Embedded Riemannian Manifolds

Colin Roberts111Ph.D. Candidate, Colorado State University, 1874 Campus
Delivery, Fort Collins, CO 80523-1874 Colorado State University, Fort Collins,
Colorado, 80523

###### Abstract

A classical result of Gelfand shows that the topologized spectrum of
characters on commutative Banach algebra is homeomorphic to the underlying
space. This fact is used in solving the Calderón problem in dimension 2 via
the _boundary control (BC) method_. To apply the BC method in dimension 3, the
algebra of complex holomorphic functions can be replaced by the space of
harmonic quaternion fields, but this space is no longer an algebra and is not
commutative. Nonetheless, it has been shown that a suitable notion of a
spectrum exists and in the case when the underlying space is convex, the
spectrum is homeomorphic to the ball. Our goal is to generalize this result to
more general manifolds in arbitrary dimension. To do so, we use Clifford
algebras of multivector fields and the set of functions we care about is the
space monogenic spinor fields. The spectrum consists of spinor valued
functionals that respect the module and subalgebra structure of the space of
monogenic spinor fields. We prove that for compact regions in
$\mathbb{R}^{n}$, the spectrum is homeomorphic to the manifold itself.
Furthermore, some essential lemmas and propositions are proven for arbitrary
compact Riemannian $n$-manifolds. Finally, we end with a Stone–Weierstrass
theorem which shows the algebra generated by the monogenic spinor fields on
arbitrary compact $M$ with boundary is dense in the algebra of continuous
spinor fields.

##  1 Introduction

This paper is motivated by the inverse tomography (or Calderón) problem for
Riemannian manifolds [1, 2, 3, 4, 5, 6, 7]. First, let us explain an important
related problem called the Electrical Impedance Tomography (EIT) problem and
note that it is equivalent to the classical Calderón problem in dimension
$n=3$. Let $M$ be a body free of interior charges and made from a (possibly
anisotropic) Ohmic material with conductivity $\gamma$. Applying a voltage
$\phi$ on the boundary $\partial M$ induces a voltage $u$ in the interior and
since the interior is free of charges, $u$ satisfies
$\mathrm{div}(\gamma\leavevmode\nobreak\ \mathrm{grad}(u))=0$. We are not
allowed access to the interior, which leaves us solely with the ability to
making measurements along the boundary $\partial M$. Given $\phi$, we can
measure the corresponding current flux $\frac{\partial
u}{\partial\boldsymbol{\nu}}$ on $\partial M$ where $\boldsymbol{\nu}$ is the
outward normal field. Hence, we have a map $\Lambda$ called the voltage-to-
current map defined by $\Lambda\phi\coloneqq\frac{\partial
u}{\partial\boldsymbol{\nu}}$. Is it possible to determine the conductivity
$\gamma$ from the voltage-to-current map $\Lambda$?

One avenue of active research seeks to solve a more general problem called the
Calderón problem for Riemannian manifolds [5, 6, 7, 8, 9]. In this setting,
one attempts to reconstruct an $n$-dimensional smooth manifold $M$ and metric
$g$ from the Dirichlet-to-Neumman (DN) map $\Lambda$ defined on the boundary
traces of harmonic differential forms, i.e., forms in the kernel of the
Laplace–Beltrami operator $\Delta$. Given the Dirichlet boundary condition
$\phi$, we have a unique harmonic $u$. The DN map is defined by
$\Lambda\phi\coloneqq\iota^{*}(\star du)$ which is valid for any $k$-form
$\phi$. Fixing $\phi$ to be a 0-form then yields the classical problem
discussed previously and in dimension 3, this is equivalent to the EIT
problem. From complete knowledge of pairs of Dirichlet and Neumann data
$(\phi,\iota^{*}(\star du))$ for forms of all grade, the geometric inverse
problem is to determine the pair $(M,g)$ up to isometry. If you would like,
please see Uhlmann’s article [2] for an excellent explanation of the
relationship between the EIT and Calderón problem.

Solutions to the Calderón problem exist in a handful of special cases, but the
$C^{\infty}$-smooth problem remains open in dimensions greater than two.
Belishev [10] and Lassas–Uhlmann [3] both show that for surfaces $S$ with one
boundary component, the classical DN map determines $S$ up to conformal class.
A better result cannot be achieved since the Laplacian is conformally
invariant in dimension 2. Belishev, Badanin, and Korikov recently achieved a
similar result for surfaces with multiple boundary components [11]. It is also
known that the DN map for forms recovers partial topological information on
arbitrary $M$ such as the Betti numbers [6], but it is not known whether the
DN operator can recover $M$ up to homeomorphism. The extension of the DN
operator to the complete DN operator in Sharafutdinov and Shonkwiler’s paper
[8] is able to recover the absolute and relative cohomologies as well.

We will mostly concern ourselves with the technique used by Belishev in his
2003 paper [10] called the _Boundary Control (BC) method_. This technique
utilizes the classical Gelfand representation for commutative Banach algebras.
That is, for a surface $S$, the spectrum (maximal ideal space) of the
commutative Banach algebra of holomorphic functions is homeomorphic to $S$.
Alongside this result, Belishev then used the complex structure of the algebra
to determine the metric up to conformal equivalence. Furthermore, it is the
Gelfand transform that allows for all of this information to be obtained from
the boundary. This led to a solution to the Calderón problem in dimension 2.
For more on the boundary control method, see [12].

To solve the problem in dimension three using the BC method, it is natural to
consider replacing complex functions with quaternion-valued functions.
Belishev and Vakulenko wrote a series of papers on this topic [13, 14, 15]. In
those papers, the authors work with the space of harmonic quaternion fields
using the language of differential forms. Their goal was to complete one
portion of the BC method by realizing a Gelfand theory for quaternion fields.
The first hurdle in defining a Gelfand spectrum on the space of harmonic
quaternion fields is that the space fails to be an algebra. Moreover, it is
not commutative. However, the space does contain commutative subalgebras and
by carefully defining a meaningful notion of a spectrum that is multiplicative
over these subalgebras, they find that for convex regions in $\mathbb{R}^{3}$,
the spectrum is homeomorphic to the ball. Is a similar result true for a more
general class of manifolds?

In this paper, we seek to faithfully capture the necessary Gelfand theory in
dimension 2 and 3 as well as generalize it to arbitrary dimension by replacing
the exterior algebra of differential forms with Clifford algebras of
multivector fields. Note that we do not lose any information with this
substitution since the exterior algebra naturally embeds into in any Clifford
algebra. Our focus will be on nondegenerate (Euclidean) Clifford algebras
$\mathcal{G}$ which we call _(Euclidean) geometric algebras_ since they
capture both complex and quaternion algebras as spinor subalgebras
$\mathcal{G}^{+}\subset\mathcal{G}$ when defined on $\mathbb{R}^{2}$ and
$\mathbb{R}^{3}$ resectively. Many of the nice properties of complex and
quaternion valued functions can be generalized to higher dimensions by
replacing them with spinor fields $f_{+}\in C^{\infty}(M;\mathcal{G}^{+})$
which are sections of the bundle of the spinor subalgebras.

A complex function on a surface can be written as a sum of even grade forms,
specifically, a 0-form $\omega$ and 2-form $\eta$. This function $\omega+\eta$
is holomorphic if and only if the generalized Cauchy–Riemann equation
$d\omega=-\delta\eta$ where $d$ is the exterior derivative and $\delta$ is the
codifferential is satisfied. Such $\omega$ and $\eta$ are called a conjugate
pair. The same is true for a harmonic quaternion fields. Instead of looking at
pairs of conjugate forms, we consider spinor fields in the kernel of the
Hodge–Dirac operator $\boldsymbol{\nabla}$. This operator is fundamental in
Clifford analysis and it is equivalent to operator $d+\delta$ on forms. Spinor
fields in the kernel of $\boldsymbol{\nabla}$ are not restricted to being only
pairwise conjugate and the associated Cauchy–Riemann equations are a bit more
general in dimensions higher than three. We refer to such fields in the kernel
of the Hodge–Dirac operator as _monogenic_.

As a side note, if we consider fields of only a single grade, if they are
monogenic then they correspond directly to harmonic fields. This lands us in
the realm of Hodge theory where it is a central result that for each grade
$k$, the space of harmonic fields is finite dimensional. Fundamentally, by
considering fields consisting of sums of even grades (i.e., a spinor), we find
the kernel to be far more rich in content. For example, in $\mathbb{R}^{2}$,
monogenic spinor fields are equivalent to holomorphic functions and in
$\mathbb{R}^{3}$, they are equivalent to harmonic quaternion fields and both
of these spaces are infinite in dimension. The real benefit of using Clifford
algebras is that it allows us to be dimension-agnostic like we can with
differential forms, but recover many of the staple complex analysis theorems
in any given dimension as well.

Within this framework, we prove a Gelfand representation for $n$-dimensional
compact regions of $\mathbb{R}^{n}$ and define the corresponding Gelfand
transform. Here, the spectrum consists of a certain type of functionals which
we call _$\mathcal{G}$ -currents_ that act on the space of monogenic spinor
fields. These functionals are valued in the base geometric algebra and respect
the structure of certain subalgebras nested inside the space of monogenic
spinor fields. Furthermore, we report a Stone–Weierstrass theorem showing that
the algebra generated by the closure of the monogenic fields is dense in the
space of continuous fields. Both of the above results answer questions posed
by Belishev and Vakulenko in [14].

The construction we use for the spectrum bootstraps from our knowledge of
surfaces: given a surface $S$, the monogenic spinor fields on the surface
$\mathcal{M}^{+}(S)$ are a commutative Banach algebra isomorphic to the
algebra of holomorphic functions. When the dimension of the manifold $M$
exceeds two, the space $\mathcal{M}^{+}(M)$ is not an algebra since products
of monogenic fields need not be monogenic. At a local scale, a monogenic
spinor $f_{+}\in\mathcal{M}^{+}(M)$ is, in essence, built out of a monogenic
fields propagated off of surfaces embedded in $M$. In fact, we can think of
these fields as a local set of variables and this lets us write power series
for $f_{+}$ locally in these variables. We find that these variables are
direct analogs to the variable $z$ in complex analysis.

The key challenge we face in this paper is to define a notion of a spectrum
for the space $\mathcal{M}^{+}(M)$. In the case of a surface $S$, characters
$\delta$ in the spectrum $\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(S)$ are the
continuous algebra morphisms $\mathrm{sp}\hskip
2.0pt\mathcal{M}^{+}(S)\to\mathbb{C}$. Since $\mathcal{M}^{+}(M)$ is not an
algebra but is a $\mathcal{G}^{+}$-module, we define characters
$\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)$ to be continuous module morphisms
$\mathcal{M}^{+}(M)\to\mathcal{G}^{+}$. Furthermore, we require that
characters are also algebra morphisms on subalgebras
$\mathcal{A}_{\boldsymbol{B}}$ (which are akin to $\mathcal{M}^{+}(S)$ for $S$
a subsurface in $M$) to a an algebra $\mathbb{A}_{\boldsymbol{B}}$ (which is
isomorphic to $\mathbb{C}$). Applying such characters to the power series
representation for $f_{+}$ in terms of the $z$ variables yields our main
theorem.

######  Theorem 1.1.

Let $M$ be a compact region in $\mathbb{R}^{n}$. For any
$\delta\in\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)$, there is a point
$\boldsymbol{x}_{\delta}\in M$ such that
$\delta(f)=f(\boldsymbol{x}_{\delta})$ for any $f_{+}\in\mathcal{M}^{+}(M)$ a
monogenic field. Given the weak-$\ast$ topology on the space of
$\mathcal{G}$-currents, the map

$\Gamma\colon\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)\to
M,\quad\delta\mapsto\boldsymbol{x}_{\delta}$

is a homeomorphism. The Gelfand transform $\mathcal{M}^{+}(M)\to
C^{0}(\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M);\mathcal{G}^{+})$ given by
$\widehat{f_{+}}(\delta)=\delta[f_{+}]$ is an isometric isomorphism onto its
image so that $\mathcal{M}^{+}(M)\cong\widehat{\mathcal{M}^{+}(M)}$.

Furthermore, we find that the closure $\overline{\mathcal{M}^{+}(M)}$
separates points which follows from the fact from Calderbank’s thesis [16].
Namely, if we know $f_{+}\in\mathcal{M}^{+}(M)$ locally, we can uniformly
extend this to a unique field on all of $M$. Given this and a theorem from
Laville and Ramadanoff’s paper [17], we prove the following Stone–Weierstass
result.

######  Theorem 1.2.

Let $\vee\overline{\mathcal{M}^{+}(M)}$ represent the minimal algebra
generated by $\overline{\mathcal{M}^{+}(M)}$. Then
$\vee\overline{\mathcal{M}^{+}(M)}$ is dense in $C^{0}(M;\mathcal{G}^{+})$.

Lastly, I hope that others find that this work warrants further investigation
on the utility of $C^{\ast}$\- and Banach-algebras of Clifford algebra-valued
functions and to this end I will finish the paper with a discussion of other
questions addressed by Belishev and Vakulenko and their relation to this work
and the Calderón problem. It is certainly a worthy endeavor to consider what
extent these theorems can be used when the information is extracted solely
from Dirichlet-to-Neumann operators.

### Acknlowedgements:

Without the unending guidance and input from my advisor Dr. Clayton
Shonkwiler, none of this work would be possible. I deeply appreciate your help
and your willingness to follow me on my own mathematical journey over these
past few years. Thank you.

##  2 Preliminaries

We address only the necessary preliminaries here, though more detail can be
found in various sources, for instance from Doran and Lasenby [18], and
Hestenes and Sobczyk [19]. Our preliminaries will include the basics of
Clifford algebras with a large motivating example, foundations of the Clifford
analysis of the Hodge–Dirac operator, and the notion of algebraic currents.

###  2.1 Clifford Algebras

To construct any Clifford algebra, take an $n$-dimensional vector space $V$
over a field $\mathbb{F}$ with quadratic form $Q$. By the polarization
identity, given a $Q$ there is a unique corresponding symmetric bilinear form
$g$. For more information on vector spaces with quadratic or bilinear forms,
please see [20].

The tensor algebra is given by $\bigoplus_{n\in\mathbb{N}}V^{\otimes^{n}}$ and
we form the Clifford algebra $C\ell(V,Q)$ by the quotient

$C\ell(V,Q)\coloneqq\bigoplus_{n\in\mathbb{N}}V^{\otimes^{n}}\leavevmode\nobreak\
/\leavevmode\nobreak\
\langle\boldsymbol{v}\otimes\boldsymbol{v}-Q(\boldsymbol{v})\rangle$ (1)

with the induced addition and multiplication from the tensor algebra. For sake
of clarity, we will think of $C\ell(V,Q)=C\ell(V,g)$ as needed, though most
define Clifford algebras only using quadratic forms. Elements of $C\ell(V,Q)$
are referred to as _multivectors_.

When we take the totally singular form $Q=0$, the corresponding Clifford
algebra is the exterior algebra $\bigwedge(V)=C\ell(V,0)$, and otherwise
$\bigwedge(V)\subseteq C\ell(V,Q)$ as a subalgebra. Note that $C\ell(V,Q)$ is
a $\mathbb{F}$-vector space of dimension $2^{n}$. For the remainder of this
paper, we take $\mathbb{F}=\mathbb{R}$.

Given linearly independent vectors
$\boldsymbol{v}_{1},\dots,\boldsymbol{v}_{k}$ and the exterior product
$\wedge$, an element of the form

$\boldsymbol{A_{k}}=\boldsymbol{v}_{1}\wedge\cdots\wedge\boldsymbol{v}_{k}$
(2)

is a _$k$ -blade_, which are the simplest multivectors. Using the $\wedge$
inherently removes lower grade elements and keeps only the highest grade
element (grade-$k$) of the product
$A=\boldsymbol{v}_{1}\boldsymbol{v}_{2}\cdots\boldsymbol{v}_{k}$ since, at the
very least, the product of two vectors yields eq. 6. This multivector $A$,
often called a _versor_ since it is a product of vectors, is a sum of
different graded elements called _$k$ -vectors_. A $k$-vector is a linear
combination of $k$-blades and we denote this subspace of grade-$k$ elements by
$C\ell^{k}(V,Q)$. Therefore, we have the direct sum decomposition

$C\ell(V,Q)=\bigoplus_{k=0}^{n}C\ell^{k}(V,Q).$ (3)

Some may refer to $k$-blades as _simple_ or _decomposable_ $k$-vectors as they
correspond to rank-1 tensors in the tensor algebra. Moreover, they are also
representative of subspaces which we discuss later. We write
$C\ell^{k\oplus\ell}(V,Q)$ to represent the direct sum space of $k$\- and
$\ell$-vectors. A basis $\boldsymbol{e}_{i}$ of $V$ induces a $k$-blade basis
by taking $\mathcal{I}=\\{i_{1},\dots,i_{k}\\}$ to be a list of increasing
indices $i_{1}<\cdots<i_{k}$ and putting

$\boldsymbol{E}_{\mathcal{I}}\coloneqq\boldsymbol{e}_{i_{1}}\wedge\cdots\wedge\boldsymbol{e}_{i_{k}}.$
(4)

Some graded elements have special names. We say grade-0 objects are _scalars_
, grade-1 are _vectors_ , grade-2 are _bivectors_ , and grade-$n$ objects are
_pseudoscalars_. Typically, objects of grade-$(n-k)$ receive the prefix
“pseudo”, for example, we have pseudoscalars of grade-$(n-0)$ and
_pseudovectors_ of grade-$(n-1)$. Given the direct sum decomposition in eq. 3,
a multivector $A\in C\ell(V,Q)$ is given by

$A=\sum_{k=0}^{n}A_{k}$ (5)

where $A_{k}\in C\ell(V,Q)^{k}$.

Multivectors also split into even and odd grades. We will work with the even-
graded elements called _spinors_ as they form their own subalgebra
$C\ell^{+}(V,Q)$. Due to this, some refer to $C\ell(V,Q)$ as a _superalgebra_
since there is this $\mathbb{Z}/2\mathbb{Z}$-splitting. Spinors may also be
defined slightly more generally (see [16]), but this notion suffices for this
paper.

One purpose of using a Clifford algebra $C\ell(V,Q)$ is to extend the exterior
algebra $\bigwedge(V)$ to include a useful interior multiplication. Given
$\boldsymbol{v},\boldsymbol{w}\in C\ell(V,Q)^{1}$, their product splits as

$\boldsymbol{v}\boldsymbol{w}=\underbrace{\boldsymbol{v}\cdot\boldsymbol{w}}_{\textrm{grade-0}}+\underbrace{\boldsymbol{v}\wedge\boldsymbol{w}}_{\textrm{grade-2}},$
(6)

where $\cdot$ is the _interior product_. However, for general Clifford
algebras, we may have _degenerate_ vectors $\boldsymbol{v}$ such that for any
other vector $\boldsymbol{w}$, $\boldsymbol{v}\cdot\boldsymbol{w}=0$. We will
want to rid of this case.

To remove degenerate vectors from the algebra, we can force the quadratic form
$Q$ to be completely nonsingular. We refer to such a Clifford algebra with
nonsingular $Q$ as a _geometric algebra_ and we write $\mathcal{G}=C\ell(V,Q)$
to denote such an algebra. To see more reasoning of calling such algebras
“geometric”, we again refer the reader to the chapter [20]. Using the vector
space basis, we can determine the coefficients of the bilinear form by

$g_{ij}=g(\boldsymbol{e}_{i},\boldsymbol{e}_{j})=\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j}.$
(7)

Let us quickly remark that if we were considering a Clifford algebra that was
not a geometric algebra, then the matrix $g_{ij}$ of the bilinear form would
be singular. We will carry on the rest of this paper working solely with
geometric algebras for this reason, so the reader can assume that we take $g$
whose matrix representations $g_{ij}$ are full rank and symmetric.

Given a $k$-vector $A_{k}$ and an $\ell$-vector $B_{\ell}$, the product is

$A_{k}B_{\ell}=\left\langle A_{k}B_{\ell}\right\rangle_{|k-\ell|}+\left\langle
A_{k}B_{\ell}\right\rangle_{|k-\ell|+2}+\cdots+\left\langle
A_{k}B_{\ell}\right\rangle_{k+\ell},$ (8)

where the brackets
$\left\langle-\right\rangle_{k}\colon\mathcal{G}\to\mathcal{G}^{k}$ denote
projection into the grade-$k$ subspace by

$\left\langle A\right\rangle_{k}=A_{k}$ (9)

when $A$ is given by eq. 5. We define the _(left) contraction_ $A_{k}\hskip
1.42262pt\lrcorner\hskip 1.42262ptB_{\ell}\coloneqq\left\langle
A_{k}B_{\ell}\right\rangle_{\ell-k}$. In general, the lowest grade term of
$A_{k}B_{\ell}$ is the interior product $A_{k}\cdot B_{\ell}=\left\langle
A_{k}B_{\ell}\right\rangle_{|k-\ell|}$ and the exterior product $\wedge$ is
the highest grade term of the product so that $A_{k}\wedge
B_{\ell}=\left\langle A_{k}B_{\ell}\right\rangle_{k+\ell}$. For a vector
$\boldsymbol{v}$ we have $\boldsymbol{v}\cdot A=\boldsymbol{v}\hskip
1.42262pt\lrcorner\hskip 1.42262ptA$ so many equations can be written with
either $\cdot$ or $\hskip 1.42262pt\lrcorner\hskip 1.42262pt$. Most will be
written with $\hskip 1.42262pt\lrcorner\hskip 1.42262pt$ as it is
algebraically and geometrically more convenient. For notational simplicity, we
also remove the subscript when projecting into the scalar subspace,
$\left\langle-\right\rangle=\left\langle-\right\rangle_{0}$, but this should
not be confused with the notation for the ideal generated by a relation used
only in eq. 1.

The _reciprocal basis vectors $\boldsymbol{e}^{i}$_ are those that satisfy
$\boldsymbol{e}^{i}\cdot\boldsymbol{e}_{j}=\delta^{i}_{j}$. Reciprocal basis
elements allow us to use the Riesz representation in order to avoid extraneous
use of dual space elements since we are able to capture this functionality
through the interior product. For sake of clarity,
$\boldsymbol{e}^{i}\cdot\boldsymbol{e}^{j}=g^{ij}$ is the matrix inverse to
$g_{ij}$ and $\boldsymbol{e}^{i}=g^{ij}\boldsymbol{e}_{j}$ are just the
“raised up” indices. For a basis blade $\boldsymbol{E}_{\mathcal{I}}$, the
reciprocal blade is $\boldsymbol{E}^{\mathcal{I}}$ and it satisfies the
equation
$\boldsymbol{E}^{\mathcal{I}}\cdot\boldsymbol{E}_{\mathcal{J}}=\delta^{\mathcal{I}}_{\mathcal{J}}$
where $\delta^{\mathcal{I}}_{\mathcal{J}}=1$ only when the sets of indices
$\mathcal{I}$ and $\mathcal{J}$ are identical.

Geometric algebras have a bilinear product
$\mathcal{G}\times\mathcal{G}\to\mathbb{R}$ called the _multivector inner
product_ which is given by

$(A,B)\coloneqq\left\langle A^{\dagger}B\right\rangle.$ (10)

This equation is given in terms of the _reverse operator_ $\dagger$ which for
$\lambda\in\mathbb{R}$ satisfies

$(A+B)^{\dagger}=A^{\dagger}+B^{\dagger},\quad(\lambda
A)^{\dagger}=\lambda^{\dagger}A^{\dagger}=\lambda A^{\dagger},\quad
A^{\dagger\dagger}=A,\quad(AB)^{\dagger}=B^{\dagger}A^{\dagger},$ (11)

and on a versor we have

$(\boldsymbol{v}_{1}\boldsymbol{v}_{2}\cdots\boldsymbol{v}_{k})^{\dagger}\coloneqq\boldsymbol{v}_{k}\cdots\boldsymbol{v}_{2}\boldsymbol{v}_{1}.$
(12)

Note that $\dagger$ acts as the adjoint in the product $(-,-)$ which follows
from the cyclic property of the scalar grade projection [21, eq. (138)]. To
see this, we take another multivector $C$ and note

$\displaystyle(CA,B)=\left\langle(CA)^{\dagger}B\right\rangle=\left\langle
A^{\dagger}C^{\dagger}B\right\rangle=(A,C^{\dagger}B),$ (13)

We define a semi-norm $|-|^{2}\coloneqq(-,-)$ called the _multivector norm_.
If $|A|=\pm 1$ we say that $A$ is _unit_. It is worth saying that for a
multivector field written in terms of basis blades
$f=\sum_{\mathcal{I}}f_{\mathcal{I}}\boldsymbol{E}_{\mathcal{I}}$ that

$f_{\mathcal{I}}=(f,\boldsymbol{E}^{\mathcal{I}}),$ (14)

so long as the quadratic form is definite (e.g., the quadratic form is the
Euclidean norm). We also have that

$\boldsymbol{E}^{\mathcal{I}}=(\boldsymbol{e}^{i_{1}}\wedge\cdots\wedge\boldsymbol{e}^{i_{k}})^{\dagger}.$
(15)

There exists a vector basis for $V$ where $p$ vectors square to $-1$ and $q$
vectors square to $+1$ and $p+q=n$. The corresponding geometric algebra is
often written as $\mathcal{G}_{p,q}$. We will focus most on the the case where
$Q$ is positive definite and to distinguish this, we can write
$\mathcal{G}_{n}$ if it is necessary. For $\mathcal{G}_{n}$, the multivector
inner product and multivector norm are both positive definite. One can see
that the multivector inner product treats the space $\mathcal{G}_{n}$ as a
$2^{n}$-dimensional inner product space with a basis given by the blades
$\boldsymbol{E}_{\mathcal{I}}$. Equation 14 just specifies that we have chosen
a set of blades orthonormal with respect to the multivector inner product.

If the basis $\boldsymbol{e}_{i}$ is orthonormal in $V$, then the set of basis
blades $\boldsymbol{E}_{\mathcal{I}}$ are orthonormal versors in
$\mathcal{G}_{n}$ since

$\boldsymbol{E}_{\mathcal{I}}=\boldsymbol{e}_{i_{1}}\wedge\cdots\wedge\boldsymbol{e}_{i_{k}}=\boldsymbol{e}_{i_{1}}\boldsymbol{e}_{i_{2}}\cdots\boldsymbol{e}_{i_{k}}.$
(16)

Their products become much clearer to compute. We have

$\boldsymbol{E}_{\mathcal{I}}\boldsymbol{E}_{\mathcal{J}}=\pm\boldsymbol{E}_{\mathcal{I}\triangle\mathcal{J}},$
(17)

where $\triangle$ is the symmetric difference of the sets $\mathcal{I}$ and
$\mathcal{J}$ and the $\pm$ is used solely due to the fact that vectors
$\boldsymbol{e}_{i}$ comprising the versors $\boldsymbol{E}_{\mathcal{I}}$ may
need to be swapped and

$-\boldsymbol{E}_{\mathcal{I}}=\boldsymbol{e}_{i_{1}}\boldsymbol{e}_{i_{2}}\cdots\boldsymbol{e}_{i_{j+1}}\boldsymbol{e}_{i_{j}}\cdots\boldsymbol{e}_{i_{k}}.$
(18)

For a concrete example, take
$\boldsymbol{E}_{123}=\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{3}$
and $\boldsymbol{E}_{124}$ both in $\mathcal{G}_{n}$, then

$\boldsymbol{E}_{123}\boldsymbol{E}_{124}=\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{3}\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{4}=\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{3}\boldsymbol{e}_{4}=-\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{2}\boldsymbol{e}_{1}\boldsymbol{e}_{3}\boldsymbol{e}_{4}=-\boldsymbol{e}_{1}\boldsymbol{e}_{1}\boldsymbol{e}_{3}\boldsymbol{e}_{4}=-\boldsymbol{E}_{34}.$
(19)

Using an orthonormal vector basis shows how nicely versors act algebraically.
Multiplication is just reduction of words in the characters
$\boldsymbol{e}_{i}$ subject to the relations $\boldsymbol{e}_{i}^{2}=1$ and
$\boldsymbol{e}_{i}\boldsymbol{e}_{j}=-\boldsymbol{e}_{j}\boldsymbol{e}_{i}$
when $i\neq j$.

######  Remark 2.1.

Though we will not cover the content here, it is worth mentioning that versors
in a geometric algebra $\mathcal{G}_{p,q}$ form a group under multiplication
called the _Clifford group_ and the unit versors define the _spin group_
$\mathrm{Spin}(p,q)$. The algebra of bivectors in $\mathcal{G}_{p,q}$ with the
commutator $[-,-]$ (often written as $\times$ as well) is the Lie algebra
$\mathfrak{spin}(p,q)$.

Geometric algebras also have a unique isomorphism
$\perp\colon\mathcal{G}^{k}\to\mathcal{G}^{n-k}$ and this $\perp$ is
equivalent to the Hodge star $\star$ in $\bigwedge(V)$. To construct this
isomorphisms, we first take an arbitrary basis for $V$ and build the volume
element

$\mu\boldsymbol{I}\coloneqq\boldsymbol{e}_{1}\wedge\cdots\wedge\boldsymbol{e}_{n},$
(20)

where the unit $n$-blade $\boldsymbol{I}$ is the _unit pseudoscalar_ and the
scalar $\mu$ represents the volume scaling. Note that $\boldsymbol{I}$
represents the subspace $V$ and for geometric algebras it defines the _dual_

$A^{\perp}\coloneqq A\boldsymbol{I}^{-1}.$ (21)

For example, if $\boldsymbol{v}$ is a vector then we have
$\boldsymbol{v}^{\perp}$ is a pseudovector representing a scaled copy of the
hyperplane perpendicular to $\boldsymbol{v}$.

Moreover, the dual allows for exchanging products

$(A\hskip 1.42262pt\lrcorner\hskip 1.42262ptB)^{\perp}=A\wedge B^{\perp}$ (22)

and the exterior product

$(A\wedge B)^{\perp}=A\hskip 1.42262pt\lrcorner\hskip 1.42262ptB^{\perp}$ (23)

and helps elucidate the geometrical meaning of $\hskip
1.42262pt\lrcorner\hskip 1.42262pt$ and $\wedge$ (see [21]). Finally, it is
worth saying that in $\mathcal{G}_{n}$ we have
$\boldsymbol{I}^{-1}=\boldsymbol{I}^{\dagger}$.

Given a $k$-dimensional subspace $U\subset V$ in a space with nonsingular $Q$,
we can put $V=U\oplus U^{\perp}$ where $U^{\perp}$ is an $n-k$-dimensional
subspace. In much the same way, given a unit $k$-blade $\boldsymbol{U_{k}}$ we
can find a decomposition of $\boldsymbol{I}$ by
$\boldsymbol{I}=\boldsymbol{U_{k}}\wedge\boldsymbol{U_{k}}^{\perp}$. Actually,
multiplication in $\mathcal{G}$ allows for projection onto subspaces using
this identification.

######  Definition 2.2.

Given an multivector $B$ and unit $k$-blade $\boldsymbol{U_{k}}$, the
_projection_ onto the subspace $\boldsymbol{U_{k}}$ is

$\mathsf{P}_{\boldsymbol{U_{k}}}(B)\coloneqq(B\hskip 1.42262pt\lrcorner\hskip
1.42262pt\boldsymbol{U_{k}})\boldsymbol{U_{k}}^{-1}.$ (24)

The projection preserves grades
$\mathsf{P}_{\boldsymbol{U_{k}}}(B_{\ell})\in\mathcal{G}^{\ell}$. A specific
application of the projection is to view spinors along planes or, eventually,
surfaces. This will be our key methodology to bootstrap from complex analysis.

######  Definition 2.3.

Let $\boldsymbol{B}$ be a unit 2-blade, then the space of _plane spinors_ are
the elements

$\mathbb{A}_{\boldsymbol{B}}\coloneqq\mathbb{R}\oplus\operatorname{Span}(\boldsymbol{B}).$
(25)

As we will see in the following example, $\mathcal{G}_{2}^{+}\cong\mathbb{C}$
and $\mathcal{G}_{3}^{+}\cong\mathbb{H}$. Furthermore, for any
$\boldsymbol{B}\in\mathcal{G}_{n}$ we also have
$\mathbb{A}_{\boldsymbol{B}}\cong\mathbb{C}$. Hence, just like the
Grassmannian of planes in $\mathbb{R}^{3}$, $\mathrm{Gr}(2,3)$, parameterizes
copies of $\mathbb{C}$ in $\mathbb{H}$ by choice of imaginary unit, plane
spinors $\mathbb{A}_{\boldsymbol{B}}$ as a subalgebra of $\mathcal{G}^{+}_{n}$
are parameterized by the Grassmannian $\mathrm{Gr}(2,n)$. If you find the
following example lacking, then Doran & Lasenby’s text [18] is very insightful
and filled with great intuition.

######  Example 2.4.

Rather than a sequence of multiple examples, it will prove to be far more
illuminating to construct one large example for which most of the
preliminaries to this point can be used in a meaningful way. As such, we shall
not rule out the utility that other researchers may gain out of using
geometric algebras with pseudo inner products even though this paper is
predominantly concerned with the positive definite case. The classical example
is the _spacetime algebra_ defined by taking $V=\mathbb{R}^{4}$ with a vector
basis
$\boldsymbol{e}_{0},\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}$
satisfying

$\displaystyle\boldsymbol{e}_{0}\cdot\boldsymbol{e}_{0}$ $\displaystyle=-1$
(26a) $\displaystyle\boldsymbol{e}_{0}\cdot\boldsymbol{e}_{i}$
$\displaystyle=0$ $\displaystyle i=1,2,3$ (26b)
$\displaystyle\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j}$
$\displaystyle=\delta_{ij},$ $\displaystyle i,j=1,2,3.$ (26c)

We refer to $\boldsymbol{e}_{0}$ as _temporal_ since its square is negative
and $\boldsymbol{e}_{i}$ for $i=1,2,3$ are _spatial_ since their squares are
positive. For this basis, the matrix for this inner product in this basis
assumes the form $\eta=\operatorname{diag}(-1,\leavevmode\nobreak\
+1,\leavevmode\nobreak\ +1,\leavevmode\nobreak\ +1)$ (often called the
_Minkowski metric_). The associated quadratic form $Q$ can be found from
$\eta$ by polarization. For a spacetime vector
$\boldsymbol{v}=v_{0}\boldsymbol{e}_{0}+v_{1}\boldsymbol{e}_{1}+v_{2}\boldsymbol{e}_{2}+v_{3}\boldsymbol{e}_{3}$,

$|\boldsymbol{v}|^{2}=(\boldsymbol{v},\boldsymbol{v})=\boldsymbol{v}\cdot\boldsymbol{v}=-v_{0}^{2}+\sum_{i=1}^{3}v_{i}^{2}.$
(27)

It is clear that the norm is definite when all vectors are spatial, but in the
case of spacetime there are null vectors $\boldsymbol{c}$ such that
$|\boldsymbol{c}|=0$. For example,
$\boldsymbol{c}=\boldsymbol{e}_{0}+\boldsymbol{e}_{1}$. The collection of null
vectors define the light cone in Minkowski space. Also, it is important to
distinguish these null vectors from degenerate vectors. Though
$\boldsymbol{c}$ is null, it is not true that for any $\boldsymbol{c}$ and all
other vector $\boldsymbol{v}$ that $\boldsymbol{c}\cdot\boldsymbol{v}=0$. This
is only true for other $\boldsymbol{v}$ on the light cone and the light cone
is not a subspace.

As the notation above suggests, the geometric algebra of Euclidean space
$\mathbb{R}^{3}$, $\mathcal{G}_{3}$, should naturally appear inside of the
spacetime algebra. The spatial _trivector_
$\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{3}$ is unit

$|\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{3}|=\sqrt{\left\langle(\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{3})^{\dagger}\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{3}\right\rangle}=\sqrt{\left\langle\boldsymbol{e}_{3}\boldsymbol{e}_{2}\boldsymbol{e}_{1}\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{3}\right\rangle}=1$
(28)

and represents the spatial subspace
$\mathrm{Span}(\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3})\subset\mathbb{R}^{4}$.
With slight abuse of notation, the projection of $\mathcal{G}_{1,3}$ onto this
subspace yields

$\mathsf{P}_{\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{3}}(\mathcal{G}_{1,3})=\mathcal{G}_{3}.$
(29)

In $\mathcal{G}_{3}$, we can specify an arbitrary multivector $A$ by

$A=a_{0}+a_{1}\boldsymbol{e}_{1}+a_{2}\boldsymbol{e}_{2}+a_{3}\boldsymbol{e}_{3}+a_{12}\boldsymbol{e}_{1}\boldsymbol{e}_{2}+a_{13}\boldsymbol{e}_{1}\boldsymbol{e}_{3}+a_{23}\boldsymbol{e}_{2}\boldsymbol{e}_{3}+a_{123}\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{3}.$
(30)

It will be pertinent later to define
$\boldsymbol{B}_{ij}\coloneqq\boldsymbol{e}_{i}\boldsymbol{e}_{j}$ for $i\neq
j$. Using this substitution, the grade projections read

$\displaystyle\left\langle A\right\rangle$ $\displaystyle=a_{0}$ (31a)
$\displaystyle\left\langle A\right\rangle_{1}$
$\displaystyle=a_{1}\boldsymbol{e}_{1}+a_{2}\boldsymbol{e}_{2}+a_{3}\boldsymbol{e}_{3}$
(31b) $\displaystyle\left\langle A\right\rangle_{2}$
$\displaystyle=a_{12}\boldsymbol{B}_{12}+a_{13}\boldsymbol{B}_{13}+a_{23}\boldsymbol{B}_{23}$
(31c) $\displaystyle\left\langle A\right\rangle_{3}$
$\displaystyle=a_{123}\boldsymbol{e}_{1}\boldsymbol{e}_{2}\boldsymbol{e}_{3}.$
(31d)

Hence, we can write a spinor as

$A_{+}=a_{0}+a_{12}\boldsymbol{B}_{12}+a_{13}\boldsymbol{B}_{13}+a_{23}\boldsymbol{B}_{23}.$
(32)

Note as well that the spatial unit 2-blades always satisfy

$\displaystyle\boldsymbol{B}_{23}^{2}=\boldsymbol{B}_{13}^{2}=\boldsymbol{B}_{12}^{2}=-1$
(33)

and we find that

$\boldsymbol{B}_{23}\boldsymbol{B}_{13}\boldsymbol{B}_{12}=-1.$ (34)

Hence, the even subalgebra $\mathcal{G}_{3}^{+}$ isomorphic to the quaternion
algebra $\mathbb{H}$ by

$\mathbf{i}\leftrightarrow\boldsymbol{B}_{23},\quad\mathbf{j}\leftrightarrow\boldsymbol{B}_{13},\quad\mathbf{k}\leftrightarrow\boldsymbol{B}_{12}$
(35)

Given a quaternion, there is an equivalent spinor $A_{+}$; the imaginary part
of the quaternion corresponds to the grade two part of the spinor
$\left\langle A_{+}\right\rangle_{2}$.

We can project down one dimension further by
$\mathsf{P}_{\boldsymbol{B}_{12}}(\mathcal{G}_{3})=\mathcal{G}_{2}$ and we can
verify quickly that

$\displaystyle\mathsf{P}_{\boldsymbol{B}_{12}}(A)=a_{0}+a_{1}\boldsymbol{e}_{1}+a_{2}\boldsymbol{e}_{2}+a_{12}\boldsymbol{B}_{12}.$
(36a)

Given that $\boldsymbol{B}_{12}^{2}=-1$ we can put $z\coloneqq
x+y\boldsymbol{B}_{12}\in\mathcal{G}_{2}^{+}$ for $x,y\in\mathbb{R}$ which is
exactly a representation of the complex number $\zeta=x+\mathbf{i}y$ in
$\mathbb{C}$ and $\mathbf{i}$ here can be thought of as the unit pseudoscalar
in the plane. Again, the imaginary part is $\left\langle z\right\rangle_{2}$.

But, the above work is not special to the starting point of
$\mathcal{G}_{1,3}$ or $\mathcal{G}_{3}$. In fact, if we take
$\mathcal{G}_{n}$ for $n\geq 2$, then there are natural copies of $\mathbb{C}$
contained inside of $\mathcal{G}_{n}$. In particular, we have the isomorphism

$\mathbb{C}\cong\\{x+y\boldsymbol{B}\leavevmode\nobreak\ |\leavevmode\nobreak\
x,y\in\mathbb{R},\leavevmode\nobreak\
\boldsymbol{B}\in\operatorname{Gr}(2,n).\\},$ (37)

which shows that complex numbers arise as plane spinors via the representation
$\zeta=x+y\boldsymbol{B}$. Given the standard basis
$\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{n}$ we have the ${n\choose 2}$ unit
bivectors $\boldsymbol{B}_{ij}$ for $j=1,\dots,n$ and $i<j$. The plane spinors
$\mathbb{A}_{\boldsymbol{B}_{ij}}$ are each isomorphic to $\mathbb{C}$.

###  2.2 Clifford Analysis

Given a smooth semi-Riemannian manifold $M$ with metric tensor field $g$ and
boundary $\partial M$, each tangent space $T_{x}M$ can be made into a
geometric algebra $\mathcal{G}_{x}M\coloneqq C\ell(T_{x}M,g_{x})$ and we call
$\mathcal{G}_{x}M$ the _geometric tangent space_. The geometric tangent spaces
are glued together to form the geometric algebra bundle
$\mathcal{G}M\coloneqq\bigsqcup_{x\in M}\mathcal{G}_{x}M$. We will call the
$C^{\infty}$ sections of this bundle $C^{\infty}(M;\mathcal{G})$ the _smooth
multivector fields_ and the continuous sections $C^{0}(M;\mathcal{G})$ the
_continuous multivector fields_. Let $\nabla$ be the Levi–Civita connection on
$M$. Then for a vector field $\boldsymbol{v}$ we have the covariant derivative
$\nabla_{\boldsymbol{v}}$ which is extended to multivector fields, e.g., by
[22]. Given local coordinates $x^{i}$ on $M$ we have the induced (gradient)
basis $\boldsymbol{e}_{i}(x)\in\mathcal{G}_{x}M$. Suppressing the pointwise
notation, $\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j}=g_{ij}$. Also in the
tangent space are the reciprocal vectors $\boldsymbol{e}^{i}$ which are Riesz
representatives corresponding to the dual basis $dx^{i}$ since
$dx^{i}(\boldsymbol{e}_{j})=\boldsymbol{e}^{i}\cdot\boldsymbol{e}_{j}$. We
will not get rid of the dual basis entirely since we still find use for the
elements as coordinate measures in integration. Basis blades
$\boldsymbol{E}_{\mathcal{I}}$ and their reciprocals
$\boldsymbol{E}^{\mathcal{I}}$ carry over to each geometric tangent space as
well.

The _Hodge–Dirac operator_ (or _gradient_) $\boldsymbol{\nabla}$ in these
coordinates is

$\boldsymbol{\nabla}\coloneqq\sum_{i=1}^{n}\boldsymbol{e}^{i}\nabla_{\boldsymbol{e}_{i}}.$
(38)

This derivative acts algebraically as a vector and

$\boldsymbol{\nabla}f=\boldsymbol{\nabla}\hskip 1.42262pt\lrcorner\hskip
1.42262ptf+\boldsymbol{\nabla}\wedge f.$ (39)

In particular, $\boldsymbol{\nabla}\wedge$ is equivalent to the exterior
derivative $d$ on differential forms $\Omega(M)$, $\boldsymbol{\nabla}\hskip
1.42262pt\lrcorner\hskip 1.42262pt$ is equivalent to the codifferential
$\delta$, and $\boldsymbol{\nabla}^{2}=\Delta$ is the Laplace–Beltrami
operator. The kernel of $d+\delta$ on $\Omega^{k}(M)$ is coupled to the
(co)homology of $M$ and so $\boldsymbol{\nabla}$ retains this property as
well. This relationship of the analysis of $d+\delta$ (and equivalently
$\boldsymbol{\nabla}$) to the absolute and relative (co)homology of $M$ is
formalized in Hodge theory and is useful for proving existence and uniqueness
for boundary value problems [23]. However, in Clifford analysis, it is
commonplace to consider mixed grade multivectors also in the kernel of
$\boldsymbol{\nabla}$. This space is far bigger since we find that grades can
“mix” together. For example, take $M$ to be the unit disk
$\mathbb{D}\subset\mathbb{R}^{2}$ and $f_{+}=f_{0}+f_{2}\boldsymbol{B}$ with
$f_{0},f_{2}\in C^{\infty}(\mathbb{D};\mathbb{R})$ where $\boldsymbol{B}$ is
the unit 2-blade field. Then we have

$\boldsymbol{\nabla}f_{+}=\boldsymbol{\nabla}\wedge
f_{0}+\boldsymbol{\nabla}\hskip 1.42262pt\lrcorner\hskip
1.42262pt(f_{2}\boldsymbol{B}).$ (40)

If we considered only singly graded elements such as the scalar fields $f_{0}$
or bivector fields $f_{2}\boldsymbol{B}$ on their own, then the only elements
in the kernel of $\boldsymbol{\nabla}$ are constant fields. On the other hand,
when we combine them together into a spinor field $f_{+}\in
C^{\infty}(\mathbb{D};\mathcal{G}^{+})$ then this $f_{+}$ now be any
holomorphic function such as $z=x^{1}+x^{2}\boldsymbol{B}$ or even any power
series in $z$ which we will revisit later. Since this space is far bigger, it
can be used to extract more topological data about $M$, namely the
homeomorphism type, as we see in theorem 3.2.

######  Definition 2.5.

Let $f\in C^{\infty}(M;\mathcal{G})$, then we say that $f$ is _monogenic_ if
$\boldsymbol{\nabla}f=0$. We denote the space of monogenic fields by
$\mathcal{M}(M)$.

Monogenic fields are the emphasis of Clifford analysis and many of the
theorems of holomorphic functions in complex analysis apply to these fields.
Once again, on the unit disk $\mathbb{D}$ take
$f_{+}=f_{0}+f_{2}\boldsymbol{B}$ then if $\boldsymbol{\nabla}f_{+}=0$ we can
use eq. 39 to derive the Cauchy–Riemann equations

$\displaystyle\frac{\partial f_{0}}{\partial x^{1}}$
$\displaystyle=\frac{\partial f_{2}}{\partial x^{2}}$
$\displaystyle\frac{\partial f_{0}}{\partial x^{2}}$
$\displaystyle=-\frac{\partial f_{2}}{\partial x^{1}}$ (41)

which shows the function $z$ is indeed monogenic. For more detail on the
relationship of monogenic spinor fields to complex holomorphic functions see
Doran and Lasenby [18, §6.3.1].

Since $\boldsymbol{\nabla}$ is grade-1, we have $\boldsymbol{\nabla}\colon
C^{\infty}(M;\mathcal{G}^{\pm})\to C^{\infty}(M;\mathcal{G}^{\mp})$ which
yields the direct sum decomposition

$\mathcal{M}(M)=\mathcal{M}^{+}(M)\oplus\mathcal{M}^{-}(M).$ (42)

We can note that each of the graded components of $f_{+}\in\mathcal{M}^{+}(M)$
are also harmonic $\Delta\left\langle f_{+}\right\rangle_{2k}=0$. Also, the
fact that a product of spinor fields is again a spinor field will make it more
convenient to work with $\mathcal{M}^{+}(M)$ instead of the whole of
$\mathcal{M}(M)$. Not only is it more convenient, but to prove the theorem
3.2, we only need $\mathcal{M}^{+}(M)$.

A key piece of the proof for theorem 3.2 will be the ability to uniformly
approximate monogenic spinor fields on open subsets $U$ by fields defined on
$M$. We will cite [16, theorem 11.7] from Calderbank’s thesis.

######  Theorem 2.6 (Calderbank).

Let $U$ be an open subset of $M$. Then any monogenic fields on $U$ may be
approximated (locally uniformly in all derivatives) by restrictions of
monogenic fields on $M$.

This will be used repeatedly. Other supplemental information can be found in
[24] as well.

From Ryan’s [25, theorem 4], we have the fact that for $\mathbb{B}$ a ball in
Euclidean space, there exists a power series representation. In fact, one can
think of this as an explicit realization of the approximation given by
Calderbank which uses the embedding $\mathbb{B}\subset\mathbb{R}^{n}\subset
S^{n}$ where $S^{n}$ is the $n$-sphere. We will construct the power series in
section 3.2. The coefficients, which we define in eq. 82, will require us to
compute integrals of multivector fields which we define next.

In order to integrate, we build differential forms from $k$-vector fields by
attaching a measure. Take the coordinate measures (dual basis in the cotangent
space) $dx^{i}$ in local coordinates and multiply by the corresponding
reciprocal vector to get _basic directed measures_
$d\boldsymbol{x}^{i}=\boldsymbol{e}^{i}dx^{i}$ (no summation implied), which
determine the _$k$ -dimensional directed measures_

$dX_{k}\coloneqq\frac{1}{k!}\sum_{i_{1}<\cdots<i_{k}}d\boldsymbol{x}^{i_{1}}\wedge\cdots\wedge
d\boldsymbol{x}^{i_{k}}=\frac{1}{k!}\sum_{\mathcal{I}}\boldsymbol{E}^{\mathcal{I}^{\dagger}}dx^{\mathcal{I}}$
(43)

An arbitrary differential $k$-form $\alpha_{k}$ is given by taking a
corresponding $k$-vector $A_{k}$ and contracting along the $k$-dimensional
directed measure

$\alpha_{k}=A_{k}\hskip 1.42262pt\lrcorner\hskip 1.42262ptdX_{k}^{\dagger}.$
(44)

Specifically,
$A_{k}=\sum_{\mathcal{I}}\alpha^{\mathcal{I}}\boldsymbol{E}_{\mathcal{I}}$ is
called the _multivector equivalent of $\alpha_{k}$_. This is a realization of
the isomorphism between $C^{\infty}(M;\mathcal{G})$ and $\Omega(M)$ as
$C^{\infty}(M)$-modules and it can be viewed as an extension of the musical
isomorphisms between vectors and 1-forms [26, chapter 13]. The multivector
equivalent of the Riemannian volume form $\mu$ is
$\boldsymbol{I}^{-1^{\dagger}}$ and $\boldsymbol{I}(x)$ represents the tangent
space $T_{x}M$. Of course, in the case where $g$ is definite, the equivalent
to $\mu$ is simply $\boldsymbol{I}$. On $\partial M$, we have the _boundary
pseuodoscalar_ $\boldsymbol{I}_{\partial}$ and dual to this the boundary
normal vector field $\boldsymbol{\nu}=\boldsymbol{I}_{\partial}^{\perp}$. As
on $M$, the boundary pseudoscalar $\boldsymbol{I}_{\partial}$ is the
multivector equivalent of the boundary area form $\mu_{\partial}$. From this
point forward, we work solely with multivector fields and contract with
directed measures to integrate. We now realize the action of
$\boldsymbol{\nabla}\wedge$ and $\boldsymbol{\nabla}\hskip
1.42262pt\lrcorner\hskip 1.42262pt$ as the equivalents of $d$ and $\delta$ by

$\displaystyle d\alpha_{k}$ $\displaystyle=(\boldsymbol{\nabla}\wedge
A_{k})\hskip 1.42262pt\lrcorner\hskip 1.42262ptdX_{k+1}^{\dagger},$
$\displaystyle\delta\alpha_{k}$ $\displaystyle=(\boldsymbol{\nabla}\hskip
1.42262pt\lrcorner\hskip 1.42262ptA_{k})\hskip 1.42262pt\lrcorner\hskip
1.42262ptdX_{k-1}^{\dagger}.$ (45)

One beautiful result in Clifford analysis is the generalization of the Cauchy
integral formula. Details for various cases are in our standard sources [18,
19, 16, 24]. For $\mathbb{R}^{n}$ with $n\geq 2$, we define the vector-valued
function

$\boldsymbol{G}(\boldsymbol{x})\coloneqq\frac{1}{\omega_{n}}\frac{\boldsymbol{x}}{|\boldsymbol{x}|^{n}}$
(46)

where $\omega_{n}$ is the area of the unit sphere in $\mathbb{R}^{n}$ and the
use of the bold $\boldsymbol{x}$ indicates that we treat this point in space
as a vector. For this work, the bold variable $\boldsymbol{x}$ will be a clear
way to distinguish when we are embedded in $\mathbb{R}^{n}$, so it is
important to keep track of this. The function $\boldsymbol{G}$ is the Green’s
function of the Dirac operator since

$\boldsymbol{\nabla}\boldsymbol{G}(\boldsymbol{x})=\delta_{\boldsymbol{x}},$
(47)

where $\delta_{\boldsymbol{x}}$ is the Dirac mass located at
$\boldsymbol{x}\in\mathbb{R}^{n}$. There is also a solution in the case $n=1$,
but it assumes a different form and we do not need it here.

Define the _Cauchy kernel_
$\boldsymbol{G}(\boldsymbol{x}^{\prime}-\boldsymbol{x})\boldsymbol{\nu}(\boldsymbol{x}^{\prime})$
by translation and multiplication by the outward normal $\boldsymbol{\nu}$ on
the boundary $\partial M$. For compact $M\subset\mathbb{R}^{n}$ and monogenic
field $f\in\mathcal{M}(M)$, we convolve the boundary value of $f$ with the
Cauchy kernel to arrive at the _Cauchy integral formula_

$f(\boldsymbol{x})=(-1)^{n}\int_{\partial
M}\boldsymbol{G}(\boldsymbol{x}^{\prime}-\boldsymbol{x})\boldsymbol{\nu}(\boldsymbol{x}^{\prime})f(\boldsymbol{x}^{\prime})\mu_{\partial}(\boldsymbol{x}^{\prime}).$
(48)

Hence, we have a method for uniquely determining a monogenic field $f$ from
the boundary values $f|_{\partial M}$. A more general notion of the Cauchy
integral exists for arbitrary manifolds with boundary (see [16]), but it seems
this also depends on a choice of embedding into a closed manifold. In the case
of our Green’s function $\boldsymbol{G}$, this corresponds to the embedding
$M\subset\mathbb{R}^{n}\subset S^{n}$ [16, proposition 9.10].

###  2.3 Currents

For the remainder, let $M$ be a smooth oriented compact Riemannian manifold of
dimension $n$ with boundary $\partial M$ and positive definite metric tensor
$g$. Let $\boldsymbol{I}$ be the unit pseudoscalar on $M$ which is the
multivector equivalent of the Riemannian volume form $\mu$,
$\boldsymbol{I}_{\partial}$ the boundary pseuodoscalar which is the
multivector equivalent of the Riemannian boundary area form $\mu_{\partial}$
and dual to $\boldsymbol{\nu}$ the boundary normal field. The space
$C^{0}(M;\mathcal{G})$ comes with the _uniform norm_

$\|f\|\coloneqq\sup_{x\in M}|f(x)|.$ (49)

Recall that since $g$ is positive definite, at any point $x\in M$ we have that
$|f(x)|^{2}=(f(x),f(x))$ is nothing but the Euclidean vector norm on
$\mathbb{R}^{2n}$, so $\|f\|$ really is a norm on $C^{0}(M;\mathcal{G})$.
Furthermore:

######  Proposition 2.7.

If $M$ is a compact Riemannian manifold, then the space $C^{0}(M;\mathcal{G})$
is a (real) $C^{\ast}$-algebra with involution $\dagger$.

###### Proof.

Note that $\mathcal{G}$ is a real $2^{n}$ dimensional Banach space with the
multivector inner product. Since $M$ is a compact Hausdorff space, it follows
that the space $C^{0}(M;\mathcal{G})$ is a Banach space (see [27]). Taking
$f,g\in C^{0}(M;\mathcal{G})$, at each point

$\displaystyle(fg,fg)=(gg^{\dagger},f^{\dagger}f),$ (50)

since $\dagger$ is the adjoint. Using the Cauchy–Schwarz inequality

$\displaystyle|fg|^{2}=(fg,fg)\leq(f^{\dagger}f,f^{\dagger}f)(gg^{\dagger},gg^{\dagger})=|f|^{2}|g|^{2}.$
(51)

The last equality follows from taking an orthonormal basis
$\boldsymbol{e}_{i}$ at any $\mathcal{G}_{x}M$ and forming the orthonormal
vector basis blades (versors) $\boldsymbol{E}_{\mathcal{I}}$ and putting
$f=\sum_{\mathcal{I}}f^{\mathcal{I}}\boldsymbol{E}_{\mathcal{I}}$. Then we
have

$\displaystyle
f^{\dagger}f=\sum_{\mathcal{I}}\sum_{\mathcal{J}}f^{\mathcal{I}}f^{\mathcal{J}}\boldsymbol{E}_{\mathcal{I}}^{\dagger}\boldsymbol{E}_{\mathcal{J}}$
(52)

from which we see that

$\displaystyle(f^{\dagger}f,f^{\dagger}f)=|f^{\dagger}f|^{2}=\sum_{\mathcal{I}}\sum_{\mathcal{J}}\left(f^{\mathcal{I}}f^{\mathcal{J}}\right)^{2}$
(53)

and finally

$\displaystyle\left(|f|^{2}\right)^{2}=\left(\sum_{\mathcal{I}}\left(f^{\mathcal{I}}\right)^{2}\right)^{2}=\sum_{\mathcal{I}}\sum_{\mathcal{J}}\left(f^{\mathcal{I}}\right)^{2}\left(f^{\mathcal{J}}\right)^{2}$
(54)

which implies that $|f^{\dagger}f|=|f|^{2}$. Taking suprema, it follows that
$\|fg\|\leq\|f\|\|g\|$ which shows $C^{0}(M;\mathcal{G})$ is a real Banach
algebra.

For $f,g\in C^{0}(M;\mathcal{G})$ and $\lambda\in\mathbb{R}$ we have that

$(f+g)^{\dagger}=f^{\dagger}+g^{\dagger},\quad(\lambda
f)^{\dagger}=\lambda^{\dagger}f^{\dagger}=\lambda f^{\dagger},\quad
f^{\dagger\dagger}=f,\quad(fg)^{\dagger}=g^{\dagger}f^{\dagger},$ (55)

by definition and at each point

$\displaystyle|f^{\dagger}f|=|f|^{2}$ (56)

as shown before. By taking suprema, $\|f^{\dagger}f\|=\|f\|^{2}$ which shows
$C^{0}(M;\mathcal{G})$ is a real $C^{\ast}$-algebra. ∎

We topologize the $C^{\ast}$-algebra $C^{0}(M;\mathcal{G})$ with the uniform
norm topology. Dually, we construct $\mathcal{G}$ valued functionals on this
space which we call currents _à la_ de Rham.

######  Definition 2.8.

The space of _$\mathcal{G}$ -currents_ is

$C^{0}_{\mathcal{G}}(M;\mathcal{G})^{\prime}\coloneqq\\{T\colon
C^{0}(M;\mathcal{G})\to\mathcal{G}\leavevmode\nobreak\ |\leavevmode\nobreak\
T\textrm{ is continuous}\\}$ (57)

Given a subalgebra $\mathcal{A}\subset\mathcal{G}$, we have the _$\mathcal{A}$
-currents_

$C^{0}_{\mathcal{A}}(M;\mathcal{G})^{\prime}=\\{T\colon
C^{0}(M;\mathcal{G})\to\mathcal{G}\leavevmode\nobreak\ |\leavevmode\nobreak\
T\textrm{ is continuous}\\}.$ (58)

The $\mathcal{G}$-currents are given the _weak- $\ast$ topology_, i.e., the
coarsest topology on $C^{0}_{\mathcal{G}}(M;\mathcal{G})^{\prime}$ where point
evaluation of fields is continuous. Specifically, for $x\in M$, the Dirac mass
$\mathcal{G}$-current $\delta_{x}\in
C^{0}_{\mathcal{G}}(M;\mathcal{G})^{\prime}$ defined by $\delta_{x}[f]=f(x)$
for $f\in C^{0}(M;\mathcal{G})$ is continuous. The $\mathcal{A}$-currents
inherit the subspace topology.

Since the target $\mathcal{G}$ of the $\mathcal{G}$-currents is itself a
$C^{\ast}$-algebra and a $\mathcal{G}$-module, we expect some currents to
respect these algebraic structures. For example, $C^{0}(M;\mathcal{G}^{+})$ is
a $\mathcal{G}^{+}$-bimodule. Given a $\mathcal{A}\subset\mathcal{G}^{+}$,
$\mathcal{G}^{+}$ and $C^{0}(M;\mathcal{G}^{+})$ are both
$\mathcal{A}$-modules and Banach algebras.

######  Definition 2.9.

Let $\mathcal{A}\subset\mathcal{G}$ be a subalgebra and let $T\in
C^{0}_{\mathcal{G}}(M;\mathcal{G})^{\prime}$ be a $\mathcal{G}$-current. We
say that $T$ is _right $\mathcal{A}$-linear_ if it is a right
$\mathcal{A}$-module homomorphism

$T[f\alpha+g]=T[f]\alpha+T[g]$ (59)

for $\alpha\in\mathcal{A}$ and $f,g\in C^{0}(M;\mathcal{G})$. Furthermore, we
say that $T$ is _multiplicative on $\mathcal{A}$_ if it is an
$\mathbb{R}$-algebra homomorphism

$T[pq]=T[p]T[q]$ (60)

for $p,q\in C^{0}(M;\mathcal{A})$. Finally, a current $T$ is _grade
preserving_ if for $h\in C^{0}(M;\mathcal{G}^{k})$ we have
$T[h]\in\mathcal{G}^{k}$.

The set of grade preserving linear multiplicative currents are the most useful
for us. It is worth remarking that currents as defined here provide an ample
setting for further study. There are plenty of tweaks that could be
interesting. One such example would be the subset of the de Rham currents
(dual to the $C^{\infty}$-smooth forms) given by the $\mathbb{R}$-currents
$C^{0}_{\mathbb{R}}(M;\mathcal{G})^{\prime}$. Here, we will make choices that
allow us to generalize the classical Gelfand result.

###  2.4 Subsurface fields

The algebra $C^{0}(M;\mathcal{G})$ is not commutative in general and the space
$\mathcal{M}(M)$ is not an algebra. This poses a direct issue for a
straightforward generalization of the Gelfand representation and Belishev’s
2-dimensional boundary control method [10], but it does not thwart the effort
completely since Belishev and Vakulenko manage to build a 3-dimensional
version [14]. Extending this approach, we will use insight on axial fields
from those two authors but make the change to think not of an axis, but of a
plane. Of course, in $\mathbb{R}^{3}$ a plane and axis are dual, but when we
extend beyond dimension-3, we will be required to use planes. If $S$ is
dimension 2, then $\mathcal{M}^{+}(S)$ is a copy of the commutative algebra of
holomorphic functions. Intuitively, we can build commutative Banach algebras
of monogenic fields for surfaces in $M$.

Let $U\subset M$ be a geodesically convex region, i.e., that all points $x\in
U$ are connected with unique shortest paths. Let $\boldsymbol{B}(x)$ be a unit
2-blade in $\mathcal{G}_{x}U$ for some $x\in U$. Since $U$ is convex, there
exists a shortest geodesic between all points in $U$ which allows us to
parallel transport $\boldsymbol{B}(x)$ to build a unit 2-blade field
$\boldsymbol{B}\in C^{\infty}(U;\mathcal{G}^{2})$. Then, at all points in $U$,
we have a projection $\mathsf{P}_{\boldsymbol{B}}$ onto $\boldsymbol{B}(y)$ in
each geometric tangent space $\mathcal{G}_{y}U$.

######  Definition 2.10.

Let $U$ and $\boldsymbol{B}$ be as before, then a continuous spinor field
$f\in C^{0}(U;\mathcal{G}^{+})$ satisfying

$f_{+}=\mathsf{P}_{\boldsymbol{B}}\circ f_{+}$ (61)

is a _subsurface spinor field on $U$_.

The definition for a subsurface spinor field on $U$ requires that
$f_{+}=\mathsf{P}_{\boldsymbol{B}}\circ f_{+}$ which means that we can put
$f_{+}=f_{0}+f_{2}\boldsymbol{B}$ where $f_{0},f_{2}\in C^{0}(U;\mathbb{R})$.

######  Definition 2.11.

Let $U$ and $\boldsymbol{B}$ be as before, then _the space of monogenic
subsurface spinors on $U$_ is

$\mathcal{A}_{\boldsymbol{B}}(U)=\\{f_{+}\in
C^{0}(U;\mathcal{G}^{+})\leavevmode\nobreak\ |\leavevmode\nobreak\
f_{+}=\mathsf{P}_{\boldsymbol{B}}\circ f_{+},\leavevmode\nobreak\
\boldsymbol{\nabla}f_{+}=0\\}.$ (62)

The _collection of all monogenic subsurface spinors on $U$_ is

$\mathcal{A}(U)=\\{f_{+}\in\mathcal{A}_{\boldsymbol{B}}(U)\leavevmode\nobreak\
|\leavevmode\nobreak\ \textrm{$\boldsymbol{B}$ parallel transported from
$\boldsymbol{B}(x)\in\mathcal{G}_{x}U$, \leavevmode\nobreak\
$\forall\boldsymbol{B}(x)\in\mathcal{G}_{x}U$}\\}.$ (63)

######  Proposition 2.12.

Let $U$ and $\boldsymbol{B}$ be as before, then the space
$\mathcal{A}_{\boldsymbol{B}}(U)$ is a commutative Banach algebra.

###### Proof.

Note that multiplication of two fields $f=f_{0}+f_{2}\boldsymbol{B}$ and
$g=g_{0}+g_{2}\boldsymbol{B}$ (dropping the subscripted $+$ on $f$ and $g$
momentarily for clarity) in $\mathcal{A}_{\boldsymbol{B}}(U)$ is commutative
and given pointwise by the familiar complex multiplication

$fg=f_{0}g_{0}-f_{2}g_{2}+\boldsymbol{B}(f_{0}g_{2}+f_{2}g_{0})=gf.$ (64)

Using the overdot notation to say which field we are taking derivatives of, we
find commutivity gives us algebraic closure since

$\displaystyle\boldsymbol{\nabla}(fg)$
$\displaystyle=\boldsymbol{\nabla}fg+\dot{\boldsymbol{\nabla}}f\dot{g}$ by the
Leibniz rule (65) $\displaystyle=0+\boldsymbol{\nabla}gf$ since $f$ is
monogenic and $fg=gf$ (66) $\displaystyle=0$ since $g$ is monogenic. (67)

Since $\mathcal{A}_{\boldsymbol{B}}(U)$ is a subalgebra of
$C^{0}(U;\mathcal{G})$, it is a commutative Banach algebra. ∎

This construction provides a notion of complex functions that are nested in
multivector fields on any manifold of dimension $n\geq 2$. In the case $n=1$,
no such fields exist and it is exactly in the 2-dimensional Euclidean case
that the complex-valued functions are just the spinor fields themselves and
the unit 2-blade field is the tangent pseudoscalar to the surface. The special
case of monogenic subsurface spinor fields serve as a realization of complex
holomorphic functions inside the more general spinor fields. If we take
$\boldsymbol{B}=\boldsymbol{e}_{1}\boldsymbol{e}_{2}$, then we have the
Cauchy–Riemann equations from $\boldsymbol{\nabla}f_{+}=0$ via eq. 41.

For example, take the case where $M$ is a compact region of $\mathbb{R}^{n}$
with the Euclidean metric. Then $M$ itself is compactly contained inside of
some ball $\mathbb{B}$ which is convex. The set of bivectors is parameterized
by $\boldsymbol{B}\in\operatorname{Gr}(2,n)$ (i.e., the possible coordinate
planes) and for each such $\boldsymbol{B}$ we can consider
$\mathcal{A}_{\boldsymbol{B}}(M)$ as a restriction of
$\mathcal{A}_{\boldsymbol{B}}(\mathbb{B})$ via theorem 2.6. Each unit 2-blade
decomposes into two orthogonal unit vectors. Let
$\boldsymbol{B}=\boldsymbol{v}\boldsymbol{w}$ where $\boldsymbol{v}$ and
$\boldsymbol{w}$ are a pair of orthogonal unit vectors and consider the
monogenic subsurface field $z\colon
M\to\mathbb{A}_{\boldsymbol{B}}\subset\mathcal{G}^{+}$ defined by

$z(\boldsymbol{x})\coloneqq\mathsf{P}_{\boldsymbol{B}}(\boldsymbol{v}\boldsymbol{x}).$
(68)

It is immediately clear that $z=\mathsf{P}_{\boldsymbol{B}}\circ z$ and in
applying the Hodge–Dirac operator

$\displaystyle\boldsymbol{\nabla}z$
$\displaystyle=\boldsymbol{\nabla}(\boldsymbol{x}\cdot\boldsymbol{v})+\boldsymbol{\nabla}(\boldsymbol{x}\cdot\boldsymbol{w})\boldsymbol{B}$
(69) $\displaystyle=0.$ (70)

We can define such a function $z$ for any choice of $\boldsymbol{B}$ and
construct new functions from polynomials in these variables. The notation $z$
should serve as a reminder of the connection to complex analysis and one may
consider $\boldsymbol{v}$ as the real axis and $\boldsymbol{w}$ as the
imaginary axis. The behavior of fields on an arbitrary convex $U$ inside an
arbitrary compact $M$ is identical.

##  3 A Clifford-Algebraic Gelfand Theorem

###  3.1 Spinor Spectrum

Through Belishev’s generalization of Gelfand’s classical result, surfaces are
determined up to conformal equivalence by the spectrum (or maximal ideal
space) of $\mathcal{M}^{+}(S)$. The naive generalization would be to seek this
out in $\mathcal{M}^{+}(M)$, but, again, this space is not an algebra! Maximal
ideals can also be identified with multiplicative functionals and this allowed
Belishev and Vakulenko to achieve their 3-dimensional result. We follow suit
with multiplicative linear currents.

######  Definition 3.1.

The _spinor spectrum_ $\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)\subset
C^{0}_{\mathcal{G}^{+}}(M;\mathcal{G}^{+})^{\prime}$ is the set of nonzero
grade preserving right linear currents that are multiplicative over the
collection of all subsurface spinor algebras $\mathcal{A}(U)$,

$\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)\coloneqq\\{\delta\neq 0\in
C^{0}_{\mathcal{G}^{+}}(M;\mathcal{G}^{+})^{\prime}\leavevmode\nobreak\
|\leavevmode\nobreak\ \textrm{$\delta$ grade preserving,}\\\
\delta(fg+h\alpha)=\delta(f)\delta(g)+\delta(h)\alpha,\leavevmode\nobreak\
\forall f,g,h\in\mathcal{A}(U),\leavevmode\nobreak\
\alpha\in\mathcal{G}^{+}\\},$ (71)

and we refer to the elements as _spin characters_.

One choice of spin character is point evaluation: if $\delta$ is defined on
$f_{+}\in\mathcal{M}^{+}(M)$ by $\delta(f_{+})=f_{+}(x^{\delta})$ for some
$x^{\delta}\in M$, then it follows that $\delta\in\mathrm{sp}\hskip
2.0pt\mathcal{M}^{+}(M)$. This shows that $M$ injects into $\mathrm{sp}\hskip
2.0pt\mathcal{M}^{+}(M)$ by the map $x\mapsto\delta_{x}$ where
$\delta_{x}[f]=f(x)$. We will find that (at least for embedded $M$) characters
defined by point evaluation are the only elements of $\mathrm{sp}\hskip
2.0pt\mathcal{M}^{+}(M)$. This shows the inclusion is surjective. In fact, the
main result is that this map is a homeomorphism.

######  Theorem 3.2.

Let $M$ be a compact region in $\mathbb{R}^{n}$. For any
$\delta\in\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)$, there is a point
$\boldsymbol{x}_{\delta}\in M$ such that
$\delta(f)=f(\boldsymbol{x}_{\delta})$ for any $f_{+}\in\mathcal{M}^{+}(M)$ a
monogenic field. Given the weak-$\ast$ topology on the space of
$\mathcal{G}$-currents, the map

$\Gamma\colon\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)\to
M,\quad\delta\mapsto\boldsymbol{x}_{\delta}$

is a homeomorphism. The Gelfand transform $\mathcal{M}^{+}(M)\to
C^{0}(\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M);\mathcal{G}^{+})$ given by
$\widehat{f_{+}}(\delta)=\delta[f_{+}]$ is an isometric isomorphism onto its
image so that $\mathcal{M}^{+}(M)\cong\widehat{\mathcal{M}^{+}(M)}$.

We prove theorem 3.2 in the following steps:

  1. i.

Utilize a power series representation for elements in a ball $\mathbb{B}$
which shows that the monogenic polynomials
$\mathcal{M}^{\mathcal{P}}(\mathbb{B})$ are dense in $\mathcal{M}^{+}(M)$.

  2. ii.

Build the elements of this series from homogeneous polynomials in variables of
the form $z$ (i.e., eq. 68). Using the fact that the spin characters are
multiplicative over the collection $\mathcal{A}(M)$, continuous, and
$\mathcal{G}^{+}$-linear we show that it suffices to determine the action
$\delta[z]$ for $\delta\in\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)$.

  3. iii.

Determine that the action $\delta[z]$ is point evaluation at some point
$\boldsymbol{x}_{\delta}\in\mathbb{R}^{n}$ by looking at the algebraic
relationships between the variables $z$ and combining this with the
multiplicativity of $\delta$.

  4. iv.

Construct a carefully selected sequence of monogenic fields on $M$ and use
continuity of $\delta$ to show that $\boldsymbol{x}_{\delta}\in M$.

The correspondence between $\delta\in\mathrm{sp}\hskip
2.0pt\mathcal{M}^{+}(M)$ is then clear and the homeomorphism follows by choice
of the weak-$\ast$ topology. The fact that the Gelfand transform is an
isometry follows directly from the fact that each character corresponds to
point evaluation.

###  3.2 Power series

Take the standard orthonormal basis fields
$\boldsymbol{e}_{1},\dots,\boldsymbol{e}_{n}$ for $\mathbb{R}^{n}$ and define
the functions $z_{ij}=x^{j}-x^{i}\boldsymbol{B}_{ij}$ where $x^{i}$ are the
coordinate functions corresponding to our basis vector fields and where
$\boldsymbol{B}_{ij}\coloneqq\boldsymbol{e}_{i}\boldsymbol{e}_{j}$. This takes
eq. 68 and multiplies by $\boldsymbol{B}_{ij}$ to match Ryan [25] and, in
effect, this is just replacing $z$ with $\mathbf{i}z$. Note that Ryan’s use of
$\boldsymbol{e}_{i}^{-1}$ become unnecessary due to our choice of a positive
definite quadratic form. Identifying $\boldsymbol{B}_{ij}$ with its plane,
note that for any compact region $M\subset\mathbb{R}^{n}$ each
$z_{ij}\in\mathcal{A}_{\boldsymbol{B}_{ij}}(M)$. Fix a natural number $k\geq
0$ and natural numbers $k_{j}$ to form the tuple $\vec{k}=(k_{2},\dots,k_{n})$
such that $k_{2}+\cdots+k_{n}=k$ with $k_{j}\geq 0$. This is often called a
multi-index with absolute value $|\vec{k}|=k$. The set of all multi-indices of
absolute value $k$ is of size ${n-2+k\choose n-2}$. For example, we can write
down a degree-$k$ polynomial in terms of the monomial variables $z_{ij}$ based
on a multi-index $\vec{k}$ by

$z_{12}^{k_{2}}(\boldsymbol{x})z_{13}^{k_{3}}(\boldsymbol{x})\cdots
z_{1n}^{k_{n}}(\boldsymbol{x}).$ (72)

But, ordering does matter. To build the _homogeneous monogenic degree $k$
polynomials_ we sum over permutations $\sigma$ which rearrange the order in
which we write the monomials but keep the total powers of each monomial the
same throughout

$p_{\vec{k}}(\boldsymbol{x})=\frac{1}{k!}\sum_{\sigma}z_{1\sigma(1)}(\boldsymbol{x})\cdots
z_{1\sigma(k)}(\boldsymbol{x}),$ (73)

where $\sigma(j)\in\\{2,\dots,n\\}$ permutes the monomials without
rearrangement. To reiterate, monomials that appear in the summand of eq. 73
with the powers given by $\vec{k}$ are just reordered from what we see in eq.
72 based on $\sigma$ which is why we do not see $\vec{k}$ explicitly appear on
the right hand side of eq. 73. Note that this is necessary since the monomials
do not commute with one another. Ryan [25, Proposition 1] shows each of these
polynomials is monogenic and linearly independent. We remark that the
polynomials $p_{\vec{k}}$ are homogeneous in the elements
$z_{ij}\in\mathcal{A}_{\boldsymbol{B}_{ij}}(M)$.

As examples, take $n=3$ and $k=2$ with $k_{2}=2$ and $k_{3}=0$ so that the
multi-index is $\vec{k}=(2,0)$. Then in coordinates
$\boldsymbol{x}=(x^{1},x^{2},x^{3})$

$\displaystyle p_{(2,0)}(\boldsymbol{x})$
$\displaystyle=\frac{1}{2!}\sum_{\sigma}z_{1\sigma(1)}z_{1\sigma(2)}$ (74)
$\displaystyle=\frac{1}{2!}z_{12}(\boldsymbol{x})z_{12}(\boldsymbol{x})$ (75)
$\displaystyle=\frac{1}{2!}(x^{2}-x^{1}\boldsymbol{e}_{1}\boldsymbol{e}_{2})^{2}.$
(76)

We can see that given our choice of $\vec{k}$, there is only one choice of
$\sigma$, i.e., the $\sigma$ such that $\sigma(1)=2$ and $\sigma(2)=2$. On the
other hand if we take the multi-index $\vec{k}=(1,1)$, then

$\displaystyle p_{(1,1)}(x_{1},x_{2},x_{3})$
$\displaystyle=\frac{1}{2!}\sum_{\sigma}z_{1\sigma(1)}z_{\sigma(2)}$ (77)
$\displaystyle=\frac{1}{2!}\left(z_{12}(\boldsymbol{x})z_{13}(\boldsymbol{x})+z_{13}(\boldsymbol{x})z_{12}(\boldsymbol{x})\right)$
(78)
$\displaystyle=\frac{1}{2!}\left((x^{2}-x^{1}\boldsymbol{e}_{1}\boldsymbol{e}_{2})(x^{3}-x^{1}\boldsymbol{e}_{1}\boldsymbol{e}_{3})+(x^{3}-x^{1}\boldsymbol{e}_{1}\boldsymbol{e}_{3})(x^{2}-x^{1}\boldsymbol{e}_{1}\boldsymbol{e}_{2})\right).$
(79)

Again, our choice of $\vec{k}$ allowed for two choices of $\sigma$ that were
not repetitive: first $\sigma(j)=j$ and the other $\sigma(1)=2$ and
$\sigma(2)=1$. Furthermore, working out the details of
$\boldsymbol{\nabla}p_{(k_{2},k_{3})}$ shows the necessity of summing over
permutations in order to ensure that each is monogenic. The collection of all
such polynomials for all multi-indices is the set of _monogenic polynomials_

$\mathcal{M}^{\mathcal{P}}(M)=\left.\left\\{\sum_{k=0}^{N}\left(\sum_{\begin{subarray}{c}{\vec{k}}\\\
{|\vec{k}|=k}\end{subarray}}p_{\vec{k}}(\boldsymbol{x})a_{\vec{k}}\right)\leavevmode\nobreak\
\right|\leavevmode\nobreak\ N\in\mathbb{N},\leavevmode\nobreak\
a_{\vec{k}}\in\mathcal{G}_{n}\right\\}$ (80)

The use of multi-index notation is also to facilitate taking higher order
partial derivatives by defining

$\nabla^{\vec{k}}\coloneqq\frac{\partial^{k}}{\partial x_{2}^{k_{2}}\partial
x_{3}^{k_{3}}\cdots\partial x_{n}^{k_{n}}}.$ (81)

In the case of a smooth manifold, the partial derivatives can be replaced with
their corresponding covariant derivatives if the need should arise. Next,
lemma 3.3, corollary 3.4, and proposition 3.5 show that for arbitrary $M$,
$\mathcal{M}(M)$ are locally uniformly approximated by monogenic polynomials.

######  Lemma 3.3.

Let $\mathbb{B}$ be a compact ball in $\mathbb{R}^{n}$, then the space
$\mathcal{M}^{\mathcal{P}}(\mathbb{B})$ is dense in $\mathcal{M}(\mathbb{B})$.

###### Proof.

Without loss of generality, suppose $\mathbb{B}$ is centered at the origin.
Then let $f\in\mathcal{M}(\mathbb{B})$ and define the coefficients
$a_{\vec{k}}\in\mathcal{G}_{n}$ by

$a_{\vec{k}}=\int_{\partial\mathbb{B}}\nabla^{\vec{k}}\boldsymbol{G}\boldsymbol{\nu}f\mu_{\partial},$
(82)

where $\boldsymbol{G}$ is the Green’s function for the Hodge–Dirac operator in
$\mathbb{R}^{n}$. By [25, theorem 4], we have

$f(\boldsymbol{x})=\sum_{k=0}^{\infty}\left(\sum_{\begin{subarray}{c}{\vec{k}}\\\
{|\vec{k}|=k}\end{subarray}}p_{\vec{k}}(\boldsymbol{x})a_{\vec{k}}\right),$
(83)

which converges uniformly to $f$ for points
$\boldsymbol{x}\in\operatorname{int}\mathbb{B}$. ∎

But, as stated previously, we have theorem 2.6 which tells us that for open
subsets in $\mathbb{B}$ we can uniformly approximate monogenic fields on those
subsets. We apply this fact to get the following corollary.

######  Corollary 3.4.

Let $M\subset\mathbb{R}^{n}$ be a compact region. Then there exists
$\mathbb{B}$ such that $\mathcal{M}(\mathbb{B})$ are dense in
$\mathcal{M}(M)$.

###### Proof.

Since $M$ is compact in $\mathbb{R}^{n}$ there exists a ball $\mathbb{B}$ such
that the closure of $M$ is contained in $\mathbb{B}$. Then by theorem 2.6, any
monogenic fields on $M$ can be uniformly approximated by monogenic fields in
$\mathcal{M}(\mathbb{B})$, and we have our result by lemma 3.3, ∎

######  Proposition 3.5.

Let $M$ be an $n$-dimensional compact Riemannian manifold and let
$f\in\mathcal{M}(M)$. Then $f$ admits a local power series representation over
finitely many open subsets.

###### Proof.

Take $f\in\mathcal{M}(M)$, let $(U,\varphi)$ a local coordinate chart such
that $U\subset M$ is an open convex region and
$\varphi(U)\subset\mathbb{B}\subset\mathbb{R}^{n}$ where $\mathbb{B}$ is some
closed ball in $\mathbb{R}^{n}$. Then
$f\circ\varphi\in\mathcal{M}(\varphi(U))$ and by lemma 3.3 and corollary 3.4,
$f\circ\varphi$ admits a power series representation. Since $M$ is Riemannian
there exists a finite covering of $M$ by convex regions and this gives us a
local power series representation over finitely many open sets. ∎

Following the details of the above proofs for a surface $S$ yields the fact
that holomorphic functions on a surface admit local power series
representations. In this case, take $x^{1},x^{2}$ as local isothermal
coordinates and define $z=x^{2}-x^{1}\boldsymbol{B}$. Then for
$f_{+}\in\mathcal{M}^{+}(S)$, we have the local power series
$f_{+}(z)=\sum_{k=0}^{\infty}z^{k}a_{k}$ where
$a_{k}\in\mathbb{A}_{\boldsymbol{B}}$.

######  Remark 3.6.

It is important to note that if $f_{+}\in\mathcal{M}^{+}(M)$, the local power
series has coefficients $a_{\vec{k}}\in\mathcal{G}^{+}$ which you can see by
eq. 82.

###  3.3 Characters

For $M$ a compact region embedded in $\mathbb{R}^{n}$ and
$f_{+}\in\mathcal{M}^{+}(M)$, we can see that for $\delta\in\mathrm{sp}\hskip
2.0pt\mathcal{M}^{+}(M)$

$\displaystyle\delta\left[f\right]$
$\displaystyle=\sum_{k=0}^{\infty}\left(\sum_{\vec{k}}\delta[p_{\vec{k}}]a_{\vec{k}}\right)$
(84)

by continuity and right $\mathcal{G}^{+}$-linearity of $\delta$ since
$a_{\vec{k}}\in\mathcal{G}^{+}$. On each monogenic polynomial,

$\displaystyle\delta(p_{\vec{k}})$
$\displaystyle=\frac{1}{k!}\sum_{\sigma}\delta\left[z_{1\sigma(1)}\right]\cdots\delta\left[z_{1\sigma(k)}\right],$
(85)

by multiplicativity over $\mathcal{A}(M)$. Hence, the action of $\delta$ is
completely determined by the action on each $z_{ij}$.

######  Proposition 3.7.

Let $M$ be a compact manifold embedded in $\mathbb{R}^{n}$ and
$\boldsymbol{B}$ a unit 2-blade field that is a parallel translation of a
coordinate plane. Then for any $\delta\in\mathrm{sp}\hskip
2.0pt\mathcal{M}^{+}(M)$ we have
$\delta(\mathcal{A}_{\boldsymbol{B}}(M))=\mathbb{A}_{\boldsymbol{B}}$.

###### Proof.

Since $\delta$ is grade preserving, it must be the case that
$\delta[z]\in\mathcal{G}^{0\oplus 2}$. Since $\delta$ is an algebra morphism,
$\delta[\mathcal{A}_{\boldsymbol{B}}(M)]\subset\mathcal{G}^{0\oplus 2}$ is
commutative subalgebra. Using linearity as well,
$\delta[\alpha+\beta\boldsymbol{B}]=\delta[1](\alpha+\beta\boldsymbol{B})=\alpha+\beta\boldsymbol{B}$
for $\alpha,\beta\in\mathbb{R}$. Hence
$\mathbb{A}_{\boldsymbol{B}}\subset\delta[\mathcal{A}_{\boldsymbol{B}}(M)]$.
If $\tilde{\boldsymbol{B}}\in\delta[\mathcal{A}_{\boldsymbol{B}}(M)]$ commutes
with $\boldsymbol{B}$, these bivectors must not intersect as subspaces except
at zero which yields the 4-vector
$\boldsymbol{B}\tilde{\boldsymbol{B}}\notin\mathcal{G}^{0\oplus 2}$. This
contradicts the grade preservation of $\delta$ and thus
$\delta[\mathcal{A}_{\boldsymbol{B}}(M)]=\mathbb{A}_{\boldsymbol{B}}$. ∎

Next, we show that the characters $\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)$
correspond to evaluation at some point in $\mathbb{R}^{n}$.

######  Lemma 3.8.

Let $M$ be a compact region in $\mathbb{R}^{n}$ and
$\delta\in\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)$. Then
$\delta(z)=z(\boldsymbol{x}_{\delta})$ for some
$\boldsymbol{x}_{\delta}\in\mathbb{R}^{n}$.

###### Proof.

Take $\delta\in\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)$ and the coordinate
planes $\boldsymbol{B}_{ij}$ and the corresponding $z_{ij}$. Applying $\delta$
to $z_{ij}$ yields $\delta[z_{ij}]=\alpha_{ij}+\beta_{ij}\boldsymbol{B}_{ij}$
with $\alpha_{ij},\beta_{ij}\in\mathbb{R}$ by proposition 3.7 and we will
collect all $\alpha_{ij}$ and $\beta_{ij}$ into matrices $\alpha$ and $\beta$
respectively. Then, since

$z_{ij}\boldsymbol{B}_{ji}=(x^{j}-x^{i}\boldsymbol{e}_{i}\boldsymbol{e}_{j})\boldsymbol{e}_{j}\boldsymbol{e}_{i}=-z_{ji}$
(86)

it follows that

$\delta[z_{ij}\boldsymbol{B}_{ji}]=\delta[z_{ij}]\boldsymbol{B}_{ji}=-\delta[z_{ji}],$
(87)

and hence

$(\alpha_{ij}+\beta_{ij}\boldsymbol{B}_{ij})\boldsymbol{B}_{ji}=\beta_{ij}+\alpha_{ij}\boldsymbol{B}_{ji}=-\alpha_{ji}-\beta_{ji}\boldsymbol{B}_{ji}.$
(88)

Therefore, $\alpha_{ij}=-\beta_{ji}$ for all $i\neq j$. Similarly, for
arbitrary $\ell\neq i$ and $\ell\neq j$ we have

$z_{ij}=z_{\ell j}+z_{i\ell}\boldsymbol{B}_{\ell j}$ (89)

so

$\delta[z_{ij}]=\delta[z_{\ell j}+z_{i\ell}\boldsymbol{B}_{\ell
j}]=\delta[z_{\ell j}]+\delta[z_{i\ell}]\boldsymbol{B}_{\ell j}.$ (90)

Expanding this yields the relationships $\alpha_{ij}=\alpha_{\ell j}$ and
$\beta_{ij}=\beta_{i\ell}$ for all $i,j,\ell$.

The relationships $\alpha_{ij}=\alpha_{kj}$ and $\beta_{ij}=\beta_{ik}$ show
that both sets of constants $\alpha$ and $\beta$ are given by $n$ numbers
since they are constant along one index. Taking this with the relationship
$\alpha_{ji}=-\beta_{ij}$ shows that both are determined by the same $n$
numbers which we call $x_{\delta}^{i}=\alpha_{ji}=-\beta_{ij}$ for
$i=1,\dots,n$, just with swapped index and magnitude. Hence there exists some
$\boldsymbol{x}_{\delta}=(x^{1}_{\delta},\dots,x^{n}_{\delta})\in\mathbb{R}^{n}$
so that $\delta[z_{ij}]=z_{ij}(\boldsymbol{x}_{\delta})$ since

$z_{ij}(\boldsymbol{x}_{\delta})=x^{j}_{\delta}-x^{i}_{\delta}\boldsymbol{B}_{ij}.$
(91)

∎

To see that the corresponding point $\boldsymbol{x}_{\delta}$ lies in the
given region $M$ for any $\delta$, we use continuity and a singular monogenic
spinor field.

######  Lemma 3.9.

Let $M\subset\mathbb{R}^{n}$ be a compact region and let
$f\in\mathcal{M}^{+}(M)$ and $\delta\in\mathrm{sp}\hskip
2.0pt\mathcal{M}^{+}(M)$. Then $\delta(f)=f(\boldsymbol{x}_{\delta})$ for some
$\boldsymbol{x}_{\delta}\in M$.

###### Proof.

To see that $\boldsymbol{x}_{\delta}\in M$, take
$f_{0}(x)\coloneqq\boldsymbol{G}(\boldsymbol{x}-\boldsymbol{x}_{0})\boldsymbol{e}_{1}$
with $\boldsymbol{x}_{0}\not\in M$. Again, $\boldsymbol{G}$ is the Green’s
function for the Hodge–Dirac operator. Then $f_{0}|_{M}\in\mathcal{M}^{+}(M)$.
By lemma 3.8 we have some $\boldsymbol{x}_{\delta}\in\mathbb{R}^{n}$ such that

$\displaystyle\delta(f_{0}|_{M})=f_{0}|_{M}(\boldsymbol{x}_{\delta}).$ (92)

Take a sequence $\boldsymbol{x}_{n}\to\boldsymbol{x}_{\delta}$ and suppose for
a contradiction that $\boldsymbol{x}_{\delta}\notin M$ and each
$\boldsymbol{x}_{n}\notin M$. Then this defines a sequence of functions
$f_{n}(\boldsymbol{x})\coloneqq
E(\boldsymbol{x}-\boldsymbol{x}_{n})\boldsymbol{e}_{1}|_{M}\in\mathcal{M}^{+}(M)$
and the sequence converges uniformly to a monogenic function
$\lim_{n\to\infty}f_{n}(\boldsymbol{x})=\boldsymbol{G}(\boldsymbol{x}-\boldsymbol{x}_{\delta})\boldsymbol{e}_{1}$.
By continuity of $\delta$,

$\displaystyle\lim_{n\to\infty}\delta(f_{n})=\lim_{n\to\infty}f_{n}(\boldsymbol{x}_{\delta}),$
(93)

which does not converge due to the singularity at $\boldsymbol{x}_{\delta}$
which contradicts the fact that the limit converges to a monogenic function.
Hence, it must be that $\boldsymbol{x}_{\delta}\in M$. ∎

One practical reason behind working with regions of $\mathbb{R}^{n}$ is that
there are clear choices of functions to use to probe whether a point is in the
region or not. For an arbitrary $n$-dimensional manifold we cannot guarantee
an embedding into $\mathbb{R}^{n}$ and the technique using the Cauchy kernel
$\boldsymbol{G}$ fails and a version of lemma 3.9 for arbitrary compact
manifolds is not obvious. Likewise, lemma 3.8 could be viewed as a local
result for characters on a coordinate patch, but it is not clear how the
restriction of a character to local coordinate patches behaves. Nonetheless,
we arrive at the proof for the main theorem.

###### Proof of theorem 3.2.

Fix $M$ a compact region of $\mathbb{R}^{n}$. It is clear that the map
$M\to\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)$ is an embedding by mapping a
point $\boldsymbol{x}\in M$ to $\delta_{\boldsymbol{x}}\in\mathrm{sp}\hskip
2.0pt\mathcal{M}^{+}(M)$ (inverse of $\Gamma$). Then, by lemma 3.9, any
$\delta\in\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)$ corresponds to
$\boldsymbol{x}_{\delta}\in M$ showing the reverse inclusion. Hence the sets
are in bijection via $\Gamma$ and under the weak-$\ast$ topology, $\Gamma$ is
also continuous and hence we have the homeomorphism $M\cong\mathrm{sp}\hskip
2.0pt\mathcal{M}^{+}(M)$.

To see that the Gelfand transform $\mathcal{M}^{+}(M)\to
C^{0}(\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M);\mathcal{G}^{+})$ is an
isometry, we note that

$\|\hat{f}\|=\sup_{\delta\in\mathrm{sp}\hskip
2.0pt\mathcal{M}^{+}(M)}|\hat{f}(\delta)|=\sup_{\delta\in\mathrm{sp}\hskip
2.0pt\mathcal{M}^{+}(M)}|\delta[f]|=\sup_{\boldsymbol{x}_{\delta}\in
M}|f(\boldsymbol{x}_{\delta})|=\|f\|.$ (94)

Hence, we have our theorem. ∎

###  3.4 Further results and discussion

Though the behavior of characters on regions has been determined, it is still
an open question whether theorem 3.2 can be extended to arbitrary
$n$-dimensional compact Riemannian manifolds with boundary. This extension is
not immediately obvious, but there is more to be said that may assist the
general case in the future. Again using motivation from complex analysis,
$\mathcal{M}^{+}(M)$ retains some necessary features but others are missing.

####  3.4.1 Stone–Weierstrass

Firstly, let us prove a Stone–Weierstrass result showing the density of
closure of the monogenic spinor fields in the space of continuous spinor
fields. The proof of the theorem will require the following lemma.

######  Lemma 3.10.

If $M$ be a compact Riemannian manifold with boundary, then the space
$\overline{\mathcal{M}^{+}(M)}$ separates points.

###### Proof.

Let $x,y\in\operatorname{int}M$ be distinct points. We want to construct some
field $f\in\mathcal{M}^{+}(M)$ such that $f(x)\neq f(y)$. Since $M$ is
compact, there exists a shortest path $\gamma\colon[0,1]\to M$ between $x$ and
$y$ and moreover by [28] this path is $C^{1}$ since both $M$ and $\partial M$
are $C^{\infty}$. Since $\gamma$ must always be $C^{1}$, $\gamma$ has a well-
defined tangent vector at each $t$ and a well-defined normal space
$N_{\gamma(t)}\gamma$ which is orthogonal to the tangent vector
$\boldsymbol{\dot{\gamma}}(t)$.

Since $M$ is compact, for all $t$ the injectivity radius of the exponential
map at $\gamma(t)$ is bounded from below by some $\epsilon>0$. Hence, we can
construct a tube $\mathbb{T}_{\gamma}$ about $\gamma$ by taking
$\mathbb{T}_{\gamma}\coloneqq\gamma\times\mathbb{D}_{\epsilon}$ where
$\mathbb{D}_{\epsilon}(t)$ is the image under the exponential map of the disk
of radius $\epsilon$ in the normal space at $N_{\gamma(t)}\gamma$. Any point
$\tilde{x}\in\mathbb{T}_{\gamma}$ is given uniquely by coordinates
$(t,\boldsymbol{v})$ where $\boldsymbol{v}\in N_{\gamma(t)}\gamma$. Given a
unit 2-blade $\boldsymbol{B}(x)\in\mathcal{G}_{x}M$, we can parallel translate
$\boldsymbol{B}(x)$ to a unit 2-blade $\boldsymbol{B}(\tilde{x})$ any point
$\tilde{x}\in\mathbb{T}_{\gamma}$ by parallel translation along $\gamma$ and
then parallel translation in the normal direction. This builds unit 2-blade
field $\boldsymbol{B}$ on $\mathbb{T}_{\gamma}$.

Then, on $\mathbb{T}_{\gamma}$, define the field
$z\in\mathcal{A}_{\boldsymbol{B}}(\mathbb{T}_{\gamma})$ using the unit 2-blade
field $\boldsymbol{B}$ following eq. 68. Then $z(x)\neq z(y)$ and
$z\in\mathcal{M}^{+}(\mathbb{T}_{\gamma})$. Since $\mathbb{T}_{\gamma}$ is a
union of open sets, $\mathbb{T}_{\gamma}$ is open in $M$, and we can uniformly
approximate $z$ by elements of $\mathcal{M}^{+}(M)$. Taking the uniform limit
of these approximations yields a function $f\in\overline{\mathcal{M}^{+}(M)}$
satisfying $f(x)\neq f(y)$. ∎

The space $\overline{\mathcal{M}^{+}(M)}$ is not an algebra, but we can
consider the minimal algebra that the space generates. Let
$\vee\overline{\mathcal{M}^{+}(M)}$ represent the minimal algebra generated by
$\overline{\mathcal{M}^{+}(M)}$, then using the previous lemma and a result
from Laville and Ramadanoff in their paper on the Stone–Weierstrass theorem
for Clifford-valued functions [17], we will get the following theorem.

######  Theorem 3.11.

$\vee\overline{\mathcal{M}^{+}(M)}$ is dense in $C^{0}(M;\mathcal{G}^{+})$.

###### Proof.

Since $\overline{\mathcal{M}^{+}(M)}$ contains 1 and separates points, it is a
candidate for the use of Laville and Ramadanoff [17, theorem 3]. In order to
use their result in all dimensions, we must have that
$f\in\overline{\mathcal{M}^{+}(M)}$ is invariant with respect to the principal
involution $f_{*}$. Since $f$ is a spinor field, $f=\sum_{2k=0}^{n}f_{2k}$ and

$f_{*}=\sum_{2k=0}^{n}(-1)^{2k}f_{2k}=\sum_{2k=0}^{n}f_{2k}=f_{*},$ (95)

so $f$ is invariant under the principal involution.

Matching our notation to Laville’s, take a basis $2k$-blade
$\boldsymbol{E}_{\mathcal{I}}$ (i.e., $|\mathcal{I}|=2k$ is an ordered list of
indices and $\boldsymbol{E}_{\mathcal{I}}$ is given by eq. 4), then given
$f\in\vee\overline{\mathcal{M}^{+}(M)}$ we can take $f\mapsto
f_{\mathcal{I}}=(f,\boldsymbol{E}^{\mathcal{I}})$ (see eq. 14) which produces
a dense subset $\vee C^{0}(M;\mathbb{R})_{\mathcal{I}}\subset
C^{0}(M;\mathbb{R})$ by the classical Stone–Weierstrass theorem. Hence, since

$C^{0}(M;\mathcal{G}^{+})=\bigoplus_{2k}C^{0}(M;\mathbb{R})\boldsymbol{E}_{\mathcal{I}}$

we conclude that $\vee\overline{\mathcal{M}^{+}(M)}$ is dense in
$C^{0}(M;\mathcal{G}^{+})$. ∎

####  3.4.2 Tomography

As discussed earlier, the work in this paper is heavily motivated by the
Boundary Control (BC) method for the inverse tomography problem. The BC method
was used in Belishev’s proof for the 2-dimensional Calderón problem in [10].
Our paper manages to show that we can determine the homeomorphism type of an
embedded manifold from the spinor spectrum, but we are missing other key facts
that would lead to a solution for the tomography problem. Essentially, we need
the following additional facts to use the BC method:

  1. i.

The Dirichlet-to-Neumann (DN) map determines the space
$\operatorname{tr}\mathcal{M}^{+}(M)$.

  2. ii.

The boundary trace map
$\operatorname{tr}\colon\vee\mathcal{M}^{+}(M)\to\operatorname{tr}\vee\mathcal{M}^{+}(M)$
where $f_{+}\mapsto f_{+}|_{\partial M}$ is an isometric isomorphism of
algebras.

  3. iii.

The space $\mathcal{M}^{+}(M)$ determines the metric structure of $M$ up to
isometry.

Given the results of this paper alongside items (i) and (ii), we would be able
to determine a compact embedded $M$ up to homeomorphism. We can view (iii) as
an extension of our result here. Namely, we have determined the homeomorphism
type of $M$ from the space $\mathcal{M}^{+}(M)$, but have yet to gain any
metric data.

Let us discuss each of the points above.

  1. i.

Using the Dirichlet-to-Neumann operator $\Lambda$ on differential forms,
Belishev and Sharafutdinov in [6] describe an ancillary boundary operator
called the _Hilbert transform_ $T$. The Hilbert transform is a classical
operator in complex analysis and it also appears in Clifford analysis as an
operator on the $L^{2}$-completion of the space
$\operatorname{tr}C^{0}(M;\mathcal{G})$. Two good sources include Brackx and
De Schepper’s paper [29] (which specifically concentrates on compact regions
of $\mathbb{R}^{n}$) and Calderbank’s thesis [16].

Given a function on the boundary
$\phi\in\operatorname{tr}C^{0}(M;\mathcal{G})$, the Hilbert transform in
Clifford analysis allows one to uniquely recover the complete boundary data of
a monogenic field $f$ with $\phi$ as a component of $f|_{\partial M}$. In
essence, the Hilbert transform yields boundary values of functions conjugate
in the generalized Cauchy–Riemann equations given by $\boldsymbol{\nabla}f=0$.
In fact, if $\phi_{k}$ is a $k$-vector, the Hilbert transform of $\phi_{k}$
contains a $k-2$-, a $k$-, and $k+2$-vector component. This is part of why we
suggest to look beyond pairwise conjugate forms.

A reasonable question to ask is: are these two Hilbert transform operators
related in some way? Moreover, Sharafutdinov and Shonkwiler extend Belishev
and Sharafutdinov’s Dirichlet-to-Neumann operator to the complete Dirichlet-
to-Neumann operator [8]. If it is not possible to relate Belishev and
Sharafutdinov’s Hilbert transform using the DN operator on forms to the
Hilbert transform in Clifford analysis, is there a related operator defined in
terms of the complete DN operator that relates to Clifford analysis?

  2. ii.

This point is essentially given as an open question by Belishev and Vakulenko
[13]. Specifically, those two ask whether it is true that the algebras
$\vee\mathcal{M}^{+}(M)$ and $\vee\mathrm{tr}\mathcal{M}^{+}(M)$ are
isometrically isomorphic. At the moment, we have a partial answer: by the
Cauchy integral formula, a monogenic field $f\in\mathcal{M}(M)$ is determined
by its boundary values therefore the map
$\operatorname{tr}\colon\mathcal{M}(M)\to\operatorname{tr}\mathcal{M}(M)$ is
an isomorphism of vector spaces and by the weak maximum principle for elliptic
operators, it is also an isometry. This of course applies to the spinor
subspace $\mathcal{M}^{+}(M)$. However, it is not clear that the algebras
$\vee\mathcal{M}^{+}(M)$ and $\vee\mathrm{tr}\mathcal{M}^{+}(M)$ are
isomorphic. In Clifford analysis, we have the _Hardy space_ $\mathcal{H}(M)$
as the $L^{2}$-completion of $\operatorname{tr}\mathcal{M}(M)$ which is
studied in both the previously referenced papers [29, 16]. Perhaps there is
more knowledge about $\mathcal{H}(M)$ that could assist in finding a proof of
fact (ii).

  3. iii.

There is likely geometric content inside the spinor spectrum and this could
lead to determining, at the very least, the conformal class of the metric.
First, it is widely known that the Hodge–Dirac operator is conformally
invariant [30]. Hence, it may be possible to construct a metric $g$ up to
conformal equivalence from the spinor spectrum given that the spinor spectrum
is already homeomorphic to $M$. This should not be shocking; Belishev in [10]
was able to do this for surfaces $S$ with single boundary component, as he
proved that the topologized spectrum is conformally equivalent to $S$ (and in
fact was determined by the classical DN operator). It could be that an
procedure analogous to Belishev’s technique for finding a conformal metric in
[10] can be performed with the spinor spectrum.

It could be that we can do better than extracting just the conformal class for
dimension $n\geq 3$. As a reminder, the 2-dimensional problem cannot determine
more than the conformal class of $g$ since $\Delta$ is conformally invariant
in dimension 2. But, if we consider subsurfaces $S$ inside of $M$, we can
collect conformal copies of the metric restricted to the surface, vary the
over a collection of surfaces passing through a point, and perhaps the
combined data from all surfaces passing through each point could produce a
metric in the isometry class of $M$.

####  3.4.3 Characters on arbitrary compact $M$

Aside from the above points, we want the results of this paper to hold true
for arbitrary compact $M$, not just compact regions of $\mathbb{R}^{n}$. We
briefly discussed the issue with our proof technique just before the proof of
theorem 3.2, but the core issue is that our proof hinged on a global power
series representation which was valid since $M$ was embedded in
$\mathbb{R}^{n}$. If the power series is only local, then we must, in some
sense, understand the restriction of spin characters to local coordinate
patches, but this is not understood.

To view the spin characters from a different perspective, it could be
interesting to take $\delta\in\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)$ and
consider $\ker\delta$. In the case for a surface $S$, the kernel of a
character is in one-to-one correspondence with the set of maximal ideals of
the algebra of holomorphic functions. Succinctly, we can match a character
$\delta_{x}$ with the class of holomorphic functions $[f]$ who vanish at the
point $x$. The maximal ideals of the space of holomorphic functions are
exactly the functions that vanish at just a single point. It is not clear that
elements in $\ker\delta$ have this property when the dimension of $M$ exceeds
2. Part of the proof for the 2-dimensional result used by Belishev in [10]
follows from [31, Exercise 26.4, pg. 205] which can be proven using sheaves.
To that end, it may be useful to think of the space $\mathcal{M}^{+}(M)$ in
the context of sheaves.

On a different note, it could also be useful to identify the
$\mathcal{G}$-currents $C^{0}_{\mathcal{G}}(M;\mathcal{G})^{\prime}$ with
$\mathcal{G}$-valued Radon measures. Additivity of measures over subsets and
the regularity of Radon measures may allow for characters to be applied to
local power series representations of the monogenic spinor fields. If this is
the case, compactness of $M$ would mean that a spin character corresponds to
evaluation at finitely many points. As a final step, we could possibly use the
fact that $\overline{\mathcal{M}^{+}(M)}$ separates points to conclude that a
character $\delta\in\mathrm{sp}\hskip 2.0pt\mathcal{M}^{+}(M)$ is evaluation
at only a single $x_{\delta}\in M$.

##  4 Conclusion

The heart of this paper is to extend the theory Gelfand on commutative Banach
algebras of $\mathbb{C}$-valued functions to noncommutative Banach algebras of
$\mathcal{G}$-valued functions. In essence, we can find copies of $\mathbb{C}$
as subalgebras of plane spinors
$\mathbb{A}_{\boldsymbol{B}}\subset\mathcal{G}^{+}$ and copies of complex
holomorphic functions as the monogenic surface spinors
$\mathcal{A}_{\boldsymbol{B}}$. Using the fact we can also locally construct a
power series for monogenic spinor fields in terms of monogenic variables $z$
with coefficients in $\mathcal{G}^{+}$, we define the characters accordingly.
Hence, we have a meaningful notion of a spectrum and achieve our main theorem.

This theory, when restricted to a 2-dimensional surface, yields the Gelfand
representation, but allows us to achieve new results in higher dimensions with
the same process. More or less, this hinges on the Cauchy integral formula for
multivector fields in the same way the Cauchy integral acts as a linchpin in
many of the classical complex analysis theorems. Hopefully this entices others
to consider the role of the Banach algebra of $\mathcal{G}$-valued functions
and the dual space of $\mathcal{G}$-currents.

We expect that more results will follow in the future. For example, we suspect
theorem 3.2 can be generalized to arbitrary compact Riemannian manifolds. If
this is to help towards a solution of the Calderón problem, this information
would all need to be extracted from the boundary and, in particular, from the
Dirichlet-to-Neumann operator. Experts in Clifford analysis and elliptic
theory may have useful insights into this problem.

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