Hodge and Gelfand Theory in Clifford Analysis and Tomography

Clayton Shonkwiler

Colorado State University

http://shonkwiler.org

August, 2022

/qipa22

This talk!

Colin Roberts

Colorado State University

Most of what I will talk about today is work of Colin Roberts.

Motivations

Calderón Problem.

Given a Riemannian manifold \((M,g)\) with boundary, determine \(g\) from the Dirichlet-to-Neumann operator \(\Lambda\).

Belishev’s 2D Solution.

On a surface, \(\Lambda\) determines the algebra \(\mathcal{A}(M)\) of holomorphic functions on \(M\), which determines \((M,g)\) up to conformal equivalence.

Clifford Algebras

If \(V\) is a vector space with symmetric bilinear form \(g\), the tensor algebra is

and the associated Clifford algebra is

\mathcal{T}(V) := \bigoplus_{j=0}^\infty V^{\otimes j} = \mathbb{R} \oplus V \oplus (V \otimes V) \oplus \dots
C\ell(V,g) := \mathcal{T}(V)/\langle v \otimes v - g(v,v)\rangle

\(\dim(C\ell(V,g)) = 2^{\dim(V)}\)

Examples

\(\bullet\) \(V = \mathbb{R}^2\), \(g(u,v) = u \cdot v\).

(a e_1 + b e_2) (c e_1 + d e_2) = (ac + bd) + ad e_1 e_2 + bc e_2 e_1 \\ = (ac + bd) + (ad - bc)e_1e_2

\(\bullet\) \(g = 0\).

C\ell(V,0) = \bigwedge(V)

\(\bullet\) If \(g\) is non-degenerate, \(C\ell(V,g)\) is called a geometric algebra. Use the notation \(\mathcal{G}_{p,q}:=C\ell(V,g)\), when \(g\) is the standard non-degenerate form of type \((p,q)\).

Gradings

If \(A \in \mathcal{G}_{p,q}\), let \(\langle A\rangle_r \in \mathcal{G}_{p,q}^r\) be the grade-\(r\) part of \(A\).

On homogeneous elements \(A \in \mathcal{G}_{p,q}^r\) and \(B \in \mathcal{G}_{p,q}^s\),

A B = \langle A B\rangle_{|r-s|} + \langle AB\rangle _{|r-s|+2} + \dots + \langle AB\rangle_{r+s}
A \wedge B := \langle AB \rangle_{r+s}
A \,\lrcorner\, B := \langle AB \rangle_{s-r}

The subset \(\mathcal{G}_{p,q}^+\) of even-graded elements forms a subalgebra, called the spinor algebra.

Examples

  • \(\mathcal{G}_{0,2}^+ \simeq \mathbb{C}\): generated by \(1\) and \(e_1e_2\), with \((e_1e_2)^2 = -1\).
  • \(\mathcal{G}_{0,3}^+ \simeq \mathbb{H}\): generated by \(1,e_1e_2, e_2e_3, e_3e_1\).
  • \(\mathcal{G}_{1,3}^2 \simeq \mathfrak{spin}(1,3)\), the Lie algebra of the Lorentz group.

Multivector Fields

\((M,g)\) smooth, compact, connected, oriented.

Definition.

The geometric algebra bundle over \(M\) is

\mathcal{G}M:= \bigsqcup_{p \in M} C\ell(T_pM,g_p).

The multivector fields \(\mathfrak{X}(M)\) are smooth sections of \(\mathcal{G}M\).

Hodge–Dirac Operator

\(g\) induces the Levi–Civita connection \(\nabla\) and covariant derivative \(\nabla_v\) on \(M\), which extend to multivectors.

Definition.

The Hodge–Dirac operator is defined in local coordinates by 

\nabla = \sum_{i=1}^n e^i \nabla_{e_i}.

For \(A,B \in \mathfrak{X}(M)\), \(\nabla(AB) = \dot{\nabla}\dot{A}B + \dot{\nabla}A\dot{B}\).

\(\nabla^2\) is the Laplace–Beltrami operator.

Examples

  • For \(A \in \mathfrak{X}^+(\mathbb{R}^2)\), \(\nabla A = 0\) iff \(A\) is a holomorphic function.
  • For \(v \in \mathfrak{X}^1(\R^3)\),
\nabla v = \underbrace{\nabla \,\lrcorner\, v}_{\operatorname{div}} + \underbrace{\nabla \wedge v}_{\operatorname{curl}}

Monogenic Fields

Definition.

The space of monogenic fields \(\mathcal{M}(M) := \ker \nabla\).

Monogenic fields

  • can be uniquely continued
  • have a Cauchy integral
  • have harmonic components

Clifford–Hodge Decomposition

Theorem [Roberts]

The space of multivector fields on \(M\) has the orthogonal decomposition

\mathfrak{X}(M) = \mathcal{M}(M) \oplus \nabla \mathfrak{X}(M),

where the Dirac fields are defined as

\nabla\mathfrak{X}(M):=\{\nabla A : A \in \mathfrak{X}(M), \left.A\right|_{\partial M} = 0\}.

Connections to EIT

\(M\) an Ohmic region in \(\mathbb{R}^3\), relate conductivity to \(g\) in the usual way.

Ohm’s Law: \(-\nabla \wedge u = J\)

Conservation: \(\nabla \,\lrcorner\, J=0\)

Define the electric DN operator \(\Lambda_E: t \mathfrak{X}^0(M) \to t\mathfrak{X}^0(M)\) by

\Lambda_E\phi = \nu \lrcorner \nabla \wedge u = \frac{\partial u}{\partial \nu}

where

\begin{cases} \nabla^2 u = 0 & \text{ in } M \\ u = \phi & \text{ on } \partial M.\end{cases}

Magnetic Analog

Define the electric DN operator \(\Lambda_E: t \mathfrak{X}^0(M) \to t\mathfrak{X}^0(M)\) by

\Lambda_E\phi = \nu \lrcorner \nabla \wedge u = \frac{\partial u}{\partial \nu}

where

\begin{cases} \nabla^2 u = 0 & \text{ in } M \\ u = \phi & \text{ on } \partial M.\end{cases}

Define the magnetic DN operator \(\Lambda_B: n \mathfrak{X}^2(M) \to n\mathfrak{X}^2(M)\) by

\Lambda_B(\nu \wedge J) = \nu \wedge \nabla \lrcorner\, B

where

\begin{cases} \nabla^2 B = 0 & \text{ in } M \\ B = \nu \wedge J & \text{ on } \partial M.\end{cases}

\(A_+ := u+B \in \mathcal{M}(M)\) is a monogenic spinor.

Geometric Generalization

Generalized electric DN operator \(\Lambda_E: t \mathfrak{X}(M) \to t\mathfrak{X}(M)\)

\Lambda_E\phi = \nu \lrcorner \nabla \wedge A

Generalized magnetic DN operator \(\Lambda_B: n \mathfrak{X}(M) \to n\mathfrak{X}(M)\)

\Lambda_B\phi = \nu \wedge \nabla \lrcorner\, A

where

\begin{cases} \nabla^2 A = 0 & \text{ in } M \\ A = \phi & \text{ on } \partial M. \end{cases}

\(\Lambda_E \times \Lambda_B\) is equivalent to the complete DN operator 

[Sharafutdinov–Shonkwiler, 2013]

Spinor DN Operator

Define the spinor DN operator \(\mathcal{J}: \operatorname{tr}\mathfrak{X}^\pm(M) \to \operatorname{tr}\mathfrak{X}^\pm(M)\) by

\mathcal{J}\phi = \nu \nabla A.

Scalar part: \(\langle \mathcal{J} \rangle = \Lambda_E + \Lambda_B\).

Theorem [Roberts]

\(\ker \mathcal{J} = \operatorname{tr} \mathcal{M}^\pm(M).\)

Connections to Boundary Control

Building on ideas of Belishev and Vakulenko, define a spinor spectrum \(\mathfrak{M}(M)\) consisting of certain grade-preserving \(\mathcal{G}_{0,n}^+\)-linear maps \(\mathcal{M}^+(M) \to \mathcal{G}_{0,n}^+\).

E.g., point evaluation \(\delta[A_+] = A_+(x_\delta)\).

Theorem [Roberts]

With the weak-\(*\) topology on \(\mathfrak{M}(M)\), the map

\gamma: \mathfrak{M}(M) \to M \qquad \delta \mapsto x_\delta

is a homeomorphism.

The Gelfand transform \(\widehat{A_+}(\delta) = \delta[A_+]\) is an isometric isomorphism, so \(\mathcal{M}^+(M) \simeq \widehat{\mathcal{M}^+(M)}\).

Questions

  1. Does the DN map determine \(\operatorname{tr}\mathcal{M}^+(M)\)?
  2. Relationship between \(\vee \mathcal{M}^+(M)\) and \(\vee \operatorname{tr}\mathcal{M}^+(M)\)?
  3. Does \(\mathcal{M}^+(M)\) determine the metric on \(M\)?
  4. Hilbert transform?

Thank you!

References

Colin Roberts, Hodge and Gelfand Theory in Clifford Analysis and Tomography, Ph.D. thesis, Colorado State University, 2022, https://hdl.handle.net/10217/235741

Colin Roberts, A Gelfand transform for spinor fields on embedded Riemannian manifolds, preprint, 2022, arXiv:2203.00118

Hodge and Gelfand Theory in Clifford Analysis and Tomography

By Clayton Shonkwiler

Hodge and Gelfand Theory in Clifford Analysis and Tomography

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