Generalised Maxwell Equations in Higher Dimensions

Math Methods in App Sciences

Bock

Gürlebeck

2018

The aim of this paper is to study complete polynomial systems in the kernel space of conformally invariant differential operators in higher spin theory. We investigate the kernel space of a generalized Maxwell operator in 3‐dimensional space. With the already known decomposition of its homogeneous kernel space into 2 subspaces, we investigate first projections from the homogeneous kernel space to each subspace. Then, we provide complete polynomial systems depending on the given inner product for each subspace in the decomposition. More specifically, the complete polynomial system for the homogenous kernel space is an orthogonal system wrt a given Fischer inner product. In the case of the standard inner product in L2 on the unit ball, the provided complete polynomial system for the homogeneous kernel space is a partially orthogonal system. Further, if the degree of homogeneity for the respective subspaces in the decomposed kernel spaces approaches infinity, then the angle between the 2 subspaces approaches π/2.

“…To study null solutions to this generalized Maxwell equation , we look at

\ker_{l} D_{1}

, ie, polynomials

bold-italicf false(bold-italicx, bold-italicu false) \in C^{\infty} false(R^{m} \times R^{m}, double-struckF; H_{1} false)

in 2 variables satisfying

D_{1} bold-italicf false(bold-italicx, bold-italicu false) = 0

and

E_{x} bold-italicf false(bold-italicx, bold-italicu false) = l bold-italicf false(bold-italicx, bold-italicu false)

, where

E_{x}

is the Euler operator with respect to x and l is a natural number. According to the argument in Eelbode and Roels, we know that for l ≥ 2,

\dim \ker_{l} {scriptD}_{1} = \dim {scriptP}_{l, 1} ({double-struckR}^{m} \times {double-struckR}^{m}, F; {scriptH}_{1}) - \dim {scriptP}_{l - 2, 1} ({double-struckR}^{m} \times {double-struckR}^{m}, F; {scriptH}_{1}) = ((\begin{matrix} m + l - 1 \\ m - 1 \end{matrix}) - (\begin{matrix} m + l - 3 \\ m - 1 \end{matrix})) \dim ({scriptH}_{1}) .

Here

bold-italicf false(bold-italicx, bold-italicu false) \in P_{l, 1} false(R^{m} \times R^{m}, double-struckF; H_{1} false)

stands for an

…”

Section: Generalized Maxwell Operator In M‐dimensional Spacementioning

confidence: 99%

“…The generalized Maxwell operator in

R^{m}

, denoted by

D_{1}

, has the following form:

{scriptD}_{1} = {normalΔ}_{bold-italicx} - \frac{4}{m} ⟨ u, {bold-italicD}_{bold-italicx} ⟩ ⟨ {bold-italicD}_{bold-italicu}, {bold-italicD}_{bold-italicx} ⟩ : C^{\infty} ({double-struckR}^{m} \times {double-struckR}^{m}, F; {scriptH}_{1}) \to C^{\infty} ({double-struckR}^{m} \times {double-struckR}^{m}, F; {scriptH}_{1}),

where

⟨ u, v ⟩ = u_{1} v_{1} + ⋯ + u_{m} v_{m}, u, v \in {double-struckR}^{m} .

According to our assumptions, we consider functions

bold-italicf false(bold-italicx, bold-italicu false) \in C^{\infty} false(R^{m} \times R^{m}, double-struckR; H_{1} false)

, which can be rewritten as

f (x, u) = \sum_{j = 1}^{m} u_{j} {bold-italicf}_{bold-italicj} (x),

with

f_{j} false(bold-italicx false) \in C^{\infty} false(R^{m}, double-struckR false)

. The equation

<...>

…”

Section: Generalized Maxwell Operator In M‐dimensional Spacementioning

confidence: 99%

“…Theorem The vector space

\ker_{l} D_{1}

has the following decomposition into irreducible modules (for the action of the orthogonal group):

\ker_{l} {scriptD}_{1} = {scriptA}_{l, 1} \oplus (‖ x {false‖}^{2} ⟨ u, {bold-italicD}_{bold-italicx} ⟩ + (m - 2) ⟨ u, x ⟩) {scriptH}_{l - 1},

where

A_{l, 1} = P_{l, 1} \cap \ker false(Δ_{x}, Δ_{u}, ⟨ D_{u}, D_{x} ⟩ false)

, with

\ker ({normalΔ}_{bold-italicx}, {normalΔ}_{bold-italicu}, ⟨ {bold-italicD}_{bold-italicu}, {bold-italicD}_{bold-italicx} ⟩) : = \ker {normalΔ}_{bold-italicx} \cap \ker {normalΔ}_{bold-italicu} \cap \ker ⟨ {bold-italicD}_{bold-italicu}, {bold-italicD}_{bold-italicx} ⟩ .

…”

Section: Generalized Maxwell Operator In M‐dimensional Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A bosonic Laplacian is a conformally invariant second‐order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher‐order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita–Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson's equation in Euclidean space is also provided. In the end, we provide Green's formulas for bosonic Laplacians in scalar‐valued and Clifford‐valued cases, respectively. These formulas reveal that bosonic Laplacians are self‐adjoint with respect to a given L2 inner product on certain compact supported function spaces.

Higher Order Fermionic and Bosonic Operators

Walter

Ryan

2019

Trends in Mathematics

This paper continues the work of our previous paper [9], where we generalize kth-powers of the Euclidean Dirac operator D x to higher spin spaces in the case the target space is a degree one homogeneous polynomial space. In this paper, we reconsider the generalizations of D 3x and D 4 x to higher spin spaces in the case the target space is a degree k homogeneous polynomial space. Constructions of 3rd and 4th order conformally invariant operators in higher spin spaces are given; these are the 3rd order fermionic and 4th order bosonic operators. Fundamental solutions and intertwining operators of both operators are also presented here. These results can be easily generalized to cylinders and Hopf manifolds as in [8].