Quaternionic Differential Equations

Generalized field equations

Generalized field equations hold for all basic fields. Generalized field equations fit best in a quaternionic setting.

Quaternions consist of a real number valued scalar part and a three-dimensional spatial vector that represents the imaginary part.

The multiplication rule of quaternions indicates that several independent parts constitute the product. In this comment, we use a suffix $_{r}$ to indicate the scalar real part of a quaternion, and we use ${\vec {a}}$ to indicate the imaginary vector part of quaternion $a$.

$c=c_{r}+{\vec {c}}=ab=(a_{r}+{\vec {a}})(b_{r}+{\vec {b}})=a_{r}b_{r}-\langle {\vec {a}},{\vec {b}}\rangle +a_{r}{\vec {b}}+{\vec {a}}b_{r}{\color {Red}\pm }{\vec {a}}\times {\vec {b}}$

(1)

The ${\color {Red}\pm }$ in equation (1) indicates that quaternions exist in right-handed and left-handed versions.

The formula can be used to check the completeness of a set of equations that follow from the application of the product rule.

The quaternionic conjugate of a is $a^{*}=a_{r}-{\vec {a}}$ From the product rule follows the formula for the norm $|a|$ of quaternion $a$.

$|a|^{2}=aa^{*}=(a_{r}+{\vec {a}})(a_{r}-{\vec {a}})=a_{r}a_{r}-\langle {\vec {a}},{\vec {a}}\rangle$

(2)

We define the quaternionic nabla as $\nabla =\left\{{\partial \over \partial \tau },{\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right\}=\nabla _{r}+{\vec {\nabla }};\quad \nabla _{r}={\partial \over \partial \tau };\quad {\vec {\nabla }}=\left\{{\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right\}$.

The quaternionic nabla $\nabla$ acts like a multiplying operator. The (partial) differential $\nabla \psi$ represents the full first-order change of field $\psi$. We assume that $\varphi =\nabla \psi$ exists in an enclosed region of the domain of $\psi$.

First order partial differential equation

$\varphi =\varphi _{r}+{\vec {\varphi }}=\nabla \psi =(\nabla _{r}+{\vec {\nabla }})(\psi _{r}+{\vec {\psi }})=\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}{\color {Red}\pm }{\vec {\nabla }}\times {\vec {\psi }}$

(3)

The equation is a quaternionic first order partial differential equation.

The five terms on the right side show the components that constitute the full first-order change.

They represent subfields of field $\varphi$, and often they get special names and symbols.

Subfields

${\vec {\nabla }}\psi _{r}$ is the gradient of $\psi _{r}$

$\langle {\vec {\nabla }},{\vec {\psi }}\rangle$ is the divergence of ${\vec {\psi }}$.

${\vec {\nabla }}\times {\vec {\psi }}$ is the curl of ${\vec {\psi }}$

$\varphi _{r}=\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle$

(4)

(Equation (4) has no equivalent in Maxwell's equations!)

${\vec {\varphi }}=\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}\pm {\vec {\nabla }}\times {\vec {\psi }}=\mp {\vec {B}}-{\vec {E}}$

(5)

${\vec {E}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ -\nabla _{r}{\vec {\psi }}-{\vec {\nabla }}\psi _{r}$

(6)

${\vec {B}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {\psi }}$

(7)

${\vec {J}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {B}}-\nabla _{r}{\vec {E}}$

(8)

Properties of the spatial nabla operator

The nabla exists in several coordinate systems. This section shows the representation of the quaternionic nabla for Cartesian coordinate systems and for polar coordinate systems.

${\vec {\nabla }}=\{{\frac {\partial {}}{\partial {x}}},\,{\frac {\partial {}}{\partial {y}}},\,{\frac {\partial {}}{\partial {z}}}\}={\frac {\partial {}}{\partial {x}}}{\vec {\hat {x}}}+{\frac {\partial {}}{\partial {y}}}{\vec {\hat {y}}}+{\frac {\partial {}}{\partial {z}}}{\vec {\hat {z}}}$

(9)

${\vec {\nabla }}a_{r}={\frac {\partial {}}{\partial {x}}}{\vec {\hat {x}}}+{\frac {\partial {}}{\partial {y}}}{\vec {\hat {y}}}+{\frac {\partial {}}{\partial {z}}}{\vec {\hat {z}}}={\frac {\partial {a_{r}}}{\partial {\rho }}}{\vec {\hat {\rho }}}+{\frac {1}{\rho }}{\frac {\partial {a_{r}}}{\partial {\theta }}}{\vec {\hat {\theta }}}+{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{r}}}{\partial {\phi }}}{\vec {\hat {\phi }}}$

(10)

Here $\{\rho ,\theta ,\phi \}$ are the coordinates with $\{{\vec {\hat {\rho }}},{\vec {\hat {\theta }}},{\vec {\hat {\phi }}}\}$ as coordinate axes.

$\langle {\vec {\nabla }},{\vec {a}}\rangle ={\frac {\partial {a_{x}}}{\partial {x}}}{\vec {\hat {x}}}+{\frac {\partial {a_{y}}}{\partial {y}}}{\vec {\hat {y}}}+{\frac {\partial {a_{z}}}{\partial {z}}}{\vec {\hat {z}}}={\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}+{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}+{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{\phi }}}{\partial {\phi }}}$

(11)

$\langle {\vec {\nabla }},{\vec {\nabla }}\rangle f_{r}={\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}{\frac {\partial {f_{r}}}{\partial \rho }})}}{\partial {\rho }}}+{\frac {1}{\rho ^{2}\sin(\theta )}}{\frac {\partial {(\sin(\theta ){\frac {\partial {f_{r}}}{\partial {\theta }}}}}{\partial {\theta )}}}+{\frac {1}{\rho ^{2}\sin ^{2}(\theta )}}{\frac {\partial ^{2}{f_{r}}}{\partial {\phi ^{2}}}}$

(12)

${\vec {\nabla }}\times {\vec {a}}=({\frac {\partial {a_{z}}}{\partial {y}}}-{\frac {\partial {a_{y}}}{\partial {z}}}){\vec {\hat {x}}}+({\frac {\partial {a_{x}}}{\partial {z}}}-{\frac {\partial {a_{z}}}{\partial {x}}}){\vec {\hat {y}}}+({\frac {\partial {a_{y}}}{\partial {x}}}-{\frac {\partial {a_{x}}}{\partial {y}}}){\vec {\hat {z}}}$

(13)

$\qquad ={\frac {1}{\rho \sin(\theta )}}({\frac {\partial {a_{\phi }}}{\partial {\phi }}}-{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}){\vec {\hat {\rho }}}+({\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}-{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{\phi }}}{\partial {\phi }}}){\vec {\hat {\theta }}}+({\frac {1}{\rho \sin(\theta )}}{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}-{\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}){\vec {\hat {\phi }}}$

${\vec {\nabla }}\times {\vec {a}}={\frac {1}{\rho \sin(\theta )}}({\frac {\partial {a_{\phi }}}{\partial {\phi }}}-{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}){\vec {\hat {\rho }}}+({\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}-{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{\phi }}}{\partial {\phi }}}){\vec {\hat {\theta }}}+({\frac {1}{\rho \sin(\theta )}}{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}-{\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}){\vec {\hat {\phi }}}$

(14)

Special formulas

The spatial nabla operator shows behavior that is valid for all quaternionic functions for which the first order partial differential equation exists.

Here the quaternionic field $a=a_{r}+{\vec {a}}$ obeys the requirement that the first order partial differential $\nabla a$ exists.

$\Delta \ {\overset {\underset {\mathrm {def} }{}}{=}}\ \nabla ^{2}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \langle {\vec {\nabla }},{\vec {\nabla }}\rangle \ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\frac {\partial {}^{2}}{\partial {x}^{2}}}+{\frac {\partial {}^{2}}{\partial {y}^{2}}}+{\frac {\partial {}^{2}}{\partial {z}^{2}}}$

(15)

$\langle {\vec {\nabla }},{\vec {\nabla }}a_{r}\rangle =\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a_{r}=\nabla ^{2}a_{r}$

(16)

$\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {a}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \nabla ^{2}{\vec {a}}$

(17)

$\langle {\vec {\nabla }}\times {\vec {\nabla }},{\vec {a}}\rangle =0$

(18)

${\vec {\nabla }}\times {\vec {\nabla }}a_{r}={\vec {0}}$

(19)

$\langle {\vec {\nabla }},{\vec {\nabla }}\times {\vec {a}}\rangle =0$

(20)

${\vec {\nabla }}\times ({\vec {\nabla }}\times {\vec {a}})={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {a}}$

(21)

${\vec {\nabla }}{\vec {\nabla }}a=({\vec {\nabla }}\times {\vec {\nabla }})a-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a===Quater({\vec {\nabla }}\times {\vec {\nabla }}){\vec {a}}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -2\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a_{r}$

(22)

For constant ${\vec {k}}$ and parameter $q=q_{r}+{\vec {q}}=\{q_{r},q_{x},q_{y},q_{z}\}$ holds

${\vec {\nabla }}\langle {\vec {k}},{\vec {q}}\rangle ={\vec {k}}$

(23)

$\langle {\vec {\nabla }},{\vec {q}}\rangle =3$

(24)

${\vec {\nabla }}\times {\vec {q}}={\vec {0}}$

(25)

${\vec {\nabla }}\,|{\vec {q}}|={\frac {\vec {q}}{|{\vec {q}}|}};\ \nabla \,|q|={\frac {q}{|q|}}$

(26)

${\vec {\nabla }}{\frac {1}{|{\vec {q}}-{\vec {q}}^{'}|}}=-{\frac {{\vec {q}}-{\vec {q}}^{'}}{|{\vec {q}}-{\vec {q}}^{'}|^{3}}};\ \nabla {\frac {1}{|q-q^{'}|}}=-{\frac {q-q^{'}}{|q-q^{'}|^{3}}}$

(27)

$\langle {\vec {\nabla }},{\vec {\nabla }}{\frac {1}{|{\vec {q}}-{\vec {q}}^{'}|}}\rangle =\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\frac {1}{|{\vec {q}}-{\vec {q}}^{'}|}}=-\langle {\vec {\nabla }},{\frac {{\vec {q}}-{\vec {q}}^{'}}{|{\vec {q}}-{\vec {q}}^{'}|^{3}}}\rangle =4\,\phi \,\delta ({\vec {q}}-{\vec {q}}^{'})$

(28)

$\nabla \ q=1-3;\nabla ^{*}q=1+3;\nabla q^{*}=1+3$

(29)

The term $({\vec {\nabla }}\times {\vec {\nabla }})f$ indicates the curvature of field $f$.

The term $\langle {\vec {\nabla }},{\vec {\nabla }}\rangle f$ indicates the stress of field $f$

Derivation of higher order equations

With the help of the properties of the spatial nabla operator follows an interesting second-order partial differential equation.

$\langle {\vec {\nabla }},{\vec {B}}\rangle =\langle {\vec {\nabla }},{\vec {\nabla }}\times {\vec {\psi }}\rangle =0$

(30)

$\langle {\vec {\nabla }},{\vec {E}}\rangle =-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\psi _{r}\rangle =-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}$

(31)

$\langle {\vec {\nabla }},{\vec {B}}-{\vec {E}}\rangle =\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}$

(32)

$\nabla _{r}{\vec {E}}=-\nabla _{r}\nabla _{r}{\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}$

(33)

${\vec {\nabla }}\times {\vec {E}}=-{\vec {\nabla }}\times (\nabla _{r}{\vec {\psi }})-{\vec {\nabla }}\times ({\vec {\nabla }}\psi _{r})=-\nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}=-\nabla _{r}{\vec {B}}$

(34)

${\vec {\nabla }}\times {\vec {B}}={\vec {\nabla }}\times ({\vec {\nabla }}\times {\vec {\psi }})={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}$

(35)

${\vec {\nabla }}\times {\vec {B}}-{\vec {\nabla }}\times {\vec {E}}={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}+\nabla _{r}{\vec {B}}$

(36)

${\vec {\nabla }}\times ({\vec {B}}-{\vec {E}})-\nabla _{r}({\vec {B}}-{\vec {E}})={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}-\nabla _{r}\nabla _{r}{\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}$

(37)

${\vec {\nabla }}\times ({\vec {B}}-{\vec {E}})-\nabla _{r}({\vec {B}}-{\vec {E}})=-(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r}){\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle$

(38)

${\vec {\nabla }}\times ({\vec {B}}-{\vec {E}})-\langle {\vec {\nabla }},{\vec {B}}-{\vec {E}}\rangle -\nabla _{r}({\vec {B}}-{\vec {E}})=\nabla ^{*}{\vec {\varphi }}=-(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r}){\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}$

(39)

${\vec {\nabla }}\varphi _{r}={\vec {\nabla }}(\nabla _{r}\psi _{r}+\langle {\vec {\nabla }},{\vec {\psi }}\rangle )=\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle$

(40)

$\nabla _{r}\varphi _{r}=\nabla _{r}(\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle )=\nabla _{r}\nabla _{r}\psi _{r}-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle$

(41)

$\nabla ^{*}\varphi _{r}=-\nabla _{r}{\vec {\nabla }}\psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\nabla _{r}\psi _{r}-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle$

(42)

$\nabla ^{*}\varphi =(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r}){\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}-\nabla _{r}{\vec {\nabla }}\psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\nabla _{r}\psi _{r}-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle$

(43)

$\chi =\nabla ^{*}\varphi =(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r})\psi +{\color {red}\nabla _{r}{\vec {\nabla }}\psi _{r}}-{\color {blue}{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }+{\color {green}\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }-{\color {red}\nabla _{r}{\vec {\nabla }}\psi _{r}}-{\color {blue}{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }-{\color {green}\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }=(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r})\psi$

(44)

Most of the terms vanish. Further

${\vec {J}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {B}}-\nabla _{r}{\vec {E}}={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}+\nabla _{r}\nabla _{r}{\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}=(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle ){\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle$

(45)

From the above formulas follows that the Maxwell equations do not form a complete set.

Physicists use gauge equations to make Maxwell equations more complete.

Second order partial differential equations

We start with the quaternionic equivalent of the Maxwell-Faraday equation.

$\nabla _{r}{\vec {B}}={\vec {\nabla }}\times (\nabla _{r}{\vec {\psi }})=-{\vec {\nabla }}\times {\vec {E}}$

(46)

Two interesting second order quaternionic partial differential equations exist.

$\zeta =(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi$

(47)

This is the quaternionic equivalent of the wave equation. It offers waves as part of the solutions of the homogeneous equation.

$\chi =\nabla ^{*}\varphi =\nabla ^{*}\nabla \psi =\nabla \nabla ^{*}\psi =(\nabla _{r}-{\vec {\nabla }})(\varphi _{r}+{\vec {\varphi }})=(\nabla _{r}+{\vec {\nabla }})(\nabla _{r}-{\vec {\nabla }})(\psi _{r}+{\vec {\psi }})=(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi$

(48)

This equation can be split into two first order partial wave equations $\chi =\nabla ^{*}\varphi$ and $\varphi =\nabla \psi$.

This equation does not offer waves as part of the solutions of the homogeneous equation.

Differential operators

$D=(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )$

(49)

This is the quaternionic equivalent of d'Alembert's operator.

$\boxdot =(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )$

(50)

This operator does not yet have a known name.

Operator $\langle {\vec {\nabla }},{\vec {\nabla }}\rangle$ represents the main part of the Poisson equation. Together with $\nabla _{r}\nabla _{r}$ this operator configures the above operators.

As is shown above, $\boxdot$ can be derived from the nabla operator. That cannot be said from the $D$ operator.