This thread is the sequel to my other thread, 'Quaternions and SR'.(adsbygoogle = window.adsbygoogle || []).push({});

My goal is to write some physical equations with quaternions.

Because I think quaternions represent the 4-dimensionality and metric used in special relativity. See:

A quaternion is a generalized complex number:

[tex]

A = a_t + ia_x + ja_y + ka_z

[/tex]

with the fundamental equation

[tex]

i^2 = j^2 = k^2 = ijk = -1.

[/tex]

Quaternions are not commutative, for instance

[tex]

ij = -ji = k.

[/tex]

Let's define

[tex]

A_3 = ia_x + ja_y + ka_z

[/tex]

and the dot and cross products as usual for 3-vectors, then the product of two quaternions is

[tex]

AB = a_tb_t + a_tB_3 + A_3b_t - A_3 \cdot B_3 + A_3 \times B_3.

[/tex]

Thus,

[tex]

\frac{1}{2}(AB - BA) = A_3 \times B_3

[/tex]

and

[tex]

\frac{1}{2}(AB + BA) = a_tb_t + a_tB_3 + A_3b_t - A_3 \cdot B_3.

[/tex]

Let's define the commutator

[tex]

\left[A,B\right] = \frac{1}{2}(AB - BA)

[/tex]

and the anticommutator

[tex]

\left<A,B\right> = \frac{1}{2}(AB + BA).

[/tex]

Now for physics. Let's define the differential operator

[tex]

\nabla = \frac{\partial}{\partial t} + i \frac{\partial}{\partial x} + j \frac{\partial}{\partial y} + k \frac{\partial}{\partial z}.

[/tex]

Then Maxwell's equations can be written

1. Coulomb's law: [tex]

\left<\nabla, E\right> = \nabla_tE - 4\pi J_0

[/tex]

2. Ampere's law: [tex]

\left[\nabla, B\right] = \nabla_tE + 4\pi J_3

[/tex]

3. Faraday's law: [tex]

\left[\nabla, E\right] = -\nabla_tB

[/tex]

4. No magnetic monopoles: [tex]

\left<\nabla, B\right> = \nabla_tB.

[/tex]

Now if we use a vector potental written as a quaternion A, which satisfies Lorentz's condition

[tex]

\nabla_t a_t - \nabla_3 \cdot A_3 = 0

[/tex]

and let

[tex]

E = -\frac{1}{2}\left<\nabla,A\right>

[/tex]

[tex]

B = \frac{1}{2}\left[\nabla,A\right]

[/tex]

then Maxwell's equations reduce nicely to two wave equations:

[tex]

4 \pi J = \frac{1}{2}\left<\nabla^2,A\right>

[/tex]

[tex]

\nabla_t B = \frac{1}{2}\left[\nabla^2,A\right].

[/tex]

That's my result so far. Any comments?

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# Quaternions and Maxwell

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