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Hi all,

just discovered the LaTeX feature, so why not play around with it a bit.

Quaternions are (sort of) generalized complex numbers where you have one real unit and 3 imaginary units i, j, and k. The basic equation is

[tex]

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

[/tex]

Now, if we have a 4-vector

[tex]

\vec{r} = (t, x, y, z),

[/tex]

we could write it as a quaternion

[tex]

R = t + ix + jy + kz.

[/tex]

Defining

[tex]

R_3 = ix + jy + kz,

[/tex]

and

[tex]

R_0 = t

[/tex]

we get

[tex]

R^2 = (t^2 - x^2 - y^2 - z^2) + 2R_{0}R_{3}.

[/tex]

Thus

[tex]

R^2{}_0 = S^2

[/tex]

Where S

Maybe it's just useless, I'm just playing around. Any suggestions?

just discovered the LaTeX feature, so why not play around with it a bit.

Quaternions are (sort of) generalized complex numbers where you have one real unit and 3 imaginary units i, j, and k. The basic equation is

[tex]

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

[/tex]

Now, if we have a 4-vector

[tex]

\vec{r} = (t, x, y, z),

[/tex]

we could write it as a quaternion

[tex]

R = t + ix + jy + kz.

[/tex]

Defining

[tex]

R_3 = ix + jy + kz,

[/tex]

and

[tex]

R_0 = t

[/tex]

we get

[tex]

R^2 = (t^2 - x^2 - y^2 - z^2) + 2R_{0}R_{3}.

[/tex]

Thus

[tex]

R^2{}_0 = S^2

[/tex]

Where S

^{2}is a relativistic invariant.Maybe it's just useless, I'm just playing around. Any suggestions?

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