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I recently completed a course on Electrodynamics, with a short introduction to special relativity (and E/B fields in special relativity).

Both in the book we were asked to study, and the final exam, the four vectors we used used real components, with the time coordinate being the zeroth component. For example:

[tex](x^\mu) = (ct, x, y, z)[/tex]

[tex](A^\mu) = (\phi / c, A_x, A_y, A_z)[/tex]

(A = vectorpotential, [itex]\phi[/itex] = scalar potential)

etc...

Also, what the book calls the 'four dimensional scalar product' has a, in my opinion, rather strange minus sign in the zeroth component:

[tex]\sum_{\mu = 0}^3 a_\mu b^\mu = -a^0b^0 + a^1b^1+a^2b^2+a^3b^3[/tex]

In the classes we were given however, we used a different convention (well, I guess it's a convention), where the time coordinate as the fourth component (counting from 1 to 4), carrying a factor of [itex]i[/itex]:

[tex](x^\mu) = (x, y, z, ict)[/tex]

[tex](A^\mu) = (A_x, A_y, A_z, i\phi / c)[/tex]

etc...

Now, the product doesn't need the pesky minus sign, because it comes from the factor [itex]i^2[/itex]...

I could only find the first convention (real vectors) in practice, I couldn't find the complex vector convention anywhere I looked, except in our classes. But, I think it's a much neater definition, because everything generalizes from 3D, without the pesky minus sign...

I was just wondering, is there any particular reason for using the real vectors, instead of the complex ones? If not, is the real vector definition really used more often, or did I just not look hard enough?

I just find it a bit strange that everything seems to be about generalization (putting time and space into a single vector, and putting vector potential and scalar potential into one vector, etc), yet an, in my opinion, beautiful use of complex vectors is not used, so that an awkward definition of the scalar product has to be used..?

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# Four vectors in complex notation

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