I tried to represent the Lorentz transform which converts a pair of space-time co-ordinates (ct,x) to (ct',x') as a function of a complex variable i.e(adsbygoogle = window.adsbygoogle || []).push({});

ct' + ix' = f(ct + ix)

Unfortunately the rules of complex algebra do not permit this because the complex product is defined as

(a + ib)(c +id) = (ac - bd) + i(ad + bc)

Consequently the required function is prohibited by the Cauchy-Riemann conditions.

I have been investigating what happens if the rules of complex algebra are changed so that the product rule looks the same in form as the Lorentz transform i.e.

(a + ib)(c + id) = (ac - bd) + i(ad - bc)

This results in a non-commutative algebra in which the Lorentz transform can be represented as multiplication by a complex constant which depends on the relative velocity.

This seems to be a convenient language for doing these problems as Lorentz invariant quantitites can be represented as complex functions of the space-time co-ordinate without reference to any particular inertial frame.

The algebra has some unusual features e.g factorisation of zero and an infinite number of square roots.

Following W.R. Hamilton I extended the algebra to four dimensions [historically the origin of the modern vector calculus] making the same modification to the quaternion algebra as above. I defined a differential calculus for this algebra and formed the first derivitave of a scalar + vector potential [this looks like a Minkowski four-vector but the time component is real and the space components are imaginary].

The result is a mathematical object with 16 partial derivative terms six of these look like the expression for the electric field, six look like the magnetic field and the remaining four look like the Lorentz gauge condition. This suggests that the algebra can be used to make field equations again using only Lorentz invariant quantities.

Has anyone seen this approach before?

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# Lorentz transform as a complex function

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