# Lorentz transform as a complex function

1. Oct 5, 2005

### quaternion

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

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?

2. Oct 7, 2005

### DrGreg

Last edited: Oct 7, 2005
3. Dec 20, 2005

### quaternion

hyperbolic numbers

Thanks for that. The algebra I have got in mind not only permits division but it is possible to define a four-dimensional differential calculus.

If I make a four-dimensional potential function and take the first derivative I get a mathematical object consisting of sixteen partial derivatives, six of these look like the expression for the magnetic field six look like the electric field and the remaining four look like the Lorentz gauge condition. Does this mean any thing to you?

It occured to me that it might be possible to write the laws of electromagnetism as four-dimensional differential equations.
The dimensionality is embedded in the algebra so the process of solving them looks like a one-dimensional equations.
As Hamilton said we could solve these problems 'without the aid of a co-ordinate system' [On Quaternions]. Do you think this makes any sense?

Last edited: Dec 20, 2005
4. Dec 20, 2005

### George Jones

Staff Emeritus
I'm not sure if this is anywhere near what you have in mind, but the complex quaternions can used to do special relativity. In this formalism: a 4-vector is a scalar plus a (3-)vector; Maxwell's equations can be written as one equation.

Regards,
George

5. Dec 21, 2005

### quaternion

Maxwell's equations

Thanks George.
Can you tell me what the equations look like in the quaternion algebra.

Do you know why all the text books use partial derivatives and vector calculus? The quaternion system seems so much simpler.

6. Dec 21, 2005

### robphy

Last edited: Dec 21, 2005
7. Mar 1, 2006