A Leonhard Euler's 4-Squares Identity & Maxwell's Equations

[Edgar53]

The 4-Squares-Identity of Leonhard Euler
(https://en.wikipedia.org/wiki/Euler%27s_four-square_identity) :

has the numeric structure of Maxwell’s equations in 4-space:

Is somebody aware of litterature about this?

 

[Vanadium 50]

has the numeric structure of Maxwell’s equations in 4-space:

It appears not to.

 

[fresh_42]

If you adjust the signs: Euler = (+,+,+,+) and Minkowski = (+, - , - , - ) in your notation, why should there be a difference? Not sure about the Maxwell equations, but the other one is just a formula. Both have symmetries and both are four dimensional, so similarities can be expected, which happens quite often in science without any deeper correlations.

 

[Dale]

has the numeric structure of Maxwell’s equations in 4-space

I am with @Vanadium 50 on this. It doesn’t seem to have the same structure to me. One is a scalar equation and the other is a vector equation.

 

[Edgar53]

Of course, it has much to do:

Leonhard Euler's four-squares identity prefigures quaternion multiplication:

It is the very reason why quaternions have a multiplicative norm, i.e. the length
of a product of two quaternions equals the product of the lengths of the
quaternions. Multiplication with a unit quaternion performs an isoclinic
double-rotation in 4-dimensional space; multiplication with an arbitrary
quaternion a double-rotation plus a stretching by the length of the quaternion.

Taking the quaternion differential

of the 4-potential A yields directly
the electromagnetic field in both components E and B:

j

k

after rearranging the terms in lines 2, 3, and 4:

i

j

k

The first line identically vanishes under Lorenz gauge

;
The terms:

are the components of the electric field (under inversion of sign, due to
the Minkowski metric (+1,-1,-1,-1));
and the terms:

;

;

are

are the components of the magnetic field.
The result is then:

i.e. the quaternion differential of the 4-potential A yields both,
the source (E) and the curl (B) part of the electromagnetic
field.
Leonhard Euler's four-square identity offers the most direct access to,
and is probably the reason for Electrodynamics!

 

[PeterDonis]

Leonhard Euler's four-squares identity prefigures quaternion multiplication

The four squares identity has squares inside the parentheses on the LHS. The quaternion multiplication equation you wrote down does not. So I don't see how your claim here is justified.

 

[PeterDonis]

This thread is overly speculative and is now closed.