# Euler's 4-squares-identity and Maxwell's equations

• A
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:

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## Answers and Replies

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

It appears not to.

fresh_42
Mentor
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
Mentor
2020 Award
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.

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!

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PeterDonis
Mentor
2020 Award
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
Mentor
2020 Award
This thread is overly speculative and is now closed.