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

  • #1
2
-1
The 4-Squares-Identity of Leonhard Euler
(https://en.wikipedia.org/wiki/Euler%27s_four-square_identity) :

upload_2017-11-19_23-4-30.jpg

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has the numeric structure of Maxwell’s equations in 4-space:
upload_2017-11-19_23-5-23.png

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Is somebody aware of litterature about this?
 

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Answers and Replies

  • #2
Vanadium 50
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has the numeric structure of Maxwell’s equations in 4-space:

It appears not to.
 
  • #3
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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.
 
  • #4
Dale
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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.
 
  • #5
2
-1
Of course, it has much to do:

Leonhard Euler's four-squares identity prefigures quaternion multiplication:
upload_2017-12-4_22-1-12.png

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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
upload_2017-12-4_22-5-30.png
of the 4-potential A yields directly
the electromagnetic field in both components E and B:
upload_2017-12-4_22-12-50.png

upload_2017-12-4_22-12-50.png

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j
upload_2017-12-4_22-12-50.png

k
upload_2017-12-4_22-12-50.png

after rearranging the terms in lines 2, 3, and 4:
upload_2017-12-4_22-14-20.png

i
upload_2017-12-4_22-14-20.png

j
upload_2017-12-4_22-14-20.png

k
upload_2017-12-4_22-14-20.png

The first line identically vanishes under Lorenz gauge
upload_2017-12-4_22-15-55.png
;
The terms:
upload_2017-12-4_22-22-5.png

are the components of the electric field (under inversion of sign, due to
the Minkowski metric (+1,-1,-1,-1));
and the terms:
upload_2017-12-4_22-18-37.png
;
upload_2017-12-4_22-18-37.png
;
upload_2017-12-4_22-18-37.png
are
upload_2017-12-4_22-20-29.png

are the components of the magnetic field.
The result is then:
upload_2017-12-4_22-25-5.png

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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  • #6
PeterDonis
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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.
 
  • #7
PeterDonis
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This thread is overly speculative and is now closed.
 

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