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

  1. Nov 19, 2017 #1
  2. jcsd
  3. Nov 19, 2017 #2

    Vanadium 50

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    It appears not to.
     
  4. Nov 19, 2017 #3

    fresh_42

    Staff: 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.
     
  5. Nov 19, 2017 #4

    Dale

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    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.
     
  6. Dec 4, 2017 #5
    Of course, it has much to do:

    Leonhard Euler's four-squares identity prefigures quaternion multiplication:
    upload_2017-12-4_22-1-12.png
    upload_2017-12-4_21-55-41.png
    upload_2017-12-4_21-55-41.png
    upload_2017-12-4_21-55-41.png
    upload_2017-12-4_21-55-41.png
    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
    upload_2017-12-4_22-12-50.png
    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!
     

    Attached Files:

  7. Dec 4, 2017 #6

    PeterDonis

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    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.
     
  8. Dec 4, 2017 #7

    PeterDonis

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    2016 Award

    Staff: Mentor

    This thread is overly speculative and is now closed.
     
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