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Electrodynamics: quaternionic potential?

  1. Feb 3, 2009 #1
    Hi!

    It's possible to construct a electromagnetic field, such that
    [tex]\vec{F}:=\vec{E} + i\cdot \vec{B}[/tex].
    Now the real part is the electric and the imaginary part is the magnetic field.
    Then, for example, the maxwell equations take the form
    [tex]\nabla \cdot \vec{F} = \rho, \qquad \rho \in \mathbb{R} [/tex]
    and
    [tex]\nabla\times \vec{F} - i \cdot \frac{\partial}{\partial t} \vec{F} = \vec{j}, \qquad \vec{j} \in \mathbb{R}^3 [/tex]
    So, it is possible to combine electric and magnetic field into one (complex) Field.

    Now my question: Is something similar possible for the electromagnetic potentials [tex]\Phi[/tex] and [tex]\vec{A}[/tex]?
    My idea is to combine the scalar and vector potential into one quaternionic potential.
    (Each quaternion consists of an scalar part and an vector part, so somehow it seems possible...)
    If possible: How do the field equations look like with such an potential?
    Or is there a different possibility to "unify" these two potentials?

    Thanks,

    David
     
  2. jcsd
  3. Feb 3, 2009 #2

    George Jones

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  4. Feb 4, 2009 #3

    dx

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    The potentials [itex] \varphi [/itex] and [itex] (A_x, A_y, A_z) [/itex] are actually the components of a single four-vector [itex] A_{\mu} = (\varphi, A_x, A_y, A_z) [/itex]. This is called the four-potential. In terms of the four potential, maxwell's equations can be written (with an appropriate choice of gauge) as

    [tex] \partial^2 A_{\mu} = j_{\mu} / \epsilon_{0} [/tex]​

    Where [itex] j_{\mu} = (\rho, j_x, j_y, j_z) [/itex] is the four-current.
     
  5. Feb 4, 2009 #4

    George Jones

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    And, in the system of complex quaternions, 4-vectors are expressed as (direct) sums of scalars plus 3-vectors.
     
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