Electrodynamics: quaternionic potential?

In summary: So, the potential (A_{\mu}) can also be written asA_{\mu} = \sum_{i=1}^4 \vec{i} where \vec{i} = (\rho_i, j_i, \vec{A}_i).
  • #1
the_viewer
3
0
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
 
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  • #2
the_viewer said:
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.

Yes.

See

https://www.amazon.com/dp/0817640258/?tag=pfamazon01-20.
 
Last edited by a moderator:
  • #3
the_viewer said:
Hi!

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

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.
 
  • #4
dx said:
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].

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

Related to Electrodynamics: quaternionic potential?

1. What is electrodynamics?

Electrodynamics is a branch of physics that studies the behavior and interactions of electric and magnetic fields. It is a fundamental theory that explains the phenomenons of electricity and magnetism, and their relationship to each other.

2. What is a quaternionic potential?

A quaternionic potential is a mathematical concept used in electrodynamics to describe the potential energy of a system. It is a four-dimensional vector quantity that combines electric and magnetic potentials in a single mathematical framework.

3. How does electrodynamics differ from classical electromagnetism?

Electrodynamics is an extension of classical electromagnetism, which only considers the behavior of static electric and magnetic fields. Electrodynamics also takes into account the effects of moving charges and changing magnetic fields, which are described by Maxwell's equations.

4. What are some practical applications of electrodynamics?

Electrodynamics has numerous practical applications in modern technology, including electricity generation, telecommunications, and electronic devices. It also plays a crucial role in understanding and developing technologies such as lasers, superconductors, and magnetic resonance imaging (MRI).

5. How has the concept of quaternionic potential impacted the study of electrodynamics?

The introduction of quaternionic potential has provided a more elegant and unified framework for understanding the behavior of electric and magnetic fields. It has also allowed for more complex and accurate calculations, leading to advancements in the field of electrodynamics and its practical applications.

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