Majorana equation in Algebra of Physical Space

In summary, the discussion is about expressing the Dirac and Majorana equations in the Algebra of Physical Space (APS). The Dirac equation has been shown to have a simple expression in APS by W Baylis and independently by the speaker in their own notation. However, the speaker is struggling with writing the Majorana equation in APS notation and is seeking clarification on the charge conjugate operator in this version of the Dirac equation. They mention the possibility of using a textbook to map the four component equations in the Weyl representation to APS notation. The question is directed towards @Jonathan Scott for any additional insights.
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Jonathan Scott
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I wasn't quite sure which forum to use for this, as it covers QM, relativity and geometric algebra, but I'm trying QM for a start.

The Dirac equation can be expressed surprisingly simply in the Algebra of Physical Space, as has been shown by W Baylis, and I worked the same thing out independently in my "Complex four-vector" notation in around 1993:

[tex]
i \overbar \partial \Psi \mathbf{e}_3 + e \overline A \Psi = m \overline \Psi ^ \dagger
[/tex]

Now someone has asked me about the Majorana equation in the same notation, but I'm so rusty on this area that I'm having some difficulty with this. I understand that starting from the conventional form of the Dirac equation the Majorana equation involves replacing the wave function on one side of the Dirac equation with its charge conjugate; this does not necessarily mean that one does the same with this alternative form, but it's a starting point.

Does anyone have a definite answer for how the Majorana equation should be written in this notation?

According to my old notes, the charge conjugate of [itex]\Psi[/itex] in this notation is probably simply [itex]i \Psi[/itex], because if this is substituted into the equation and the sign of [itex]A[/itex] is changed, you get the same equation but now referring to the opposite charge. Can anyone confirm whether this is indeed the recognized charge conjugate operator in the APS version of the Dirac equation?

If all else fails, I can go to a textbook which contains the Majorana equation in the Weyl representation, work out the four component equations and map them back into APS notation, but so far I have not yet had the patience and clarity of mind to follow through on this approach.
 
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1. What is the Majorana equation in Algebra of Physical Space?

The Majorana equation in Algebra of Physical Space is a mathematical equation that describes the behavior of a fermion, a type of subatomic particle, in a physical space. It was developed by Italian physicist Ettore Majorana in the 1930s and is based on the principles of quantum mechanics.

2. How does the Majorana equation differ from other equations in physics?

The Majorana equation is unique in that it accounts for the spin and charge of a fermion simultaneously, whereas other equations in physics typically only account for one or the other. Additionally, the Majorana equation is a relativistic equation, meaning it takes into account the effects of special relativity.

3. What is the significance of the Majorana equation?

The Majorana equation is significant because it allows for a more complete understanding of the behavior of fermions in physical space. It has been used to explain phenomena such as neutrino oscillations and could potentially have applications in fields such as quantum computing.

4. Can the Majorana equation be applied to other types of particles?

No, the Majorana equation is specifically designed for describing the behavior of fermions. Other types of particles, such as bosons, have their own equations that govern their behavior.

5. How is the Majorana equation used in current research?

The Majorana equation is still an active area of research in the field of particle physics. It has been used to make predictions about the behavior of neutrinos and has also been applied in the study of dark matter. Researchers are also exploring its potential applications in quantum computing and other areas of technology.

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