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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.
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.