- #1

Tio Barnabe

The Dirac Hamiltonian is essentially ##H = m + \vec{p}##. I found a issue with this relation, because we know from relativity that ##E^2 = m^2 + p^2## and there seems to be no way of ##E = \pm \sqrt{m^2 + p^2} = m + p##. To get out of this issue, I tried the following.

I considered ##E## as a Euclidean vector, such that ##\vec{E} = (m, \vec{p})## and ##\vec{E} \cdot \vec{E} = m^2 + p^2##. We can identify ##E^0 + E^1 = m + \sum p_i## as a scalar representing the total energy, for ##m## and ##p## are both energies themselves.

However, I do not like this way of avoiding the initial issue. Is there a better explanation for the Dirac Hamiltonian being of that form? Is there a reason for why we can add the two energies in that way?

I considered ##E## as a Euclidean vector, such that ##\vec{E} = (m, \vec{p})## and ##\vec{E} \cdot \vec{E} = m^2 + p^2##. We can identify ##E^0 + E^1 = m + \sum p_i## as a scalar representing the total energy, for ##m## and ##p## are both energies themselves.

However, I do not like this way of avoiding the initial issue. Is there a better explanation for the Dirac Hamiltonian being of that form? Is there a reason for why we can add the two energies in that way?

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