Undergrad Trying to understand Dirac Hamiltonian

[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?

 

[hilbert2]

The $p$ in that equation is not just a scalar multiplying an identity matrix. There's nondiagonal elements too, and they affect how it operates on a spinor.

 

[Tio Barnabe]

Yes. I'm aware that $p$ is actually $\vec{p}$, a 3x1 matrix, the 3-momentum, in the first equation.
In the second equation (that for $E^2$) it's just the inner product of it with itself.
But, I think this doesn't affect what I said.

You may take a look at my edited post.

 

[Fra]

The problem is related to the fact that you are starting from an equation of classical mechanics, and the electron is a fermion without classical correspondence.

What you prefer, heuristics or axiomatic approaches is a matter of preference. To what extent you can "understand" what an electron is, starting from classical mechanics can be discussed. Whats clear is that no classical image will do. But you might be able to make it in an abstract interpretation. But what you can do is to
1. start from the second order KG equation, of a spinless classical particle, make the heuristic formal quantization
2. Then you will see when the relativistic interpretation of this becomes strange or non-physical and particle looking to go backwards in time etc.
3. Now, the KG equation has two linearly independent solutions and you can make a change of depdent variable here, that involves the pauli matrices. You end up with a first order equation instead, but vector valued. And you can identify the electron/positron, and this "form" which can be seen as formally equivalent if you work this out, has also the right interpretation. And therefore is preferred.

So that in a way you can "interpret the electron" as in a mathematical sense dual to a picture where you have a spinless particle with the same mass, but seems to behave strangely. And it might behave strange enough that if it was seen, it would be interpreted by the observer as the electron. Sounds familiar to something else, doesn't it, but perhaps its a coincidence?

/Fredrik

 

[Fra]

if it was unclear, here is the transformation you can try
\vec{\psi} =\frac{i\hbar}{mc}\left(A^{\mu}\partial_{\mu} +B\frac{mc}{i\hbar}\right)\phi
And phi is the solution to KG, and psi solves the dirac equation. you can view this transformation simply as a change of dependent variable, and transforming the second order KG into a first order system of differential equations instead, and this "reinterpretation" of the system. A and B are 2x4 matrices that just have elements of 1 and i, i don't type all details but they are related to gamma matrices
A^{\mu}=i\gamma^{\mu} B


A bit "philosophy" or "interpretations":

This is one way of motivating the dirac equation. But the peculiar thing here is that you start with a boson and end up with a fermion. But that's exactly the trick we need to understand the electron in the first place.The part which you can thinkg about is if there is a physical interpretation of this "transformation"? This is the interesting part, as the transformation is an interesting mix between internal and spacetime.

When I first went though this, I viewed this as an inference process. Suppose you have the prior idea that you are seeing a relativistic boson, then the expectation as per the KG logic, can be "recoded" into the dirac form, loosely speaking without changing the information. IMO while this interpretation in physical tends to be messy and not 'they way its taught IMO, you can't help by to associate this to the idesa of supersymmetry? If we see supersymemtry as a change of coding on the observer side? But where once coding is "preferred" due to reasons you can ponder on about?

/Fredrik

 

[Tio Barnabe]

Thank you Fredrik for trying to help

I have got a decent solution to the issue. I realized how the thing works!