Hamiltonian for Fermionic Fields

If the Jaynes-Cummings equation describes the Hamiltonian for bosons, what equation describes the Hamiltonian for Fermion fields?

 

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I may have found the equation:

\(H = \sum_{k} (\epsilon_k - \mu) c^{\dag} c_k\)

Problem is, I don't know what epsilon is. Nor mu. Is one of them an interaction on the field, as I would have expected this since there is an interaction term of the boson field in the Jayne-cummings equation \(\hbar \lambda (\hat{\sigma}_{+}\hat{a}+\hat{\sigma}_{-}\hat{a}^{\dagger})\)

 

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epsilon is the frequency and mu is the muon particle. So there you have your summation of energy for the muon on the field H. No idea what equation could be for fermions, but If I had to guess I would say the Dirac equation

 

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Mmm... yes. I had an incline the mu was for muon particles, but I was not sure. I also suspected the Dirac Equation since they are directly attributed to spin 1/2 particles. But I am not sure what formalism expresses it in the conventional way i.e. \(\mathbb{H}\).

 

I found it I think http://electron6.phys.utk.edu/qm2/modules/m9/dirac.htm near the top

 

It's called the Dirac Hamiltonian (of course

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) - Seems to be expressed as

\(H= \alpha \cdot pc + \beta Mc^2\)

where all the usual suspects would be expected in the Dirac Equation.

 

Which leads me to my next problem... I'll need to type it. Guh, more latex.

 

The irony of G.D. thinking that asking a question about "what" equation describes fermions leads one to the conclusion that the asker doesn't perhaps know that much about fermions.

Or maybe it's just me.
I'm reasonably sure, though I'm no Ph.D., that bosons are found in a few more equations than the Jaynes-Cumming.

 

I don't see why there should be any irony. I did suspect the Dirac Equation, but it wasn't evidently clear to me why I should suspect it. As for the Jaynes-Cumming equation, it is probably the most well-known, which is why I said that equation, and why it would bother you other equations are out there, and why I never chose them, is beyond me.

ps. I never asked just ''about fermions'' - I meant there total energies. I know fine well the Dirac equation describes fermions.

 

According to Dr. Valone (where? [1]) the zero point fluctuations (a field in its own right) contributes to electron energy. The energy which is contributed to the electron is of a magnitude of \(\frac{e^2 h}{4Mc\alpha^2}\) with an upper bound of \(hf=15MeV\). Since a true Hamiltonian is of the total energy of the system, it only seems fair to take this vacuum energy into account, but the Dirac Hamiltonian, as far as I understand it does not do this for fermions, if that is indeed the usual Hamiltonian we attribute to electrons. Therefore, why should it not be consistent to assume the total energy of the system takes on the form of:

\(H= \alpha \cdot pc + \beta Mc^2 + \frac{e^2 h}{4Mc\alpha^2}\)

Is there any specific reason why this is not considered? Was Dirac aware of the energy of the ZPF contribution to electrons?

[1] Thomas F. Valone http://books.google.co.uk/books?id=...t energy contribute to a hamiltonian?&f=false

 

Yes, well Arfa. I find it ironic you have never heard of the Klein Gorden equation, if you are sure other equations describe bosons.

 

How do you know I haven't heard of the Klein-Gordon equation?
Do you know what it says about fermions? Do you know why there are different equations for bosons than for fermions? Do you know what a Hamiltonian is?

p.s. science is about making claims and expecting them to be challenged. It's what happens when a paper is submitted for publication, for instance. You must be expecting your posts to be challenged if you only attempt to convey a vague grasp of the fundamentals rather than something more solid. Your questions give this away.

 

Of course you know the KG equation, just as much as I know about the Dirac Equation. But in the same sense, you were unsure off-hand, or you would have mentioned it, as much as I was unsure if the Dirac Equation equated the Hamiltonian. It doesn't pay to get smart sometimes.

 

Arfa

Nice sly modification you did there on your post. For your information, I don't consider sciforums as a place where I would expect peer reviews. As for my knowledge on the statements, it's no more than a question, and I expect nothing more than an answer.

 

''Do you know what it says about fermions? Do you know why there are different equations for bosons than for fermions? Do you know what a Hamiltonian is?''

Arfa, what is this? I am the one who asked the question, for gods sake.

 

Well, the answer is that you need to understand a bit more, then you can ask questions that don't give away your lack of understanding.

 

I'll tell you what. Once my question is answered, I'll answer this question. I know exactly what the Dirac Equation has to say about fermions, nearly every single bit. But not Pauli Matrices, hands up. But I've made that clear here before.

 

Getting smart again are we?

Answer the question posed if you can, then I will answer yours.

 

Just in case you don't know, we moved up one question:

According to Dr. Valone (where? [1]) the zero point fluctuations (a field in its own right) contributes to electron energy. The energy which is contributed to the electron is of a magnitude of \(\frac{e^2 h}{4Mc\alpha^2}\) with an upper bound of \(hf=15MeV\). Since a true Hamiltonian is of the total energy of the system, it only seems fair to take this vacuum energy into account, but the Dirac Hamiltonian, as far as I understand it does not do this for fermions, if that is indeed the usual Hamiltonian we attribute to electrons. Therefore, why should it not be consistent to assume the total energy of the system takes on the form of:

\(H= \alpha \cdot pc + \beta Mc^2 + \frac{e^2 h}{4Mc\alpha^2}\)

Is there any specific reason why this is not considered? Was Dirac aware of the energy of the ZPF contribution to electrons?

[1] Thomas F. Valone http://books.google.co.uk/books?id=...t energy contribute to a hamiltonian?&f=false