Would like here to explore nonzero mass eigenvalues
posed as
Nabla psi = mu psi
basically Klein Gordon, but not usually posed this way.
where mu si basically mmcc/hbar hbar
and the left hand side nabla is basically the wave equation operator.
The only obvious case is of course mu=0, the zero mass case when any function f(x-ct) is applied on the left.(zero mass boson like the photon)
Questions are, is it valid to pose the question in this way, can we expect to find nontrivial finite mass solution as eigenvalue? Or is this mathematically impossible ? It should not escape attention that the more usuall approaches seem to fail here.
I am not sure if I understand the question correctly. Are you asking whether there is a way to formulate the Klein-Gordon equation as an eigenvalue equation, i.e. in an Hamiltonian form?
If so, then the answer is yes. Done in the 50s by some Japanese authors Sakata/Taketani), but there's a review by Feshbach/Villars on this: Article Elementary Relativistic Wave Mechanics of Spin 0 and Spin 1/...
It would be interesting to see some actual finite mass prediction
out of this; or we are not at this stage yet?
Well, the Klein-Gordon equation does not make any mass predictions for elementary particles if this is what you mean! Neither does any other fundamental equation in QM. In addition, the Hamiltonian form does not provide any new physics or have any other solutions than the "normal" KG equation has.
The Hamiltonian form only make much more natural the transition from nonrelativistic to relativistic QM, and the failure of the latter (because it needs to be superseded by QFT).
Maybe if you tell me more about the background of your question, I might be able to point you to the right direction.
Yes, that was my impresion, I wanted to make sure.
Im not sure QFT is in a position to supersede QM yet.
More talk than hard evidence.
Perhaps one has to suppplement such QM equations with something more, but as you say, no one suceeds yet.
"Im not sure QFT is in a position to supersede QM yet.
More talk than hard evidence."
But you are familiar with QFT, aren't you? The successes of QFT? The renormalization group? And so forth...?
QM might be a nice, mathematically well-founded theory. Nevertheless the domain of validity is also well-known, and limited. There is no relativistic QM! And it is superseded by QFT, no question about that! The issue with QFT is the mathematically very weak foundation, granted. So let's put more efforts into luring mathematical physicists into that topic...
There are whole books written about relativistic QM
The Dirac equation is in this subject, predicts spin and antiparticle. You can see how Klein gordon (or Einstein relativity) is recovered with the free Dirac equation.
H=c alfa.p +beta mcc
alfa and beta are the Dïrac matrices. You just have to square H
So I see QFT more of a playground for new ideas. Needs consolidation.
I have dealt with parts of QFT. Only trust (rare) exact methods so far, not that perturbation stuff.
The limitation of relativistic QM is mainly to keep to one particle. That is why you go to second quantization and the like.
I guess the problem is solutions found for any value of m, makes the eigenvalue problem sensless for mass eigenvalues.
You just get
kk -(omega omega/cc) = mmccc/hbar hhbar
where k=2 pi/lambda
omega is angular frequency, lambda is wave length
This is the solution from a simple wave exp(i(kx-omega t)
One can substitute in turn lambda = vT using
omega =2 pi/T , to get other well known conclusions.
Nothing new
One cannot however discard the possibility of other types of solution?
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