In a 2011 paper, Gersten and Moalem derive among other things photon wavefunction and wave equations for massless particles of any spin. Is it possible to derive quark masses too using their approach?
http://iopscience.iop.org/1742-6596/330/1/012010/pdf/1742-6596_330_1_012010.pdf
Unfortunately, Victor, your question does not make any sense. Questions like this one only create confusion. One should always critically think and evaluate one's questions before posting them!
Wave equations for (free) particles of any spin have been derived by Pauli and Fierz, many, many years ago. After Wigner's work, one may view this task as a kind of exercise in the representation theory of the Poincaré group, supplemented by the principle of locality.
The wave mechanics of the free electromagnetic field ("photon wave function") and the perturbative treatment of interactions of atoms and molecules with the quantized electromagnetic field have been developed by Jordan and Dirac in the late twenties. Their theory has nothing to do with quarks and their masses. Deep explanations of why masses of quarks and leptons have the values they have are missing!
No. Their approach is group theoretical and is presented in Bogoiiubov and Shirkov's book. It is possible to write the equations of motion for fields that tranform in the spin-1 representation of the Lorentz group into terms of the irreducible spin-1/2 representations; but this is mathematics, not physics (and goes back to Weyl). This doesn't imply that the spin-1/2 components can be identified with other fields, in particular because the charges don't match.
The "photon wave function" is not new at all. Various authors have bumped into this formulation. But it is not relativistically covariant, the correct one is the one of Hestenes in space time algebra. The photon mass is zero from symmetry considerations, and the quark mass should be zero too because of the gauge symmetry. In the standard model, it takes the Higgs mechanism to give mass both to elementary bosons and fermions.
The representation of the "photon wave function" is [1/2, 1/2] and not [1/2, 0] + [0, 1/2] like for two spin 1/2. The two components are helicity components, that is, spin 1 parallel or anti-parallel to the momentum.
Since the number of degrees of freedom is the same, the two representations are related by a linear transformation-it's just a change of basis-so they describe the same physics. This is well-known mathematics.
Although the representations [1/2, 1/2] and [1/2, 0] + [0, 1/2] are both 4-dimensional, they are not equivalent as representations of the Lorentz group -- or actually of its covering group SL(2,C). In the first case a rotation by 2 pi is represented by unity, in the second case it is represented by minus the unity. Hence particles transforming according to the first (second) representation must be bosons (fermions). That is physically a very big difference.
It's recommended to read the corresponding sections in Bogoliubov and Shirkov and, also, to recall that the spin-1 representation of the Lorentz group is reducible. These are mathematical operations and are well-known. The identification with fermions and bosons is a totally different issue, completely unrelated to these aspects. For the latter cf. Wightman's article, https://eudml.org/doc/122216.
Victor
,In principle, YES
the pre-Big-Bang energy singularity consisted of a Bose-Einstein State of Photons. You have my reference. These were the building blocks for electrons and photons (neutron = photon + electron). Since photons are derived to consist of quarks, these also must be formed from photons.
I do not know the paper by Gersten and Moalem, and there approach. However, any property of photons could be contributing to the formation of quarks. By the way: Do quarks have spin, and if yes, spin number and + or -? Ingo
Manuel, I gave the context as the standard model. The Higgs mechanism is part of the standard model, it is a fact. Whether the standard model really describes Nature is another problem.
I would not try to deduce quark masses from photon wave function!;-) Photon propagator codes information about various virtual pairs to which it can decay but even if one could abstract out this information about quark masses, it would not explain them.
Of course, also Higgs mechanism fails to explain quark masses: it only reproduces them. This is often forgotten. Also it is forgotten that LHC only proved the existence of Higgs, not Higgs mechanism. One obtains fermionic couplings of Higgs proportional to mass of fermion just by assuming gradient coupling. The little bit too low value of Higgs mass causing cosmic existential problems in future could be seen as one signature about something being wrong in the theoretical picture.
The spin-1 representation is irreducible under the rotation group, where only integer values are allowed. However, as is well-known, angular momentum can take half-integer values. And it's a standard exercise that when we add two spin 1/2's, we obtain a spin-0 and a spin-1; and this goes in both directions. Therefore it's another exercise to show that a 4-vector can be represented by bispinors. There shouldn't be any need to spell out facts that can be found in all books.
Adding two spin 1/2, we get the 1/2 x 1/2 representation of SO(3), which is equivalent to the 0 + 1 representation, that is, a reducible one. There is an analog thing for the Lorentz group SO(3,1).
The spin 1 representation of SO(3) is irreducible, but the [1/2, 1/2] representation of SO(3,1) is reducible if restricted to one of the SU(2) groups that compose the covering group SL(2C) of SO(3,1), or to its rotation subgroup SO(3).
To make things clearer, spin-1/2 representations aren't representations of SO(3), but of SU(2), its covering group.
Unfortunately, Victor, your question does not make any sense. Questions like this one only create confusion. One should always critically think and evaluate one's questions before posting them!
Wave equations for (free) particles of any spin have been derived by Pauli and Fierz, many, many years ago. After Wigner's work, one may view this task as a kind of exercise in the representation theory of the Poincaré group, supplemented by the principle of locality.
The wave mechanics of the free electromagnetic field ("photon wave function") and the perturbative treatment of interactions of atoms and molecules with the quantized electromagnetic field have been developed by Jordan and Dirac in the late twenties. Their theory has nothing to do with quarks and their masses. Deep explanations of why masses of quarks and leptons have the values they have are missing!
Dear Victor,
I saw your question this morning and I must say that I agree with the arguments raised by Martin Frohlich. Even though I am less sharp in judging the paper you have mentioned as a mere exercise, I should remark that no Physics is contained in it. The derivation of quark masses requires the hypothesis on a mechanism of interaction and an additional field inducing the coupling. The mechanism could be that proposed by Higgs and then extended by Weinberg, or any other phenomenological model.
The point is that any derivation of the masses will face with the free parameters of your theory and at the moment I do not see any way out!
Sorry but this is the way is it
Pino
Dear Dr. Martin Frohlich and Dr. Giuseppe Dattoli: ok i see your points. Thanks and best wishes
Dear Dr. Martin Frohlich: i apologize if my question sounds irrelevant. Allow me to describe shortly my rationale, correct me if i am wrong: i recall that QCD lagrangian equals to Dirac lagrangian plus additional term, so it seems to me possible - at least in theory- that some kind of modified photon wavefunction may explain the symmetry underlying hadron and quarks. But you are right that the deep explanation of quark masses is missing. Best wishes
I too agree with Frolich.. first of all the meaning of mass to quarks are not well understood.
@Giuseppe, Manuel, Asoke and Vinodkumar: you are right, my question is out of the wall of standard picture. But i just found this paper by Rosen in MPLA 2007, in essence he suggests that quark and lepton mass and structure may be derived from Dirac-Goldhaber theory. At least he is able to derive top quark mass. See http://home.comcast.net/~gerald-rosen/heuristicmpla.pdf
http://home.comcast.net/~gerald-rosen/heuristicmpla.pdf
@Prof. El Naschie: thank you for your answers. If you wish, you can send your paper to my email at [email protected]. I will find time to read it. Thanks
Link to the paper you asked about is not working. But I believe it is possible, I worked on a semi-classical model for undergraduates that does the same thing.
https://www.academia.edu/879621/Symmetry_In_Pair_Production_and_The_Importance_of_Quantum_Spin_in_Particle_Formation
@Charles: thank you for your reference. Have you been able to derive all quarks, baryons and mesons? Best wishes
Just the Up and Down quarks. The model was to made to introduce advanced physics concepts at the undergraduate level, which for a semi-classical educational model is not bad, but I never tried to push it to see if it was more robust.
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