To the best of my knowledge, because the Dirac equation explains very well the gyromagnetic ratio of the Electron, it is typically considered to be an equation for the Electron. There seems to be no such thing as the Dirac equation for the Proton or Neutron despite them being spin-1/2 particles. After the calculation that I conducted in the present paper, I would like to believe that the Dirac equation can be thought of as an equation for any general spin-1/2 particle. I certainly will be happy to hear your thoughts on this very important matter.
Preprint On the Extra Anomalous Gyromagnetic Ratio of the Electron an...
Of course the Dirac equation describes the proton, since it's an electrically charged, spin-1/2 particle. And this description has been extensively tested using nuclear magnetic resonance techniques, most recently with traps in, for instance, http://arxiv.org/pdf/1201.3038v2.pdf
The only inputs for deducing the Dirac equation are Lorentz invariance and the fact that the spin is 1/2. (This isn't how it happened, historically, but that doesn't matter.) Mass and spin are the Casimirs of the representation of the Poincaré group, so the fact that the proton is not an elementary particle is irrelevant: if it can be described as a massive, spin-1/2 particle, its wavefunction satisfies the Dirac equation with all that implies.
There's a subtlety, however, regarding electric charge and that affects the neutron. which is electrically neutral. This is the issue, how to describe spin-1/2 neutral particles, that was resolved by Majorana, generalizing Dirac's approach. It's thus possible to describe neutrinos, too. The Dirac equation does not imply that parity is a good symmetry-if it is, it can describe the properties of the corresponding particles, if it isn't, it can, too. For instance, the electron has weak interactions and its properties are described by the Dirac equation-the two chiralities have different properties, that's all.
Now, of course, it's known that the proton and neutron are bound states of quarks, that are charged, spin-1/2, particles, so the Dirac equation applies to them, when the description in terms of quarks is appropriate. In nuclear magnetic resonance experiments the quarks are not relevant, it's the proton and the neutron-since they do carry spin-that are affected by magnetic fields. For the proton this is easy to understand, for the neutron it takes a bit more work-but these issues are now understood and taught, hopefully, everywhere.
Protons and Neutrons are made up of sub atomic particles (quarks), Dirac equations describes all spin-½ massive particles, for which parity is a symmetry, such as electrons and quarks, and is consistent with both the principles of quantum mechanics and the theory of special relativity
Source: http://en.wikipedia.org/wiki/Dirac_equation
Bhushan, if you say all spin-1/2 particles, why does it not give the correct gyromagnetic ratio for the Proton, Neutron, and atomic Nuclei?
May be because protons , neutrons are not sub atomic particles they are made up of quarks , for quarks Dirac equations work it seems.
I am not sure though , will do some research and get back to you.
Of course the Dirac equation describes the proton, since it's an electrically charged, spin-1/2 particle. And this description has been extensively tested using nuclear magnetic resonance techniques, most recently with traps in, for instance, http://arxiv.org/pdf/1201.3038v2.pdf
The only inputs for deducing the Dirac equation are Lorentz invariance and the fact that the spin is 1/2. (This isn't how it happened, historically, but that doesn't matter.) Mass and spin are the Casimirs of the representation of the Poincaré group, so the fact that the proton is not an elementary particle is irrelevant: if it can be described as a massive, spin-1/2 particle, its wavefunction satisfies the Dirac equation with all that implies.
There's a subtlety, however, regarding electric charge and that affects the neutron. which is electrically neutral. This is the issue, how to describe spin-1/2 neutral particles, that was resolved by Majorana, generalizing Dirac's approach. It's thus possible to describe neutrinos, too. The Dirac equation does not imply that parity is a good symmetry-if it is, it can describe the properties of the corresponding particles, if it isn't, it can, too. For instance, the electron has weak interactions and its properties are described by the Dirac equation-the two chiralities have different properties, that's all.
Now, of course, it's known that the proton and neutron are bound states of quarks, that are charged, spin-1/2, particles, so the Dirac equation applies to them, when the description in terms of quarks is appropriate. In nuclear magnetic resonance experiments the quarks are not relevant, it's the proton and the neutron-since they do carry spin-that are affected by magnetic fields. For the proton this is easy to understand, for the neutron it takes a bit more work-but these issues are now understood and taught, hopefully, everywhere.
Yes, the Dirac eqn. is valid for any elementary fermion and composite fermion also as far we are not talking about such properties that are dependent on its composition.
Regards,
Rajat
Rajat, I am not saying it is invalid, but, that the fact that is predicts correctly only the gyromagnetic ratio of just one particle, the Electron. What about the others? Why does it not give the correct values? This paper merely say, yes, in principle, the Dirac equation does account for the gyromagnetic ratio of these particles if one considers the second order terms as has been done in the present paper.
The statement that the Dirac equation predicts the gyromagnetic ratio of the electron only is incorrect. This is textbook material (cf. Itzykson and Zuber, ``Quantum field theory'', for instance). The gyromagnetic ratio for any particle, of spin 1/2,in a fixed external field, is equal to 2-that's a consequence of the Dirac equation-and quantum fluctuations modify this value giving rise to what's called the ``anomalous magnetic moment'', denoted (g-2)/2. This is sensitive to the contribution of unkown particles, so its measurement can lead to signals of their presence. All known particles, of spin 1/2, in fact, have an anomalous magnetic moment, since all are subject to quantum fluctuations. And this can be computed and measured for the proton and the neutron, as well as for the electon. However the fluctuations themselves do not give rise to universal contributions for all particles. For the proton and neutron the calculation and measurement are much more difficult due to the contribution of fluctuations due to the strong interactions, that are strong at the energy scales, where the magnetic moment is measured.
Strong interactions (i.e. the fact that they are composite particles) spoils the extension of Dirac's considerations to proton and neutron which, in fact, have an anomalous magnetic moment. This, clearly, doesn't imply that there are no antiparticles which are, in fact, perfectly well known to exist and are commonly produced in high energy accelerators
Dirac equation is for elementary particle spin 1/2. It is a one-particle equation of relativistic quantum mechanics, not taking into account any interaction, i.e. effects of Quantum Field Theory.
WE believe that states describing elementary particles are representations of Poincare group and these representations are labeled by two Poincare invariants - mass and spin. If you attempt to write relativistic equation (which transforms correspondingly when switching to another reference frame) for a spin 1/2, only one you will get is a Dirac equation.
It is not an equation for a composite system. So, if you consider, e.g. neutron as a elementary particle, equation for it will be the Dirac equation. But, as said above in comments, this description is not very good. For leptons (electron, muon, tau, neutrinos) it seems to be all right.
@Dr. Christian Baumgarten: thank you for the link to math-ph/0210055, it discusses spin 3/2 particle. Some time ago i also found a paper by S.I. Kruglov which discusses a square root of Proca equation and it perhaps works for 3/2 spin particles too. What do you think?
http://arxiv.org/abs/hep-th/0405088
The Dirac equation describes spin 1/2 point-like particles, which are Fermions. This equation can be derived from the group of SL(2,C) transformations, which is the Lorentz transformation group plus parity. Eigenstates may have both positive and negative energy. Because negative energy states are occupied, we get anti-particles and the probability density is always positive here. Dirac equation gives correct gyromagnetic ratio for the electron. However for proton and neutron the natural form of Dirac equation does not predict a correct gyromagnetic ratio. Dirac equation is thought to be the correct theory for electrons because for leptons deeper substructures are absent. That is leptons can be treated as point particles. It also works fairly well for muon and the tau particle: for muon the gyromagnetic ratio differs from 2, which is the theoretical prediction calculated from Dirac equation, by 0.1%.
the compositeness of the nucleon prevents a simple use of the Dirac equation to describe it
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