Talk:Dirac equation

message: The mathematical formulation section I think is quite conversational. In particular it begins with historical developments including labelling psi as a wave-function in one of the first few sentences: I think this could be misleading for newcomers. Perhaps it might be better to call this the historical development section, and have a mathematical formulation section which is more reference-like? I'm not sure what the ordering of the sections should be.  — Preceding unsigned comment added by Zephyr the west wind (talk • contribs) 09:50, 8 June 2022 (UTC)
replies: text: Personally I wish the math section was written assuming that I don't know what a bispinor is. But Wikipedia is always like this these days, requiring the reader to have the equivalent of a grad degree in math to follow what is otherwise simple diff eq.71.65.253.231 (talk) 02:19, 22 August 2022 (UTC)
thread_title:  Mathematical formulation 

message: So I am using the android wikipedia app, and for some reason when I browse in dark mode, the equation boxes in this article just appear as white boxes with no text. They also seem pretty unnecessary as the equations inside them can just be displayed like the other equations in the page 105.182.127.188 (talk) 08:12, 22 May 2023 (UTC)
thread_title:  The equation boxes are unnecessary and also illegible in dark mode 

message: Section External Links, item Pedagogic Aids to Quantum Field Theory contains the statement "click on Chap. 4 for a step-by-small-step introduction to the Dirac equation, spinors, and relativistic spin/helicity operators." The problem here is that https://www.quantumfieldtheory.info/ has no clickable link to Chap. 4. David Spector (talk) 13:19, 31 July 2023 (UTC)
replies: text: Perfectly true, so I have removed it. JBW (talk) 14:07, 16 November 2023 (UTC)
thread_title:  "Pedagogic aids" 

message: I strongly recommend against modifying the current article with this content. There are so many technical problems with it. The first is that if E is the energy eigenvalue of the Hamiltonian operator associated with a conservative (some times called closed) physical system, then it ought to correspond to the total energy. The total energy in Special Relativity (SR) is not m\,c^2, but for the zero-field rest frame, which has v=p=0. The identity,
E = \gamma\,m\,c^2 = \sqrt{|p|^2\,c^2 + m^2\,c^4},
must be preserved at all times, and with the understanding that p = \gamma\,m\,v. Here of course, \gamma(v) = \left(1 - \frac{|v|^2}{c^2}\right)^{-1/2}. Breaking this identity will violate the theory of SR. Moreover, it is possible to define an operator of multiplication in momentum space, \gamma(p) and to study the spectrum of this operator. It has all the correct properties. In the case of no external fields, the operators \gamma(p), and \gamma(v) will be equivalent. In general, they are not the same, and one is able to study the differences between these linear operators, as one possible introductory study of Zitterbewegung.
I also agree that there is much redundancy, modulo the technical discrepancies, and that this may constitute a pass at original research, which should not be included in Wikipedia.
Additional technical problems involve the fact that every notion of correspondence principle always has a counterexample. All known counterexamples make use of the fact that on the classical symplectic space, canonical coordinate changes do not preserve the spectra of linear operators on (infinite) dimensional Hilbert space. The, perhaps best known, attempt to systematize this is called Geometric quantization. But it is fraught with its own challenges.
Moreover, the relation v_j=c\alpha_j is extremely questionable for particles with any mass, m > 0. For this to be true, the \alpha_j's would need to be less than 1. But in the simplest most well known representation of the \alpha_j's every non-zero entry has complex modulus of 1. It makes no sense! There is likely no experiment that measures the eigenvalues of a velocity operator. This statement is so contentious that one really must include some sort of reference to an actual reproducible experiment.
The so-called equation (3) is not relativistic, and it is missing minus the imaginary unit, -i. One should not confuse the linear independent time variable in a partial differential equation with that of the relativistic proper time. The proper time is the correct relativistic generalization of the Newtonian time, which acts as a continuous parameter. Parametrization is very different from a coordinate. If the signs were corrected, the conclusion of the (would-be) equation (3) is that the 4-velocity contracted into the 4-momentum would give the contradiction that m\,c^2 = -m\,c^2. The only solution of x=-x is x=0, for x\in\{\text{All Known Mathematical Sets}\}. Now before anyone tries to invoke colossal confusion by thinking they can adjust the signature of the Minkowski geometry to avoid this, first be aware that the Hamiltonian operator is strictly positive because it is the total energy, the mass times the identity operator is strictly positive, and so are their squares. The momentum operator squared is strictly non-negative, and so the relativistic invariant mass squared must also be a non-negative operator. Second, the assertion (3) would imply that \beta\,\alpha_j = \alpha_j for \beta = \alpha_0 \neq \bf{1}, which is yet another contradiction!
Finally, there is also the logical contradiction that the velocity operator, in fact, does commute with the free Hamiltonian, since there is no Zitterbewegung in the free case. To see this, one may simply define again, p(v) = \gamma(v)\,m\,v, or equivalently (for the free Hamiltonian only) v(p) = [\gamma(p)]^{-1}\,p / m, as operators of multiplication.
I want to emphasize that these are purely factual and logical objections, and they are purely in the context of encyclopedic standards for Wikipedia. Nothing here should be construed as a human is bad, nor that someone's ideas are bad.  MMmpds (talk) 21:51, 2 May 2024 (UTC)
replies: text: I encourage contributors to this discussion to include references because nothing will be added without them. Johnjbarton (talk) 22:11, 2 May 2024 (UTC)
thread_title:  Some Healthy Dissent