Talk:Dirac equation/Archive 3

Validity of interpreting the solution to Dirac equation as a probabilistic wavefunction for a single particle
I understand that Dirac originally thought that the Dirac equation described a wavefunction in a similar way to the Schrödinger equation, but actually it described a classical field which is subject to quantization. I notice the article doesn't say anything about this; maybe it should. This was discussed above at "Two kinds of Dirac field" but it is still a problem with the article. The interpretation of the Dirac equation is very important and needs to be discussed. Count Truthstein (talk) 17:58, 16 December 2012 (UTC)


 * This is not specific to the Dirac equation. See first quantization and second quantization. JRSpriggs (talk) 19:09, 16 December 2012 (UTC)
 * The interpretation of the Dirac equation is relevant to the article on the Dirac equation. If the explanation of its interpretation is covered under more general explanations in other articles, maybe a brief mention is worthwhile along with a link to more detail in other articles. Count Truthstein (talk) 19:53, 16 December 2012 (UTC)


 * In first quantization, the "wave function" (or spinor) ψ is interpreted as a probability amplitude for the presence of a particle in a certain state. This can be construed as quantum mechanical theory (relative to classical physics) or as a classical field theory (relative to quantum field theory).
 * In second quantization, that "wave function" is replaced by the annihilation operator $$a\,$$ (which operates on the real wave function $$\Psi\,$$). So ψ -> $$a\,$$ and the probability that a particle is at a certain point ψ*ψ becomes the expectation value of first annihilating and then re-creating the particle $$\Psi^{*} a^{*} a \Psi \,$$ at that point. So although the theory becomes much more complicated, the meaning of ψ has not actually changed that much. JRSpriggs (talk) 05:15, 17 December 2012 (UTC)

Remove Dirac equation in curved spacetime to a new article
See the talk there, and the history of this talk page and the history of the article, around 2007-2012. I think it's sensible, there is Maxwell's equations in curved spacetime and Dirac equation in the algebra of physical space. M&and;Ŝc2ħεИτlk 23:01, 20 April 2013 (UTC)

This article has become a disgrace
We have people who are NOT EVEN PHYSICISTS doing major edits and the result is now hopeless confusion, grotesque errors, and chaos. I wrote the original article. I am a published theoretical physicist. I am entitled to attempt such a project. A logician, or a student, is NOT. We need a ruling from Wikipedia LOCKING THIS ARTICLE DOWN so I can attempt to repair it. — Preceding unsigned comment added by Antimatter33 (talk • contribs) 00:45, 21 April 2013 (UTC)


 * ...Surely, I reverted to your version and others have made edits after? I haven't made any major edits, just that reversion and cleaning up equation brackets, the boxes, added links, added references to places where citations were requested...
 * In any case, are you disagreeing with the Dirac equation/Dirac equation in curved spacetime split? M&and;Ŝc2ħεИτlk 06:27, 21 April 2013 (UTC)


 * To Antimatter33: 'Locking down' an article for the exclusive use of someone who claims to be an expert is not the way we do things here at Wikipedia; see WP:OWN. If you have specific complaints about the article, please state them. However, I would note that when I compared your last version to the current article, I saw that you had inconsistent units in the energy–momentum relation. So apparently you are not infallible. JRSpriggs (talk) 11:07, 21 April 2013 (UTC)


 * Why are you, someone who admits to having no advanced training in physics - and it's abundantly clear from what you contribute, that you don't - doing editing an article on something as basic as the Dirac equation? Why don't you stick to your areas of expertise? 76.105.101.35 (talk) 16:52, 21 April 2013 (UTC)

Agreed. Continuing from above: The newest significant changes seem to include: Antimatter33, you have to accept people like these editors will make additions in good faith without intending to "vandalize", be them physicists, mathematicians, professionals, students, interested laymen... There is no way to "lock down" pages.
 * Akhmeteli introduced the 4th order PDE in one component (sourced),
 * Count Truthstein tried to clarify the status of the Dirac field in QFT (which has raised queries before multiple times),
 * Chris Howard introduced various links including the massless case (the Weyl equation - which wasn't even mentioned before),
 * others did technical fixes...

It's understandable that you would not like your excellent hard work and prose to be deteriorated, and (except for devoted lunatics) we all agree with that. We know you are an expert researcher with ~30 years of experience and an excellent author of this article. But it's "NOT" just about one expert per article. As irritating as it may be, anyone is as equally "entitled" to editing as you are, even if it's counter to your plans.

If you list all of the worst errors here then we can correct mistakes and blend the prose, i.e. keeping your lean prose style and letting others have their say, i.e. collaborate and not fight.

If not... then considering past frustrations, you probably can't assume we are editing in good faith (in particular "the latest green students" like me?), so don't joke yourself.

(P.S. Please stop starting new posts in old sections at the top, else they can be buried in new posts further down and archived and no-one will read what you have written. Just press "new section" at the top and fill it in). M&and;Ŝc2ħεИτlk 13:10, 21 April 2013 (UTC)


 * I am not personally affronted. The purpose of an encyclopedia article is to stick to the subject and stick to facts, well-known and commonly accepted, about that one subject. There is no fourth-order equation of any sort associated with the Dirac equation! That is just plain balderdash! A subject this important needs a treatment that is done by an expert, and I am one (note: I got to be one over decades, not days), and needs to be well-written and needs to STICK TO FACTS. There are an abundance of facts to report. 76.105.101.35 (talk) 16:52, 21 April 2013 (UTC)


 * WIth all due respect, your statement "There is no fourth-order equation of any sort associated with the Dirac equation!" is not supported by any facts. On the other hand, I published an article in a decent peer-reviewed journal (the Journal of Mathematical Physics - see the reference in the Wikipedia article) where I derived such fourth-order equation for one component, and this equation is indeed equivalent to the Dirac equation. The referees of the article and the editor of the journal agreed with me. Of course, that does not necessarily mean that my derivation is correct, but until you show where exactly I screwed up, why should I (or anyone else) believe you on your word?

Akhmeteli (talk) 08:10, 23 April 2013 (UTC)


 * Let me ask you: Why are you "recommending" to people what they can and cannot edit? And if you are an expert, why haven't you made a single contribution to the article (prior to 21/04/2013) yet stating the article has to "STICK TO FACTS" given their abundance? You're OK to complain about petty things instead of just doing it yourself with an edit summary. M&and;Ŝc2ħεИτlk 17:13, 21 April 2013 (UTC)

Recent revert (13/08/2013)
Now-blocked, a WP:sockpuppet of indef-blocked , edited this article so I reverted to the version prior (D10 had generally bad formatting skills, see any contribution). Afterwards, tried to reinstate the good faith and correct edits by other editors, most edits should be recovered, if I've missed anything feel free to add. M&and;Ŝc2ħεИτlk 08:11, 13 August 2013 (UTC)

Pauli
An ip editor has been roaming around in the lead claiming that Dirac has plagiarized Pauli (first sentence). User BigDwiki reformulated the sentence as to read as of present. To what extent did Dirac base his work on Pauli's? To the extent that Pauli should be mentioned in the first sentence? I'd say probably not. YohanN7 (talk) 23:38, 17 November 2013 (UTC)


 * I've removed the rather vague statement as it stood from the lead, as there have been no objections. This has been going on for quite some time (claims that Dirac "maliciously plagiarized" from Pauli).  An IP editor with some apparent axe to grind has been inserting rather anti-Dirac sentiments into related articles without citations.  So be on the watch for POV edits.  Sławomir Biały  (talk) 17:45, 26 December 2013 (UTC)

Validity of interpreting the solution to Dirac equation as a probabilistic wavefunction for a single particle (continued)
Note: The first four posts are copied from an old thread. YohanN7 (talk) 16:01, 4 November 2013 (UTC)

I understand that Dirac originally thought that the Dirac equation described a wavefunction in a similar way to the Schrödinger equation, but actually it described a classical field which is subject to quantization. I notice the article doesn't say anything about this; maybe it should. This was discussed above at "Two kinds of Dirac field" but it is still a problem with the article. The interpretation of the Dirac equation is very important and needs to be discussed. Count Truthstein (talk) 17:58, 16 December 2012 (UTC)


 * This is not specific to the Dirac equation. See first quantization and second quantization. JRSpriggs (talk) 19:09, 16 December 2012 (UTC)
 * The interpretation of the Dirac equation is relevant to the article on the Dirac equation. If the explanation of its interpretation is covered under more general explanations in other articles, maybe a brief mention is worthwhile along with a link to more detail in other articles. Count Truthstein (talk) 19:53, 16 December 2012 (UTC)


 * In first quantization, the "wave function" (or spinor) ψ is interpreted as a probability amplitude for the presence of a particle in a certain state. This can be construed as quantum mechanical theory (relative to classical physics) or as a classical field theory (relative to quantum field theory).
 * In second quantization, that "wave function" is replaced by the annihilation operator $$a\,$$ (which operates on the real wave function $$\Psi\,$$). So ψ -> $$a\,$$ and the probability that a particle is at a certain point ψ*ψ becomes the expectation value of first annihilating and then re-creating the particle $$\Psi^{*} a^{*} a \Psi \,$$ at that point. So although the theory becomes much more complicated, the meaning of ψ has not actually changed that much. JRSpriggs (talk) 05:15, 17 December 2012 (UTC)


 * The question Count Truthstein raises is conceptually very relevant. With special relativity, the interpretation of the wave function as a probability amplitude is in principle impossible. This conclusion can be reached without appealing to QFT or Second quantization. At the risk of stirring up a storm; Dirac's original rationale for deriving his equation was wrong, and his interpretation of it was, at least initially, wrong. Ironically, the Dirac equation is in QFT superseded - by the Dirac equation - but with a different interpretation of its solutions. It is unclear whether Dirac ever realized its true nature. He should have, because he was very productive within QFT as well as RQM. But still he seems to have clung on to the probability interpretation (and also to hole theory) for a very long time. The original interpretation still lingers in the literature. Here is a snippet from the preface to Steven Weinberg's "Lectures on Quantum Mechanics":


 * There is one topic I was not sorry to skip: the relativistic wave equation of Dirac. It seems to me that the way this is usually presented in books on quantum mechanics is profoundly misleading. Dirac thought that his equation was a relativistic generalization of the non-relativistic time-dependent Schrödinger equation that governs the probability amplitude for a point particle in an external electromagnetic field.
 * ... .The right way to combine relativity and quantum mechanics is through the quantum theory of fields, in which the Dirac wave function appears as the matrix element of a quantum field between a one-particle state and the vacuum, and not as a probability amplitude.
 * ... .The right way to combine relativity and quantum mechanics is through the quantum theory of fields, in which the Dirac wave function appears as the matrix element of a quantum field between a one-particle state and the vacuum, and not as a probability amplitude.
 * ... .The right way to combine relativity and quantum mechanics is through the quantum theory of fields, in which the Dirac wave function appears as the matrix element of a quantum field between a one-particle state and the vacuum, and not as a probability amplitude.
 * ... .The right way to combine relativity and quantum mechanics is through the quantum theory of fields, in which the Dirac wave function appears as the matrix element of a quantum field between a one-particle state and the vacuum, and not as a probability amplitude.


 * This, admittedly, refers to QFT, but if only the impossibility of the probability amplitude interpretation is to be shown, QFT isn't needed. It suffices to use the finiteness of the speed of light to show that instantaneous measurements of position is impossible making time uncertain, and that pair creation occurs, rendering the number of particles present uncertain. The statement "Ψ(x,y,z,t)*Ψ(x,y,z,t)dV is the probability of upon measurement finding the particle in dV at time t " lacks meaning because the emphasized parts lack meaning. See, for instance, Landau and Lifshitz, Quantum Electrodynamics.


 * I can't say the article is "wrong" according to Wikipedia guidelines, because it does reflect what is presented in secondary sources. But perhaps the complementary view could be presented alongside the standard view. YohanN7 (talk) 15:36, 4 November 2013 (UTC)


 * The complimentary view needs to be phrased in a way that makes some sense to the layperson, at least on the surface. What the quotation does is to rubbish an interpretable statement and to replace it with something that has no surface meaning beyond "another mathematical formalism". In particular, it does not replace it with an interpretation, but basically says: the probability amplitude interpretation is wrong, because it is a quantum field and not a wavefunction.
 * As to the specific criticism of the probability interpretation statement, perhaps there is a related statement that does make sense, which should then be given? Perhaps the statement that a particular measurement will find a certain particle density, i.e. the expected number of particle interactions (with a suitably defined measuring particle field) in a 4-space volume element dt∧dx∧dy∧dz? Or is the idea of probability incompatible with QFT, in which case the Copenhagen interpretation is presumably incompatible with QFT?  — Quondum 19:14, 4 November 2013 (UTC)


 * The idea of probabilities is of course compatible with QFT. Probability densities aren't as far as I know, because of the way QFT models reality. My claim (really not my claim of course) is that this is true already at the level of RQM for the reasons I stated above.
 * I don't know much about the compatibility of the Copenhagen interpretation other than that most fields (and wave functions) are not believed to be measurable in any sense.
 * I gave the Weinberg quote to demonstrate that Dirac's interpretation can be criticized. I don't mean it to go into the article. If you read it between the lines, ..., well, it is a pretty strong verdict coming from someone who should know. Then I don't agree with you that Weinberg rubbishes an interpretable statement and replaces it with air. It's the other way around. (The Dirac wave function is still a wave function b t w, not a quantum field (operator).) But then if you mean that QFT is harder to understand than QM 101, then you are right. Is this a reason to pretend that the original Dirac interpretation is correct? YohanN7 (talk) 22:33, 4 November 2013 (UTC)
 * My QM/RQM/QFT books vary in the treatment of this issue. Some use (mostly the "pure" QM with "Dirac chapter", like Shankar) the Dirac interpretation, others point out that it is wrong but then proceeds to use the same probability terminology (e.g. Greiners RQM). Yet others (e.g. Landau and Lifshits QED) point out that it is wrong in several respects (and why), and then never use the phrase "probability density". YohanN7 (talk) 19:55, 4 November 2013 (UTC)
 * Actually, L&L calls the conserved quantity for the Dirac equation the particle density. This makes some sense and is not directly connected to measurements as is probability. But they don't define what they mean by "density", at least not what I can see right now.


 * As for the explanation JRSpriggs gave above of the probability of particles being somewhere. It works in non-relativistic QFT but it seems to suffer from the same problems as the Dirac interpretation when relativity is included. YohanN7 (talk) 23:23, 4 November 2013 (UTC)
 * Should correct myself here. There is no problem with JRSprigg's explanation. It uses the propagator formalism in a degenerate way (same spacetime point for disappearance an reappearance of particle). But no probability density can reasonably be obtained this way. For this one would need an extended region of space (and time) to consider. In that chosen region of space and time, all sorts of things can, and will happen, not merely the disappearance and reappearance of one single particle. YohanN7 (talk) 18:08, 5 November 2013 (UTC)
 * Even if considering only a single spacetime point, there should be an amplitude for pair creation to occur there at that very event? If so, one has to take these possibilities into account. It seems hard (if possible at all) to define a true one-particle probability density in QFT if the term probability density is to be carried over from ordinary QM. YohanN7 (talk) 18:08, 5 November 2013 (UTC)


 * Aha! "the conserved quantity for the Dirac equation [is] the particle density" is exactly the kind of description that would work perfectly, even if it takes some interpretation to make exact (and is what I was trying to describe). I suspect that antiparticles manifest as negative density for this to hold, which might be worth mentioning too.  What this boils down to is conservation of single-family lepton number for the Dirac equation.  It would be good if some variant of this were put in as a "more correct" contrasting interpretation than the statement that you (quite rightly) are unhappy with.  — Quondum 16:06, 5 November 2013 (UTC)


 * Yes, we might be getting close to something reasonable. That antiparticles come with negative density is correct for scalar particles. I'll quote an example from Franz Gross, Relativistic Quantum Mechanics and Field Theory. Shoot a normalized (norm = 1) wave package with strictly "positive components" into a potential barrier. After the collision, norm = 1 still holds (because we have a conserved quantity). But, the wave function now very obviously have spatially separated components moving away from each other. The example uses the Klein-Gordon equation, but I'm pretty sure that (qualitatively) the same result is obtained with the Dirac equation, only a bit trickier to solve. The transmission and reflection coefficients behave in a paradoxical (Klein paradox?) manner in the corresponding experiment for electrons, but I need to refresh my memory here. It's in Greiner's book, but I don't have it with me now.
 * It still remains to show, in a reasonably way, that even without interactions, the probability density interpretation has problems. This might be best done using some paradoxical results, like the Zitterbewegung, that appears when one insists on a one-particle interpretation. (This is already mentioned in the article b t w.)
 * Naturally, it should be emphasized too that in the non-relativistic limit, the probability amplitude interpretation is perfectly fine.
 * Best would be if an expert took care of this. The Dirac equation is very pretty, and people seem to feel very strongly (sometimes too strongly) about it from what I have read, here and elsewhere. YohanN7 (talk) 17:19, 5 November 2013 (UTC)
 * THIS IS NOT THE PLACE TO DISCUSS SUCH MATTERS. It is an encyclopedia article, not a vehicle for your doubts or your research. DO YOU GET IT??? This discussion has NO PLACE in the exposition of the Dirac equation as it existed when he wrote it. Antimatter33 (talk) 01:43, 3 June 2014 (UTC)

Lagrangian (density)?
There seems to be no mention of the Lagrangian (density) that generates the equation. It can be found (at the very least) in the "DeMystified" books on QFT (2008, McMahon) and Supersymmetry (P. Labelle, 2010), to be (quoting from each respectively):


 * $$\mathcal{L} = \bar{\psi}(i\hbar \gamma^\mu \partial_\mu - mc) \psi = \bar{\psi}(\gamma^\mu P_\mu - mc)\psi$$

where the overbar is the Dirac adjoint spinor:


 * $$\bar{\psi} = \psi^\dagger\gamma^0$$

I'd add this, but then not, given previous threads (now archived)... Just a thought - maybe someone with thorough expertise may (or may not) add such a section... M&and;Ŝc2ħεИτlk 20:37, 22 December 2012 (UTC)

I agree with you, I also think there should be a mention of the Lagrangian. — Preceding unsigned comment added by 206.45.23.24 (talk) 02:47, 6 August 2014 (UTC)

The lead
In my personal opinion, the lead needs to be toned down a bit.

But there are also errors. The DE wasn't first to incorporate SR (the KGE was), and it doesn't account fully for QM + SR (QFT does). Moreover, spin is not a consequence of QM + SR. You don't need QM (the classical EM field has spin 1) and you don't need SR (the rotational symmetry allowing spin is present in QM with Galilean relativity). YohanN7 (talk) 16:00, 7 November 2013 (UTC)


 * Try telling that to Antimatter33, who simply cannot and will not accept edits from others. Despite claiming the article is full of errors by "green students", he has repeatedly failed to give long lists of concrete examples of incorrect edits (only vague, petty and insignificant edits which can be easily corrected). The whole article as of 13 Feb 2011 before recent edits was more or less written by Antimatter33. M&and;Ŝc2ħεИτlk 05:58, 8 November 2013 (UTC)


 * With the exception of the last edits to move a few sections around, I'm not inclined to change the content of the article.


 * Other editors of this article may be interested in some threads at Talk:Relativistic quantum mechanics. M&and;Ŝc2ħεИτlk 06:08, 8 November 2013 (UTC)


 * I have also come to the conclusion that I'm not touching it. If I was to add what should be added, then the article would become badly internally inconsistent telling two very different stories, one from 1928 and one from 2013. YohanN7 (talk) 10:44, 8 November 2013 (UTC)


 * The article is NOT about QFT, it is about the Dirac equation and its implications. This is a completely well-defined story in 1928 OR 2013. And yes, spin does emerge directly from the marriage of relativity and the Dirac version of quantum mechanics. It's his goddamn equation, after all, and who the hell are you to doubt him? This is an encyclopedia article, not a vehicle for the personal expression of your refined worldview. Got it?? Antimatter33 (talk) 01:37, 3 June 2014 (UTC)


 * You treat the subject more like religion than science. Such a disgrace. Dirac would have wanted to strangle you. YohanN7 (talk) 03:59, 23 August 2014 (UTC)

New section Dirac equation in one spatial dimension
Is there any value in this?

There are no references. I question the name "Dirac equation" as well because in two spacetime dimensions, the underlying physics would presumably obey $O(1, 1)$-symmetry, not $O(3, 1)$-symmetry (Lorentz symmetry). YohanN7 (talk) 13:33, 6 July 2016 (UTC)

I removed it. If the author want it back, references are needed, not only for the equation, but also for why it would be usefull to consider it in the context of this article. YohanN7 (talk) 10:03, 7 July 2016 (UTC)

And a brand new account,, reverted with the observation that it is better to be constructive than destructive. This is exactly what you'd expect as the first edit, right? Reverted back. YohanN7 (talk) 10:43, 11 July 2016 (UTC)

Apologies, "green student" here, did not notice the persistent personal agenda of the section's previous author, or its recency. I would agree with you however, the link to the article on the Hopfield dielectric overlaps with the author's own research and could use additional clarification. While an intuitive description of spin in varying dimensions does appear to emerge, the intro misleads the layman into believing a simple case of the Dirac equation will be presented rather than a new form on tenuous footing. Perhaps a short line and reference, if one can be found, will suffice. — Preceding unsigned comment added by Gawonni (talk • contribs) 01:36, 21 July 2016 (UTC)

Dirac equation and the correspondence principle
The Dirac equation can be proved with the help of the correspondence principle. The energy and momentum of a particle can be expressed by the equation
 * $$E^2=p_1^2c^2+p_2^2c^2+p_3^2c^2+m^2c^4$$

This equation can be divided into $$E$$ on both sides. We obtain
 * $$E=\frac{v_1}{c}p_1c+\frac{v_2}{c}p_2c+\frac{v_3}{c}p_3c+\frac{v_0}{c}m\,c^2\,\,\,\,\,\,\,\,\,\,(1)$$

where $$v_0=\sqrt{c^2-v^2}$$, and $$v^2=v_1^2+v_2^2+v_3^2$$  ;

Really $$\frac{p_1c}{E}=\frac{mv_1c(c/v_0)}{mc^2(c/v_0)}=\frac{v_1}{c}\,$$ and so on, and $$\frac{mc^2}{E}=\frac{mc^2}{mc^2(c/v_0)}=\frac{v_0}{c}\,$$;

The Dirac equation has the form
 * $$i\hbar\frac{\partial\psi}{\partial t}=(\alpha_1\hat p_1c+\alpha_2\hat p_2c+\alpha_3\hat p_3c+\alpha_0 \,m\,c^2)\psi\,\,\,\,\,\,\,\,\,\,(2)$$

where $$\alpha_j$$ is matrix $$(j=0,1,2,3,)$$. We obtain from (1) and (2): $$v_j/c\rightarrow \alpha_j$$.

In fact, in quantum mechanics it shows that the relativistic velocity operator $$v_{\nu}=dx_{\nu}/dt;(\nu =1,2,3)$$ has the form $$v_{\nu}=c\alpha_{\nu}$$, ie is a matrix operator (see textbook LA Borisoglebsky "Quantum mechanics", Minsk, publishing house "University", 1988, p.340-342).

Really
 * $$\frac{dx_{\nu}}{dt}=\frac{\partial x_{\nu}}{\partial t}+[H,x_{\nu}]$$

where
 * $$H=\alpha_{\nu}p_{\nu}c+\alpha_0m\,c^2$$

Since the operator $$x_{\nu}$$ does not depend on time, it will be $$dx_{\nu}/dt=[H,x_{\nu}]$$. We get
 * $$\frac{dx_{\nu}}{dt}=[(\alpha_{\mu}p_{\mu}c+\alpha_0m\,c^2),\,x_{\nu}]$$

The matrix $$\alpha_{\mu}$$ commutes with $$x_{\nu}$$, so that the matrix $$\alpha_{\mu}$$ can be factored out. Finally we have
 * $$dx_{\nu}/dt=c\alpha_{\mu}[p_{\mu},x_{\nu}]=c\alpha_{\mu}\delta_{\mu\nu}=c\alpha_{\nu}$$

The eigenvalues of the matrix of the velocity operator equal to $$\pm c$$, but as the speed of the operator does not commute with the Hamiltonian operator, then the experience is always measured by the average value of the relativistic velocity operator, and it is less than $$c$$.

Thus, the correspondence between equations (1) and (2) is confirmed. Alexander Klimets (talk) 10:55, 28 July 2016 (UTC)

Note moved from article to talk
In this edit, User:Dharam Vir Ahluwalia added the following note to the lead. Since that was an inappropriate place for discussion of article content, I'm moving the note to this page, which is an appropriate place. A note added by D. V. Ahluwalia: I am opting to leave the above paragraph unedited. However, in my opinion, spin is a consequence of Poincare space-time symmetries -- and even then, one must remember that spin is not a Casimir of the underlying algebra but it is related to the eigenvalues of the squared Pauli Lubanski pseudo vector. Spin may be incorporated in a quantum formalism and how to do it is beautifully described in Steven Weinberg's classic on the theory of quantum field.

 &mdash; jmcgnh  (talk) (contribs)  19:38, 18 May 2017 (UTC)


 * It is indeed correct that relativity alone is sufficient for allowing spin. For instance, the classical electromagnetic field has spin 1. (In fact, ordinary rotational symmetry would suffice for allowing spin and can be used to derive the Pauli equation with the same amount of rigor (and ignoring relativity) as the Dirac equation.) It does, however, not imply the existence of particles with any particular given spin, whether combined with QM or not. YohanN7 (talk) 09:59, 26 May 2017 (UTC)


 * That said, there's more substance to the claim that SR + QM (at least in full QFT) predicts the existence of the positron given that the electron exist. YohanN7 (talk) 10:04, 26 May 2017 (UTC)

Derivation of the Dirac current
The IP’s edit is seemingly right, but we should replace—in the following paragraph—subtraction with summation now, shouldn’t we? Incnis Mrsi (talk) 15:17, 28 August 2019 (UTC)

Pedagogy of "equation"
I was listening to a new podcast interview between Lex Fridman and Eric Weinstein. Eric maintained that if the curious audience member isn't even up to speed on the Dirac equation, now nearly 100 years old, the discussion is not going to progress much beyond superficial metaphor.

Eric also put forward the view that the best and brightest in the modern world have forgotten how to believe in ourselves (excepting mainly Elon Musk) with the implication of "just be brave, and dive in".

I have (barely) enough math not to capsize into the electron sea at the first cross-breeze, and so I decided to show up and attempt to make what I could out of this discussion as one of Eric's insanely intrepid curious cats, for whom the curly nabla is a mysterious yet enticing ball of string.

In this formulation, the Dirac spinor field equation (which are four complex equations, and so eight equations in total) is converted into an equivalent system of two real vector equations (which are two 4-dimensional equations, and so again eight equations in total).

For about three seconds I thought that maybe this was a careless grammatical error. But then it occurred to me that the Dirac equation in one notation could well turn into a set of Dirac equation elements expressed in another notation (each element being a mathematical equation in having its own equality sign).

Here's an analogy from computer science which might cast light on the problem (or not).

Clearly one assignment: b := a;

One parallel assignment or two implied sub-assignments? c,d := a,b;

One parallel assignment or two implied sub-assignments? a,b := b,a;

In the third case, don't try separating out those two sub-assignments as sequential statements: a := b; b := a;

One of the problems for the uninitiated in following a page such as this is that the word "equation" is narratively contextual, and does not line up with the presence of equal signs in the notational expressions as simplistically as one might expect.

Depending on the notion of the moment, the physical Dirac equation is variously one mathematical equation, or a collection of mathematical equation elements.

In this formulation, the [physical] Dirac spinor field equation (which [comprises] four complex equation [elements], and so eight equation [elements] in total) ...

To the intrepid reader for whom the nabla is an enticing ball of string—a naive reader in desperate need of finding any initial handrail at all—one suspects this insider compression is not helping the cause. &mdash; MaxEnt 16:36, 21 April 2020 (UTC)


 * You may want to read spinor first, and also perhaps refresh your knowledge of the nabla, of curl and divergence, and maybe even vectors. Spend some quality time with complex numbers and the Schroedinger equation. Eric is not quite correct, it's not the Dirac equation that is turning things upside down, its the spinors and the Clifford algebra that is at the root of present-day upheavals. 67.198.37.16 (talk) 08:06, 22 November 2020 (UTC)

'see also' section
In WP, 'see also' is used to provide links to related WP articles, so that section of the article should be renamed to avoid confusion.--Brian Josephson (talk) 21:36, 25 January 2021 (UTC)
 * I've looked again at the article and see that in fact there is a proper 'see also' section, and the only problem is that the first item, beginning 'The Dirac equation appears on the floor of Westminster Abbey', is not like the other items a WP article, so this should be moved elsewhere. Perhaps if there isn't a suitable section for it, a new one can be created. --Brian Josephson (talk) 21:43, 25 January 2021 (UTC)
 * Yes, it definitely shouldn't be there, I moved it to the lead. Tercer (talk) 09:06, 26 January 2021 (UTC)
 * Thanks, that's good!--Brian Josephson (talk) 09:16, 26 January 2021 (UTC)

Non-encyclopaedic self-citation
I'm removing the following text from this article:
 *  As a differential equation in one real component 


 * Generically (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component. Furthermore, this remaining component can be made real by a gauge transform. 

There are several problems with this text. I don't believe this is encyclopaedic -- this is not textbook content, this is an unusual and non-standard derivation. The idea of a one-component Dirac field is somewhat misleading, although the precise statement might actually be true: it is plausible that one can take four coupled linear differential equations and replace them with one non-linear equation of the fourth order. However, mathematics usually moves in the opposite direction, taking something complicated and non-linear, and trying to factor it into simpler linear parts. Which is what Dirac accomplished. A different problem is that this is a self-citation, added by User Akhmeteli on 14:49, 7 October 2012‎, who otherwise doesn't edit WP, except to add this self-citation to this article. :-( 01:35, 1 December 2020 (UTC)
 * I suggest that the removed text be restored. First of all, the critique of the text is factually incorrect: the equation for one component of the Dirac spinor, which is equivalent to the Dirac equation, is not non-linear, it is linear with respect to the remaining component. Second, not every system of first-order partial differential differential equations (PDE) for several unknown functions is equivalent to a single PDE for one unknown function (although the converse is true), and I believe it is obviously important that the Dirac equation is equivalent to a PDE for just one component of the spinor. Of course, I am biased, but that does not necessarily mean that the important fact about the Dirac equation should be removed. Neither is my (low) activity with Wikipedia is relevant to whether this text belongs in the article on the Dirac equation. Let me also add that my result was properly published in the Journal of Mathematical Physics. If there will be no objections, I will restore the text in a month.Akhmeteli (talk) 05:55, 2 June 2021 (UTC)
 * No way. Wikipedia is supposed to present the mainstream point of view, not on obscure new ideas. Also, Wikipedia is not the place to promote your own work. Clearly you're WP:NOTHERE to build an encyclopedia, you have been editing for almost a decade, with laser-like focus on adding a citation to your paper to this article. Tercer (talk) 08:40, 2 June 2021 (UTC)
 * The text I suggest to restore is fully mainstream: it was published in a decent journal (Journal of Mathematical Physics) and cited, e.g., in a book on the Dirac equation (V.G. Bagrov, D. Gitman, The Dirac Equation and its Solutions, de Gruyter, 2014, p. 25, citation[2]). According to https://en.wikipedia.org/wiki/Wikipedia:Reliable_sources, "When available, academic and peer-reviewed publications, scholarly monographs, and textbooks are usually the most reliable sources." So what I suggest is not "new" for Wikipedia's purpose, as it is fully sourced. If you believe the text is incorrect, I guess you are qualified enough to pinpoint an error. Until then what you write about me is just irrelevant ad hominem critique. So I only suggest content I know well, I am not active at Wikipedia, that does not mean I cannot suggest any content, and there is no reason why I cannot suggest properly sourced content that I created. The content I suggest is clearly relevant and important for the Dirac equation: the equation can be written as just one equation for one function, whereas traditionally it is written as four equations for four functions.Akhmeteli (talk) 00:47, 7 June 2021 (UTC)
 * Yes, you are allowed to suggest the addition of content you created, and we are allowed to say no. The problem with your paper is not correctness, but relevance. Your paper is rather obscure, being cited by yourself or by V.G. Bagrov (that you thank in the paper itself) doesn't count for anything. Tercer (talk) 08:13, 7 June 2021 (UTC)
 * Thank you for not questioning my right to suggest the text or correctness of the text. Neither do I question your right to say "no", but that does not mean I have to accept your "no" as the last word in this matter. Others considered the matter (you can look at the early discussions in this Talk) and agreed with me, so this text had been a part of this article for several years until it was removed based on factually incorrect arguments. Let us consider your arguments. You question the relevance of the text. I find this highly unusual (as I guess you are quite knowledgeable in quantum theory): the Dirac equation can be written as just one equation for one function, and this is not relevant for an article named "Dirac equation"? I don't think this argument holds water. I can agree that my paper is "rather obscure", although it was cited at least 9 times by others, but how does this "obscurity" disqualify the text from being included in Wikipedia? Can you give a reference to Wikipedia policies or guidelines supporting your point of view? The text is correct, properly sourced, and highly relevant. I just don't see valid objections.Akhmeteli (talk) 11:29, 7 June 2021 (UTC)
 * 9 times? I see here only 7 citations, of which only one is not by yourself or your collaborators. Ultimately, you have to convince other editors that the citation is relevant, and you haven't managed to do that. Tercer (talk) 13:17, 7 June 2021 (UTC)
 * At https://scholar.google.com/scholar?oi=bibs&hl=en&cites=4453516567009192033 there are 8 citations by others, plus the book by Bagrov/Gitman that I mentioned. Yes, I do know some of the people who cited my article, but none of them is my collaborator. But again, this is not an issue, what is important is the relevance. In your book, the fact that the Dirac equation can be written as just one equation for one function, rather than four equations for four functions, is not relevant to an article named "Dirac equation"? This is strange.

Dirac denounced Quantum Field Theory
At 42:30  into  this  video,  Dirac  denounced  Quantum Field  Theory  as  an  abomination. Should not  this  be  reflected  in  Wikipedia's  article? https://www.youtube.com/watch?v=jPwo1XsKKXg — Preceding unsigned comment added by 47.201.179.7 (talk) 04:35, 16 January 2017 (UTC)

Others such as E.T. Jaynes also criticized QFT, but this article is about the Dirac equation and is not a good place for a discussion of those criticisms.36.2.1.44 (talk) 06:29, 22 October 2017 (UTC)

How do explain the time evolution of the spatial operator without QFT? --130.149.50.205 (talk) 13:22, 26 July 2018 (UTC)

WHY is this comment in here?
This is not the place (WP) to discuss such things. It's utterly against the idea of an encyclopedia article, which is to present known facts in the form in which they are known to the experts in the subject.

This article is in woeful shape. It is what is left of what I wrote some years ago, and it has been rendered almost incoherent with pointless asides and plain bad English prose. For something so important to be so poorly represented on WP makes me embarrassed to have been a part of it. — Preceding unsigned comment added by Antimatter33 (talk • contribs) 04:10, 3 April 2019 (UTC)


 * Antimatter33's last version was from Feb 2011, it was this version and it was revived in Sept 2012 in this version Indeed both of those versions seem to be elegant. However, after a super-fast skim of the current version, it does not seem so bad, either, so I'm not clear about what the objections really are.


 * Articles that are popular, especially with students, get edited by students, and, no matter how lucid that may have started out being, tend to go down in quality in the days before midterm and final exams. That's how Wikipedia works. You ain't seen nothing until you've experienced the repeated, relentless, systematic vandalism of Diego Rivera by right-wingers still fighting a senseless anti-commie crusade from the 1930's. We are lucky that Dirac did not belong to the Communist Party, else this page would be only a stub.


 * The youtube video is from the Royal Society, a lecture about the history of physics/math/whatever. I didn't watch, but the imprimatur should imply that the contents are credible. 67.198.37.16 (talk) 08:36, 22 November 2020 (UTC)

Antimatter is right, the article is a shambles. I gave up on it when the statement about its relationship to the Klein-Gordon equation was removed. cheers, Michael C. Price talk 05:57, 14 September 2021 (UTC)