Talk:Dirac equation

Mathematical formulation
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)
 * 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)

The equation boxes are unnecessary and also illegible in dark mode
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)

Electron hole != Positron
Um, an electron hole is not a positron. This article should not claim so in the Hole theory section. — Q uantling (talk &#124; contribs) 22:18, 19 June 2023 (UTC)
 * I've edited the article. — Q uantling (talk &#124; contribs) 01:19, 20 June 2023 (UTC)
 * I don't think there is any dispute that, in the modern understanding that these two concepts are distinct; but | your edit was to the section explaining the historical development of the Dirac equation -- and in that context, the "holes" Dirac had been postulating were in fact positrons -- and, more to the point, this is the language that is used in popular and historical accounts. For instance, this is how Roger Penrose describes it in section 24.8 of The Road to Reality, which I think lines up pretty well with the previous version of the article:
 * At first Dirac was cautious about making the claim that his theory actually predicted the existence of antiparticles to electrons, initially thinking (in 1929) that the ‘holes’ could be protons, which were the only massive particles known at the time having a positive charge. But it was not long before it became clear that the mass of each hole had to be equal to the mass of the electron, rather than the mass of a proton, which is about 1836 times larger. In the year 1931, Dirac came to the conclusion that the holes must be ‘antielectrons’—previously unknown particles that we now call positrons. In the next year after Dirac’s theoretical prediction, Carl Anderson announced the discovery of a particle which indeed had the properties that Dirac had predicted: the first anti-particle had been found!
 * I know I have seen versions of this where the distinction between the solid-state electron hole and a "hole in the Dirac sea" is noted -- which seems a better approach than just deleting the well-supported (if not clearly-sourced) reference to the 'holes' all together, but I don't have a source for that handy, and without clarifying I'm not sure the new version is an improvement. ShadyNorthAmericanIPs (talk) 21:30, 20 June 2023 (UTC)
 * Yes, if that is how it went historically then we should mention it. I ask only that it be clearly explained that this is not the modern understanding. — Q uantling (talk &#124; contribs) 01:52, 21 June 2023 (UTC)

Positron, Dirac hole, and electron hole
The Hall effect shows that some materials have carriers of current that are positive charges, which are described as electron holes. These carriers are not positrons, right? Is it that a Dirac hole, unlike an electron hole, is actually a positron? Whether yes or no, would someone who knows what they are talking about please edit the article to clarify the relationship of these three concepts? — Q uantling (talk &#124; contribs) 14:39, 21 July 2023 (UTC)


 * I agree with clarifying here the distinction between the two concepts known as "holes", but going into an explanation of all the different types of positive charge carriers feels like it would just confuse things: as it stands today, I don't believe any reader would come away with the impression that "holes" (of either type) are the only type of positive charge carrier, or that all positive charge carriers would be described by the article's subject (the Dirac equation).
 * What in particular do you find confusing or lacking in this article (aside from the clarification that Dirac sea holes are not solid state electron holes)? If the Hall effect article is unclear about which charge carriers it is referring to, then it seems like it should be fixed there, right? Adding the clarification here isn't going to help readers there.
 * As an aside, apologies for the misleading edit message on my last edit, which (as you noticed) was in fact relevant to this topic; I wasn't trying to sneak the positron note in: evidently a month-old draft got revived when I went to do the trivial update in the lede, and I should have checked before submitting. I'll add in a ref for the existing edit at least. ShadyNorthAmericanIPs (talk) 23:40, 21 July 2023 (UTC)
 * If only Dirac holes but not electron holes are positrons, I'd like that said explicitly. I don't want the reader to come away thinking that all holes that mark the absence of an electron are positrons. — Q uantling (talk &#124; contribs) 16:49, 24 July 2023 (UTC)
 * Is the closing paragraph of that section nor sufficient in this regard? This is the text right now:
 * In certain applications of condensed matter physics, however, the underlying concepts of "hole theory" are valid. The sea of conduction electrons in an electrical conductor, called a Fermi sea, contains electrons with energies up to the chemical potential of the system. An unfilled state in the Fermi sea behaves like a positively charged electron, though it is referred to as a "hole" rather than a "positron". The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.
 * ShadyNorthAmericanIPs (talk) 16:57, 24 July 2023 (UTC)
 * 'Refered to as a "hole" rather than a "positron"' is too weak, and hence confusing, IMHO. It needs to say "is not a positron."  Thanks — Q uantling (talk &#124; contribs) 17:21, 24 July 2023 (UTC)
 * I would agree. A positron is a distinctive particle, with unique behavior, including being antimatter. Holes in electronics are a convenient concept, but NEVER show the behavior of positrons. This, in my opinion, is not just a difference that needs more emphasizing, but is also a bit of a confusion in physics itself. If Dirac's holes don't work, then what does work to explain the negative E solutions of Dirac's equation? David Spector (talk) 13:25, 31 July 2023 (UTC)

"Pedagogic aids"
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)
 * Perfectly true, so I have removed it. JBW (talk) 14:07, 16 November 2023 (UTC)

The Dirac equation and the correspondence principle
The Dirac equation can be justified using the correspondence principle. In the special theory of relativity, the energy and momentum of a particle are expressed through the relation $$E^2=p_1^2c^2+p_2^2c^2+p_3^2c^2+m^2c^4$$

It can be divided by $$E$$ on both sides and converted to the following form

$$E=\frac{v_1}{c}p_1c+\frac{v_2}{c}p_2c+\frac{v_3}{c}p_3c+\frac{v_0}{c}m\,c^2\,\ ,\,\,\,\,\,\,\,\,(1)$$

where the value is $$v_0=\sqrt{c^2-v^2}$$, and $$v^2=v_1^2+v_2^2+v_3^2$$ ; Indeed, \frac{p_1c}{E}=\frac{mv_1c(c/v_0)}{mc^2(c/v_0)}=\frac{v_1}{c}\, etc. and also $$\frac{mc^2}{E}=\frac{mc^2}{mc^2(c/v_0)}=\frac{v_0}{c}\ ,$$;

The Dirac equation (in the absence of an electromagnetic field) 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 $$\alpha_j$$ are matrices, $$(j=0,1,2,3,)$$.

It follows from the correspondence principle between equations (1) and (2) that $$v_j/c=\alpha_j$$ or $$v_j=c\alpha_j$$.

Indeed, it has been shown in quantum mechanics that the relativistic velocity operator $$v_{\nu}=dx_{\nu}/dt;(\nu =1,2,3)$$ ; has the form $$v_{\nu}=c\alpha_{\nu}$$, that is, it is a matrix operator. Indeed, the velocity operator is found according to the general rules for differentiating operators with respect to time

$$\frac{dx_{\nu}}{dt}=\frac{\partial x_{\nu}}{\partial t}+[H,x_{\nu}]\,\,\,\ ,\,\,\,\,\,\,(3)$$ where $$H$$ is the Hamilton operator

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

Since $$x_{\nu}$$ - the coordinate operator - does not explicitly depend on time, then $$dx_{\nu}/dt=[H,x_{\nu}]$$. Substituting the Hamilton operator here, 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 the matrix operator can be bracketed (put out of brackets). 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 velocity operator are equal to $$\pm c$$, but since the velocity operator does not commute with the Hamilton operator, the average value of the relativistic velocity operator is always measured experimentally and it is less than $$c$$.

Thus, the correspondence between equations (1) and (2) is confirmed. 178.120.7.190 (talk) 04:33, 17 August 2023 (UTC)


 * This is very interesting information, but don't you want to add it to the article? You have added it only to the Talk page for the article. Almost no one will see it here. David Spector (talk) 12:32, 17 August 2023 (UTC)
 * Add if you think it's necessary.178.120.7.45 (talk) 02:06, 20 August 2023 (UTC)
 * I would hold off until we can get a better consensus here.  Thanks — Q uantling (talk &#124; contribs) 15:18, 20 August 2023 (UTC)
 * @David Spector We already discuss in the article that Dirac was trying to achieve $$E^2 = p^2 + m^2$$ by taking the square root of the right-hand side. How does what you write here give additional understanding of the Dirac equation?  Thanks! — Q uantling (talk &#124; contribs) 13:02, 17 August 2023 (UTC)
 * I don't understand your question. I merely suggested you edit the article instead of the Talk page. David Spector (talk) 13:16, 17 August 2023 (UTC)
 * Would agree w/ Quantling that the IPv4 User's remark may not add anything new to the article: the relation of the alpha matrices to the velocity is already covered (with citations) in the Zitterbewegung section, and it's not clear that this is what is typically meant by the "correspondence principle" -- this is (re-)identifying the terms in the Dirac equation with their corresponding classical terms, not showing how the Dirac equation reproduces classically-observed phenomena in the limit of large quantum numbers (in fact, it's possibly the most famous case where the obvious interpretation is starkly at odds with observed behavior)
 * 2600:1702:6D1:28B0:A5EB:8E02:3D7C:8484 (talk) 15:25, 17 August 2023 (UTC)
 * I'm not understanding this issue, so I will stop posting here. I mistakenly thought that a post with my name in it was directed to me, when I guess it was not. David Spector (talk) 15:33, 17 August 2023 (UTC)
 * Since I swooped in as an uninvited 4th party / second anonymous user, I'll try to summarize:
 * 1. IPv4 user posts Original Research on talk page that overlaps with content already present in the article
 * 2. User David Spector suggests it may have been mistakenly posted on the talk page rather than the article, and invites IPv4 user to add it to the article
 * 3. User Quantling notes that the content may be redundant if added to the article
 * 4. User David Spector replies to post #3 with confusion (possibly misinterpreting #3 and #1 being the same author), repeats invitation to add content of #1 the article
 * 5. IPv6 user (me) butts in to agree with #3 and offers some additional unsolicited critiques of #1
 * Hope this thread hasn't been too alienating. I'm not a subject-matter expert or key stakeholder on this page, I just wanted to reiterate that, in my opinion, the original post would need a lot of work before it would be a constructive addition to the article.
 * 2600:1702:6D1:28B0:A5EB:8E02:3D7C:8484 (talk) 23:09, 17 August 2023 (UTC)
 * Thank you for your clarifying summary of what happened here. Unfortunately, the actual issue involves the Dirac Equation, one of many parts of QM that I have never studied and do not understand. (I'm imagining that it is a version of conservation of energy that somehow complies with special relativity. I have a good imagination, but not an accurate one.) So I do not see how I can be of any help in resolving whether these beautiful equations (#1) should be discarded or somehow modified to fit the article. If they truly are original research, then they need to be published and vetted, not posted in WP, per WP policy. Good wishes to all, and to all a good night. David Spector (talk) 00:12, 18 August 2023 (UTC)

The principle of correspondence has different formulations. I am impressed by Dirac's formulation.“The correspondence between quantum and classical theories consists not so much in the extreme agreement at $$ \hbar \rightarrow 0 $$, but in the fact that the mathematical operations of the two theories obey in many cases the same laws.” The principle of correspondence in the methodology of science is the statement that any new scientific theory must include the old theory and its results as a special case. Alexander Klimets (talk) 13:15, 30 November 2023 (UTC)


 * There are broadly 3 forms of the correspondence principle, one historic due to Bohr, one modern which amounts to the classical limit concept, and sometimes the third the common sense idea that new theories must still explain old experiments. See Correspondence principle. Johnjbarton (talk) 14:53, 2 May 2024 (UTC)



Some Healthy Dissent
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)


 * I encourage contributors to this discussion to include references because nothing will be added without them. Johnjbarton (talk) 22:11, 2 May 2024 (UTC)

Suboptimal structure
With all due respect to previous contributors, I think this article begins with a much too formal mathematical exposition. The "historical" section is most well composed and should be promoted to the beginning. I have successfully taught undergraduates the basics of the Dirac equation using his original papers - it is hard to improve upon the clarity of his own presentation - and it matches well with what they learned in non-relativisic quantum mechanics.

Inroduction to the Dirac equation in Wikipedia should not presume familiarity with "bispinors", notation of modern quantum field theory, or undefined "natural units", whatever those are.

One doesn't get a second chance to make a first impression. IMHO the introduction should be minimally accessible to undergraduates who know some quantum mechanics, and, preferably, begin with statements of Dirac's motivation and intent for proposing a description of quantum mechanics consistent with special relativity.

I would undertake this myself, but I have read accounts of the flame wars on this article, and the present historical section - with which I have no connection - seems to me to do the job quite well as it is. If it was promoted to the front as it is, this article would be greatly improved. Charles Clark (talk) 02:37, 5 May 2024 (UTC)


 * I agree.
 * Seems like the section "Comparison with related theories" belongs under the "History". Johnjbarton (talk) 02:56, 5 May 2024 (UTC)
 * I move the sections, please review. Johnjbarton (talk) 03:04, 5 May 2024 (UTC)
 * A great improvement IMHO.
 * I shall have to look again, but I believe that Dirac found that alpha and beta, if represented by matrices, necessarily were of minimal dimension 4 x 4. That was a consequence of matrix mathematics under the unitary constraints, I believe, not a perceived precondition that the vectors had to be "bispinors." If I am correct, that is another towering achievement of Dirac: the minimal manifestation of the electron is as a spin 1/2 particle!
 * Surely there are few other examples of so much being done for so many by so few! Charles Clark (talk) 01:38, 6 May 2024 (UTC)
 * The article as a whole lacks references, including the History section. It would be very helpful if more inline references could be added. Johnjbarton (talk) 02:16, 6 May 2024 (UTC)