Talk:Dirac equation

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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)

@105.182.127.188 agreed : please stop boxing. 92.184.105.168 (talk) 07:01, 14 August 2024 (UTC)

The Dirac equation and the correspondence principle

More information

, It can be divided by ...

Please see WP:NOTFORUM

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

$E^{2}=p_{1}^{2}c^{2}+p_{2}^{2}c^{2}+p_{3}^{2}c^{2}+m^{2}c^{4}\,\ ,\,\,\,\,\,\,\,\,(*)$

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

$E={\frac {v_{1}}{c}}p_{1}c+{\frac {v_{2}}{c}}p_{2}c+{\frac {v_{3}}{c}}p_{3}c+{\frac {v_{0}}{c}}m\,c^{2}\,\ ,\,\,\,\,\,\,\,\,(1)$

where the value is $v_{0}={\sqrt {c^{2}-v^{2}}}$ , and $v^{2}=v_{1}^{2}+v_{2}^{2}+v_{3}^{2}$ ; Indeed, ${\frac {p_{1}c}{E}}={\frac {mv_{1}c(c/v_{0})}{mc^{2}(c/v_{0})}}={\frac {v_{1}}{c}}\,$ etc. and also ${\frac {mc^{2}}{E}}={\frac {mc^{2}}{mc^{2}(c/v_{0})}}={\frac {v_{0}}{c}}\ ,$ ;

Here relation (*) should not be considered a notational condition (meaning that the entries $E^{2}$ and $p_{1}^{2}c^{2}+p_{2}^{2}c^{2}+p_{3}^{2}c^{2}+m^{2}c^{4}$ have the same meaning). Relation (*) is considered as a real (meaningful) equality indicating the relationship between $E(v)$ , $p(v)$ and the independent variable $v$ .

To return to the original equation (*) (as if "squared"), equation (1) must be multiplied by $E$ on both sides.

The Dirac equation (in the absence of an electromagnetic field) has the form $i\hbar {\frac {\partial \psi }{\partial t}}=(\alpha _{1}{\hat {p}}_{1}c+\alpha _{2}{\hat {p}}_{2}c+\alpha _{3}{\hat {p}}_{3}c+\alpha _{0}\,m\,c^{2})\psi \,\,\,\,\,\,\,\,\,\,(2)$

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

$H=\alpha _{\nu }p_{\nu }c+\alpha _{0}m\,c^{2}$

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

${\frac {dx_{\nu }}{dt}}=[(\alpha _{\mu }p_{\mu }c+\alpha _{0}m\,c^{2}),\,x_{\nu }]$

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 $c$ .

From the above it is also clear that in Dirac's theory the components of the 4-velocity $v_{j}$ and the corresponding coordinates $x_{j}$ obey the Clifford algebra, i.e. they anticommute. — Preceding unsigned comment added by 178.120.7.149 (talk) 05:14, 15 March 2025 (UTC)

Thus, the correspondence between equations (1) and (2) is confirmed. And, in general, see my article https://philpapers.org/archive/ALXOTF.pdf 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 | 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 | 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.

Paul Dirac developed significant portions of the new quantum theory in the second half of the 1920s. While he did not apply Bohr's correspondence principle, he developed a different, more formal classical–quantum correspondence. Dirac connected the structures of classical mechanics known as Poisson brackets to analogous structures of quantum mechanics known as commutators: $\{A,B\}\longmapsto {\frac {1}{i\hbar }}[{\hat {A}},{\hat {B}}].$ By this correspondence, now called canonical quantization, Dirac showed how the mathematical form of classical mechanics could be recast as a basis for the new mathematics of quantum mechanics.

With the help of this correspondence it is shown how the mathematical form of classical mechanics can be transformed into the basis of the new mathematics of quantum mechanics, which is shown by the first author. 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)

Close

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)

Spurious i factor in d/dy terms of four coupled equations

In the expansion into four coupled linear equations, the d_y term appears to have a spurious factor i. Either the preceding i should be deleted, or the wave function components should be multiplied by -i to compensate. This is because the sigma matrix for the y direction is imaginary. My guess is someone calculated it correctly but subsequently someone added the "missing" i to make the pattern look more consistent. I'll let someone else decide whether I'm right and if so how to fix it. Jonathan A Scott (talk) 13:12, 5 September 2024 (UTC)

I think the right fix 1) move to Math section (it's not history), 2) put the gammas back from the reference. I did #1. Johnjbarton (talk) 16:07, 5 September 2024 (UTC)

My point was that the "i" before the \partial_y should not be there in each of the four lines! 2A02:C7C:D442:9200:5D29:7CEE:FDF7:B9A3 (talk) 19:54, 5 September 2024 (UTC)

Sorry, wasn't logged in. I was more worried about the spurious factor of i. Jonathan A Scott (talk) 19:56, 5 September 2024 (UTC)

The decision I wasn't sure about is whether to remove the i for d_y or to multiply each of the components in the following column by -i. Jonathan A Scott (talk) 19:57, 5 September 2024 (UTC)

In my opinion the equation from the reference should be presented. Then we don't need to decide. Johnjbarton (talk) 20:11, 5 September 2024 (UTC)

I found the reference in Google Books, page 7, equations (2.12) through (2.15). The quote in this Wikipedia article has been reorganised to improve the readability and is not in the same layout but the original clearly did not have a factor i on the d_y terms. So my opinion is that the i should be removed - a 1-character correction. But I'm a bit concerned that someone might look at the "pattern" and think that the i has been accidentally left out (which I suspect is what happened previously). Jonathan A Scott (talk) 08:40, 6 September 2024 (UTC)

I haven't really been following this discussion, but ... perhaps add a sentence (or even just a clause of a sentence) just before or after the equation to explain to the reader why the "expected pattern" is not followed. It will be both educational and serve as defensive editing so that folks don't try to "fix" it to something else. —

Q

uantling (talk | contribs) 12:35, 6 September 2024 (UTC)

I've removed the spurious "i". Is there a way to include a comment in the source without it appearing on the page so anyone trying to edit it will see it? Jonathan A Scott (talk) 13:08, 6 September 2024 (UTC)

I've added a Markdown comment to call attention to the fact that the lack of "i" for the \partial_y term is deliberate. Jonathan A Scott (talk) 16:42, 6 September 2024 (UTC)

Does the source verify the content?

In [this https://en.wikipedia.org/w/index.php?title=Dirac_equation&diff=prev&oldid=1304947548] the source

Paul Adrien Maurice Dirac (1930). The Principles of Quantum Mechanics 1st edition by Paul Adrien Maurice Dirac b19020808 d19841020 [1930] {530.1--loc}.

was added for the content:

The algebraic structure represented by the gamma matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Clifford's ideas had emerged from the mid-19th-century work of German mathematician Hermann Grassmann in his Lineare Ausdehnungslehre (Theory of Linear Expansion).

However as far as I can tell neither Grassmann nor Clifford are mentioned in Dirac's book. Thus the source does not verify the content. The edit was done by @Kanu Parasrampuria with a newcomer tag and an edit summary:

Added citation to Dirac’s The Principles of Quantum Mechanics for the original matrix form of the Dirac equation, including α and β matrices and their properties.

However that does not match the content with the reference. Normally I would just revert the change but I want to encourage adding sources: maybe this is a misplacement or misunderstanding? Johnjbarton (talk) 16:14, 9 August 2025 (UTC)

@Johnjbarton You are right. Thanks for the mention. Kanu Parasrampuria (talk) 18:09, 9 August 2025 (UTC)

Some suggested improvements

This article seems to have a few structural problems.

Too many short subsections, which sometimes only contain a sentence or two, making the table of contents very confusing.
Lack of focus at times, repeating information (in too much depth) found on other articles.
Lack of in-line citations.
The writing style in the later half of the article differs in having many 1/2 sentence paragraphs, so establilshing a more coherent tone may be preferential.
Repetition throughout the article; for example the History section has a section on the covariant formulation, which firstly why is this in the history section, and second is a repeat of the Mathematical formulation section, where I'd imagine it should be.

Minor point:

A section titled Other formulations follows from Mathematical formulations seemingly to imply that it's a non-mathematical formulation which is obviously nonsense.
History section starts with no context. It doesn't really even have any real history in it...

I'm sure there are other issues. But basically I'm declaring these here now and I may start improving this article. Such issues tend to be a problem with many long Wikipedia articles written over decades, so this is hardly the only one. Any additional input/objections would be appreciated. OpenScience709 (talk) 11:04, 29 October 2025 (UTC)

I agree with all of your points. Lack of sources is the #1 problem. Too much emphasis on math overall. Johnjbarton (talk) 15:40, 29 October 2025 (UTC)

A list of shortfalls is a good start. From there I'd start listing concrete steps on how to improve things. Or when there is a decent chance that the changes would be welcome, which I venture is the case, please jump straight into making edits to the article. Thank you —

Q

uantling (talk | contribs) 13:32, 30 October 2025 (UTC)

This page is genuinely terrible the more I looked at it. At this point I'm basically rewriting the whole thing. I will give a detailed breakdown of the issues soon. OpenScience709 (talk) 10:05, 18 November 2025 (UTC)

Article rewrite

As stated above, the old version of this article had major problems. Here is an incomplete rundown of them, justifying why a complete rewrite was necessary. Here the Section X.X is the old version section index:

Section 1.1 (Making the Schrödinger equation relativistic): is largely gone since it mainly deals with the deficiencies of the Schrodinger equation, most of which were actually not historically relevant for Dirac (see citations) and which information can be found on the Schrodinger equation page, so is redundant here.
Section 1.2 (Dirac's coup): reduced some redundant the information here, reducing the number of equations, making the argument a bit more historically in line with Diracs presentation, but my new version of this retains the key content, just more in words (and expanding a lot on other points).
Section 1.3 (Covariant form and relativistic invariance): Merge this with the mathematical formulations section, relevant Lorentz invariant section, and remove the unecessary discussion on \gamma^5 which does not see to here tie in with the Dirac equaiton in this version (the information can be found elsewhere on Wikipedia on more relevant pages). Parity discussion is elsewhere now.
2.1 (Pauli theory): Removed tangent into Stern-Gerlach experiment. The derivation of the Pauli equation from the Dirac equation should be in either the Pauli or the Dirac equation page, not both. I think it makes logically more sense for it to be in the former, not the latter.
2.2 (Weyl theory): Two sentence section. Moved any relevant info later down the page.
3.1 (Identification of observables): Don't quite see the relevance/point/validity of this section. Feel free to enlighten me.
3.2 (Hole theory): Largely covered in the new history section so no need to have this. Last paragraph found in history section. Penultimate one discusses problems with the hole theory, which is already one step removed from the Dirac equation, so was cut. Additionally, I feel the focus on the discussion of the Dirac sea/hole theory should be the corresponding pages rather than the Dirac equation page. It should only be included here to the extent that it is pertinent to the equation and the history, which the new history section does. To that end i moved some of the relevant comments to the new solutions part in Properties. The new Interpretation section is much more coherent interpretation of the equation itself imo.
3.3 (In quantum field theory): Tiny section with the information just found within the new text.
4.0 (Mathematical formulation): Kept most of info, but got rid of explicit rewrite in terms of 4 PDEs as its unecessary clutter.
4.1 (Dirac adjoint and the adjoint equation) and 4.2 (Klein–Gordon equation): Information kept but not in its own subsection.
4.3 (Conserved current): Wikipedia is not a repository of proofs, especially ones that do not provide any new information for the topic at hand.
4.4 (Solutions): Kept info.
4.5 (Lagrangian formulation): Kept info.
4.6 (Lorentz invariance) and 4.7 (Further discussion of Lorentz covariance of the Dirac equation): Combined with earlier information on Lorentz transformation properties into one single section on this topic, eliminating the many redundancies. Eliminated most of 4.7 with it being a textbook presentation format rather than a encyclopedic one. Its also in a notation thats not consistent with the other section. Also corrected the statement of 4.6 that the Dirac equation is invariant; its not. Its covariant. (4.7 did say this fortunately). Elimintated the fibre bundle discussion since thats just a feature of spinors, so belongs to those and other articles, and is less pertinent to this one (if you disagree, please let me know).
More generally on the Lorentz properties, one could make a strong argument for moving most of this (thus part describing why and how the spinor transforms) to the Dirac spinor article. It is a small detour from the equation itself. On the other hand it is a key property, and most new readers will probably meet spinor representations in this article.
Section 5 (Other formulations): Collapsed into one section in new version discussing relevant generalizations/other features of the Dirac equation.
6.1 (Vector symmetry): Repetition of stuff found earlier.
6.2 (Gauging the symmetry): Unecessarily long, although process of gauging is relevant. Scalar electrodynamics comment is completely unecessary since that has nothing to do with the Dirac equation.
Rest of 6 (Extension to color symmetry, etc): Information kept in essence.
More generally I somewhat rewrote section 6 but otherwise kept it as it is relevant. For now I think that the process of gauging the Dirac equation is most relevant to be kept in the page for Dirac equation.
Key new things: History section is basically all new. New section on related Dirac equations, on classical, QM, and QFT interpretations, spacetime symmetry currents, and some other smaller things.
Rewrote everything so its in one voice, using a unified notation across the article.
Removed a *lot* of repetition that was found in the old version, hence why this article is shorter.
Removed unecessary detours that were occasionally taken in the article.
New version has a significantly more logical content structure in History, Formulation, Properties, Relation to other equations, Gauging. Easy to know where to find stuff, while old version was all over the place.
I'm sure there is much to improve with this version, especially wording and any minor errors I missed along the way. So please do contribute to improving that!

OpenScience709 (talk) 13:59, 19 November 2025 (UTC)

Sorry but this summary is not useful because the neither the original nor the rewrite has numerical section numbers. Johnjbarton (talk) 17:02, 19 November 2025 (UTC)

Numerical by table of contents. OpenScience709 (talk) 17:31, 19 November 2025 (UTC)

I guess you posted the list for feedback. My feedback is "Please use section headings to identify sections."

However, for me and I guess most editors, using the edit summary is far superior to such a list. You made your edits in series, great thanks! You listed the changes, good. Please just combine these in one step: embed the list in the edit summaries for the series of edits. I believe most edit tools include the section heading with the edit summary if you edit by section, so -- presto -- that problem is also solved.

Posting a summary to the Talk page is great. Especially important for me is any comments about sources removed. I think every source deletion should be explained. Johnjbarton (talk) 18:26, 19 November 2025 (UTC)

I only uploaded the sections in series. But these edits did not correspond to the removal of other sections from the old version (cause imo the old version was a structural mess at times). So I couldnt discuss what the issues were with the (only occasionally) corresponding old draft sections. Hence why I did that here.

As for source removal, I don't quite agree. Sources are just the tools used to reference the information to show it is true/supported. I didnt include many sources from the old version simply because I found other soruces for the information in the new version. Whether you use one source or another doesnt matter as long as they are both reputable. What does require comment is information removal and the reasoning for why some information would be removed, which I did above. OpenScience709 (talk) 22:51, 19 November 2025 (UTC)

Also I disagree with your edit "Delete one sentence about what Dirac did not do wrt to the Dirac equation.Off topic and does not fit here, its about Dirac.": I guess there are two things here. One is his follow up paper, the second is what he did afterwards.

The follow up paper specifically deals with the Dirac equation and its implications, so is relevant.
I can see why maybe the statement concerning Dirac not working on the equation/relativistic wave mechanics, may appear less relevant. But it is of historical value nonetheless, primarily cause it is suprising. Like he found this amazing equation and was like "yeah, anyway" for two years and went to do other stuff. Also, you can't dissasociate Dirac the person from the equation in this section. The literature does not do that, so Wikipedia shouldnt do that either.

OpenScience709 (talk) 17:44, 19 November 2025 (UTC)

I put a different version of the sentence back so I think this is done. Johnjbarton (talk) 16:40, 21 November 2025 (UTC)

Interpretation

The section "Interpretation" is confusing. Right off there is "interpretation" which in QM is unavoidably bound to questions about realism. This section is about different equations not different interpretations on one equation. Maybe "Other formulations"?

The section is also structured oddly. The classical field theory is a niche case and should be last. The other three should be presented as 3 not 1:2. Johnjbarton (talk) 01:05, 22 November 2025 (UTC)

No. It's effectively the same equation, just acting on different things. This has nothing really to do with QM interpretations. Its addressing the question of what the equation is describing, which is different depending on whether its describing a wavefunction, a classical field, operators, or path integral variables. But the equation, so its formulation, is the same. Its not "other formulations" either since its not a different formulation. Its all the same formulation, mathematically speaking. Other formulations require a different mathematical way to describe things, so how in GR one has different formulations (like tetrad formulation, Newman-Penrose formulation, etc). We could rename the section to "physical interpretation" if that helps?

Also the classical field theory view isn't a niche way to think about things, in that almost the entire article is formulated in classical field theory way. Its just that thats not a good physical interpetation, as fully classical spinor fields don't really arise in our world. But that interpreation is, as the source says, a "useful fiction".

The motivation for the original structuring was in the usual pedalogocial approach taken to physics; classical, QM, QFT(x2). Now the order doesnt make much sense. Its not an order of "most correct" since then it would be QFT, QM, classical. So what is the idea behind this order? OpenScience709 (talk) 13:57, 22 November 2025 (UTC)

Also I expanded on the part on classical spinor field theories since it is important to state what their symmetry group is, and how they are fundamentally different from quantum theories based on Hilbert spaces. It addresses common confusions about what spinors are and why they are valid reps despite not being reps of the Lorentz group. OpenScience709 (talk) 14:46, 22 November 2025 (UTC)

But I also agree with your addition to explain how classical spinor fields do not arise from an attempted classical limit of quantum ones. OpenScience709 (talk) 14:48, 22 November 2025 (UTC)

This section is still eluding us. Think in terms of a general reader. How can we explain this content more clearly?

This has nothing really to do with QM interpretations.

Exactly my point. Thus using this heading is confusing because many readers approach the subject as the step beyond QM.

It's effectively the same equation,...

So it's not the same equation. Something is the same but we've not figured out what that is.

I disagree with your claim about the usual approach to Dirac equation. Even the preface for the Năstase book goes out of its way to explain the niche between classical and quantum that it fills. The Dirac equation is typically considered a relativistic step beyond QM and the first step towards QFT. That is the historic path and one commonly employed in less advanced settings. I think "notability order" or "historic order" makes more sense, but I don't feel as strongly about the order as the section title.

How about "Spinor fields"? Would you agree to start with something like "The Dirac equation can be applied to different definitions of the spinor field"? Johnjbarton (talk) 18:20, 22 November 2025 (UTC)

I guess my fundamental point is that I don't see that "interpretations of QM" has a monopoly on the word "interpretation" in QM related pages, especially when its the most accurate descriptor for a section. But I can see your worry that some people approaching the subject may possibly get confused just from the title.

My problem with "spinor fields" is that its more than that. Its the framework itself that changes. A section titled "spinor fields" sounds like itll be about the mathematics of spinor fields, such as rep theory, so seems like an even more misleading section title. "Frameworks"? That could do the trick? "Physical frameworks"? There is something unsatisfying about this tho. Could also rename the entire set of sections to 2. "Mathematical details" (this is not quite the best title since its not like the mathematical details are sequestered to this section alone. Maybe "Definition"? Some other equation pages use this term to describe the analogous section. But there is a bit more than just definition here.), 2.1 "Covariant form", 2.2 "Dirac action", 2.3 "Formulations"/"Physical formulations"? (Formulations can have two meanings; either a mathematical formulation (original usage), or a physical formulation (new usage?)). Would either of these work and be better than the current 2.3 "Interpretations"?

I still think its the same equation. (I realized I made a sign error in the original version, in case thats the point of conflict... :P ) The Dirac equation is the statement that

i(\gamma ^{\mu }\partial _{\mu }-m)\psi =0

. Thats the Dirac equation. No matter what

\psi

is specifically, its the Dirac equation, and is refered to as such in the literature. Its the context in which it is being applied that differs.

(I assume you mean preface of chapter 37): Nastase goes out of his way to make clear that quantum spinor fields do not reduce to classical spinor fields, which is an important point. He then goes on to explain that nonetheless a lot of the usage of the Dirac equation is as a classical field, which is a "useful fiction". As a result, a lot of the math is done in the context of classical field theory. Thats what I mean; practically one frequently deals with spinors as classical fields, even if in the end they are not that. (Side point that Nastaste is technically wrong to say that spinors are fermions as spinors are simply a type of reps; supercharge are an example of spinors that are not fermions; but this is an aside.)

I could see an argument for historical order. Although on the other hand I'm trying to isolate the history to the history section alone. A more intrinsic, property, order seems more encyclopedic to me.

P.S. Sorry for long text I tend to always write whenever we chat. Clearly I always have a lot to say :P. I am apologising, but don't expect it to change any time soon... OpenScience709 (talk) 23:41, 22 November 2025 (UTC)

I'm just reading the section:

In quantum mechanics, the Dirac spinor...
In the second quantization form of quantum field theory the Dirac spinor is ...
In the path integral formulation of quantum field theory, the spinor field ...
The Dirac equation also arises in describing the time evolution of a spinor field ...

Four paragraphs, each about Dirac spinor field. The section is about the role of the Dirac spinor field (the thing that varies between paragraphs since you insist that the equation does not). "Dirac Spinor Fields" just says what we will read. Maybe the Dirac on front avoids other expectations. Johnjbarton (talk) 00:24, 23 November 2025 (UTC)

Its four paragraphs about the meaning of the solution

\psi (x)

, each of which is a Dirac spinor field, sure. But its still ultimately the framework that is changing, hence why each paragraph opens with stating the framework. I just don't see how a section titled "Dirac spinor fields" isn't giving the impression of being about the mathematical details of the Dirac spinor field, as opposed to the framework under considerations (QM, QFT, classical), and hence the meaning of the fields and what the equation describes. OpenScience709 (talk) 00:54, 23 November 2025 (UTC)

"Meaning of the Dirac fields" then Johnjbarton (talk) 02:42, 23 November 2025 (UTC)

I guess that could work for lack of a better option. OpenScience709 (talk) 11:53, 23 November 2025 (UTC)

So I think that narrows it down, no? We are varying the framework/formulation/theory (within which the Dirac equation means something different and describes a different object). Do we agree? The question is what is the best title for this. OpenScience709 (talk) 00:59, 23 November 2025 (UTC)

Some Healthy Dissent

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