Talk:Dirac equation

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)

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. —Quantling (talk | contribs) 22:18, 19 June 2023 (UTC)

I've edited the article. —Quantling (talk | 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. —Quantling (talk | 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? —Quantling (talk | 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. —Quantling (talk | 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 —Quantling (talk | 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)

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

$E^{2}=p_{1}^{2}c^{2}+p_{2}^{2}c^{2}+p_{3}^{2}c^{2}+m^{2}c^{4}$

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 ${\frac {mc^{2}}{E}}={\frac {mc^{2}}{mc^{2}(c/v_{0})}}={\frac {v_{0}}{c}}\ ,$;

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 $c$.

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)