Talk:Dirac equation/Archive 1
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|Archive 1||Archive 2|
- 1 "This equation is more intelligent than me"
- 2 Taking another shot at making this a decent page
- 3 Missing: Antiparticle Discussion
- 4 Observables section
- 5 Comment
- 6 Noninteracting sea?
- 7 Electromagnetic Interaction
- 8 Gamma matrices
- 9 Upper and lower psi functions
- 10 Dirac bilinears
- 11 Links
- 12 Interesting and praiseworthy treatment
- 13 math/latex
- 14 single particle theory??
- 15 Adjoint spinor isn't explicitly defined
- 16 Constructive comment re. a problem with this article
- 17 Jim Al Khalili 'Everything and Nothing" BBC4
- 18 Dirac equation as an equation for just one component
- 19 Grammar in the intro paragraph
- 20 Clean up and clarify
- 21 Dirac a physicist?
- 22 Probability current? KG Equation?
- 23 spelling
- 24 A small neutrality quibble
- 25 "This article is just crap again"...
"This equation is more intelligent than me"
Taking another shot at making this a decent page
Since this topic is so important to all particle physics, it should be a first rate page. I set out to make it that but bad grad students chopped my work to shreds. I'm going to try again to make it right but without the assistance of the editors I don't have a lot of hope that I will succeed. Antimatter33 (talk) 18:19, 11 February 2011 (UTC)
- OK I've removed all the irrelevant comments (many my own) and all comments here in the nature of a physics discussion. I am going to make this page tight. Bear with me, it will take some time. Let's keep this Talk page limited to discussion of the article and its clarity or lack thereof. If you see errors point them out here. I will be monitoring. Antimatter33 (talk) 18:45, 11 February 2011 (UTC)
Missing: Antiparticle Discussion
The intro paragraph mentions that one of the chief triumphs of the Dirac Equation is its prediction of antiparticles. However antiparticles are not mention again. May I ask the talented people working on this article to elaborate on this important aspect of the Dirac Equation? Thank you. —Preceding unsigned comment added by 126.96.36.199 (talk) 20:57, 12 December 2007 (UTC)
- This problem is still present a year and a half later. The article contains the statement "As we shall see below, it brings a new phenomenon into physics—matter/antimatter creation and annihilation." but there is no further mention of matter/antimatter. Mollwollfumble (talk) 00:23, 9 July 2009 (UTC)
I know a lot is made about how the Dirac equation predicted antimatter, but didn't the Klein-Gordon equation sort of do that 2 years before? (States of negative energy interpreted as antimatter states to get round the problem of states decaying to states of lower energy indefinitely). It's just that nobody had the correct interpretation until Dirac came along, and since he had his own equation too it was attributed as coming out of that. Even if the Dirac equation and all quantum mechanics beyond it had never been written down, we'd still have at least predicted antimatter by now, if not discovered it. — Preceding unsigned comment added by 188.8.131.52 (talk) 18:00, 29 February 2012 (UTC)
I was really impressed by the completeness of this article and it seems that it could be used as an excellent introductory chapter in a textbook. AS to that section - Identification of observables- I would like to know what the continuation is... Please whoever was writing it..don't let us hanging..
"Thus the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and we must take great care to correctly identify what is an observable in this theory. Much of the apparent paradoxical behavior implied by the Dirac equation amounts to a misidentification of these observables. Let us now describe one such effect. (cont'd)"
Changed the priority to "Top". Rationale: the Dirac equation is the basis of QED as we know it.
Aoosten 20:58, 12 December 2006 (UTC)
There is a mistake in the free (anti-)particle solution:
is a spinor operator (2x2 matrix), not a spinor component. I leave it as an exercise to the author to fix it :-)
Aoosten 20:58, 12 December 2006 (UTC)
I think the whole idea of introducing the nonrelativistically covariant notation first before manifestly covariant notations in many topics, including the Dirac equation, is merely a reflection of historical inertia, of students being taught noncovariantly in turn teaching noncovariantly later... Phys 21:53, 15 Nov 2003 (UTC)
- That's a little presumptuous. The advantage of the non-covariant notation is that it has the form of a Schrodinger equation, which emphasizes that the Dirac equation is a quantum mechanical wave equation. -- CYD
- If you assume the Dirac equation is the first-quantized equation for a particle (But then, you'd have to explain the Dirac sea). But you know the correct interpretation for it is as a second-quantization of a classical relativistic field equation! Phys 18:22, 16 Nov 2003 (UTC)
- To be precise, the Dirac field theory is obtained by the first quantization of a classical field equation; or, alternatively, the second quantization of the Dirac wave equation. I don't think either approach has any great advantage over the other. -- CYD
- Unfortunately electrons are fermions, so introducing it initially as the quantization of a classical relativistic field equation means that you have to start out by introducing the students to the concept of a classical anticommuting field of Grassman variables, which could be pretty intimidating unless they are mathematicians... --Matt McIrvin 03:42, 17 Oct 2004 (UTC)
- By necessity, hole theory assumes that the negative-energy electrons in the Dirac sea interact neither with each other nor with the positive-energy electrons. Without this assumption, the Dirac sea would produce a huge (in fact infinite) amount of negative electric charge, which must somehow be balanced by a sea of positive charge if the vacuum is to remain electrically neutral. However, it is quite unsatisfactory to postulate that positive-energy electrons should be affected by the electromagnetic field while negative-energy electrons are not.
While it's true there appears to be a problem with an infinite negative chage density, the early pioneers of QED assumed the charges of the proton sea would cancel out the charges of the electron sea. It was never assumed the negative energy electrons are not affected by the electromagnetic field. Otherwise, a hole (positron) would not be deflected in the opposite direction by an electromagnetic field. The positive energy electrons also interact with the negative energy electrons. This is necessary for computing the vacuum polarization. Phys 02:57, 14 Jan 2005 (UTC)
Yes, I don't know what I was thinking when I wrote that. Thanks. -- CYD
You can add to this the fact that the negative-energy electrons in the Dirac sea should interact among each other. Come to think of it they should behave like a metal. Some serious shielding of electric fields should be going on. Bound states of electrons and holes should occur, etc. etc. The Dirac sea is a fascinating thought but untenable.
Aoosten 21:16, 12 December 2006 (UTC)
Positive and negative solutions to the Dirac equation have opposite parity. Obviously, a missing electron from an otherwise fully occupied "sea" of states would constitute a state with the same parity as the original electron. The notion of a Dirac sea is inconsistent with parity.
The section about hole theory should better be deleted or downgraded to a historical section.
Aoosten 19:35, 12 December 2006 (GMT+2)
The last paragraph deserves some comment. The equation that describes protons, neutrons and other non-leptonic fermions is not mentioned. And what is the basis for the claim that quarks ARE described by the Dirac equation? I don't think anybody knows that their g-factors are equal or very close to 2.
Aoosten 21:16, 12 December 2006 (UTC)
Just noticed that the Pauli-Dirac Gamma matrices (Well... the article uses alphas) at the beginning are different from how they're specified in the 'Gamma matrices' Wiki article. Shouldn't the four components in the bottom left be negative w.r.t. what they are currently?
- They aren't wrong, although the situation can be confusing. The relationship between the alphas and the gammas is explained in the "Relativistically covariant notation" section towards the bottom of this article. Unfortunately, "Dirac matrices" can refer to any of these matrices, which becomes a problem when the non-covariant introduction of this article links to Gamma matrices out of context. Melchoir 23:52, 16 April 2006 (UTC)
Are you sure they are right? Unless I've multiplied them incorrectly they all square to give the identity matrix so they are not a representation of the Clifford algebra.
- The alpha matrices (often times alpha_0 is simply called betha) are not supposed to be a representation of the Clifford algebra. The Gamma matrics are the ones that are a represntation of the Clifford algbraDauto 02:37, 30 May 2007 (UTC).
Should the sentence regarding the similarity transform be changed to: A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other (up to the overall sign) by a similarity transformation [User: rarsn] 18:52, 17march2011 (PST) Rarsn (talk) 01:56, 18 March 2011 (UTC)
In the section Covariant Form and Relativistic Invariance, the equation psi' = U * psi implies that the new psi after a Lorentz transformation is related to the original psi by a unitary transform. However, I don't believe this is generally the case. The transform relating the psi's is unitary for a rotation, but for a velocity boost I don't think it is. Consistency of the probability interpretation is maintained across Lorentz frames not by unitarity but by the fact that the current, psibar gamma psi, transforms as a four vector. and that the four divergence of the current vanishes, as discussed in the article.Rarsn (talk) 05:38, 19 March 2011 (UTC)
Upper and lower psi functions
The two upper psi's in the spinor represent the spin states of the electron in an external field, while the two lower ones the spin states of the positron in the same field.
But where do these positron energies and wavefunctions COME from? They basically disappear when electron kinetic energies are non relativistic, and Dirac reduces to Pauli. Okay, so the positronic components represent a relativistic effect.
Looking at their magnitude I have come to the conclusion (correct me if wrong) that the "relativistic effect" is that the positronic psi's simply represent half the increase in energy (mass) due to motion. If the electron's total energy is 1.4 M (where M is the rest mass) and kinetic energy is therefore 0.4 M, we will find that the upper psis have energy of 1.2 M and the lower ones now 0.2 M.
So my conclusion is that the origin of the positronic psi content in Dirac is really straightforwardly "simple": Basically, the positronic component of the wavefunction appears so that the momentum of the wave can increase greatly, without the assocated CHARGE increasing. Charge must be Lorentz invariant, so the only way to increase the momentum of a wave greatly without increasing its associated charge-density, is to have it a mix of particle and oppositely charge antiparticle. And that's what happens. THAT is where the virtual positronic component that appears comes from. It's half the mass-increase, basically.
I haven't seen it explained anywhere quite this way, although in any texts it's noted that as total energy of the electron makes it to 3M, the upper components get 2M and the lower components now get up to M, and we have enough energy available to produce a real positron, should we have a system available to offload the momentum properly. But in lower energy relativistic states where the positronic contribution is less than M and the positron is somewhat virtual, I don't think I've seen it pointed out that it's always just enough to cancel the electron's extra charge-density which would ordinarily result from the increased relativistic momentum of a matter-wave.
What do you think? Can we open the math section on interpretation of this spinor with a little plain English explanation of what's going on? Steve 02:18, 24 June 2006 (UTC)
In this section the tensor matrix σμν is not defined. I believe it is (1/2)(γμγν - γνγμ)
- It is standard to have an i in the numerator Xxanthippe 12:00, 11 October 2006 (UTC)
There, fixed it.
Both of the links under "Selected Papers" are broken as of 29th April 2007. Does anyone know of an alternative source so they can be fixed? 184.108.40.206 23:33, 29 April 2007 (UTC)
Interesting and praiseworthy treatment
I just wanted to pass along a word of praise for those who worked on this article. It is quite rare (frankly, I've never encountered it before in WP) to see an article that focusses so well on the motivation for an equation, e.g., the problematic situation that gave rise to a new formulation, as well as the challenges faced by early investigators. The effect is to make the article exciting, rather like an adventure story, and that without in any way decreasing its seriousness. Well done! --Philopedia 01:57, 31 October 2007 (UTC)
- I agree. Most of these math articles read like mini textbooks on the topic. The reference -- Fisher, Arthur. (July 1986) Popular Science. New ferment in the mirror world of antimatter-antigravity. Volume 229; Page 54. -- has some information that would look good in the history section. Some other material that may be of interest: * Calkin, M. G. American Journal of Physics (August 1987) Proper treatment of the delta function potential in the one-dimensional Dirac equation. Volume 55; Page 737. * Amado, R. (January 1984) Physics Today. Dirac equation. Volume 37; Page S40. -- Jreferee t/c 16:48, 16 November 2007 (UTC)
Currently, wikipedia is generating very different images for \phi on its own versus \partial \phi (for example). The problem seems to be a difference in fonts depending on some automatic choice of whether to inline a small font equation versus displaying a larger pretty equation. On other pages this may have no effect, but in the context here it confusingly appears as though the two are intended as completely different symbols. Can someone escalate this bug? 220.127.116.11 (talk) 02:20, 17 April 2008 (UTC)
single particle theory??
I am a little concerned with the following comment in the history section: "The Dirac equation describes the probability amplitudes for a single electron." While this may have been Dirac's original goal, as far as I can see this is incorrect on account of the fact that, as the author(s) of this article him/her-self states, one must postulate an infinite sea of particles to fill the negative energy states-i.e. you are immediately pressed into a multi-particle theory. While am by no means an expert on this matter though, and could very well be wrong, I think the the consistency of this interpretation deserves some discussion in the article. —Preceding unsigned comment added by 18.104.22.168 (talk) 17:33, 25 October 2008 (UTC)
Adjoint spinor isn't explicitly defined
If \Psi^dagger should be clearly defined as the complex conjugate and transpose of the vector \Psi in the definition of the adjoint spinor.
Constructive comment re. a problem with this article
I believe that this article is of limited value to anyone who does not have an advanced degree in physics. I'll tell you up front, I have only an M.S. in physics, but I am very interested in relativistic quantum mechanics, and I understand a fair amount about it. I have purchased over a dozen books that include the subject of the Dirac equation, and only one have I found which actually explains to an intelligent person with a reasonably strong background in physics and mathematics what the Dirac equation is actually doing. This one book is how I came to understand the Dirac equation.
I think that in this wikipedia article there are things left out of the explanation that should be there, for the sake of a person that is trying to learn something, not just re-read what he/she already knows. I found a web site which shows much (not all) of the the left-out details I'm talking about. The url is
What is being left out of this wikipedia article is the explicit presentation of the four 4x4 matrices that are the coefficients in the equation, and the four separate differential equations that result for Ψ(r,t). And the article does not explain (at least not in an explicit, straightforward manner) one of the most important outcomes of the Dirac equation: that when you combine these four equations, through substitutions, into one second-order differential equation (in the field of a proton, I think?) you get five separate terms: Two of them are the non-relativistic Schroedinger equation, one is (or is similar to?) a relativistic correction, one is the spin-orbit energy, and the last is a relativistic correction to the potential called the "Darwin term". The "Darwin term" effect, I have read, was completely unknown at the time Dirac published the equation. Not long after it was experimentally verified to exist. This is so interesting ... why is it not mentioned?
I think it is a shame that wikipedia would leave out such basic explanations, not to mention fascinating moments in the history of science. I hope that someone who is an expert might want to address this. I will not be so presumptuous, because I'm not qualified to do it.
- The goal of this page is to present the Dirac theory in a concise way. That can't be done without assuming the reader has some knowledge of physics beyond the basic level. The point of an encyclopedia is to gather knowledge in one place and so stimulate the reader to learn on his own, not to teach a subject. Antimatter33 (talk) 18:35, 11 February 2011 (UTC)
Jim Al Khalili 'Everything and Nothing" BBC4
If, as he says, the universe is the debris remaining after matter and antimatter from the Big Bang anihilated itself, does that mean that the matter antimatter particles continually appearing in a vacuum will also leave 'debris', ie new particles to add to our universe? He never addressed this issue.
Dirac equation as an equation for just one component
If there are no objections, I would like to make the following addition to the article:
In a general case (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.
Source: Journal of Mathematical Physics, 52, 082303 (2011) (http://jmp.aip.org/resource/1/jmapaq/v52/i8/p082303_s1 or (free access for personal use) http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf )
Comment: I am certainly biased, but it seems to me that this addition may be interesting and useful for many readers.
Grammar in the intro paragraph
Looking at the final sentence of the lede:
- Although Dirac did not at first fully appreciate what his own equation was telling him, his resolute faith in the logic of mathematics as a means to physical reasoning, his explanation of spin as a consequence of the union of quantum mechanics and relativity, and the eventual discovery of the positron, represents one of the great triumphs of theoretical physics, fully on a par with the work of Newton, Maxwell, and Einstein before him.
There's some faulty grammar there, and anyway it's a bit of a run-on. Because I'm a complete amateur in my interest, though, so I wanted to run this proposal by other editors:
- Although Dirac did not at first fully appreciate the implications of the equation, his faith in mathematical logic as a means to physical reasoning proved itself: the equation explained spin as a consequence of the union of quantum mechanics and relativity, and it contributed to the eventual discovery of the positron. The Dirac equation thus represents one of the great triumphs of theoretical physics, on par with the work of Newton, Maxwell, and Einstein.
This last claim (on par with Newton, Maxwell, and Einstein) may also benefit from citation of a reliable source – it could be seen as WP:OR or WP:SYN – but I'm primarily on a copyeditorial mission here. Any thoughts or recommendations welcome. /ninly(talk) 16:58, 4 November 2011 (UTC)
Clean up and clarify
Little issues which could be resloved now:
- Bits here and there need tidying up, like brackets in equations.
- Also it should be clearer what some equations mean to readers who do not understand index/tensor notation - inluding the gamma matrix form of the equation (the most compact). It is possible to state what both forms say without loss of information, in fact it would illustrate the use of index notation.
- The initial equation uses x for position instead of r, this isn't a problem but r is more clearer and universally understood to be the spatial position in 3d, the appearance of x makes it look more like a vector in the x-direction.
- Furthermore the more familiar vector and matrix notation (in boldface) for the Dirac matricies and the current density should be used, everything looks like a plain italic scalar (for quantities without the indicies - those which have are vector components interchangably understood as the full vector, and appearances of the Einstein summation) etc. Boldface was used for x but nowhare else, which is a bit strange. It is clearer for those who have had exposure to some level of vectors and matricies (who will have seen boldface vectors and matricies), which will engage them into the meaning of the equation sooner. All the complicated-to-understand though simpler-to-write index notation should come later, for the more experinaced reader to read further, and so less experianced readers are not befuddled and switched off at first sight of all those indicies.
- A few bits of text could be written a bit better, few more links could be added...
I know this is just me being pernickity, but i'll do these now. And yes sources will be added in case someone is checking my contributions inside out: if I added resources, provided edit summmaries, wrote on the talk (well here that is) etc...
- Hi---sorry I had to revert your change. I'm fairly new to this, so maybe that was a horrible breach of etiquette, so, once again: sorry. However, standard notation might boldface the alpha matrix vector, but not its components, even though they are matrices. Also, the line which explained the use of a curly-d derivative operator was actually wrong. Curly-d down-mu is d curly-d by curly-d x up-mu, not curly-d by curly-d x down-mu. I.e. "The gradient with respect to a contravariant position-time four-vector x up-mu is itself a covariant four-vector," pg 226, Introduction to Elementary Particle Physics, David Griffiths. It's also explained in numerous other books, that's just the one I happen to have to hand first.
- You also changed single words for either less-helpful or pretty much identical words all the way through, which didn't add anything, and in some cases detracted from the article. I certainly found it much easier to read before.
- There probably is a lot of scope for improvement, and please do do it! But those pernickety points aren't so helpful. For example, the point of this article is not to explain index notation. It should use the most clear way of explaining it, but that is with index notation, in most cases. 22.214.171.124 (talk) 20:09, 29 November 2011 (UTC)
- Firstly, I can't see how I was emphasising so much on index notation in the actual article. I mentioned it to death above here on the talk page, but not in the article. There is nothing wrong with mentioning what symbols and notation mean: loads of physics articles have statements like "where ∇ is the del operator", "where * denotes complex conjugate", "where † denotes hermitain conjugate (complex conjugate transposed)", "where ż dentotes diff. of z wrt time", "where zxxx denotes the 3rd order partial derivative of z wrt x" etc. A typical reader will not understand all this index notation. If they were wrong, you could of corrected them, but given the other problems I geuss it doesn't matter.
- I'll at least add the referance again for the initial equation. It could be written in a less clumsy form:
- but i'll leave that for now. Also the notation x should be r, its clearer that the equation is true in 3-d. That much can be done.
- In addition the brackets still need cleaning up in all subsequent equations again.--F=q(E+v^B) (talk) 06:58, 30 November 2011 (UTC)
- Sorry, I should have just corrected that and left the rest. I guess I felt I didn't have time to, and that it would be irresponsible to leave it, but then I wrote loads here instead. Should probably not have reverted your changes :(. 126.96.36.199 (talk) 17:46, 1 December 2011 (UTC)
In addition to adding links and an extra source. I decided to clean up notation. There is a hint of a time derivative as
but then spatial derivatves as
I'm not saying in any way that this subscript notation is wrong, it is more compact - just that its best for the 1st 1/2 of the article to use the more familiar and full notation so less advanced readers, who still know partial differentiation, can settle into article better. -- F = q(E + v × B) 20:39, 28 December 2011 (UTC)
Dirac a physicist?
- Where did you read that from? He was definitley an electrical engineer but became (one of the greatest ever) mathematical physicist - go and read him up.--Maschen (talk) 21:41, 8 January 2012 (UTC)
I appreciate the effort put in to write about the 4-probability current, but the current context should really be moved to the main article on probability current, and in the Dirac equation article it can be linked and mentioned before its use in the subsequent formalism. By no means am I saying to delete, just reduce a little padding and move context is all. =)-- F = q(E + v × B) 08:43, 12 January 2012 (UTC)
A small neutrality quibble
Dirac's work is extremely important, but the line "...fully on par with the work of Newton, Maxwell, and Einstein before him" is a little silly. — Preceding unsigned comment added by 188.8.131.52 (talk) 15:43, 1 February 2012 (UTC)
- As a luddite in this argument, and having spent the last 2 years finally understanding what the fuck Dirac's utterly amazing equation is - something I was only capable of grasping after the surrealism of Einstien's theory, I'm disappointed that this prose has been removed. What Einstein suggested was a bit crazy (a satellite travels in a straight line through curved space. Wake up idiots, you're speaking all three dimensionally). Dirac's equation is crazier. It suggests that cause can happen after effect, and it is only the work of a purist mathematition who could blur out "reality" to return reality to us on a very confusing plate. It is well on a par.
"This article is just crap again"...
Michael C Price, please may you explain why??? If you come out with a comment like that in the article edit history - you should also explain. If its something I have done then say it - I can take it. -- F = q(E + v × B) 22:11, 7 February 2012 (UTC)
- The edit of Michael C Price contradicts to the MoS which discourage links in section titles. Also, ASCII substitutes have to be replaced with en dashes. So, who introduces and preserves a "crap"? F=q(E+v^B), make your job and do not become insulted by every run-by comment. Incnis Mrsi (talk) 13:01, 8 February 2012 (UTC)
- To answer, it something that AntiMatter deleted. I have restored the relevant lost material which shows the relationship between the KG and DE in a few lines. -- cheers, Michael C. Price talk 22:48, 8 February 2012 (UTC)
- Unless anyone objects I'll move the new "Comparison with the Klein-Gordon equation" section from "backgrouund and deveopment" into the "mathematical formulation" section. Seems better suited there. -- cheers, Michael C. Price talk 16:32, 17 February 2012 (UTC)
|This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.|
|Archive 1||Archive 2|