Talk:Dirac equation - Misplaced Pages

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"This equation is more intelligent than me"

I can't find the right place in this page to insert this rather incisive quote of his. — Preceding unsigned comment added by 93.97.48.95 (talk) 02:11, 25 September 2011 (UTC)

The beginning! -- cheers, Michael C. Price 20:34, 4 February 2012 (UTC)

Insert at the end of the very first paragraph? Thanks for finding it, but there is no referance. Someone will just delete it without a referance. --Maschen (talk) 10:35, 8 February 2012 (UTC)

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)

I've succeeded in restoring the article from the beginning well into the section on Dirac's ansatz. It is now both technically and historically correct and should stay that way. Antimatter33 (talk) 09:55, 12 February 2011 (UTC)

Alright I have got the beginning mostly repaired with much irrelevant and distracting information removed. More later. Antimatter33 (talk) 11:20, 12 February 2011 (UTC)

OK the section "Mathematical Form" may be considered complete. Antimatter33 (talk) 20:51, 12 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 130.212.215.27 (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 gave up in frustration because my work was vandalized. I'm going to try again. Antimatter33 (talk) 18:26, 11 February 2011 (UTC)

Observables section

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)"

Comment

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:

${\frac {\mathbf {\sigma } \cdot \mathbf {p} }{E+m}}$

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)

True, and we plan to stick to the historical facts anyway here. This is not a page on second quantization or field theory - it's just for the Dirac equation. Antimatter33 (talk) 18:47, 11 February 2011 (UTC)

Noninteracting sea?

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)

Electromagnetic Interaction

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)

History is important. The Dirac equation emerged before any of this was known. It will be mentioned but the treatment is of the Dirac electron and positron, by the nature of things. Antimatter33 (talk) 18:43, 11 February 2011 (UTC)

Gamma matrices

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?

I haven't changed them, as I'm not really sure if they're wrong or not. —The preceding unsigned comment was added by 81.179.121.11 (talk • contribs) 23:38, 16 April 2006.

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 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)

Dirac bilinears

In this section the tensor matrix σ is not defined. I believe it is (1/2)(γγ - γγ)

It is standard to have an i in the numerator Xxanthippe 12:00, 11 October 2006 (UTC)

There, fixed it.

What are the references (e-references will be great) for the multivectorial definitions of the couplings? Nuewwa (talk) 08:29, 30 November 2008 (UTC)

Links

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? 172.141.125.200 23:33, 29 April 2007 (UTC)

thanks for the heads up , will investigate Antimatter33 (talk) 18:41, 11 February 2011 (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)

I heartily concur with Philopedia. One of the best articles I've encountered at WP. Bravo! 71.188.252.208 (talk) 21:30, 7 April 2008 (UTC)

Any plans for nomination to GA status? Venny85 (talk) 21:35, 7 April 2008 (UTC)

Thanks I'm going to try again to make this page tight without much hope of succeeding before grad students with no knowledge vandalize it. One must try, as Dirac himself said. Antimatter33 (talk) 18:30, 11 February 2011 (UTC)

math/latex

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? 150.203.48.127 (talk) 02:20, 17 April 2008 (UTC)

I agree, but we can't do much about it at this level. Antimatter33 (talk) 18:38, 11 February 2011 (UTC)

Just replace \phi with \phi\, to get a consistent font. Dauto (talk) 04:08, 18 March 2011 (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 128.230.246.37 (talk) 17:33, 25 October 2008 (UTC)

It is wrong and will be edited out shortly. Antimatter33 (talk) 18:25, 11 February 2011 (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.

It was once :( I'll fix it in time. Antimatter33 (talk) 18:33, 11 February 2011 (UTC)

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

  http://electron6.phys.utk.edu/qm2/modules/m9/dirac.htm

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.

Thanks for your consideration of this ... Worldrimroamer (talk) 01:27, 4 August 2009 (UTC)

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)

The goal is also to provide links and pointers to sites where the reader can teach themselves the subject. -- cheers, Michael C. Price 19:17, 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.

(||||) This is how my tilde key appears when typed —Preceding unsigned comment added by 86.4.87.71 (talk) 15:16, 2 April 2011 (UTC)

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.

Akhmeteli (talk) 04:09, 21 August 2011 (UTC)

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...

--F=q(E+v^B) (talk) 21:12, 28 November 2011 (UTC) — Preceding unsigned comment added by F=q(E+v^B) (talk • contribs)

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. 163.1.231.41 (talk) 20:09, 29 November 2011 (UTC)

No worries.

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 z_xxx 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:

\left(c{\boldsymbol {\alpha }}\cdot \mathbf {\hat {p}} +\beta mc^{2}\right)\psi =i\hbar {\frac {\partial \psi }{\partial t}}\,\!

\psi =\psi (\mathbf {r} ,t)\,\!

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 :(. 163.1.231.41 (talk) 17:46, 1 December 2011 (UTC)

Its cool. Please don't worry or apologize!--F=q(E+v^B) (talk) 18:57, 2 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

{\frac {\partial \psi }{\partial t}}\,\!

but then spatial derivatves as

\partial _{x}\psi ,\partial _{y}\psi ,\partial _{z}\psi \,\!

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?

He was a mathematician. — Preceding unsigned comment added by 71.198.36.167 (talk) 05:46, 8 January 2012 (UTC)

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)

Probability current? KG Equation?

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)

I intend to just do it now. It does detract slightly from the article so there shouldn't be any objections.-- F = q(E + v × B) 08:49, 12 January 2012 (UTC)

spelling

I believe that the plural of the word 'index' is 'indices', not 'indicies'. Bo Jacoby (talk) 20:46, 21 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 149.254.224.248 (talk) 15:43, 1 February 2012 (UTC)

This has been slightly fixed recently.-- F = q(E + v × B) 22:11, 7 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.

— Preceding unsigned comment added by 93.97.48.95 (talk) 00:53, 12 February 2012 (UTC)

"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)

Responded at your talk page here.-- F = q(E + v × B) 20:24, 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 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 16:32, 17 February 2012 (UTC)

Two kinds of Dirac field

There's an important subtlety to the Dirac equation that I don't think this article captures. Historically, Dirac introduced his equation as a relativistic version of Schrödinger's equation, with the wave function interpreted as a quantum amplitude giving a probabilistic description of fermions. The current version of this article does a pretty good job of describing this. But the Dirac field has a completely different significance in Quantum Field Theory, in which it is not a quantum amplitude but rather something akin to a classical field with a quantum description as a field operator. This is described briefly at the article on fermionic fields. I am not sure how best to introduce or describe this distinction in a way that works with this article, but think it's important to do so. Any other thoughts on this?-Dilaton (talk) 19:18, 8 February 2012 (UTC)

You sound like you're talking about the Schroedinger picture where the time-dependent quantum weirdness (wave behavior) goes into the wave equation, vs. the Heisenberg picture where the the fields are classical and the quantum weirdness (the wave-nature) and time-dependence is put into the operators. Any quantum equation can be expressed with either picture, including the Dirac one. They are mathematically equivalent, but sometimes one picture is easier to use and sometimes another. S B H arris 19:48, 8 February 2012 (UTC)

FYI the Schrödinger equation is not a (spacial) probabilistic description too. It is Copenhagen interpretation which is. Incnis Mrsi (talk) 19:26, 8 February 2012 (UTC)

I think Dilaton is talking about second quantization. In QFT, fields aren't at all like wavefunctions - they're like classical variables that get quantized. They're the things the wavefunction is a function of. In QFT you compute the probability amplitude as a function of the configuration of fields, just as in QM you compute the probability amplitude as a function of (say) the position x. Put yet another way, in QFT the path integral is over fields configurations, while in QM it's over position.

So Dilaton is correct, but it's not going to be easy to explain to a non-specialist. Waleswatcher (talk) 14:21, 10 February 2012 (UTC)

In QFT you can actually use either formalism mentioned above. The Heisenberg picture is more often used in QFT because it goes more naturally with the Lagrangian treatment than the Hamiltonian treatment (what Schroedinger introduced for QM), and QFT is more often formulated in Lagrangian terms. The Klein-Gordon equation (for example) can be written for classical fields (provided they are bosonic) and then quantized just once (what is sometimes called second quantization, but since this is quantization of the field, if you start with a free-field equation it really only needs doing once). This can be done using either the Schroedinger formalism where the operators are time-independent, or the Heisenberg one where the operators are time-dependent (this is the more familiar treatment in QFT but not the only one). The solutions are the same, provided some assumptions are made about what the answers mean. I have a QFT text (Peskin and Schroeder) that does this in Chapter 2.

The same is true of the Dirac equation and fermionic fields. THIS wiki article treats the Dirac equation with the Schroedinger operator formalism, but in QFT the same equation looks somewhat different because it is set up with Lagrangians and Feynman propagators, in order to more easily quantize fields for interacting particles. The path-integral formulation is Lagrangian and so you only see it represented in what we call the Heisenberg picture, but in theory it could be done the other way (as the above text demonstrates when they introduced QFT with the Klein-Gordon equation applied to "free fields" (no interacting or bound particles). In any case, to address the question above, field quantization ("second quantization" of QFT) is already there to be used with either type of operator (time-independent or time-dependent) with the Dirac equation (or any relativistic equation)-- it's not something new, that Dirac invented. And yes, it's not discussed much here because this article is about the development of the Dirac equation, not its later use in QFT, where rewriting it as a quantum Lagrange-Euler equation makes it look different. S B H arris 17:32, 10 February 2012 (UTC)

With respect, we're not talking about Schrodinger versus Heisenberg pictures here. That's simply a question of whether you put the time dependence into the operators or the state, it's a fairly trivial change. What's at issue here is that Dirac fields (and scalar fields and vector fields and every kind of quantum field) are not probability amplitudes. They aren't wavefunctions at all, they aren't anything like the "psi" in non-relativistic quantum mechanics. They're actually like the "x" in quantum mechanics. So the Dirac equation is on a very different footing from the Schrodinger equation - it's more closely analogous to Newton's equation F=m x dot dot than it is to the (better) "quantum-state vectors in Hilbert space", we still are referring to probability densities in the DE, just as we are in the Pauli equation that the DE reduces to in the low-energy limit. And indeed, both DE and Pauli equation reduce to the Schroedinger equation (SE) at low energy, and if there is no external field that makes spin-states non-degenerate. You've said the SE has a psi that isn't like the psi in Dirac. But how can one equation reduce to another in some limit, if it's not talking about the same mathematical object? At what point does this magic happen? Finally, if Dirac's psis are not about proability density (for particle location or any other observable), why did Dirac sweat so much about making the sum of them unitary (something only important in figuring probability amplitudes of observables)? The texts are full of ordinary psi solutions of the Dirac equation (DE). I'm looking at one for an electron AT REST. As a spinor it has four parts, each of them a simple exponential.

Now look, I'm claiming no expertise on this issue, but I have a lot of texts written by experts, and I read the notation well enough to know what they are saying at least some of the time (and if not, I can usually tell). Yes, it's quite true Schrodinger equation. In fact, the wavefunction of any QFT satisfies the standard Schrodinger equation in the standard form H psi = i hbar psi dot, not the Dirac or Klein-Gordon equation - but as Dilaton points out, that psi has little to do with the Dirac field (it's a functional of the Dirac field). Waleswatcher (talk) 05:01, 11 February 2012 (UTC)

It does look like the article needs to be re-written in places. For one thing, it refers to the Dirac psi as a wavefunction, which is very bad notation (that may be what Dirac thought it was at first, but that's not what it actually is). I notice that the article on fermionic fields is better. I'll take a shot at this article in a few days if no one else does first. Waleswatcher (talk) 13:59, 11 February 2012 (UTC)

It seems a bit odd to be arguing with somebody who thinks that Dirac thought his psi's (or columns of psis) were wavefunctions, when they really wasn't. I would think Dirac could tell a function from an operator. Whether we call these psis "wavefunctions" or that when QFT methods are invoked, the fields are quantized with creation and annihiliation operators-- that's almost the definition of QFT's approach. But it's a separate procedure and it's not "in" the DE or part of it. Rather it must be done as a second and deliberate step, and the DE "forces" it (or encourages it) more than other equations, because if you don't do it, you end up solving the DE for one-electron, producing all the odd effects like negative-energy states, Zwitterbewegung, Klein paradox behavior, and so on. But this is the fault of QM plus relativistic energies, not the DE itself (again, since low energies the DE becomes the Pauli, etc, obviously this stuff is not "in" the Dirac EQUATION). The DE might be said to predict a small amplitude for ONE positron for that one electron (if read correctly), but not an infinite number of both (that idea came out of Dirac's imagination, not his equation). The problem is not in the DE itself, but in the fact that classical QM and relativity are not compatible without an infinite number of particles in every problem, which is exactly what QFT presumes.

One of the reasons to create the creation/annihilation operator formalism of QFT has nothing to do with the DE, because it applies just as well to the Schroedinger equation, although with minimal impact when slow particles are treated. But one of my texts (Peskin and Schroeder) has a cute section in chapter 2 where they look at the "classical" QM probability of a particle going from x to x1 faster than light. Even using relativistic expressions for momentum and energy, that probability turns out to be positive (but small) for a particle to "tunnel" outside its own light-cone and thus violate relativistic causality. The only prescription for fixing this, is to create antiparticles that propagate also in a way to cancel all probabilities of observable particles to appear outside their light-cones. But this is not a problem only "in" the Dirac equation and it really has no place in an article on the DE per se. It appears as a problem for any QM equation describing a single particle and a classical (non-quantized) field. QFT, with all of its infinite-particle-pair soup instead of the classical field, is the only prescription to fix it. That is not Dirac's fix (who instead invented a Dirac sea that is, again, not implicity in his equation and represents the vacuum, not things like the EM field), that is QFT's fix (where something like the Dirac sea appears by operator magic, out of the vacuum OR any field). And to answer the question above, that is why the Dirac field looks different in QFT formalism. All fields look different-- not just Dirac's. This infiniite number of particles that come from these operators and various fields can only be shown to provide unitary probability densities for observables, after renomalization, which Dirac certainly never attempted. Dirac famously asked Feynman in a seminar in about 1947, if Feynman's new theory for QED was "unitary." Feynman didn't (at that time) even know what the word meant. But it took some time to show that QFT theories of various types were unitary, and that was why QED in complete form came a generation after the Dirac equation. And one of the reasons why Dirac himself didn't invent it. S B H arris 19:32, 12 February 2012 (UTC)

Dirac equation is indeed an equation, not a physical theory. These are articles like fermionic field where such things as n-particle states, creation operators and observables, should be explained in details. When doing such improvements, please, take a look on a discussion at Talk:Quantum superposition, this is an important point (in distinguishing fields from wave functions) too. Incnis Mrsi (talk) 15:21, 12 February 2012 (UTC)

"Whether we call these psis "wavefunctions" or (better) "quantum-state vectors in Hilbert space", we still are referring to probability densities in the DE" - Again, from a modern point of view the ψ in the Dirac equation is neither a wavefunction nor a vector in Hilbert space, and its square (with the gamma 0) is not a probability density (it's a charge density or mass density, depending on what you multiply it by). The Dirac field is an operator. As for why the Dirac equation reduces to the Schrodinger equation in the NR limit, it's because the Dirac field acting on the vacuum creates a single particle state, and that is a vector in the Hilbert space. Because the field satisfies the Dirac equation, so does that state. So I suppose the ψ in the Dirac equation can be interpreted as representing the state one gets after acting on the vacuum once with a Dirac field, but that's not what the notion ψ (with no bracket) usually refers to in modern treatments. In case you simply don't believe me, google brought me a very reliable online source, notes from David Tong (a full professor at Cambridge U. and among the world's experts on quantum field theory). http://www.damtp.cam.ac.uk/user/tong/qft/five.pdf Here's a quote:

Let’s pause our discussion to make a small historical detour. Dirac originally viewed his equation as a relativistic version of the Schrodinger equation, with ψ interpreted as the wavefunction for a single particle with spin.....This is a very diﬀerent viewpoint from the one we now have, where ψ is a classical ﬁeld that should be quantized.

It's possible that this is too technical/subtle of a distinction to try to make in a wikipedia article. And I suppose there's an issue as to whether we want to describe the Dirac equation as Dirac thought of it, or whether we want to describe the Dirac equation as it's taught in modern quantum field theory courses. I'm not sure of the answer to that. Waleswatcher (talk) 02:11, 20 February 2012 (UTC)

Well, both. As long as the article doesn't become a chapter of a QFT textbook and concentrates more on the Dirac equation than anything else (hence title) its all fine. The article does state the version proposed by Dirac, and later in the article everything becomes advanced. I see no problems with your addition of "fermionic fields", but I have come across the terminology that its a "4-component spinor wavefunction", in a few of the sources given in the article (and cited). Anyway thanks for the tweaks. =)

I don't know QFT inside out (yet - will some day...) so I will not become too involved in the terminology, or even the article anymore. Just thought to add my answer. -- F = q(E + v × B) 13:41, 20 February 2012 (UTC)

OK, that's very reasonable. I think the best course of action is probably to have a small section describing the intellectual history of how Dirac's ψ has been interpreted over the years. That will make it less important how it's referred to in the rest of the article. But writing such things in a way that's both correct and comprehensible to a general reader is extraordinarily hard. Waleswatcher (talk) 14:50, 20 February 2012 (UTC)

Indeed its hard to write/edit maths/physics articles so laypeople can understand them. Its the solution to the equation - so there is nothing wrong with the inclusion. The only boundary condition is not to talk too much about quantum fields, even if Dirac's theory did activate much of QFT. Anyway go for it - in the Background and development section. =) -- F = q(E + v × B) 16:43, 20 February 2012 (UTC)

On second thought I will come back to this breifly. A heading or two will be added in the properties section, and once again will try to shorten the length of the initial description of probability current and this will be moved into the properties subsection Adjoint spinor and conservation of probability current. It still somehow seems detracting, and not 100% relavant (perhaps ~60%, not so sure). Anyone who disagrees may revert. -- F = q(E + v × B) 21:52, 20 February 2012 (UTC)

It is good to see this issue being taken seriously, and to see the improvements. Maybe there is hope. I'll see what I can do. Sorry if I step on any toes in the process.-Dilaton (talk) 19:55, 23 February 2012 (UTC)

Most of the edits were positive definite. =) Although I disagree with the removal in this edit . As I said way up above, I tried everything to make the index notation as clear as possible - why did you not think so? I'm not fussed; it doesn't matter either way - just asking.

Also why did you remove the subsection in this edit instead of merging with the Square root of the KG equation section? If anything the larger section (still standing) should be cut down and merged into that section. Again - I'm not going to be too involved (actually have no time either - unfortunately procrastinating as usual here on wikipedia instead of uni work), simply wondering.-- F = q(E + v × B) 12:43, 25 February 2012 (UTC)

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