Talk:Dirac equation

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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 15:31, 10 June 2007 (UTC) This article, and some that it references, are full of inaccuracies and often outright wrong statements. It should be entirely rewritten, which I plan to do. I have started with the section "Covariant Form". The section on quaternions was removed as being patently wrong. The section on the derivation of the equation is also mostly wrong. Identification of the physical content ranges from poor to completely absent. This article should be merged with those on Dirac spinors and gamma matrices, with appropriate references to the theory of representations of the Lorentz group and Clifford algebras. DeSitter33 12:27, 18 August 2007 (UTC) The opening of the "Details" section was modified with proper attention to the historical context and the mathematical basis of the equation. DeSitter33 08:00, 21 August 2007 (UTC) The sections on covariance was extensively expanded and sections comparing the Dirac theory to the Schroedinger and Pauli theories were added. WIP. DeSitter33 12:30, 22 August 2007 (UTC)

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Help with this template Comments: edit – history – watch – refresh Geometry guy 15:31, 10 June 2007 (UTC) This article, and some that it references, are full of inaccuracies and often outright wrong statements. It should be entirely rewritten, which I plan to do. I have started with the section "Covariant Form". The section on quaternions was removed as being patently wrong. The section on the derivation of the equation is also mostly wrong. Identification of the physical content ranges from poor to completely absent. This article should be merged with those on Dirac spinors and gamma matrices, with appropriate references to the theory of representations of the Lorentz group and Clifford algebras. DeSitter33 12:27, 18 August 2007 (UTC) The opening of the "Details" section was modified with proper attention to the historical context and the mathematical basis of the equation. DeSitter33 08:00, 21 August 2007 (UTC) The sections on covariance was extensively expanded and sections comparing the Dirac theory to the Schroedinger and Pauli theories were added. WIP. DeSitter33 12:30, 22 August 2007 (UTC)

[edit] 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)

[edit] Antimatter etc.

Hello there Dirackers,

I am responsible for the content down to the "cont'd" in the "Observables" section - I've been meaning to resume my presentation and hopefully will get to it shortly. I've been busy with research and haven't done any Wiki-ing lately. Patience please! This section will be greatly fleshed out. This is where we will discuss antimatter, and it is the proper place for that, I think.

The section of the Dirac equation in curved spacetime should go toward the end or moved to a separate article, as this is not an umambiguous issue, and brings up the entire issue of the meaning of spinors on curved manifolds, a difficult issue worthy of a separate treatment.

-drl

Antimatter33 (talk) 16:12, 16 February 2008 (UTC)

[edit] 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)"

[edit] 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:

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)

B = ×A Is that correct? I think there was a mistake and I tryed to correct it, but I do not know if it is correct.

Plàcid 21:33, 27 Jan 2004 (UTC)

That's correct.

It would be good to add a section with the solution to the hydrogen atom. The way that the spin-orbit interaction and Thomas precession terms drop out of the Dirac equation is, to me, the most interesting and accessible result from the equation.DKREBS 10:59, 25 September 2006 (UTC)

[edit] Moved from article

I moved the following text from the article.

All upper explanations are old and wrong.The relativistic electron has only one positive energy and two different motions: one in forward and other in backward as which ot them contain spinning in right and in left. Terefore fore components of total function describe fore motions with equal energy.Three matrixes describe three strongly correlated oscillations in three mutually perpendicular directions. In result of this fermion strongly correlated motion (Zitterbewegung) there is no any difergence in electrostatic interactions known in classical approximation and there is only magnetic interaction between magnetic intensities of own magneric field and magnetic dipole moment of the relativistic quantized electron. In result of thie interactions is obtained self-energy of the relativistic quantized electron.

Comments, anyone?

Anville 23:39, 2 Nov 2004 (UTC)

What could it mean? Aoosten 21:41, 9 June 2007 (UTC)

[edit] 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)

[edit] 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)

[edit] Dirac Equation in Curved Spacetime

I miss something about the Dirac-Equation in curved space. Can someone add it? --141.63.56.202 08:29, 14 Apr 2005 (UTC)

What are you missing?

Here is the place to give an introduction to the work of Fock and Ivanenko who formulated the Dirac equation in a curved Spacetime allready in 1930. Here a link is needed to a page on Fock-Ivanenko bivectors.

This section appears to have a lot of mistakes. The claim that the origin of the deviation in gyromagnetic ratios of neutrons is due to the curvature of spacetime is implausible. What if I am considering these particles in flat space in the absence of gravity? Should I not get the gyromagnetic ratio predicted by the standard Dirac equation ? Also the modified gamma matrices, the gamma-bar matrices are defined to be constants. Specifically, they dont have any dependence on the metric that describes the local curvature and this is unusual becuase one would expect the equation to reduce to the original Dirac equation in the appropriate limit. I cant be certain here but it seems to me that neither is the curved space Dirac equation correctly stated nor is the curvature in any way responsible for the magnetic moment deviation.

It seems to me that the last part of this section is completely incorrect. The derivations of the equations in the linked article are filled with misunderstandings, calculational errors and false assumptions. For example, the author claims that the general form of a metric tensor is a constant tensor (with all the entries of order unity). This part should certainly be deleted.

I think too that this section is a mistake, because I had a final graduate work about the gravitomagnetic fields to study the Lense-Larmor-Thirring effect, and just for curiosity I derivate the Dirac Equation for a weak newtonian potencial generatad by the particle, and the I add the newtonian and gravitomagnetic potencials of the Earth as a external field. By the fact the gamma matrices under a curved spacetipe should depend of the newtonian or any high-order relativistic gravitational potencial, like I make it. Then I add the magnetic field to the hamiltonian of the Dirac equation with a newtonian field geenrated by the test particle, and the direct calculation of my formula shows the predicted gyromagnetic ratio of the neutrons and protons don't change so much as a predicted by the standard Dirac Equation. Even he put the first order quantum elecytodynamics correction, it shows that the gravitation correction of the gyromagnetic ratio of the protons and neutrons is 35 orders of magnitude smaller of the electrdynamics correction !!! For the electrons is 42 orders of magnitude, respetively. By the fact, the gravity has a infinitesimal influence of the gyromagnetic ratio, but can be ignored, or we make a fooled measurement like measure the diameter of the entire Universe with a error smaller of the size of an electron!
To do so, the formula of the gyromagnetic ratio is: g = 2 (1 + %Alfa / (2 * %pi) (1 - (4*%Alfa_g) / %pi), where %Alfa is the fine structure constant, whick %Alfa=1/137 and %Alfa_g is the gravitational coupling constant, which: %Alfa=m_e²/M_p², is the square of the ratio of the particle' mass and the Planck mass.
This exercise shows that the gravity field plays a very,very,very minir role in particle physics.

There is also an error, presumably by the same author (capital Neutrons and Protons), when he states: "In the case of the Neutron, yes it is clearly not a fundamental particle since it does decay into a Proton, Electron and Neutrino". This is not true, since also heavy quarks decay into lighter ones through the weak interaction, although they are so called fundamental particles (=point-like). This is actually what happens in the case of neutron->proton decay also, where d-quark goes to u-quark, electron and anti-neutrino. All that prevents particles from decaying into lighter ones are the conservation laws of quantum numbers. Maybe the whole section should be erased. Or maybe someone more professional, for example the previous guy, could write there something about the modification of gamma matrices and so on.

[edit] Relativistic Quantum Chemistry

Quantum chemists solve the Dirac equation for molecules containing heavy atoms. These are non-perturbative many-particle applications. The article is incomplete at an elementary level without a paragraph on relativistic quantum chemistry. Aoosten 21:48, 9 June 2007 (UTC)

[edit] 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).

[edit] 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)

[edit] Dirac bilinears

In this section the tensor matrix σ^μν is not defined. I believe it is (1/2)(γ^μγ^ν - γ^νγ^μ)

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

There, fixed it.

[edit] Where to write more about spinors?

I'd like to write much more about the four-spinors that show up when solving the Dirac equation. The u for particles and v for anti-particles have many properties that are discussed in various books, but it'd be too much information to add to an already long article here.

So what title should I give to an article to be specific about these spinors which show up in the dirac equation? (Dirac spinor redirects to Spinor which looks more math-y and not so much physics-y) JabberWok 05:05, 3 March 2007 (UTC)

== I'm a newby here. Can anyone tell me how one can add E (energy) and m (mass) in the definition of the Dirac Spinor: i.e. [phi; phi*sigma.p/(E+m)] ?

Thanks in advance. —The preceding unsigned comment was added by 72.140.140.66 (talk) 03:27, 6 March 2007 (UTC).

[edit] Can all you math whizes

Who really work with this equation a lot and understand its guts, please take a look at the upper and lower psi function explanation above (in the section dated June, just two above this one), and give me some feedback on my understanding ??

THANKS!! SBHarris 23:23, 27 March 2007 (UTC)

The charge associated with a Dirac wave function is q(u^2+v^2), so u and v both contribute with the same sign, while to the energy they contribute with an opposite sign. For electrons you _manually_ put q=-e, for positrons q=+e. This is just one example of a major paradox, another being the 'zitter bewegung'. See e.g. Sakurai, Advanced QM. These paradoxes must be eliminated before it is possible to decide whether your statements make sense! You're on your own, don't expect positive referee reports but get cracking! Aoosten 22:08, 9 June 2007 (UTC)

Both Greiner and Foldy (of Foldy-Wouthuysen) see [1] agree for for non-relativistic particles, the upper spinor energy components for a free particle with non-relativistic kinetic energy are large, and the lower ones much smaller, on the order of v/c smaller. They are NOT the same. Obviously with the v/c dependence the lower components are connected to being relativistic mass terms, and that relativistic mass-energy has something to do with the virtual positronic component in any electron solution with kinetic energy (even if only a small one). Pauli had a way of getting rid of the lower component spinors for non-relativistic particles to get back to his Pauli equation, but as Foldy points, out, his transformation didn't preserve unitarity. Foldy and Wouthuysen found one that did, and continued to do so into the relativistic realm. I don't have the reference, but I've seen somebody's paper in which they point out that after the FW transformation in the relativistic realm, the upper energy terms are rest energy plus half the kinetic energy, and the other half of the kinetic energy for the lower terms. Which is another way of saying that half the electron's kinetic energy is available to make a positron, and if the kinetic energy is large enough (2mc^2) this half exceeds the positron rest mass, and then there's enough to make a "real" positron, if conditions are right to do it (the rest of the energy goes to make a new electron, plus of course the original electron that carried the energy). [2] SBHarris 01:04, 13 December 2007 (UTC)

[edit] 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)

[edit] 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)

[edit] Coupling to an electromagnetic field - units - mixed conventions

Can someone please clarify the coupling of the momentum to the electromagnetic field. I do not see why the coupling term has a 1/c factor in there. Should it not be:

p -> p - eA

I arrive at this taking the units of the magnetic vector potential to be Tesla meters:

[A] = Tesla L

Only then are the units right ie:

[p] = [eA] = ML/T

Looking at multiple sources to understand this I came to think that there may exists an old/different definition of the magnetic vector potential which makes the units right. It would have to be:

[A] = Tesla L^2/T

I would appreciate if someone could clarify this.

Thanks! —Preceding unsigned comment added by Miobrad (talk • contribs) 15:40, 30 March 2008 (UTC)

I found that in the past the curl of the magnetic vector potential was defined to give the magnetic field strength H. It seems it has something to do with that. But I still have not found exactly where the 1/c is coming from (I am missing a factor of epsilon_0 which could be combined with mu_0 to give the c..). —Preceding unsigned comment added by Miobrad (talk • contribs) 10:55, 31 March 2008 (UTC)

[edit] 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)