Talk:Dirac equation

From Wikipedia, the free encyclopedia

	Physics Portal This article is within the scope of WikiProject Physics, which collaborates on articles related to physics.
B	This article has been rated as B-Class on the assessment scale.
Top	This article is on a subject of Top importance within physics.
Help with this template Comments: edit – history – watch – refresh Contents 1 Low Importance 2 Moved from article 3 Noninteracting sea? 4 Electromagnetic Interaction 5 General Relativity 6 Gamma matrices 7 Upper and lower psi functions 8 Dirac bilinears 9 Where to write more about spinors? 10 Can all you math whizes [edit] Low Importance Who rated this article "Low Importance" ? It should of course be "Top Importance".

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)

[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)

[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] General Relativity

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

[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)