Dirac equation as one equation for one function

[A. Neumaier]

I insist that the set of solutions for the full covariant spinor ψψ\psi does not depend on the choice of $\xi$.

Yes but this is trivial since you start with a covariant equation and reformulate it algebraically. Of course the set of solutions must be the same as that of the original equation.

In any case, I have nothing new to say. Good luck with your notion of covariance!

 

[akhmeteli]

Yes but this is trivial since you start with a covariant equation and reformulate it algebraically. Of course the set of solutions must be the same as that of the original equation.

Trivial or not, it is still relativistically covariant, whereas there is no meaningful covariance in your equation.

In any case, I have nothing new to say. Good luck with your notion of covariance!

Thank you very much for your time and input.

 

[A. Neumaier]

it is still relativistically covariant

But non manifestly so, since you only get one linear combination of the spinor, and only the full spinor is covariant. Manifest is only the spurious covariance that is also manifest in my nonrelativistic equations.

 

[akhmeteli]

But non manifestly so

You can argue that the word "manifestly" in the title of my work is not appropriate, but then maybe FODE is in a good company, as one can also say that the original Dirac equation is not manifestly covariant: first, if it is manifestly covariant, then why authors of most books I've read (say, Itzykson, Zuber) believe it is necessary to prove relativistic covariance of the original Dirac equation; second, to prove that, you need to know the not so simple Lorentz-transformation properties of the Dirac spinor (and these properties are not manifest in the original Dirac equation).

since you only get one linear combination of the spinor, and only the full spinor is covariant.

Let me make a remark on terminology first: we do not discuss linear combinations of spinors, as any linear combination of spinors is also a spinor (four complex numbers per spacetime point), unlike components of spinors in FODE (one complex number per spacetime point), which are scalars (for a fixed eigenvector).

Then let me note that one can consider FODE not as an equation for one component of the spinor, but as an infinite set of equations for all components of the spinor, and I would say the set of all components of the spinor is covariant. And again, all these equations have the same set of solutions for the spinor.

Manifest is only the spurious covariance that is also manifest in my nonrelativistic equations.

As I said, the comparison of FODE and your equation seems downright frivolous, as the set of solutions of your equation changes with the change of your vector $\xi$.

 

[ftr]

akhmeteli , does your equation give a different answer/perspective on the problem of relativistic particle in a box. Thank you.

 

[akhmeteli]

akhmeteli , does your equation give a different answer/perspective on the problem of relativistic particle in a box. Thank you.

I am afraid the equation of my work (the fourth-order Dirac equation) cannot be derived for the box potential, as electromagnetic field inside the box identically vanish, so the "transversality" condition (requiring that some component of electromagnetic field does not vanish identically) is not satisfied. One can criticize the fourth-order Dirac equation for failing to describe a free particle (or a particle in a box), but this does not seem to be a real problem, because if you have at least one charged particle in the Universe, you have electromagnetic field everywhere, and, however weak that field may be, one can derive the fourth-order Dirac equation.

As for some new perspective... Let me note that the equation is of the fourth order, so one may be tempted to consider (for example, for some specific electromagnetic field) an analogy with elasticity equations.

 

[ftr]

one can derive the fourth-order Dirac equation

So what is this real function represent. Does it describe the electron as electromagnetic source or something like that?

 

[akhmeteli]

So what is this real function represent. Does it describe the electron as electromagnetic source or something like that?

Let me first explain how I understand your question (if I am wrong, please let me know).

So I showed that the Dirac equation is generally equivalent to a fourth-order equation for one of the components of the Dirac spinor, and this component can be made real by a gauge transform. So I assume that "this real function" in your question is this component after the gauge transform.

So what does it represent? Difficult to say. Note that you can choose the component pretty arbitrarily, and no component seems any better than others, so if one of the components represents something specific, then what do all other represent? One can say, however, that this component represents a solution of the Dirac equation, and this seems important and unexpected, as it means that a charged particle can be represented (or described) by a real function. While Schroedinger made such conclusion for the Klein-Gordon particle long ago, expanding this conclusion to the Dirac particle was not trivial.

So I am not sure what the real function represents, but I cannot resist the temptation to speculate:-) As a basis for speculation, I am going to use the results for the Klein-Gordon particle. It turned out that the Klein-Gordon field, after it is made real by a gauge transform following Schroedinger, can be algebraically eliminated from the equations of scalar electrodynamics (describing Maxwell field, Klein-Gordon field, and their minimal interaction). Furthermore, the resulting equation for the electromagnetic field describe its independent evolution (see, e.g., my article published in European Physical Journal C, Section 2). Thus, if you wish, the Klein-Gordon particles can be considered "ghost" particles:-) Can one make a similar conclusion for the Dirac particle and spinor electrodynamics? Yes, but so far it was done in a less satisfactory manner (see Section 3 of the same article. I am trying to improve the result, and some interim material is presented in a recent arxiv article). And, of course, the "ghost" interpretation is not the only one possible.

 

[ftr]

"ghost" interpretation

I think your "a large (infinite?) number of particles" is more interesting.

Can you extend the idea to two particle Dirac equation

https://en.wikipedia.org/wiki/Two-body_Dirac_equations

 

[PeterDonis]

Moderator's note: This thread was closed for review, but has been reopened.

 

[akhmeteli]

I think your "a large (infinite?) number of particles" is more interesting.

Can you extend the idea to two particle Dirac equation

https://en.wikipedia.org/wiki/Two-body_Dirac_equations

Yes, this interpretation may be interesting. I don't have any ideas on adapting the results to the two-body Dirac equations. Transition to many-particle theories is considered in Section 4 of the EPJC article. I mentioned elsewhere that coherent fermionic states (K.E. Cahill and R.J. Glauber. Phys. Rev. A, 59:1538, 1999.) can be used, but I don't have a clear idea. You may also wish to see what Barut did for two electrons (see my post https://www.physicsforums.com/threads/qed-lagrangian-lead-to-self-interaction.245242/#post-1806603 )