State-Observable Duality (John Baez series)

[dextercioby]

As a side comment, A. Barut also wrote a splendid little book on classical field theory and electrodynamics. I'm quite surprised that someone putting preprints on arxiv on field theory hasn't heard of Barut or his famous group theory book.

 

[Eckard]

It might be sufficient in principle but forcing physics into the Procrustes bed of banning C would make many things very tedious - from the Fourier transform to creation and annihilation operators. How would one write the canonical commutation relation [q,p]=i hbar without using complex numbers?


As already the title of my essay "Continuation Causes Superior but Unrealistic Ambiguity" indicates, C is excitingly superior to R which is on its part superior to R+ while R is once redundant, and C is twice redundant, i.e. fourfold copy of reality if we obey the undeniable property of all measurable functions of time to be restricted to what already is or at least will become past.
From this restriction follows that R+ and cosine transform are sufficient, in principle.

So called "verschaffte Quantisierungsbedingung" can be written as 2 pi pq/h - 2pi qp/h = i. Planck's constant h has nothing to do with i and nothing with non-commuting matrices.

Both the imaginary unit and the property to not commute are redundant artifacts due to complex Fourier transformation from one-sided reality into complex domain with Hermitian symmetry. Notice: Fourier transformation requires arbitrary analytic continuation, and it is further based on an arbitrary omission. This inevitable implies redundancy and ambiguity, which would vanish with correct return into the one-sided and real domain of reality.

Once again, who can point me to a clarification by John Baez concerning the notions number and continuum?

 

[akhmeteli]

As significant progress (relevant to this thread) has been made recently, let me offer a brief followup.

1) My article that I quoted throughout this thread was published in IJQI at last (see the postprint at http://akhmeteli.org/akh-prepr-ws-ijqi2.pdf )

2) The 4th order PDE for one real function, which is generally equivalent to the Dirac equation, was written in an explicit form and published in the Journal of Mathematical Physics (http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf )

3) The spinor field was eliminated from the equations of spinor electrodynamics (the Dirac-Maxwell electrodynamics) (with some important caveats), and the results of the IJQI article for scalar electrodynamics were extended to spinor electrodynamics (see http://arxiv.org/abs/1108.1588). One can make different conclusions from these results though: spinor electrodynamics can be replaced by equations for either: a) complex electromagnetic 4-potential, or b) real electromagnetic 4-potential plus one real function (the gradient of which equals the imaginary part of the complex electromagnetic 4-potential of a) ). There is still also a possibility that the above mentioned caveats can be removed altogether, but this has not been proven right or wrong yet.

 

[marcus]

Congratulations on your publication of such interesting materials. Thanks for telling us about the IJQI paper. I think I recognize some PF people in the acknowledgments!

 

[akhmeteli]

Congratulations on your publication of such interesting materials. Thanks for telling us about the IJQI paper. I think I recognize some PF people in the acknowledgments!

Thank you for your kind words

 

[qsa]

As significant progress (relevant to this thread) has been made recently, let me offer a brief followup.

1) My article that I quoted throughout this thread was published in IJQI at last (see the postprint at http://akhmeteli.org/akh-prepr-ws-ijqi2.pdf )

2) The 4th order PDE for one real function, which is generally equivalent to the Dirac equation, was written in an explicit form and published in the Journal of Mathematical Physics (http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf )

3) The spinor field was eliminated from the equations of spinor electrodynamics (the Dirac-Maxwell electrodynamics) (with some important caveats), and the results of the IJQI article for scalar electrodynamics were extended to spinor electrodynamics (see http://arxiv.org/abs/1108.1588). One can make different conclusions from these results though: spinor electrodynamics can be replaced by equations for either: a) complex electromagnetic 4-potential, or b) real electromagnetic 4-potential plus one real function (the gradient of which equals the imaginary part of the complex electromagnetic 4-potential of a) ). There is still also a possibility that the above mentioned caveats can be removed altogether, but this has not been proven right or wrong yet.

you said this in your paper

may be important for interpretation of quantum theory

would you like to tell us about it ?

 

[akhmeteli]

you said this in your paper

may be important for interpretation of quantum theory

would you like to tell us about it ?

It's a long story...

I just obtained some mathematical results, which do not determine some specific interpretation, but can be used in several different interpretations. For example, in my IJQI article I wrote:

"For example, in the Bohm (de Broglie-Bohm) interpretation (Refs. 5;6;7), the electromagnetic field can replace the wave function as the guiding field. This may make the interpretation more attractive, removing, for example, the reason for the following criticism of the Bohm interpretation: "If one believes that the particles are real one must also believe the wavefunction is real because it determines the actual trajectories of the particles. This allows us to have a realist interpretation which solves the measurement problem, but the cost is to believe in a double ontology. 8"

So the results may be useful for the Bohm interpretation. On the other hand, some people may wish to use my results to adopt an interpretation without matter field altogether, just electromagnetic field. Yet another, quite different interpretation is possible (http://arxiv.org/abs/quant-ph/0509044, second paragraph on p. 4).

In general, the results seem to enable local realistic interpretations (see the discussion of the Bell theorem, using other people's arguments, in Section 5 of the IJQI article).

 

[qsa]

I will have more to say about the interprtation. But can you say anything about the nature of spin.

 

[akhmeteli]

I will have more to say about the interprtation. But can you say anything about the nature of spin.

Beyond some formal results, I am not sure I can say something new about the nature of spin. I think this issue depends on the interpretation, and, as I said, my results do not fix one and only interpretation (though they can make some of the interpretations much more attractive).

On the other hand, some of the formal results seem most relevant to spin, for example, the surprising fact that the Dirac equation (which is a greatest, if not the greatest source of information on spin 1/2) is equivalent (up to "transversality") to just one 4th order PDE for just one function (complex or, if you don't mind a fixed gauge, real). Or the fact that the spin 1/2 field can be naturally eliminated in some sense from spinor electrodynamics, turning the latter into a system of equations for a complex electromagnetic 4-potential.

Maybe somebody else will be able to mine more information about spin and charge, using my results.

 

[qsa]

here is what you claim in your paper

http://arxiv.org/PS_cache/quant-ph/pdf/0509/0509044v1.pdf

It seems that there may exist a somewhat different
interpretation of real charged fields: the one-particle Ψ-
function may describe a large (infinite?) number of particles moving along the above-mentioned trajectories. The
total charge, calculated as an integral of charge density
over the infinite 3-volume, may still equal the charge
of electron. So the individual particles may be either
electrons or positrons, but all together they may be regarded as one electron


this is the same picture which I get from my own idea (though it is derived from totally different concept). but I am surprised (but not too much) that nobody wants to touch such conclusions, although in QED such picture is accepted as long as you call them virtual i.e. not real. but I guess it is a confusing strange conclusion. I do like to hear some opinions.

 

[akhmeteli]

here is what you claim in your paper

http://arxiv.org/PS_cache/quant-ph/pdf/0509/0509044v1.pdf

It seems that there may exist a somewhat different
interpretation of real charged fields: the one-particle Ψ-
function may describe a large (infinite?) number of particles moving along the above-mentioned trajectories. The
total charge, calculated as an integral of charge density
over the infinite 3-volume, may still equal the charge
of electron. So the individual particles may be either
electrons or positrons, but all together they may be regarded as one electron


this is the same picture which I get from my own idea (though it is derived from totally different concept). but I am surprised (but not too much) that nobody wants to touch such conclusions, although in QED such picture is accepted as long as you call them virtual i.e. not real. but I guess it is a confusing strange conclusion. I do like to hear some opinions.

Just wanted to say that I have published another article (online so far) in the European Physical Journal C - http://link.springer.com/article/10.1140/epjc/s10052-013-2371-4 , and it includes the passage you quoted along with the results described in my post 73 in this thread, item 3. It is my understanding that there will be open access to the article in a few days. Meanwhile you can look at the preprint version of the article (http://arxiv.org/pdf/1108.1588v3.pdf ). The preprint version differs from the journal article in two respects: first, some errors have been corrected in the journal article, e.g., a nasty typo in the metric tensor; second, the preprint version reflects some further development (not much of it though, as I have been busy with other (experimental) projects over the last two years): I added two longish paragraphs in the conclusion on the possible modification aimed at inclusion of Barut's self-field electrodynamics (SFED). This is important from the point of view of comparison with experiments, as SFED seems to reproduce QED effects with high precision. Probably, some additional modifications will be needed to fully include SFED.

 

[akhmeteli]

I noted earlier that in a general case three out of four components of the Dirac spinor function can be algebraically eliminated from the Dirac equation, and the remaining component can be made real using a gauge transform.

An update: http://arxiv.org/abs/1502.02351

Abstract:

Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac spinor function. This was done for a specific (chiral) representation of gamma-matrices and for a specific component. In the current work, the result is generalized for a general representation of gamma-matrices and a general component (satisfying some conditions). The resulting equivalent of the Dirac equation is also manifestly relativistically covariant and should be useful in applications of the Dirac equation.

 

[marcus]

Reference [8] in the April 2013 paper ( http://arxiv.org/pdf/1108.1588v3.pdf ) to work by "nightlight" appears to have a garbled URL. I think the intended URL might be:
http://sci.tech-archive.net/archive/sci.physics.research/2005-08/msg00455.html

however this does not get me to anything I can read.

 

[akhmeteli]

Reference [8] in the April 2013 paper ( http://arxiv.org/pdf/1108.1588v3.pdf ) to work by "nightlight" appears to have a garbled URL. I think the intended URL might be:
http://sci.tech-archive.net/archive/sci.physics.research/2005-08/msg00455.html

however this does not get me to anything I can read.

http://sci.tech-archive.net/Archive/sci.physics.research/2005-08/msg00455.html
(case matters!)
My sincere apology to you and nightlight - I screwed up with LaTex :-(