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Sergey Fedosin: on 10/4/12 at 5:49am UTC, wrote If you do not understand why your rating dropped down. As I found ratings...

Sergey Fedosin: on 10/2/12 at 8:46am UTC, wrote After studying about 250 essays in this contest, I realize now, how can I...

Benjamin Dribus: on 10/2/12 at 6:13am UTC, wrote Dear Andrew, Thanks, this helps a lot! Take care, Ben

Andrew Norton: on 9/29/12 at 2:20am UTC, wrote Dear Ben Thank you for the Qns.... re. Q1 and Q2: The solutions in the...

Benjamin Dribus: on 9/26/12 at 18:39pm UTC, wrote Dear Andrew, Thanks for the fascinating read! This is indeed (as far as I...

Andrew Norton: on 9/22/12 at 8:40am UTC, wrote Dear Andrey Thank you for the reference to your paper. Yes, it may well...

Andrew Norton: on 9/22/12 at 8:08am UTC, wrote (ctd. from last post, due to web form error) and absorb infinite energy...

Andrew Norton: on 9/22/12 at 7:46am UTC, wrote Hi Ken Only the short path-memory experiments support the retrocausal...


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Dissolving Quantum Paradoxes
The impossibility of building a perfect clock could help explain away microscale weirdness.

May 21, 2019

CATEGORY: Questioning the Foundations Essay Contest (2012) [back]
TOPIC: On Quantum Foundations and the Assumption That the Lorentz Equation of Motion Defines a Fully Resolved Electrodynamics by Andrew H. Norton [refresh]
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Author Andrew H. Norton wrote on Aug. 30, 2012 @ 12:08 GMT
Essay Abstract

The derivation of an equation of motion (EOM) for a particle-like field source is still an open problem for a general classical field theory. However, for the case of Maxwell field theory formulated in Minkowski spacetime, the answer is believed to be known: up to small radiative corrections, the worldline of a charged particle satisfies the Lorentz EOM. This belief is based on the implicit assumption that the mathematical methods used to derive the EOM from the field theory lead to an EOM that is fundamental in the sense that the charge-center of the particle is not very much smaller than the spatial resolution of the dynamical description, so that the EOM defines, as far as is possible, a fully resolved dynamics. Standard methods for deriving an EOM do not satisfy the above assumption. They are resolution limited, leaving open the possibility that the derived EOM is not fundamental. Typically, this is due to use of a point-multipole (eg., Lienard-Wiechert) approximation for the field of the particle that becomes applicable only at some distance from the particle. In the case of Maxwell field theory, it can be proved that the Lorentz EOM is not fundamental--it provides only a lowest resolution approximation to an electrodynamics that exhibits multiple length scales. Evidence for a multiscale electrodynamics can already be found in the physics literature, for example, in Hestenes' Zitterbewegung interpretation of Dirac electron theory. In this essay I collect together this evidence in the form of an ansatz for the quantum physics of the electron. This ansatz has been used to derive a 4th order EOM for the electron that not only models electron spin dynamics, but remains well defined throughout the full range of length scales over which quantum electrodynamics applies.

Author Bio

Andrew Norton has had research positions in numerical relativity (University of New England, Australia, and the University of Canberra), nonlinear optics (University of Sydney), and electrodynamics and photonic crystals (University of Technology, Sydney). His last position was a 2 year postdoc at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam, Germany. Since April 2011 he has been "self funded" while writing up research he started in Germany. He currently resides in Sydney and will be looking to rejoin paid academia in the not too distant future.

Download Essay PDF File

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Vladimir F. Tamari wrote on Sep. 2, 2012 @ 14:56 GMT
Dear Dr. Norton

I tried to undertand your paper, but it was too technical for my understanding. Reading between the lines I could feel how difficult it is to calculate the behavior of particles using the available formulations. Is Nature itself that complicated? Perhaps I am too naive, but I feel that General Relativity is so complicated because it is framed in the assumptions of Special...

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Author Andrew H. Norton replied on Sep. 2, 2012 @ 22:58 GMT
Dear Vladimir

Yes, I think sometimes Nature really is that complicated. However, one of the reasons I like general relativity is that Nature also seems to be very geometrical. So even when the equations are so huge that they can only be studied with computer algebra, there is still often some relatively simple picture that goes with it. Your water-clock representation for time reminded me of something I came across on the web last week that I thought was really cool (I too like science-based artwork),

The Water Clocks of Bernard Gitton


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Andrey Akhmeteli replied on Sep. 4, 2012 @ 02:17 GMT
Dear Dr. Norton,

Thank you for an interesting essay.

Your fourth-order EOM for electron may have something in common with my recent result: the Dirac equation in electromagnetic field is generally equivalent to a fourth-order PDE for just one component of the Dirac spinor (
2303_1.pdf , published in J. Math. Phys.) I also argue elsewhere that electromagnetic field, rather than the wavefunction, can be the guiding field in the de Broglie - Bohm interpretation.

Thank you

Andrey Akhmeteli

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Vladimir F. Tamari replied on Sep. 5, 2012 @ 07:00 GMT
Thanks Author A. for the fascinating page about the water clock.

Hmm Nature appears complicated and may well be so, so it needs all the help it can get from smart physicists like Einstein to simplify its workings...NOT!

Best wishes


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M. V. Vasilyeva wrote on Sep. 6, 2012 @ 20:23 GMT
Andrew, I found your essay very interesting and very well written, even though, admittedly, it was a bit over my head, since I am just an analyst and have a top down approach to physics, rarely bothering with details. I gave you a high grade all the same and think you will do well.

My essay is a fun read. I wonder what you think: The Nature of Space.

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Author Andrew H. Norton replied on Sep. 9, 2012 @ 02:56 GMT
Thank you for your comments. I enjoyed reading your essay and left some feedback there for you. Andrew

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Sergey G Fedosin wrote on Sep. 11, 2012 @ 09:59 GMT
Dear Andrew,

You use simple charged sphere model of the electron. There is another model of electron where it is in the form of disk. You can see it in § 14 of the book: The physical theories and infinite nesting of matter. Perm: S.G. Fedosin, 2009-2012, 858 p. ISBN 978-5-9901951-1-0. If to suppose the disk model is more real can you use it for calculations?

Sergey Fedosin

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Hoang cao Hai wrote on Sep. 19, 2012 @ 14:35 GMT

Very interesting to see your essay.

Perhaps all of us are convinced that: the choice of yourself is right!That of course is reasonable.

So may be we should work together to let's the consider clearly defined for the basis foundations theoretical as the most challenging with intellectual of all of us.

Why we do not try to start with a real challenge is very close...

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Member Ken Wharton wrote on Sep. 21, 2012 @ 23:56 GMT
I've finally had time to go through your essay in detail, and have been pleased to see at how many points our thinking intersects. But I'm still trying to figure out where you see the retrocasuality in the bouncing droplet experiments -- although I'm hoping you're right, as it would shift these experiments from justifying the Bohmians to justifying us retrocausalists! (I also found your point about deBroglie intriguing... is anything written concerning the distinctions between deBroglie and Bohm when it comes to this issue?)

Some questions about your essay:

1) In 2.1, you talk about the "quantum wavefunction" as an epistemic construct, which is all good, but I assume you mean the standard wavefunction psi, used by everyone? If so, how can all the trajectories that make it up conform to the future measurement? If an electron will be measured at position X, sure, I can build an epistemic construct out of all possible paths that might take that electron to X, but that construct will look nothing like psi, as psi doesn't "know" about X.

2) You make it sound like it is an objective fact whether a given potential is retarded or advanced, but this would only be possible if one could associate particular "portions" of a field with particular particles. Given that a given field potential might be emitted from one particle and absorbed by another, or perhaps even never interact with a particle at all, means that this retarded/advanced distinction is somewhat subjective (you can always turn one into the other by adding a free field). That said, your point that the particle emits and absorbs the same energy/momentum is more objective -- but since this isn't exactly correct (as pointed out in 2.6), how is it possible to objectively impose the needed condition?

3) Can you further explain to me the source of the incoming (converging) wave in the bouncing droplet experiments? I'm particularly interested to understand what happens if the droplet is in the vicinity of a barrier (as it's passing through a double-slit for example.)

4) What are your thoughts concerning double-slit experiments for *photons*? Will it be a similar description to electrons, or wildly different?

Please do keep me on your mailing list as you develop all this further -- I'm quite interested to learn more!

All the best,


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Author Andrew H. Norton replied on Sep. 22, 2012 @ 07:46 GMT
Hi Ken

Only the short path-memory experiments support the retrocausal picture (footnote 3 in paper). But on the other hand, I don't think any of the bouncing droplet experiments support Bohmian mechanics. The Qn is whether the pilot-wave is a phase wave (Bohmian Psi), or a standing wave that is distinct from the Psi wavefunction. In the bouncing droplet system, the pilot-wave is a standing wave.

If pilot-waves are standing-waves, then retrocausality is involved just because standing-waves have an advanced component. Of course, real bouncing droplets only simulate having an advanced field. But they do it well enough to show how things like double-slit experiments could work for electrons.

Everyday standing-waves (drum skins, guitar strings, skipping ropes, etc) arise because of reflections from boundaries. Retrocausality is irrelevant for these. But the standing wave surrounding a "deBrogle clock" exists only because the "ticks" of the clock propagate on both the forward and backward light-cones. The "inward" waves are not reflections from boundaries----they are "retro-sourced" directly by the particle.

For distinctions between de Broglie's theory of the Double Solution and Bohmian mechanics see, for example, de Broglie ref [25], Chapter 8. However, I'm pretty sure de Broglie never mentions standing waves. Likewise, his thesis is about phase waves, not standing waves (the reference for an English translation is given in [23]).

I imagine the wavefunction (a phase-wave) applying for an ensemble, with standing-waves generated by individual particles. The connection would be something like Huygens principle but I haven't given much thought to it yet. In fact, my main point for Section 2.1 was that if there is a URT, then I can ignore all the messy and difficult problems associated with interpreting the wavefunction until we get the dynamics of an individual electron right.

Re. your qns:

(1) I meant the future measurement process, not the measurement result.

I had in mind natural boundary conditions for the experiment. Eg. the electron hits a screen => the worldline terminates in the plane x(final)=0, leaving y(final) and z(final) as free boundary conditions for the variational problem. If the Lagrangian is 2nd order then the natural boundary conditions associated with the free data are non-trivial.

It seems proper that only those worldlines that satisfy whatever (if any) future boundary conditions that are imposed by the experiment should count towards working out probabilities. But I don't know what that implies, if anything, for the wave-function.

(2) Let A_ret and A_adv be the retarded and advanced Lienard-Wiechert fields for a point-charge current source in Maxwell theory. Then in free space (ie., no matter, no other currents, no boundaries) the potential due to the electron (in its point-like approximation) has the general form

A = (A_ret + A_adv)/2 + rho (A_ret - A_adv)/2 .

The 2nd term here is a free radiation field (everywhere non-singular and satisfying the source-free Maxwell eqns), and rho is an arbitrary constant.

If the electron is not in free space then one can expect A to be scattered and reflected, giving rise to an additional radiation field A_env, due to the response of the environment to the presence of the electron. If nonlinear materials are involved, then A_env will not even be linear in the charge q of the electron, nevertheless A_env --> 0 as q --> 0, so A_env should be regarded as part of the potential of the electron.

There are situations (eg., in a photonic crystal) where A_env needs to be taken into account because it radically changes how the electron radiates. These are special cases though. The best we can do if deriving an EOM for an electron in a given external potential A_ext, is to assume that A_env = 0.

When the approximation A_env = 0 is not good enough, then I think no EOM can be derived (though perhaps an iterative procedure may be applicable, starting with A_env=0). Possibly, one needs to solve the full Maxwell field equations with moving boundary conditions for a worldtube surrounding the electron (as proposed in Kijowski's formulation of electrodynamics -- GRG 26, 1994, 167-204).

Assuming that the approximation A_env = 0 is OK, the only freedom in A is our choice for the constant rho. (For example, the Lorentz-Dirac equation is derived with rho=1, which gives A = A_ret.)

If the electron has a persistent circular spin motion then it would emit infinite energy (to future null infinity) for rho>0 and absorb infinite energy (from past null infinity) for rho

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Author Andrew H. Norton wrote on Sep. 22, 2012 @ 08:08 GMT
(ctd. from last post, due to web form error)

and absorb infinite energy (from past null infinity) for rho lessthan 0.

So Maxwell theory says rho=0 for all electrons always (if they have spin as suggested). So we need something better than linear Maxwell theory----a soliton theory that allows rho to change from its free particle value of rho=0 just while the electron emits or absorbs a photon.

How rho changes (or even if rho=rho(t) is a good idea) will depend on the soliton theory, which we don't actually know. The best clue we have is that the free electron spin and mass must be "solitonic" ---- ie. be restored to their proper values after any interactions.

(3) In the droplet experiments, the ultimate source of the incoming wave is the power delivered by the vertical driving force that oscillates the bath. More abstractly, one can think of the incoming wave as a reflection of the outgoing wave from a nonlinear medium with gain. Probably best to read [22].

Figs 5 and 6 of [22] are interesting. In fig. 6 you can see that it really is a standing wave that is generated.

In [23] a nice figure is given that shows numerical simulations for droplet paths as they go through a slit.

(4) I'm guessing that photons are just the e.m. waves that are radiated while electrons make transitions between limit cycles. In that case, there is nothing quantized about the photon that does not follow from the dynamics of the electrons that are involved in emitting and/or absorbing the photon. So I would have to say that double-slit for photons = double-slit for waves.



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Member Benjamin F. Dribus wrote on Sep. 26, 2012 @ 18:39 GMT
Dear Andrew,

Thanks for the fascinating read! This is indeed (as far as I know) an assumption few have even thought to challenge. Your challenge is all the more interesting in consequence. Let me ask a few questions.

1. I am trying to figure out exactly what you are suggesting concerning U(1) symmetry. On page 2, you write, "If so, then this underlying theory could have important technological applications: it says that the spin phase of an electron can be experimentally detected and manipulated. This possibility is precluded in quantum theory by the exact U(1) phase symmetry that is built into the complex formalism, and that corresponds to an approximate spin-phase invariance in the underlying realistic theory." This seems to suggest that if your "underlying realistic theory" would preclude U(1) symmetry. However, you refer to the "realistic electron theory" as "underlying QED." Are you referring to QED in the general sense (as in "a quantum theory of electrodynamics"), or are you referring to "the" current theory called QED, which involves U(1) symmetry? I must assume the former, since you expect technological implications.

2. If U(1) symmetry is not exact, then what replaces it?

3. What effects would you expect in the field of quantum computing?

If you want to know why it is important for me to know the status of the gauge symmetry in your approach, it is because my own ideas involve deviations from symmetries expected in conventional physics. Of greatest relevance for me are the external symmetries (from the Poincare group, etc.), but similar deviations involving internal symmetries would also be interesting. If you are interested, I discuss this in my essay here On the Foundational Assumptions of Modern Physics. If you do read it, please bear in mind that I do not necessarily disagree with your use of "retrocausation," since I mean something slightly different than usual by the "binary relation underlying the causal order."

Good luck with the contest, and take care,

Ben Dribus

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Author Andrew H. Norton replied on Sep. 29, 2012 @ 02:20 GMT
Dear Ben

Thank you for the Qns....

re. Q1 and Q2:

The solutions in the underlying realistic theory (URT) are worldlines for the electron charge-center. These are spacetime helices that spiral around an average worldline (the center-of-mass worldline for the electron). The spacetime helix describes a circular spin motion in some spin-plane. Free initial data for the 4th order EOM consists of a center of mass position, a spin-plane direction, a spin phase angle, and a boost to the momentum rest frame of the electron. There are 3 cases to consider:

1. All interaction length scales for the problem are large compared with the size of the helix (which has diameter = Compton wavelength). In this case the initial data for the spin direction and the spin phase are more-or-less irrelevant---solutions with different spin data still spiral around the same average worldline and this average worldline is well described by the Lorentz EOM.

2. In this case at least some interaction length scales for the problem are comparable to the electron Compton wavelength. Typically, the average worldline is no longer well described by a soln of the Lorentz equation, but the average worldline is still more-or-less independent of initial spin phase angle. Changing the initial phase angle just twists the helical solution around its average worldline onto another helix that has approximately the same center-of-mass worldline. This is the U(1) invariance of QM (ie, of Schrodinger eqn, Dirac eqn, and of QED). In QM this approximate U(1) invariance is taken to be exact and built into the complex formalism. This is a simplifying approximation that is almost always a good approximation. By encoding spin phase as a complex phase, real valued results are by definition spin phase invariant.

3. The atypical case is that the experimental conditions are contrived so that the center-of-mass worldline of the electron becomes sensitive to the spin phase initial data. In such an experiment, the assumed U(1) invariance of QM is broken so that it becomes apparent that this invariance is only an approximation. An example is the experiment described in ref [7], designed to detect the helical structure of the worldline of the electron charge-center.

re. Q3. An interesting Qn. I suppose knowing the electron dynamics could be useful in designing a quantum computer based on electron spins, but I don't see spin phase being manipulated any time soon. Perhaps the most important contribution will be just a new way to think about maintaining a quantum state in terms of temporal isolation and retrocausality.

I hope you find something here that may be of use in your research.



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Member Benjamin F. Dribus replied on Oct. 2, 2012 @ 06:13 GMT
Dear Andrew,

Thanks, this helps a lot! Take care,


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Sergey G Fedosin wrote on Oct. 2, 2012 @ 08:46 GMT
After studying about 250 essays in this contest, I realize now, how can I assess the level of each submitted work. Accordingly, I rated some essays, including yours.

Cood luck.

Sergey Fedosin

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Sergey G Fedosin wrote on Oct. 4, 2012 @ 05:49 GMT
If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is
was the quantity of people which gave you ratings. Then you have
of points. After it anyone give you
of points so you have
of points and
is the common quantity of the people which gave you ratings. At the same time you will have
of points. From here, if you want to be R2 > R1 there must be:
In other words if you want to increase rating of anyone you must give him more points
then the participant`s rating
was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

Sergey Fedosin

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