State-Observable Duality (John Baez series)

[Careful]

And I do just that, although implicitly, to obtain the values of the second temporal derivative of 4-potential at time point x0 based on the values of 4-potential and its first temporal derivative in the entire 3-space at the same time point x0 (actually, I need this input in an infinitesimally small spatial area to get the second derivative in (the vicinity of) some point).

I believe I correctly posed the Cauchy problem for 4-potential, and it’s an epitome of causality! Let me repeat: I showed that if you know 4-potential and its first temporary derivative in the entire 3D space at time point x0, you can calculate the second derivative of the 4-potential in the same 3D space at the same time point. That means that you can integrate the system of PDE (at least locally). Again, I cannot vouch that there is no faster-than-light propagation in the model, but the dynamics of the model is still local and causal. And again, faster-than-light propagation, if any, is inherited from scalar electrodynamics, so the trick I use does not seem to add any serious problems and may solve some.

But that has nothing to do with causality! You confuse the cauchy problem, which is a characteristic trait of deterministic theories, with causality. Within your theory, you do not only have to show that the cauchy problem is well posed, but that your solution only depends upon ''initial-data'' within the past lightcone (the whole discussion hinges upon what you mean with that)! For example, if I were to pick in Barut's theory half the sum of the retarded and advanced Green's function instead of the retarded one, I would still have a well posed cauchy problem but I doubt it whether there exists any physicist who would call such solution causal (causality is usually only well under control for first order PDE's in a background spacetime - that's the reason why for example causality gets havoc for complex Klein Gordon fields (see http://www.lorentz.leidenuniv.nl/~vanbaal/FT/lect.pdf ), but is restored in QFT where you have a first order ODE). Therefore, we eliminate lot's of initial data which *formally* still would satisfy the causality demand, but which would introduce non-local correlations between fields at different spacetime points (through negative energy solutions, and that is where the first order aspect creeps in). Moreover, I don't see how first order initial data in time are sufficient for you since your final equation contains *fourth* order derivatives in time. So typically causality dies such as happens in the Abraham-Lorentz equation for a point particle electron: people are still writing about that (for a recent good review, see Eric Poisson).

I cannot prove that it is going to work for spinor fields. You cannot prove that it isn’t. Let me emphasize though that almost all results of Schrödinger’s work in Nature for scalar field hold true for spinor field, as shown in http://arxiv.org/PS_cache/arxiv/pdf/1008/1008.4828v1.pdf . In particular, seven out of eight real functions comprising the Dirac spinor function were eliminated from spinor electrodynamics (the Dirac-Maxwell electrodynamics), and the resulting system of equations is overdetermined, which gives a reason to hope that the eighth component can be eliminated as well.

Well, for starters, you would not be able to eliminate the complex numbers. Second, even if you would disregard that, you would have a system of four coupled quadratic polynomials, you may be able to solve even that but it aint going to look pretty .

 

[akhmeteli]

But that has nothing to do with causality! You confuse the cauchy problem, which is a characteristic trait of deterministic theories, with causality.

I stand corrected. However, I did not say that there can be no faster-than-light propagation in the model (see my posts 37 and 58 in this thread). And I did not say there can – after all, this is a relativistic model (certainly, this argument is not conclusive).

 

[Careful]

I stand corrected. However, I did not say that there can be no faster-than-light propagation in the model (see my posts 37 and 58 in this thread). And I did not say there can – after all, this is a relativistic model (certainly, this argument is not conclusive).

I repeat what I said before: the core idea of what you try to do is nice. So make the excercise anew and this time correctly (as well physically as mathematically). You can send it by private mail to me and I will give my opinion on it before you hand it into some editor who is not asleep :-)

 

[akhmeteli]

As you probalby figure, as I understand this, they are not in contradiction, they are complementing each other. Rather than saying that the collapse somehow "violates" the unitary time evolution, I would put it like this:

During the real information input, it makes not sense to refer to expectations, as we collect the REAL feedback. So clearly the real input makes the notion of "expected evolutions" moot. Why refer to "expectations" when we have the real stuff? A real measurement.

On the contrary the unitary evolution is merely an expected extrapolation between the real inputs. So in that respect, for me it's clear that the measurement is far more important than the expected evolution in between measurements. So at the measurement, the unitary evolution is not violated, it's moot.

Sorry, It has taken me quite some time to reply.

So I don't quite see the difference between "violated" and "moot": you question unitary evolution anyway. Note that you clearly prefer measurement to unitary evolution, whereas I tend to think that unitary evolution never stops. So, as I said, we just disagree.

Here I deviate from mainstream though: I don't see the unitary evolution as a description of the actual evolution; I see it as the observers _expected evolution_ and in between measurements/corrections, the observers actions is in consistency with this expectation. The case where the epxected evolutions agree with actual future, corresponds to a form of equilibrium, that we see mainly in controlled, tuned setups where we study subsystems. These are the domains where we QM formalism as we know it is justified.

Some people extrapolate into arbitrary cases without hesitation. I don't belong to those though.

In a sense one can also think of the unitary evolution as geodesic flows, and what an observer sees when this is violated is NOT an "expected non-unitary evolution", they will see a new force, or new interaction. Once the expectation is correctly update the new expected evolution will again be unitary. However this picture will still be constrained by complexity. I don't think it's generaelly possible to always resolve new interactions with larger symmetries that respect unitarity, because that pictures is physically too complex to encode in an arbitrary observer. So what I think happens is NOT a new non-linear QM, I think there is intrinsic loss of decidability and that the models needs to be complemented by another layer darwin-style evolution.

Still this mechanis, can be used by a sufficiently large observer, to product expectations not only on general unitary motion, but of the specific hamiltonians. So I see a potential for first principle explanation of the choice of symmetries we see. But this means that the entire standard model is scale dependent in a deeper sense (a sense much deeper than regular renormalisation)

As far as I can see, the above is just your opinion. What criteria am I supposed to use to decide whether to agree with your opinion or not? On the one hand, I don't think this opinion can be confirmed by experiment, and on the other hand, esthetically this opinion does not appeal to me, so I don't have to agree and I don't want to. Sorry. No offense meant.

 

[akhmeteli]

If I think about what you said, and while I doubt I can say you are directly "wrong" even from my perspective, the best word that comes to my mind is that your reasoning to me seems "irrational".

The one thing where I could imagine "constructing" more like a proof against your view is to look at the action of a system interacting with another system, and when A is informed about B, A might se a collapse; which should genereally change it's action. This would be observable. Thus the collapse event, helps explain interactions. Without this, the entire hamiltonian or lagrangian of the compositve system (including the interaction) must be a priori put it - it can't be "constructed" but piecwise interactions of parts. I see this as a less explanatory.

But again, your view could explain this as well, in a different way that invalidates these types of arguments, but it seems to me like a less systematic and less rational way to approach the problem, that's the main thing I see.

Ultimate I think the discriminator is which mode of reasoning that is the more effective learner. The structural realist is focused more on finding ontological realisties without strategies of HOW, and my view is more focused on the strategy and process whereby progress and fitness is increased, without reference to ontological objective things or what has to be found in the infinite end.

/Fredrik

Dear Fredrik,


Imagine for a moment that we live in a Newtonian world, so the ontological picture is good enough. Your arguments would be equally strong or equally weak in that world as they are now strong or weak in our quantum world, however, limiting ourselves to discussions of measurement results would have made physicists' life in the Newtonian world much more difficult than it could be, at least that is what I think. Therefore, the only reasonable argument in favor of such approach, putting our brain on a leash, would be the impossibility of such ontological picture. So I suspect you are sure there cannot be a satisfactory ontological picture, whereas I am not. For example, in spite of any no-go theorems, the Bohm interpretation offers such a picture. I don't like this picture, but if I had to choose, I would prefer it to your approach.

 

[akhmeteli]

I repeat what I said before: the core idea of what you try to do is nice. So make the excercise anew and this time correctly (as well physically as mathematically). You can send it by private mail to me and I will give my opinion on it before you hand it into some editor who is not asleep :-)

Thank you very much.

 

[Fra]

You're right it's just my choice of reasoning. I merely wanted to make the comments to your discussion for perspective. I find you to be outstandingly polite (bonus points;) so no offense taken of course, no need to even think about it.

So I don't quite see the difference between "violated" and "moot": you question unitary evolution anyway.

To just address this briefly:

In a sense I question unitary evolution yes. But to me, the difference is wether you think

- there is predictable non-unitary evolution
or
- there is a predictable unitary evolution
or like I think
- the expected evolutions are ALWAYS unitary (for reasons that en expectation by definition takes place on a given - fixed/conserved - information), but that these expectations need corrections in a real interaction, and these corrections are not unitary nor decidable.

However since the expected evolutions is observer dependent, it's quite possible for one observer B to described a observer A experiencing collapses with respect to another systems - in a unitary way. There is no conflict here. The only thing that prevents this from been a universal solution is that for B to predict more than A can, is that B has more information - meaning B has to physically be more complex in order to encode and process more information. When you are considering a case where both A and B are inside observers, the information bound on B suggestes that this breaks down.

I think your idea is that there is always a more complex perspective from which the evolution IS truly unitary. In the above sense, I accept that idea. But as I see it, this scheme is limited in the sense I tried to put forward. I could imagine that no-go theorems or other things may be constructable or may even exists that check these things, but I haven't made any it seems intuitive enough to me.

So as I see it the "reason" for unitary expectations inbetween mesurements is almost on par with why in the absence of external forces a systems follows a geodesic. The prior information defines our geodesics. But the geodesics are a funtion of information, which then also updates when information changes - but this information process also contains inertia.

/Fredrik

 

[Fra]

So I suspect you are sure there cannot be a satisfactory ontological picture, whereas I am not.

Actually I make no judgement on that. So on this particular thing, perhaps we can agree.

My view does not actually ban it's existence. It's that my view doesn't focus on what exists and what doesn't in the ontological sense, I focus on what we can infer. It is in fact possible! that incomplete inferences are CONSISTENT with an ontological objective realiy in the sens you probably mean - so I do not ban in, but in my terminology such a thing would be a "conincidence", or equilibrium. And in the effective sense that's how objectivity as we know it is inferred. All our experimental researtch all give "incomplete" pictures, but they all seem to point towards something that is a objective consistence. At least in the steady state sense - we still lack argumetns to claim anything such as "eternal law" etc. That is just wild unjustified extrapolations.

But as I see it, this is not something we need to assume. It doesn't mean it's impossible though. But assuming it, and using it as a constrain risk making our inference irrational (or at least it's how I see it).

But indeed it's just my private non-authorative opinon.

/Fredrik

 

[Eckard]

Whether this is more or less convenient is a secondary question, but in principle you can do without complex numbers (or pairs of real numbers) in quantum theory.

I quoted in http://www.fqxi.org/community/forum/topic/833 those who claimed that complex numbers are indispensable in quantum theory while R is sufficient in principle for all other disciplines of physics. According to my own reasoning, even R+ should be sufficient as to describe reality, in principle. I faced John Baez angrily denying that without any explanation.

The reason for me to ask for help in this thread is a hint by an other fqxi contest contestant. He told me that John Baez clarified a matter that I am suggesting: reinstating more precisely the Euclidean notion of number as the measure of distance from zero and not as introduced by Dedekind a point. Who can point me to such clarification?

I also suggested to reinstate the old notion of continuum, which was still defined by Charles S. Peirce as something every part of which has parts. I already gave some reasons for both suggestions.

What I learned from this thread is the sentence: "Alas, nothing much came of this."
I hope my very basic suggestions are better and will help to get rid of arbitrariness and ambiguity.

 

[A. Neumaier]

I quoted in http://www.fqxi.org/community/forum/topic/833 those who claimed that complex numbers are indispensable in quantum theory while R is sufficient in principle for all other disciplines of physics.

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?


Here is also some information about an earlier part of the thread:

I never heard about Barut.

Asim Barut was a very creative mathematical physicist. He wrote a nice book
Barut and Raczka,
Theory of group representations and applications,
Warszawa 1980.
from which I learned a lot about how to use representation theory in physics. He also worked on alternatives to QED, and found a reformulation which predicted a Lamb shift to some accuracy. But it seems to differ at high orders, and never was picked up by mainstream physicists. See the report
http://streaming.ictp.trieste.it/preprints/P/87/248.pdf
and the papers
Phys. Rev. A 34 (1986), 3500–3501
Phys. Rev. A 34, (1986) 3502–3503
Int. J. Theor. Phys. 32 (1993), 961-968.

 

[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 :-(