State-Observable Duality (John Baez series)

[MTd2]

"I’m writing a paper called “Division algebras and quantum theory”, which is mainly about how quantum theory can be formulated using either the real numbers ℝ, the complex numbers ℂ, or the quaternions ℍ — and how these three versions are not really separate alternatives (as people often seem to think), but rather three parts of a unified structure.

This is supposed to resolve the old puzzle about why Nature picked ℂ when it was time for quantum mechanics, while turning up her nose at ℝ and ℍ. The answer is that she didn’t: she greedily chose all three!

But sitting inside this paper there’s a smaller story about Jordan algebras and the Koecher–Vinberg classification of convex homogeneous self-dual cones. A lot of this story is ‘well-known’, in the peculiar sense that mathematicians use this term, meaning at least ten people think it’s old hat. But it’s still worth telling — and there’s also something slightly less well-known, the concept of state-observable duality, which is sufficiently lofty and philosophical as to deserve consideration on this blog, I hope.

So I’ll tell this story here, in three parts. The first is just a little warmup about normed division algebras. If you’re a faithful reader of This Week’s Finds, you know this stuff. The second is also a warmup, of a slightly more esoteric sort: it’s about an old paper on the foundations of quantum theory written by Jordan, von Neumann and Wigner. And the third will be about the Koecher–Vinberg classification and state-observable duality."

http://golem.ph.utexas.edu/category/2010/11/stateobservable_duality_part_1.html

 

[Fra]

This is supposed to resolve the old puzzle about why Nature picked ℂ when it was time for quantum mechanics, while turning up her nose at ℝ and ℍ. The answer is that she didn’t: she greedily chose all three!

She did? I'm not convinced. All along is the premise that nature choose uncountable sets. As far as I see, the similarity between these things is larger than the differences. All those three number systems are uncountable. The dimensionality seems to me to be less of a problem, as long as the dimensionality is finite.

Did nature really choose an uncountable number system?

Almost every single paper or elaboration that tries to probe the issue of "why complex numbers" misses the prior question; why an uncountable number system?

/Fredrik

 

[Fra]

Maybe it's reasonable to say that nature is indifferent to which of those three to choose (and thus could choose all three), but wether the choice IS among those at all is another interesting but still open question IMHO.

/Fredrik

 

[arivero]

Adler

 

[qsa]

Integers plus logic (bigger than and smaller than ...). Nature does not have a mind of its own. Just like my system shows.

 

[MTd2]

Adler

What`s up with that?

 

[MTd2]

"This is the second part of a little story about the foundations of quantum mechanics.

In the first part, I introduced the heroes of our drama: the real numbers ℝ, the complex numbers ℂ, or the quaternions ℍ. I also mentioned their crazy uncle, who mainly stays locked up in the attic making strange noises: the octonions, O.

When our three heroes were sent down from platonic heaven to tell the world about the algebraic structure of quantum mechanics, they took on human avatars and wrote this paper:

Pascual Jordan, John von Neumann and Eugene Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29–64.
That’s what I’ll tell you about this time.

Then, in the final episode, we’ll meet the Koecher–Vinberg classification of convex homogeneous self-dual cones, and see how it’s really all about state-observable duality."

http://golem.ph.utexas.edu/category/2010/11/stateobservable_duality_part_2.html

 

[lumidek]

"I’m writing a paper called “Division algebras and quantum theory”, which is mainly about how quantum theory can be formulated using either the real numbers ℝ, the complex numbers ℂ, or the quaternions ℍ — and how these three versions are not really separate alternatives (as people often seem to think), but rather three parts of a unified structure.

Dear MTd2,

it may sound politically correct to argue that Nature chose R,C,H (and O as well?) to be on equal footing, except that She didn't. In quantum physics, all questions are answered by calculating the probabilities, and all elementary terms contributing to probabilities always have to be squared absolute values of probability amplitudes - which are, by definition, complex numbers.

There is absolutely no democracy here, and if you don't like this special role of C, you will have to ask for asylum in a different multiverse because our multiverse is based on C. R,C,H have different properties, so for particular questions, they are unequally appropriate. To describe probability amplitudes or matrix elements of observables or whatever else, one always has to work with complex numbers.

Real numbers would be too small because they wouldn't allow us to have Schrodinger's equation - ihd/dt psi = H.psi - which has the imaginary unit and has to contain one. (One also needs the complex numbers to remember whether a wave goes to the left or right.) Also, the commutator of operators e.g. [x,p] is equal e.g. to i.hbar and contains the imaginary unit, too.

On the other hand, the quaternions H would be too large. Schrodinger's equation only has one coefficient which has to be an "imaginary unit". This renders all the remaining imaginary units of quaternions - e.g. j,k - redundant. Every quaternion can be written as A+Bj where A,B are complex numbers. Schrodinger's equation would evolve the A,B pieces separately. So in effect, one would deal with a density matrix mixing the "A" and "B" parts of the wave function.

The general multiplication table of the quaternions would never be actively used in physics because the no-j and with-j pieces would remain forever decoupled.

While quaternions couldn't introduce any physics that is not contained in complex numbers, octonions would be totally unacceptable because they're not associative so the octonionic matrices cannot be interpreted as operators. And observables have to be operators that are associative - because if we make N successive measurements, only the ordering matters. There is no "bracketing" of the pairs of the measurements that could possibly influence the final result.

R,C,H,O may be viewed as a sequence of "similar objects of increasing size" but that's only a superficial perspective. Whenever a specific and well-defined enough question is studied, the status of the four structures is totally different and incomparable. They have various roles in theoretical physics but these roles are very different for the four structures, both when it comes to their reach and their type. One may learn R,C,H,O as a sequence in maths - that's the first encounter with the bigger structures. However, this analogy immediately goes away once a mathematician or physicist studies the contexts in which they're relevant. The unity is fictitious.

In particular, [URL='https://www.physicsforums.com/insights/author/john-baez/']John Baez[/URL] and many others jump to the misconceptions based on numerology. They only want to "double the dimensions" all the time. But the algebraic structures are not given just by their dimension: they are defined by their multiplication tables of the imaginary units. If these multiplication tables are not fully used in a physical context, the algebraic structure doesn't really operate there - and arranging numbers into octonions etc. would be just a fake administrative bookkeeping trick. In particular, the octonions have 7 imaginary units and G_2 is the automorphism of the multiplicative table. So whenever octonions really play role somewhere, G_2 has to play the role, too. But the G_2's role in physics is very rare. G_2 holonomy 7-dimensional manifolds are a rare example.

Cheers
LM

 

[arivero]

What`s up with that?

http://arxiv.org/abs/hep-ph/0505177 section 9. Also the book `Quaternionic Quantum Mechanics and Quantum Fields'' published by Oxford University Press in 1995

 

[Careful]

http://arxiv.org/abs/hep-ph/0505177 section 9. Also the book `Quaternionic Quantum Mechanics and Quantum Fields'' published by Oxford University Press in 1995

Well as far as I know, the first foundations were layed out by Finkelstein even back in 1962 or so. I remember some papers of Princeton mathematicians solving the tensor product problem in this approach. If someone is interested in that, I can look up the references.

 

[akhmeteli]

it may sound politically correct to argue that Nature chose R,C,H (and O as well?) to be on equal footing, except that She didn't. In quantum physics, all questions are answered by calculating the probabilities, and all elementary terms contributing to probabilities always have to be squared absolute values of probability amplitudes - which are, by definition, complex numbers.

There is absolutely no democracy here, and if you don't like this special role of C, you will have to ask for asylum in a different multiverse because our multiverse is based on C. R,C,H have different properties, so for particular questions, they are unequally appropriate. To describe probability amplitudes or matrix elements of observables or whatever else, one always has to work with complex numbers.

Real numbers would be too small because they wouldn't allow us to have Schrodinger's equation - ihd/dt psi = H.psi - which has the imaginary unit and has to contain one. (One also needs the complex numbers to remember whether a wave goes to the left or right.) Also, the commutator of operators e.g. [x,p] is equal e.g. to i.hbar and contains the imaginary unit, too.

Dear lumidek,

I have nothing to say about quaternions or octonions, but I wonder if the case of real vs. complex may be less straightforward.

You offer some reasonable arguments, but there may be some minor details making your arguments less than watertight.

For example, you appeal to the Shroedinger (sorry, cannot produce the umlaut with tex) equation. The problem is that THAT very Schroedinger had a very different opinion on this issue. In his beautiful but little known article (Nature, v.169, p.538(1952)) he wrote, discussing the Klein-Gordon-Maxwell electrodynamics: "That the wave function ... can be made real by a change of gauge is but a truism, though it contradicts the widespread belief about 'charged' fields requiring complex representation." And he did not have in mind a replacement of a complex function by two real ones!. Indeed, if you have a solution (\psi, A^\mu) of the Klein-Gordon-Maxwell equations (Klein-Gordon field interacting with electromagnetic field), then a gauge transform would give you (at least locally) an equivalent solution in the unitary gauge (\phi, B^\mu), where \phi is real, \phi^2=(\psi*)\psi, and the electromagnetic field is the same for A^\mu and B^/mu.

It does not matter (in principle) that you cannot get rid of complex numbers in this way for the Schroedinger equation, because Nature definitely prefers the Klein-Gordon equation. You may say: but Nature prefers the Dirac equation even more, and you cannot simultaneously make four complex components of the Dirac spinor real with a gauge transform?

True, however, the Dirac equation with an arbitrary electromagnetic field is equivalent (at least locally) to one partial differential equation of the fourth order for one of the components of the Dirac spinor function (provided a certain function of electromagnetic field does not vanish) (see http://arxiv.org/abs/1008.4828 ; see also comment http://www.akhmeteli.org/Comment_on_ref._8.html ). The other components of the Dirac spinor function can be expressed as functions of that one component and its derivatives. I cannot be sure this result has not been previously published elsewhere, but neither I nor people I discussed this issue with so far are aware of any such publication. However, this is not important for this discussion. What's important is you can replace the Dirac equation with the equation of the fourth order for one complex component and make that component real by a gauge transform.

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.

 

[Careful]

Dear lumidek,

I have nothing to say about quaternions or octonions, but I wonder if the case of real vs. complex may be less straightforward.

You offer some reasonable arguments, but there may be some minor details making your arguments less than watertight.

For example, you appeal to the Shroedinger (sorry, cannot produce the umlaut with tex) equation. The problem is that THAT very Schroedinger had a very different opinion on this issue. In his beautiful but little known article (Nature, v.169, p.538(1952)) he wrote, discussing the Klein-Gordon-Maxwell electrodynamics: "That the wave function ... can be made real by a change of gauge is but a truism, though it contradicts the widespread belief about 'charged' fields requiring complex representation." And he did not have in mind a replacement of a complex function by two real ones!. Indeed, if you have a solution (\psi, A^\mu) of the Klein-Gordon-Maxwell equations (Klein-Gordon field interacting with electromagnetic field), then a gauge transform would give you (at least locally) an equivalent solution in the unitary gauge (\phi, B^\mu), where \phi is real, \phi^2=(\psi*)\psi, and the electromagnetic field is the same for A^\mu and B^/mu.

It does not matter (in principle) that you cannot get rid of complex numbers in this way for the Schroedinger equation, because Nature definitely prefers the Klein-Gordon equation. You may say: but Nature prefers the Dirac equation even more, and you cannot simultaneously make four complex components of the Dirac spinor real with a gauge transform?

True, however, the Dirac equation with an arbitrary electromagnetic field is equivalent (at least locally) to one partial differential equation of the fourth order for one of the components of the Dirac spinor function (provided a certain function of electromagnetic field does not vanish) (see http://arxiv.org/abs/1008.4828 ; see also comment http://www.akhmeteli.org/Comment_on_ref._8.html ). The other components of the Dirac spinor function can be expressed as functions of that one component and its derivatives. I cannot be sure this result has not been previously published elsewhere, but neither I nor people I discussed this issue with so far are aware of any such publication. However, this is not important for this discussion. What's important is you can replace the Dirac equation with the equation of the fourth order for one complex component and make that component real by a gauge transform.

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.

Sure but I thought that the point was that you cannot do this with the multi-particle wave function. But you would then argue that entanglement does not exist and has never been measured and so on and so forth I guess. You can do that with me, but Ludimek would be less forgiving I guess :-)

 

[akhmeteli]

Sure but I thought that the point was that you cannot do this with the multi-particle wave function. But you would then argue that entanglement does not exist and has never been measured and so on and so forth I guess.

Dear Careful,

I am very glad you've resumed posting to physicsforums. Thank you for your remark.

First of all, it was not lumidek's point "that you cannot do this with the multi-particle wave function." So your remark seems to suggest that more sophisticated arguments are needed to prove that real numbers are not enough for quantum theory. It may well be they are not enough, but in that case it's interesting to know why exactly.

Let us discuss your remark.

Entanglement? I have recently taken part in a very lengthy discussion on nonlocality, entanglement and what not in another forum (https://www.physicsforums.com/showthread.php?t=369328 ), so maybe I should not restart that discussion here. Let me stick to constructive arguments.

In my paper accepted for publication in the International Journal of Quantum Information (there is a preprint at http://www.akhmeteli.org/akh-prepr-ws-ijqi2.pdf ; see also comment http://www.akhmeteli.org/Comment_on_ref._6.html ), I start with the equations of (non-second-quantized) scalar electrodynamics. They describe a Klein-Gordon particle (a scalar particle described by the Klein-Gordon equation) interacting with electromagnetic field (described by the Maxwell equations).

It is shown that this model is equivalent (at least locally) to a local realistic model – modified electrodynamics without particles, as the matter (particle) field can be naturally eliminated from the equations of scalar electrodynamics, and the resulting equations describe independent evolution of the electromagnetic field (electromagnetic 4-potential). Furthermore, using some nightlight's ideas, this evolution is shown to be equivalent to unitary evolution of a certain (second-quantized) quantum field theory. The model uses real fields only (actually, electromagnetic 4-potential only), but is equivalent (at least locally) to scalar electrodynamics. Does it describe entanglement? The answer is somewhat longish (https://www.physicsforums.com/showpost.php?p=2813903&postcount=576 ).

So if a local model with real fields can have the same unitary evolution as a quantum field theory, maybe it is not quite obvious that multiple-particle wave function cannot be described in this way?

I should note that I don't have similar results for spinor electrodynamics (yet:-) ).

 

[Careful]

Dear Careful,

I am very glad you've resumed posting to physicsforums. Thank you for your remark.

First of all, it was not lumidek's point "that you cannot do this with the multi-particle wave function." So your remark seems to suggest that more sophisticated arguments are needed to prove that real numbers are not enough for quantum theory. It may well be they are not enough, but in that case it's interesting to know why exactly.

Let us discuss your remark.

Entanglement? I have recently taken part in a very lengthy discussion on nonlocality, entanglement and what not in another forum (https://www.physicsforums.com/showthread.php?t=369328 ), so maybe I should not restart that discussion here. Let me stick to constructive arguments.

In my paper accepted for publication in the International Journal of Quantum Information (there is a preprint at http://www.akhmeteli.org/akh-prepr-ws-ijqi2.pdf ; see also comment http://www.akhmeteli.org/Comment_on_ref._6.html ), I start with the equations of (non-second-quantized) scalar electrodynamics. They describe a Klein-Gordon particle (a scalar particle described by the Klein-Gordon equation) interacting with electromagnetic field (described by the Maxwell equations).

It is shown that this model is equivalent (at least locally) to a local realistic model – modified electrodynamics without particles, as the matter (particle) field can be naturally eliminated from the equations of scalar electrodynamics, and the resulting equations describe independent evolution of the electromagnetic field (electromagnetic 4-potential).

Hmmm, I always thought it was the other way around here, that you eliminate the gauge field and express it in terms of the (scalar) field. There is another problem with that idea because as far as I remember a charged, self interacting spinless field is (classically) unstable.

Furthermore, using some nightlight's ideas, this evolution is shown to be equivalent to unitary evolution of a certain (second-quantized) quantum field theory. The model uses real fields only (actually, electromagnetic 4-potential only), but is equivalent (at least locally) to scalar electrodynamics. Does it describe entanglement? The answer is somewhat longish (https://www.physicsforums.com/showpost.php?p=2813903&postcount=576 ).

Ok, can you make it simple here? Do you say (a) I believe the predictions of QM to be wrong and Bell violating correlations are never measured (b) I think the predictions of QM and local realism to be true but somehow there is a fundamental reason why those correlations cannot be obtained in practical experiments. Concerning your comments about retrieving a second quantized theory from this : (a) how does Planck's constant creep in ? (b) I assume you support on the work of Kowalski and use his Hilbert space formalism to express solutions of non-linear PDE's?. But you never ever retrieve QFT in this way since you have no superpostion principle (since the relevant equations are real and nonlinear), all you have done is expressed the solution to your nonlinear equation in a QFT like language. But your creation and annihilation operators are not ''free'' in any sense, all you have is an equation of motion for one state and I guess the evolution ''operator'' is unitary because you have some classical conserved current (because you work with a complex scalar field) right?
Actually, it cannot be an operator in the proper sense since your equations are nonlinear.

This would also complicate a collaps of the wavefunction postulate since a collapse would modify the dynamics (unless you of course express the dynamics in it's most universal form and simply put some terms in the intial state to zero).

So if a local model with real fields can have the same unitary evolution as a quantum field theory, maybe it is not quite obvious that multiple-particle wave function cannot be described in this way?

I should note that I don't have similar results for spinor electrodynamics (yet:-) ).

As said before, I think you make a severe interpretation mistake here unless I have overlooked some very important thing. But I don't think so. So, my answer is no, you won't get entanglement out unless you take some acausal propagator, like half the sum of the retarded and advanced green function or something like that (or half the difference of the advanced and retarded green's function). And something like that would give you at best non-local effects, it would not give you entanglement in the usual sense, like we understand this for *linear* theories.

 

[akhmeteli]

Dear Careful,

Thank you very much for your remarks.

Hmmm, I always thought it was the other way around here, that you eliminate the gauge field and express it in terms of the (scalar) field.

Maybe I don't quite understand what you mean. If somebody prefers to eliminate the gauge field, like Barut, does this mean that the matter field cannot be eliminated instead? Anyway, is this some criticism or not? So far I don't quite understand what seems to be the problem.

There is another problem with that idea because as far as I remember a charged, self interacting spinless field is (classically) unstable.

This model is equivalent to scalar electrodynamics (with some caveats), so I guess both the model and scalar electrodynamics must be either stable or unstable. So if there is a problem with the model, neither is scalar electrodynamics free from this problem. However, I believe scalar electrodynamics is a reasonably decent theory. Of course it has obvious problems, e.g. it does not describe spin, and of course the model inherits this weakness. But I would think the model still is of some interest, at least as a toy model. Of course, similar results for spinor electrodynamics would be much preferable, but I cannot offer such results.

Ok, can you make it simple here? Do you say (a) I believe the predictions of QM to be wrong and Bell violating correlations are never measured (b) I think the predictions of QM and local realism to be true but somehow there is a fundamental reason why those correlations cannot be obtained in practical experiments.

I believe I discussed this issue in Section 5 of the article using other people's arguments (see also the comment).

Briefly:

1. No loophole-free violations of the Bell inequalities have been observed experimentally yet (I guess you know that, but I offer the relevant quotes in the article).

2. A typical proof of the violations of the Bell inequalities in quantum theory uses two mutually contradictory assumptions: unitary evolution and theory of quantum measurements (say, the projection postulate): unitary evolution cannot provide irreversibility or destroy a superposition, whereas the projection postulate does just that.

Therefore I conclude that there are reasons to doubt that the Bell inequalities are violated in Nature or in quantum theory (or quantum theory is self-contradictory, then local realistic theories just cannot be worse than quantum theory:-) ).

Concerning your comments about retrieving a second quantized theory from this : (a) how does Planck's constant creep in ?

The Planck's constant is inherited from the Klein-Gordon equation, as the matter field mass is not zero.

(b) I assume you support on the work of Kowalski and use his Hilbert space formalism to express solutions of non-linear PDE's?.

Yes.

But you never ever retrieve QFT in this way since you have no superpostion principle (since the relevant equations are real and nonlinear), all you have done is expressed the solution to your nonlinear equation in a QFT like language.

While the strict principle of superposition is abandoned, there is a "weak (or approximate) principle of superposition". Indeed, let us start with two different states in the Fock space corresponding (via the Kowalski - Steeb procedure) to two different initial fields in 3 dimensions ksi(t0,x) and psi(t0,x) (so these states are not the most general states in the Fock space). We can build a "weak superposition" of these states as follows: we build the following initial field in 3D: a ksi+b psi, where a and b are the coefficients of the required superposition. Then we can build (using the Kowalski - Steeb procedure) the Fock state corresponding to a ksi+b psi. If ksi and psi are relatively weak, only vacuum state and a term linear in ksi and psi will effectively survive in the expansion of the exponent for the coherent state. However, what we typically measure is the difference between the state and the vacuum state. So we have approximate superposition, at least at the initial moment. However, as electrodynamic interaction is rather weak (this is the basis of QED perturbation methods), so nonlinearity of the evolution equations in 3D can be expected to be rather weak, this "superposition" will not differ much from the "true" superposition of the states in the Fock space. At least, one can expect this, until this issue is studied in detail.

But your creation and annihilation operators are not ''free'' in any sense, all you have is an equation of motion for one state and I guess the evolution ''operator'' is unitary because you have some classical conserved current (because you work with a complex scalar field) right?

I did not understand the point about “freedom" of creation/annihilation operators. Why and in what sense indeed must they be “free”? And I don’t quite understand, why “all I have is an equation of motion” only for one state? You can start with any state in 3D (the 4-potential and its first temporal derivatives can be pretty much arbitrary in 3D at time t0), it’s pretty much like classical electrodynamics, and there is a state in the Fock space corresponding to this state in 3D. Indeed, not every state in the Fock space has a corresponding state in 3D, but how can you be sure every state in the Fock space can be realized in Nature? Furthermore, strictly speaking, we have to deal only with one actual state of the Universe.
I think you are right about the conserved current (there is no doubt it is conserved though), but I am not sure about its relation to the unitarity of the operator. Anyway, until you explain why you think this is all about only one state, it is not easy to pinpoint what seems to be the problem.

Actually, it cannot be an operator in the proper sense since your equations are nonlinear.

I don’t quite get it – the equations are not linear in 3D, but they are indeed linear in the Fock space.

This would also complicate a collaps of the wavefunction postulate since a collapse would modify the dynamics (unless you of course express the dynamics in it's most universal form and simply put some terms in the intial state to zero).

It is my understanding that there is little if any experimental evidence of collapse. I tend to believe that there are no gaps in unitary evolution. Of course, nobody cares what I believe, but the following quote from Schlosshauer’s paper (M. Schlosshauer, Annals of Physics, 321 (2006) 112-149) seems to suggest that I am not alone (though I have doubts about his conclusion (iv) – my take is different):
“(i) the universal validity of unitary dynamics and the superposition principle has been confirmed far into the mesoscopic and macroscopic realm in all experiments conducted thus far;
(ii) all observed ‘‘restrictions’’ can be correctly and completely accounted for by taking into account environmental decoherence effects;
(iii) no positive experimental evidence exists for physical state-vector
collapse;
(iv) the perception of single ‘‘outcomes’’ is likely to be explainable through decoherence effects in the neuronal apparatus.”
Anyway, do you think there is experimental evidence of collapse?

I am not sure, maybe we could agree on what you say in parenthesis, but I am not sure I fully understand that phrase

As said before, I think you make a severe interpretation mistake here unless I have overlooked some very important thing. But I don't think so. So, my answer is no, you won't get entanglement out unless you take some acausal propagator, like half the sum of the retarded and advanced green function or something like that (or half the difference of the advanced and retarded green's function). And something like that would give you at best non-local effects, it would not give you entanglement in the usual sense, like we understand this for *linear* theories.

I explained (by referencing an earlier post) in what sense there is entanglement in the model and in what sense there is not. The real question is whether the model is experimentally hopeless or not. Maybe it is, but I don’t quite see the relevant argument in your remark.

Again, thank you for great questions.

 

[Fra]

It is my understanding that there is little if any experimental evidence of collapse. I tend to believe that there are no gaps in unitary evolution.

Are there experimental evidence of information updates?

To deny an information update, aren't you forced to consider much MORE information, relative to which the apparent "information update" is again expected according to an deterministic evolution?

This solution is to me flawed because the removal of the collapse, only works for a different context, and an information update is by definition context dependent.

This logic of resolving the issue seems to necessarily self-inflate in complexity to the point where I think it becomes impossible to represent and compute. So what is the gain here?

/Fredrik

 

[Careful]

Dear Careful,

Thank you very much for your remarks.



Maybe I don't quite understand what you mean. If somebody prefers to eliminate the gauge field, like Barut, does this mean that the matter field cannot be eliminated instead? Anyway, is this some criticism or not? So far I don't quite understand what seems to be the problem.

There is a definite reason why Barut did it in this way, that is: (a) the equations of the gauge field are just linear hyperbolic equation with a source term, so you can explicitely solve them in integral form and keep causality under control by using the correct Green's function (b) eliminating the scalar field explicitely cannot work for all I know, the second order terms are indeed an ordinary d'Alembertian, but there are gauge field dependent first derivative and mass terms. These types of equations do not have an explicit integral representation as far as I know and causality will be much harder to control.

This model is equivalent to scalar electrodynamics (with some caveats), so I guess both the model and scalar electrodynamics must be either stable or unstable. So if there is a problem with the model, neither is scalar electrodynamics free from this problem. However, I believe scalar electrodynamics is a reasonably decent theory. Of course it has obvious problems, e.g. it does not describe spin, and of course the model inherits this weakness. But I would think the model still is of some interest, at least as a toy model. Of course, similar results for spinor electrodynamics would be much preferable, but I cannot offer such results.

As far as I remember, Schroedinger wrote a paper about the instability of the self interacting scalar field and also I made some computations showing that. Don't ask me to look up the reference, since I would have to dig into thousands of papers to find it

I believe I discussed this issue in Section 5 of the article using other people's arguments (see also the comment).

Briefly:

1. No loophole-free violations of the Bell inequalities have been observed experimentally yet (I guess you know that, but I offer the relevant quotes in the article).

2. A typical proof of the violations of the Bell inequalities in quantum theory uses two mutually contradictory assumptions: unitary evolution and theory of quantum measurements (say, the projection postulate): unitary evolution cannot provide irreversibility or destroy a superposition, whereas the projection postulate does just that.

Therefore I conclude that there are reasons to doubt that the Bell inequalities are violated in Nature or in quantum theory (or quantum theory is self-contradictory, then local realistic theories just cannot be worse than quantum theory:-) ).

Ok, so basically you take my option (a). In contrast to you, I do not see the absence of a loophole free experiment so far as an indication against QM, nor as an argument pro local realism. The reason is that in each experiments, something else goes wrong: (a) at long distance, you have to use photons and there you have dector problems, but at short distances people used massive particles AFAIK and bell inequality violations have been measured. Unfortunately, in case of (b) causality was not under sufficient control, but no realist up to date has offered a single theory which could explain both experiments. Moreover, there is nothing wrong with unitary evolution versus collaps of the wave function.

The Planck's constant is inherited from the Klein-Gordon equation, as the matter field mass is not zero.

Silly of me, it was 3 am for me, so I am entitled to a stupidity then

While the strict principle of superposition is abandoned, there is a "weak (or approximate) principle of superposition". Indeed, let us start with two different states in the Fock space corresponding (via the Kowalski - Steeb procedure) to two different initial fields in 3 dimensions ksi(t0,x) and psi(t0,x) (so these states are not the most general states in the Fock space). We can build a "weak superposition" of these states as follows: we build the following initial field in 3D: a ksi+b psi, where a and b are the coefficients of the required superposition. Then we can build (using the Kowalski - Steeb procedure) the Fock state corresponding to a ksi+b psi. If ksi and psi are relatively weak, only vacuum state and a term linear in ksi and psi will effectively survive in the expansion of the exponent for the coherent state. However, what we typically measure is the difference between the state and the vacuum state. So we have approximate superposition, at least at the initial moment. However, as electrodynamic interaction is rather weak (this is the basis of QED perturbation methods), so nonlinearity of the evolution equations in 3D can be expected to be rather weak, this "superposition" will not differ much from the "true" superposition of the states in the Fock space. At least, one can expect this, until this issue is studied in detail.

I did not understand the point about “freedom" of creation/annihilation operators. Why and in what sense indeed must they be “free”? And I don’t quite understand, why “all I have is an equation of motion” only for one state? You can start with any state in 3D (the 4-potential and its first temporal derivatives can be pretty much arbitrary in 3D at time t0), it’s pretty much like classical electrodynamics, and there is a state in the Fock space corresponding to this state in 3D. Indeed, not every state in the Fock space has a corresponding state in 3D, but how can you be sure every state in the Fock space can be realized in Nature? Furthermore, strictly speaking, we have to deal only with one actual state of the Universe.
I think you are right about the conserved current (there is no doubt it is conserved though), but I am not sure about its relation to the unitarity of the operator. Anyway, until you explain why you think this is all about only one state, it is not easy to pinpoint what seems to be the problem.

I don’t quite get it – the equations are not linear in 3D, but they are indeed linear in the Fock space.

The point is that that in your theory, the initial values determine a very small class of states (namely coherent states): this is the huge contrast with ordinary QFT and this is logical since you have not an infinite number of particle degrees of freedom. This is what second quantization does for you and it is most adequately expressed in the Schroedinger picture of QFT where you have to use *functional* derivatives and *functional* integration. In my first response, I had another scheme in mind (that's why I said that U would be nonlinear, actually my viewpoint had a nonlinear U but a linear (initial) state superposition, while Kowalski's viewpoint has a linear U but a nonlinear ''superposition'' principle) but to use the same language I briefly looked to Kowalski's way of phrasing it again. However, my conclusion with the measurement problem remains the same: you cannot collapse to a one or two particle state since that is *not* a coherent state and therefore, you would have to extend your theory and leave the purely classical domain. Another minor remark (but you can easily surpass this) concerns the particular creation operators Kowalski uses, I hope you did a Fourier transform because his creation operators are defined in position space and not momentum space. So your math is formally correct, but you might miss (a) the essential mathematical differences with ordinary QFT (b) as well as the physical distinctions.

Moreover, I definately think it is the conserved current which has to be responsible for unitarity, there is no a priori reason within Kowalski's scheme why the operator should be unitary.

It is my understanding that there is little if any experimental evidence of collapse. I tend to believe that there are no gaps in unitary evolution. Of course, nobody cares what I believe, but the following quote from Schlosshauer’s paper (M. Schlosshauer, Annals of Physics, 321 (2006) 112-149) seems to suggest that I am not alone (though I have doubts about his conclusion (iv) – my take is different):
“(i) the universal validity of unitary dynamics and the superposition principle has been confirmed far into the mesoscopic and macroscopic realm in all experiments conducted thus far;
(ii) all observed ‘‘restrictions’’ can be correctly and completely accounted for by taking into account environmental decoherence effects;
(iii) no positive experimental evidence exists for physical state-vector
collapse;
(iv) the perception of single ‘‘outcomes’’ is likely to be explainable through decoherence effects in the neuronal apparatus.”
Anyway, do you think there is experimental evidence of collapse?

It depends upon what you mean with collapse. I think there is no way you can deny observation and decoherence does not explain observation. The problem I have with decoherence is that it is far more nonlocal than the collapse is : it requires the observer to actually know some details of the state of the universe which he cannot know by any means. So yeh, I believe the collapse mechanism is still the best thing proposed so far.

I am not sure, maybe we could agree on what you say in parenthesis, but I am not sure I fully understand that phrase

I explained (by referencing an earlier post) in what sense there is entanglement in the model and in what sense there is not. The real question is whether the model is experimentally hopeless or not. Maybe it is, but I don’t quite see the relevant argument in your remark.

Again, thank you for great questions.

So we agree there is no entanglement in your theory in the ordinary quantum mechanical sense. That's what I meant with my original comment, if you take an ordinary entangled two particle state, you cannot eliminate the complex numbers by means of a gauge field.

 

[MTd2]

http://golem.ph.utexas.edu/category/2010/11/stateobservable_duality_part_2.html#comments

State-Observable Duality (Part 3)
Posted by John BaezThis is the third and final episode of a little story about the foundations of quantum mechanics.

In the first episode, I reminded you of some basic facts about the real numbers ℝ, the complex numbers ℂ, and the quaternions ℍ.

In the second episode, I told you how Jordan, von Neumann and Wigner classified ‘formally real Jordan algebras’, which can serve as algebras of observables in quantum theory. Apart from the ‘spin factors’ ℝn⊕ℝ and the Jordan algebra of 3×3 self-adjoint octonionic matrices, h3(O), these come in three kinds:

The algebra hn(ℝ) of n×n self-adjoint real matrices with the product a∘b=12(ab+ba).
The algebra hn(ℂ) of n×n self-adjoint complex matrices with the product a∘b=12(ab+ba).
The algebra hn(ℍ) of n×n self-adjoint quaternionic matrices with the product a∘b=12(ab+ba).
In every case, even the curious exceptional cases, there is a concept of what it means for an element to be ‘positive’, and the positive elements form a cone. In this episode we’ll explore that further: we’ll meet the Koecher–Vinberg classification of convex homogeneous self-dual cones, and see how it’s really all about state-observable duality.

 

[arivero]

Well as far as I know, the first foundations were layed out by Finkelstein even back in 1962 or so. I remember some papers of Princeton mathematicians solving the tensor product problem in this approach. If someone is interested in that, I can look up the references.

I think I remember to have read and appreciated these papers too, but more than 20 years ago, so I can not tell anything useful :-(

 

[MTd2]

Who are thiese guys Barut and Kowalski? What did they do? Who is this Finkelstein and what foundations did he lay? About what?

I am clueless

 

[Careful]

Who are thiese guys Barut and Kowalski? What did they do? Who is this Finkelstein and what foundations did he lay? About what?

I am clueless

You learn to know these people only when you try to solve problems in the standard paradigm and if you are clever enough to rediscover what they thought about too. That is, irrespective of whether you agree with them or not. Barut for example was I believe a good friend of Abdus Salam, has been a professor for many years at the famous institute for theoretical physics in Trieste and was a local realist. He has illuminated the tiny differences there exist between the *observed* predictions of QFT and a nonlinear classical self interacting field theory. Almost all successes of QED like the Lamb shift, vacuum polarization and the g-factor can be explained by classical means.

Kowalski is a Polish Biophysicist I guess who has written a very nice little book about a Hilbert space formalism to solve nonlinear ODE's and PDE's.

 

[MTd2]

I am still clueless about who they are. Kowalski and Finkelstein are quite common surnames. I never heard about Barut.

 

[Careful]

I am still clueless about who they are. Kowalski and Finkelstein are quite common surnames. I never heard about Barut.

We live in the time that you can easily google:
for example, google on Asim Barut, David Finkelstein and K. Kowalski use words such as phyisics, Hilbert space, nonlinear and so on.

 

[arivero]

But the algebraic structures are not given just by their dimension: they are defined by their multiplication tables of the imaginary units. If these multiplication tables are not fully used in a physical context, the algebraic structure doesn't really operate there - and arranging numbers into octonions etc. would be just a fake administrative bookkeeping trick.

As a marginal note, in the topological avatars of these structures the spheres "S0", S1 and S3 are the only ones whose fundamental group is abelian, and then they admit a Lie Group structure (I read it here, please nobody takes me as an expert ). But there is not a Lie group with the topology of S7, and this fact is other way for the algebraic structure to appear and to put the octonions in a different foot that the other three division algebras. Some people calls S7 a "soft Lie group"

EDIT: and of course there is a standard and "democratic" role of division algebras in supersymmetry and string theory, as told in week 158" of Baez, references 5-8 there, and some other topic we spoke in sci.physics.research ten years ago. It is unrelated, or seems to be, to the question of foundations of quantum mechanics.

 

[Careful]

As a marginal note, in the topological avatars of these structures the spheres "S0", S1 and S3 are the only ones whose fundamental group is abelian, and then they admit a Lie Group structure (I read it here, please nobody takes me as an expert ). But there is not a Lie group with the topology of S7, and this fact is other way for the algebraic structure to appear and to put the octonions in a different foot that the other three division algebras.

To add a bit of fuel to the discussion, why limit yourself to associative division algebra's? I agree with Lubos that associativity is a necessary requirement, but what I don't see is why division algebra's should be mandatory. It appears to me that a (complex) involutive algebra with a well defined trace-functional is actually sufficient.

 

[MTd2]

Arivero, did you recall that RCHO Ideal Algebra he finds the bosons of the standard model by squaring the O^2?This the same way John says these are used to get around non commutativity.

 

[arivero]

MTd2, yep, is a standard manipulation with octonions, so I did not paid attention, but now that you mention it, could it be relevant to relate fermions and bosons? Who knows. Remember that the problem with O is non _associativity_

 

[MTd2]

Are bosons some non associative beings?

 

[lumidek]

Dear lumidek,

I have nothing to say about quaternions or octonions, but I wonder if the case of real vs. complex may be less straightforward.

You offer some reasonable arguments, but there may be some minor details making your arguments less than watertight.

For example, you appeal to the Shroedinger (sorry, cannot produce the umlaut with tex) equation. The problem is that THAT very Schroedinger had a very different opinion...

Hi, I was answering the question from the viewpoint of science which builds on scientific arguments, not from the viewpoint of a mindless religious groupthink that worships (fake) authorities. What Schrödinger wrote about this issue is clearly complete nonsense.

The "i" factor in the Schrödinger's equation is completely fundamental - and its counterpart appears in any equivalent description of quantum mechanics, too. In Heisenberg's picture, we have -i.hbar.d operator/dt = [H,operator], you see the factor of "i" again. In Feynman's path integral approach, the histories are weighted by exp(i.S/hbar) and again, "i" in the exponent is absolutely essential and equivalent in origin to the "i" in the other pictures.

Schrödinger's mixing of Schrödinger's equation and Klein-Gordon equation was partly an artifact of the early stages of QM, partly of the fact that he has never really understood quantum mechanics. Schrödinger's equation - with a general Hamiltonian - is a completely general equation that describes any quantum system, including a system of Klein-Gordon quanta. It's just that one has to use a more complicated Hamiltonian - the total energy of the Klein-Gordon field - in this case.

On the other hand, the "real" Klein-Gordon equation is real, indeed. But it is not a quantum equation. It is a classical equation. The only consistent way to obtain a working quantum system out of the "real" but classical Klein-Gordon equation is to (second) quantize it, e.g. to add hats to all the fields. That ends up with the Schrödinger's equation that controls a wave functional describing the second-quantized Klein-Gordon field. It is a standard Schrödinger's equation - with the crucial "i" coefficient and with a complicated field-theoretical Hamiltonian.

Schrödinger has never truly understood Born's probabilistic interpretation of QM: he has never accepted that the "wave" in his equation shouldn't be interpreted as a classical field but as a probability wave. That's the source of all his confusions mixing classical fields - which may be naturally real - with wave functions and amplitudes - that always have to be complex.

 

[Careful]

I told you, akhmeteli

 

[Fra]

(ii) all observed ‘‘restrictions’’ can be correctly and completely accounted for by taking into account environmental decoherence effects;

This mode of explanation fails at some point because a given observer simply can't take into account the entire environment, because of limited information capacity and computational capacity.

The environment beeing an infinite information sink is a kind of "in principle" argument that I think fails becuase it ignores what should be obvious that no given, but still arbirary, observer can register and process infinite information.

And register and process information, and produce an expectation should be the essence of science. Any "in principle" argumetns that violate this, are IMO not quite scientifically sound.

One might be tempted to say that the collapse is the result of the incomplete observer, but this is not a technical problem, it's a fact that any physical observer IS incomplete. Why deny this?

The conincidental success of this picture so far, is because we do constrain ourselves to studying small subsystems where these information constraints doens't limit. But in cosmological models this would be grossly violated, and there is also a problem with this when trying to understand the actions chosen by nature even in the case of subsystems (Standard Model), that leads then to sets of possible (or consistent) actions.

/Fredrik

 

[arivero]

Are bosons some non associative beings?

No, and because of that I said "who knows?", I do not see the relation clearly. But it is interesting that some approaches near to division algebras, like Connes's C+H+H+M3(C), have been unable to define supersymmetry, while other studies (Evans) do establish a strong link between division algebras, triality and supersymmetry. Still, it is a RCHO topic, not a topic on fundations.

 

[MTd2]

OK, I will recast the discussion on that thread. Let's meet there again!

 

[arivero]

weighted by exp(i.S/hbar) and again, "i" in the exponent is absolutely essential and equivalent in origin to the "i" in the other pictures.

Equivalent, but perhaps even more fundamental that the "i" in Schroedinger eq. and the "i" in Heisenberg [x,p]. The pivotal work here was from Dirac, on Contact Transformations (P.A.M. Dirac, The Lagrangian in Quantum Mechanics, Phisik. Zeitschr. der Sowjetunion, 3, p. 64, 1933) and it was influential in Feynman.

Also, but without direct link -afaik- with this work, let me note the "i" in a exponential is the hallmark of Fourier transform and then of Dirac's delta. So, if we want to build a measure \delta_{f'} concentrated in the minimum of f(x), we can write

<br /> &lt;\delta_{f&#039;} | g(x) &gt; = \int \int \lim_{y\to x} e^{i z {f(y)-f(x)\over y-x}} g(x) dz dx<br />

 

[arivero]

<br /> &lt;\delta_{f&#039;} | g(x) &gt; = \int \int \lim_{y\to x} e^{i z {f(y)-f(x)\over y-x}} g(x) dz dx<br />

Now, if you buy this...

<br /> \epsilon = { y - x \over z}<br />

<br /> \int \int ... dz dx = \int \int \lim_{\epsilon \to 0} e^{{i\over \epsilon} (f(y)-f(x))} g(x) dx {dy \over \epsilon}<br />

Note that actually we are evaluating g(x) in the points which are a minimum of the function f(x). So it seems as if f were a classical potential V(x) and g an "observable" O(x), in a sort of zero dimensional classical mechanics:
<br /> ... = \int \int \lim_{\epsilon \to 0} {1 \over \epsilon} e^{{i\over \epsilon} ( V(y)-V(x)) } O(x) dx dy<br />

Anyway, the point here is that we see that the main role of the complex numbers is to provide a "dirac delta" via usual theory of Fourier transform. Were we to use quaternions or reals, we should need an alternative "quaternionic dirac measure" or similar beast; perhaps it exists, I have never heard of it, but mathematics is big forest.

Also, from above we have an intuition of other role of the complex conjugation: were we able to separate O(x) in a product Q(x)Q*(y), we could formally write
<br /> ... = \left(\int \lim_{\epsilon \to 0} {1 \over \sqrt \epsilon} e^{{i\over \epsilon} V(q) } Q(q) dq \right) \left(\int \lim_{\epsilon \to 0} {1 \over \sqrt \epsilon} e^{{i\over \epsilon} V(q) } Q(q) dq\right)^*<br />

 

[akhmeteli]

Are there experimental evidence of information updates?

To deny an information update, aren't you forced to consider much MORE information, relative to which the apparent "information update" is again expected according to an deterministic evolution?

This solution is to me flawed because the removal of the collapse, only works for a different context, and an information update is by definition context dependent.

This logic of resolving the issue seems to necessarily self-inflate in complexity to the point where I think it becomes impossible to represent and compute. So what is the gain here?

/Fredrik

I admit that I don't understand every word you're saying, so my response is based on what I think I understood, therefore the response may be completely off mark, in which case I apologize.

You seem to object against my rejecting the collapse, whereas I have to reject it in spite of your objections, although I sincerely respect your opinion. I gave my reasons for that. Besides my reasons, my formal justification is there is no experimental evidence of collapse. You're telling me rejecting collapse makes calculation impossible. I'm saying Nature does not give a damn about our problems.

So you're asking what's the gain? I'd say, ideally, a simpler, no drama quantum theory. What you see though is inflation in complexity. So I guess we just disagree.

 

[akhmeteli]

Thank you very much for your remarks.

There is a definite reason why Barut did it in this way, that is: (a) the equations of the gauge field are just linear hyperbolic equation with a source term, so you can explicitely solve them in integral form and keep causality under control by using the correct Green's function (b) eliminating the scalar field explicitely cannot work for all I know, the second order terms are indeed an ordinary d'Alembertian, but there are gauge field dependent first derivative and mass terms. These types of equations do not have an explicit integral representation as far as I know and causality will be much harder to control.

Again, I am not sure I quite understand your argument. So Barut did something for some reasons. I did something else for some other reasons. I guess there’s nothing wrong with it, per se.

Problems with causality? I am afraid I just fail to see any serious problems. What’s happening actually? I just use boring algebraic (algebraic!) elimination of the scalar field, so the resulting model is pretty much equivalent to scalar electrodynamics. Furthermore, I show that in this model (here I am making some obvious modifications of what’s written in the article), if you know the 4-potential and its first temporal derivative in a small spatial vicinity of spatial point x at time point x0, you can calculate the second temporal derivative of the 4-potential pretty much in the same spatial vicinity and at the same time point. Therefore, the equations can be integrated, at least locally, so the model is local. Do you think there may be problems with faster-than-light propagation? I cannot exclude such a possibility, but, first, the model is still local, and second, faster-than-light propagation, if any, must be also in scalar electrodynamics, because it is pretty much equivalent to the model. If the model stinks, the scalar electrodynamics stinks as well, so the model is in good company. Again, spinor electrodynamics is certainly better, but I don’t have equally good results for it.

As far as I remember, Schroedinger wrote a paper about the instability of the self interacting scalar field and also I made some computations showing that. Don't ask me to look up the reference, since I would have to dig into thousands of papers to find it

I see. Strictly speaking, maybe I could just shrug off this objection, as it is not specific enough, but let me try the following argument: I am not sure that this objection is relevant, as what I have is a self-interacting vector field, not scalar field. And again, instability, if any, must also be present in scalar electrodynamics, so the model is in good company.

Ok, so basically you take my option (a). In contrast to you, I do not see the absence of a loophole free experiment so far as an indication against QM, nor as an argument pro local realism. The reason is that in each experiments, something else goes wrong: (a) at long distance, you have to use photons and there you have dector problems, but at short distances people used massive particles AFAIK and bell inequality violations have been measured. Unfortunately, in case of (b) causality was not under sufficient control, but no realist up to date has offered a single theory which could explain both experiments.

Well, you see, I don’t think I am against QM, as I fully embrace what I think is its crucial part – unitary evolution. What I question is collapse. But do you think collapse is part and parcel of QM?

Moreover, there is nothing wrong with unitary evolution versus collaps of the wave function.

Well, I gave my reasons. I don’t think I have anything meaningful to add

The point is that that in your theory, the initial values determine a very small class of states (namely coherent states): this is the huge contrast with ordinary QFT and this is logical since you have not an infinite number of particle degrees of freedom.

I don’t quite understand what you mean by “particle degrees of freedom”, but I do have an infinite number of degrees of freedom. Whether this is enough to describe reality, I don’t know.

However, my conclusion with the measurement problem remains the same: you cannot collapse to a one or two particle state since that is *not* a coherent state and therefore, you would have to extend your theory and leave the purely classical domain.

First of all, I hope you don’t demand that I solve the measurement problem – that would be a tall order:-). Second, I am not sure I have to extend the theory. At least this is not obvious. Let me ask you first, what are we talking about? One or two matter particles or one or two photons?

Another minor remark (but you can easily surpass this) concerns the particular creation operators Kowalski uses, I hope you did a Fourier transform because his creation operators are defined in position space and not momentum space. So your math is formally correct, but you might miss (a) the essential mathematical differences with ordinary QFT (b) as well as the physical distinctions.

I agree. I have not tried to compare what I call “a” quantum field theory, where the model is embedded, with, say, QED or experiments. Nevertheless, it seems noteworthy to me that a local theory can be naturally embedded into a QFT.

Moreover, I definately think it is the conserved current which has to be responsible for unitarity, there is no a priori reason within Kowalski's scheme why the operator should be unitary.

Perhaps.

It depends upon what you mean with collapse. I think there is no way you can deny observation and decoherence does not explain observation.

Are you sure you said what you wanted to say? How can “observation” explain ”observation”? Anyway, could you explain what you mean?

The problem I have with decoherence is that it is far more nonlocal than the collapse is : it requires the observer to actually know some details of the state of the universe which he cannot know by any means. So yeh, I believe the collapse mechanism is still the best thing proposed so far.

Well, decoherence is no relative of mine :-), so I don’t worry about any problems with decoherence. My position is there are no deviations from unitary evolution, so we can do without collapse, even if it is “the best thing so far”.

So we agree there is no entanglement in your theory in the ordinary quantum mechanical sense. That's what I meant with my original comment, if you take an ordinary entangled two particle state, you cannot eliminate the complex numbers by means of a gauge field.

On the other hand, as I said, “my” states have non-zero projections on the two-particle subspace in the Fock space. It is not quite obvious that this is not enough.

 

[akhmeteli]

Hi, I was answering the question from the viewpoint of science which builds on scientific arguments, not from the viewpoint of a mindless religious groupthink that worships (fake) authorities. What Schrödinger wrote about this issue is clearly complete nonsense.

Dear lumidek,

Thank you for your reply.

Of course, you don’t have to agree with Schrödinger’s “party line”. Does this mean I have to agree with your “party line”?
Your arguments are reasonable and correct, but are they universally correct?

The "i" factor in the Schrödinger's equation is completely fundamental - and its counterpart appears in any equivalent description of quantum mechanics, too. In Heisenberg's picture, we have -i.hbar.d operator/dt = [H,operator], you see the factor of "i" again. In Feynman's path integral approach, the histories are weighted by exp(i.S/hbar) and again, "i" in the exponent is absolutely essential and equivalent in origin to the "i" in the other pictures.

I have no problems with “i” or complex numbers in general. They are indeed extremely convenient. The question is, however, can we do just with real numbers? “Can”, “not “should”?

Let me give you an example. We can write the Dirac equation as what you call “Shrödinger” equation, with “i” and all. Does that mean that we cannot do just with real numbers in this case? No. As I said (and I gave a reference to the derivation), we can write a pretty much equivalent equation (of the fourth order) for just one real function. My conclusion? While “i” may be prominent and convenient, that does not mean it is necessary.

You may insist that in this case the Dirac equation is classical, not quantum. While I admit that this is a pretty standard parlance, let me note that the equation contains the Planck’s constant, so in fact it is significantly quantum.

However, I certainly agree that more developed theories, such as QED, are second-quantized, unlike Klein-Gordon or Dirac equation. On the other hand, I mentioned (and gave a reference) that, for example, scalar electrodynamics in its real form, while being classical (if we use your definition), can be embedded into a (second-quantized) quantum field theory.

So I still tend to think that, in general, appearance of “complexity” does not necessarily eliminate “reality”. Is some specific real form useful? This is a different question.

 

[Fra]

Hello Akhmeteli,

It could be that we simply agree and we could leave it at that but here is a few more follow up comments.

What I tried to say is: What exactly is it that you reject? ie. what do you mean by collapse?

To me, the collapse is nothing but an information update. What "collapses" is simply the prior information state. And it's weird to deny the information update, it seems quite fundamental in any real measurement.

It seems you're idea is the a deterministic unitary evolution from initial conditions holds and is never interrrupted by collapses? The unitary evolution is basically nothing but an expectation of the future, based on the past. Then I wonder: Doesn't your expectations update at any point where you acquire New information?

It seems to be what you say about rejection is that: There is no new information, and all information is already at hand, and not information updates are necessary? And that maybe the idea is that the "apparent collapse" is only due to the limits of a real observer?

But as I see it in universe all there is, is collections of rela observers.

I agree nature doesn't care about armchair problems, but I think the implications seem deeper than that. I don't mean that it just becomes a "technical problem" and "hard" to compute, I mean that it may be physically impossible for one subsystem of the universe, to infer and hold, the complete information about the remainder so as to avoid a collapse. Another effect of hte same objection is problems of renormalization, where infinites can be interpreted as a symptom of the excess information that simply doesn't fit consistently.

If a subatomic partible interacts with it's environment as if the environment was perfectly unitaryilyu evolving with respect to the subatomic observer, then this would imply that an very large(large referring to the scaling of the environment) of information has to exists in the subatomic particle - which would then imply other weird stuff that we don't see - like gravitational fields.

The "computational problem" and representation problem are imo intimately related to renormalisation problems, zero shifts etc. This is somehow the problem you get for fully embracing violation of unitarity throughout the process. As I see it, it's only "expected evolutions" which are by construction unitary. There is no logic to me why unexpected event (like information updates) should be unitary.

/Fredrik

 

[Fra]

Well, you see, I don’t think I am against QM, as I fully embrace what I think is its crucial part – unitary evolution. What I question is collapse. But do you think collapse is part and parcel of QM?

Interesting perspective.

To me, the "core/hearth" of QM (that will survive even if the exact QM formalism as we know of today wont) is the observer-biased measurement perspective, because real observers are always biased, and this makes a difference to interactions.

Components of this is generally expectations as a function of prior information.

This has several components whose logic I think we should try to understand

- the logic of forming an expectations given current given information

- expectations of the self-evolution/action of information no measurement on system is performed (typically a self-evolution is IMPLIED by self-consistency, of inertial systems).

- The logic of revisiing or updating of expectations as NEW information arrives (for whatever reason; anyone trying to seek an "explanation" as to WHY new information arrives has IMO completely missed the point)

All these things; "to form and encode and expectation" requires a context, which I abstractly think of in terms of information processing. Sometimes the constraints of representation and processing, introduces a cutoff that limits the expectations, and also the actions following in line with expected evolution.

I think tha the unitary evolution of QM, is simply due to the fact that it's clear that unless new information arrives, there is no rational ground to change the expectations (EXCEPT the SELF-evolution IMPLIED by consistency of sets of non-commutative information: FLOWS are implicit; this is time; And it's this self-evolution that is always unitary). So I fully agree that _expected evolution_ as seen by any real inside observer, is always unitary. BUT, this does not mean that the actual evolution as future will show, will be ON EXPECTATION. When it's not, the observer is effectively exposed to a deforming force or interaction.

So I am personally extremel convinced tha the collaps and information updates are important for understanding how different forces and interactions emerge and encode. The undecidability implied in this picture I suggest may possible be cruciual for understanding unification too.

So the unitarity in QM, is IMO just a simple component, not in any way the core.

But a lot of people may choose to disagree.

/Fredrik

 

[Careful]

Thank you very much for your remarks.


Again, I am not sure I quite understand your argument. So Barut did something for some reasons. I did something else for some other reasons. I guess there’s nothing wrong with it, per se.

Problems with causality? I am afraid I just fail to see any serious problems. What’s happening actually? I just use boring algebraic (algebraic!) elimination of the scalar field, so the resulting model is pretty much equivalent to scalar electrodynamics.

Algebraic elimination? What the hell are you talking about, you have a not so easy PDE for the scalar field which one cannot solve exactly. I mean Barut had to solve Maxwell's equations and by doing so, he automatically chose the correct boundary conditions (in other words he picked the retarded green's function and did shut off the vacuum EM field). You really are talking nonsense here: you have to *solve* the equation for the scalar field in terms of the EM gauge potential (which is far less easier than solving Maxwell's equations) and you have to put the correct boundary conditions. Alas, the boundary conditions for the scalar field are not that easily chosen since you cannot shut it off in the infinite past. Really, this is by far (together with the instability) my most serious objection (because your math just isn't correct here). Look carefully at what Barut did, and you will understand what I say.

Do you think there may be problems with faster-than-light propagation? I cannot exclude such a possibility, but, first, the model is still local, and second, faster-than-light propagation, if any, must be also in scalar electrodynamics, because it is pretty much equivalent to the model. If the model stinks, the scalar electrodynamics stinks as well, so the model is in good company. Again, spinor electrodynamics is certainly better, but I don’t have equally good results for it.

From what you say here, I am rather damn sure that you will have problems with faster than light propagation. And yes, scalar electrodynamics stinks And therefore, your model is not in good company, we all know spinors to be of primordial importance.

Well, you see, I don’t think I am against QM, as I fully embrace what I think is its crucial part – unitary evolution. What I question is collapse. But do you think collapse is part and parcel of QM?

Oh yes, I am damn sure about it. Like I said, decoherence is no option for me, too nonlocal.

I don’t quite understand what you mean by “particle degrees of freedom”, but I do have an infinite number of degrees of freedom. Whether this is enough to describe reality, I don’t know.

What don't you understand? The point is that your theory as only 5 *local* degrees of freedom (1 real number for the scalar field -after you used gauge invariance- and 4 local degrees of freedom from the gauge field itself). QFT has an *infinite* number of *local* degrees of freedom because an operator is an infinite dimensional object. Your counting of infinities is just wrong.

First of all, I hope you don’t demand that I solve the measurement problem – that would be a tall order:-). Second, I am not sure I have to extend the theory. At least this is not obvious. Let me ask you first, what are we talking about? One or two matter particles or one or two photons?

That is really semantics, since whatever elimination trick you chose (and I am sure you pick the wrong one), the conclusion remains the same.

I agree. I have not tried to compare what I call “a” quantum field theory, where the model is embedded, with, say, QED or experiments. Nevertheless, it seems noteworthy to me that a local theory can be naturally embedded into a QFT.

That is indeed by itself a nice result and may serve as a new way of constructing QFT by making the necessary extensions I alluded to.

Are you sure you said what you wanted to say? How can “observation” explain ”observation”? Anyway, could you explain what you mean?

Well, observation is something which makes that we know only one possibility. Decoherence does not contain such ingredient, all it tells you is that the environment may cause a pure density matrix to (approximately) evolve into a mixed one.

On the other hand, as I said, “my” states have non-zero projections on the two-particle subspace in the Fock space. It is not quite obvious that this is not enough.

But that is not the point ! You cannot *observe* those projections because they are nonphysical in your framework.

 

[arivero]

Probably the point with Dirac is that locally you can do a heavy use of the gamma matrices to reimplement a lot of the properties of complex numbers. There is a strong relationship between division algebras and clifford algebras. Furthermore, akhmeteli, in your paper you rely in electromagnetic field, it is not really a discussion of Dirac or KG equations in a pure form. You could not recover the non relativistic limit, for instance (perhaps with an electrostatic potential?).

I have no problems with “i” or complex numbers in general. They are indeed extremely convenient. The question is, however, can we do just with real numbers? “Can”, “not “should”?

From my comparison, above in #34-35, between regularized classical mechanics and quantum mechanics, I'd say that the complex numbers are needed:

1) to use its conjugation to separate the double integral in a product of two integrals, each equivalent to having a measure of an observable

2) to have a dirac delta or an object similar to Fourier transform. This is probably the main role. Note for example that if you ask an exponential of combination of physical quantities to be dimensionless, you need to have a constant similar in units to Planck constant, to cancel the units of the product <x | p>.

If all we need for 1,2 is a conjugation, then all the four normed division algebras have it. But if we need further properties in the conjugation, it is a different history.

Is there a theory of Fourier transforms with octonions, quaternions or reals?

 

[Careful]

Probably the point with Dirac is that locally you can do a heavy use of the gamma matrices to reimplement a lot of the properties of complex numbers. There is a strong relationship between division algebras and clifford algebras.



From my comparison, above in #34-35, between regularized classical mechanics and quantum mechanics, I'd say that the complex numbers are needed:

1) to use its conjugation to separate the double integral in a product of two integrals, each equivalent to having a measure of an observable

2) to have a dirac delta or an object similar to Fourier transform.

If all we need for 1,2 is a conjugation, then all the four observable algebras have it. But if we need further properties in the conjugation, it is a different history.

Is there a theory of Fourier transforms with octonions, quaternions or reals?

You have seen some interesting points (I refer to my widening of the discussion by going over to involutive algebra's with a trace functional) Congratulations! I know there is a Fourier theory for (general) Clifford algebra's, so yes I guess the quaternions and the reals do classify in this category (it is just so that the class of Fourier integrable functions is going to look very different, and that is quite an important point :-)). For, the Octonions, I would not know how to do this.

But I guess you might google for Clifford, Fourier theory, Sommen and Soucek

 

[akhmeteli]

Algebraic elimination? What the hell are you talking about, you have a not so easy PDE for the scalar field which one cannot solve exactly. I mean Barut had to solve Maxwell's equations and by doing so, he automatically chose the correct boundary conditions (in other words he picked the retarded green's function and did shut off the vacuum EM field). You really are talking nonsense here: you have to *solve* the equation for the scalar field in terms of the EM gauge potential (which is far less easier than solving Maxwell's equations) and you have to put the correct boundary conditions. Alas, the boundary conditions for the scalar field are not that easily chosen since you cannot shut it off in the infinite past. Really, this is by far (together with the instability) my most serious objection (because your math just isn't correct here). Look carefully at what Barut did, and you will understand what I say.

Dear Careful,

Your comments are reasonable and generally correct, but that does not necessarily mean that I lied or screwed up, as this issue hinges on specifics, rather than on generalities.

Bear with me for a moment.

I would like to draw your attention to two crucial circumstances.

First.

I agree that I “have a not so easy PDE for the scalar field which one cannot solve exactly.” Nevertheless, I stand by my word that I eliminated the scalar field algebraically. How can that be, if our statements seem to contradict each other? Note that I did not say from which equation I eliminated the scalar field. You are absolutely right, I could not eliminate it from the Klein-Gordon equation, but I actually eliminated it from the Maxwell equation, where the right-hand-side is just the current for the scalar field. The standard expression for this current is given in Eq. (10) of http://www.akhmeteli.org/akh-prepr-ws-ijqi2.pdf (accepted for publication in IJQI). Still, it may be not obvious how one can "extract" the scalar field from its current. So here comes the second circumstance.

Second.

I eliminate the scalar field AFTER the gauge transform. Following Schrödinger, I perform a gauge transform to the unitary gauge, where the scalar field is real. What happens to the current in Eq.(10)? The part in brackets vanishes, and the current acquires a new form (ibid., Eq. (13)). Both Eq.(10) and Eq.(13) are pretty much copied from the Schrödinger’s paper in Nature, there is nothing new about them. But now you can indeed algebraically eliminate the real scalar field using Eq.(13) and the Maxwell equation Eq.(12)! I hope now it is obvious (the elimination is performed somewhat ”cleaner”, without any square roots, in http://arxiv.org/PS_cache/arxiv/pdf/1006/1006.2578v1.pdf ).

Two remarks.

First.

One could argue that the scalar field was not eliminated completely, as part of it (the phase) entered the 4-potential through the gauge transform. On the other hand, we are still left with electromagnetic field only.

Second.

One could argue that the elimination is not completely algebraic, as I used the Klein-Gordon equation later to obtain causal equations for the electromagnetic field. I don’t know. I think the elimination is still technically algebraic. But maybe this is not important. We are still left with causal equations for the electromagnetic field.

I’ll try to reply to your other comments later.

 

[akhmeteli]

I told you, akhmeteli

:-)

Well, I guess I always try to read other people's posts exactly as they are written, not trying to read their mind:-)

 

[akhmeteli]

This mode of explanation fails at some point because a given observer simply can't take into account the entire environment, because of limited information capacity and computational capacity.

The environment beeing an infinite information sink is a kind of "in principle" argument that I think fails becuase it ignores what should be obvious that no given, but still arbirary, observer can register and process infinite information.

And register and process information, and produce an expectation should be the essence of science. Any "in principle" argumetns that violate this, are IMO not quite scientifically sound.

One might be tempted to say that the collapse is the result of the incomplete observer, but this is not a technical problem, it's a fact that any physical observer IS incomplete. Why deny this?

The conincidental success of this picture so far, is because we do constrain ourselves to studying small subsystems where these information constraints doens't limit. But in cosmological models this would be grossly violated, and there is also a problem with this when trying to understand the actions chosen by nature even in the case of subsystems (Standard Model), that leads then to sets of possible (or consistent) actions.

/Fredrik

Dear Fredrik,

I think I know what you mean, but I just don't want to agree with you, and I don't feel I have to accept your arguments. I would have to accept them if there was some experimental evidence of collapse or some experimental evidence against unitary evolution. As far as I know, there is no such evidence, and while I respect your philosophy, I don't feel I have to accept it. I tend to believe that Nature can exist without any observers, however unscientific that may look to you. Therefore, I tend to think that Nature can be described without any reference to any observers. Of course, this is just my opinion. This is a matter of philosophy, and I try to avoid philosophical discussions in this forum. What seems to be important is that my rejection of collapse does not seem to go against experimental evidence.

 

[akhmeteli]

What I tried to say is: What exactly is it that you reject? ie. what do you mean by collapse?

For example, the projection postulate. According to it (and I am cutting some corners here), if an observation gives us a certain value of some observable, the system collapses into an eigenstate of this observable with the relevant eigenvalue. I don’t accept this, as this postulate contradicts unitary evolution.

To me, the collapse is nothing but an information update. What "collapses" is simply the prior information state. And it's weird to deny the information update, it seems quite fundamental in any real measurement.

It seems you're idea is the a deterministic unitary evolution from initial conditions holds and is never interrrupted by collapses? The unitary evolution is basically nothing but an expectation of the future, based on the past. Then I wonder: Doesn't your expectations update at any point where you acquire New information?

It seems to be what you say about rejection is that: There is no new information, and all information is already at hand, and not information updates are necessary? And that maybe the idea is that the "apparent collapse" is only due to the limits of a real observer?

But as I see it in universe all there is, is collections of rela observers.

I agree nature doesn't care about armchair problems, but I think the implications seem deeper than that. I don't mean that it just becomes a "technical problem" and "hard" to compute, I mean that it may be physically impossible for one subsystem of the universe, to infer and hold, the complete information about the remainder so as to avoid a collapse. Another effect of hte same objection is problems of renormalization, where infinites can be interpreted as a symptom of the excess information that simply doesn't fit consistently.

If a subatomic partible interacts with it's environment as if the environment was perfectly unitaryilyu evolving with respect to the subatomic observer, then this would imply that an very large(large referring to the scaling of the environment) of information has to exists in the subatomic particle - which would then imply other weird stuff that we don't see - like gravitational fields.

The "computational problem" and representation problem are imo intimately related to renormalisation problems, zero shifts etc. This is somehow the problem you get for fully embracing violation of unitarity throughout the process. As I see it, it's only "expected evolutions" which are by construction unitary. There is no logic to me why unexpected event (like information updates) should be unitary.

Again, I think I understand your point of view. Does it mean I have to accept it? No.
My take is quite different: I believe that, whether there is information update or not, unitary evolution “never calls in sick.” Again, nobody cares what I believe. What’s important, my point of view does not go against experimental evidence, as far as I can judge. Therefore, I don’t feel I have any moral duty to accept your point of view.

 

[akhmeteli]

Probably the point with Dirac is that locally you can do a heavy use of the gamma matrices to reimplement a lot of the properties of complex numbers. There is a strong relationship between division algebras and clifford algebras. Furthermore, akhmeteli, in your paper you rely in electromagnetic field, it is not really a discussion of Dirac or KG equations in a pure form.

I agree, but I don’t think there is such thing as pure Dirac or KG, at least there is no such thing in Nature: as soon as you have a charged particle, you have electromagnetic field as well. So I am not sure this is a weak point of the article.

You could not recover the non relativistic limit, for instance (perhaps with an electrostatic potential?).

This is certainly a good question, but to recover it with what purpose? To check if the model correctly describes experimental evidence? I am not sure this is very important, as the model is pretty much equivalent to scalar electrodynamics, and I think we have some idea about where scalar electrodynamics “rules” and where it “sucks”. Of course, we can introduce external conserved currents to describe, e.g., a KG particle in Coulomb field. Similar calculations were performed by Barut in his SFED (there are two differences: Barut uses the Dirac equation rather than KG, and he “smuggles in” second quantization in some form). The calculations are really cumbersome, and the results are pretty much what one would expect.
You’ll appreciate that any solution of the equations of scalar electrodynamics yields a solution for the model (after a gauge transform).
What would be really interesting is to consider the Aharonov-Bohm experiment in the model (again, you have to add external conserved currents). As electromagnetic field evolves independently in the model, I speculate that the interference would be influenced by penetration of electromagnetic field within the solenoid – the solutions of the model are sensitive to weak fields. On the other hand, the interference effects may be a result of the choice of the gauge.
However, it is not easy to study solutions of such complicated (not “complex”:-) ) nonlinear equations, even numerically.

From my comparison, above in #34-35, between regularized classical mechanics and quantum mechanics, I'd say that the complex numbers are needed:

1) to use its conjugation to separate the double integral in a product of two integrals, each equivalent to having a measure of an observable

2) to have a dirac delta or an object similar to Fourier transform. This is probably the main role. Note for example that if you ask an exponential of combination of physical quantities to be dimensionless, you need to have a constant similar in units to Planck constant, to cancel the units of the product <x | p>.

If all we need for 1,2 is a conjugation, then all the four normed division algebras have it. But if we need further properties in the conjugation, it is a different history.

Maybe I don’t quite understand your 1), that’s why it looks to me that it’s just about a method of calculation, not about a possibility of presenting quantum theory in terms of real numbers (but not just replacing complex numbers with pairs of real numbers).

As for your 2), you don’t need complex numbers to present delta-function: you can just use cosines, as delta is an even function

Is there a theory of Fourier transforms with octonions, quaternions or reals?

For reals, you can expand a real function in cosines and sines.

 

[Fra]

Hello Ahmeteli, of course you have no obligations to accept my argument if you don't find them convincing. I also obviously respect your position. Also, there are plenty of others that disagree with me as well.

Again, nobody cares what I believe. What’s important, my point of view does not go against experimental evidence, as far as I can judge. Therefore, I don’t feel I have any moral duty to accept your point of view.

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

 

[Fra]

For example, the projection postulate. According to it (and I am cutting some corners here), if an observation gives us a certain value of some observable, the system collapses into an eigenstate of this observable with the relevant eigenvalue. I don’t accept this, as this postulate contradicts unitary evolution.

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.

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)

/Fredrik