# State-Observable Duality (John Baez series)

Gold Member
"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

## Answers and Replies

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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
Gold Member

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

Gold Member
Whats up with that?

Gold Member
"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.

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

"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, John Baez 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
Gold Member
Whats 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

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.

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.

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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 :-)

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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:-) ).

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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.

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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.

Careful said:
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.

Careful said:
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:-) ).

Careful said:
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.

Careful said:
(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.

Careful said:
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.
Careful said:
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.
Careful said:
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.
Careful said:
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
Careful said:
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

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 definetly 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.

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Gold Member

State-Observable Duality (Part 3)
Posted by John Baez

This 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
Gold Member
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 :-(

Gold Member
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

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.

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

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
Gold Member
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.

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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.

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