State-Observable Duality (John Baez series)

[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

 

[Careful]

:-)

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

Too one dimensional for me, like this you miss all the fun

 

[Careful]

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.

I just looked at those equations, I didn't give it a single moment of deep reflection, but shouldn't you just not try to avoid precisely this kind of stuff?

For example, your scalar field is a non-local expression because of the square root of the d'alembertian and then, strictly speaking, you should still substitute this monstrocity in your equation (11) to solve for the gauge field. I have no idea at first sight how to control causality here! So yeh, I guess my conclusion remains here that you are screwing up mathematically (I never ever implied you lied or were dishonest though).

Moreover, it appears to me that this ''technique'' is not going to work for spinor fields.

 

[akhmeteli]

Too one dimensional for me, like this you miss all the fun

I guess I had all the fun I needed (and then some) when my ex-wife read into my words whatever she wanted:-)

 

[akhmeteli]

I just looked at those equations, I didn't give it a single moment of deep reflection, but shouldn't you just not try to avoid precisely this kind of stuff?

Dear Careful,

Thank you very much;
actually, you did all I asked you to do (and I do appreciate it), and you saw everything I expected you to see.

Now, first things first.

Before I try to reply to your specific comments, let me ask you an important question: does equation (15) in http://www.akhmeteli.org/akh-prepr-ws-ijqi2.pdf qualify as algebraic elimination of the scalar field, in your book?

 

[Careful]

Dear Careful,

Thank you very much;
actually, you did all I asked you to do (and I do appreciate it), and you saw everything I expected you to see.

Now, first things first.

Before I try to reply to your specific comments, let me ask you an important question: does equation (15) in http://www.akhmeteli.org/akh-prepr-ws-ijqi2.pdf qualify as algebraic elimination of the scalar field, in your book?

Sure, but that was hardly the point. However, this elimination will get problematic in realistic physical situations since the argument will become negative.

 

[akhmeteli]

Sure, but that was hardly the point. However, this elimination will get problematic in realistic physical situations since the argument will become negative.

Again, generally, this may be a very reasonable comment, but there is no such potential showstopper in the improved version in Section II of http://arxiv.org/PS_cache/arxiv/pdf/1006/1006.2578v1.pdf , where no square root is needed (I rewrote the Klein-Gordon equation in terms of \Phi=\phi^2 (where \phi is the scalar field) and eliminated \Phi using Eq.(13) of that preprint, where there are no square roots).

 

[Careful]

Again, generally, this may be a very reasonable comment, but there is no such potential showstopper in the improved version in Section II of http://arxiv.org/PS_cache/arxiv/pdf/1006/1006.2578v1.pdf , where no square root is needed (I rewrote the Klein-Gordon equation in terms of \Phi=\phi^2 (where \phi is the scalar field) and eliminated \Phi using Eq.(13) of that preprint, where there are no square roots).

Correct, but now you pay another price: B_{\mu} B^{\mu} may become zero which blows up your equations too Moreover, there is something else here, you take einstein summation, but your equation (12) gives 4 independent equations (I did not mention that before) which should all give the same result ! This imposes a severe constraint, so it is actually sufficient that one B_{\mu} vanishes.

Anyhow, as I told you before, the physics of your elimination ''stinks'' (no insult intended). In my experience, if the latter condition is fullfilled, the mathematics usually gets wrong too.
If I were you, I would redo the whole thing and reframe it in the Barut elimination: that one is physically and mathematically sensible.

 

[akhmeteli]

For example, your scalar field is a non-local expression because of the square root of the d'alembertian

First, it may be non-local in some sense, but it is certainly local in some sense, because the scalar field in some point is fully defined by 4-potential and its first temporal derivative in an infinitesimally small spatial vicinity of that point, at the same time.

Second, as I said, I can do without the square root, in the first place – see Section II of http://arxiv.org/PS_cache/arxiv/pdf/1006/1006.2578v1.pdf (the derivation there is “cleaner” than in the IJQI article), although I appreciate that you may dislike the quotient there as well.

and then, strictly speaking, you should still substitute this monstrocity in your equation (11) to solve for the gauge field.

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

I have no idea at first sight how to control causality here! So yeh, I guess my conclusion remains here that you are screwing up mathematically (I never ever implied you lied or were dishonest though).

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

Moreover, it appears to me that this ''technique'' is not going to work for spinor fields.

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

 

[akhmeteli]

Correct, but now you pay another price: B_{\mu} B^{\mu} may become zero which blows up your equations too

That is so, but this is a much lesser price, as the set of points where B_{\mu} B^{\mu}=0 has fewer dimensions than the set of points where the expression under the square root is negative. So one can hope that it would be possible to perform something like extension by continuity. Anyway, I make the caveat “at least locally”. To summarize: this does not look like a big problem.

Moreover, there is something else here, you take einstein summation, but your equation (12) gives 4 independent equations (I did not mention that before) which should all give the same result ! This imposes a severe constraint, so it is actually sufficient that one B_{\mu} vanishes.

I do not agree that this imposes a severe constraint. Again, what you say is logical and generally correct, and you are raising a very good question again, but, as I said, this issue hinges on specifics, not generalities.

Please bear with me again for a moment, as this is another crucial point.

I agree, equation (12) gives 4 independent equations. But! One of them, that with subscript \mu equal to zero, is “more equal than the others”, and it is this equation that must be used for elimination of \Phi.

How could that be? You see, I am trying to pose the Cauchy problem. I assume that I know B^{\nu} and their first temporal derivatives in the entire 3D space at time point x0. Only the component of equation (12) with subscript \mu equal to zero allows us to calculate \Phi based on this input data only, as this is the only component of Eq.(12) that does not include any second temporal derivatives of \B^{\nu} (as the second temporal derivative in the Dalambertian is canceled with a term from B^{nu}_{,\nu 0}. So I use the zeroth component of Eq.(12) to eliminate \Phi and substitute the relevant expression for \Phi in the first, second, and third components of Eq.(12). So why do I NOT have “severe constraints”? Because these components contain and now define the second temporal derivatives of B_1, B_2, B_3.

What do we still need to pose the Cauchy problem? We need the second temporal derivative of B_0. How do we get it? The short answer is by using the current conservation and the Klein-Gordon equation.

Anyhow, as I told you before, the physics of your elimination ''stinks'' (no insult intended). In my experience, if the latter condition is fullfilled, the mathematics usually gets wrong too.

I don’t quite see why physics “stinks”. At least it doesn’t stink more than physics of scalar electrodynamics. I do have causality and locality as I have a well-posed Cauchy problem. And I don’t know why mathematics stinks. I just don’t see any showstoppers.

If I were you, I would redo the whole thing and reframe it in the Barut elimination: that one is physically and mathematically sensible.

Again, I think the elimination I use is physically and mathematically reasonable. Furthermore, I am not too shy to say that I strongly prefer algebraic elimination.

 

[Fra]

I just don’t see any showstoppers.

I skimmed your paper, and in what context does this belong, and what's the problem to which this paper aims to be suggestions towards a solution?

Is your objective to restore realism because it may seem according to cetain ways of reasoning to be preferred by an "objective science"? And that abandoning realism is like lowering the ambition?

Or is there some other agenda in which is a piece of a bigger puzzle?

/Fredrik