An anonymous reader wrote an interesting comment about my own pet theory (which I called theory of incomplete measurements, or TIM for short). This anonymous comment exemplifies both what I like and what I dislike about physics discussions these days:

- On one hand, it gives honest feedback, both positive and negative, and this hints at the kind of exchange of ideas that I enjoy when discussing with knowledgeable people. For instance, the anonymous reader cites a number of relevant sources, and makes a number of intelligent objections to my work.
- On the other hand, I cannot imagine any good reason why this reader would choose to remain anonymous, only bad ones. Too much work? The risk of backlash if anybody knew this person dedicated any time to fringe physics? A fear of true, bi-directional discussion? Whatever his reasons were, I suspect that I would not find it admirable.

I first wrote a quick reply, but I think that this comment demands a more prominent answer.

Below are excerpts from the anonymous reader’s comment, which I will hereby refer to as “Dr. Anonymous”, with my thoughts (not necessarily answers) inserted.

#### A theory, or just a descriptive framework?

1. The title would be better as “A general framework for describing physical measurements”, as this is all that is provided. Theories are rather more specific than what is given, and make predictions.

The obvious implication here is that my choice of title is a bit too grandiose and that the article delivers too little to merit such a title. But is it really true that the article only provides a framework for describing physical measurements? Is it true that it makes no predictions?

Granted, I never intended the TIM to be some theory of everything. What it is is a theory of *measurements*. What you need for this is to make predictions about measurements, not about gravitation or bananas. Otherwise, it would be called a theory of gravitation or a banana theory… However, with respect to measurements, the TIM does a number of predictions, starting I believe with equation (9) in the article. Most of these predictions may be qualitative or physical instead of mathematical or numerical. But that does not make them any less predictive, and it does not make them any less falsifiable, which is the most important thing for progress in science.

In particular, I can think of at least three predictions that are apparently suprising, if not shocking, to most physicists who commented about my paper. Specifically, according to the TIM:

- All our measurements of space-time can be reduced to properties of electromagnetic interactions (roughly, counting wave-fronts of electromagnetic waves).
- It does not make sense to normalize the wave-function on a continuum ranging from “minus infinity” to “plus infinity”, the correct way being to normalize it on a discrete sets of points that are in a time-like connected region. In particular, there is no such thing as an “instantaneous collapse” of the wave function.
- The laws of general relativity and quantum mechanics are not absolute, but depend on specific choices of physical measurement. In particular, two measurements with distinct resolutions may give distinct metrics. This is a generalization of the “scale relativity principle” dear to Laurent Nottale.

theories can either can be fitted or not be fitted into the particular framework given.

Isn’t it useful enough to have a relatively simple framework where both general relativity and quantum mechanics can be fitted?

Other frameworks are often less general, including, for example, ‘quantum’ logic (measurements represented by propositions), W*-algebras (measurements represented by algebras with a ‘complete’ set of idempotent elements), and convex set theory (states represented by convex linear measures) – see, eg, Hans Primas, “Chemistry, quantum mechanics and reductionism” (Springer), and Stanley Gudder’s “Stochastic quantum mechanics”, for overviews. However, more general approaches, at similar levels to your own, are the operational frameworks of Guenther Ludwig and of Charlie Randall and David Foulis – see, eg, the references at http://plato.stanford.edu/entries/qt-quantlog/ .

All these are interesting references. I’ve only scanned through those available on the web. But the main articles of Randall and Foulis for example require a subscription that I don’t have.

#### Personal gratification, or improvement of the collective understanding?

2. The paper is sincerely and honestly written, and often quite well written, and clearly the ideas have provided you with a personally satisfactory way of thinking about physical phenomena.

It is true that I was looking for an understanding of physics I would be satisfied with. But having made some progress, I hope to offer a way of thinking that is also satisfactory to others. It’s a worthy goal too. Last time I checked, there were over 2 million Google hits on “quantum measurement problem”. That can only mean that I’m not alone being dissatisfied with the “mainstream” way of thinking about physical phenomena.

However, there is often little more than this to be achieved by embedding known theories into a particular framework unless they (i) suggest better theories, and/or (ii) allow better ‘understanding’ of known theories.

The TIM does suggest better theories. It suggests a theory of relativity that would explicitly incorporate an additional parameter specifying how the measurement is done. This is pretty much what Nottale’s scale relativity or doubly-special relativity boil down to when looked at from a TIM point of view. The TIM also suggests an extended quantum mechanics where space-time coordinates do not commute and are discretized. Aren’t these suggestions enough for starters?

More imporantly in my view, the TIM also allows better understanding of known theories. Or, more precisely, it provided *me* with a so far more satisfactory illusion of better understanding, but as I wrote in my reply, it may just be that ignorance is bliss… Anyway, if I understood things correctly, the TIM shows for example that the “collapse of the wave function” is a necessary consequence of the fact that we demand repeatability from our measurements. One of the meanings of “theory of incomplete measurements” is that it suggests a theory for when measurements do not collapse instantaneously, or perfectly. That is yet another “better theory” that the TIM suggests.

Point (i) is the important one, as point (ii) is quite subjective – for example, one person’s interpretation of QM is very often another person’s anathema!

I would argue that “one person’s interpretation of QM is very often another person’s anathema” is a *bad thing*. Isn’t it embarassing for physics that QM is not defined sharply enough to leave room for multiple conflicting interpretations? After all, there is very little room for interpretation in Newton’s second law, or in Ohm’s law. So why do we need an interpretation of quantum mechanics at all?

Here, I apparently disagree somewhat strongly with Dr. Anonymous. Point (ii) is probably the most important one for quantum mechanics, because QM is not clearly understood yet, and that ought to be fixed. The TIM might be a step in the right direction, or maybe not. But at least, it’s an attempt to address this problem.

Dr. Anonymous is in very good company here. The problem of unambiguously interpreting quantum mechanics has resisted physicists for 80 years. So now it’s more commonly qualified of “philosophical” or “uninteresting”. In other words it became what Douglas Adams called Somebody Else’s Problem… Well, maybe it does not interest mainstream physicists. Maybe, actually, you need to be able to ignore the problem to become a mainstream physicist. But it definitely interests me. And it is definitely relevant to modern physics.

Since no better theories are in fact suggested, the content of the paper has only subjective significance – this is not necessarily a bad thing, but it does make it more of interest to philosophers rather than to mainstream physicists such as myself. Similar comments apply to the operational frameworks mentioned above.

I explained why I disagree with the assertion that no better theories are suggestd. But it is interesting that the final objective of all this discussion is to kick the TIM outside of “mainstream physics” into the lower realm of “philosophical interest only” with severely restricted “subjective significance only”. This is a pretty good way to invalidate the whole thing.

But the TIM has a physical significance that I think is not so easily dismissed. It shows that practically any equation in quantum mechanics or general relativity is only an approximation. And it tells physicists how to fix that. That’s a pretty broad challenge, and you cannot address this challenge by saying that the problem is only subjective. It is not a subjective issue that our laws of physics were not rewritten when we changed the definition of the metre from a reference solid to some distance travelled by light during such and such time. Why not? What makes a 1/r^2 law relative to a solid rod remain a 1/r^2 law relative to a ray of light? Seriously, is this a subjective question?

#### Is it OK to disagree with Einstein on general relativity?

3. I have little to remark on the relativity sections, as my only disagreements are at the ‘philosophical’ level – eg, I would have given Einstein a little more credit for physical justification of GR (principle of equivalence, and the recognition of curvature effects in SR from a mass rotating about a fixed point);

Actually, I do give credit to Einstein for these “physical justification”, only to explain why I believe they are not valid. So, despite a lot of admiration for Einstein, I simply point out a couple of logical flaws in the reasoning. It’s not flaws in the mathematical results, fortunately, but conclusions that do not follow from the initial statements.

The recognition of curvature effects in special relativity from a mass rotating about a fixed point is the easiest to explain. If a mass rotates about a fixed point, there will be a contraction along the path it follows, but not along the radius joining the mass to the central point. In other words, the ratio of circumference to radius is not 2pi, it’s a little less. See this page for a more modern discussion than my reference [12].

My objection to this is that you define a 3D surface that is curved, namely the 4D helicoid corresponding to the area that the radius covers over time. But we all know that you can find a 2D surface, like an helicoid or a sphere, in a 3D space, without necessarily inducing a “curvature effect” on the 3D space. In other words, the rotating mass in special relativity is not sufficient to imply a 4D non-euclidean geometry.

Again, there is nothing “philosophical” about this. It’s a discussion of whether there is a flaw of logic in the original reasoning, as I think there is, or not.

writing down the local Minkowski metric or geodesic equation does not give them any physical significance per se;

That is true. In the context where they are written in the TIM, the only point is to connect the traditional notation to the TIM notation, and to show why they are compatible.

the two-slit experiment is a poor justification for curved spacetime

It is part of a series of observations to show that light rays are a poor foundation to build an euclidean geometry on. I think that it is a better justification to say: “we don’t know how to build an euclidean geometry because the closest we have is not euclidean in this and that case” than to incorrectly infer that a 3D curved surface implies a 4D curved space-time. But that’s just the biased opinion of someone who saw a logical flaw in the generally accepted justification, and spent some serious time finding a justification he thought might be less flawed to salvage the beautify theory of general relativity.

and the existence of the Planck length could just as well be used to argue against instead of for scale invariance of physical laws.

This is how it is used in the TIM. If there is a scale-invariant constant in physics, a scale-invariant physics requires a pretty serious rethinking, similar to what Laurent Nottale attempted with his scale relativity. Again, this is a pretty serious issue that one cannot just paper over with a “philosophical question – dismissed” sticker… 🙂

But the majority of “mainstream physicists” are just not ready to make their space-time fractal, as this kind of theory apparently demands. And they are probably right that a lot of assertions in the current presentation of scale relativity sound ad-hoc, that the mathematics is still a bit inconsistent and immature, and so on. But that only means the solution is not completely satisfying yet, certainly not that the problem is not there. In my opinion, this issue is a thorn in XXI^{st} century physics about as painful as the constancy of the celerity of light implied by Maxwell’s equations was for XIX^{th} century physics. Which is a pretty long-winded way of saying it’s worth solving.

#### Linearity and complex Hilbert spaces

The objections Dr. Anonymous makes to the quantum mechanics part are by far the most interesting ones. They are also the most technical ones, and it’s possible that my answers will be unconvincing, if only by lack of space. Expect more iterations here, and possibly some serious changes in the paper to make things more precise or easier to understand.

4. I think that the quantum sections are a little misleading to a careless or non-expert reader(though not intentionally so!), both in terminology and presentation

It is definitely not my intention to mislead. I think that a reason they may be misleading is because it might seem like I attempt to deduce all of quantum mechanics, when in reality there are things I know I deduced, and others I simply borrow because we know from quantum mechanics that they work. The line is a bit blurred, not the least because I tried to push as much as I could into the “demonstrated” half, and sometimes failed (for the moment at least).

the arguments given to introduce linear operators and wavefunctions and the like have little substance,

This is pretty vague, but fortunately specific examples are given later, which I will address then. I acknowledge that the TIM allows only to derive certain properties of QM, but not all of them. What is missing may simply be because I don’t know how to derive it, because it derives from other reasons, or because it’s not necessary for a working physics. I found cases in each category.

- Linearity or real-valuedness appear to be not necessary for a working physics, because the TIM highlights counter-examples. So they are considered as added hypotheses that make some cases easier to solve. For example, the “principle of superposition” is demoted to a simple “hypothesis of superposability” which, when applicable, makes more theorems and results applicable.
- The evolution law, for example the Schrödinger equation, is something that I have not entirely been able to derive. There are multiple derivations of Schrödinger, but none of them seems to apply too well in the TIM. I have, however, been able to offer a quick qualitative “sanity test”, which requires an ad-hoc, but not unreasonable hypothesis. So that’s all there is in the TIM article, and I would certainly understand if Dr. Anonymous or anybody else qualified this section of “having little substance”, because it does.
- The collapse of the wave function, on the other hand, can entirely be deduced from the need to have repeatable measurements. I have several lines of reasoning that all lead to the same result, so I’m pretty convinced that this is solid.

and do not lead even to the usual Hilbert space description.

By this, I think that Dr. Anonymous means primarily that only a real-valued HIlbert space is implied, not a complex-valued one. Since she makes the objection more explicit below, I’ll address it there. Another possible interpretation is that it’s always a finite-dimension Hilbert space, even for space-time coordinates. But this is intentional.

Now, if the TIM is not deriving the “correct” Hilbert spaces, isn’t it easy to show that it makes incorrect predictions? In other words, instead of not being a theory, it would simply be a wrong theory. So these little differences are what makes things interesting and challenging.

The best one can say is that standard QM fits into your general framework, but that very little of it is implied by this framework.

If it were only the collapse of the wave function, it would already be an interesting result. But I think that more is implied, specifically practically all the axioms but the evolution law. If it’s not implied by my reasoning, it is important to tell me why not. The specific remarks that follow are not entirely convicing to me.

#### Probabilities and evolution

4a. One can certainly, and trivially, rewrite measurement probability distributions p=(p1, p2, …, pn), which live on the n-simplex, as “unit vectors”, psi=(psi1, psi2, …,psin), which live on the surface of an n-ball (where psi1=sqrt[p1], etc).

This rewrite, made in section 4.1 of the TIM, is indeed mathematically “trivial”, but a lot is trivial in hindsight. The really difficult question is: if it’s so trivial, why aren’t we taught more often that the state vector in quantum mechanics is the only form a state describing predictions of future measurement results can take? It’s possible someone else wrote it before me, but I know this is not something I’ve been taught, and I tried to find references to such a result after discovering it. Honestly, it was so “trivial” I thought impossible no one else would have made this observation. But it is still not a standard staple in QM teaching.

However, there is no reason whatsoever in the TIM framework to postulate that measurements and evolution are linear with respect to psi, rather than being linear with respect to p (or indeed with respect to any any other function g=(g1, g2, …, gn) of the probabilities – eg, g1=exp[p1]).

Indeed. This is why it is not postulated in the TIM.

For measurements, it is only observed that *if* a measurement is real valued, and *if* psi is the probability vector as defined in the TIM, then one *can* define an operator that 1) is linear in psi, 2) has the measurement results as eigenvalues with specific vectors as eigenvectors, and 3) can be used to compute the right expectation value. So from these properties, one has to deduce that it’s the “observable” of quantum mechanics.

For the evolution, I have to introduce it ad-hoc, and I think it is pretty clear in the article. I can’t say that I’m happy with it, and I honestly don’t have a satisfying answer to offer yet.

And it is merely ex post facto analysis in the light of QM to suggest such postulates are even reasonable!

It is normal to use “ex post facto analysis in the light of QM” to find the results of QM, just like it is reasonable to use “ex post facto analysis in the light of metrology” to find a definition of time that is based on the TIM, but fits our standard definition of time pretty accurately. So this is not cheating, this is simply verifying that QM “fits”.

One needs something else to ‘derive’ a physical significance for linearity in psi (eg, Kaehler manifolds).

Here, I’m really lost. I don’t see how Kaehler manifolds can give a physical significance to anything. The physical significance for linearity in psi is that if you construct a linear operator on psi with the right eigenvalues and eigenvectors, it trivially allows you to compute the right expectation value. What else do you need?

In the absence of this something else, it would seem far more reasonable, for example, to postulate that the evolution and operators should be linear with respect to the probability vector p !

I don’t think so. To get the eigenvectors, you need the nxn matrix. Then, to get the right expectation value (i.e. average observed value), it cannot be linear in p.

#### Complex or real Hilbert spaces?

4b. Even when one formally writes down a vector psi as above, it only inhabits a real Hilbert space, rather than the complex Hilbert space needed for QM (eg, to explain the physical double-slit experiment, the Aharanov-Bohm effect, the singlet state, etc).

The argument that complex Hilbert spaces are needed for QM is a frequent one. If that is true, if QM probabilities are somehow special and dependent on complex numbers, how can we get a “double slit experiment”, i.e. interference, with classical waves over water? Only classical probabilities are at play in that case.

In reality, complex numbers are only a tool in the computation of probability, to sum probabilities that are not independent from one another. In the water interference scenario, like in the double slit experiments, the Aharonov-Bohm effect or the singlet state, you have probabilities that are “entangled”. It’s simpler to explain with the water waves, but it’s fundamentally the same mathematics in all cases. The waves can propagate through two slits in a wall, but they come from the same source. The height of water at one point (hence the probability of presence of water particles) can be computed by summing the contribution of the wave coming from one slit and from the other. But these contributions are not independent from one another. They are always at a constant phase from one another, which for water waves is a function of the difference in distance between the point where you compute the probability and each of the two slits.

I tried to illustrate how complex numbers emerge in the subsections “Trajectory measurement” and “Normalization of the wave function” in the TIM paper, but to be honest, these are some of the sections I consider the least satisfying of the whole article… It attempts to demonstrate that having a field of “probability of presence” is the correct way to represent the predictions about an experiment where the question is “where is a single particle”, but it does not make a very good case that this is the same thing as the wave function in traditional quantum mechanics.

Hence, while the formulas written down “look” quantum mechanical, much more is needed.

A little more may be needed in terms of explanations. But f the formulas look quantum mechanical and behave like quantum mechanical formulas, why can’t I legitimately use the quantum mechanical toolset to solve them? And if the predictions and the formulas are quantum mechanical, why is “much more” needed?

#### Can one use an operator notation for a non-injective and non-linear operator?

4c. The notation M^|psi> for general nonlinear operators, and M^|psi>=m|psi> for eigenvectors, is misleading. One does not have, for example, the property

(M^ + N^) |psi> = M^|psi> + N^|psi>

in general, even though the notation suggests it.

The TIM article is very careful to not use the property you cite, and goes to extraordinary lengths to explain that it does not hold in the general case. That’s probably one of the key point of the whole article. Isn’t it a bit disingenuous to say that the notation suggests something when the text explicitely says the opposite?

The “hat-M” notation is an intermediate step in reaching the “inverse-hat-M” or “check-M” notation, which does have the property you cite. It would be very confusing to change the notation meaning “apply an operator” just because some operators are linear and some are not, don’t you think? It’s a bit as if I insisted that the notation f(x) implies a smooth function because I’m used to sin(x) or cos(x), and then required that you use a different notation for any discontinuous functions…

It would be more accurate to represent nonlinear operators as functions mapping the sphere to itself, eg, M(psi), and to refer to vectors satisfying the property M(psi) = m psi as ‘fixed points up to scaling’ or similar.

It would not really be more “accurate”, it would be a change of notation that would obscure the fact that hat-M and check-M are cousins, hat-M being constructed to be a linear operator based on the eigenvectors (or fixed points) identified with hat-M. My notation would only be inaccurate if I did not carefully explain that the notation “hat-M” is used for a non-linear operator, or if I used properties that implicitly require linearity. I do not think that I do either.

As I point out in my short answer, in the general case, M(psi) would be even worse as a choice of notation, because it implies that hat-M is a function, and in the most general case hat-M is not injective, something which the text also insists on. We have a mathematical object which is a “random function with fixed points”, and this strange thing will do strange things to your notation. The best I can do is to take the closest notation, and the closests / simplest notation for what I wanted to express is the operator notation.

#### Collapse limited by the speed of light, or normalization limited by the speed of light?

4d. The QM of entangled systems (more than one particle) is not easily explainable by postulating that collapse is limited to the speed of light, and does not appear to be compatible with recent experiments by Gisin et al (eg, http://prola.aps.org/abstract/PRA/v63/i2/e022111 and http://lists.paleopsych.org/pipermail/paleopsych/2006-April/005093.html,).

The article does not postulate that collapse is limited to the speed of light. It only focuses on how to normalize the wave function, and it points out that collapse is not instantaneous, which is not the same thing. Our computation of probabilities must obey constraints set of the experiment. One of these constraints is that we cannot collect data or setup entanglement between particles faster than the speed of light. Consequently, the space-time points that are summed in a single probability normalization may not be simultaneous in a given reference frame. The experiments listed would only tend to prove that this is the correct approach, not to invalidate it.

For example, suppose Alice and Bob live in spacelike separated regions and share a singlet state (or ensemble thereof), and each decides to measure spin in directions a or b, at random. The probability for Bob to measure spin -up is 1/2 in any run. Now, if collapse is at the speed of light, then his result cannot depend on whether Alice has found spin up or spin down for her measurement – it can only depend on local variables in his vicinity. But in fact Alice and Bob find perfect correlations when they happen to make the same choice of measurement direction – which cannot be explained via such local variables (Bell’s theorem). So, whatever entanglement is, its effects cannot easily be explained, and indeed are exacerbated in strangeness, by restricting collapse to the speed of light.

First, there is a somewhat ambiguous use of Bell’s theorem in your sentence. Bell’s theorem does not prove that there are no hidden variables in quantum mechanics, it shows how to detect if there are any. Only experiments using Bell’s theorem such as those of Alain Aspect have led us to think that there are no hidden variables by applying the theorem. I think that this is what you meant…

Second, in your experiments, Alice and Bob’s particle can only be entangled if we made them entangled to start with. For example, they can be the result of two particles generated by the same event. These particles then travel at most at the speed of light. So entanglement itself is carried at most at the speed of light.

More to the point, when the experiment is performed, no correlation can be established until the data of both Alice and Bob can be collected in a single point. All the TIM is suggesting is that the normalization condition, if seen as a sum of probabilities being one, can only be collected at that point, where both Alice and Bob are in the past light cone. What will matter is that Alice and Bob’s events are in the past light cone. It does not matter if they occured at the “same time” or not, which is good, because the notion of “same time” is quite obsolete in relativity. One of the experiments I found funny, because it’s done in such a way that both Alice and Bob think they made the experiment first. Well, it does not really matter, they both did it in the past for the guy collecting the data!

#### The final word…

I apologise for corresponding anonymously, but after a friend pointed me to your paper and said that you were keen for feedback, I thought this might be helpful as a once-off thing.

That makes me sad. It all started as a helpful and intelligent discussion. But all of a sudden, it’s no longer a discussion, it’s just a one-way, once-off “thing”, a quick attempt to bring the poor fool out of his erroneous ways before moving on to more important “things”.

Oh well… Better having anonymous discussions than no discussion at all… 🙂