@Renan, I'm very impressed that you managed to get a 22 page article into PRL! ;) More seriously, this looks like a rather significant piece of work that I'll go through in more detail later - thanks for posting the link.

@Lev, I agree that commutivity has meaning independent of the passage of time, however this is a very clear way (in my view) of understanding much of the significance of non-commuting observables.

I suppose one of the frustrating things for me here (and I'm not really cross at you, it is a general trend, and one that I think is probably interesting for discussion here or in a related forum) pertains to the aspect of your question that highlights *general physical*, which you claim that most respondents aren't responding to. Here we must ask what does 'physical insight' constitute as opposed to 'mathematical insight'. In most cases, I expect (and please correct me if I'm wrong) that people are actually asking for a 'classical interpretation'. I can give a good example of a very dear friend and colleague who when teaching refers to 'quantum double think' when he gives a nice, classical, easy to understand classical explanation/analogy of quantum phenomena. I'm always frustrated by this, because the analogies usually fail at some level of approximation that he's not telling you about. From my point of view, I now routinely find the classical interpretations more bizarre and difficult to grasp than the quantum ones, so that quantum insight (informed by mathematics) seems far more real and physically motivated than the alternatives.

When it comes to Einsteinian 'smell', it would be quite arrogant for me to suggest that I have a fraction of his insight, but one of my goals as a teacher/supervisor is to help my students (undergraduate and postgraduate) to develop their intuition in the quantum space. Having said that, if 'the ship were going down', we'd have some clear evidence for a failure of quantum physics, and conversely (and perhaps surprisingly), quantum mechanics is far more powerful and important that ever and the proof of this can be seen from the fact that the 'spooky' effects are all being demonstrated and resulting in new devices.

In my point of few this is an intrinsic feature of space-time. Variables which do not maintain any dependency on path or time have no conjugate value.

Please refer to the related question...

https://www.researchgate.net/post/Which_physical_variables_do_not_exist_in_conjugate_pairs_and_do_not_have_an_uncertainty_relationship_in_quantum_mechanics?ev=tp_feed_post_xview

Georg, are you suggesting that the dependance on the path or time is this "general physical reason"?

And what is this mysterious "intrinsic feature of space time"?

However, you still didn't answer the question: What ('physically') makes two physical entities conjugate?

Here is a copy of the answer Andrew Greentree gave in the space for a related question:

"Dear Lev,

The argument regarding the physical insight comes from the time-ordering that is implied by the commutation properties of the two observables. Every time you perform a measurement you must project the system into an eigenstate of the measurement apparatus. If you perform sequential measurements using two different observables, if the measurements commute, i.e. are in some sense on different properties of the system, then the order is unimportant and no generalised uncertainty relationship holds. Conversely, if the observables do not commute, then the measurements are, again in a very real sense, interfering with each other.

In the case of position and momentum, a position measurement forces the particle into a definite position eigenstate. But to define such a position eigenstate requires a summation over all momentum eigenstates, and vice versa. This follows immediately from Fourier theory.

This to me is the physical insight behind the relationship."

Andrew, if I understood you correctly, the reason has to do with the way "variables" interact temporally, isn't it?

Dear Lev, please see my answer to the other questions in order to resolve the "mystery"...

In my prospect the physical fundamental principle for these relations is the quantization of space-time through interactions between quantum systems (please see my answer to the linked question). Every interaction means an action on both partners, which are quantum systems. Thus these quantum systems do have a smallest delta in time (dt) and accordingly in space (ds). These smallest differences between two interactions define the norm in space-time, which is the constant speed of light c. Thus c equals to the ratio of ds/dt. In that way conjugate physical variables are bound to each other by the norm of space-time and do not commute.

...

https://www.researchgate.net/post/Which_physical_variables_do_not_exist_in_conjugate_pairs_and_do_not_have_an_uncertainty_relationship_in_quantum_mechanics?ev=tp_feed_post_xview

Andrew, here is a part of today's Diederik Aerts' answer to a related question which does not quite agree with yours:

"Hence, commutativity (and non commutativity) have in depth a physical meaning that is independent of considering time. There is also an effect related to time consecutive applications of the two measurements, but that is rather a secundary effect. This can be seen also very well in reference to classical physics. There all observables commute, since they are represented by functions on state space, and not by operators in Hilbert space. However, consecutive measurements also in classical physics often do not commute, in the sense of 'giving rise to the same outcomes when executed in reverse order'."

Georg,

"Thus c equals to the ratio of ds/dt. In that way conjugate physical variables are bound to each other by the norm of space-time and do not commute."

Thanks for you answers.

However, as one compares all the answers, including yours, to my and to the related question (by Derek Abbott), I still can't see any reasonably clear answer to the posed question.

OK Lev, I understand. The case is a little bit tricky since your question aims at the core problem of current physics. I just took a look at your chapter: "Nature is Fundamentally Discrete but Our Basic Formalism is Not" and found very familiar positions to me. E.g. chapter 4: "The need for a fundamentally new formalism to elucidate the “discreteness” and to guide the development of new physics." is coherent with my statements in my upcoming paper "the realistic principle of quantum theory" which will be uploaded by the end of this week to ResearchGate. There my proposition is to derive everything what is existent from the bare existence itself. Surprisingly one can find a formal system of existence which can be interpreted as the platform where physics happens. And this platform is totally discrete!

Georg,

My question is not really "tricky". Here is the background.

With the development of our new formalism (ETS), or new formal language (see introduction to it at http://www.cs.unb.ca/~goldfarb/BOOK.pdf), I started looking more carefully at the state of physics, including of quantum mechanics. And I found that during the last century, instead of facing honestly the discovered fundamental difficulties---related to the inconsistency between the continuous formal apparatus and the ubiquitous discreteness---we have been continuing on the Ptolemaic epicycles-like path (see https://www.researchgate.net/post/Are_you_concerned_about_the_central_inconsistency_in_physics---between_the_old_continuous_formalism_and_discovered_in_the_last_century_ubiquitous_discreteness_in_Nature).

The ETS formalism proposes a fundamentally new approach to "physical reality", including a constructive approach to the *concept of discreteness* in Nature, as opposed to the current and I believe futile attempts to reshape the classical continuous formalism, including the concept of space. The reason for the futility is (briefly) that any formal language has an intrinsic structure which cannot be modified without destroying its integrity.

Of course, the big (human) problem is this: How many physicists are prepared to start from the very beginning? ;--) Yet, how else can we begin to do physics if the basic math. language needs to be changed?

Hi,

I agree with Andrew Greentree's answer. In fact I just published a paper on exactly this topic : http://prl.aps.org/abstract/PRL/v109/i19/e190403 (http://arxiv.org/abs/1105.4014)

An important part of this discussion is the observation that classical mechanics can be expressed in terms of commuting observable operators and classical wave-functions in the Hilbert space as already pointed out by Koopman and von Neumann. Therefore the essential difference between quantum and classical mechanics is the commutativity.

Non-commutativity means that the order of measurements is important. In the following preprint these same ideas are shown to be valid in special relativity http://arxiv.org/abs/1107.5139, where we show how classical mechanics can be expressed in terms of spinor fields.

Renan,

As you can see from the above there is some unsatisfactory disagreement.

Moreover, you still didn't answer the original question. Please read it again.

I'm surprised: in almost all answers, people don't pay reasonable attention to the question. ;--)

Hi,

Two variables with irreducible measurement conflict implies that their observable operators do not commute. When such commutator is a constant proportional to i (not possible for finite dimensional systems), we call that pair of operators "conjugate" with each other. The proportionality constant indicates the degree of conflict in the measurement.

@Lev, you are right, how many scientists are open enough and will spend a lot of their valuable time to build a new basis for things which are apparently working well, in terms of quantitative formalism.

I have just uploaded the current draft of a possible basis called the formal system of existence. In my opinion one do not need to go far until one reaches the basic conclusion of our existence, which I tried to make understandable in the "analytic argument" at the very beginning, but the impact of this conclusion is game changing.

Georg: "how many scientists are open enough and will spend a lot of their valuable time to build a new basis for things which are apparently working well, in terms of quantitative formalism."

Georg, all serious scientists are, obviously, concerned about their "valuable time": after all, we have just one life to contribute. The difference among them is that they are concerned about different scales of time (at which their contributions are evaluated). Of course, *very* few can 'smell'---relying on the Einstein appellation to his "nose"---when the 'ship' is going down. ;--)

Renan,

You are still quoting the formal (math.) definition of conjugate variables via their non-commuting Hermitian operators, while I'm asking for the 'physical' reasons / intuition why they are "conjugate", independent of the Hilbert space formalism.

OK Lev, thus the conclusion should be to open a worldwide interdisciplinary institute for building a new foundation of science.

Georg, I'm not sure if this is the right way to go, although, today, it has been a popular path (under various names) but with very little to show for it.

I believe we need to start with a concrete proposal: although it is true that the change is in the air, we need a concrete proposal on the table to be critically evaluated and to suggest us in these scientifically unprecedented (very transformative) times what has been missing from the present scientific picture.

I definitely agree with you Lev, the only question is, how do we conduct the elaboration of such a concrete proposal, since it can not be worked out by only few people, because of its extent and for reasons of acceptance too. Since I am both, scientist and entrepreneur, I also think in terms of feasibility.

Georg, this is an important practical question. I will address this as it relates to the ETS formalism

Since I'm more competent in the field of information processing, in this area I have more definite projects in mind.

However, in physics, as always, there appears to be two general directions to pursue, theoretical and experimental.

First, it is the reformulation of one of the old known laws in the new language, the language related to the proposed structural representation ("struct"). Of course, one can always attempt to look at the new law relating energy to the size of the struct.

Second, the experimental direction is related to the verification and discovery of the *event* structure of some basic physical processes (light, etc.): we must verify if all real processes conform to the proposed event structure, e.g. which concrete sequence of events is associated with the propagation of light.

Hi Lev,

nice discussion :-) and funny answers :-)

I don't know your formalism (ETS) right know, but with respect to conjugacy it might be interesting for you to go back to the (geometrical) definition of conjugate variables (if I remember correctly, you should at least go back to polar geometry in the 19th century ;-))

Time and energy are needed to complete/saturate the (four-)vector formalism used nowadays for "relativistic" descriptions; commutativity and noncommutativity are subsidiary concepts as in planar (i.e. standard complex) geometry you'll always have commutativity, and in higher dimensional "curved" geometry you'll automatically find noncommutativity dependent on the objects you are going to base your description on.

The main reason (at least to my understanding) for this type of conjugacy is the polar relation of \vec{x} and \vec{p} (or better \vec{v}) with respect to circles or spheres, i.e. to a certain quadric. So the question seems always to be the identification of poles and polars. And of course you'll find such structures repeatedly in physics by the very construction of the objects which are used to describe the observables...

Rolf,

Do you consider what you described a *physical* insight? ;--)

Depends on how you understand "physics"-if you want to relate (general perceivable) observations to mathematical models, you will be forced to use geometry and geometrical models to formulate any "physics". So my answer is "yes"! :-)

OK, Rolf, let me clarify my point.

When we decide to introduce in physics the concept of "conjugate" variables---as it relates to time and energy, to take an example---this concept is supposed to reveal something really important about the physical relationship between the two variables. The relationship between two *important* variables is supposed to have some deeper / wider physical meaning and not just some formal expression. What is this wider physical meaning?

An important point: conjugacy arises and is important in classical mechanics. Suppose for example that you have a large cloud of particles on the plane, filling out a region of a certain extent in the variables q1, q2. It is clearly possible to let the particles move using frictionless dynamics and get them to fill a lesser region of the plane. This is, in my opinion, the basic indication of the fact that the two coordinates are *not* conjugate.

Now take a large cloud of particles in one dimension, and describe them by their position and momentum. In this case, they still occupy a finite region in two dimensions, but now the coordinates correspond to momentum and position. In this case, one finds that no amount of letting the particles move through frictionless motion allows any variation in the area of the cloud. This is a nontrivial but standard consequence of the nature of frictionless motion. This incompressibiity is an expression of the conjugacy of position and momentum: any attempt to focus the particles in space (bringing them together) leads to a defocussing in momentum, that is, to a greater variation, and viceversa.

A similar treatment of energy and time is possible, but not illuminating, and in any case, not really needed until one considers relativity. What I meant to point out is that the basic characteristic of conjugacy of two variables can be taken to be the impossibility of reducing an initial spread in both variables simultaneously by any kind of frictionless motion. In quantum mechanics, this then leads to the impossibility of defining jooint probability distribution functions for any pair of conjugate variables.

@Renan, I'm very impressed that you managed to get a 22 page article into PRL! ;) More seriously, this looks like a rather significant piece of work that I'll go through in more detail later - thanks for posting the link.

@Lev, I agree that commutivity has meaning independent of the passage of time, however this is a very clear way (in my view) of understanding much of the significance of non-commuting observables.

I suppose one of the frustrating things for me here (and I'm not really cross at you, it is a general trend, and one that I think is probably interesting for discussion here or in a related forum) pertains to the aspect of your question that highlights *general physical*, which you claim that most respondents aren't responding to. Here we must ask what does 'physical insight' constitute as opposed to 'mathematical insight'. In most cases, I expect (and please correct me if I'm wrong) that people are actually asking for a 'classical interpretation'. I can give a good example of a very dear friend and colleague who when teaching refers to 'quantum double think' when he gives a nice, classical, easy to understand classical explanation/analogy of quantum phenomena. I'm always frustrated by this, because the analogies usually fail at some level of approximation that he's not telling you about. From my point of view, I now routinely find the classical interpretations more bizarre and difficult to grasp than the quantum ones, so that quantum insight (informed by mathematics) seems far more real and physically motivated than the alternatives.

When it comes to Einsteinian 'smell', it would be quite arrogant for me to suggest that I have a fraction of his insight, but one of my goals as a teacher/supervisor is to help my students (undergraduate and postgraduate) to develop their intuition in the quantum space. Having said that, if 'the ship were going down', we'd have some clear evidence for a failure of quantum physics, and conversely (and perhaps surprisingly), quantum mechanics is far more powerful and important that ever and the proof of this can be seen from the fact that the 'spooky' effects are all being demonstrated and resulting in new devices.

Hi Andrew,

Let me deal *briefly* with some of the issues you brought up.

1. "I agree that commutativity has meaning independent of the passage of time, however this is a very clear way (in my view) of understanding much of the significance of non-commuting observables."

Don't you realize the inadmissible ambiguity of such statement?

2. "Here we must ask what does 'physical insight' constitute as opposed to 'mathematical insight'."

I already hinted that I don't consider the highly visible trend---emerging with the classical field theory but taken to the extreme in the modern fundamental physics---of relying more and more on mathematics as purely *calculational*, rather than structural, machinery, which takes us far away from the perceptual data and does not give us a satisfactory insight into the nature of things. Of course, I realize that once the road has been taken---and it was, several centuries ago---there is no turning back. As a mathematician by education, I realize that physics has been successful beyond all our expectations, but my point is that now we have reached the stage of diminishing returns. (However, I do have a positive proposal, see the refer. in my earlier answers.)

3. ”… if 'the ship were going down', we'd have some clear evidence for a failure of quantum physics, and conversely (and perhaps surprisingly), quantum mechanics is far more powerful and important that ever and the proof of this can be seen from the fact that the 'spooky' effects are all being demonstrated and resulting in new devices.”

We will see such evidence as we begin to pursue the alternative formalisms.

When I said that “the ship is going down”, I meant that it is going down because of its acquired ‘weight’, for which it was not designed. I.e. the *structure* of the machinery is becoming less and less useful for understanding the nature of reality (and we should not buy the usual cop-out that "the reality is like that").

===============================================

In contrast to what I see now in physics, the constructions in pure math. have a beautiful logic. But since these constructions do not refer to reality (Einstein), it is the responsibility of physicists to decide when the conventional math. game cannot be ‘repaired’ and need to be abandoned for an entirely new formal language. This is the most important role of the physicist's "nose" that Einstein referred to.

Lev,

In reply to your last answer:

1. "We will see such evidence as we begin to pursue the alternative formalisms. " For that to happen, you either need experiments to suggest something that can't be explained by QM, or you need to come up with predictions from the alternative formalisms that you suggest which can violate QM. Till then, I'm placing all my bets on the present formalism of the theory.

2. "In contrast to what I see now in physics, the constructions in pure math. have a beautiful logic."

wrt the 'beauty' that you speak of, I am left with no option but to quote the greatest of modern physicsts, Richard Feynman: “It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong”.

That is why physics is different (and in my opinion, harder and more interesting) than math - because we can't say whatever we want, we are constrained by the fact that it must agree with what we see in nature.

3. I see that you are overtly inspired by Einstein about concepts of reality. With all due respect to Einstein, I would not base my (or the general phycisist's) understanding of reality, especially after the violation of Bell's inequalities have been observed in experiment, and the EPR paradox issue has been nearly resolved once and for all. After all, reality, the way Einstein wanted to think about it, is not 'real' - we do have spooky action at a distance, and EPR's question, 'Can quantum mechanical description of reality be considered complete' has been answered in the affirmative.

4. About the deeper physical meaning to conjugate observables that you talk of, let us take the example of the quadratures of the electromagnetic field, E_0 and E_{\pi/2}. They are both electric fields, and the physical insight is that these quadratures do not commute, hence their rms deviations have a product greater a certain value. What deeper physical insight can one ask for about the two quadratures of the field, which are the same physical quantity in most senses?

Avik,

1. "For that to happen, you either need experiments to suggest something that can't be explained by QM, or you need to come up with predictions from the alternative formalisms that you suggest which can violate QM."

I will address this part under my other question

https://www.researchgate.net/post/How_concerned_are_you_about_the_central_inconsistency_in_physics---between_the_old_continuous_formalism_and_the_ubiquitous_discreteness_in_Nature1

2. "wrt the 'beauty' that you speak of, I am left with no option but to quote the greatest of modern physicists, Richard Feynman: “It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong”.

That is why physics is different (and in my opinion, harder and more interesting) than math"

I'm afraid you missed my point completely. ;--)

What I am suggesting is that the physical theories become mathematically unattractive precisely because the basis of the old marriage (between math & physics) has 'degenerated'

3.. "After all, reality, the way Einstein wanted to think about it, is not 'real' - we do have spooky action at a distance, and EPR's question, 'Can quantum mechanical description of reality be considered complete' has been answered in the affirmative."

I'm convinced that, in this respect, Einstein has been misunderstood. He brought up this issue to point to the much bigger problem that physics has encountered and which QM has not adequately addressed (Just because QM *allows* for that does not mean it has adequately addressed it.)

4. My general point about the conjugacy is simply to draw, once more, attention to the pervasive feature of modern physics---which it picked up *with vengeance* from the classical physics---to introduce concepts (including variables) with dubious, or non-enlightening, role. By this I mean that they do not help us in our quest to understand the *structure* of 'reality'. Again, I am suggesting (as Einstein has also been contemplating most of his professional life, see http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1457116 ) that the marriage between physics and the present mathematics has ended in separation, which physicists pretend not to notice since they have not yet selected the next suitable 'partner'.

Hi,

One more try. Given two measurable physical quantities we can find that:

a) there is no fundamental observable limit on how precise simultaneous measurements can be carried out.

b) there is a fundamental limit that cannot be surpassed and in fact, the measurements depend on the sequential order. In this case there there is no way to represent the system with a commutative algebra successfully used in classical mechanics. One possible way is to represent those quantities through non-commuting operators in the Hilbert space. However, there many are other alternatives, including the representation in terms of Moyal brackets and there may be others yet to be found with exactly the same physical content.

At this point I would like to quote Julian Schwinger:

"Quantum Mechanics: Symbolism of Atomic Measurements"

http://books.google.com/books?id=DmVXSy5bhkEC&source=gbs_navlinks_s

There is actually (many answers here are excellent, so this is just a quick &sloppy 2 cents' worth)

Here's an example: the color of a piece of paper AND its surface area are conjugates. Let's say the piece of paper is green. Now start whittling down its area again and again with a pair of shears: there will come a point when you cannot discern anymore that the color is green - for all kinds of reasons, your piece of paper will have now reduced to a tiny dot of undistinguishable color, probably dark greyish.

The point at which the surface area becomes indistinguishable depends on the color itself : there is a 'conjugation' between paper surface area and relative color. You cannot know both simultaneously with an arbitrary degree of accuracy.

You can generalize this approach to any number of situations, although I'd much favor the mathematical approach.

Hmmm, sorry for missing the discussion for a while, but in the parallel discussion on conjugacy, James Wheeler (to my opinion) made the point. As long as you use formalisms whch derive from a Hamiltonian/Lagrangian (or which use this principle), you will ever have to talk about cinjugacy as this is an intrinsic ingeredient of this approach. So whatever you use to represent variables, you'll find conjugate relationships if you have suitable variable/state reps.

From what I understand the conjugate variables can describe phase space for any given system completely like momentum and position, energy and time. Also quantum mechanically these are related by Heisenberg Uncertainty Principle.

There are a lot of answers based on analogy here. Understanding the physical meaning of these things is something I have also struggled with. I think down at the fundamental levels of physics we are missing the explanation of why things are the way they are. We think we know that they are correct, but we don't know why they are correct. This doesn't trouble some people, they say "shut up and calculate" . I'm not one of them.

I'm not sure I can answer your original question, but if two variables are conjugates then isn't this just telling us that they are not independent of each other, but are related by the mechanisms we use to measure them.

Dear Lev,

Excellent discussion. What is even more excellent that not only sharpness of ideas is honed but also that the discussion reveals the different science personalities.

What I mean by the first is that we have somewhere the reality (assuming that such exists and we, as physicists, have very little choice on that issue, so leave it to philosophers) but also we try to explain it in a formal language of math (whether math is apart of a real world or not I also leave to philosophers). We really struggle with the definitions and meanings in the "real" world and struggle with their representations (intended parallels to the representation theory).

What I mean by the second is that different classes of peoples have different minds, and there are these that think in imagery of the "real world" and others in formal constructs of the "imaginary/math world".

Unfortunately, (or maybe fortunately) we are limited to the language we use.

So now there comes my three cents in the discussion.

My impression is that use of conjugate variables is very unfortunate (and probably comes from the math people). I strongly prefer use of dual. Let me explain why. As with simple electro-magnetic to Maldacena dualities when you look up close there is a notion of if one is the other not and vice versa. What it practically means is that for instance in electro-magnetic duality if a charge moves we can have or not have the magnetic component depending on the inertial system we are in. The same is true in every case of duality I looked at. So in essence we have some independent way of spotting movement therefore we can create arbitrarily new variables. If the system is at rest we have an x but when it moves we have p. The same with kinetic energy if a body is at rest the energy is null when it moves you create it. So the duality means that there are different description of the same as the laws of nature should not depend on the inertial system. So the movement is generally a relative thing (since the different variations on the relativity issues).

On the other hand we know two more things, (1) that energy is conserved so if it is it must be an overriding principle as almost any other quantum or classic entity translates into energy change. And at the same time from the Noether theorem we connect conservation laws with symmetries of the space. Therefore we created a space out of nothingness out of energy by the requirement that the notion of the inertial system exist. Therefore everything comes back to the duality of are we moving or not (Mach would have a ball now). The problem of quantization is a very natural consequence of this type of approach but I will come back to it later.

Asking for a physical understanding of a mathematical statement is not separating maths from physics, it is joining them together.

Jared,

Equations *alone*---without the concepts behind the basic variables involved in the equations---should not be the sole subject of physics, although judging by the history of physics that seems to become more and more the case.

Reading a beautiful and important book by the late Milič Čapek "Philosophical Impact of Contemporary Physics" (1961), it became quite clear to me that, historically, to 'save' the science physicists have been prepared to go to a great length to modify the meanings of the basic physical concepts such as mass, space, time, energy, etc.

It should be clear, however, that this process cannot continue indefinitely, without turning physics into a measurement-based 'accounting' science, which will rob us of any *structural*, as opposed to the numeric (equational), insight into the nature of reality.

Dear Jared, David and many others,

It really feels like deja vu. I just stated above that this discussion highlights different peoples' personalities.

Math is not physics.

Greeks Arabs, Persians, Chinese, Hindu had their own physics varieties. They could describe motion and forces and make marvelous engineering devices. A descriptive and qualitative language is a part of every human endeavor. What created modern natural sciences is the drive to objectivization by mathematization, therefore, quantitation. Only quantitative predictions are making a modern science so compelling. However, Math as a formal language can or does not need to be considered a part of physical reality. It is like art: the objects produced for instance a painting is a part of our physical reality but the creative though does not to be. Without an interpretational underpinning or the descriptive level (what the math actually means) the discipline is loosing its meaning. The same equations describe physical, chemical economic, population dynamics, and God knows what else.

Lev question is VERY VALID because it asks about the interpretational side of conjugacy. If you read my other threads you immediately realize that modern physics is incomplete. Is incomplete not because the math is defective but because some basic understanding of physical reality is escaping us. Equally well we can ask what is the physical interpretation of the meaning of limitation on how fast the objects can move (speed of light). Or we can ask why only four elementary forces or something on this vain. Math will never answer these questions. And besides math as a formal language, there is math that belongs to the real world, like unfinished proofs, half computer proofs or concepts that nobody knows how to crack and people working on them. But as a product we get Goedel theorems or "proofs" of Fermat conjecture. For me it is just amazing that the structure of human Mind (brain) has any tools to visualize and describe the world at all. So let's focus on the subject.

Note, please, that I just slightly clarified the question description.

I agree with the point made earlier by David Benton:

"I'm not sure I can answer your original question, but if two variables are conjugates then isn't this just telling us that they are not independent of each other, but are related by the mechanisms we use to measure them."

David,

In light of the phenomenon of quantum entanglement (and the Big Bang), It appears that practically all variables are "not independent" but to various degree.

However, IMO the deeper underlying issue raised by my original question is that it is the limitations of the present math. language that prevents us from addressing the nature of the "variables interrelationships", similar to the analogous situation with the entanglement.

Although this may appear not to be of much relevance to the mature physicists, it is still worth noting that the "variables" themselves don't really 'exist' in a proper 'physical' sense of the word and are necessitated by the logic of the entire numeric, or continuous, formalism (hence, in particular, the Heisenberg's indeterminacy relations).

I think to ask this question is just like to ask:

Is there a *general physical* reason why the speed of light in vacuum is the same in all inertial frame?

Of cause we what to know why, we can even guess some answers. But right now we probably we have no solid evidence to back up these answers.

Just like the speed of light, De Broglie's relation is an experimental fact. It relate particle (E, p) to wave (\omega, k). SO, it is a basic assumption. (We also need one more assumption, Born's probability interpretation of wave function.) Once we have (\omega, k), we can write down the (plain) wave immediately. Now we have the conjugate (Fourier) pair (\omega, t) and (k,x), in other words, (E,t) and (p,x).

If we want to know where is these conjugate pair comes from, we are actually looking for a theory more fundamental than QM, which of cause out of the scope of QM.

Zhen,

If I may -I respectfully can't agree with your post at all - see e.g. an earlier example about the reason for the 'physical' reason underpinning conjugates (e.g. color and surface area of something, etc.) We don't need at all a'more fundamental' theory than QM to explain conjugates - regardless of whether there actually IS such a theory. Both Heisenberg's & Schrödinger's equations are ample to explain everything ....and there is solid straightforward experimental evidence for both.

Also - there is actually a physical reason for the speed of light being an invariant.

Dear Prof. Ransford,

Thank you very much for your response.

What I'm trying to say is that QM is built up so that de Broglie relation satisfied. If one invent a theory which violate de Broglie's relation, then it won't be a correct theory.

Special Relativity is built up so that the invariance of speed of light is satisfied.

Also, could you explain what's the "physical reason" for the speed of light being an invariant?

Thank you!

Andrew Greentree:

"I suppose one of the frustrating things for me here (and I'm not really cross at you, it is a general trend, and one that I think is probably interesting for discussion here or in a related forum) pertains to the aspect of your question that highlights *general physical*, which you claim that most respondents aren't responding to. Here we must ask what does 'physical insight' constitute as opposed to 'mathematical insight'. In most cases, I expect (and please correct me if I'm wrong) that people are actually asking for a 'classical interpretation'. "

Andrew, first of all, I'm not "asking for a 'classical interpretation' " *at all*. And this reminded me what I missed to say when I asked about the "general physical reason".

Although I was asking about some underlying physical intuition, but as I said in the question description, I had a hunch that there is none.

However, the important question is "Why is there none *presently* " ?

I believe that the answer to this question has to do with the following very general situation: we need a radically different, non-numeric, (data) representation formalism. It is true that we are bound to rely, in our "physical" explanations, on the basic math. formalism: there is no way out of this. So, as Zhen-Hua Zhang and other suggested, we need a radically new formal language to deal with the "physical" reality, and we have proposed one (see for example, http://www.cs.unb.ca/~goldfarb/FQXi_5.pdf) which is supposed to provide a better link between the "reality" and the "formality".

Dear Lev,

It seems perhaps slightly odd to be coming back to this discussion after so long - but I see that there have been some very interesting discussions, that I won't be able to comment on, but I'll try to reply to some recent comments and hopefully will say something that has not been too well covered above.

Your previous post to me seems far more general than asking solely about conjugate variables, and is therefore far deeper than that restricted comment. Indeed many of the other comments on this post are reaching out to that deeper understanding of quantum.

The essential problem we have with quantum (and it is a legitimate problem) is when we compare it to the other great frameworks of physics.

Relativity says that the laws of physics are the same in all directions and for all frames of reference (no preferred frame of reference), and that cause must precede effect. From that, everything else follows. Beautiful, elegant, physically motivated from which mathematical rigour follows.

Thermodynamics is also very nice from this point of view: Law 0 - temperature exists, Law 1: heat differences can be used to perform work, Law 2: Entropy increases unless you do work on a system, Law 3, you don't get to absolute zero in finite time. (My favourite statement of this is, apart from the zeroth law, You can't win, you can't break even, you can't stop playing). From these principals we can formulate mathematical equations and relationships.

Quantum mechanics, on the other hand, is very different. Glancing at Bransden and Joachain, we have "Postulate 1. To an ensemble of physical systems one can, in certain cases, associate a wave function or state function which contains all the information that can be known about the ensemble. This function is in general complex; it may be multiplied by an arbitrary complex number without altering its physical significance." and so on. This effectively starts with the mathematics, which is far more powerful than the language, and goes from there.

There is a very active (and important) approach to reformulating the postulates on quantum information grounds, with new postulates being things like: "A quantum state cannot be cloned", "No information can be signalled faster than the speed of light", and "No bit commitment". This is an unfinished and important project that at worst will give us a new way of understanding quantum mechanics, but may actually provide qualitatively new insight or phenomena (if we're lucky).

But there is still the important question: why should we expect our limited language and our limited comprehension to be able to make sense of the quantum world? This is why I come back to my question about physical insight and whether it actually means classical insight (and I know it does for many people). What we often 'want' is to understand quantum processes in terms of easily packaged concepts and quantities that really mean something independently of how we see them. We want to see particles as spinning tops, we want to believe that they really do move in some Galilean/Newtonian sense when we aren't looking at them. Why on earth should we expect that us, and in particular our brains, evolved over billions of years from the earliest cells, and shaped in more recent times to cope with the perils avoiding predators, foraging for food, hunting prey and domesticating plants and animals (all classical tasks) should in any way fit us to explain quantum phenomena.

Perhaps the problem that we have is based on the fact that we are evolved to do physics, we are specifically evolved to do classical physics so that we can coordinate our bodies relative to other bodies.

That a mathematical description exists, and what is more, works, is truly amazing and personally surprising to me. Mathematics certainly appears to be a human construct, and there is a lot of true mathematics (ie mathematics that follows from the axioms) that appears to have no basis in the physical world whatsoever. Mathematical treatments of classical physics are good examples of this! But we should be exceedingly grateful for the insights from mathematics and build on them.

It is not at all obvious to me that we need another fundamental formalism that departs from the mathematical rigour. The maths works - we need to deal with that, and we make more errors in trying to impose new preconceptions onto quantum physics, than we do by applying the mathematically rigorous treatments to challenging situations. I would be happy with a more physically motivated set of postulates, but those postulates must enable the derivation of at least the postulates as they stand (and the postulates put forward by people like Fuchs do exactly that).

Sigh, this turned into another rambling rant again, didn't it!

Andy

Dear Andy,

Thanks for the reply!

1. "It is not at all obvious to me that we need another fundamental formalism that departs from the mathematical rigour."

Who said it should depart from rigor? ;--)

2. " I would be happy with a more physically motivated set of postulates, but those postulates must enable the derivation of at least the postulates as they stand (and the postulates put forward by people like Fuchs do exactly that)."

I don't believe this will happen. It looks to me that we are in for a surprise of shocking proportions. I guess it is hard to anticipate a scientific tsunami when, historically, we have never seen one. ;--) But some of us see the tell-tailed signed.

From various discussions here and elsewhere, I can sense that, quite naturally, such superficial and brief chats cannot change anyone's mind about their favorite formalisms (even if it will come to a divorce with the spouse). ;--)) However, despite this, judging even by the interest expressed in some questions posted here at RG, it appears that we probably still need to find a way how to discuss these deeply 'troubling' issues.

Zhen,

Let me try and illustrate briefly a physical reason why the speed of light is an invariant. I'll reason by "ad absurdum" - proof by contradiction.

There's an accident at twelve noon in a location A, and an observer at C . The observer at point C knows of the accident at time 12 noon plus a tiny fraction of time which we'll call ε, which is the time needed by light to travel from the crash scene to the observer's retina.

It so happens however that there is another observer at point B, midway beween A and C:

A____________________B______________________C

This observer at B is not immobile, as C is, but mobile, and it so happens that she passed at point A precisely at 12 noon, at the very time when the crash took place. She took a instantaneous digital picture of the crash at A when it happened, and she is travelling towards C at almost the speed of light (it does not matter at which exact speed because we'll move B all the way to A in a minute).

When she reaches B, she beams an image of the picture of the crash to C. This image is sent by radio waves.

Thus, the digital image taken by the second observer in A is sent from point B at the speed of light towards C, by an observer travelling herself at the speed of light.

Hence, the speed of the image travelling between B and C is twice the speed of light: this image arrives at C at a speed of half ε (the time necessary to travel from A to B, half the time ε necessary to travel twice that distance between A and C) plus one fourth ε, which is the time needed to travel between B and C at twice the speed of light, yielding an overall total time of ¾ ε .

In other words, the digital image of the crash arrives at C before the direct eyewitness observation at C. Looking at the picture, the observer in C knows of the crash before she can see it.

So far, so good, this scenario is not obviously impossible, so let's keep going in a recurrent fashion: onboard the vehicle by which the second observer travels, there is a small rocket with another observer. This third observer fires off the rocket in B and starts towards C at the speed of light, and from this rocket sends by radiowave a copy of the digital image.

A simple calculation shows that this image now arrives in C at time ⅔ ε , faster than the earlier ¾ ε and also quicker than ε.

We can keep going recurrently, both by moving the point B all the way to A (the digital photo is sent as it is being taken) and by having a series of rockets which all serially launch themselves from their immediate foregoer. We end up in a scenario whereby the observer in C receives an image of the crash at the exact same time when it occurs (which would be equivalent to the speed of light being infinite.)

Now we can move the point C all the way to, say, Alpha Centauri (some 44 trillion kilometers away from Earth) and observer C still sees the crash as it happens.

We reached however this conclusion - that the speed of light is infinite - by making the reasonable earlier assumption that that the speed of light was finite (it took time ε to reach C) and spread outwards from whatever event gave rise to it, so we're seemingly ending up in a place where the speed of light is both finite and infinite. Something has to give. Since separate experiments show very clearly that the speed of light is not infinite, as it anyway stands to reason (how could anyway an event happening locally be visible everywhere at once throughout the universe) then it ensues that the only implicit assumption in the above reasoning - that the speed of light is not an invariant - is false.

Ergo, the speed of light is unchangeable regardless of any circumstance - an invariant.

I hope this helps

Dear Prof. Ransford,

You layout a very interesting thought experiment, and it helps me a lot. Thanks first.

According my understanding, you showed that if the speed of light is not invariant, then it can equals to any speed , even infinit. This obviously contradicts with experimental results.

To sum up what you want to say(correct me if I misunderstood you): the variant of speed of light contradicts experimental fact. This is not a physical reason to me: we still do not know why the speed of light is invariant, we just know it is invariant.

I used to read of some paper talking about that the finiteness of the light speed may reflect space-time is discret (although the finiteness of c is slightly different from the invariant of c, but just as your thought exp. shows, they are closely related). To me, the discret of space-time is a "physical reason" of the finiteness of light's speed. This reason maybe is right or maybe wrong, we need experiment to test it, which is really another story.

Finally, I want to quote Lev Goldfarb:

"From various discussions here and elsewhere, I can sense that, quite naturally, such superficial and brief chats cannot change anyone's mind about their favorite formalisms (even if it will come to a divorce with the spouse). ;--)) However, despite this, judging even by the interest expressed in some questions posted here at RG, it appears that we probably still need to find a way how to discuss these deeply 'troubling' issues."

;-).

Not sure if I understand your post Zhen, so please bear with me.

If the speed of light is not invariant, then all kinds of material contradictions crop up: that's what i showed here.

You are looking - if I understand correctly - for a physical reason why light speed is finite. You are looking for explanations, such as mayhap the discreteness of spacetime, and so on.

But there are 2 very separate issues here, which you appear to meld into one?

First, the fact of the finiteness itself: you don't need to seek further - if something - anything - happens in a local point anywhere in spacetime, there is no reason whatsoever why a 'signature' of that event should be instantly available anywhere or anywhen else throughout spacetime. The fact of the local origin of the event in itself leads to a finite time needed to reach other locations in spacetime, regardless of the measure of that finite time.

Second, the value itself - the numbered value - of the finiteness. Here, indeed, there has to be a physical explanation why the speed of light is 300,000km a second and not, say, 2 million km/s . It may very well be that the discreteness of spacetime plays a role in how this values comes about, or something lse.

This is a good example of why we need *general physical* descriptions :)

The problem is that such inevitably heavy reliance on equations for the interpretation of Nature was built-in from the very beginning but especially after the Scientific Revolution. However, the main problem is that during the last century and a half, following the original trend, the "general physical" (as opposed to mathematical) intuition has almost disappeared.

Again, I want to emphasize that this was an *inevitable* trend.

So I am convinced that if we want to get closer to 'physical' reality, we have to seek *radically* new (structural) formal tools of non-numeric nature.

Don, thanks for your view.

However, as John Bell mentioned, Bohr was to a considerable degree an obscurantist. And it appears that you also like that hazy state of affairs (glorifying it). For example, you say:

"The particle-wave duality has enhanced our understanding, that was previously inadequate."

The "particle-wave duality" was a necessary cop-out, since the corresponding phenomenon *is neither wave nor particle*.

As historically has always been the case, today as never before, when we are faced with the need to radically revise our basic formalisms and understanding , instead we take the easy way out and resort to the obscurantist renaming of corresponding phenomena (the ostrich syndrome).

I guess, different scientists as all different human beings, react quite differently to a critical state of affairs in their field. ;--)

Dear Lev,

I know we don't always see eye to eye, but I would like to amplify your comment about particle-wave duality. You are absolutely correct that it is a cop-out, and to be precise, describes neither particles nor waves correctly. The only advantage of it is that it allows us to preserve classical notions of particles and waves and highlights when these descriptions are more (or less) valid.

Holding on to these descriptions is what leads directly to the 'paradoxes' such as delayed choice, Hardy's paradox and interaction free measurement. I placed 'paradoxes' in inverted commas precisely because these effects are not paradoxes from the quantum point of view, but only from the classical viewpoint, and serve to confound (helpfully) our classical interpretations.

Equally you are correct in highlighting issues with the intellectually unsatisfying complementarity principle that gives a classical/quantum boundary without actually specifying how or why the boundary comes into being.

However, (and this is where I am going to restate everything I have said before, so you can tune out now!) the resolutions to the above issues do exist and are mathematical. We have the wave equations and extensions that allow us to treat decoherence. We have an understanding of collective phenomena and entanglement that allows us to quantify some of Bohr's intuition, and also confound it. These great successes do not arise from just renaming the phenomena, but in seriously confronting the implications of the theory and providing tests to those concepts.

For me, the big breakthroughs in understanding quantum theory have arisen because concepts such as entanglement, complementarity etc., ceased to be philosophical questions, but instead became testable ideas.

Bell showed us that the EPR experiment was testable. Wheeler's delayed choice experiment highlighted the dangers of identifying concepts like 'wave' or 'particle' as elements of reality. Gisin's research on collapse in space-like separated frames of reference illustrates the importance of no-signalling and non-locality. I could go on. The key is that the successes have been precisely due to testable predictions, which ultimately boils down to mathematical results.

The issue remains (as I said above) - although it would be nice to have a 'general physical' view, it is not clear to me what is meant by that. Typically what is desired is a pithy explanation based on comforting classical ideas (I am going through your ESD paper and I see you are attempting more than that, Lev). Well, we have no right to expect such a solution, and the testable rigour that we get from mathematical descriptions may well be the only means we have to understand quantum physics.

You mentioned in your post above about the heavy use of equations, and especially the growth in equation based physics following the Scientific Revolution. Equally, this was the period when rigour was established and nuance could be quantified. And again, if we want theories that are falsifiable, then ultimately they must be quantifiable, and that means Reductio ad mathematicas.

Andrew,

First, you might say that I'm a pure mathematician by education and a great fan of the Bourbaki view on the critical role of mathematical *structures*. So Let's not worry about the rigor here: it is a must.

When I talk about a radically new formalism, it does not imply that the rigor has to be sacrificed in any shape or form, as you have repeatedly and mistakenly imputed to me. ;--)

By a radically new ("relational") formal language, I mean the shift (in representing reality) mainly from numbers to structs, which are new kinds of formal entities that can be viewed informally as streams of ‘temporally’/'causally' interconnected structured events of purely ‘informational’, i.e. non-spatial, nature and should be understood as such. It appears that not numbers but structs are the blueprints for the spatial instantiation of events. Yes, it appears that informational reality precedes the spatial one.

It was already accepted in the last century that the Universe is all about various “events”. Yet the basic questions that remain are these: Which kinds of events (or what is the nature of these events)? and What is the right formal language for representing them? So far, in the end, we rely on numbers for representing the states of the system, and I'm suggesting that the actual events have non-numeric structure and what should considerably simplify our view of Nature (e.g. no wave-particle duality) is this new structural language that is suppose to *mirror* the structure of actual events in Nature.

In particular, it suggests that 'particles' might simply be the streams of events, each of particular structure: there are no 'particles' and no 'waves', just spatial instantiations of the corresponding structs. Now, this already is a falsifiable statement suggesting something radically new in physics---that, over time, photons, electrons, etc. are streams of structured events---and we should be able to verify its validity experimentally. We are not talking here about Feynman's diagrams, which are simply auxiliary tools *for calculations*.

My apology for referring you here again to, for example, my last paper http://www.cs.unb.ca/~goldfarb/FQXi_5.pdf which was written for physicists (but with strict space limitations).

Conjugation becomes an objective property of dependent measurables, or simply, it reflects (natural dependency between two actions). If so, then one can not find a unique, finite and bounded, i.e., physical, properfunction that can describe both actions, mutually and deterministically.

More simpler, you can't separate the action of the energy of any system from it's time evolution; nor the linear momentum of a particle from its position.. etc. because such pairs of measurables naturally depend on each other.

So, in a sense, it is a simple statement about the interdependence of some variables, which is not surprising at all. However, the adjective "conjugate" does not sound very appealing or helpful. ;--)

Still, the non-mathematical nature of this dependency---if it is well understood---is not clearly spelled. out.

The *adjective* *conjugate* has a mathematical meaning, rather than physical. Surprisingly, it appears that a mathematical interpretation is unavoidable to explain and point your question out :)

I guess that was the point of the question: as I mentioned two days ago here, following the foundations set in place during the Scientific Revolution, we have gradually relied more and more on the numeric formalism (including the concept of variables), to the point that the pure numeric interrelations completely substituted 'reality'.

However, as I have repeatedly suggested, we should not assume, as many even leading physicists have done, that we are in the last stages in the development of science. I believe this is naive. On the contrary, we might well be at the threshold of a completely new beginning of science, where the new, non-numeric and non-point-based, formalisms will allow us to come much closer to the 'reality, and to see its completely new, structural, side.

I there will be, perhaps soon.

Andrew and Lev,

Not having contributed to this disussion, I went back to December and quickly tried to absorb some of the wisdom loaded down.

I'm not concerned about the physics-mathematics language mismatches and even less about the physics-philosophy language mismatches. What concerns me more is that many physicists don't want to accept that the world of particle physics is quantum-mechanical and not classical. We are mammoths, our senses can only distinguish crude phenomena, that's why we struggle to find a way to replace quantum commutators by something classiscal. But in reality the classical world does not exist except in the limit of the stubborn observer/mammoth.

So the question of why quantum operators commute or don't commute should not be searched for in the classical limit where their physical properties have lost their essential properties, where complex quantities have been stripped of their phases and replaced by their real magnitudes.

Dear Matts,

I am suggesting something beyond what you are suggesting.

1. You are quite right: there cannot be two physics.

2. However, the critical point is that the formal machinery of QM itself is an outgrows of the classical machinery, so that there cannot be a clean separation between them: after all the basic physical concepts came before QM.

3. Finally and most importantly, the situation with QM strongly suggests---as many leading physicists of the last century have also observed---that we are at the threshold of a completely new and unknown to us era in physics. And partly, the latter can be intuited precisely because of the standoff between the classical mechanics and QM. Scientifically, this is a *very* inadequate state of affairs. Attempts to placate this situation which we have witnessed throughout the last century have not worked.

There is a difference between the observer and the predictor. The laws of conservation that we take for granted - such as conservation of energy - were not always considered obvious, but were brought about by observation, and observation on a macroscopic scale. At one stage conservation of energy was fairly contentious. We did not predict this and then search for it. Lev used the phrase "the latter can be intuited precisely because of the standoff between the classical mechanics and QM" . It is the intuit that I am interested in. Intuition is a very valuable tool and it comes from having a mental picture of how the world works. We can make connections between disparate things using intuition, and you don't need mathematics to do that - but you do need mathematics to be convincing! Also it is not necessarily the case that being able to calculate something means that we understand it. It would be interesting to conduct a poll to see what proportion of news ideas arise from mathematics first, or from mental conception first.

For me *general physical" principles are ones that I can intuit, but I appreciate that not everyone thinks this way.

David, what I meant by "intuit" *in this particular case* is that based on our millennia-old experience in science we expect our basic models of reality to be 'reasonably transparent', at least form the formal point of view. In that sense, the standoff between the classical mechanics and QM creates a non-transparent situation, which is exactly the reason why Einstein, Schrodinger, and some others couldn't accept the situation.

In general, what most of us in science are psychologically unprepared for is the possibility that this might be the time in history of science we may need to start almost from the beginning. The main psychological difficulty is also conditioned by the extremely advanced states of both mathematics and physics.

Lev,

"Einstein, Schrodinger, and some others [who} couldn't accept the situation of their time" knew next to nothing about physics, it is useless to refer to them This we can state in retrospect, now that single quanta can be observed, that entangled states of macroscopic dimensions have been verified, that matter in the Universe is known to be dominated by a dark, unknown component, that Einstein's equation has been generalized in hundreds of ways merely to explain the accelerated expansion.

It may be true "that this might be the time in history of science we may need to start almost from the beginning". This sounds prophetic, but I don´think that the work of theoretical physicists is influenced by it. In cosmology one has to explain the problems at hand: Inflation, Dark Enerhy, our Future, be the explanation a Multiverse, a cyclic universe, a higher dimension manifold, a particular string model , etc.

Matts,

Without commenting *your* incredible perception that "Einstein, Schrodinger, and some others knew next to nothing about physics", I would like to remind you that the history of science has never cared for the interests of the majority of scientists in the pre-revolutionary period. The important thing in science is to catch the 'direction', which has nothing to do with the various fads of the pre-revolutionary time period. That is why it is so important, especially for young scientists, to 'leave the ship in time when it is beginning to sink'. ;--)

Lev and Arno,

Yes I was provokative. Remember that Einstein introduced the cosmological constant in order to make the Universe static. Although he later admitted that it was a blunder - when Hubble in 1929 showed him that the Universe expanded - it was not at all a blunder, because the entire Universe known until then comprised only the Milky Way which is not expanding. The Universe known today is some 170 000 times larger than the Milky Way, so one should not extrapolate Einstein's statements about our Galaxy to the entire Universe.

Since then the cosmological constant has been reintroduced, not to Einstein's credit, and for a completely different reason than to make the Universe static.

The Black Holes' solution to Einstein's equations was discovered by Schwarzschild, and I don't know whether Einstein believed such objects existed. This exemplifies, that progress in science is not based on what someone believes or not, nor on whether someone is famous or not. Scientific results have to be repeatable, theories falsifiable, and published in the scientific literature.

I could carry out a similar discussion on Schrödinger, but I don't have the time just now.

Matts,

You are still missing the simple point: as far as the *big* scientific picture is concerned (not the myopic ones) the issues that you mention are just 'details'. And the basic question is this: Do you like (trust) this big picture or not? Do you feel that the outlines of the 'temple' are basically finished? Here only your scientific (Pythagorean) 'nose', to rely on the Einstein's appellation to his nose, can guide you. Of course, the answer to the question whose scientific nose was right we can get only after 'the fact', some time (how much?) later.

However, this is were the truly big scientific stakes are made.

Wow! a relatively simple (hmm.. to me:-) question quickly mushroomed into grand-philosophical discussion... I'll try (if anybody is still interested:-) to answer the original question. First, "conjugation of a pair of variables in quantum mechanics" is not just an exclusively quantum mechanics "resident" (it just happened that QM is too popular...); it is known in many other areas (some of them perhaps not that glamorous). Indeed, as any engineer in the informatics knows that "DELTA omega (x) DELTA time >= O (1)" (i. e. the shorter the time of observation, the less determined is the frequency of an observed signal). Multiply both parts of this eqn by "hbar", and you get a more famous uncertainty (conjugation) between energy and time: "DELTA energy (x) DELTA time >= O (1)" . Hope, this points to a "more general" physics. Now, what is the "physical" reason/explanation for that? Oh, well, it is simply due to our imperfect choice/defentition of terms. The notion of "frequency" implies actually an idealized situation whereby we need essentially infinite time to make sure we have a perfect sinusoide with infinitely repeating identical cycles. Since it cannot be done in reality, we never know what happened with that presumably sinusoidal form beyond our observation. Thus, the shorter observation time, the less certain are we about the frequency we are trying to measure. Similar explanations can be found for other conjugated pairs of variables. E.g. the pair "momentum -- distance" runs into the same problem -- to determine a well-defined , single-valued momentum/velocity, we have to trace our particle all over its infinite trajectory; the shorter the stretch of of our measurement base, the less certain we are about the momentum. Well, hopefully it may go for an explanation...:-) But I suspect the esteemed audience is by now too far into the philosophy...:-):-)

The physical reason of the conjugate variables was already known to Pythagoras. If you play a string of a finite length, the sound emitted is integer multiple of a fundamental frequency given by the inverse of the length. In this case the conjugate spectrum is 'quantized'. This example contains the inner nature of quantum mechanics as I discuss in my paper. See my Researchgate page.

Dear Alexander and Dolce,

You are both absolutely (without any fringe notions) right, correct and convincing. The only remark, I would have is that, if you read the entire discussion you would have noticed that the discussion evolved from a relatively trivial level of "it is because it is" (which unfortunately both of your answers qualify) to the more involved discussion about the foundation of why the things we see are what we see them. It is highly nontrivial to note that you need an infinite time to define well the frequency. This notion branches into the definition of time (what it is) and the definition of motion (what is it). Since the time unit is arbitrary we would have had something in complete stillness (never happening when the unit of time would be long enough) or in a very instance be out of the real universe if the unit were too short. Every case of conjugacy or duality (as I prefer) couples the dynamic with dimensional parameter in which this dynamic event happens and ties it in with the definition of time. So any conjugacy tells you something about the nature of time. If somebody believes that time is an external parameter with a linear flow (as you both seems to believe) then the entire discussion is just trivial and not worth spending time on (because the things are what they are). If however one believes that time needs defining and is intricately connected to what happens then the discussion is highly nontrivial. It is highly nontrivial that QM behaves at the same time as wave mechanics with its Fourier coupling between regular and reciprocal spaces and at the same time as discrete dynamics with transitions and quantization. My take on it (as I expressed it on other threads) is that the notion of time and space is breaking down at the certain level and math of continuous functions is misfitting the reality. This breakdown is coupled to how we define time and space and that I do not believe that time and space exist as a separate background notion that exists without the dynamics that happens in the foreground. That short remark bypasses the notion that the entire world is connected even though the EM signal would have had no chance to connect them so in classical relativity they should not be correlated. As you see, things can be simple and complicated at the same time depending on the point of view. We have to be grateful to RG that allows us to vent our thoughts that otherwise wouldn't have had ever a chance to be voiced, even if they sound a bit strange.

Alexander,

In addition to what Boguslaw said, here is where *my* 'problem' (I guess you call it "grand-philosophical discussion"). I will skip, for now, the reason why it is not a "philosophy", but can come back to it if you would be interested. ;--)

So, on the one hand, different people here and in the related discussion

https://www.researchgate.net/post/Which_physical_variables_do_not_exist_in_conjugate_pairs_and_do_not_have_an_uncertainty_relationship_in_quantum_mechanics_Can_you_list_which_pairs_exist_and_list_those_variables_that_dont_exist_in_pa

have given several to some extent *different* 'explanations', including the last one pointing of all people to Pythagoras. And if the matter was as trivial as you suggest, we would have seen a single simple answer. Why is this not the case?

Thus we come to the core of my problem with this. For several centuries, there has been a continuous terminological attempt (or may be self-deception) to present any concrete theory in such a way as if it is dealing with a true and meaningful physical reality at some deeper level inaccessible to our senses. Yet we still don't even have proper or *independent* (of each other) definitions of most basic concepts, e.g. of mass, charge, spin, etc. We basically don't know what they are outside the *computational* setting that is being updated from time to time.

Alexander, I'm not sure if you realize that the reason you gave for the conjugacy as "physical" is actually also computational.

Now, if physics is presented as an *advanced* computational science, capable of various (powerful) *computational* predictions, I would have no problem at all. But if the theory is presented as capturing the "reality", then when I'm told of conjugate variables, I'm supposed to ask: What does it really mean?

As some physicists suggested, including Boguslaw, the problem is that the *spatialized* time is not the 'right' time, and hence its relation to space and *all* other 'meaningful' connections with reality break down: just think of all the meaningless derivatives wrt time. (Incidentally, I did proposed a radically different approach to time).

Put differently, when I'm given a 3x5 board that has to be used to cover some opening, I do not ask: What is 3x5 or what does it mean? But when I'm told of deep physical meaning of time and energy, and that they are "conjugate", what am I supposed to think? Now, if this is not a subtle form of 'propaganda', or 'brain washing' (self-deception?) then may be I'm missing something very basic.

I am sorry but this thread seems to keel towards the Medieval discourse on the question how many angels could be seated on the point of a needle.( Also) a needless discussion about a non-existing problem

Why do we need non-mathematical physics and for what existing questions.

Lev can you give us "unsolved" items that cannot be explained by the existing QM?

QM was the answer for unsolved items for specifically chemists who were and are very pleased with it to explain and understand characteristics of molecules

Dear Harry,

I fully recognize your general statue and position in sciences but your remarks are at least not courteous. If you read this and some associated threads more carefully you would know that coming to the discussion from a point of view of academic authority does not work here. Every modern science has a descriptive and quantitative element. Physics is a very well developed science therefore it has a large quantitative element but this is certainly not the whole physics. That existence of a quantitative element does not annul the descriptive element as the same math can describe so many different processes that without direct association with a descriptive element it is almost without a base (read nonsense).

As all modern sciences containing the quantitative elements are young, physics is young too, but the roots are very old, probably as old as the human abstract thought itself. This old-new science has pretty good answers about what we experience everyday but for only less than 4% of observable universe., i.e. what we detected or argued that we can predict as existing. I do not even mention things that we did not discovered yet. So basically claiming that science is great because explains everything is just silly. We have no comprehensive theory or stability for complex systems. Quantum chemistry is weak in predicting course of individual reactions. Predicting protein's activity or even a conformational ensemble is beyond the scope and most likely will remain beyond the scope of any technique for conceivable future. Even the discovery of classical complex systems has less than 150 years. There is an enormous arrogance to call this discussion medieval?

QM as any other stepping stone in development of sciences (at least in Kuhnian understanding) is defective and new theory must emerge. Asking questions like Einstein asked (what happens when we sit on a photon or why the inertial mass is equal gravitational mass) are not popular anymore. If questions are not asked in official literature because it got muzzled by an effective but hindering progress peer review system, the questions must be asked somewhere. In Newton times there was no differentiation between math and physics and both fed each other. Nowadays it is much more difficult to be recognized by both communities. Physics is not math as math is not physics. A quantitative part of physics is math but the rest is our curiosity and discovery.

Your rhetorical questions about unsolved problems or why we need nonmathematical physics are so condescending. QM cannot answer even the most elementary problems. If you do not believe me then try to build a brick (clay brick we use to build houses) using QM. When you succeed please let me know if I am still alive. Another example, observation of movement of stars preceded by several millions years the computational (predictive) part of astronomy (physics).

OK, Harry,

Just for you. ;--)

Since you asked "for unsolved items for specifically chemists", let me address the problem related to the structure of molecular classes, i.e. of classes of 'similarly" structure molecules, e.g. proteins, etc.

The problem is this: Why do we see only certain more or less stable and *relatively* small number of molecular classes, compared to an *incomparably* larger set of the structural possibilities allowed by QM? In other words, why do molecular structures tend to fall into a *relatively* small set of such classes?

To translate the same problem into another field, e.g. astronomy, we can ask the same question wrt to the classes of galaxies: Why do we observe only a very small number of classes of structurally similar galaxies out of an incredibly large number of such possible structures allowed by the GR and QM?

Lev

OK now you're talking. But you could have mentioned that somewhere in the thread and in the question itself. The uncertainty relations of the conjugation has been answered to full satisfaction just the last 2 days

Boguslav

Where I come from we are used to be direct. I absolutely do not consider myself an authority in this respect at all, but then I have studied "philosophy of science".

What struck me was that nobody gave examples of why a new physics without math would be needed.

As for the medieval analogon: I indeed wanted to provoke a down to earth discussion why a new physics is needed. As for QM I can appreciate it and in a practical sense it was the break-through for atomic and molecular physics also in an applied sense.

discussions a al Bohr versus Schrodinger/Bohr are of less importance because chemical reactivity can be explained by the wave functions. So for me they (or rather their square) are "real".

To finalize: I followed this thread from begin to end because I am very interested and hope this provocation also helps to get people in the field.

Harry, you took words from my mouth...:-) It looks like in some parts the discussion got to an almost religious dispute level... Sure, one can call it "philosophical"; but personally from my student years I got a profound disregard for the "philosophy of science". Of course, it may be my damn personal business:-), but it was heartening for me to see the same attitude in Feinman's lectures (see his collection of notes for physics students...) (Anybody here to accuse me in "academic authority" and "general statue"?:-)-) Would be nice to get some recognition this way...:-)

OK, Alex,

Since you yourself asked for the "recognition", I will oblige. ;--))

The problem with many of today's science 'workers' (and I do mean *science*, not philosophy, Alex) is that they have learned the set of standard skills and decided that that is all you need to be a good scientist. But what about the future?

As all of us, especially those working in science, *should* know is that---as is the case with any intellectual/cultural undertaking---scientific achievement can appropriately be evaluated only several centuries after the fact. (If you are interested why I can explain it later). Again, science as probably the greatest achievement of the human mind, is all about its contribution to the *future* of our civilization. Ironically, in this sense, its contribution to the present does not matter very much.

So . . . I'm reasonably sure that with the attitude like yours (and if it will make you feel better I can add Feynman to the company)---correct me if I'm wrong---who is going to get involved in a 'subversive' or radical scientific work? And yet without such work the next scientific revolution is *absolutely* impossible. The problem is that many science 'workers' think we have arrived at the "promised land", and we just have to be reasonably patient. They are reasonably happy about the state of science and don't see any serious reasons not to be.

The only thing I can suggest is this: at least don't be 'militant' and arrogant about your 'satisfaction', because, I can assure you that, even if you can't see it, we are still at a very primitive state of science. (Recall Einstein's: "One can best feel in dealing with living things how primitive physics still is.") And if you don't see that, at least don't 'laugh' at those who are struggling to do something about it. ;--)

Although, alas, as we also know, the self-satisfied cannot understand the seekers. ;--)

Oh, by the way, Alex, without some serious philosophical thinking no radical transitions in science are possible. So, in this respect, don't look at Feynman as an example, but look at such critics as Einstein and Schrodinger.

Carlo Rovelli wrote:

"It's sort of the fashion today to discard philosophy, to say now we have science, we don't need philosophy. I find this attitude very naïve for two reasons. One is historical. Just look back. Heisenberg would have never done quantum mechanics without being full of philosophy. Einstein would have never done relativity without having read all the philosophers and have a head full of philosophy. Galileo would never have done what he had done without having a head full of Plato. Newton thought of himself as a philosopher, and started by discussing this with Descartes, and had strong philosophical ideas.

But even Maxwell, Boltzmann, I mean, all the major steps of science in the past were done by people who were very aware of methodological, fundamental, even metaphysical questions being posed. When Heisenberg does quantum mechanics, he is in a completely philosophical mind. He says in classical mechanics there's something philosophically wrong, there's not enough emphasis on empiricism. It is exactly this philosophical reading of him that allows him to construct this fantastically new physical theory, scientific theory, which is quantum mechanics.

The divorce between this strict dialogue between philosophers and scientists is very recent, and somehow it's after the war, in the second half of the 20th century. It has worked because in the first half of the 20thcentury, people were so smart. Einstein and Heisenberg and Dirac and company put together relativity and quantum theory and did all the conceptual work. The physics of the second half of the century has been, in a sense, a physics of application of the great ideas of the people of the '30s, of the Einsteins and the Heisenbergs.

When you want to apply thes ideas, when you do atomic physics, you need less conceptual thinking. But now we are back to the basics, in a sense. When we do quantum gravity it's not just application. I think that the scientists who say I don't care about philosophy, it's not true they don't care about philosophy, because they have a philosophy. They are using a philosophy of science. They are applying a methodology. They have a head full of ideas about what is the philosophy they're using; just they're not aware of them, and they take them for granted, as if this was obvious and clear. When it's far from obvious and clear. They are just taking a position without knowing that there are many other possibilities around that might work much better, and might be more interesting for them.

I think there is narrow-mindedness, if I might say so, in many of my colleague scientists that don't want to learn what is being said in the philosophy of science. There is also a narrow-mindedness in a lot of probably areas of philosophy and the humanities in which they don't want to learn about science, which is even more narrow-minded. Somehow cultures reach, enlarge. I'm throwing down an open door if I say it here, but restricting our vision of reality today on just the core content of science or the core content of humanities is just being blind to the complexity of reality that we can grasp from a number of points of view, which talk to one another enormously, and which I believe can teach one another enormously. "

Once we are on the subject of importance of philosophy, I might as well translate what Friedrich Engels said about the philosophy 'haters':

"Some scientists imagine that they free themselves of philosophy when they ignore or scold it. But since without thinking they cannot make a single step, and for thinking one needs logical categories, they uncritically borrow those categories

1) either from the everyday thinking of the so-called educated people who are in fact under the influence of the long-dead philosophical systems

2) or from the compulsory university courses on philosophy (which not only represent fragmentary views, but are also a miss-mash of views mostly belonging to bad schools of thought )

3) or from the uncritical and unsystematic reading of all kinds of philosophical works,

at the end they still end up relying on the philosophy but for the most part of the worst kind, so that those who most scold philosophy end up as the slaves of the worst vulgarized remnants of the worst philosophical teachings."

Wow! turned out the religious disputes haven't been invoked for nothing -- now we've got preached upon from a high altar by the High Priests of Eternal Science ! It is up to them now, whom of the great physicists we should worship -- and whom not... I am in trouble! "Forgive me, Father, for I sinned!" Well, here you've got an unrepentant sinner, folks... This sinner has never worshiped anybody, and he even believes that worshiping and science don't belong together. Furthermore, as a working stiff of science, he also believes in a simple principle -- "shut up and calculate" (Mervin's motto?). And as far as preaching... well, folks, as they say in the streets where I came from, "show me your money" -- show me your published results on a par with those on behalf of whom you are preaching, and then I'd listen... Meanwhile, this discussion's got a bit philosophical/religious to my taste, sorry.

My apologies Alex for distracting you from the important work of "shut up and calculate" kind. ;--)

Enjoy life the way you like it!

Dear Harry and Alex,

It is so nice to come back to the old nest of arrogant physicists. Bad will that emanates from your posts that is represented by misreading other peoples relatively clear communications can only be attributed to this incurable disease.

Harry, nobody on this or any other thread claims that physics can exist without math (or quantitative element). Actually, if you follow philosophy of sciences, you notice that mathematization as a realization of the objectivity requirement, would produce, and it is producing, more math in chemistry, biology and any other modern science. Lev as a pragmatic of science, who is more interested in its philosophical underpinnings just asks provocative questions that you kind of like. Why then deny this by stating that he proposes something without math. Even his philosophical writings have some math in it.

Alex, you just sin with real hutzpa. You write: (Anybody here to accuse me in "academic authority" and "general statue"?:-)-) Would be nice to get some recognition this way...:-). You take a statement and reverse it. Nobody accuses you of anything and you take a genuine reverence as a reverse. Besides, sometimes it is good to check the records. Einstein said that only theory can determine what is measurable. The reverse is also true without a theory you would not be able to calculate anything. New physics cannot be created by math alone, ask Einstein about it. So if you are so fluent in your math, show me where the quantization of charge is coming from. Why force is proportional to acceleration (maybe there is a correction)? Almost every foundational statement of physics is of a religious nature. Viva dialecticism.

Guys take it easy. This is no fight for grant money.

Boguslaw,

You might be surprised that I would prefer to talk about a concrete formalism we are developing than about any philosophy. But, unfortunately, since I couldn't get people to understand it, I had to spend a lot of time thinking why. ;--)

Lev,

I red some of it. If you have it further developed I will continue. I personally believe in simpler computational schemes taking over more cumbersome (like with money).

Boguslav

My comment was that the original question keeled away from the direct question on the "conjugacy/cation of a pair of variables in QM".

And for sure I do not understand the remark: none of our variables exist in Nature"

I paraphrase you by mentioning "Nature was there before man" and I add will be there after us.

As for your own example: QM cannot answer even the most elementary problems; try to build a brick using QM.

My reply QM is and was not meant to explain the building of a house or even the production of one piece of a human artifact, but to explain unresolved in the classical sense issues on the properties of very limited and thus very well defined systems thus isolated atoms/molecules in vacuum. There the job is still done but nothing more than that. But in that filed I at least are very satisfied that we have a concept of phenomena like light emission and absorption by matter in a isolated system and can actually PREDICT for an analogous system by way of math. Not more and not less.

In this respect physics describes very defined ideal(sed) systems not "real life/nature".

Harry: "And for sure I do not understand the remark: "none of our variables exist in Nature". "

Harry, what I'm suggesting is that there is no such *basic* entities in Nature as, for example, energy, mass, etc. (they are derivative concepts). Why did I mentioned it in connection with my original question?

On the way of winding down this discussion, I'd like to summarize my observations.

I think conjugacy is *conceptually* (not computationally) non-transparent because it is related to deeper ('temporal') interconnection between the 'abstract' variables involved. Moreover, I feel, the main reason we can't 'see' in a more transparent manner this interconnection has to do with the inadequacy of the numeric formalism as a whole. I believe we need a more structural (non-numeric) formalism to get at such non-transparent interconnections. By "structural" I mean a non-numeric form of data representation: instead of numbers, new kinds of, structured, entities, such as, for example, "structs" proposed by us. I want to emphasize that such structural forms of data representation will require different kind of "measurement" tools and are supposed to capture previously inaccessible side of physical processes.

Now, what do I mean by "variables don't exist in Nature"?

First, the concept of field has become the central one during the last century, but it is a spatially motivated concept. As some of you know, during the last two decades the primacy of space has been questioned by a number of prominent physicists and there are a number of projects under way (related to the efforts in quantum gravity) to change this situation. But what practically all physicists are unprepared for is that this is not going to be a business as usual affair, since basically all the physical and mathematical concepts, including "variables", are spatially motivated. So does a weight of a tree exist as the basic entity? To me the obvious answer is no: it is a derivative concept based on other more basic, structural, concepts (and I don't mean Higgs boson here since it is also spatially motivated concept). I have discussed these general issues in the above paper. Incidentally, it is not that difficult to intuit that energy and time are not actually spatial concepts, but we have to treat them as such because of our numeric, or spatial, formalism.

So again, our main difficulties are going back all the way to the ancient measurement practices, which today, for the first time in the history of science, probably have to change radically. And obviously, our busy lives are not very conducive to such radical rethinking.

Lev

I like to ask you what kind of new measurements you think of and other variables than those you mention by which we describe (some parts of) our surroundings and events and which have been around for centuries and by which we can communicate

Harry,

I used phrase "different kind of 'measurement' tools" not to suggest "new variables".

Let me try to explain.

As we know, one of the basic conceptual ideas of QM brought to physics is that it proposed to view physical processes as sequences of transformations, where *each* concrete transformation (from the initial to the final state) cannot be observed, only some numeric characteristics of the initial and final states.

'Fundamental transformations' are those with respect to which all other transformations can be decomposed. However, Schrodinger equation, for example, is not a satisfactory tool for approaching this transformational reality:

1) it does not describes the transformation itself (we are not allowed to talk about what is happening during the transformation)

and

2) when one "particle" is transformed into one or several others we basically have to 'stitch' together the equations for various "particles" involved.

In the formalism proposed by us, it is the *structure* of transformations themselves and their interconnections that become the primary object of study (something like generalized Feynman diagrams), but the form of description of a single transformation is not numeric but 'structural'.

In particular, what this formalism (ETS) suggests is that, consistent with QM, any elementary "particle" is a sequence of transformations, where each transformation has a particular *structure*. If this turns out to be a satisfactory hypothesis, we are in for a different kind of physics. It suggests, for example, that a photon (or an electron) is a sequence of some (still unknown to us) transformations, which we have to discover (via new "measurements").

So when I talk of new "measurements", I'm talking about processes and devices involved in the identification of the structure of the transformations involved.

(For a general introduction see http://www.cs.unb.ca/~goldfarb/FQXi_5.pdf)

Lev

I've noticed your Feynman pictures and start to see a bit what you are after. For in the F formalism QED is visualized much better and in that sense more intuitive to follow, I agree. My own experience with 2-photon absorption for instance. Still we can only define initial and final state not the transformation.

Question: how would you describe transitions induced by EM-radiation in your strucs?

As for the existence of an electron as mass entity or transformative unit I myself do not stick to any convention as far as it describes and predicts observed phenomena.

Here we start to disagree I like to see the observations first and then the formalism to have a "human" explanation for it in whatever form and a way to predict other observations. That to me is what Einstein is for me the paragon of intuitism. Heisenberg for instance is somewhat of a trial-and-error approach of how math can be used in physics, because he had just studied hard in matrix-algebra. His insight was that he "saw" that it matched observational phenomena.

By the way he is an example of doctor evil as you might know, and the absolute contrast of an objective scientist being a true Nazi. This may also be the way too much intuitism may lead is my fear

Lev

I now notice that most of what I contributed is already presented (by others) in your sister entry in this topic. In that entry you even postulate your PERSONAL need for a new description of nature/reality; that sounds quite subjective in a field that attempts to be objective, in the sense that more people have the same intuition. Here the initial question is the concrete conjugation of variables / operators in QM. So, I switch to the thread there