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Questioning the Foundations Essay Contest (2012)
[back]

TOPIC: Can Differentiable Description of Physical Reality Be Considered Complete? by Wang Xiong [refresh]

TOPIC: Can Differentiable Description of Physical Reality Be Considered Complete? by Wang Xiong [refresh]

How to relate the physical \emph{real} reality with the logical \emph{true} abstract mathematics concepts is nothing but pure postulate. The most basic postulates of physics are by using what kind of mathematics to describe the most fundamental concepts of physics. Main point of relativity theories is to remove incorrect and simplify the assumptions about the nature of space-time. There are plentiful bonus of doing so, for example gravity emerges as natural consequence of curvature of spacetime. We argue that the Einstein version of general relativity is not complete, since it can't explain quantum phenomenon. If we want to reconcile quantum, we should give up one implicit assumption we tend to forget: the differentiability. What would be the benefits of these changes? It has many surprising consequences. We show that the weird uncertainty principle and non-commutativity become straightforward in the circumstances of non-differentiable functions. It's just the result of the divergence of usual definition of \emph{velocity}. All weirdness of quantum mechanics are due to we are trying to making sense of nonsense. Finally, we proposed a complete relativity theory in which the spacetime are non-differentiable manifold, and physical law takes the same mathematical form in all coordinate systems, under arbitrary differentiable or non-differentiable coordinate transformations. Quantum phenomenon emerges as natural consequence of non-differentiability of spacetime.

WANG Xiong has obtained his B.Sc. majoring in math from the Shanghai Jiao Tong University, China. Currently, he is a research student at City University Hong Kong at the Centre for Chaos and Complex Networks. From undergraduate time, he continue an independent solitary quest for a unified foundation for mathematics and physics.

Hello Wang Xiong

Thank you for a stimulating article. I have a feeling that it will need some work to make some mathematicians content with it. Somewhere it seems to be assuming something about a measure on space-time. Anyway there seems to be a deep idea in it.

Regards

Alan H.

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Thank you for a stimulating article. I have a feeling that it will need some work to make some mathematicians content with it. Somewhere it seems to be assuming something about a measure on space-time. Anyway there seems to be a deep idea in it.

Regards

Alan H.

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Hello Wang Xiong,

I gravitated to your paper because of my familiarity with the work of one of the authors you cite. In section 4.4 you mention that Laurent Nottale did promising work in a similar direction, but did not arrive at a satisfying conclusion.

He has since published his book "Scale Relativity And Fractal Space-Time" which delves into some of the questions you raise with considerable rigor.

The new book repairs most of the deficiencies in the Math of his earlier works, attempting to both correct errors and expand upon the earlier work you cite. I was privileged to see the book in manuscript form, and the published version is quite the Magnum Opus for Nottale - over 750 pages in all! It is obviously recommended reading, considering your subject. You will find many points of agreement.

I read your paper very swiftly, and I will need to re-read it to make meaningful comments. Best of luck!

Jonathan

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I gravitated to your paper because of my familiarity with the work of one of the authors you cite. In section 4.4 you mention that Laurent Nottale did promising work in a similar direction, but did not arrive at a satisfying conclusion.

He has since published his book "Scale Relativity And Fractal Space-Time" which delves into some of the questions you raise with considerable rigor.

The new book repairs most of the deficiencies in the Math of his earlier works, attempting to both correct errors and expand upon the earlier work you cite. I was privileged to see the book in manuscript form, and the published version is quite the Magnum Opus for Nottale - over 750 pages in all! It is obviously recommended reading, considering your subject. You will find many points of agreement.

I read your paper very swiftly, and I will need to re-read it to make meaningful comments. Best of luck!

Jonathan

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One more thing;

Nottale suggests that a continuous but non-differentiable spacetime is necessarily fractal. Do you agree, or assume this?

All the Best,

Jonathan

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Nottale suggests that a continuous but non-differentiable spacetime is necessarily fractal. Do you agree, or assume this?

All the Best,

Jonathan

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Dear Jonathan,

In fact, there lacks a rigorous definition of fractal.

So what we really means is continuous but non-differentiable spacetime

also there is a subtle issue about continue, there are also discussion about to give up continue, such as levy flight kind of path.

We still need some new math for this problem.

Thanks

Regards,

Xiong

In fact, there lacks a rigorous definition of fractal.

So what we really means is continuous but non-differentiable spacetime

also there is a subtle issue about continue, there are also discussion about to give up continue, such as levy flight kind of path.

We still need some new math for this problem.

Thanks

Regards,

Xiong

I like your essay. It was very informative, and not too difficult to understand. Your point was very clear.

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Dear Wang Xiong,

"All weirdness of quantum mechanics are due to we are trying to making sense of nonsense." Let me admire first your excellent style of writing. I also found out that non-differentiability is an important issue. I would merely appreciate if you could explain more in detail:

"But by analogy, we can say that quantum equation must of order 1.5 as for x. So there is less than two, but more than one independent variables. This is why the x and dx/dt are somehow not independent to each other, but are conjugate variables with uncertainty relation between them."

Perhaps you will be among those who are rather shocked by my very basic criticism. It seems unlikely to me that we will easily agree on the logical consistency of SR. Nonetheless I invite you to challenge in detail what I wrote here and also here .

Regards,

Eckard

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"All weirdness of quantum mechanics are due to we are trying to making sense of nonsense." Let me admire first your excellent style of writing. I also found out that non-differentiability is an important issue. I would merely appreciate if you could explain more in detail:

"But by analogy, we can say that quantum equation must of order 1.5 as for x. So there is less than two, but more than one independent variables. This is why the x and dx/dt are somehow not independent to each other, but are conjugate variables with uncertainty relation between them."

Perhaps you will be among those who are rather shocked by my very basic criticism. It seems unlikely to me that we will easily agree on the logical consistency of SR. Nonetheless I invite you to challenge in detail what I wrote here and also here .

Regards,

Eckard

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Wang Xiong,

I found your essay interesting, as it has a relationship with my own. Whereas you state a solution may exist examining non-differentiability, my essay postulates that we have been anti-differentiating the Newtonian field incorrectly, which leads to the incorrect form of the Einstein field equation. If a function F_{1} is Newtonian field strength, then F_{1}' is gravitational force. I state that we have probably been mistaking F_{1}' for (C-F_{2})' so that the Einstein field equation with the cosmological constant should read similarly, i.e. where the constant term is equated to a potential for energy of the vacuum and L_{uv} is equated to a residual dynamic stress energy tensor. This would allow a large magnitude for the constant while still appearing like attractive gravity but also allowing a repulsion after a certain radius. Solves the old and new cosmological constant problems.

I will peruse your paper more in depth to see if there are places from which I can learn.

Regards,

Jeff Baugher

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I found your essay interesting, as it has a relationship with my own. Whereas you state a solution may exist examining non-differentiability, my essay postulates that we have been anti-differentiating the Newtonian field incorrectly, which leads to the incorrect form of the Einstein field equation. If a function F

I will peruse your paper more in depth to see if there are places from which I can learn.

Regards,

Jeff Baugher

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Wang,

Interesting essay. I agree that differentiable manifolds are probably "too good to be true," and are assumed mostly because they are easy to work with mathematically. I also agree that the path integral formulation of quantum theory provides key insight, and that fractals, scale-dependent phenomena, self-similarity, etc. deserve more attention. I have a couple of questions.

1. What exactly do you mean by a "nondifferentiable manifold" and a "fractal manifold?"

2. How do time and causality fit into your picture?

To see what is motivating these questions, you might look at my essay

On the Foundational Assumptions of Modern Physics

I also reject differentiable manifolds and emphasize path integrals, but there are some differences too. Take care,

Ben Dribus

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Interesting essay. I agree that differentiable manifolds are probably "too good to be true," and are assumed mostly because they are easy to work with mathematically. I also agree that the path integral formulation of quantum theory provides key insight, and that fractals, scale-dependent phenomena, self-similarity, etc. deserve more attention. I have a couple of questions.

1. What exactly do you mean by a "nondifferentiable manifold" and a "fractal manifold?"

2. How do time and causality fit into your picture?

To see what is motivating these questions, you might look at my essay

On the Foundational Assumptions of Modern Physics

I also reject differentiable manifolds and emphasize path integrals, but there are some differences too. Take care,

Ben Dribus

report post as inappropriate

Dear Ben Dribus,

I am very happy to have math background reader like you,

your questions are very important!

1. What exactly do you mean by a "nondifferentiable manifold" and a "fractal manifold?"

this is a big problem,indeed a big subject of math, impossible to address in such a short paper,

roughly, you can think nondifferentiable manifold just as manifolds without additional differentiable structure.

Because we add this differentiable structure without physical reasons, so as to allow calculus to be done on manifolds.

2. How do time and causality fit into your picture?

this is an important problem, which I still need to clarify.

I think the first step is to get this picture fit with SR

Xiong

I am very happy to have math background reader like you,

your questions are very important!

1. What exactly do you mean by a "nondifferentiable manifold" and a "fractal manifold?"

this is a big problem,indeed a big subject of math, impossible to address in such a short paper,

roughly, you can think nondifferentiable manifold just as manifolds without additional differentiable structure.

Because we add this differentiable structure without physical reasons, so as to allow calculus to be done on manifolds.

2. How do time and causality fit into your picture?

this is an important problem, which I still need to clarify.

I think the first step is to get this picture fit with SR

Xiong

Wang,

Please read Nobel lecture Gerardus t' Hooft

http://www.nobelprize.org/nobel_prizes/physics/laureates/199

9/thooft-lecture.pdf

p.360 about same problem.

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Please read Nobel lecture Gerardus t' Hooft

http://www.nobelprize.org/nobel_prizes/physics/laureates/199

9/thooft-lecture.pdf

p.360 about same problem.

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Hello, Wang Xiong! Profound dialectical essay. But: "the truth is to be drawn and is provided an unlimited number of viewers" (A.Zenkin "Scientific counterrevolution in mathematics"). You could draw and ontologically justify the structure of space-time by Wang Xiong? Sincerely, Vladimir Rogozhin

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Dear Wang Xiong,

interesting essay, in particular I like your style.

We disagree about the importance of differentiability (see my essay). I would also ask the same questions then Ben.

As I understand your approah in a right sense, then fractal means non-smooth. But from the math point, it is wrong. Consider Alexanders horned sphere: it is a wild embedding of the 2-sphere with fractal dimension (between 2 and 3). But it is a smooth embedding inducing a smooth metric (from the embedding space).

Therefore I think one can save differentiablity and the concept of a manifold.

4-manifolds have special properties.

Best

Torsten

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interesting essay, in particular I like your style.

We disagree about the importance of differentiability (see my essay). I would also ask the same questions then Ben.

As I understand your approah in a right sense, then fractal means non-smooth. But from the math point, it is wrong. Consider Alexanders horned sphere: it is a wild embedding of the 2-sphere with fractal dimension (between 2 and 3). But it is a smooth embedding inducing a smooth metric (from the embedding space).

Therefore I think one can save differentiablity and the concept of a manifold.

4-manifolds have special properties.

Best

Torsten

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Dear Wang Xiong

The assumption that space-time is a smooth differential manifold is a convenient but fundamentally metaphysical assumption that you are right to identify as an implicit assumption made in physics. It is an assumption that I make in the theory mentioned in my essay and I am happy to explicitly acknowledge that it is metaphysical. After all, the search for new physics theories has an element of looking for a set of metaphysical assumptions that reproduce known results - we just need to be willing to be explicit about those assumptions.

However, my assumption of a smooth differential manifold turns out to perhaps not quite be what it seems in physically measurable terms. This is because in my extended GR manifold (S3*S7) the compactified S7 dimensions effectively provide the discrete physical units in which spatial distances and times are measured - a unit system of the Planck scale. Other theories with smooth space-time and compactified dimensions can have the same feature. Although the limit dx->0 is technically possible for the assumed smooth manifold, in measurement terms the limiting procedure runs into the finite Planck scale in which the distances are measured. Have you considered such scenarios where even the assumption of a smooth manifold may not fully transfer to physically measurable terms?

Michael

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The assumption that space-time is a smooth differential manifold is a convenient but fundamentally metaphysical assumption that you are right to identify as an implicit assumption made in physics. It is an assumption that I make in the theory mentioned in my essay and I am happy to explicitly acknowledge that it is metaphysical. After all, the search for new physics theories has an element of looking for a set of metaphysical assumptions that reproduce known results - we just need to be willing to be explicit about those assumptions.

However, my assumption of a smooth differential manifold turns out to perhaps not quite be what it seems in physically measurable terms. This is because in my extended GR manifold (S3*S7) the compactified S7 dimensions effectively provide the discrete physical units in which spatial distances and times are measured - a unit system of the Planck scale. Other theories with smooth space-time and compactified dimensions can have the same feature. Although the limit dx->0 is technically possible for the assumed smooth manifold, in measurement terms the limiting procedure runs into the finite Planck scale in which the distances are measured. Have you considered such scenarios where even the assumption of a smooth manifold may not fully transfer to physically measurable terms?

Michael

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Dear Michael,

Thanks for your question, it's very important question

even the assumption of a smooth manifold may not fully transfer to physically measurable terms,

yes, but I think there is no need to limit it to smooth case

anyway, the general manifold will be very complicated and rich in content

Xiong

Thanks for your question, it's very important question

even the assumption of a smooth manifold may not fully transfer to physically measurable terms,

yes, but I think there is no need to limit it to smooth case

anyway, the general manifold will be very complicated and rich in content

Xiong

Dear Wang,

Our languages are somewhat different. I am not highly trained in math. However I can see your points clearly.

You explain non differentiability as the reason why gravity theory cannot come to grips with quantum phenomena. Yes, I agree completely, only I would phrase it differently: The common assumption that an object can have a velocity at a point is a fallacy, v can never be dx/dt no matter how many epsilons are thrown at it. It is great math and rotten physics.

You seem to favor a fractal picture of reality, do yo think this can be related to Feynman's sum over histories approach?

I like your thoughtful clearly written essay.

Don L.

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Our languages are somewhat different. I am not highly trained in math. However I can see your points clearly.

You explain non differentiability as the reason why gravity theory cannot come to grips with quantum phenomena. Yes, I agree completely, only I would phrase it differently: The common assumption that an object can have a velocity at a point is a fallacy, v can never be dx/dt no matter how many epsilons are thrown at it. It is great math and rotten physics.

You seem to favor a fractal picture of reality, do yo think this can be related to Feynman's sum over histories approach?

I like your thoughtful clearly written essay.

Don L.

report post as inappropriate

For better clarification my approach

I sending to you Frank 3 keen articles

http://ctpweb.lns.mit.edu/physics_today/phystoday/Ab

s_limits393.pdf

http://ctpweb.lns.mit.edu/physics_today/physt

oday/Abs_limits393.pdf

http://ctpweb.lns.mit.edu/physics_toda

y/phystoday/Abs_limits400.pdf

All the best

Yuri Danoyan

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I sending to you Frank 3 keen articles

http://ctpweb.lns.mit.edu/physics_today/phystoday/Ab

s_limits393.pdf

http://ctpweb.lns.mit.edu/physics_today/physt

oday/Abs_limits393.pdf

http://ctpweb.lns.mit.edu/physics_toda

y/phystoday/Abs_limits400.pdf

All the best

Yuri Danoyan

report post as inappropriate

Dear

Very interesting to see your essay.

Perhaps all of us are convinced that: the choice of yourself is right!That of course is reasonable.

So may be we should work together to let's the consider clearly defined for the basis foundations theoretical as the most challenging with intellectual of all of us.

Why we do not try to start with a real challenge is very close and are the focus of interest of the human science: it is a matter of mass and grain Higg boson of the standard model.

Knowledge and belief reasoning of you will to express an opinion on this matter:

You have think that: the Mass is the expression of the impact force to material - so no impact force, we do not feel the Higg boson - similar to the case of no weight outside the Earth's atmosphere.

Does there need to be a particle with mass for everything have volume? If so, then why the mass of everything change when moving from the Earth to the Moon? Higg boson is lighter by the Moon's gravity is weaker than of Earth?

The LHC particle accelerator used to "Smashed" until "Ejected" Higg boson, but why only when the "Smashed" can see it,and when off then not see it ?

Can be "locked" Higg particles? so when "released" if we do not force to it by any the Force, how to know that it is "out" or not?

You are should be boldly to give a definition of weight that you think is right for us to enjoy, or oppose my opinion.

Because in the process of research, the value of "failure" or "success" is the similar with science. The purpose of a correct theory be must is without any a wrong point ?

Glad to see from you comments soon,because still have too many of the same problems.

Regard !

Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY

August 23, 2012 - 11:51 GMT on this essay contest.

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Very interesting to see your essay.

Perhaps all of us are convinced that: the choice of yourself is right!That of course is reasonable.

So may be we should work together to let's the consider clearly defined for the basis foundations theoretical as the most challenging with intellectual of all of us.

Why we do not try to start with a real challenge is very close and are the focus of interest of the human science: it is a matter of mass and grain Higg boson of the standard model.

Knowledge and belief reasoning of you will to express an opinion on this matter:

You have think that: the Mass is the expression of the impact force to material - so no impact force, we do not feel the Higg boson - similar to the case of no weight outside the Earth's atmosphere.

Does there need to be a particle with mass for everything have volume? If so, then why the mass of everything change when moving from the Earth to the Moon? Higg boson is lighter by the Moon's gravity is weaker than of Earth?

The LHC particle accelerator used to "Smashed" until "Ejected" Higg boson, but why only when the "Smashed" can see it,and when off then not see it ?

Can be "locked" Higg particles? so when "released" if we do not force to it by any the Force, how to know that it is "out" or not?

You are should be boldly to give a definition of weight that you think is right for us to enjoy, or oppose my opinion.

Because in the process of research, the value of "failure" or "success" is the similar with science. The purpose of a correct theory be must is without any a wrong point ?

Glad to see from you comments soon,because still have too many of the same problems.

Regard !

Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY

August 23, 2012 - 11:51 GMT on this essay contest.

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Wang

I appreciated your approach to finding a unified foundation. I found one from a different approach, but will need validation from yours. I hope you might give me your view. I do not feel well qualified to comment in detail on yours, but was anyway impressed with the parts I fully followed.

Best wishes, and thanks

Peter

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I appreciated your approach to finding a unified foundation. I found one from a different approach, but will need validation from yours. I hope you might give me your view. I do not feel well qualified to comment in detail on yours, but was anyway impressed with the parts I fully followed.

Best wishes, and thanks

Peter

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Dear Wang Xiong,

You raise an interesting issue: differentiability in Nature. But whereas you resort to non-differentiable manifolds of Spacetime to correct existing mathematical models of the Universe, I argue in my essay, “The Metaphysics of Physics”, all mathematical models of 'what is' the Universe are in essence metaphysical and will ultimately fail. Knowing truly 'what is' the Universe is like knowing another person; what they think, how they feel and how they will behave. We can only know ourselves truly. All knowledge in this sense is therefore self-knowledge.

The question of differentiability in Nature can be considered in the following very simple context, for example. When we take a thermometer reading, is the final reading on the thermometer “asymptotically approximated” (as would be the case if the underlying physical processes were differentiable) or will the final reading be exact (up to the limits of our instruments).

In my view, there is a simple explanation that does not require differentiability and is able to explain the reading we get would be exact and not asymptotic. The physical view here can be described as follows: “energy propagates continuously as a wave while it manifests discretely in interactions”. Thus, we can view Nature as being 'continuous' and explain the 'manifestation' of Nature (as readings in our instruments, for example) as being 'discrete'.

Constantinos

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You raise an interesting issue: differentiability in Nature. But whereas you resort to non-differentiable manifolds of Spacetime to correct existing mathematical models of the Universe, I argue in my essay, “The Metaphysics of Physics”, all mathematical models of 'what is' the Universe are in essence metaphysical and will ultimately fail. Knowing truly 'what is' the Universe is like knowing another person; what they think, how they feel and how they will behave. We can only know ourselves truly. All knowledge in this sense is therefore self-knowledge.

The question of differentiability in Nature can be considered in the following very simple context, for example. When we take a thermometer reading, is the final reading on the thermometer “asymptotically approximated” (as would be the case if the underlying physical processes were differentiable) or will the final reading be exact (up to the limits of our instruments).

In my view, there is a simple explanation that does not require differentiability and is able to explain the reading we get would be exact and not asymptotic. The physical view here can be described as follows: “energy propagates continuously as a wave while it manifests discretely in interactions”. Thus, we can view Nature as being 'continuous' and explain the 'manifestation' of Nature (as readings in our instruments, for example) as being 'discrete'.

Constantinos

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See my discussion with George Ellis

http://fqxi.org/community/forum/topic/1337#addPost

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http://fqxi.org/community/forum/topic/1337#addPost

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If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is and was the quantity of people which gave you ratings. Then you have of points. After it anyone give you of points so you have of points and is the common quantity of the people which gave you ratings. At the same time you will have of points. From here, if you want to be R2 > R1 there must be: or or In other words if you want to increase rating of anyone you must give him more points then the participant`s rating was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

Sergey Fedosin

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Sergey Fedosin

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