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FQXi FORUM

February 23, 2018

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: The Amazing Effectiveness and Usefulness of Mathematical Models in Physics by Ed Unverricht [refresh]

TOPIC: The Amazing Effectiveness and Usefulness of Mathematical Models in Physics by Ed Unverricht [refresh]

This essay presents visual images of the mathematical modelling of elementary particles of the Standard Model. These images demonstrate the effectiveness and usefulness of the mathematical model by enhancing our understanding of the concepts and our ability to make predictions. You will see massive particles built from swirling vortexes of energy. Massless particles are built from expanding and contracting harmonic oscillators zipping through space. These models cement the relationship between the complex numbers, matrices, probabilities of the quantum world and the observations seen in the experiments of physics.

Ed Unverricht has been actively involved in the field of programming computer animated motion based on particles and forces for many years. Recent projects completed include gravity animations based on Newtons laws. Rotate, zoom and walk through views of the main molecular structures stored at the protein data bank, http://www.rcsb.org. Modelling of the ideal gas law PV = nRT, an Ising model of ferromagnetic interactions and numerous other scientifically interesting structures.

Dear Ed Unverricht,

I found your colorful essay quite interesting. I have essentially the same model of the electron and my Z and W bosons are close to yours, but my model of quarks in hadrons is somewhat different, as described in*The Chromodynamics War* ISBN–13:978–0–9791765–7–9 in 2009. Your very nice illustrations certainly helps readers to visualize these fundamental particles.

Have you programmed any dynamic processes, or attempted to calculate the particle masses from these models? What distinguishes the three families of particles in your model?

I invite you to read my current essay and comment on it.

Best regards,

Edwin Eugene Klingman

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I found your colorful essay quite interesting. I have essentially the same model of the electron and my Z and W bosons are close to yours, but my model of quarks in hadrons is somewhat different, as described in

Have you programmed any dynamic processes, or attempted to calculate the particle masses from these models? What distinguishes the three families of particles in your model?

I invite you to read my current essay and comment on it.

Best regards,

Edwin Eugene Klingman

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Thank you for the comments, much appreciated. I would love to compare hadron models, I like to compare any model that allows for the understanding and modelling of complex spin as in spin 3/2 particles.

Regarding your questions, although outside the scope of what I wrote in the essay, families of particles are distinguished by their windings. All massive particles with both transverse and longitudinal components of spin can have different amounts of the two components. Ie. a particle can turn once transverse for each longitudinal spin, or two transverse spins for each longitudinal spin. The model does not limit the standard model to only three families.

Regarding programming dynamic processes, yes I have done a number. There are a lot more to go as this style of model is built to allow easy modelling of more complex structures, both electronic bonding with discrete energy levels and nuclear bonding with magic numbers are natural extensions of this model.

I have read your essay and like you, I disagree with the statement that "No local model can reproduce QM prediction". I will gather my thoughts and comment more directly on your essay.

Regarding your questions, although outside the scope of what I wrote in the essay, families of particles are distinguished by their windings. All massive particles with both transverse and longitudinal components of spin can have different amounts of the two components. Ie. a particle can turn once transverse for each longitudinal spin, or two transverse spins for each longitudinal spin. The model does not limit the standard model to only three families.

Regarding programming dynamic processes, yes I have done a number. There are a lot more to go as this style of model is built to allow easy modelling of more complex structures, both electronic bonding with discrete energy levels and nuclear bonding with magic numbers are natural extensions of this model.

I have read your essay and like you, I disagree with the statement that "No local model can reproduce QM prediction". I will gather my thoughts and comment more directly on your essay.

Yes, indeed, this essay represents visual images of the mathematical modelling of elementary particles of the Standard Model.

Great work!

Sincerely,

Miss. Sujatha Jagannathan

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Great work!

Sincerely,

Miss. Sujatha Jagannathan

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Dear Ed,

Your way of representing the symmetries of particles is quite illustrative and has pedagogical interest. How do you define the genus 0 (sphere) and 1 (torus) of your pictures? Thanks.

Michel

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Your way of representing the symmetries of particles is quite illustrative and has pedagogical interest. How do you define the genus 0 (sphere) and 1 (torus) of your pictures? Thanks.

Michel

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Hi Ed,

I quickly read through your essay (I'll reread it more carefully later) and I found it**pretty interesting**! It has many features which readily resonate with my own ideas in TOEBI. Very nice essay indeed and I'll get back to it.

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I quickly read through your essay (I'll reread it more carefully later) and I found it

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Hi,

The essay indeed raised some thoughts which I need to ponder in peace. Nevertheless, interesting essay.

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The essay indeed raised some thoughts which I need to ponder in peace. Nevertheless, interesting essay.

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Dear Ed Unverricht,

Interesting visualisations, but I'm struggling to work out how you've done them. Take the SO(3) spinning particle. Could you please explain what is plotted and how it relates to SO(3) rotation?

Best wishes

Michael Goodband

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Interesting visualisations, but I'm struggling to work out how you've done them. Take the SO(3) spinning particle. Could you please explain what is plotted and how it relates to SO(3) rotation?

Best wishes

Michael Goodband

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Thank you for your comment.

Using a program like MSPaint is a way of illustrating which objects make up an SO(3) group. If you rotate and object 180 degrees and compare it to the same object that was "flipped" vertically, you can visually see that some objects end up being the same (they are members of S(3) group) but some objects are not the same, ie. they look reversed in a mirror (they are members of the SO(3) group).

Using a program like MSPaint is a way of illustrating which objects make up an SO(3) group. If you rotate and object 180 degrees and compare it to the same object that was "flipped" vertically, you can visually see that some objects end up being the same (they are members of S(3) group) but some objects are not the same, ie. they look reversed in a mirror (they are members of the SO(3) group).

Dear Ed Unverricht,

As can be expected from someone programming computer animated motions based on particles and forces, your essay provides clear illustrations with required explanations that any one can comprehend what the physicists are saying about the particles of the standard model. The picture is thus very clear, the mathematics is right. But the whole thing is founded on wave-particle duality. If wave-particle duality is correct, then the standard model is correct. As argued in my essay, the interpretation based on mathematical relation can be tricky; the wave-particle duality is a wrong interpretation; bodies cannot remain in two forms at any given instant.

The model proposed by me (not explained in my essay) views light as streams of particle-pairs; the pair rotates as it moves forward, thus path of each particle is a helix. The axis of the helix is circular and the pair is thus confined to circular path of very large radius. Such a motion creates small variations in their speeds, and this creates a varying charge in them, and hence a varying electromagnetic field. I think, based on how you explained the models in the essay, that you would have got a clear mental picture of my model. It is another mathematical model, but devoid of the meaningless concept of duality. Please visit my site: finitenesstheory.com.

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As can be expected from someone programming computer animated motions based on particles and forces, your essay provides clear illustrations with required explanations that any one can comprehend what the physicists are saying about the particles of the standard model. The picture is thus very clear, the mathematics is right. But the whole thing is founded on wave-particle duality. If wave-particle duality is correct, then the standard model is correct. As argued in my essay, the interpretation based on mathematical relation can be tricky; the wave-particle duality is a wrong interpretation; bodies cannot remain in two forms at any given instant.

The model proposed by me (not explained in my essay) views light as streams of particle-pairs; the pair rotates as it moves forward, thus path of each particle is a helix. The axis of the helix is circular and the pair is thus confined to circular path of very large radius. Such a motion creates small variations in their speeds, and this creates a varying charge in them, and hence a varying electromagnetic field. I think, based on how you explained the models in the essay, that you would have got a clear mental picture of my model. It is another mathematical model, but devoid of the meaningless concept of duality. Please visit my site: finitenesstheory.com.

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Thank you for the comments. As I argued in the essay, I believe the wave-particle duality is in play and that the standard model is correct. The visual representation of the particles is meant to provide additional understanding to the model and increase its "usefulness".

The representation of photons is meant to highlight the Heisenberg uncertainty principle. Modelling a photon as expending and contracting, allows you to know where it is when it is contracted but you dont know its wavelength. When expanded, you know the wavelength of the photon, but you dont know where it is, ie. the photon is spread out over its entire wavelength. The visual is meant to highlight the concept that Delta-x times Delta-p

The representation of photons is meant to highlight the Heisenberg uncertainty principle. Modelling a photon as expending and contracting, allows you to know where it is when it is contracted but you dont know its wavelength. When expanded, you know the wavelength of the photon, but you dont know where it is, ie. the photon is spread out over its entire wavelength. The visual is meant to highlight the concept that Delta-x times Delta-p

Hi Ed,

Thanks for reading my essay, answering one of the questions posed (actually, the one my girlfriend thought I should omit), and watching the Digital Physics movie trailer. To answer your question, the movie will eventually be available over the internet, but I'm hoping to have some public screenings at film festivals first. Please sign up for the mailing list if you want to be updated.

As for your essay, I wish I had better knowledge of the particle zoo so I could better appreciate your visual model. Do you think your visual model represents the reality of particles more closely, or do believe it just is another "model" that offers a different perspective and perhaps a more intuitive insight?

Related to this thought, I disagreed with one of the quotes you referenced. If two models both fit the data, I would be ok with preferimg one of them to the other if the one was more elegant. This view would go along with Occam's razor and Leibniz's notion that to understand something is to compress the experimental results into a more simple theory. You wouldn't accept a theory that just added each experimental result as an axiom, so to speak, would you?

Jon

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Thanks for reading my essay, answering one of the questions posed (actually, the one my girlfriend thought I should omit), and watching the Digital Physics movie trailer. To answer your question, the movie will eventually be available over the internet, but I'm hoping to have some public screenings at film festivals first. Please sign up for the mailing list if you want to be updated.

As for your essay, I wish I had better knowledge of the particle zoo so I could better appreciate your visual model. Do you think your visual model represents the reality of particles more closely, or do believe it just is another "model" that offers a different perspective and perhaps a more intuitive insight?

Related to this thought, I disagreed with one of the quotes you referenced. If two models both fit the data, I would be ok with preferimg one of them to the other if the one was more elegant. This view would go along with Occam's razor and Leibniz's notion that to understand something is to compress the experimental results into a more simple theory. You wouldn't accept a theory that just added each experimental result as an axiom, so to speak, would you?

Jon

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Hi Ed,I have forgotten to copy my answer to your blog.

Your essay is excellent. Full of great pictures. You present an approach that I was waiting for. I mean: a visual language, independent, universal and baggage-free description. The last notion is explained in Max Tegmark’s very inspiring letter “The Mathematical Universe” (arXiv:0704.0646v2) where he says: “a description to be complete, it must be well-defined also according to non-human sentient entities (say aliens or future supercomputers) that lack the common understanding of concepts that we humans have evolved, e.g., “particle”, “observation” or indeed any other English words. Put differently, such a description must be expressible in a form that is devoid of human “baggage”.

I think that visual imaging could be the answer.

You say: Usefulness is also high if the model can be represented in a number of different ways and leads to a predictive power. My essay delivers the predictive power that five out of Thurston geometries remain to be uncovered in nature. These are five exotic Riemannian manifolds, which are homogeneous but not isotropic: the geometry of S2 × R, H2 × R, the universal cover of SL(2, R), Nil geometry and Solv geometry. Maybe you could prepare visualizations of these geometries?

Obviously our, humans, problem is that some geometrical structures we are not able to imagine in our brains e.g. S3 and H3. Unfortunately it needs to take a look from R4 perspective. But we can create some projections on R3 or cross-sections. You have tried to help to imagine quarks. It is really interesting… Do not stop in your useful work. Your rating is unfair. You deserve much more.

Regards,

Jacek

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Your essay is excellent. Full of great pictures. You present an approach that I was waiting for. I mean: a visual language, independent, universal and baggage-free description. The last notion is explained in Max Tegmark’s very inspiring letter “The Mathematical Universe” (arXiv:0704.0646v2) where he says: “a description to be complete, it must be well-defined also according to non-human sentient entities (say aliens or future supercomputers) that lack the common understanding of concepts that we humans have evolved, e.g., “particle”, “observation” or indeed any other English words. Put differently, such a description must be expressible in a form that is devoid of human “baggage”.

I think that visual imaging could be the answer.

You say: Usefulness is also high if the model can be represented in a number of different ways and leads to a predictive power. My essay delivers the predictive power that five out of Thurston geometries remain to be uncovered in nature. These are five exotic Riemannian manifolds, which are homogeneous but not isotropic: the geometry of S2 × R, H2 × R, the universal cover of SL(2, R), Nil geometry and Solv geometry. Maybe you could prepare visualizations of these geometries?

Obviously our, humans, problem is that some geometrical structures we are not able to imagine in our brains e.g. S3 and H3. Unfortunately it needs to take a look from R4 perspective. But we can create some projections on R3 or cross-sections. You have tried to help to imagine quarks. It is really interesting… Do not stop in your useful work. Your rating is unfair. You deserve much more.

Regards,

Jacek

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Hi Ed, I see unfair, outrageous votings also on your essay so I have to compensate this undervaluated work and give you the highest. I have planned a little bit less but there is no choice. Your essay is one of the best...

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Dear Ed,

Greetings. The images are very nice as a visual aid (in so far as they refer to the particle aspect of the quantum object), and we hope to use some of them as teaching aids, as also the videos listed in your references [which we haven't seen yet, but surely will].

When you refer to the Higgs boson as a swirling vortex of energy, what does it really mean? Is it not too classical (non-quantum) a picture?

Thus, we felt a bit concerned that the graphics, while nicely emphasising the particle aspect of wave-particle duality, seemingly omits the wave aspect, which you will agree is an equally important aspect. Thus one could do set up, at least in principle, a double slit interference experiment with a beam of Higgs bosons. A Higgs in flight is a wave, which passes through both the slits, and then eventually collapses to a point after reaching the screen. Once it has collapsed onto the screen, we are comfortable thinking of it as a particle and using your visual aids. But what during the flight - how to form a visual aid, if at all possible?

Thanks again for the instructive graphics.

Regards,

Anshu, Tejinder

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Greetings. The images are very nice as a visual aid (in so far as they refer to the particle aspect of the quantum object), and we hope to use some of them as teaching aids, as also the videos listed in your references [which we haven't seen yet, but surely will].

When you refer to the Higgs boson as a swirling vortex of energy, what does it really mean? Is it not too classical (non-quantum) a picture?

Thus, we felt a bit concerned that the graphics, while nicely emphasising the particle aspect of wave-particle duality, seemingly omits the wave aspect, which you will agree is an equally important aspect. Thus one could do set up, at least in principle, a double slit interference experiment with a beam of Higgs bosons. A Higgs in flight is a wave, which passes through both the slits, and then eventually collapses to a point after reaching the screen. Once it has collapsed onto the screen, we are comfortable thinking of it as a particle and using your visual aids. But what during the flight - how to form a visual aid, if at all possible?

Thanks again for the instructive graphics.

Regards,

Anshu, Tejinder

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Thank you for the comments. Unfortunately space limitation restricted the essay and made it impossible to spend much time on wave/particle duality. This is my second essay and if FQXI comes up with a contest that has anything to do with wave/particle duality, expect to find my essay there.

BTW, there is a great essay by Matt Visser here that reflects many of my views on the subject. A quote from the essay dealing with the concept of “tunnelling”/“barrier penetration”:

*Despite yet more common misconceptions, tunnelling is a simply wave phenomenon; it is not (intrinsically) a quantum physics phenomenon. Under the cognomen “frustrated total internal reflection”, the classical tunnelling phenomenon has been studied and investigated for well over 300 years, with the wave aspects (the “evanescent wave”) coming to the foreground approximately 150 years ago ..*

BTW, there is a great essay by Matt Visser here that reflects many of my views on the subject. A quote from the essay dealing with the concept of “tunnelling”/“barrier penetration”:

Dear Ed,

You have obviously put a lot of effort into trying to put a visual model for quantum phenomena, and the visualizations are very pleasing. Your models remind me a little of DeBroglie's standing wave model, but they are prettier, and the extension to a torus gives your model greater expressive power.

But let me ask you these questions:

1. How do you reconcile your visual model with, say. that of the wavefunction in the position basis of a single particle spread over a region of space? I have the impression that you advocate something like an upward scaling proportional to the wave-length (at least your discussion of photons suggests this), but this way of visualizing quantum objects seems to run into difficulties when the wavelength in question approaches mesoscopic scales. How do you visualise its the process of its collapse to a small region upon a measurement? Is it the reverse scaling process only much faster?

2. Can you derive novel predictions (ideally testable) from your model? Can you predict any particle masses currently not theoretically predictable? Can you predict the neutrino masses?

I enjoyed your visualizations and wish you all the best,

Armin

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You have obviously put a lot of effort into trying to put a visual model for quantum phenomena, and the visualizations are very pleasing. Your models remind me a little of DeBroglie's standing wave model, but they are prettier, and the extension to a torus gives your model greater expressive power.

But let me ask you these questions:

1. How do you reconcile your visual model with, say. that of the wavefunction in the position basis of a single particle spread over a region of space? I have the impression that you advocate something like an upward scaling proportional to the wave-length (at least your discussion of photons suggests this), but this way of visualizing quantum objects seems to run into difficulties when the wavelength in question approaches mesoscopic scales. How do you visualise its the process of its collapse to a small region upon a measurement? Is it the reverse scaling process only much faster?

2. Can you derive novel predictions (ideally testable) from your model? Can you predict any particle masses currently not theoretically predictable? Can you predict the neutrino masses?

I enjoyed your visualizations and wish you all the best,

Armin

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Hi Armin,

Thank you for the comments, most of the comments on my essay have had to do with wave properties. Unfortunately, my answer is that it would require another essay to properly present it. To give you an idea of the type of models that can be used to model wave nature, consider:

All particles can exhibit wave properties. A particle that expands and contracts as it moves, will behave differently if it hits another particle when expanded compared to when it is contracted. In 1924, a physicist named Louis deBroglie proposed that the electron would exhibit a wave-like nature based on the electrons kinetic energy. His theory, together with the Davisson-Germer experiment done in 1927, established that an electron, accelerated by an electric field does have wave properties. Based on deBroglie's calculations, an electron, accelerated through a field of 54 volts, has a wavelength of 0.167 nanometers. Based on this wavelength, the electron takes 5.6 attoseconds to complete one expansion/contraction cycle. When these electrons are shot at a surface of nickel atoms where the spacing between atoms is a similar size, the electrons show a recoil pattern that allowed the spacing between the nickel atoms to be calculated.

The wave nature of electrons is only visible when the electrons are accelerated to very high speeds. For photons, mostly all you see is the wave property for understanding of refraction, defraction and interference. For neutrinos, the wave nature is even more important as the neutrino type also changes over time.

Best regards,

Ed

attachments: collision.jpg

Thank you for the comments, most of the comments on my essay have had to do with wave properties. Unfortunately, my answer is that it would require another essay to properly present it. To give you an idea of the type of models that can be used to model wave nature, consider:

All particles can exhibit wave properties. A particle that expands and contracts as it moves, will behave differently if it hits another particle when expanded compared to when it is contracted. In 1924, a physicist named Louis deBroglie proposed that the electron would exhibit a wave-like nature based on the electrons kinetic energy. His theory, together with the Davisson-Germer experiment done in 1927, established that an electron, accelerated by an electric field does have wave properties. Based on deBroglie's calculations, an electron, accelerated through a field of 54 volts, has a wavelength of 0.167 nanometers. Based on this wavelength, the electron takes 5.6 attoseconds to complete one expansion/contraction cycle. When these electrons are shot at a surface of nickel atoms where the spacing between atoms is a similar size, the electrons show a recoil pattern that allowed the spacing between the nickel atoms to be calculated.

The wave nature of electrons is only visible when the electrons are accelerated to very high speeds. For photons, mostly all you see is the wave property for understanding of refraction, defraction and interference. For neutrinos, the wave nature is even more important as the neutrino type also changes over time.

Best regards,

Ed

attachments: collision.jpg

Dear Ed,

I enjoyed your essay and the beautiful examples and illustrations concerning the standard model.

I particularly enjoyed the quotation in the summary:

When one accepts one theory and rejects another which is equally consistent with the phenomenon in question, it is clear that one has thereby blundered out of any sort of proper physics and fallen into mythology – Epicurus, Letter to Pythocles

Well it does beg the question of how we decide which theory is physics and which theory is mythology. In fact it is quite likely that both theories have merit, for example the theory of gravity as expounded by Newton and Einstein.

I find myself in the difficult position of having a theory (the Spacetime Wave theory) (see Solving the Mystery) which seems at odds with the Satandard Model of particle physics. It treats the fundamental particles (electron, proton and neutron) as looped wave disturbances of spacetime. Further it shows that the idea of a quark does not fit in with this new theory as an independent stable particle.

I am not saying that the standard model is wrong. What I am saying is that it does not provide a good unified description of reality. When we do have such a complete unified description of reality (including all fundamental forces, mass and charge) I expect that it will be possible to explain why the standard model is so effective in using ideas of symmetry to describe the particles whose real nature is described elsewhere.

This must sound like mythology (or even heresy?).

Best regards

Richard

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I enjoyed your essay and the beautiful examples and illustrations concerning the standard model.

I particularly enjoyed the quotation in the summary:

When one accepts one theory and rejects another which is equally consistent with the phenomenon in question, it is clear that one has thereby blundered out of any sort of proper physics and fallen into mythology – Epicurus, Letter to Pythocles

Well it does beg the question of how we decide which theory is physics and which theory is mythology. In fact it is quite likely that both theories have merit, for example the theory of gravity as expounded by Newton and Einstein.

I find myself in the difficult position of having a theory (the Spacetime Wave theory) (see Solving the Mystery) which seems at odds with the Satandard Model of particle physics. It treats the fundamental particles (electron, proton and neutron) as looped wave disturbances of spacetime. Further it shows that the idea of a quark does not fit in with this new theory as an independent stable particle.

I am not saying that the standard model is wrong. What I am saying is that it does not provide a good unified description of reality. When we do have such a complete unified description of reality (including all fundamental forces, mass and charge) I expect that it will be possible to explain why the standard model is so effective in using ideas of symmetry to describe the particles whose real nature is described elsewhere.

This must sound like mythology (or even heresy?).

Best regards

Richard

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Dear Ed,

I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

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I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

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Hi Ed,

I enjoyed your impressive visualizations of the Pythagorean faith in the logical beauty of nature. "The laws of nature are described by beautiful equations", as Dirac put it, and you are wonderfully showing that. May be, you would be interested to read our essay where we are doing metaphysical conclusions from this primacy of mathematical beauty.

Good luck in this contest!

Alexey Burov.

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I enjoyed your impressive visualizations of the Pythagorean faith in the logical beauty of nature. "The laws of nature are described by beautiful equations", as Dirac put it, and you are wonderfully showing that. May be, you would be interested to read our essay where we are doing metaphysical conclusions from this primacy of mathematical beauty.

Good luck in this contest!

Alexey Burov.

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Dear Ed,

Thanks again for the kind comments of my essay. I finally got to read your interesting and visually impressive essay. I agree it is very useful to find visual models of particles, and that if they are found to contradict reality, should be replaced. Cool movies and animations!

Best wishes,

Cristi

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Thanks again for the kind comments of my essay. I finally got to read your interesting and visually impressive essay. I agree it is very useful to find visual models of particles, and that if they are found to contradict reality, should be replaced. Cool movies and animations!

Best wishes,

Cristi

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Dear Ed Unverricht,

Your essay is very good. It deserves high score. Perhaps you could help to model the relationship that is important in my essay.

If we define x=classical electron radius / proton Compton wavelength = 2.13252558524. Using this ratio, we can define the following dimensionless value:

DeltaP =2-1 / (2pi * x + 2) =1.935060944. The model would help to understand the possible physical significance of DeltaP. I would be very grateful.

Regards,

Branko

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Your essay is very good. It deserves high score. Perhaps you could help to model the relationship that is important in my essay.

If we define x=classical electron radius / proton Compton wavelength = 2.13252558524. Using this ratio, we can define the following dimensionless value:

DeltaP =2-1 / (2pi * x + 2) =1.935060944. The model would help to understand the possible physical significance of DeltaP. I would be very grateful.

Regards,

Branko

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Dear Ed Unverricht,

In my search for pieces to the Great Cosmic Puzzle and for mathematical toolsets with which to assemble a Cosmological Model based on a True Understanding. Your paper impresses me as being 100% relevant and MOST useful to Quantum Mechanics. I hope to apply these understandings to my idea of Combinatorial Quantum-wave Mechanics (CQM).

Your paper stands above...

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In my search for pieces to the Great Cosmic Puzzle and for mathematical toolsets with which to assemble a Cosmological Model based on a True Understanding. Your paper impresses me as being 100% relevant and MOST useful to Quantum Mechanics. I hope to apply these understandings to my idea of Combinatorial Quantum-wave Mechanics (CQM).

Your paper stands above...

view entire post

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