Dear Sofia,

You are right that the discontinuity of a wave function is with mathematical and physical problems. Mathematically its derivatives are not defined and therefore they cannot be solutions of Schrödinger equation, which is the main condition for their formulation. But the physical observables as momentum, energy, etc cannot also be defined at least in some intervals of space and time.

Dear Daniel,

What I found attractive in Gao's interpretation was the fact that, indeed, the electron charge has to be at a given time concentrated in one point. Otherwise, if the electron charge is distributed, we would have electrostatic repulsion between different parts of the electron charge. To my best knowledge, QM regards the electron as a point particle and point charge.

This argument of Gao made me feel less convinced that there is no particle, only a wave.

However, I suppose that you understand in such things better than I. It seems to me that QFT says something on self-interaction. What is known to you on this?

About your comment, indeed, Gao's interpretation is incomplete. He says that the particle jumps from one position to another, though this is not done absolutely at random, but respecting the probability density of the wave-function. On the other hand, Gao does not tell us what is the wave-function, and how it acts on the particle for imposing the constraint of the density of probability, on the jumps of the particle. You see, to my understanding, the wave-function has to be continuous and derivable. I agree that Gao doesn't say this.

Gao said that it would tackle the problem of the cause that makes the particle jump, "in a future work". I consider this attitude not serious.

Another problem, Gao's interpretation requires a preferred frame of coordinates. Since the simultaneity is relative, by one frame the particle jumps instantly from one wave-packet to another, but by another frame, the time of disappearence from one packet and the time of appearence in another packet, may differ. Meanwhile, the particle either disappeared from the universe, or it is present simultaneously in two different places.

There is also a problem with the entanglements, which requires a preferred frame too.

Dear Sofia,

If you want to consider the self-energy problem for the electric charge presented in the Relativistic Quantum Mechanics for the quantized electromagnetic and Dirac fields, you are coming back to the beginning of the thirties of the past century. That was a big problem in Classical Electrodynamics worked by Lorentz and Poincarè among others. The concentration of the electric charge in one point leads us to infinite mass but its spatial dispersion can make weaker the interaction but without avoiding its repulsion.

Fortunately, mainly due to Schwinger, Feynman and Tomonaga (nobel prizes) we can explain how this energy can be finite and converge within a physical context of QED, i.e. within a polarization of the vacuum and the renormalization group. This makes us to calculate properly the cross section of a charged particle in agreement with experiment and to use a pointlike picture for an electric charge, but this is just a model. Nobody can tell how the electric charge distributes in one electron, for example. But the idea that the charge of a particles is spacially distributed doesn't like me because its value is always constant and following the Coulomb law with a great accuracy, in the static case. Thus I still follow with the idea that we have particle-wave picture as a complementary physical point of view.

My Dear Sofia,

1- As you know I believe that the motion is a discontinuous motion, and I put a theory based on this concept, and I listed the theory of Shan Gao in my list of references.

2- Yes in physics we deal with continuous functions like velocity, acceleration, etc..

So I think the better way to deal with discontinuous motion is to suppose that the mean of discontinuous values must equal the continuous values to be compatible with the classical physics, for example in one dimension we can pose:

ε is the duration during which a moving particle exists before disappearing and

that μ is the duration of the particle's disappearance from

our world before it reappears later, so if the particle at time t1 appears in the location m1 and at time t2 appear in location m2 we can say:

classical velocity = (m2 - m1)/(ε+μ)

and so on...

3- Shan Gao deal with the velocity as a spontaneous velocity I mean with infinite value, but in my theory I didn't use this assumption and instead of it I use a new idea that supposes that the particle didn't feel the duration of its disappearance μ from our world simply because it wasn't in our world (as it was) during this

phase !, and using the special relativity I can calculate ε and μ etc..

With best regards.

Dear Mazen,

Shan Gao did not suppose infinite velocity. He claimed that for a movement that consists only in jumps between different positions, the concepts of velocity and acceleration are meaningless.

By the way, to your attention, the velocity does not have the dimension of mass divided by time. So, I advise to reconsider your formula (m1 - m2)/(ε+μ).

About disappearence of particles, as long as the wave-function describes a state with a fixed number of particles, there is no disappearence. This is indeed a weak point in Gao's proposal, and can be fixed only by assuming a preferred frame. I would rather ask Gao to review his proposal than admit a preferred frame just for saving Gao's proposal, which anyway is incomplete, as I said in my answer to Daniel.

Dear Daniel,

Yes, I just read in Wikipedia the page about self-interaction,

https://en.wikipedia.org/wiki/Renormalization

I understand that the effect exists, but this particular doubt of mine about Gao's work, was relaxed. Indeed, the word 'self-interaction' that Gao used is just inappropriate. He wanted to say that assuming the electron charge distributed within the electron cloud, would entail repulsion between different parts of the cloud. So, assuming the electron charge localized at one time at a certain position, avoids the repulsion. Gao did not get into the problems of infinite mass-energy and renormalization - I understand that these problems appear at the scale of 10-13cm, which is the nuclear radius or less.

You see, I am asking myself if we can afford to reject the idea of a particle which at each time occupies a given position. Let's take a proton instead of the electron. Did somebody find the quarks at some positions, and the gluons at some distances from the quarks? The quarks and gluons are confined particles. They have to be together.

On the other hand, my other objections remain: the lack of explanation of what is the wave-function, how it acts on the particle constraining it to pick positions according to the wave-function squared modulus, which meaning has the linear momentum, and the need of a preferred frame, remain.

Dear Sofia,

The main problem of the discontinuities is that don't allow differential equations and also the quantum physical operators. On the other hand it would be necessary to justify their existance. How could you introduce the discontinuities of the states?

The problem of the mass and the charge divergences for fundamental particles is within the core of Quantum Field Theory and it was for a long time the main issue to solve. For overcome all these results it is necessary to have a good argument or theory.

My Dear Sofia,

1-

Yes you are right he didn't define directly the velocity but

In:

http://philsci-archive.pitt.edu/775/1/qm-dm.PDF

Gao Shan said:

"Thus it seems that the objects must move spontaneously in order to exist."

and said too:

"Thus during the infinitesimal interval near any instant the object will always move in a random and discontinuous way."

So even he didn't work on the concept of velocity, it seems (or at least that's what I understand from his words) that the time interval between two consecutive spontaneous moving is going to zero,

so based on this subject I can say that the velocity (if we calculate it as its general definition) go to infinite.

2-

About velocity:

you said:" the velocity does not have the dimension of mass divided by time "

sure!, I didn't say that, I said before:

" appears in the location m1 and at time t2 appear in location m2 we can say"

3-

You said:

" About disappearence of particles, as long as the wave-function describes a state with a fixed number of particles, there is no disappearence. "

On the contrary, I think the disappearance idea is the power point of the discontinuous motion ... for example in my theory:

a- In non-relativist case when the wave-function describes a state with a fixed number of particles the duration μ almost zero so the duration of the particle's disappearance is almost zero so we have a conservation of number of particles.

b- In relativist case when μ become comparable with ε,

so yes, in this case, the number of the particles is not conserved (and this is compatible with relativist quantum theory)

c-But, in this case, the beautiful result from the disappearance duration is that we can resolve the vacuum catastrophe and we can calculate with very good approximation the density of Dark Energy,

I put this result in this paper:

Research Proposal The origin of dark energy

With best regards.

Dear Mazen

Gao explicitly said that for a particle that at any time appears in another place, by jumping from one place to another, NOT traveling continuously between them, the concepts of velocity and acceleration have no meaning.

The "general definition" of velocity is

lim Δt → 0 [r(t + Δt) - r(t)]/Δt.

You have to know that a function r(t) which is NOT continuous, it does NOT admit a derivative. This is the situation in the present case. Since the particle is NOT continuously present in the space-interval between two points r(t) and r(t1) there exists no series of points r(t + Δt) with smaller and smaller Δt, between r(t) and r(t1). So, you cannot calculate the limit of a series that does not exist.

Also, please don't use consacrated notations. By m one denotes mass, not position.

"On the contrary, I think the disappearance idea is the power point of the discontinuous motion"

If you want to build another theory than the quantum theory (QT), it's your right. But it is irrelevant to my question. My question is about Gao's interpretation, i.e. if it is compatible with QT. I am not looking for theories other than QT. As to dark energy, I am not competent in that.

Good luck with your paper

My Dear Sofia,

1-About:" please don't use consacrated notations. By m one denotes mass, not position. "

Yes, you are right, I am sorry.

2- About:" concepts of velocity and acceleration "

The concept of velocity exists in classical mechanics and even exists in quantum mechanics, the concept of velocity need the concept of the path of the particle, in classical mechanics, we have this concept (I mean path or trajectory) but in quantum mechanics, we don't have this concept, nevertheless the concept of velocity still a survivor in quantum mechanics!

How we can define the velocity even we didn't have a well define path?

For example in discontinuous motion we can do as Feynman did in path integral formulation:

At t1 the particle exists in point (x1,y1,z1) and at t2 the particle exists in (x2,y2,z2), so we can take all imaginary (or virtual) paths that connect the point (x1,y1,z1) to point (x2,y2,z2) and work on continuous velocities on them.

3- For point:" I am not looking for theories other than QT "

Yes, I agree with you, but I think the vacuum catastrophe is very related to the kernel of quantum mechanics because it is related to the uncertainty principle, but this up to you if you didn't want to deal with this subject.

With best regards.

Dear Mazen,

There is no trajectory in the quantum formalism. The formula I mentioned in my previous post was the classical formula, because the way you spoke of velocity was as if you define it classically.

"So even he didn't work on the concept of velocity, it seems (or at least that's what I understand from his words) that the time interval between two consecutive spontaneous moving is going to zero, so based on this subject I can say that the velocity (if we calculate it as its general definition) go to infinite."

There is no "go to" without a series, you cannot calculate a limit without a series.

About the quantum formula for velocity, you won't escape the need of a series. Didn't you learn about derivatives of operators? The velocity operator is the derivative of position operator. Look at the formula of this derivative and see what it requires.

About vacuum catastrophe I know the following: for understanding the calculus with integrals one should first understand simple algebra. So, first of all the basic concepts have to be elucidated, and after that one would go to vacuum catastrophe.

Daniel Baldomir

Daniel, my wise friend,

The problems you point to are the same as I object about. We think the same.

"The main problem of the discontinuities is that don't allow differential equations and also the quantum physical operators. On the other hand it would be necessary to justify their existance. How could you introduce the discontinuities of the states?"

Now, some clarification. The differental equations you speak about are on the wave-function. Gao acknowledges the quantum formalism, but please recall that this formalism is on the wave-function. The particle of which Gao speaks is a substructure of QM. Bohm also proposed a substructure with particles, though they were supposed to follow continuous trajectories. However, it was proved that those continuous trajectories lead to a contradiction with the quantum formalism. So, Gao proposed jumps instead of continuous trajectories, hoping that this proposal won't contradict the quantum formalism.

However, Gao forgot to tell us how he imagines himself the wave-function. As a real wave? As epistemic? He spoke something of realism, but without answering what he thinks that the wave-function is. Neither did he explain in which way does the wave-function act upon the particle. Bohm proposed a quantum potential, Gao proposed that maybe in a future work he would address this problem. (Impressing proposal isn't it? So, Gao build a whole work, a whole book, the main clue of it would be eventually treated in a future work.)

With kindest regards from me, as always

Dear Sofia,

After reading the chapters 6 and 7 of the Gao's book, I must say that my ignorance of what is an electron (with electric charge, spin and being a fermion following the Pauli's exclusion principle) is infinite. I cannot follow the experiments of a distribution of an electric charge in a his classical and quantum systems. That is not the usual picture of an electron that I have and in the chapter 7 he uses these results for justifying his discontinous motion or observables (less the spin). Frankly to undestand a motion as it is represented in fig.7.3 for one electron (charge distribution, momentum, ....) it is necessary much more considerations to do, for instance, the motion of a set of points only is defined if there is a "center of mass" or something like this. In quantum mechanics the things are even worse.

It would be very helpful if Gao could discuss with us these things in your thread.

Daniel Baldomir

My dear Daniel,

My opinion on Gao's book is very bad. Indeed, this book deserves strong criticism. It is full of hocus-pocus. It is not acceptable to say that the charge density produced by an object performing random jumps obeys the absolute square of a wave-function, that the charge current is equal to the probability current given by the wave-function, without explaining what is the wave-function, and how it controlls the jumps of the particle. Without a control, such jumps would produce a chaotic tableau, having nothing to do with the wave-function. Bohm for the difference, said that the wave-function guides the particle, and proved that the Bohmian particle's trajectory obeys the equation of continuity. Gao proved nothing similar.

Also, if you read in section 7 Gao's explanation about entanglements, I supposed you would have noticed the abracadabra between the concept of quantum superposition and the idea of a definite position at a definite time. Section 7 is full of self-contradictions. Functions of a position vector which changes discontinuously, should also have discontinuous values - but in Gao's book those functions are contiuous and even derivable.

I want indeed to invite him to answer questions, but an article containing strong criticism would be also desirable. You see, there are many people who write all sort of fantasies which simply don't deserve attention. But about Gao's idea of jumps, I feel that can't be easily rejected. Moreover, it would solve some hard problems. Therefore, as Gao is a very knowleageable fellow, I would expect from him to imbed his idea into a much more rigorous material.

I'd like to ask you a few additional things. Tomorrow with God's help, now is quite late in my country.

Kind regards!

My Dear Sofia,

1-

About: "There is no "go to" without a series, you cannot calculate a limit without a series."

I think we have two approaches here:

a- If between two appearances we have always at minimum one appearance so we have always an infinite number of jumps, so I think for this reason Gao Shan noted that we have in this case what he called "Dense set".

In this case, I think the best mathematical formulation is the fractal model as Laurent Nottale did in his theory "Scale Relativity", but in this approach, I think that the idea of disappearance and appearance is not suitable, and in this theory only we have the idea that the path of the particle is not differentiable, etc..

b- But If we arrive at two appearances that we don't have another appearance between them, so in this case we need to suppose in general that we have two durations ε and μ as I did in my theory.

c- For me, I think that the infinity does not exist in nature, all thing have a finite natural number ... so I follow the second approach.

2-

About: "The velocity operator is the derivative of position operator. Look at the formula of this derivative and see what it requires."

Yes sure, but I think we didn't have any contradiction between the disappearance and appearance idea and the derivative concept because the wave function is continuous, and the continuity of wave function comes from the potential that the particle can appearance in any time! so the continuity of time give us this result and this result does not contradict the disappearance and appearance idea

, sure I think too that space and time do not continue but this another thing...

With best regards.

Dear Mazen,

Shan Gao's work is wrong. He contradicts himself all the time. I don't know whether it is salvageable or not, anyway it's not my duty to improve it.

The idea that a particle jumps from one place to another has to be compatible with the wave-function. A function of a discontinuous variable is discontinuous. Therefore it is also non-derivable. The Schrödinger equation is for wave-functions derivable up to the 2nd derivative. Gao's offers some hocus-pocus with the jumps, i.e. wording, that despite the discrete positions, the density of probability is continuous.

A jump should be instantaneous, otherwise one has to admit that the particle disappears for a while from the universe. Since a wave-function as |1> is an eigenfunction of the number-of-particles operator, such a disappearence is in contradiction with QM. As I said, I don't look for theories which are incompatible with QM. Gao did not say that he looks for such a theory, he believes that his interpretation is compatible with QM. I think that his interpretation is hocus-pocus.

Daniel Baldomir

Dear Daniel,

I have a very bad opinion about Gao's interpretation. The fellow should learn what is a rigorous proof. Most of his claims in the book are not proofs but wording, abracadabra. So the particle jumps from position to position, but, by Gao, the wave-function (w-f) is continuous. I never heard that a function of a discontinuous variable can be continuous. The Schrödinger equation requires not only continuity of the w-f, but also derivability up to the 2nd derivative.

That's not all. The fellow required that the tableau of positions picked randomly by the jumping particle, reproduce the density of probability provided by the absolute square of the w-f. But he didn't care to explain in which way an external field influences the jumps of a particle supposed to jump at random. Also, the w-f has not only an absolute square but also a phase. Gao didn't care to define such a thing.

The abracadabra is especially irritationg when he tries to explain how a single particle occupying at each time a well-defined position, can create an interference pattern. Such a pattern is the result of two objects, two waves which meet.

The truth is that the idea of a particle that at a given time can't be in two positions at once, seems to me unavoidable, but Gao's abracadabra is not what I would like to see.

I am writing you also a message - please be kind and read it.

With best regards

particles cannot be 1 point: uncertainty. in my opinion a point is less than itself :)

Dear Sofia,

I fully agree with you and what proves that you are right is that he doesn't take the opportunity that your thread could be for explaining about the results presented in his book. For example, if he were open to a free discussion here in your thread, then I would try to read with more care to read it and I suppose many people as me.

Dear Paul,

It is not the same to be a point (mathematical concept that cannot exist in the reality) than to say that the fundamental particles as the electron behaves as if they were a point within our measurements and calculations at present. Perhaps in the future with more accurate techniques and theories we could get to distinguish many details that nowadays are lost.

My Dear Sofia,

1-You said:

" A function of a discontinuous variable is discontinuous. Therefore it is also non-derivable. The Schrödinger equation is for wave-functions derivable up to the 2nd derivative "

As we know the wave-function give us in any time the probability to find the particle in any position in space.

the wave-function is continuing in space and time...

Is this contradict the jumping idea?

No...

why?

because the continuity of wave-function tells us that the particle can be found at any time in any position (sure with respecting of Schrodinger equation solution).

And also the jumping idea tells us that the particle can appear (by jumping from its original position) at any time in any position (sure with respecting of Schrodinger equation solution).

Always In real experience, we found the particle in a specific location and in a specific time, we didn't find any continuity!

The continuity only means that the particle can appear at any time.

2- you said:

" A jump should be instantaneous, otherwise one has to admit that the particle disappears for a while from the universe. Since a wave-function as |1> is an eigenfunction of the number-of-particles operator, such a disappearence is in contradiction with QM. As I said, I don't look for theories which are incompatible with QM. "

I think Mr. Gao doesn't have a problem with the number-of-particles operator because the jump is instantaneous in his theory.

But I think the problem is with me :)

But I said to you before: In non-relativist case, my theory is compatible with QM because the duration of disappearance is almost zero, and in relativist case, my theory is compatible with QM because in this case, the number-of-particles is not conserved.

With best regards.

Dear Daniel, and Paul,

I absolutely agree with Daniel. By the QFT the electron cannot be a geometrical, dimensionless point. I am no specialist in QFT, I only read what the explanations, so, QFT says that a bare electron has less mass than the mass obtained from experiment, and around the electron there are virtual photons. Well, a photon should have some wavelength. So, taking the electron for a point seems to be done in QM just for simplicity in calculi with the wave-function. By the way, so are taken the proton, or even atoms.

Dear Mazen,

You have to distinguish between assumptions and conclusions. If the wave-function is generated, as Gao says by jumps of the particle, not by a continuous motion, the set of positions, dense as it may be, is countable, it may contain an infinite number of positions, but it is not a continuum.

Gao says that this countable set of positions covers the pattern of the wave-function. It may be but the wave-function is continuous, while this coverage is not.

An infinitesimal duration of separation, for a huge number of jumps may sum up to a finite interval of time. QM does not permit. Neither does it permit instantaneous jump, which would mean that the particle is simultaneous in two positions, i.e. it becomes two particles.

But, please, examine deeper your arguments before sending them to me. I am busy.

IMHO, the only consistent interpretation of QM is in terms of response. (Anyone interested please look at the series of papers we published over years with late Stig Stenholm; a place to start is "Quantum theory of an electromagnetic observer: Classically behaving macroscopic systems and the emergence of the classical world in quantum electrodynamics", PRA 92, 022122, 2015). The minimal formal framework for this interpretation is quantum electrodynamics. I therefore have grave doubts about any "interpretation" based on the Schroedinger equation in a potential.

dear Sofia, dear Daniel, it is exactly what I said: didn`t You see the irony?

In the Hilbert Book Model, all elementary particles hop around in a stochastic hopping path that recurrently regenerates a coherent hop landing location swarm. A location density distribution describes this swarm and a stochastic process generates the subsequent hop landing locations. This process owns a characteristic function that equals the Fourier transform of the location density distribution. The location density distribution equals the squared modulus of what physicists call the wavefunction of the elementary particle. Elementary particles act as elementary modules. Together they constitute all other modules that exist in the universe. The book "The Mathematics of Physical Reality" contains a chapter with title "Action" that treats the relation of the stochastic hopping path and the path integral.

Hans van Leunen

Hi, Hans,

What is "Hilbert Book Model"? I never heard of such a thing.

"all elementary particles hop around in a stochastic hopping path that recurrently regenerates a coherent hop landing location swarm. A location density distribution describes this swarm and a stochastic process generates the subsequent hop landing locations. This process owns a characteristic function that equals the Fourier transform of the location density distribution. The location density distribution equals the squared modulus of what physicists call the wavefunction of the elementary particle."

I don't succeed to figure out to myself this picture. Do you mean that in that model the particle is jumping instead of a continuous trajectory? If positive answer, then I see a problem. The particle would visit a (practically) infinite number of positions, one after another. So, the set of points visited is a countable set. However, the volume occupied by the wave-function is a continuum, is not a countable set.

The situation is even worse. The wave-function has a phase. This is also a function of position and time. From the density of probability you cannot extract the phase of the wave-function.

Third issue is that the picture of a storm does not fit the picture by which Feynman obtained the path integral. You see, Feynman also thought of many paths, most of them of a crazy form (though continuous). But the particle supposed to travel all the possible paths between two space-time points (t1, r1) and (t2, r2), was considered as traveling these paths SIMULTANEOUSLY. i.e not in series (one paths after another), but in parallel. This is what permitted to him to do summation of the phases of all these paths and obtain the path integral. An ordinary particle cannot travel on many paths simultaneously. So, Feynman's particle is not an ordinary particle. Though Feynman's path integral works excellently.

Sofia,

The Hilbert Book Model is a purely mathematical model of physical reality that starts from a trustworthy foundation. It is also the name of a personal project that I gave in 2011 that name because the model steps with universe wide steps though a read-only repository. So it behaves like a book. The repository consists of a huge number of Hilbert spaces. This explains the name "Hilbert Book Model".

The model and a survey of the of the project is contained in "The Mathematics of Physical Reality"; you can freely acces this pdf file on http://vixra.org/abs/1904.0388. I regularly update the file.

If you take this file to a local print shop, then his service can produce an affordable ring band book. The file contains a front cover and a back cover. In my village the print shop adds transparent sheets to guard the content of the book.

The book is controversial and unorthodox. It contains new physics and new mathematics. You can also use the file as an e-book.

A docx version of the pdf file is accessible at http://www.e-physics.eu/HBMPSK.docx. I use MathType to configure the formulas in MS Word documents. Also see: http://www.e-physics.eu/__QuaternionicDifferential.rtf

That file contains formulas of quaternionic differential equations that are usable at discussion sites like ResearchGate.net

Dear Hans,

In my reply to you I posed a problem that you didn't refer to at all. But, it is the main problem. Is the topic of path integral familiar to you? If it is, can you refer to that problem?

I am repeating the problem here. The particle that Feynman imagined is not a usual particle as in the classical mechanics. It does not visit different points serially, i.e. one after another, but in parallel, i.e. simultaneously. So, it is at the same time in different places, on different trajectories. The particle follows between (t1, r1) and (t2, r2) all the possible trajectories, in parallel, not serially. That means, not one trajectory after another, but all of them simultaneously.

I don't see a way to reconcelliate a serial movement of jumps with a movement in parallel. DO YOU? If you can offer a way of reconcelliation, it would be great.

You see, if these trajectories are not travelled simultaneously, one cannot expect that their phases interfere, as happens in Feynman's path integral. If my problem is not clear to you please tell me.

With kind regards

My Dear Sofia,

1-You said:" If the wave-function is generated, as Gao says by jumps of the particle, not by a continuous motion, the set of positions, dense as it may be, is countable, it may contain an infinite number of positions, but it is not a continuum. "

I think we have a misunderstanding here, I said before that the jumping idea doesn't contradict the continuity of wave function but I didn't mean as Gao did in his theory.

From my point of view, any appearing (I mean when the particle did one jump) will cause the wave function collapse.

so if for example, the particle exists in (t1, r1) so the particle does one jump to be in the new coordinates (t2, r2) and in this moment t2 the wave-function will collapse.

So the continuity of wave-function tells us that the particle can be found at any time, And also the jumping idea is compatible with this thing and tells us that the particle can appear (by jumping from its original position) at any time.

2- You said:" An infinitesimal duration of separation, for a huge number of jumps may sum up to a finite interval of time. QM does not permit. "

Sorry, I don't understand what you mean by "QM does not permit."

3- You said:" Neither does it permit instantaneous jump, which would mean that the particle is simultaneous in two positions, i.e. it becomes two particles. "

Yes this happened with the theory of Gao, but in my theory, I didn't have this problem because always the total velocity of jumping is: distance/(ε+μ) < c (I mean by the distance the minimum length between r2 and r1), in other words, I mean that the interval between the two appearances of the particle is not a space-like interval.

4- You said:" But, please, examine deeper your arguments before sending them to me. I am busy. "

First I highly appreciate your time that you spend with us on the subject of Quantum Foundations, very few people still care about this subject in In this time, for me, I try to explain my ideas as I can, so please forgive me if I failed some times, again thank you for your time that I highly appreciate.

With best regards.

Mazen,

The elementary particle hops around in an ongoing stochastic hopping path that recurrently regenerates a coherent hop landing location swarm, which is described by a continuous location density distribution that equals the squared modulus of the wavefunction. The hop landing locations are generated by a stochastic process that owns a characteristic function, which is the Fourier transform of the location density distribution of the swarm. A displacement generator is attached to this characteristic function. It controls the movement of the swarm as a whole, which moves as a single unit. Where the particle hops violently will the swarm move smoothly. This is due to the very large number of hop landings that are contained in the swarm. The fact that the hop landings are controlled via Fourier space causes that the swarm also acts as a wave package. However, it is no genuine wave package because it is recurrently regenerated at a very high regeneration rate.

Still the particle shows particle behavior AND it shows wave behavior.

My lovely Mazen,

You are so sweet. But, I have great reluctance in accepting something that differs from QM. You, see, it's easy to propose a different theory than QM. But one who proposes, has to make sure that his/her theory agrees with ALL the experiments in QM. Different people come with proposals, if I would check for them whether they don't run into conflict with some experiment, I won't be able to do anything else in my life.

You have to understand me.

By QM does not permit, I mean that a wave-function of a fixed number of particles (in particular the wave-function of a single particle), are eigenfunctions of the number-of-particles operator. A jumping particle, the jumps of which are not instantaneous, disappears from the universe for some times, s.t. it violates the number-of-particles conservation. On the other hand, if the jump is instantaneous, the particle appears at once in two places, so, we have two particles. Again violation of the number of particles.

Worse than that, if in one frame of coordinates the particle appears as jumping instantaneously, in another frame there would be an interval of time between the disappearence in one point and appearence in another point. So, it's total inconsistence with the number of particles.

Now, the human resources are limited. When someone sees that he/she cannot and cannot explain the problems in QM, he/she leaves those problems. I believe that I see a gleam of logic in some problems, otherwise probably I would do the same.

However, despite all the things that I told you above, please visit my new question

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

With warm regards

Sofia

Dear Hans van Leunen,

I think you mean that if we apply the statistical set of momentum (which is the Fourier transform of the location density distribution) to the statistical distribution of particle position we can generate the new location density distribution and again we apply the new statistical set of momentum (which is the Fourier transform of the new location density distribution) and so on..

But I think you know that the big difference between the classical mechanics and quantum mechanics is exactly this Fourier transformation between the position and momentum...

With best regards.

Dear Mazen Khoder , Hans van Leunen , and all the users,

I invite everybody to read my new question

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

Everybody knows the great success of Feynman's path integral theory. So, it is plausible that it is correct. The only advantage of the idea of "hops", is that it allows a particle to jump from one wave-packet to another wave-packet. But I have doubt that this is what physically happens, I mean, that this is the explanation of how a single-particle wave-function can have more than one wave-packet.

I invite everybody to participate to my new question mentioned above, and maybe we have some chance to clarify the issue.

With best regards

My Dear Sofia,

First thank you for your kind reply,

You said:

" By QM does not permit, I mean that a wave-function of a fixed number of particles (in particular the wave-function of a single particle), are eigenfunctions of the number-of-particles operator. A jumping particle, the jumps of which are not instantaneous, disappears from the universe for some times, s.t. it violates the number-of-particles conservation. On the other hand, if the jump is instantaneous, the particle appears at once in two places, so, we have two particles. Again violation of the number of particles. "

We have an important issue here,

- First, the interval between the two consecutive appearances of the particle is a time-like interval.

- For example, the velocity of jumping is: distance/(ε+μ) < c (If we choose to calculate the velocity using the direct line between the two positions of appearances)

- in non-relativist quantum mechanics, we have μ the number-of-particles is conserved

- But we didn't have the situation that one observer sees two particles at the same time

Why?

because (and this is an important thing) the observation cause the collapse of wave function I mean that the particle reset the time of existence ε, in other words for each observation (before the particle disappear) the particle start the duration ε again, so if we still do the observations of particle then the particle didn't can disappear that means the particle didn't can move.

and this what we called quantum Zeno effect.

so always the observer feels all durations, so we didn't have any space-like interval.

With best regards.

I do not think such a scheme can work, but it might be necessary to see how Gao makes it to work in a formal sense, if indeed he does have a formal proposal, and not just merely an intuitive idea.

There exist many forms of the path integrals. The classical one, by Feynman, has continuous paths in configuration space, which are not differentiable and so have no well-defined velocity. There are versions in which the particle moves in phase space, that is, it is assigned simultaneously a position and a momentum at various time steps. These paths are then in fact discontinuous, thereby capturing the idea that we cannot sensibly find a continuous evolution of a quantum particle simultaneously in position and momentum.

However, your description appears to describe discontinuous paths in configuration space: that does not seem right.

Best wishes,

Francois

F. Leyvraz

Hi, Francois,

I am glad that you decided to show yourself on RG. In general, you are a rare guest here.

About Gao, I sent him strong objections, but he answered me that he is on travel now.

I didn't see a formalism in Gao's work, only wording. He did not define a wave-function, but said that all the set of positions visited by the particle, cover the space spanned by the wave-function, with the appropriate probabilities. The concept of the wave-function cannot be defined only from probabilities. The wave-function has also phase.

He also contradicts himself. On one hand he says that since this is a motion of jumps, the concepts of velocity and acceleration have no meaning, but on the other hand he says, in another section of his book, that position and velocity are both defined. In any case, with an incomplete definition of the wave-function, he cannot say on what act the QM operators. Worse than that, Gao does not say how the jumps of his particle, declared by him as being random, are influenced by fields or other types of environment, if velocity and acceleration have no meaning.

It doesn't seem to me a serious work. I read practically all his book and there is plenty of objections from my side.

But, since yo are in RG, please be kind and visit also my last question:

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

I believe that it's the most difficult question I ever asked on RG. It goes straight to the heart of the superposition principle. Please see whether you can give an answer.

With kind regards,

Sofia

It seems that the picture of an ongoing hopping path that recurrently regenerates a coherent hop landing location swarm brings lots of confusion. The swarm is a discrete set, which can be described by a continuous location density distribution. That distribution equals the squared modulus of what physicists call the wavefunction of the particle. Its phase plays no role. Only its modulus is used. The dynamic location density distribution is a real function and has a quaternionic parameter space. This function has a Fourier transform. For that reason it can be considered as a wave package. However, it is not a normal wave package, because its subject, the hop landing location swarm is recurrently regenerated. Normal wave packages disperse when they move. This special wave package does not do this.

A private stochastic process keeps generating the hop landing locations of the stochastic hopping path. The swarm moves smoothly as one unit and can be characterized with a position, a speed and an acceleration. It contains a huge number of hop landing locations. The footprint of an electron hops about 1020 times per second and contains about 1010 hops. In the hopping path, light speed plays no role. The hops cause deformation of the embedding field. There light speed enters its influence.

Hans van Leunen

Dear Hans,

I don't understand what you neam by 'swarm'. But, in my modest opinion, we don't have to do with a particle as Gao thinks. It's a wave. Only a wave can be present simultaneously on different trajectories, and only a wave has phase.

I think that Feynman mislidingly used the word particle. Well, in his time the quantum theory was in process of being built, it is understandable that there were confusions.

In the 2slit experiment you have no 'swarms', the quantum object travels simultaneously through the two slits. When reaching a point of a screen behind the slits, the phases of the wave-packets comming through the two paths add up. It may be that the sum is null, in which case the point remains dark, or it may be that the sum is different of zero, in which case the point is bright.

Anyway, there is no jump between the two paths as long as they don't overlap. One can ensure that by passing the two wave-packets through optic fibers. Thus, each wave-packet travels exclusively through its fiber. The quantum object advances toward the screen simultaneously through the two fibers.

The quantum object is rather a wave, not a particle.

Of course, one would ask, if we place detectors on the two paths, why only one of the the detectors fires in a given trial of the experiment. Well, this is another whole story.

With kind regards

Sofia,

We can only catch the quantum world in a model. Most aspects of that world cannot be perceived, not even when observation is aided by sophisticated equipment. Several scholars in the early decades of the twentieth century have tried to discover which model could fit. One of them was John von Neumann. He doubted long between Hilbert spaces and projective geometries and finally he selected Hilbert spaces. Many quantum physicists followed him in that choice. Then Schrodinger came with his equation that used a wavefunction. Apart from him Heisenberg came with his matrix theory. Then Dirac showed that both theories are principally identical and could be represented in a Hilbert space. The wavefunction is a hypothetical object. It is never observed. Quantum objects behave in ways that can be explained by similarity to wave behavior. Their detection patterns show interference behavior. Nobody ever saw a quantum object as a wave or as a wave package. In similar sense show quantum objects behavior that is similar to the behavior of point-like particles. However, no instrument can directly observe a point-like particle. So the model that must explain the behavior of quantum objects must offer an explanation of this wave-particle dualism.

A separable Hilbert space can archive the dynamic geometric data of a point-like particle when it applies quaternions for the eigenvalues of its operators. However, the eigenspaces of the operators in a separable Hilbert space are countable. These eigenspaces can only store an ongoing hopping path. The squared modulus of the wavefunction is a detection probability distribution. As a storage medium the separate Hilbert space can only implement the wavefunction or better its squared modulus as the descriptor of a coherent hop landing location swarm. The ongoing hopping path can recurrently regenerate such hop landing location swarm when the hopping object hops around in a stochastic manner such that after each cycle the path is closed. If the object can be characterized by its wavefunction, then each cycle must generate a coherent swarm that represents a closed hopping path, while each time the location density distribution of the hop landing location swarm is practically identical. This can only happen if a stochastic mechanism generates the hop landing location in a sufficiently controlled way. This means that the mechanism must be a stochastic process that owns a characteristic function, which is the Fourier transform of the location density distribution that describes the hop landing location swarm that the mechanism recurrently reproduces. This fact implies that the swarm can show interference patterns. That is why the object shows wave behavior. Thus the separable Hilbert space, the stochastic process and the operator that manages the storage of the dynamic geometric data of the hopping object are all private to the quantum object. Together, they form the platform on which the object lives. The eigenspace of the operator archives the complete life story of the quantum object.

In quantum physics many of such objects exist and each of these elementary particles owns its own platform.

A Hilbert space applies a private version of a quaternionic number system to define the inner products of pairs of vectors. It uses that same version for the specification of the eigenvalues of its operators. Further it applies a vector space from which the Hilbert space retrieves its vectors. Many separable Hilbert spaces can share the same vector space. The selected version of the quaternionic number system is also used as the private parameter space of the Hilbert space. A dedicated reference operator manages this parameter space in its eigenspace. However, in a separable Hilbert space, that eigenspace only stores the rational values.

One separable Hilbert space acts as a background platform. Its reference operator manages the background parameter space of the model. All other separable Hilbert spaces float with the geometric center of their private parameter space over the background parameter space. In this way will all elementary particles float over the background parameter space.

So where Dirac used only one Hilbert space, a better suited model will apply a myriad separable Hilbert spaces that all share the same underlying vector space.

The elementary particles act as elementary modules. Together they constitute all other modules that exist in the model. Some modules are modular systems.

This sketches the main structure of an applicable model.

The main message is that the hopping paths of the particles recurrently regenerate a similar coherent hop landing location swarm that show particle behavior as well as wave behavior. The particle lives on a platform that moves as one unit.

The map of the swarm on the background parameter space moves smoothly, while the image of the point-like particle hops violently.

Be aware, elementary particles are very complicated constructs!!!

A possibly relevant direction of research :

Article Casting Loop Quantum Cosmology in the Spin Foam Paradigm

(It is my own work - which is perhaps why I was paired with this question by researchgate)

The short story is that if you follow the standard path integral derivation for a system with a configuration variable having a discrete spectrum (in the paper discrete spectrum of volume in quantum cosmology) the path integral has precisely the structure of being a sum over trajectories that are constant almost everywhere aside from a discrete number of jumps.

No longer active in the field, but I tried to deposit most / all of my thoughts on the subject in my dissertation in the (small) chance that others would be interested.

to Hans van Leunen

>"All other separable Hilbert spaces float with the geometric center of their private parameter space over the background parameter space. In this way will all elementary particles float over the background parameter space."

Can it be that we can build some topological approach describing this in terms of two distance metrics:

1) distance in the wavefunction space or perhaps, if I understand right "their private parameter space over the background parameter space".

2) distance in observable, measurable paremeters mapped to the same "their private parameter space over the background parameter space".

So it is dual distance metrics graph, e.g. like say Facebook (distance exist both in sense of 'handshakes' and km)

What is really tempting here is to explain 1/f noise, which is still not really understood and for which

a) variance of observable parameters diverges roughly as log(time)

b) changes of what should be independent observable parameters tend to correlate

So we can guess that 1/f noise is some case of quantum walk where exist rather complicated densely interconnected graph of wavefunction states (or maybe some other 'private' graph of states).

That seem to be the case as 1/f noise tend to arise in systems having many degrees of freedom and some weak non-linearity which presumably cross-links the 'wavefunction states'.

Nikolay Pavlov

A parameter space is a flat field. A field is described by a function that applies such parameter space. Both the field and the parameter space can be represented by eigenspaces of normal operators that reside in a quaternionic Hilbert space. This holds both for separable and non-separable quaternionic Hilbert spaces.

A Hilbert space is defined as a vector space that uses an inner product that gets its values from a selected version of a number system. That number system must be an associative division ring. Only three suitable associative division rings exist: the real numbers, the complex numbers, and the quaternions.

A huge number of separable Hilbert spaces can apply the same underlying vector space. This construct can act as a dynamic base model that can be used to model quantum physical systems. Separable Hilbert space offer countable eigenspaces. Non-separable Hilbert spaces can support continuum eigenspaces. Continuum eigenspaces can represent continuous fields.

Quaternions are ideally suited as storage bins for dynamic geometric data of point-like objects. Quaternionic fields can define dynamic fields and quaternionic differential calculus describes the dynamic behavior of these fields.

Quaternions cannot store spacetime coordinates! They only store Euclidean space and time as a combined pair of data. At the observed event, spacetime coordinates become Euclidean coordinates. Thus the Hilbert space based archive only holds observable events. Observers perceive in spacetime coordinates. The observed data do not fit in Hilberts structured archive.

The stochastic processes that generate the hop landing locations of elementary particles are spatial Poisson point processes. That indicates their relation to shot noise. The spatial Poisson point processes are combinations of a genuine Poisson process and a series of binomial processes that are implemented by spatial point spread functions. The spatial Poisson point processes are controlled by a characteristic function that is the Fourier transform of the point spread function.

The book

"The Mathematics of Physical Reality" http://vixra.org/abs/1904.0388 gives a detailed description of a possible mathematical model.

Also see "Steps to the Hilbert Book Model" http://vixra.org/abs/1906.0395

``Random jumps'' by itself doesn't mean anything. Quantum mechanics, in the canonical formalism, provides a very precise way for computing the probability that any ``jump'' occurs.

Feynman's path integral provides a completely different way for computing the same probability that any ``jump'' occurs.

There isn't any ``discontinuous motion'' occurring. The position, as a function of time, that's part of any given sample, is a continuous function of time-it's not a differentiable function of time. However it is known how to consistently define the correlation functions of the velocity, nonetheless. In the path integral formalism it's clear that there aren't any new issues with the quantum mechanics of a particle, that aren't, already, present for the case of a classical string, in equilibrium with a heat bath.

None of the accepted physical theories has a an accepted foundation. Without such foundation only observations can reveal something of the behavior of physical reality. Observations never explain that behavior. So they also cannot verify the truth of any explanation of a behavior of physical reality. At the utmost physical theories can describe the observed behavior of physical reality. That suffices for applied physics. It does not help theoretical physics.

It is possible to observe low dose beams of particles. it is possible to measure the signal to noise ratio in those beams. Image intensifier devices can intensify the images of detected quanta, such that they become visible. Look whether you can observe discontinuous motion in intensified images. See: http://www.e-physics.eu/#_What_image_intensifiers reveal

Only "random jumps" cannot replace, or be equivalent, to QM because this contradicts Bell, who's results prohibit classical statistical theories as a substitute for QM. Both in Feynman's initial formulation of probability amplitude and in phase-space path integrals there is "something else". In the former, it's (a) the concept of probability amplitude itself which is strictly nonclassical, and (b) the measure which is not even real. In the phase-space path integrals, it is the measure, which, while real, is nonpositive.

To add, continuity of trajectories is a secondary issue. The primary one is that we cannot have classical stochastic description of trajectories.