My answer to the question "Probability density of what?", is: QM predicts only probabilities of results of measurements. For example, the probability of getting a detection of a quantum object in some region of the space.

About "Why does the squared modulus of the wavefunction describe a probability density distribution?", this is an analogy, and it works. The absolute square of the amplitude of the classical wave defines its local intensity. But, if we have to do with a single particle in the wave, then the local intensity becomes the local density of probability to find the particle there.

I think that sometimes the heavy math is easier to understand than the concepts of a theory. Even ordinary things like "energy" are hard to explain in it's more fundamental levels.

About "Why does the squared modulus of the wavefunction describe a probability density distribution?" I would add the fact that is a postulate in quantum mechanics, so we can't show that, only attest it a posteriori.

What is the relation of these distributions with fields?

I don't know how exactly to answer this question, but since in QFT the particles are treated as excited states of the field, we investigate the behavior of particles via transition amplitudes, which has more similarity with the way we treat the energy levels in QM. Besides that, the fact that in QFT the number of particles is not conserved (in general), we have problems to define the probability density and the probability current in this context.

How do they interact? Do you know the mathematics that describes these interactions?

At least in QFT, the Feymann Diagrams are the hammer in our toolbox, but of course we need much more. I think that the first step for someone who wants to venture in this world is the complex calculus.

Obviously physical reality is ruled by stochastic processes and not by all kinds of strange forces and force carriers.

Well, not a question but a affirmation. And I have no reasons to agree with it.

For which particles we may speak of a quantum field, and which ones are "too much classical" to be represented by a field? For instance, for atoms we may speak of a field, or only for elementary particles?

I think that depends on the convenience and the regime. QM is not relativistic, so in the relativistic limit, QM will lose accuracy. But I never heard about treating atoms like fields. I think that the field treatment is only for what we currenty understand as elementary particles but it's a really good discussion. Can we think, for instance, in a 0-spin atom as a excited state of a bosonic field, or of a fermionic field if its spin one-half? Can we think about ionization as a decaying process, for example? I don't know. It's a really good question.

One time, talking to my advisor, I said that I don't understand QFT. He replied: "Don't you understand QFT??? Well... NOBODY DOES.

The problem that bothers me is not what is a field in general, that we all know more or less, but how looks like a quantum field. I don't ask about mathematics, we all are ready to say that in a quantum field there are no particles, but excitations. My problem is that I am trying to imagine how this field and these excitations look like, how they are spread in space.

Here is a problem that I presented within a couple of questions - see the figure.

A single particle wave-packet, describing a proton, lands on a beam-splitter and exits as a superposition (1/√2) ( |a> + i|b>). Then, each wave-packet passes through the electrostatic field of a condenser - the two condensers are identical.

Since there is only one proton, one would expect that in a given trial of the experiment, only the wave-packet containing the proton be deflected by the condenser field, i.e. either |a> be deflected, or |b>, but not both.

Though the things go differently: bringing |a> and |b> to cross one another, after many trials an interference pattern appears, which indicates that in each trial and trial both |a> and |b> were deflected, each by the respective condenser field.

That seems to say the strange thing that in each trial of the experiment we have two protons, one in |a> and one in |b>.

But, where from? What impinged on the beam-splitter was a wave-packet with a single proton. The beam-splitter is not a proton generator.

Here is the problem with fields. In this experiment one has a proton-field, in which the single proton carried by the initial wave-packet, is an excitation. But what happens with this excitation at the output of the beam-splitter? Does it become double (two protons, one in each wave-packet)? Alternatively, does it go to one of the wave-packet? In short, where it goes and how it evolves?

Hans,

My problem is about excitations of quantum fields.

How are these excitations distributed in space? Do they stay in a small volume? Do they occupy all the space? Is their distribution in space dictated by a wave-function? Can they disappear in one region of the space and re-appear in another region?

Next, do we have quantum fields of atoms too? An atom is also an excitation of a field? Of which field? Does there exist a vacuum of, say, Li atoms, and one Li atom is an excitation of this field?

Best regards

Dear Hans,

I refer here to your ideas above. Just please take in consideration that, unfortunately, I never had the occasion to learn QFT systematically. When I was a student we were only taught Dirac's equation - one lesson.

Your idea is attractive, but there are problems. Let's go step by step.

"elementary particles hop around in a stochastic hopping path, and after a while this hopping path has formed a recurrently regenerated hop landing location swarm that can be described by a location density distribution which is the Fourier transform of the characteristic function."

"In fact the characteristic function can be considered to contain a gauge factor that acts as a displacement generator that describes the movement of the swarm as a whole. The fact that the coherence of the swarm is controlled from Fourier space . . . . As a consequence the particle hops violently, but the geometric center of the swarm moves smoothly."

Before going to mathematics - gauge factors and so - let's see the feasibility. What happens if the wave-function is of the form

(1) |ψ> = α|a> + β|b> + γ|c> + ζ|d> + η|e>,

and each wave-packet flies to another lab-station? Assume also that the stations are far from one another. How can your particle be present in each one of the 5 wave-packets? Does it jump over very big distances?

I will stop here with discussing your proposal. If we can solve this issue (very difficult, in my opinion), we can go on to the rest of your comment. You see, I had some other idea, close to your one, but it lead me to the same difficulty: what happens when we have more than one wave-packet, and the wave-packets are distant from one another.

Klaus Kassner proposed the following solution: one should not ask how many particles contains the wave-function, before the wave-packets are tested. So, I understand that he inclined to admit that there may be a particle in each wave-packet, and at the measurement, 4 of the particles disappear.

But I consider his solution impossible. The 5 wave-packets may be produced from a single Fock wave-packet, |1>, by a series of beam-splitters. A beam-splitter is not a producer of particles. As to the initial Fock state, we get it trivially from a down-conversion pair: one particle in the pair is detected separately and serves as herald, the other particle goes to the beam-splitters.

With kind regards,

Sofia

Marcos,

Your answer is very lucid, at each point and point. I am sorry that I didn't see it until now.

Now, I want to refer to your words

"the fact that in QFT the number of particles is not conserved (in general), "

Please consider the problem I pose in my previous reply to Hans. It is in the regime of low velocities, i.e. non-relativistic QM. To this problem the formalism of QM won't answer.

Another issue: you say

"I never heard about treating atoms like fields. I think that the field treatment is only for what we currenty understand as elementary particles . . ."

I have the same opinion. I never heard of a vacuum for atoms. And, though, there exist coherent beams of atoms, and even atom lasers. The number of particles in such a beam is not constant. I ask myself whether this fact doesn't hint that the formalism of the field is correct for the atoms too.

In fact, I saw this formalism applied to atoms, in a work of the Aspect group

Gomes, Perrin, Schellekens, Boiron, Westbrook, and Belsley, "Theory for a Hanbury Brown Twiss experiment with a ballistically expanding cloud of cold atoms", arXiv:quant-ph/0606147v2.

Hans,

Yo are trying to help, but my problem is not there where you give explanations. You didn't read my last answer to you. Please read it. It essentially asks: how many particles we have in the wave-function (1)? Is there a particle in each wave-packet? Or, alternatively, does the particle jump from one packet to another, despite the big distances between the regions where the packets travel?

Please leave those quaternions and concentrate on my problem if you want to help me. I don't ask about mathematics but about phenomenology.

Hans,

Is the third time that I repeat that what is interesting to me is whether in my wave-function

(1) |ψ> = α|a> + β|b> + γ|c> + ζ|d> + η|e>,

there is a particle in each one of the 5 wave-packets, i.e. we have 5 particles, or, one and the same particle jumps from one wave-packet to another.

I don't ask about modules, quaternions, quarks, Fourier transforms, color confinement, etc.

I know that my question is very difficult and it's hard to have an answer. But sending me, instead of that, a theory of everything, is of no worth to me. I am sorry, but I don't have time to read irrelevant things, I have an enormous material to deal with.

Best regards

Hans,

My formula (1) is a honest coherent superposition of wave-packets, acknowledged by the QM. Take any two of the 5 wave-packets, bring them to cross one another, and you'd get an interference pattern. Speaking in space representation or in momentum representation is equivalent. So, what's your problem? Did you never see a multi-packet wave-function?

My problem is whether in each wave-packet there is a particle. THIS IS THE QUESTION. I am not interested that my question be changed, neither am I interested in displacement generators, internal oscillators, quaternions, quarks, elementary modules, etc., etc.

I appologize a lot, but I am so busy that I can't afford reading again and again, things not connected to my question. You have a thread of yours, you can speak there about whatever you want.

With appologies again,

Sofia

I have a very simple question to ask and comment in answer to the intro question:

"Why does the squared modulus of the wavefunction describe a probability density distribution? Probability density of what? ":

What are the dimensions of the numerical answer obtained when plugging real values into the wave function?

If somebody takes the time to do the calculation and carefully traces back what these dimensions must be, maybe the analysis of these dimensions of the answer will give definite hints as to the identity of what the squared modulus of the wave function is an expression of the density of.

I comment here the second part of Hans' post. He says,

" . . . the probability of detecting the owner of the wave function at the location and time of the parameter value. For a point-like object it can be the location where the object actually was, is, or will be. . . . The swarm may reflect the hop landing locations of a cycle of a stochastic hopping path of a point-like object. The mystery lies thus in the fact that the wavefunction is a smooth function, while it may represent a swarm of discrete locations."

The idea is appealing. After all, we have a single quantum object in each trial of the experiment, and this particle feels where are the walls, if any, fields, if any, etc. Without comming in contact with them, how can the quantum object "know" of their existence, and behave accordingly?

Moreover, this idea coincides with Feynman's idea in his "Path integral" theory, that is so much successful. So, why the wave-packet is smooth, I think that Feynman's mathematics explains. Let's not forget that the paths of the particle are endowed with phases that undergo superposition, cancelling one another, or enhancing one another.

My problem is, however, how can cover a single quantum system, a wave-function consisting in a couple of wave-packets, traveling in regions widely separated from one another. To my best knowledge, Feynman didn't tackle this situation.

Some people answered me that what we have in our apparatus is not a wave, not a particle, but a FIELD. Regrettably, throwing magic words doesn't help me. So, I asked the question what is a field. There followed a magic phrase: a "particle" is an excitation in the field.

Dear Hans

I see.

I think you hit the major snag in making heads or tails of QM when you write:

"The problem is that this smooth distribution in fact describes the location density distribution of a swarm of locations that may be hop landing locations."

As formulated, this does not specify whether both representations (smooth distribution) and (location density distribution of ... locations) are "additive" representations or "alternate" representations.

Smooth distribution of "what"?

Location density distribution of "what"?

Let's recall that initially, Schrödinger introduced the wave function to represent a theoretical "volume" within which all of the momentum energy of the electron stabilized in the hydrogen ground state orbital was guaranteed to be contained, which does not allow visualizing this momentum energy other than being evenly distributed within this theoretical volume. So the smooth distribution refers by definition to the total momentum energy that the electron might be associated with at any given moment.

So the smooth distribution aspect would be the smooth distribution of the total momentum energy of the electron within the volume that can be defined by the wave function at any given moment.

The location density distribution of possible locations of the point-like behaving electron on its part can only refer to Heisenberg's statistical spread of all possible such locations of the electron within the volume defined by the wave function.

This would establish both representations as being "alternate representations" and not as "additive representations".

Sofia rightly mentions that Feynman's path integral is also a player in this picture, that is, a third "alternate representation", that I wouldn't be surprised if what you just wrote about what the location would relate to, that is, where the point-like behaving electron "was, is, or will be", might have been what inspired him the path integral representation:

"This function specifies the probability of detecting the owner of the wave function at the location and time of the parameter value. For a point-like object it can be the location where the object actually was, is, or will be."

The "owner" would be the point-like behaving electron, or more precisely, the location where the point of application of its momentum energy physically applies.

This last comment of yours makes me immediately think of a time varying trajectory, possibly introducing a vectorial direction of the point-like behaving electron to be related to any possible location, which is exactly what the path integral introduces in the mix: the plenum of all possible trajectories of the electron within the volume defined by the wave function.

So maybe if these three representations are visualized as being alternate representations rather than additive representations, coherent dimensions could be assigned to each representation without introducing the confusion that seeing them as additive seems to lead to, which prevents clearly assigning dimensions to what the "density distribution" of all possible instantaneous locations of the electron within the volume defined by the wave function.

Does this make any sort of sense to you?

Best Regards

André

Dear Hans

Interesting.

I haven't gone as far as you in tackling all cases, nor explored in the same direction. In fact, I kept my attention only on the initial case that led to the development of QM. All else remains to be explored from this perspective.

What I did was going back to de Broglie and Schrödinger original project of trying to define the hydrogen ground state orbital as if it involved the electron as being captive in some form of resonance state.

We know, as Feynman justly remarked, that QM is not in complete sync with electromagnetism. This led me to try to see what would be the behavior of "the owner", as you say, of the wave function, if it was due to electromagnetic resonance properties that a localized electron would have.

This led to this description that allows seeing the localized electron as captive on an axial resonance trajectory within the wave function defined volume allowed by the path integral, along which trajectory all possible locations in the swarm of possible locations allowed by the Heisenberg statistical spread could be visited over time, and that accounted at any given moment for the precise momentum energy that the electron must have as a function of its instantaneous distance from the proton, possibly providing an explanation to the fine structure of the hydrogen spectrum that Sommerfeld associated to a hypothetical orbit that the electron would follow, in relation with the fine splitting of the main spectral lines.

This recently published paper analyzes this first draft analysis, titled "The Hydrogen Atom Fundamental Resonance States":

http://file.scirp.org/Html/17-7503469_84158.htm

Best Regards

André

Dear Prof. Weschler, Leunen and others,

Isn't it enough to show that where there is a String Theory in a spacextime resolution of spacetime Relativity, which happens in Stock Market Econophysics Field of Archimedean Stock and Flow, because of spacextime resolution of stock market physical actions, String condensation is enough to demonstrate a field like the Higgs Field experimentally.

Soumitra Kumar Mallick, CA

for the RHMHM School of Mathematical Sciences Thinking,

USA, Japan, India

Dear Hans

You write: "Inside atoms the electric charges seem to couple to the platforms on which the elementary particles reside and these platforms seems to bind via the electrical attractions into mostly neutral conglomerates."

Yes. All elementary charges tend to stabilize into axial least action electromagnetic equilibrium states. But even when so stabilized in overall "neutral" configurations, the Coulomb force stemming from Maxwell's first equation (Gauss's equation for the electric field) still remains in full adiabatic action between each pair of these charges. See next:

You wrote: " As long as a stochastic process stays in the same oscillation mode it does not use energy to stay in that mode."

Absolutely right. This is due to the adiabatic nature of energy induction in elementary charges as a function of the distances separating them. This is analyzed in this paper:

https://www.omicsonline.org/open-access/on-de-broglies-doubleparticle-photon-hypothesis-2090-0902-1000153.pdf

You also wrote: "To arrange that this photon keeps its integrity the corresponding oscillation must stop during the emission or absorption action. If the Einstein-Planck relation means something, then this halt must last a full generation cycle of the hopping path. Thus, the emission duration of all photons must correspond to this cycle time."

This concept that the emission duration of an EM photon must correspond to one full cycle of the emitted photon frequency is related to the standard definition of what a transverse wave can be.

Historically and by definition, transverse waves (such as shear waves in the Earth crust) are deemed to propagate only in solid materials due to the fact that the transversely oscillating motion of the material required for the wave to propagate needs a restoring force that fluids or absence of material cannot provide. It can be observed that such transverse wave transmission involves no longitudinal motion of the oscillating "material", only transverse oscillation of the "material".

Isn't it interesting that these are precisely the properties that the theoretical "ether" must have if electromagnetic transverse "electromagnetic waves" as theorized, are to physically exist and propagate at the submicroscopic level?

Seems to me that this would involve that the "ether" be made of a "substance" that could transversely oscillate while being unable to move longitudinally, but that such an "ether substance" never was detected.

A contrario, de Broglie`s conclusions about his double-particle photon hypothesis seem to imply that localized EM quanta would behave as if they would move "longitudinally point-like" while their local energy "substance" transversely oscillates in standing mode between electric and magnetic states, which implies practically instant separation of the "longitudinally point-like behaving" photon energy quantum upon emission.

This requires no underlying carrying medium behaving as a solid material if the longitudinally point-like behaving localized quantum provides its own momentum energy, which seems to be the case for the de Broglie double-particle photon, as definable in full accordance with Maxwell's equations.

You will find the description of this de Broglie photon here:

https://www.omicsonline.org/open-access/on-de-broglies-doubleparticle-photon-hypothesis-2090-0902-1000153.pdf

I think that de Broglie's conclusions involving a physically existing energy quantum whose "substance" would transversely oscillate in "standing mode", and propagating without any longitudinal component as mandated by the standard transverse wave definition should be analyzed further and could possibly provide a long sought after solution to the "ether" conundrum.

I think that this de Broglie inspired new understanding of how electromagnetic quanta can self-propel and default self-guide in straight line in vacuum due to their default equal densities local inner standing E and B fields, without the need of any underlying medium is likely to completely refocus the whole perspective.

Best Regards

André

Dear Hans

You wrote: "As long as a stochastic process stays in the same oscillation mode it does not use energy to stay in that mode. This is especially clear for the electron shells that follow the solutions of the Helmholtz equations. Only if the process changes its oscillation mode, then a string of one-dimensional shock fronts is absorbed or emitted that represents a photon, which transports the corresponding energy."

This is precisely where I am at in my project. That is, trying to establish the actual progressive sequence of resonance emission and absorption of EM photons that de Broglie and Schrödinger were initially after, but from the electromagnetic perspective.

I am currently beefing up my math and in search of all math and data that could help my project along.

Best Regards

André

These questions can only be answered once one is situated within

a given theory situation, for example

Copenhagen

Here the wave function is physically nonexistant, only a mathematical

convenience, not a physical field. Then add Born postulate

Bohm

The wave function is physically real, atttracting the particle to the place where the square modulus is largest. There is a noisy background which

kiks the particle around, so the whole thing conspires to look like qm prediction.

It is probably very inconvenient to mix up physical and mathematical conceps this much, but current theories are not very good yet. Be sceptical.

Ari,

I don't think that somebody would be interested in a point whose only activity/effect is to wait, no matter for what. Thus, nobody would ever express it. In our daily life we have to do with fields that do other activities than to wait.

Dear Hans

Your experimental observations and what you describe in your last post and your linked documents are not in contradiction with what I understand of electromagnetic photons actually being longitudinally point-like behaving "kinetic-energy" quanta "physically-existing-as-a-substance", that would self-propel and electromagnetically self-guide as the de Broglie double-particle photon inner structure reveals. The same for the free-moving electron (different from the electron captive in some least action resonance state, that mandates the use of a volume-defining-wave-function/density-spread-statistical-representation).

I wouldn't know however how to reconcile their Maxwell equations compliant electromagnetic LC equations descriptions with your own mathematical representations.

I need to seriously beef up on my math before I can address this. A study in progress.

Thank you for your links.

Best Regards

André

Conservation of matter is easily deduced from Schrodinger.

Dear Hans

Thank you so much for your new calculus. I will definitely get into studying it.

The recurrent "spherical pulses" that you mention act practically like the cyclic spherical parallel magnetic interaction that prevents the electron from crashing on the hydrogen nucleon from the electromagnetic perspective.

If you read sections 20 and 21, in relation with Figures 9 and 10 in this paper, I would highly appreciate your comment as to whether you see a similarity and if you think your math could bridge the gap between our views:

http://file.scirp.org/pdf/JMP_2018042716061246.pdf

I have the same question for the cyclic spherical repulsion that can explain the electron Zitterbewegung in Sections 18 and 19 with reference to Figures 7 and 8.

Looking forward to find sufficient lee time to study your math in depth.

At first glance, the cyclic nature of the magnetic spheres interaction seems to be the key to relating both views.

Best Regards

André

Follow

Hans

It is true that some elementary particles only have a short lifetime

and can be recreated into others, but at each vertex total energy and momentum

are conserved. I believe that is all that can be said empirically, apart from

which reactions are allowed and which not.

However , in any case im at loss with your terminology, for example what is your definition of module? I know it as a mathematical algebraic term,

nothing to do with physics. So your exposition is too far from the main

trend to be understandable.

Juan