Sofia D. Wechsler This question reminds me of Bell‘s theorem which relates to hidden variables. The theorem includes the assumption of causality and locality and shows that the results of QM cannot be explained by new hidden variables.

Is this what you mean by new axioms?

I am trying to recall the current axioms of QM and I think for the Copenhagen interpretation we would be listing:

- wave / particle duality

- the uncertainty principle

- the correspondence principle

- quantum superposition?

- the exclusion principle

Are these the starting axioms to which you would consider adding a further axiom for completeness?

Richard

Richard Lewis

Dear Richard, thank you for following my question. Yes, this is what I mean. Though, there is one more axiom: the description of quantum objects by states in Hilbert spaces with an algebra of complex numbers.

For example, Bohm's mechanics violates the uncertainty principle, by assuming the existence of particles following definite trajectories on which both position and linear momentum are well defined at each time. And I proved, in a previous article that this assumption leads to the situation in which Bohm's mechanics fails to predict the same predictions as the quantum mechanics (QM).

As you saw in my article, the most popular interpretations of the QM each one makes an assumption that contradicts the principles of the QM or of the general physics, and therefore fails.

Sofia D. Wechsler Dear Sofia. I would not be trying to add additional axioms to QM but instead would challenge the existing axioms.

Starting with wave/particle duality which I instinctively dislike because it suggests an object is neither one thing or the other. We need a clear unified description of fundamental physics.

To take the example of light waves this is clearly a wave but when light is emitted by an atomic electron it is always emitted in discrete bundles of energy. So light is a wave which is quantised because that is the way it is emitted.

Wave particle duality in the context of the electron has been used as for example in the Bohr model to explain the different energy levels of for example the hydrogen atom.

By treating the electron as purely a wave it is possible to explain the energy levels of the hydrogen atom in a much better way. ( see my paper on the hydrogen bond). This explains why the electron is quantised and uses only classical ideas in the explanation. Then the quantisation of the electron also explains the quantisation of light.

I can go on to challenge the other axioms of QM if you would like me to. My view to answer the original question is that we should not be trying to correct QM by adding axioms because QM does not provide a correct description of physical reality.

Richard

Richard Lewis

Dear Richard, yes, I would like to see how do you challenge the axioms of the QM, but give me a few days. I am reading a long article, referring exactly the derivation of the QM from basic axioms.

By the way, you say,

"By treating the electron as purely a wave it is possible to explain the energy levels of the hydrogen atom in a much better way. ( see my paper on the hydrogen bond). This explains why the electron is quantized and uses only classical ideas in the explanation. Then the quantization of the electron also explains the quantization of light."

I disagree. The energy levels in the atom are not sharp, each level has a certain width. As to the photon, it may be emitted not only by an atom but also by a black body - so Planck found that the electromagnetic energy is quantized.

With best regards

Sofia D. Wechsler Yes I will wait for you to follow those references to the original axioms. i think when you look at black body radiation you will find that ultimately the radiation is coming from light emitted by atomic electrons. In the first instance, they imagined little oscillators emitting radiation at specific frequencies. Those oscillators are actually atoms which emit radiation at different frequencies depending on temperature.

Richard

Sofia D. Wechsler When I refer to the precise energy levels of hydrogen I am referring to the emission and absorption lines discovered by Joseph con Fraunhofer in 1814. He found that the spectrum of light from the sun had very precisely located dark lines at specific frequencies.

The dark lines associated with hydrogen were precisely measured and in 1885 Johann Balmer deduced a mathematical formula for the wavelength and frequencies of the hydrogen lines.

In the early 20th century, Niels Bohr developed a model of the hydrogen atom which explained these energy levels but he assumed that the energy of the electron was partly in the mass of a particle and partly wave energy.

Now if you discard wave/particle duality, all the energy of the electron is wave energy given by E=hf. Then you have the model as described in the paper on the hydrogen bond.

Richard

Richard Lewis

No, Richard, the radiation from atoms looks like spectral thin lines. The black body radiation is a spectral continuum. The behavior of this spectrum lead to Planck's discovery.

https://en.wikipedia.org/wiki/Black-body_radiation

QM is uncomplete . At 0K there is no radiation from a black body and the hole density of energy should be equal to zero according to Stefan law: this is wrong, the density of energy should be not equal to zero at 0K so somebody should introduce the vacuum energy density at 0K of a block body. Planck had made an essay in 1911 but he fall. I will do soon.

Sofia D. Wechsler The way I understand the curves in the Wikipedia article on black body radiation it shows the relationship between radiation intensity at different wavelengths. So for example the green line at 4000K represents the intensity of radiation at different wavelengths at a temperature of 4000K.

If you now imagine the black body device which is an enclosed box heated to 4000K the black body radiation is measured at an opening. The source of the radiation is from the atoms in the walls of the enclosure and each atom is like a little oscillator giving of light at some frequency. So my claim that the light is coming from individual atoms is correct.

In the black body case we do seem to have a range of possible frequencies of emission. In the case of the Fraunhofer lines we have very specific frequencies of absorption.

In the particular case of hydrogen the absorption frequencies are very well defined and are used in astronomical observations to measure redshift.

Richard

Richard Lewis

Richard, the radiation of the black body comes from atoms but it does not leaves the opening of the black body at the frequencies produced by the atoms. It is split and absorbed by the molecules of the body and re-emitted to the black body, and split and re-emitted again, until it reaches the spectrum specific to that temperature.

Sofia D. Wechsler Yes, I agree there must be a lot of absorption and emission going on within the cavity of the black body device.

Have you made any progress in constructing a list of the fundamental axioms of quantum theory as a reference for considering new axioms?

Richard

Richard Lewis

No, Richard, I just finished reading that long article that I told you. Its title is "Quantum theory from 5 axioms". But I think that it is wrong.

Now, please tell me, did you delete some of your comments? I want to refer to the comment in which you gave a list of 5 or 6 features of the quantum mechanics, but I don't see that comment.

Sofia D. Wechsler Hi Sofia. No I haven’t deleted any comments but if you scroll down to the bottom of the page you will see numbers 1and 2 inside blue circles. Click on 1. I don’t know why RG has started doing this - it confused me the first time I encountered it.

The list of 5 or 6 items was from memory so don’t take it as definitive.

Best wishes

Richatd

Richard Lewis

Dear Richard, yes, I did as you said and now I see the five features. Thank you for the advice.

Now, that long paper of L. Hardy that pretends to derive the quantum theory from 5 postulates, did not mention the uncertainty principle and the superposition principle, neither any principle that would include these two features.

You see, the quantum formalism includes more than the features/principles that you mentioned. It includes the algebra of Hilbert spaces which in turn uses the group theory. In addition, there is the so-called Born rule for calculating amplitudes of probabilities. For instance, the transactional interpretation of the quantum theory contains an additional hypothesis which in fact is in disagreement with the Born rule.

Sofia D. Wechsler Yes, what I was referring to was the Copenhagen interpretation of quantum theory which is covered here in Wikipedia:

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

There is a whole mathematical formalism of quantum theory which you describe and the various interpretations try to relate that mathematical model to physical reality. This is where the problems arise and you get comments from physicists like Richard Feynmann saying "no-one understands quantum theory". I think what he is saying is that we have a mathematical model and a procedure for working out the possible results of experiments but it doesn't make any physical sense.

At the heart of this problem is the one you have highlighted which is the "ontology of particles". It seems that when you think in terms of particles that you get contradictions as you have pointed out in your research.

I would like to illustrate my point by referring to the Bohr model of the hydrogen atom in which there is a "particle" with definite mass and momentum which is in orbit around the nucleus but constrained by quantum rules to be in specific orbits or energy levels. There is a wave function associated with this electron "particle" which allows us to calculate the probability of finding the particle at any point. This is a difficult description of physical reality and is the sort of thing that gave Albert Einstein such concerns about quantum theory. There were other attempts to improve the model by considering a mixture of particles and waves (pilot wave models) but these were not much better.

What I am saying is that the electron is a pure wave. There is no localised "particle" in there. If you imagine the particle of the Bohr model with its mass and momentum and then think of the track of that particle around the nucleus then this is the path of the wave. The mass and momentum and charge are spread out along the entire path of the wave.

So why is it that when we experimentally test the electron of a hydrogen atom, we find what seems to be a "particle". Think of the experiment using a laser which knocks the electron out of the orbit of the hydrogen atom and then detects the electron to find its original position at the time of collision with the laser pulse. This experiment is going to give a point location result even though the actual collision was between two waves.

I hope this explains my point of view that I don't think you will resolve anything by adding more axioms to the mathematical model when the actual problem is in the interpretation of quantum theory.

Richard

Adding axioms or deleting axioms is only possible if we know “what is going on”.

For example, the cosmic microwave background radiation (CMBR) shows a dipole (Doppler effect) that quantifies the relation between the CMBR and the observer (solar system). It shows that the solar system has an absolute velocity and a clear direction of motion in respect to the CMBR. The observations are confirmed recently (https://iopscience.iop.org/article/10.3847/2041-8213/ac6f08). However, the characteristics of the CMBR are equal to a rest frame.

The dynamical part of our universe is the universal electric field and the corresponding magnetic field. The consequence is that the electromagnetic field is in rest in relation to the motion of all the observable and detectable phenomena (particles, etc.). That is obvious because if 2 electrons move in opposite direction, it is impossible to imagine how the electromagnetic field can be the creator of both particles and is also in motion in every direction: modern QFT (https://arxiv.org/ftp/arxiv/papers/1204/1204.4616.pd).

The conclusion has to be that the properties of the electromagnetic field are part of Euclidean space as a rest frame and can be described accordingly.

Another example is probability. It is not a mathematical influence because probability is a pure physical effect. Actually there is still no mathematical foundation that underlies probability. But we cannot compare probability with e.g. thermodynamics because probability shows to be a universal dynamical property of the universe (it influences everything). But mutual influences are restricted to the electromagnetic field thus probability is part of the properties of the electromagnetic field (Euclidean space). Moreover, probability is in line with the universal conservation laws (energy and momentum).

Unfortunately it is thought that the speed of light is the maximum propagation of dynamics (changes) in space. But the entanglement of 2 polarized photons shows that the velocity of a mutual influence is not restricted to the speed of light. The distinct mutual influence is part of the properties of the electromagnetic field thus we have to conclude that the speed of light is restricted to the transfer of “tangible” energy in space (quanta). The universal electric field and the magnetic field are corresponding fields thus a local change of 1 quantum of energy generates a corresponding magnitude to a local vector by the magnetic field and visa versa. But vectors “act” instantaneous. The influence of a vector within the spatial structure of the magnetic field is determined by the magnitude of (resultant) vectors in the same direction and vectors in the opposite direction (just addition and subtraction of magnitudes).

If a sensor “absorbs” the local energy of an electromagnetic phenomenon, the local corresponding configuration of vectors within the magnetic field will vanish immediately. Unfortunately this description represents phenomenological physics (the limited point of view we used in classic physics). The vectors don’t vanish because of the conservation of momentum. Thus it is a transformation into another local configuration. But because of the instantaneous “velocity” of vectors, it shows like a collapse. In my opinion "the collapse" has a duration of ≈ 6 x 10-23 second to determine the local change within the electric field for 1 transferred quantum of energy.

Personally I have the opinion that QM don't need more axioms or less axioms. It is a model with its own formalism that is derived from experiments. It is not a theory to explain the underlying mechanism of physical reality.

With kind regards, Sydney

Explications of Sydney Ernest Grimm signifies that QM mechanics need new axioms. The idea is that we can't transfer energy at any rate: there is always a duration not equal to zero to transfer any amount of energy even thought in the photo-electric phenomena. For a photon the ratio between the energy of the photon and the duration of its absobation is an universal constant i.e. the power of all photons are constant. We can find the the theory which justify this idea and it is in Restreint relativity by adding a new four vector with this new universal constant and so we will get three dualities which deseappears in a system of unities where the constant c , the Planck constant h and the new universal constant are equal to one. A simple calculation of this new universal constant from the classical model of electric dipole gives us an universal constant equal to E-13 watts wich agree aprroximatively with the duration given by Sydney Ernest Grimm .

Richard Lewis

Dear Richard, as far as I know the electron position in an atom was never located. You say

"So why is it that when we experimentally test the electron of a hydrogen atom, we find what seems to be a "particle". Think of the experiment using a laser which knocks the electron out of the orbit of the hydrogen atom and then detects the electron to find its original position at the time of collision with the laser pulse. This experiment is going to give a point location result even though the actual collision was between two waves."

I believe that such an experiment cannot be done. If you use so energetic photons as fit for seeing the electron position, not the whole electron cloud, the linear momentum of the photon exiting the laser in very poorly defined. So, the photon will fly from the laser in all the directions. You won't be able to say anything about the location where the photon and the electron met.

With best regards,

Sofia

I think that QM does not need new axioms, but a new meaningful interpretation. I tried to give it -- see my little article (no formulas) `Subquantum leapfrog`

Preprint Subquantum leapfrog

Sean Carroll wrote an article about understanding QM for everyone to read (The NewYork Times, Sept. 7, 2019: https://www.nytimes.com/2019/09/07/opinion/sunday/quantum-physics.html).

The article was a motivation for other theorists to write a paper about the foundations of QM (how to interpret the formalism). For example Lee Smolin, Gerard ‘t Hooft, Sabine Hossenfelder and of course Sean Carroll himself. But none of these papers made every theorist happy although the supposed existence of super determinism by Sabine Hossenfelder is a promising “new” approach (https://arxiv.org/abs/2010.01324).

This RG discussion about the completeness of QM is just 1 week old but has already resulted into a “fresh” discussion: https://www.researchgate.net/post/What_are_the_basic_postulates_of_the_quantum_mechanics.

However the underlying “topic” is still the interpretation of QM. But is there any proof that we have to hypothesize new axioms to solve theoretical problems? As far as I know in theoretical physics we always have solved problems with the help of the reduction of the number of different concepts. And an axiom is just a concept without an underlying “theoretical structure”. Mostly an axiom represents a certain point of view to formulate a proposed aspect of physical reality.

Personally I never cared about the interpretation of QM because it is a limited theory and that is why they started to replace QM by QFT some 70 years ago. Moreover, I am not responsible for the created formalism so I don’t feel any responsibility to focus on the theory to solve the conceptual problems. Nevertheless the formalism of QM represents physical reality and even in QFT the interpretation of physical reality is problematic. Not at least because QM is partly incorporated into QFT. In other words, these problems have to be solved, no matter if it is QM or QFT.

But if we want to understand physical reality at the smallest scale size we have to create some “tangible” concepts as a kind of tools to build up physical reality as we know it in phenomenological physics. Now there is a real problem because there were RG discussions about the “tangible” reality of Planck’s constant – the quantum of energy – and the result of these discussions doesn’t surpass the limited description in the text books. In other words, if there is no “tangible” conceptual framework why do we hope to get a realistic model to interpret the formalism of QM?

Thus the whole problem is about methods. How do we solve these problems if our present method isn’t well suited to get the desired result.

With kind regards, Sydney

A fact of great importance is that in QM the spatio-temporal and impulse-energy representations are connected by the Fourier transform. From this immediately follows the uncertainty principle.

And why are they related by the Fourier transform? I think because in the base of physical world lyes projective geometry with its fundamental principle of duality. This duality manifests itself in the form of wave-particle duality. The Fourier transform reflects the pole-polar correspondence of projective geometry.

Some decades ago there was a lot of public discussion about the theory of everything. Because a lot of theorists dreamed openly about the ultimate equation that describes “everything” in our universe. But to describe “everything” in one equation is a bit troublesome because our scientific method was at the basis of QM. The result is that the formalism of QM describes the results of experiments in an accurate way. Obvious because the formalism was “build up” with the help of the outcome of many, many experiments. Experiments that show the mutual relations between the distinct phenomena.

For example the quantum of energy (h). The detection of its existence shows that Planck’s constant is a basic building block that emerges within the range of mutual relations between matter object (unfortunately every measurement instrument represents matter). But we don’t know if Planck’s constant exists as a basic building block or as an effect that represents change (energy) from the viewpoint of the universe itself. A point of view that is supposed to be independent from the existence of matter.

Thus if we dream about the ultimate equation we are dreaming about a mathematical notation of concepts that only represents mutual relations. That is why our description of the properties of elementary particles represents just mutual relations. Comparable with the determination of the individual weight of objects with a pair of scales in classic physics.

There were objections against the dream of the ultimate equation and one of the arguments was that the result needed too much “sophisticated” mathematical formalism to describe everything. Thus at the moment that every theorist is jubilant about the ultimate equation we face the problem that we don’t know the “tangible” reality behind the equation. Like the present problems to determine the “tangible” reality of the formalism of QM.

So I doubt that “sophisticated” mathematical representations of physical reality (even Fourier transform) can solve the problem. Because all these mathematical tools represent simplifications of composite concepts. If we want to solve the problem with the help of mathematical physics it is reasonable to use matching fundamental mathematics like set theory and topology.

Nevertheless, it doesn’t solve the problem that we need “tangible” concepts to understand physical reality at the smallest scale size. Thus from this point of view I understand the motivation of Sofia Wechsler to discuss the axioms at the basis of QM.

With kind regards, Sydney

>>>A few decades ago there was a lot of public discussion about the theory of everything. Because many theorists have openly dreamed of a final equation describing "everything" in our universe.

The physical world is the realization of some complex mathematical structure. The problem is to find out which one. In my opinion, it is not just one equation. (The Fourier transform, I hope, will be only a small detail in this complicated structure.)

I agree that the underlying structure that is responsible for the observed/determined physical reality must correspond with a mathematical model.

But if we propose that physical reality at the smallest scale size must be simple because of the reduced complexity, the mathematical model that describes physical reality at the smallest scale size must be simple too. For example all the geometrical changes can be described in an accurate way with the help of addition and subtraction of elements with variable properties under invariance. Actually, in 3D set theory and topology every addition is a subtraction at exactly the same moment, in line with the universal conservation laws in physics. A conservation of properties in physical reality that is dominated by the dynamics of the electromagnetic field.

However, in mathematics there is no difference between the mutual relations and absolute properties. We don’t describe the transformation of the shape of an object under invariance with the help of a limited sequence of changing variables. While in theoretical physics nearly the whole conceptual framework is “build up” with the help of mutual relations. Although universal conservation laws and universal constants represent properties that exist everywhere within the volume of the universe, these universal properties don’t reveal the underlying structure of the universe (c.q. QM). At least not without of a lot of research during a lot of centuries.

The consequence is that your suggested mathematical approach will be more successful than the results we have obtained with the help of the scientific method. Although it seems obvious that we need the descriptions of theoretical physics to keep us “down to earth”.

With kind regards, Sydney

We are in captivity of old dogmas. We talk about conservation laws and immediately imagine the symmetries of the Euclidean space. And what will the conservation laws look like in the Lobachevsky space?

Set theory in physics? Leave it to the dogmatists-Burbakists.

Time will tell.

With kind regards, Sydney

Oh sure.

Thank you. It's a pity that no one else wants to express their opinion.

Sorry, but my short response is obvious because reality – like we observe it in daily live – is Euclidean. Like daily live shows that it must have a structure too, because everything we can observe has structure. And everything that has a structure can be described with the help of set theory. Like every element of a set that changes its geometrical properties under invariance can be described with the help of topology. Universal conservation laws represent invariance.

Non-Euclidean geometry is directly related to the mutual relations between the phenomena in our universe. But our universe is the cause (“the creator” in QFT) and not the effect of everything we can observe/detect. The idea that observable reality is the source that determines everything represents the conceptual framework of classic physics (Nikolai Lobachevsky lived in the 19th century).

The discovery of the universal scalar field by Peter Higgs proves that space itself is perfectly flat. We even have the observation that there exist no pre-CMB radiation because of the black body characteristics of the CMBR. That means that in the early universe space was without any doubt 100% Euclidean. Thus non-Euclidean geometry cannot be the foundational model that underlies physical reality. For example I need a dominant vector field in relation to a matter object to get a hyperbolic description of the energy distribution in vacuum space around. Without these requirements there is no hyperbolic geometry to be find.

Anyway, the discussion is about the basic properties of the electromagnetic field (QM) and the question if the known (axiomatic) properties describe it all.

With kind regards, Sydney

Sydney Ernest Grimm, Lev Verkhovsky , Alaya Kouki , Richard Lewis ,

Antoine Suarez ,

and all the other participants to this discussion,

I examined the most popular interpretations of the QM. All of them added supplementary principles to the QM, and it turned out that these supplements contradicted one or another principle of the quantum mechanics. For example, Bohm's mechanics added to the quantum formalism the axiom that there exist particles which follow continuous trajectories guided by the wave-function. I proved in

Chapter A Non-Relativistic Argument Against Continuous Trajectories ...

that this axiom leads to a contradiction with the QM predictions for an experiment discussed in the above article.

Also, for instance, the transactions interpretation contains the additional axiom of the "hand-shake". I proved that it contradicts the Born principle.

An analysis of the most popular interpretations of the QM can be found in

Article The Quantum Mechanics Needs the Principle of Wave-Function C...

That would still be an open question. An afirmative answer would be irresponsible from a scientific point of view, but there are those prone to orthodoxy in this matter, closed minded in my view.

Since a couple of decades high energy particle physicists are convinced that particles are local "excitations" of the basic quantum fields (QFT). The used term "excitation" in the literature is a bit crazy because the "mechanism" contradicts the conservation of energy and momentum because it describes only half the process of its origin.

We can only get a local concentration of energy (the particle) if this amount of energy is supplied by the structure of the electromagnetic field from vacuum space around (set theory). Thus a particle is a local surplus of energy that is surrounded by a large volume that has a deficit of energy. And the deficit of energy is equal to the surplus of energy.

The same reasoning can be used for the magnitudes and amount of involved vectors because the universal electric field and the magnetic field (vector field) are corresponding fields and the momentum of every "free quantum" represents corresponding vector(s) within the structure of the magnetic field.

Now it is easy to examine the relevance of some "axioms" that are underlying the formalism of QM. Unfortunately there are "axioms" that cannot be weighted with the help of the simple description above. The consequence is that the simple description must be replaced step by step by a more detailed description. But at the moment we are "constructing" a detailed description the whole problem about the relevance of axioms and their mutual contradictions is no longer important.

Because the question about the relevance of the axioms had the aim to understand the “tangible” reality that underlies the formalism of QM. The detailed description resolves the problem because it represents the "tangible" reality.

With kind regards, Sydney

Quantum Mechanics is simply a statistical and mathematical system regarding predictions of the quantum world. It is not a theory. For this there are dozens of proposed theories to explain quantum mechanics. The most famous of these is called the Copenhagen interpretation, and the most ridiculous theory IMO is called the Many Wolds Interpretation. And of course one can always propose another theory or addendum to one of dozens of theories.

There is evidence that in the early universe there was a period that there was no matter in the universe. Thus the Higgs field was perfectly flat, the electromagnetic field had still its amplitudes and the gravitational field didn’t exist because there was no matter. If we think about it for a while there arises the intriguing question if there existed electromagnetic waves in that period.

The image above shows a particle (1), a local surplus of energy. The particle is surrounded by a volume (2) that has a deficit of energy that is equal to the surplus of energy of the particle (see my previous comment). The particle and the region around (2) are part of vacuum space (3) with an average energy distribution. I have drawn in a schematic way the structure of the electromagnetic field.

I can draw the situation with the help of an overlay diagram and A (green) is the average energy distribution within the structure of the electric field, B (blue) is the deficit of energy within the spherical volume (2) and C (red) is the surplus of energy of the particle (1).

If we suppose that the particle decreases its energy (the change to a lower state) the released energy by the particle is an electromagnetic wave. Why?

The particle emerged by the redistribution of energy within the structure of the electromagnetic field. The stability of the creation must be an active influence/property of the electromagnetic field. Thus if the particle emits energy the relation between the surplus of energy and the deficit of energy cannot be changed.

The consequence is that the surplus of energy (C) that represents the particle decreases with exactly the same amount of energy that represents the local deficit of energy (B). And this amount of energy is Planck’s constant. Thus the frequency of an electromagnetic wave represents the step by step (1 h) decrease of the local surplus and deficit of energy. In other words, without stable local concentrations of energy in the universe there exist no electromagnetic waves. There are only amplitudes, related to the structure of the electromagnetic field.

The size of a particle must be a number of units/elements of the structure of the electromagnetic field otherwise the particle cannot have a spin. That means that the metric of the structure of the magnetic field must be at least ≈ 0,27 x 10-15 m (the average size of rest mass carrying particles is 0,8 x 10-15 m). And the particle cannot exists without a corresponding wave structure (B). Thus the question why a particle has a corresponding wave form is really simple.

So think about this method to get a “tangible” description of physical reality at the lowest scale size and the possibility to verify proposed "axioms".

With kind regards, Sydney

Sofia,

You claim that you have proved the Transactional interpretation wrong. you state in your manuscript , (PDF) The Quantum Mechanics Needs the Principle of Wave-Function Collapse—But This Principle Shouldn’t Be Misunderstood.

When speaking of TI the manuscript states, " It is a misunderstanding of the Born rule,the latter is the inner product of two waves (consisting of integration of the arithmetic product, over all its variables)." you have observed an often discussed criticism of TI, but this inner product is in Hilbert space. Cramer has addressed this issue with this latest manuscript:

J. Cramer, C. A. Mead, Symmetry, Transactions, and the Mechanism of Wave Function Collapse. Symmetry.; 12(8):1373. (2020).

Best Regards

Peter Tanguay

Peter N Tanguay

Dear Peter, their article is 48 pages long. Can you tell me which section from that article to read?

Dear Sofia,

The entire manuscript is a worthy read, but I recommend : the abstract to help you find what you are looking for , and Sec. 12. on pgg 33-34. I highly recommend reading up to Sec. 12 if you want to fully make a thoroughly researched conclusion. Best Regards, Peter Tanguay

Peter N Tanguay

Dear Peter, first of all thanks for calling my attention on this article. I began to read it.

To tell you frankly, this theory with detectors sending waves backwards to the past seems to me science fiction. Nevertheless, I am reading the article.

Also, the detectors are macroscopic objects, they don't obey wave-functions. So, it's not clear what type of waves they send to the past. Moreover, in the case of entanglements the waves sent by the detectors have to do handshake with the waves emitted by the sources according to the correlations predicted by the entanglements. It is not clear to me how does it happens that the waves from the detectors are so smart as to do handshake according to the correlations.

Thus, I expect to find answers to these questions in the article you sent me. Let's see if there are answers.

With kind regards and thanks again,

Sofia

Sofia,

I also have difficulty with some aspects of the theory. However, if I think of the Wheeler-Feynman standing waves as atemporal ,then I find TI (with some upgrades and modifications) is an interpretation that offers the most promise of a phenomenological understanding, IMHO.

Perhaps, after completing your research, you can state your observations and opinions on TI, using your excellent logical skills, along with your intuition. You can state your ideas without necessarily "proving". Proving has a very high standard. as opposed to a body of evidence, or many complications. Best regards, Peter Tanguay

Sofia,

the problems you mention with TI are also some of my own. In fact, I have a few more! At some points, Cramer refers to the handshake as pedagogical, and at other times phenomenological. I think of an absorber and emitting particle as constantly sending out a signal saying, I can absorb and I can emit. When these match up in spacetime, we have a transaction. not necessarily a retrocausal handshake, but a constant potential of probabilities. All those fields create a background. It is the geometry of the waves across spacetime that appeals to me most.

I do not mean to detract from your discussion , only add to it. I need not take up your discussion on this topic further, so if you have questions or ideas feel free to research gate email me. Peter

Peter N Tanguay

Peter, I am reading the article. About what you say, "an absorber and emitting particle", there is no absorber particle. The detection occurs in a detector, where there are many particles. A single absorbing particle (abs) would generate with the incident particle (inc) an entanglement: (α|inc>s |abs>s+ β|inc>ns|abs>ns), where s stands for scattered, ns for non scattered, and |α|2+|β|2=1. This entanglement is still a microscopic item, it needs a detector for detecting whether we have got a detection, |inc>s |abs>s , or no detection |inc>ns|abs>ns.

For more clarity maybe you'd have a look at the section 2 of my article

Article In Praise and in Criticism of the Model of Continuous Sponta...

Look at figure 2. You can see how the detection occurs: it needs many particles for forming an avalanche. The avalanche is a macroscopic signal and the circuitry of the detector can feel it. The multitude of particles in the detector breaks the entanglement, so we either detect the compound particle, or, the particle was not detected.

These people with their TI should spend some time in a physics lab.

Sofia,

Resorting to personal attacks when your research knowledge is found lacking is unbecoming of a professional. Professor Cramer has bent more time in a lab then you. http://faculty.washington.edu/jcramer/experiment.html

I will no longer engage in such a childish discussion

Peter N Tanguay

Peter, how do you know how much I was in a lab?

I also don't want to engage in such a discussion.

Nature of statistics or probability theory will depend on the cases of its applications ?. Perhaps it may provides very good analogy for average behavior ! But this cant be true at all..!. can concept of trueth define to the nature of statistics or probability theorem? . Statistical methods are have same meaning for all its applications whatever we apply.

QM is over complete, its axioms are contradictory.

Claude Pierre Massé

Please make your reply clearer. Please give a specific example of postulates of the quantum mechanics which are contradictory.

Contradictions in quantum mechanics:

A black body at zero kelvin have an energy density equal to zero (as given by Planck theory), however the black body is an infinate oscillators with a fundamental energy for everyone not equal to zero so the contradiction.

@Kouki: Wrong. The energy density zero of a black body at zero Kelvin is just the energy differerence between the energy of the state and the ground state. The energy of the ground state (which is infinite, if you assume the space to be infinite, otherwise it is finite) is not directly observable and has nothing to do with the "observable" energy of the actual state (which is zero in this case). So there is no contradiction here.

Sorry, have to correct myself. The ground state energy is infinite also in a finite volume, unless an ultraviolet cutoff is introduced, such as the Planck length.

K. Kassner

The energy density of a black body as given by Planck formulae is proportionnal to the forth power of the temperature (Stephan law). If the temperature is T=0K than the energy density is equal to zero and nothing can't move. The Planck black body theory is based on an infinate oscillators which exchange energy with the electromagnetic field in the body and at T=0 those oscillators will not move at all so the density of energy is equal to zero. But this is in contradiction with quantum mechanics which predict an energy of 1/2 hv for the ground state of every oscillator and at T=0K those oscillators are in their ground states so the density of energy in the black body is not equal to zero as predicted by Planck theory than the contradiction.

Of course the energy of th ground state of the oscillators at T=0K should be finite because the black body is finite. Another proof that the energy of th ground state is finite is the density of vacuum energy as given by General Relativity: it is proportionnal to the constant lambda and this constant have a value equal to 1E-52 m-2. If the energy of the ground state is infinite there is no Universe: it will explose rapidly.

Quantum Mechanics luck an universal constant: it is the average of the energy of the fundamental state of quantum field (or the electromagnetic field, or the ZPE average at zero kelvin). The black body theory of Planck is incomplete we should add to the Planck formulae a similar formulae for the Zero Point Energy and so we introduce a new universal constant in the second member. At T=0K the first member vanishes and the integration of the second member gives us exactly the energy density of vacuum as given by General Relativity: the way for unification and eventually extracting energy from vacuum is open: we undrestand vacuum.

My take on the question:

First, I would like to point out that the notion of completeness of a physical theory as used by Einstein and in present-day discussions is different from the one implied in the question. The idea of completeness meaning that "no additional axiom" can be added to a theory, seems pretty meaningless, as I will argue. Essentially, it leads astray, away from pertinent questions.

So what did Einstein mean by completeness of a theory? He asserted that in order for a theory to be complete, it is necessary that there is a correspondent in the theory for each "element of physical reality". Admitting that a general definition of "element of reality" may be difficult, he gave a *sufficient* condition for the existence of such an element. He stated that if we can predict a physical quantity with certainty, without disturbing the system, then it is real, i.e. there must be an element of reality determining this result of the prediction (and being present without any measurement, of course). Most people would follow him on that path. Even Bohr seems to have accepted this premise. When discussing a particular experiment, we may not necessarily know, just *what* the element of reality is, but we know that one must be there, if we can predict the outcome with certainty without having disturbed the system.

In the Einstein-Podolsky-Rosen paper [1], the authors discuss a particular quantum state, describing two entangled particles and show that it satisfies their condition for an element of reality to be present, for which, however, there is no corresponding element in the theory. In particular, the state considered does not contain information on this element of reality. Since they have shown it to be present, they believe to have demonstrated quantum mechanics to be incomplete. The wave function they consider is a (1D) superposition of product states with opposite momenta of the particles, so it describes a state of total momentum zero. (The operators "sum of the two particle momenta" and "difference of their positions" commute, hence these observables can in principle be measured simultaneously.) They then note that by measuring the momentum of particle 1 they can predict with certainty the momentum of particle 2, without disturbing that particle (which is supposed to be far away, so it does not interact with the apparatus measuring particle 1), therefore there must be an element of reality corresponding to the momentum of the second particle (i.e. it must have a certain value), and there should be a descriptor in any complete theory for it. They further note that by measuring instead the position of particle 1, they can predict with certainty the position of particle 2, without disturbing particle 2, therefore there must be an element of reality corresponding to the position of the second particle, too (i.e. it must have a certain value), and there should be a descriptor in any complete theory for it. But quantum mechanics states, via the Heisenberg uncertainty relation, that there is no description of particle 2, in which both its momentum and position have definite values. So quantum theory must be incomplete. EPR finish their paper by expressing their belief that a completion of quantum theory be possible. This would mean to add variables to the description which are not given by the wave function ("hidden variables") and which would allow, if known, to calculate the unknown values corresponding to the elements of reality momentum and position.

As an aside, Bohmian mechanics, which is an attempt to add the appropriate variables to render the quantum mechanical description of reality complete, does not seem to achieve this goal. While the position variables that Bohmian mechanics adds to the wave function do describe, according to the interpretation of the theory, the real positions of particles and would therefore, if known, allow to predict observed positions with certainty, the momentum variables appearing in Bohmian mechanics don't allow to predict observed momenta with certainty, in general. This can be easily seen by considering a standing plane wave state, say, ψ(x)=cos(px/ℏ). The Bohmian particle in this has the "real" velocity and, hence, momentum zero (because the wave function is real and the velocity expression involves its imaginary part). But a measurement will give either +p or -p, as this is a superposition of two plane waves with these momenta. So knowing the hidden variable Bohmian momentum, which is exactly zero, we cannot predict with certainty the momentum of the particle that would be obtained in a measurement. According to EPR, Bohmian mechanics would still be an incomplete theory, albeit less incomplete than quantum meachanics based on the Schrödinger equation alone. (*)

Thus, "completeness" of a theory in the sense of EPR (and, of course, also that of modern authors discussing the question) refers to *variables* (or parameters) corresponding to reality and means that the theory already has all the variables needed to describe reality, with no need for the addition of further variables. (We could add more variables, but they would be superfluous.)

The question, on the other hand, suggests that completeness refers to *axioms*, not variables. And it requires for completeness of a theory that "no axiom can be added to it". Now it seems to me that whether an axiom can be added to a theory is just a question of the imagination of the persons trying to improve that theory and can therefore never be excluded. For example, we could add to special relativity the axiom that, in addition to a maximum velocity for energy transport, there is a maximum angular momentum per mass unit. If this is chosen large enough, it will not affect any of the predictions of the theory that have been the subject of observation so far. For large enough radii, it will enforce maximum velocities smaller than the speed of light and thus lead to different predictions from special relativity, but if the limit radius exceeds that of the known universe, these would not have been observable in the past. Once we can see further out, we might decide between special relativity and "angular momentum special relativity"... This is an ad-hoc example that I made up (just to show that all that is required is a bit of imagination).

But there are historical cases where additional axioms have been added to a theory to modify it or at least its interpretation. Brans and Dicke added a scalar field to general relativity in order to make the theory satisfy Mach's principle more generally. This gave rise to a scalar-tensor theory, in which we have two entities describing the properties of spacetime, a metric field (a second rank tensor) and the scalar field coupling to it (a zeroth rank tensor). If the coupling parameter between the fields goes to infinity, the theory reduces to general relativity. Since it turns out experimentally that the coupling parameter is pretty large, most experts in general relativity believe that general relativity without the scalar field is already the correct theory, but of course, it is never possible to exclude the existence of the scalar field, because its effects become arbitrarily small, if the parameter is chosen arbitrarily large.

My third example is Bohmian mechanics, in which the equations of motion of the particles constitute an additional axiom beyond the Schrödinger equation. In this case, the predictions of the theory and non-relativistic quantum mechanics are indistinguishable, as long as the initial probability distribution of Bohmian particles is given by the absolute square of the initial wave function.

In principle, it should be possible to add axioms to any theory, the only difficulty being to avoid that the new axioms contradict the old ones. (This is very simple in theories that postulate a Lagrangian as the fundamental source of equations of motion. Adding another Lagrangian that respects the symmetries of the system will give a consistent theory with an additional axiom that in the limit where the added Lagrangian is negligible reduces to the original theory.)

For reasons of thought ecomomy, the new axiom also should be independent of the old ones, but that is not an absolute necessity.

Then there are two possibilities. Either the theory with the additional axiom(s) makes different predictions from the one without them (Brans-Dicke theory). Then it should be possible to refute either the old or the new theory by a crucial experiment. But in cases where parameters are chosen in a way to make the effects of the new axiom very small, it may not (presently) be possible to distinguish between the theories. In any case, what would the possibility to add an axiom have to do with completeness of the theory? It appears that axioms can always be added to cover cases, to which the current theory has not yet been applied. The second possibility is that the additional axiom does not lead to different predictions (Bohmian mechanics). In this case, *no* experimental distinction between the theories is possible and one would normally prefer the theory without the additional axiom, on the basis of Occam's razor. Or else, if one is philosophically inclined not to attribute much importance to Occam's razor, the preference might be based on different ideas, such as compatibility with realism or determinism. It might also be just a matter of taste...

In any case, I do not see how a completeness criterion based on the *impossibility* of adding an axiom can be meaningful. This impossibility never holds, or so it seems at least.

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[1] A. Einstein, B. Podolsky, N. Rosen, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?, Physical Review 47, 777-779 (1935)

(*) A Bohmianist might object to this description that the measurement of momentum changes the momentum, due to interaction with the measuring apparatus, from the true value zero to one of the two possible measurable values, so the fact that these do not agree with the "true momentum" before measurement does not affect the realism of the theory. To which one could answer that in the EPR experiment the momentum of particle 2 can indeed be predicted with certainty after measurement of the momentum of particle 1, so it must be the true momentum of particle 2, but the Bohmian description, wave function of the total system and positions of the two particles, does not contain a variable describing this true momentum. Moreover, even if we knew the wave function and the particle positions (which constitutes a complete state description in Bohmian mechanics), we could not predict the momentum of particle 2 with certainty *before* the measurement on the first particle. Bohmianists of course have an answer to that as well. In Bohmian mechanics, the condition of no disturbance of the second particle by measurement of the first is not satisfied. (In fact, part of Bohr's answer to Einstein was that this is also true in standard quantum mechanics. The disturbance just is not a mechanical one, rather it consists in a restriction of definability of one quantity by measurement of another one.)

The wave function is a constitutive part of reality and it collapses into a momentum eigenstate of the first particle after the measurement. So the measurable momentum of the second particle is not an element of reality *before* measurement of the first particle's momentum, but it becomes one *afterwards* and this is due to a nonlocal interaction of the two particles mediated by the wave function. This is fine as long as the Bohmianist does not belong to the subcategory of Bohmianists denying that the wave function collapse is real. They say that you "can throw away the empty branch of the wave function" after measurement of the first particle, so the collapse is not a physical process but just a convenience of description. This is a problem, however.

If you just "can throw away" the empty branch of the wave function, you are not obliged to do so. You could also keep it, it should not matter, render calculations somewhat more complicated at best. But then the wave function could be taken unchanged in the spatial region where the second particle is, and then it would not contain information about the momentum of the second particle, meaning again that an element of reality is missing from the theory. Therefore, the collapse of the wave function is a *must* (not a *can*) in Bohmian mechanics, if it is to be a complete theory.

However, the rules for application of the collapse are the same in Bohmian mechanics as in standard qm -- just project the wave function on the eigenstate corresponding to the measured quantity and renormalize its modulus to one. This is not the description of a dynamical process. So Bohmian mechanics would be incomplete at least in the sense that it does not describe just *how* the collapse evolves in time. However, the wave function is considered a real object in Bohmian mechanics, so a description of its dynamics should be provided not only for ordinary quantum evolution (given by the Schrödinger equation) but also for the process during a measurement (collapse, not described by the Schrödinger equation). I don't know of any such detailed description, but then I am not an expert on Bohmian mechanics.

K. Kassner

A. Most of the comment of K. Kassner is out of topic. My question questions the completeness of the set of axioms of the QM, not what can we measure/infer from measurements.

The postulates of the QM are the following:

1) All the information about a quantum object is contained in the wave-function.

2) The wave-function is a state belonging to a Hilbert space possessing all the properties of an algebra over a field of complex numbers.

3) To each physical quantity - named in the quantum language observable - is associated a Hermitian operator and the values the observable displays at a measurement is one of the eigenvalues of the operator.

4) The values displayed by two observable measured on the same wave-function obey the uncertainty relations.

5) The frequency with which an observable takes a certain value is equal to the absolute square of the amplitude of probability of that eigenvalue, obtained from the wave-function.

B. All of the most popular interpretations of the QM supplement the QM with additional postulates. As discussed in my work

Preprint The Wave Function Cannot be a Real Wave -Then, Can We Speak ...

the added postulates lead to predictions that disagree with the QM predictions. In particular, de Broglie and Bohm tried to add to the QM the postulate that the quantum object consists in a particle moving along a continuous trajectory guided by the wave-function. It is proved in

Chapter A Non-Relativistic Argument Against Continuous Trajectories ...

that the assumption of a continuous trajectory leads to predictions contradicting the QM predictions.

I tried for a long time to prove that the QM set of axioms is complete, i.e. any axiom added, and contradicts the axioms 1 - 5, will lead to predictions that contradict the quantum predictions. So far I didn't succeed to do such a proof, so I am open to suggestions.

Dear Sofia,

I just wanted to point out that the discussion (in the scientific community) of completeness of quantum mechanics is *not* about axiomatic completeness but about completeness of the description of reality that quantum mechanics wants to provide. Of course, you can ask any question you like, but if you ask questions that are off topic (regarding the most interesting ongoing issue) or that can be trivially answered, your contribution has a tendency towards irrelevance.

As I already told you before, your proof regarding the non-existence of continuous trajectories in quantum mechanics is incorrect and cannot, in particular, exclude Bohmian trajectories. (This is a statement about logic, not implying that I personally believe in Bohmian trajectories as paths of quantum particles.) Bohmian mechanics does make the *same* predictions as nonrelativistic standard quantum mechanics with respect to *all* possible experimental observations, despite having both additional variables (the Bohmian particle positions) and two additional axioms (the equations of motion of those particles and the quantum equilibrium postulate). I have given an explicit proof of this on my science education pages (https://www.researchgate.net/project/Science-education-on-ResearchGate), and this proof is correct.

Now, the question of axiomatic completeness of quantum mechanics as a theory of everything is easy to answer. The theory is incomplete. Its most explicit formulation is the standard model with quantum fields for every known elementary particle and the Lagrangian and hence field equations for all fields. It does not contain dark matter nor dark energy, for which there is observational evidence. It states neutrinos not to have mass, but we know that at least one neutrino type must have mass. So we need modifications of the model (i.e., new axioms) and additional variables (new fields).

Moreover, we have Goedel's incompleteness theorems about axiomatic theories that are complex enough to include the theory of the natural numbers. This means that there are theorems that are true but not provable inside the theory, no matter how many axioms we add. If we believe that the axioms of a physical theory should include those of the mathematics that is used in the description of the theory, then a physical theory cannot be axiomatically complete (at all), if it contains the theory of the natural numbers. (Whether this is true for our physical theories, I am not sure. Maybe we can get by with a reduced version of the theory of real numbers that does not include laws about infinite sets arising from the theory of natural numbers.)

Anyway, it is clear that any discussion of the completeness of quantum mechanics that refers to the very restricted set of axioms that you indicate (even leaving out the Schrödinger equation or another equation of motion for the wave function) does not really mean completeness of the theory, but at best completeness of the framework for theories provided by quantum mechanics. That is, the question is: should we believe that our final "theory of everything" (if we ever get there...) can have this kind of structure (a Hilbert vector that is a complete state description, etc.) or do we need more structure for the theory to be complete?

Even then, the question is not whether we cannot add an axiom, but whether we *must* add stuff, be it axioms or structural elements. It is always possible to add axioms to a physical theory without changing its empirical content. You can add structure to quantum mechanics to obtain Bohmian mechanics and the predictions of the two theories for experiments are indistinguishable. You can add an ether to special relativity to transform it into Lorentzian ether theory and the predictions for experiments are indistiguishable. The recipe is roughly the same: add new elements to the theory (Bohmian particles, the ether) and add axioms that make sure that they remain unobservable (in Bohmian mechanics all the predictions regarding experiments can be obtained from the wave function alone, the equations for the particle positions are constructed so that they cannot contradict these; in Lorentzian ether theory, time dilation and length contraction conspire in a way as to make the absolute time unobservable). Making a theory more complicated by adding stuff without adding empirical predictions is always possible. Completeness would mean that it is not *required*, not that it is *impossible*.

So you cannot prove completeness by showing that any additional axiom contradicts those that are present. Because one can *construct* axioms so that they do not contradict the ones that are already present.

Since reality seems to contain "stuff" for which we do not yet have a full description, a discussion of the completeness of the quantum mechanical framework is best done by looking at a subset of reality that we know well, at least in principle. That is, we should consider experiments with well-known objects, such as electrons or photons, rather than take into account unknown dark matter fields. This is what EPR did and what is done in discussing modern experiments on violations of the Bell inequalities.

One reason why the framework might be incomplete, is a strange property of entangled states: having maximum knowledge of an entangled two-particle system by knowing its wave function, we do not have maximum knowledge of the subsystems, the two separate particles. They do not *have* a wave function, i.e. they do not have a quantum state.

It *is* possible to describe the complete statistics of the outcome of all experiments that can be done on one of the particles only via a density operator describing a "mixed state". However, a mixed state is not a state but an ensemble of states with probabilities assigned to each of them. As long as some of the probabilities are different from one and zero, this is to be interpreted as different states being present with these different probabilities, and clearly constitutes incomplete knowledge, as complete knowledge would require to know which state is realized and not just what probability there is for one of several realizable ones.

The question then is: can we call a framework complete, in which the maximum possible knowledge of a system does not imply maximum possible knowledge of its subsystems? If not, can we make it complete by addition of elements to the theory (hidden variables, axioms to describe them)?

These would be meaningful questions. The one posed by yourself is either meaningless or trivial.

which we need is a description of a system without asking the question is it a corpuscle or a wave. the question which sould be asked is what we want to measure ?. The problem in quantum mechanics should be treated as thermodynamics. The key for quantum mechanics as a "thoery of everything" is energy.

Here an important article from www.futura-science.us speaking about this:

"Quantum entanglement is a fundamental phenomenon in quantum mechanics discovered by Einstein and Schrödinger in the 1930s. Two physical systems, such as two particles, are found to be in a quantum state in which they form a single system in a certain subtle way.

Any measurements on one of the systems will affect the other irrespective of the distance between them. Before entanglement, two non-interacting physical systems are in independent quantum states, but after entanglement these two states are in a way "entangled" and it is no longer possible to describe them independently.

This is why, as indicated above, non-local properties appear and a measurement on one system instantly influences the other system, even at a distance of light-years. The entanglement phenomenon is one of the most disturbing in quantum mechanics and is the basis of its Copenhagen interpretation .

Quantum entanglement is at the heart of the famous experiments known as the EPR paradox and Schrödinger's cat or Wigner's friend. The entanglement phenomenon is based on the mathematical and physical principles of quantum mechanics. That is to say the notions of state vectors and tensor products of these state vectors on the one hand, and the principles of superposition of states and reduction of the state vector on the other hand.

Remember that in quantum mechanics, which is the extension of Heisenberg's matrix mechanics and Schrödinger's wave mechanics, there is a complete reworking of the kinematics and dynamics of the physical and mathematical quantities associated with observable phenomena and physical systems.

Quantum mechanics, even though it deals with wave-particle duality, is not a theory that can be reduced to particle wave mechanics.

The dual nature of matter and light shown in the case of charged particle theory and electromagnetic radiation theory is only a consequence of the reworking of the differential and integral laws associated with physical phenomena and a physical system.

The introduction of the concept of wave function for a particle is then only a very special case of the introduction of the concept of state vector for a physical system with dynamic variables giving rise to a measurable phenomenon, whatever this system and these variables, as long as a notion of energy and interaction between this system and a classical measuring instrument exists.

It is because the differential and integral laws describing the change in space and time of an observable quantity in classical physics naturally have the form of the kinematic laws of a discrete or continuous set of material points, that correspondences are found between the general quantum formulation of these laws and the quantum laws of electrons and photons.

It is important to remember that in classical physics already a phenomenon is measured and defined from the modification in the kinematic and dynamic state of a particle of the material being tested.

An electromagnetic field is defined by its effect on a charged test particle of matter at a point in space and therefore, in particular, a field of light waves.

Temperature can also be defined by the dilation of a material body at one point, and here too, an observable quantity is, in the last analysis, defined by the kinematics of a material point and the sum total of the energy and momentum exchanges.

The solution to the wave-particle duality problem therefore lies in the two central ideas in the Copenhagen interpretation and quantum mechanics in the form given by Dirac, Von Neumann and Weyl from the work of Bohr, Heisenberg and Born.

-in nature there is fundamentally neither wave nor corpuscle in the classical sense. These concepts are only useful and are still used in the theory because they must necessarily establish a correspondence between the form of the quantum laws and the form of classical laws that must emerge from the former.

Just as a test particle serves to define an electromagnetic field, a classical measuring instrument serves to define a quantum system by the way in which this quantum system will affect the measuring instrument. Inevitably, the kinematic and dynamic description of this instrument will involve the classical wave and particle concepts.

Quantum formalism must therefore express both all of this and the fundamental non-existence of the classical particle and wave, just as relativity is based on the non-existence of absolute space and time. This property of formalism is largely satisfied by the Heisenberg inequalities.

-the wave-corpuscle duality is not derived from any subtle association of particles and waves, i.e. there are no special laws restricted to the laws of motion and the structure of particles of matter and to the waves of interaction fields (electromagnetic, nuclear etc.), but there are laws of change in time and space of any physical quantity which are modified, and in particular the general form of a differential law and an integral law.

It is because this framework is quantified that it necessarily applies to any physical system at all. It is very important to remember in this connection that the existence of an energy is an essential property in all the laws of physics. The universality of energy and the fact that any definition of the measurement of a phenomenon is based, in the final analysis, on an interaction with energy automatically ensures that the laws of quantum mechanics apply when describing the change in any arbitrary system.

This is why wave mechanics, which finally is based largely on the existence noted by de Broglie of a strong analogy between Maupertuis' principle for the motion of a particle of matter and Fermat's principle for a light beam, is merely a very special case of quantum mechanics since the latter does not finally apply to the laws governing the motion of particles in space and time but to the change in all directly or indirectly measurable physical quantities.

In particular, the laws of quantum mechanics naturally contain the possibility of creating or destroying a particle and of its transformation into another particle, which is not a phenomenon that can be described using the Fermat or Maupertuis principles.

The construction and form of quantum theory are thus based on the ideas that:

-the laws of physics do not fundamentally apply to something in space and time.

-particles and waves are not fundamental structures but approximations of the form of the laws and objects of the physical world.

-energy is at the heart of the quantification process and ensures/explains the universal character of quantification (the quantification of certain classical dynamic variables, probability amplitudes for observing these values).

However, the laws of quantum mechanics emerged historically and can be introduced for teaching purposes as a first approximation with the wave and matrix mechanics of particles in classical space and time. But it is central to understand as quickly as possible that these mechanics are not the true structure of quantum mechanics.

The way we proceed is reminiscent of thermodynamics which functions independently of whether or not the physical system has any atomic structure. The total energy of the system, called an equation of state of the system, is considered and there is a set of fundamental variables called variables of state related by the energy function and other equations of state of the thermodynamic system. The system is defined as a black box (what is inside is not important) and only the sum totals of input and output energy and the values of the variables of state are measured.

Nevertheless, quantum mechanics does achieve a synthesis of the wave and corpuscular structure for the change of physical values. In particular, this means that the physics and mathematics of waves and fields must appear in the form of these laws such that, when they are applied to particular systems such as classical electrons, protons and electromagnetic fields, we find the wave mechanics of these systems.

Thus the principle of the superposition of fields in electrodynamics and optics must reappear to describe the state of a quantum system. The entire structure of Fourier analysis must especially be present.

Similarly, the structure of analytical mechanics with the Hamiltonian function of the energy of a classical mechanical system must be kept and play a central role.

Bearing in mind the above considerations, the way in which quantum mechanics is constructed starts to become clear.

The observable variables Ai and a total energy H called the Hamiltonian are associated with a physical system.

In the case of a particle having momentum variables Pi and position variables Qi placed in a potential V(Qi), the function H of the particle is written:

H=T(Pi)+V(Qi)

where T(Pi) is the kinetic energy of the particle.

In its initial form, the Schrödinger equation for such a particle involved an object called an energy operator H, derived from the previous function, and giving rise to a differential equation for a function Ψ (Qi) called the wave function of which the square gives the probability of measuring the particle with the value Qi of its position.

The formulation of quantum mechanics makes use of all this and generalises it. We still have an energy operator H but the wave function is merely a special case of the state vector (think of thermodynamics) of any physical system.

To clearly show the departure from the concept of wave function this vector is denoted by Ι Ψ >. This is Dirac's vector notation for introducing Fourier analysis abstracted from Hilbert's functional analysis for linear partial differential equations.

An observable dynamic variable A, transcribed in the form of a linear operator A, can then have a series of values an during a measurement. Experience shows that there is a probability IcnI 2 of observing each value an, and that the state vector of the system is written as a vector sum of the base vectors associated with each value an such that:

Ι Ψ > = ∑ cn Ι an >

where

∑ ΙcnΙ 2=1 with n=1,2 ....

as required for introducing probabilities.

The base vectors Ι an > and the values an are called the eigenvectors and the eigenvalues of the linear operator A .

It is in this sense that we speak of a superposition of states in quantum mechanics. The coefficients cn are complex numbers of which the square gives the probability of finding the system in the state cn Ι an > of its dynamic variable A. This can be the position, the speed and quantum state variable that can be associated to express the characteristics of the system.

In the case of electrons, the phenomena of diffraction and interference which they display depend precisely on this principle of superposition of states applied to their states of position. Except that it is not a question of a series of discrete values xn for Q1=x=A1 but a continuous distribution. It is also for this reason that, generally speaking we refer to probability amplitudes for cn by analogy with light waves where the square of an amplitude gives the intensity of the light at a given point.

Schrödinger's equation in its general form is then an equation of change written:

(ih/2π) d Ι Ψ >/dt = H Ι Ψ >

If we have correctly understood the long arguments developed above we should not be surprised that as soon as we can define an energy and physical variables for any system, Schrödinger's equation above will apply and is absolutely not confined to notions of the change in space and time of a particle in a potential.

In particular, if the system were a quantum animal that could take the form either of a quantum whale or a quantum dolphin, in the sense where there would be two energy states for the same physical system, such as a quantum aquatic mammal, Schrödinger's equation would apply!

And this is what happens in neutrino or K meson oscillation phenomena, and also in the multiplets of isospin such as quarks and leptons in the electroweak theory and in QCD.

It is clear that this has nothing to do with notions of wave-corpuscle duality and wave mechanics.

During a measurement the state vector makes a quantum jump to now consist only of I an >. By analogy with a superposition of plane waves in a wave packet, we speak of reduction of the wave packet for the wave function of the position of a particle and, generally speaking, of reduction of the state vector for a quantum system.

With these fundamental notions in mind, we can study the phenomenon of entanglement in somewhat more detail.

Consider a simple quantum system, a quantum coin in a game of quantum heads or tails.

The base state vectors will be Ι f > and Ι p > for heads and tails. The coin can be in a state of quantum superposition such that its state vector is:

Ι Ψ > =c1 Ι f >+c2 Ι p >

where Ic2I2 will give the probability of observing the coin in the state of heads, for example.

If we use two coins A and B; we will then have two state vectors:

Ι ΨA > = c1a Ι fa >+c2a Ι pa > et Ι ΨB > =c1b Ι fb >+c2b Ι pb >

The two coins are considered as initially having no interactions, which means that we will have two independent Hamiltonians Ha and Hb.

Let H be the Hamiltonian of the system made up of these two coins and I psi > its state vector.

Then H =Ha+Hb and the state vector of the complete system and the most general form of the solution to the Schrödinger equation is a rather special product called a tensor product (χ) of the state vectors of each coin.

Thus

Ι Ψ > = ( c1a Ι fa >+c2a Ι pa > ) (χ) ( c1b Ι fb >+c2b Ι pb > )

= c1a c1b Ι fa > (χ) Ι fb >+c1a c2b Ι fa > (χ) Ι pb >+ c2a c1b Ι pa > (χ) Ι fb >+ c2a c2b Ι pa > (χ) Ι pb >

This is just the abstract re-transcription of the technique of separation of variables in a partial derivative equation.

If the Hamiltonian can no longer be broken down into a sum of Hamiltonians of coins with no interactions, during a brief instant when the coins might be electrically charged for example, the state vector of the system can no longer be described exactly as a tensor product of the state vectors of its parts with no interactions.

And this is exactly what we call the entangled state!

But this requires some important explanations. The state vector is always the sum of tensor products of the base states, heads or tails of a coin with no interaction, but the coefficients giving the amplitudes of the probabilities of finding the results of observations of the two coins can no longer be broken down into products of the amplitudes of the states of each coin before interaction, i.e. entanglement.

If the two entangled coins are separated and transported to opposite antipodes, a measurement on one will instantly affect the quantum state of the other. This means that the results of measurements on the second coin will no longer be independent of measurements made on the first one.

The EPR paradox and Bell's inequalities are essentially based on an analogous situation with physical systems formally giving rise to the same mathematical equations.

Here we see the full power of the abstract formulation of quantum theory, and above all the nature of quantum theory itself, in the sense that general principles are at work in a large variety of different physical systems and they result in mathematical equations that are largely independent of the form and of the physical system and of the physical variables of the system.

So that if we wish to analyse any given quantum phenomenon, the principles of quantum mechanics can be tested with the physical system and the type of dynamic variable that are the easiest to produce experimentally.

And indeed, the EPR paradox was initially formulated with variables of position and momentum for a pair of particles. But it keeps its essential meaning if we take the spin variables of a pair of particles, whether they be electrons or photons, for example. This is the reason why David Bohm proposed to test the paradox in this form, and this is what Alain Aspect did in 1982 with a pair of entangled polarised photons."

I would like to mention that Scientific Research Publishing (SCIRP) is on Beall's list of Potential predatory scholarly open‑access publishers, which means that the Journal of Quantum Information Science might be a predatory journal.

Scientific Research Publishing is a fraudulent,low-quality, and deceptive journal and has no Clarivate impact factor,which means that SCIRP is below average both mathematically and physically and has no credibility.

Sadly,a lot of of Sofia's lifetime that she spent writing pseudoscience papers for publication went to waste.  

The wave function may seem impossible to be a real wave, There are so many amazing things that happen in the wave function. But I assure you it is a real wave. You have just never seen it, really no one has except for me, but you can see the entire wave function happen in my work with your own eyes.

QM is still something in expansion, so we dont know what other rules may apply eventually. I dont think details of how the wave function interacts with

EM fields are currently expressed well.

Juan Weisz,

Please can you elaborate on what you mean by "interaction" between the wave function and the electromagnetic field?

Issam

Yes, for example in the Bohm Aharonov effect you have a phase shifting effect of a magnetic field and the phase of the wave function , supposedly through the magnetic vector potential, showing up in interference experiments.

This interaction is sometimes described with a KE term of form

(p-qA).(p-qA)/ 2m is QM which includes fields. An electric field may be included as -xqE term in the Hailtonian, for example.

Even gravitational potentials also have influence of the phase of a wave function.