Great question Hans,

But as is obvious from the length of the question, any answer would likely have to be several books long.

A great compendium, covering many different and even contradictory viewpoints (from wave function realism and monism through to reductionism) would be 'The Wave Function', by Alyssa Ney and David Z Albert (editors).

Spiros, thanks for an interesting reply

1- I thought that the issue of hidden variables had been largely put to rest, in part because of Gleason's theorem, and also because Renato Renner and Roger Colbeck in Zürich have convincingly demonstrated that no possible hidden variables hypothesis could possibly improve the significant outcomes produced by current theories, even in those cases where Gleason's theorem would possibly not apply.

2- You are right that the answer is actually very simple : the wave function is a function that satisfies Schrödinger's, period.

But its interpretation is still controversial, in part I believe because some find it hard to accept its consequences (such as, there is no expression of spatial separation between different variables bound up within a valid Schrödinger, which leads to the odd situation confirmed by Bell's inequality and experimentally by Aspect, Gisin, et al.)

I would not dare to put such a question without having thought about the answer. I come to the conclusion that a kind of Alice's wonderland exists in the lowest levels of physics. Quite many restrictions are put by the observable universe on what could exist in the non-observable world.

My own model (that I would not dare to call a model of physics) is based on first principles that I derived from quantum logic. Quantum logic was first introduced in 1936 in a paper by Garret Birkhoff and John von Neumann. For me that paper is more important than everything that Einstein has written. For me it represents the foundation of my model from which I derive everything via pure deduction. However, I must admit that here and there I took some guidance of what we know from reality.

I have formatted my project as a game in which everybody can participate. Any implementation of the game starts with well selected first principles. From that foundation a model is deduced via trustworthy mathematical methods. The target of the game is to produce a model that contains many features and phenomena that we know from reality.

To answer the question it could be useful to know to what is related the "wave function". As QM tells to any system is associated a Hilbert space such that a state of the system is represented by a vector. It seems that in the present case the system is a particle. And indeed if you kake the most elementary model of particle, with a trajectory represented by a square integrable map x(t), it is a vecor of a Hilbert space : the state is the map itself. Because any measurable map is based on a finite number of measures, there is an unavoidable uncertainty about the map x. It is not too difficult to show that the probability to measure a given map is linked to the square of the vector representing the map.So we have some probability distribution to mesure the presence of the particle at a given time and a given location. But the usual representation by a "wave function" muddles the problem : there is wave which could propagate, and it is not related to any physical phenomenon. The Schrödinger equation can be proven if the system meets some basic requirements.  All this can be mathematically proven (there is no exotic assumtion required), this is the logical consequence of how we model physical phenomena. And there is no mystery.

This can be extended to the GR context with spinor. Then particles are sections of a fiber bundle : basically they belong to some "general solutions" of differential equations, among which the actual solution is defined by the initial condition. So we have a section defined at any point (the "wave") associated to a trajectory, measured with some probability. This can be proven with some assumtpiion which support the use of spinor to model the motion of material body.

Jean Claude,

What you state is indeed the traditional interpretation of the wave function. The measure that you talk about comes close to the probability density operator/matrix that is becoming more popular.

The question that is put on top is about a quite different interpretation of the wave function. There the possibility is explored to interpret the wave function as a continuous descriptor of a dynamic fine grain structure. This description smoothens and thus filters the information befor it is processed (read: inverstigated in its full content)

It is quite possible that such a fine structure exists. If this possibility exists, then it is the task of scientists to investigate that possibility. This duty is counteracted by the fact that physicists have cultivated the attitude that any relevant physical statement must be experimentally verified. All investigations of what can exist below the wave function show that probably the fine structure and its dynamics cannot be observed. Only the smoothened and averaged effects of the fine structure and its behavior have shown to be observable. As long as physicists do not change their attitude, this dilemma will stay.

If you publish about non-measurable facts, then you are categorized as a philosopher. May be these philosophers are the new physicists!

Thancks Hans

Well, there are actually two dilemna, which are related. The first, usual, comes from the capcity to observe some phenomena, at some scale, and the second come from the way we model these phenomena. It is clear that when we compute, organize data, that is do actual and practical physics, we work with some specifications of the mathematical objects that we have introduced : when dealing with a trajectory, by necessity we choose some, simpler, specification. This intriduces a bias, which is distinct from the usual uncertainty of any measure. And this bias can be mathematically formalized. At marcoscopic level it does not matter much, because it is always possible to increase the precision and adopt morecomplicated specifications, but when dealing with objects that cannot be easily observedn this bias is important, and most of the usual paradoxes of QM come from that.

Of course one cannot discard the possibility that the physical world is more complicated, and that it involves a fine structure. But before assuming new phenomena it is better to use all we can get with the more common, and checked, physical objects.

There is indeed some philosophy involved, notably the relation between science and reality. On these matters you can see my paper on common structures of scientific theory on this site.

Jean Claude

My highest interest goes to the origins of the phenomena that we can observe. I have come to the conclusion that this origin is hidden in structures and behavior that we cannot observe. For me it is not important whether this observation barrier is temporary or fundamental. In both cases an approach can help in which the restrictions of the tools of physicists are eliminated. That approach uses completely deduced models. Taking that approach immediately generates a conflict with the attitude of many physicists that all relevant physical statements must be experimentally verifiable.

For that reason I have formatted my research project as a modeling game. The models in this game are not positioned as physical models. Each of the participating models must start with well selected first principles. This foundation must be extended to a higher level model by using trustworthy mathematical methods. The target of the game is to reach a level in which the model shows many features and phenomena that we know from observing physical reality. Even at that level the model is still not considered as a physical model, but a discussion can be started whether the resulting model is suitable as a model of physics.

The Hilbert Book Model (HBM) is an instance of the game. It already reaches a level in which it shows many features and phenomena that we know from physical reality. Is also explains the origins of these phenomena. Further, the smoothened forms and the averaged effects of the fine structured features and behavior are similar to the smoothened data that are used to by the tools of contemporary physics.

So scientists must now decide whether they can ignore these discoveries. 

The first principles of the HBM are based on a 1936 paper of Garret Birkhoff and John von Neumann in which they introduced quantum logic and its relation to a separable Hilbert space. I do not interpret quantum logi as a logic system. What were interpreted as propositions is interpreted by me as construction elements. So the most compact formulation of the first principles of the HBM is:

At any progression instance, all discrete objects in universe can be represented by a closed subspace of an infinite dimensional separable Hilbert space.

I do not know this paper of Birkhoff and von Neumann. Actually one can go further than their conclusion : any model using quantitative variables that belong to infinite dimensional vector spaces can be represented in Hilbert spaces (see the proof in my paper on Hilbert spaces in systems modelling). What is interesting is that from where one can introduce observables, the laws for the evolution of systems (including the Schrödinger law) and many results on interacting systems. These results do not involve any physical assumptions : they hold even in economics !

Of course we must precise what is the status of such models in scientific theorires, and this require some epistemology, and defining criteria to choose the right theory.

@Jean Claude

Isn't it true that any function that's absolutely integrable, like a density of probability can be represented as psi* psi, where psi is square integrable? I think you can give a sensible answer to this question.

If your answer is yes, then we must have that i\partial_t \psi = \hat h \psi where \hat h is an hermitian operator. I would say that behind the wave function there is a tautology. The actual form of the Hamiltonian is obtained from the principle of insufficient reason, based on the symmetries of space, because \hat h must be isotropic. We just don't know what is going on down there.

The concept of particle was introduced to describe the motion of an extended object as a whole in space, by means of a mathematical point. The last time I checked on Euclid's elements, the idea of a point was obscure: Definition 1: A point is that which has no part. When we describe Jupiter's motion we use the concept of particle, however Jupiter has some parts.

As Einstein pointed out:

"In the present treatment, geometry is related to actual things (rigid bodies), and its theorems are statements concerning the behaviour of these things, which may prove to be true or false. ... There are advantages in isolating that which is purely logical and independent of what is, in principle, pure empiricism. This is satisfactory for the mathematician. He is satisfied if he can deduce his theorems from axioms correctly, that is, without errors of logic. The question as to whether Euclidean Geometry is true or not does not concern to him. But for our purpose it is necessary to associate the fundamental concepts of geometry with natural objects; without such association, geometry is worthless for the physicist."

Can somebody please tell me in what way can we describe the position of an electron by means of a mathematical point, except, approximately, when the electron is detected?

A point-like particle can hop around in a location swarm. The hopping path may be stochastic, but the swarm can still characterize the particle via its symmetry and statistical characteristics. If the swarm is a coherent distribution of locations, then it can be described by a continuous location density distribution. Via normalization this function becomes a probability density distribution and corresponds the squared modulus of the wave function of the point-like object. In this way the point-like object has no internals, but it is characterized by its externals.

The swarm elements can be location values that are eigenvalues of a location operator. The corresponding eigenvectors span a subspace of the Hilbert space. This subspace is eigensubspace of a swarm operator that adds extra eigenvalues, which are based on the symmetry and statistical characteristics of the swarm and the hopping path. The eigensubspace represents the point-like object.

To Oscar,

One can prove that Hilbert space structures appear in general quantitative models,  whenever the variables belong to infinite dimensional vector spaces. They are not necessarily functions, even if this the most common case. Hilbert spaces of functions are well known in analysis, see any book on the subject.

Point : it is clear that this is the concept of a location (since the Greeks) and because in Relativity we assume that it exists in a space-time, it shall require 4 parameters to be identified. There is no need to call for a "frame" with an origin and 4 vectors. Actually the simple fact that you need 4 parameters defned in a chart, such that you can pass from one chart to another, is enough to say that the universe (the container) has the structure of a manifold. Think how practically you locate something : you can give its address, or its GPS coordinates, but I doubt that you would use a frame (even if it possible). The structure of manifold (as postulated in GR) is much more natural and physical that the affine vector space of galilean or SR geometry.

Solid : the trouble enters when one consider several points together. The paradigm is the rigid body, a solid. Because then you have to tell how you go from one point to another. The common reasoning is based upon the rotation of frame, the derivative of one element of the Lie group of displacement (the Poincaré group in SR). But this is very  artificial (mathematically correct but not physically accurate) : there is no rigid body in Relativity, and because the universe is not isotropic you do not jump from one point to another when you ccompare locations : the path matters. So to define the motion (translation and rotation) of a particle we need to introduce something different, through a fiber bundle, and this is how enter the spinors (see my paper Particles and fields for more).

To Hans :

One can conceive a particle (a location), a solid body and by extension some "rotation" of a body. It is clear that in any physical model of particles their trajectories (yes, you have trajectories, such as in a collider) are assumed to belong to some families of curves defined by a finite (small) number of parameters, which are general solutions of a least action problem. So you work actually with sections of a vector bundle, which, by definition, are defined all over the universe (but a specific solution is given by the nitial condition). There is no need to call for a new concept (a swarm).

Every Hilbert space owns a Gelfand triple, which offers operators that have continuum eigenspaces. The Gelfand triple emerges from the Hilbert space in a way that is similar to the way that the real numbers emerge from the rational numbers.

Both the Hilbert space and the Gelfand triple are structure storage places. The data are contained in eigenspaces of operators. The structure uses the hierarchy that is offered by vectors and subspaces. Apart from eigenvectors also eigensubspaces exist. The usage of the subjects is shown above.

The eigenspaces of operators that reside in the Hilbert space can be mapped (read embedded) into the eigenspaces of corresponding operators that reside in the Gelfand triple. This need not be a perfect map. A stochastic difference may occur. This fact is the reason of existence of quantum physics.

At least one operator pair delivers a perfect fit. Their eigenspaces are suitable as parameter spaces. Together with other operators these parameter space operators can define functions.

Squared integrable functions can also form a Hilbert space, but then these functions act as Hilbert vectors.

(The Gelfand triple was introduced in the sixties, long after the first usage of Hilbert spaces in quantum physics.)

However, distributions, like the Dirac delta function, where already introduced in the thirties. Gelfand triples where introduced in order to enable the combined management of discrete and continuous spectra.

@Jean Claude

The swarm elements hop and as a consequence the swarm moves. The swarm elements form a stochastic micro-path and the swarm as a whole moves along a trajectory. The micro-path can implement a quasi-rotation.

With respect to a frame that moves with the swarm, the swarm stays at rest. It means that the micro-path is closed. With respect to a frame that does not move with the swarm the micro-path is open. In the most extreme case all swarm elements are located along a line. In that case the swarm has reached its maximum speed.

To Hans,

I guess that what you call a Guelfand triple is the triple defined by a positive kernel. Indeed positive kernels are a key ingredient in the definition of the Hilbert space associated to any model (they give the correspondance between the variables, as seen in the model, and the vectors of the Hilbert space). However a triple needs some set, upon which the arguments of functions are defined. This is not a mathematical problem, but a physical one : I assume that these arguments can be the location of a particle. Then we come back to the idea of families of trajectories (for particles).

I do not have objections to this construct, but if we are in physics, you need to relate precisely mathematical objects to physical objects and their properties. Physics do not come from abstract mathematical construts, but from this association of physical phenomena to mathematical representations.

That's what I meant with my citation of Einstein. I know there are no rigid bodies. Actually there are no absolutely rigid bodies of any kind. At the same time, considering that we have no direct sensorial information of an electron's structure, we cannot form a mathematical concept to directly describe its motion. It might be objected that the de Broglie- Bohm theory is successful in explaining some phenomena using the concept of particles moving along well defined paths. However, from the continuity equation of quantum mechanics we can only tell where the sources of the current of probability are, information we can use to determine the current, but for the addition of a solenoidal field. That there is something wrong with that picture follows from the fact that in the base state of hydrogen the phase of the wave function is independent on the spatial coordinates. (An objection that was raised by Einstein.) The assumption that the electron is there just seated at rest in the system of the center of mass is obviously absurd.

I feel that the whole of particle Physics has got out of control, because it has been made too complicated. 

It got too complicated because of the Copenhagen Agreement.  Bohr's idea that matter at and below particle Physics was some sort of wave of stuff (matter or energy?) flowing in orbits, a bit like we see jet streams in the atmosphere these days.

They should mostly join up with themselves in some form of loop. These loops must be at least a subset of the knots that exist in Knot Theory. Photons and energy spreading out from the first Big Bang are the exceptions because they form open strings. Closed strings (loops) and open strings form the basis of  string theory, but it also fails because it does not address some of the fundamental errors.

There seems to be no compelling reason why the Probability Density was adopted.

• There are other ways that have yet to be thoroughly explored. The probability density concept worked quite well when considering the orbits of electrons around protons etc, where how it flowed was not as important as the energy it has when it is flowing. The latter can be predicted using the Probability approach. The detail can NOT be predicted.

However, as soon as we begin describing finer details concerning how these probabilities arise, the Physics seems to float on its own hot air. The standard model is in trouble; particle masses, charges etc are all introduced as constants and it is agreed that there are too many.

Coming back to some of the results we think are almost 'God given'. If we replace probability density by a simple mass/energy density, Schrodinger's equation has the double time differential, which makes it cleanly Relativistically Invariant. Spin becomes a bit of a problem, but I am assured that this can be resolved.  Anyway, when we look at  Quantum Theory, Spin is really described like a black box flight recorder in an airplane; No one knows what is inside it. It exists and works. We can describe it with a small matrix stuck on the top left-hand corner of appropriate tensors, but that is about all.

When we cannot explain the finer details in Quantum Theory, we revert to the 'duality of matter.',  a giant fudge if ever I saw one. 

I think we would get a much more coherent description of what lies under the 'wave equation' if we abandon the concept of Probability Density.  Such a change wont destroy is the huge advances that Quantum Mechanics has brought to use.

• The majority of the advances resulting from Quantum Mechanics come from Helmholtz's half of the theory., and we should pay more attention to that. 
• Notice that the concept of 'Curl' as defined in Vector Analysis by Gibbs and Heavyside is an approximation, requested by Maxwell, because it got rid of the horrible complexity of working with quaternions, which he hated.

There is a way of achieving this, but this note is getting too long, so I will stop here.

@Jean Claude

Hilbert spaces can be defined such that the values of their superposition coefficients, eigenvalues and inner products are taken from a suitable division ring. Three such division rings exist: the real numbers, the complex numbers and the quaternions.

Using these numbers the Hilbert spaces and their Gelfand triple are no more and no less than structured storage places.

For example, they do not contain a mechanism that ensures dynamical coherence.

Via the eigenspaces of operators the Hilbert spaces and their Gelfand triples introduce geometrical concepts.

Progression emerges in a different way.

@Krasnoholovets

The skeleton relational structure that forms both quantum logic and the set of closed subspaces of an infinite dimensional separable Hilbert space can be used as the foundation of a model that can be extended such that after a series of development steps shows many aspects that we also know from observing physical reality.

@Gilbert

Quaternionic functions are much simpler than the combinations of spinors and matrices that are used to simulate quaternionic behavior with complex number based functions.

Quaternionic differential equations can be formulated much more compact than their real number based equivalents.

The following quaternionic equation couples two normalized continuous quaternionic functions ψ and φ. It holds for any pair of such functions.

Φ = ∇ψ = m φ

∫|ψ|² dV= 1

∫|φ|²dV= 1

∫|Φ|² dV= ∫|∇ψ|² dV= m²

In quaternionic format the Dirac equation for the free electron runs ∇ψ = m ψ*

The Dirac equation for the free positron runs ∇*ψ* = m ψ

Φ = ∇ψ represents a differential continuity equation

∇={∂/∂τ, ∂/∂x, ∂/∂y, ∂/∂z}

∇ is the quaternionic nabla.

m is a real number.

I have no experience with Quaterions, so I just believed Maxwell;-)

There is hardly any difference. Quaternions is mathematics. Maxwell equations add a physical interpretation.

The quaternionic equations can be split into real valued terms and imaginary terms.

Both categories can contain scalars as well as vectors. A quaternion is a real scalar combined with a real 3D vector.

@Krasnoholovets

I try to do the same (developing new science) in a different way.

See: http://vixra.org/abs/1405.0340

Dear Hans,

You ask essentially whether a finer structure can underlie the wave-function. I will refer directly to the Bohm Mechanics (BM), for being specific. It's indeed the most widely and systematically investigated finer-structure. It seems to me that at the present time, to speculate on a general non-local theory of finer-structures is too great a task for us, and we'd better start with particular cases.

So, it non-plausible that experimentally it's impossible to say whether Standard QM (SQM) or BM is correct, because they are based on hypotheses that differ. If QM admits a finer-structure, then indeed QM is incomplete and has to be completed.

However, I claim that we didn't try enough seriously to find an experimental fact that would differ in consequence of the difference in hypotheses of SQM and BM. Some scientists, and to my regret, people of a high wisdom and experience, just refute the BM without having in hand a rigorous proof that it is wrong.

Let me tell you that BM is in conflict with the Special Relativity (SR). That is an immediate consequence of Hardy's thought-experiment - known in literature as Hardy's paradox. The adepts of BM think that the micro-world admits a preferred frame of reference. It's not a pleasant idea, but the fact that we don't like a certain consequence it not a sufficient reason for not investigating and clarifying the issue. The idea that the microscopic world is non-deterministic is also unpleasant. So is the idea that a quantum system is wavelike, though miraculously, at the contact with the macroscopic detector, it becomes particle-like.

Best regards,

Sofia

Sofia

The swarm follows nearly directly from quantum logic when that relational structure is taken as foundation of quantum physics. See the attached link.

If quaternions are used to represent the space-progression model that emerges from this foundation, then the swarm acts in concordance with relativity.

http://vixra.org/abs/1502.0186

Hans, I apologize, I only now saw your message from April. I can't spend much time with details - I regret. Well, I have an objection. You say in your (too much long) question

"After a while the set of locations looks like a swarm."

The movement of a quantum system has to obey the continuity equation. This equation has two components, current and density, and they have to be consistent with one another, i.e. satisfy the equation. You don't have much freedom with your "storm". If you start at a certain position and describe a trajectory - this is what I see that you suggest - you'd better think of a Bohmian trajectory, otherwise you may get into inconsistencies. The Bohmian trajectory permits NO STORM because such a trajectory DOES NOT intersect itself. For instance, in a plane wave, it is a straight line.

Different people suggest all sort of interpretations of the quantum mechanics (QM). Typically, they describe only general lines, not a debugged formalism. The Bohmian mechanics (BM) is a thoroughly debugged and consistent formalism. So, unless you undertake the task of building an alternative formalism, at the same level of consistency, you'd better stick to BM and accept its constraints.

Dear Hans,

What exists beneath the wave function is the various resonace states of the electron:

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

Best Regards

André

@André Michaud,

It's not a good procedure to indicate an article without, first, describing in a few lines the idea in the article.

The square of the modulus of the wave function describes a location density distribution. On its turn this distribution describes a swarm of locations where the owner of the wavefunction has been lately. This swarm was generated by a stochastic process, which owns a characteristic function. That characteristic function equals the Fourier transform of the location density distribution of the swarm. This is how the stochastic process ensures the coherence of the location swarm. Thus the owner of the wavefunction hops around in a stochastic hopping path, whose hop landing locations constitute the hop landing location swarm. See: Article How gravitation works

The wave-function is indeed a coherent "swarm", but we don't know "swarm of what?"

The wave-function tests all the environment, even beyond the location where we find the particle, but, again, we don't know from which material is made the wave-function.

The wave-function doesn't admit a substructure of a particle following a certain path through the volume occupied by the wave-function. However, at each point in that volume are displayed all the properties of the type of particle: mass (if any), charge (if any), internal structure (if any), spin (if any), etc. In consequence, if the wave-function consists in a couple of wave-packets, each point, in each wave-packet, displays these properties.

Also, if the wave-function consists in a couple of wave-packets, it is not correct to say that the 'particle' travels withing one of the wave-packets, because there is no such substructure, a particle following some trajecory ( the trajectory of one of the wave-packets). One cannot say that before the detection 'the particle was at the place where the detection is reported'.

See also my question

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

Sofia,

Elementary particles are elementary modules. All massive objects are modules or modular systems and composed modules are constituted from elementary particles. Elementary particles are generated by a private stochastic process that owns a characteristic function. The composed modules own their own stochastic process, which owns a characteristic function that is a dynamic superposition of the characteristic functions of the components of the module. The characteristic function of the stochastic process of the elementary particle equals the Fourier transform of the location density distribution of the generated hop landing location swarm. With other words superposition takes place in Fourier space rather than in configuration space.

If you want more details, then you must dive deeper in the Hilbert Book Model Project. See papers in http://vixra.org/author/j_a_j_van_leunen

From Einstein's first formulation of relativity, M=E/c2, or slower energy/light energy (the fastest energy of all forms aka structure), all matter, including particles, is stored or slowed energy formed from the very curvilinear or least action elliptical path light is known to travel, in 3D a conical or swarm shape and 4D, a figure 8 shape -- note the linear portions of 8, also two cones facing each other from the smallest entry point or Planck length to maximal expanse (180 degrees, a line again; Dirac function). Once the fastest energy of all, linear light, reaches curvature along the figure 8, various matter forms, again being of light, conically or helically or in a spiral 'swarm' (quanta, elements, DNA, populations, planets, galaxies) of varying densities depending solely on the volume of curved space, all predictively distributed logarithmically according to the elliptical distribution or Ulam's spiral of prime numbers as the acceleratory rate light-squared (speedXspeed) travels, thus explaining the indivisibility of both light and corresponding primes, which thereby would physically reach their maximum at 1, or when light reaches linear (180 degrees; Dirac function) vacuum light in curvilinear space along the figure 8 and beyond the Calibi-Yau manifolds.

Cj,

The formula E=mc² comes from the very beginning of relativity, and is misunderstood. The mass never changes, so the concept of "mass at rest" is irrelevant. It happens that, in discontinuous processes, where particles appear or disappear, there is a global change of the mass, which is related to the total balance of energy in the process.

How is defined the "curvilinear or least action path" along which light travels ? This is a fact that there are privileged lines of propagation of a signal, at least for the EM field, which is the basis for the definition of the "speed of light", however these lines do not come from the Maxwell's equations, in General Relativity it is generally said that light follow geodesics, defined through the covariant derivative of the gravitational field, which assume both a direct action of the gravitational field on the EM field, and the existence of a gravitational field itself which is absent in the Einstein's theory. It is clear that the metric should be involved, but it requires an additional assumption.

Jean,

In 2D light follows an elliptical path, in 3D yes a geodesic. It is agreed light is the fastest phenomenon of ALL, so follows the shortest known path, a geodesic curve, or thee 'gravitational field' intrinsically emergent from the very nature of light itself and thus, Einstein's long sought UFT uniting general with special relativity, the discrete (quanta) and continuous (wave), the one and the many, and all the other binaries in a conserved, asymmetrical (motion small (min) to large (max)/expansion-contraction/change differential-speed derivative-tangent line/rate-ratio; propulsion)-symmetrical universe, one verse.

"light is the fastest phenomenon of ALL, so follows the shortest known path," the problem is that a geodesic, defined through the covariant derivative of a connection, is the shortest path only if the connection is symmetric. The proof does not hold for a connection other than the Levy-Civita connection. Moreover there is no relation between the speed and the length of the curve...

Symmetry is preserved globally, a geodesic, where there is stasis, not locally ALONG the geodesic at the derivative or relative change where the relation between speed and length is time.

"the fastest phenomenon of all is the nonlocal connection between measurement results in quantum entanglements".

(I just wanted to remind the people that the question is about the wave-function - the comments about the speed of light and geodesics are understood to me, but don't seem related to the question. If somebody addresses another issue than the question, he/she would better open another thread. It's not useful for the poster of the question and for those interested in it, when people fill pages about something else.)

The fastest phenomena in physics are quite probably the hopping steps in the hopping path of elementary particles. The generation rate of these steps is at least 1020 Hertz. The step width is on average the Compton wave length. See "Generating Mass from Nothing"; Article Generating Mass from Nothing

Advice: see the distribution of the location swarm as a Gaussian distribution.

Dear Sofia,,

I understand. I happened to wander on this forum by chance... I do not believe in the wave function or the entanglement, or in any of the quantic magic and I do not want to disturb the believers of the true religion of the Quanta of the last days, certainly not to try to convince them to come back to science... I know that the certainties of the Academic life are too much comfortable.

Have a good day dear Sofia

Dear Jean Claude,

There is sadness in your answer. I wonder why? Is my answer the cause to this? I wouldn't want that, and I am sorry.

You say you don't believe in the wave function? But it works perfectly. We just have no idea from which material it is made. I am affraid that we are D O O M E D not to know, despite the extraordinary efforts done for finding out. Indeed, we have only access to results of measurements with macroscopic apparatuses that certainly disturb the wave-function.

Recently I read an article of Salvatore Savasta and his team, that I liked very much. To put the things in short, as if we didn't have enough puzzles about the wave-function, he proved one more puzzling thing: the wave-function of one single particle, does not remain with only one particle all the way from preparation to detection. No, in between pop out from the vacuum and back into it, virtual particles similar with the "real" one.

So, one more question was added to the many questions about the wave-function: when we speak of the single-particle wave-function, how many particles are there in fact?

But, all this "balagan" (disorder, irreasonable behavior) is no serious reason to be sad.

Have a joyful day!

Sofia

Hans,

Your question is rather a whole article. I just want to comment on the following

"How the wave function is generated:

Take a particle.

Embed it in a continuum.

At the next instance embed it at a slightly different location.

That location is NOT KNOWN IN ADVANCE.

After a while the set of locations looks like a swarm.

The stochastic characteristics of the process are constant.

Thus after a while the continuous location density distribution that describes the swarm no longer changes."

No, it's not exactly a swarm. The wave-function has a coherence time τ equal to the length d of the wave-packet divided by the group velocity. If there were a swarm from any position to any position, in repeated experiments we could detect the particle at very different arrival times, the spread in the arrival times would be much wider than the coherence time.

According to Feynman's path integral, yes there is a storm, but it is not a real particle that storms, and not with velocity less that the light velocity. What storms there, we have no idea.

Dear Hans,

Thanks you, I will order the book

Dear Sofia,

A bit of sadness...perhaps. I am 71 years, I do not expect fame or anything,, I just go on my quest for understanding and, as I can, give advice or answers. The issue of Quantum Physics for me is clear. What I have written on the subject, and tested many times, is both true (they are mathematical theorems, they do not need experimental checking) and operational (they justify most of the computations usually done, but in a safer environment). So we have answers to many questions but people are still lost in tortuous explanations, piling mysteries and interpretations which are not necessary, meanwhile there are true topics which should occupy true scientists. I am working on a GUT, the topic is difficult, but I hope to succeed in the next months. What is interesting is that, using my theorems, it is quite easy to prove that there should be "elementary particles", with the properties that we know... and I come back on the concept of measure which is actually at the core of the gauge theories.

Have a good day

Dear Jean Claude,

I can say that I 'breath' quantum mechanics, I work on it extremely much. Neither do I expect fame, I really want to understand, as you want.

Now, although I insist on taugh mathematical rigor, I am a phenomenologist. I ask, as many of us do, what is behind the formulas we write. You see, when we are trying to build in our mind a model about what can be the wave-function, we have to test this model against all the known experiments. It's a hard job but unavoidable.

I did this. And, I am displeased to say that what remains after such a pedant cross-check, is that the concept of a particle floating inside the wave-packet, is WRONG. There are a couple of very strong arguments against this concept.

So, unfortunately, we can rather say what the wave-function IS NOT, than what it YES is.

But, at least, due to the wave-function, the life is not boring.

With kind regards, and please banish the sad thoughts!

Sofia

Sofia, If you decide for mathematical rigor, then try the Hilbert Book Model. It explains perfectly how the wave model goes together with the particle model and it settles with several lies that still reign contemporary physics. However, you need solid knowledge of number systems, Hilbert spaces and differential calculus. Try to read "Structure of Physical Reality" and possibly other papers at http://vixra.org/author/j_a_j_van_leunen . If you doubt the mathematical rigor of the Hilbert Book Model, then check http://www.e-physics.eu/#_Challenge.

Dear Sofia,

Thanks you for tour kind words, they are welcome in this violent world.

To understand my approach of QM, I have no background as experimental scientist (I have been Director of R&D in a big company so I know more the world of engineers that I respect deeply). My background in research comes from Economy : Economists are very familiar with models. They would (for most of them) never believe that their model IS the reality. Only an efficient way to understand and forecast what happens (as the engineers do). And this is the problem with physicists : they believe that their variables are physical objects, on the ground that these variables are their data, obtained at great cost in experiments. But they forget that before collecting the data, they must format their organization, and after collecting their data they must use statistical tools to draw a conclusion. And these 2 phases have their own impact on the whole process of understanding and representing the real world. One of the problem is that the mathematical level of many physicists is poor, so they are at lost when trying to understand what happens. The more complicated the model, the more acute is the distance between reality and the data, and the perverse effects of modelization. I have never seen a full list of the equations of the Standard Model, but it involves very technical mathematical concepts, that very few can master, so most of the papers are dedicated to explore some obscure results, usually by the introduction of mysterious physical phenomena to patch the inconsistency of the model. The typical example is the Higgs boson : the model is mathematically inconsistent so one add a particle to bring the gap. When I read posts on this thread they are full of creative tricks which are a mix between high mathematics, usually not understood, and bizarre physical objects with unexpected properties. But this is fashionable, and scientifically correct with respect to the Academia, so why to worry ? I do, because for 50 years smart people have dedicated their life for nothing :they cannot show the simplest result, they are still in the mysteries... What a waste !

Have a good day, dear Sofia.

Dear Jean Claude, read at least the the introduction of "structure in Physical Reality"; in http://vixra.org/author/j_a_j_van_leunen. There it is claimed that at its basics, physical reality is very simple. It can be captured with rather simple math. The theory involved is lattice theory. More important is the statement that this basic structure works as a seed and that it applies the power of mathematical restrictions to ensure that it only develops into the kind of physical reality that we can observe. To follow this development requires the knowledge of mathematical restrictions and that is not the strength of most physicists. This is why only some mathematicians can follow the path upward to more complicated levels of the structure of physical reality. Physicists and criticasters rely too much on their intuition and their observations

Questions are quests, open-ended searches; not self-preserving, ego attention seeking, false humility echo chambers according to cyclically-diverted irrelevance. Enjoy.

Dimiter,

The reason that a stochastic process generates the location density distribution is not the most interesting fact. The fact that this process is controlled by a characteristic function and why the stochastic process does what it does are the really interesting facts.

What professor Nelson does is very revolutionary. In fact he states that stochastic processes control the universe. The Hilbert Book Model follows the same strategy. The general trend is the belief that forces rule the binding of particles.

Dimiter,

In the HBM the elementary particles can zigzag in time. At the(time) reflection points, observers perceive pair production or pair annihilation events.

https://en.wikiversity.org/wiki/Hilbert_Book_Model_Project#Zigzag

As is shown in the paper “The Dynamics of Wave-Particle Duality”, arXiv:1701.01168v9 (present also in ResearchGate), any Helmholtz-like wave equation is associated with exact sets of stationary trajectories, fully determined by the launching conditions and coupled by a suitable “Wave Potential” function. The very presence of wave diffraction and interference is due to such a trajectory coupling.

In full analogy with the classic mechanical case of particles launched (under assigned initial conditions and with a well defined energy) into a stationary external field, the corresponding wave-mechanical case is adequately described by the stationary (and therefore energy-dependent) Schroedinger equation, which is itself a Helmholtz-like equation. The particles are simply “piloted” by the relevant de Broglie matter waves along the coupled Helmholtz trajectories, thanks to the “gentle” energy preserving action of the Wave Potential, independently from the number of particles involved in the process.

Both in the classical and in the wave-mechanical case the trajectories are the same (in a stationary external field) for ever and ever. The wave-mechanical case requires, however, the assignment of the launching conditions on each point of a surface. To each starting point of the launching surface there corresponds, then, a single trajectory (for ever and ever).

This photo might illustrate, because the connection is through geometry and also matter. The theoretical introduction of force systems in mechanics do not require matter. The rigid body mechanics is based on distributed matter, and there is an angular velocity, which is derived from a motion in space.

What puzzled me today is that the roads are so Cartesian, and the buses and large transport vans do not fit so well anymore, since they increased in size, while the 90 degree bend remain.

Physicists accept the idea of a 3 dimensional Euclidean space, and they discourse at length about "inertial frames", but the stay stuck in an unidimensional world. Take the momentum, immediately we have p=mv, as if the only motion was transversal, that is unidimensional. And what of rotations, for which you bring your frame ? Hum.. they can be defined, but this is complicated, and there is the spin which is... a quantic effect. And the rotational momentum ? It can be defined by an axial vector (?) and the cross product, so that one come back to a vector... And of course the wave function is worse : it is reduced to a scalar.

Physicists must give relief to their vision of the world. All objects, including particles, have not only a location (they are at a point of space and time) but also an arrangement which can be measured by a rotation with respect to a frame. And the representation of their motion must account for both. The spinors are a path in this direction, but as they come out of the blue in the Standard Model they are useless out of particles physics.

For more you can have a look at my latest post on this site : Great unification theory, a solution, which answers most of these questions without extra dimension, exotic particle or an all power full vacuum. And it is perfectly in line with everyday physics.

In section 1 of the working paper of the ongoing project

https://www.researchgate.net/project/Mathematical-Concept-of-Proximity-and-its-Consequences-for-the-Interpretation-of-Quantum-Mechanics

three possible "origins" of the "wave function" are sketched. With regard to an ontological respect the first one seems to agree partly with Hans' idea of a "hop landing location swarm" of an elementary particle or bound electron.

The idea that stochastic processes govern the generation of detection location probability distributions and that these processes own a characteristic function and can be described as the combination of a genuine Poisson process and a binomial process, which is implemented by a spatial point spread function returns regularly in physics. The concepts of the Optical Transfer Function and its modulus, the Modulation Transfer Function and the concept of the Detective Quantum Efficiency are based on these ideas. The OTF is the Fourier transform of the Point Spread Function and depends on the distribution of color (energy), angular distribution and phase homogeneity of the detectable objects.

To Christian,

I agree with you, at the beginning is the concept of measure, which is an experiment in which one compares data between a system that we know and a system that we do not know This is the starting point of gauge theories. And in a gauge the first thing that Physics represent is the motion of a material body, including both the transversal and the rotational motion. The right representation is then by the Spin group, which lays in a Clifford algebra. Then the momentum (both translational and rotational) comes as a representation of a Clifford algebra, that is a spinor. See my paper on Great unification theory for more.