@Nicolis: "Concepts of quantum mechanics were developed, historically, from those of classical mechanics-but this is history, not logic."

Bohr would disagree. Of course, it is history, but it is also logic. According to Bohr's interpretation, to which most physicists nowadays at least pay lip service, you need classical notions to interpret quantum mechanics. Coleman says otherwise, but there is no need to be convinced by that. So it is a matter of interpretation.

"Of course it's possible to learn quantum mechanics first and deduce classical mechanics as a limiting case-there's no logical obstacle."

There is. How do you define the notionsof position or momentum without having a classical limit? The only way you could do quantum mechanics without classical mechanics would be to work entirely with undefinded notions first. Results of measurements would acquire meaning only in the limit. I doubt that you could call giving the mathematics of the theory without assigning meaning to the mathematical terms (there is no meaning to a wave function without a classical world behind...) teaching. So I repeat my claim that you cannot teach quantum mechanics before classical mechanics. You can babble about incomprehensible concepts at best. A theory is different from the mathematics behind.

On the other hand, you can define all the terms of relativity that require definition without having to know classical mechanics. You cannot do that with quantum mechanics.

Now, every theory including classical mechanics contains a number of undefined concepts of which you must have preconceived ideas. Force is such a concept in Newtonian mechanics that is roughly as undefined as points and straight lines are in Euclidean geometry. But it is visualizable and the theory can be understood without a precise definition of force. The concept becomes more precise as you go along with the theory. The only undefined concepts on which you can build in quantum mechanics are classical concepts. So you need classical mechanics before quantum mechanics.

"The interpretation of quantum mechanics doesn't depend on classical mechanics-it's the other way around."

That is a nice idea, but it seems unsubstantiated to me.

Note: Classical mechanics is not the limiting case of v 0). For each case of this separation into classical and quantum, there is an orthogonal separation into non-relativistic and relativistic. Not that Nature care about such classification.

The point Coleman raises in the second part of his talk is the question of how a fully quantum observer would experience the world. How, and to which extent, does a classic experience arise as a limiting case of a true quantum experience? Formulated and analysed in the language of quantum mechanics (with a rather limited form of an experience operator, if I recall correctly).

Like Wittgenstein's  question of how it would seem like to us, if it seemed like the Earth rotates around the Sun.

That depends on the worldview. Classical mechanics is definitely a crude hack. No wonder it does not work in microworld. It cannot be the thing. Is quantum mechanics the thing? It is not known. One has to believe and no number of successful calculations can give definite answer to the general world view question. If one believes that quantum mechanics is the thing, then one should be curious how the hunter-gatherer's mind reads it. 

No, Kåre,

You criticize me for saying that classical mechanics is the limiting case of v

Sofia> You criticize me...

I only tried to tell you how these words are used, at least by Coleman and others in the field (including me). You can find the 2x2 matrix classification I indicated in textbooks. It is with this meaning of  words you must interpret what Coleman says.

I did not think of the passing from quantum to classical as a physical process. Only as a mental one, depending of whether one wants to analyse the fine structure of the hydrogen atom, or the orbit of Saturn. But the process you mentioned is indeed an interesting example of how a quantum description suddenly seems to be turned into a classical one. 

In other cases, delayed choice experiments makes it difficult to pinpoint when a wave-packet is really "destroyed", instead of just being entangled with increasingly many, complex and massive objects. 

No, the classical mechanics of macroscopic bodies is not the non-relativistic limit of the relativistic mechanics of macroscopic bodies-that statement is, simply, meaningless, since the terms aren't defined.

Planck's constant and the speed of light are completely independent quantities. And ``macroscopic'' doesn't mean anything by itself. 

Classical, non-relativistic, physics, describes effects where quantities that have the dimensions of angular momentum  are large with respect to Planck's constant and quantities with the dimensions of velocity are small, compared to the speed of light. These approximations have consequences: the description involves solutions of the equations of motion.

In classical, relativistic, physics, the description, still, involves solutions of the equations of motion; only, now, these are different. In particular, it's not possible to describe a fixed number of particles in interaction, unless the motion is integrable. 

In quantum, non-relativistic, physics, the description involves all possible configurations, whether they solve the equations of motion, or not. 

So an observer is a quantum system, if it can be found in a superposition of states; it's a classical system if it can't. 

In quantum, relativistic, physics, there are further consequences and further constraints.

What matters is that classical, non-relativistic, physics is a limiting case of classical, relativistic physics: and a  completely different limit  of quantum, non-relativistic physics. The two limits don't have anything to do with each other. 

Said differently: a classical rigid body is a non-relativistic approximation of a classical, deformable body, that can be described, under certain conditions, in a relativistically invariant way; it's, also, a, totally different, approximation of a quantum rigid body.

A photon is a quantum object, because it obeys Bose-Einstein statistics and a relativistic object, because it is a massless particle. The former property is relevant when considering systems of many photons. However, when considering ideal experiments involving photons, then it's possible to limit the description to that of a non-relativistic, two-level system. The special properties of photons, as excitations of the electromagnetic field aren't relevant in this case. 

So it turns out to be irrelevant for the subject at hand that the particles used are photons-they could have been non-relativistic electrons in a magnetic field, that probes their spin, that's, in this approximation, just a label. 

If the detector doesn't click (``remains silent'') this implies something definite about the outcome, that's all. This means that the outcome of the measurement is described by a projection on the space that's consistent with the fact that the particular detector remained silent. 

There is no doubt that classical mechanics is obtainable as the limit v/c -> 0 from relativistic mechanics in the absence of gravity. (In the presence of gravity we need an additional weak-field limit.) These are both classical theories, they have no problem with assigning objective meaning to concepts appearing in equations; quantum mechanics is simply not involved.

The relationship between quantum mechanics and classical non-relativistic mechanics is much more complex.

In priniciple, it would be possible and meaningful to teach relativity as the basic theory without the intermediate step of Newtonian mechanics and then derive the latter as a particular limit. This is not true for quantum mechanics, because concepts of quantum mechanics (position, momentum) are based on classical mechanics, and without having them first, there is no viable interpretation of quantum mechanics. So you cannot meaningfully teach quantum mechanics first and then derive classical mechanics as a limiting case.

Concepts of quantum mechanics were developed, historically,  from those of classical mechanics-but this is history, not logic. Of course it's possible to learn quantum mechanics first and deduce classical mechanics as a limiting case-there's no logical obstacle. 

The interpretation of quantum mechanics doesn't depend on classical mechanics-it's the other way around. 

In the canonical formalism there's a phase space, whose elements are the states of the system, that's even-dimensional (for finite number of degrees of freedom); position and momentum are derived notions, whether in classical or quantum mechanics. 

For an infinite number of degrees of freedom things are correspondingly more involved, but it's the infinity of degrees of freedom, that's the issue, not whether the system is classical or quantum. Of course the distinction does matter. 

I think that it is a pedagogical mistake try to present classical mechanics as a limit of quantum mechanics or vice verse. If you do that you loose how they born and what was the main experimental needs for each one. Studying Kepler, Newton or Lagrange's mechanics is very different than do it with Bohr, Born, Heisenberg or Schrödinger. In the first case the motion of the planets where the main physical background using cartesian geometry and infinitesimal calculus. Meanwhile quantum mechanics born as the impossibility to relate electromagnetic radiation (ultraviolet catastrophe) with matter (Kirchhoff, Wien, Reyleigh, Planck). This makes them very different theories in Physics although they can be related obviously.

The same happens with special relativity theory which has born by seen that the electromagnetic waves motion was independent of the mechanical properties of the medium (no aether) and also independent of the state of motion for one observer. In fact Maxwell equations were no possible to be transformed by the Galileo group of transformations used in classical mechanics, and no body wanted to tell it even knowning the difficulties in optics (Fresnel) or electrodynamics (Lorentz, Poincaré).

Again, it is possible to put quantum mechanis and relativity in very fundamental form as has done Dirac and it has reached a wonderful development in quantum field theory, but all these theories have a very different phenomenogical context that would be a crime to avoid to teach to the students of Physics.

==> (Kåre) "I did not think of the passing from quantum to classical as a physical process. Only as a mental one, depending of whether one wants to analyse the fine structure of the hydrogen atom, or the orbit of Saturn."

Mental? Oh, Kåre! A nuclear reaction is intially a quantum process, but when it grows to bulks of material, it may become a nuclear catastrophe. (Do you recall Tchernobil?)

==> (Kåre) "delayed choice experiments makes it difficult to pinpoint when a wave-packet is really "destroyed", instead of just being entangled with increasingly many, complex and massive objects."

Yes, this is the problem. If the wave-packet represents a photon which ionizes atoms of the type present in the detector, and if the detector is ideal, then there should occur ionization in the detector. But if the detector produces no recording, something impedes a massive movement. (Suspect thing!!!)

The problem is, we would like to be quantum observers, but we are not.  

Raul> we would like to be quantum observers, but we are not.

Please explain how a quantum observer would be, which we are not.

I agree with Coleman's sentence: "we shouldn't try to explain QM in base of ClaM, but ClaM in base of QM", I cannot see the other way around, since  CM emerges from Quantum Physics.

Chemstry is all based on QM and cannot be explained otherwise unless we resort to the strange euristic tools of the 1800's.  As long as we remain in statistical mechanics (thermodynamics) which is still CM and Physical Chemestry which mixes it with QM, it is probably less difficult in this case to see how QM is more fundative and how the Chemestry emerges from QM.

All the electronics we know is strictly based on Fermi-Dirac statisics of the electron which is Quantum Physics. The electronic devices are based on components (transistors) of different kinds, relying on accurate and very complicated models (which I had to study) of solid state Physics, a branch of Quantum Physics. There we can see very clearly that the curves of the voltage and currents, for each device, follow exaclty the behaviours predicted by the Solid state Physics. Impossible to explain the tunnel effect classically. There is no classical counterpart full stop. Either it is accepted or it is refused, direct consequence of the  Heisenberg uncertainty principle. This is the most evident and widespread success of Quantum Physics nowadays including TCP/IP protocols implemented by exclusively by electronic devices.

"the issues with the passing from quantum to classical, by Feynman's theory."

I see instead very helpful the path integral approach of Feynman in explaining how reality we experience emerges from the quantum world. The resort to the "Action" is really genial and is the most satisfactory approach I know, merrying the macroworld CM with the microworld QM.

Regarding the example of the Sun, Colemean is a bit confusing, did not catch the right example I guess or messed things up..

The formulas provide a complete mathematical description and are the foundation for experiments. Classical systems are described by the solutions of their equations of motion-in phase space or configuration space-quantum systems by superpositions and measurements involve projections on the subspaces that realize the constraints, described by the measurement.  That's all there is. There aren't any theoretical ambiguities and there have been real experiments that can probe all aspects of the theoretical description. 

If the observer is, also, a quantum system, no problem: the complete system is a composite system of two subsystems and their superpositions can be described, also. If the observer is not a quantum system, this just means that its evolution is defined by its classical equations of motion. So an electron in an external electromagnetic field can be described by the Schrödinger or Dirac equation, as appropriate, while the gauge field, that defines the electromagnetic field, is taken as a fixed field configuration, e.g. a constant field. 

Or a classical particle in a quantum electromagnetic field can, also, be described. 

A many body system of electrons can be described by different limits of the Dirac Lagrangian where the fermionic fields are quantized, but the electromagnetic field is classical. And the case of classical particles interacting with a quantized electromagnetic field, also. No problem there, either. 

The technical ways these exercises can be solved involve personal taste-but the results of the calculations, namely the transition probabilities for the quantum system and the appropriate invariant quantities of the classical system don't depend on the technique used to find them. They only matter.

Any statement, in words, can and must be, checked against the consequences of the mathematical description. That's how it's possible to deduce that it is, indeed, possible to describe quantum detectors interacting with quantum particles and distinguish this case from that of classical detectors interacting with quantum particles and so on for the two other cases. That's, also,  how it's possible to deduce that the objections presented are not correct.

In the case at hand, as mentioned many times, the quantum particles can be described by two-level systems, describing the polarization states, while the detectors can, also, be described by two-level systems (``click'', ``didn't click''). If the detectors can't be found in superpositions, they're classical, if they can, they're quantum. If the polarization states can't be superpositions, one is dealing with classical objects, if they can, with quantum objects. 

One specifies an interaction, corresponding projection operators  and computes, by standard methods, whatever one wants. Projections imply that it's not possible to undo them-subsequent evolution and measurements take place in the subspace defined by the projection. 

Raul, Kåre,

==> (Kåre to Raul) "Please explain how a quantum observer would be, which we are not."

Feynman vs Coleman. Of course we are not quantum observers. I think that Coleman, brilliant as he is, didn't though use all his knowledge.

Just try to write the wave-function of a man. The action-functional divided by ħ, varies terribly quickly from path to path, s.t. Euler-Lagrange equations are suitable for the man's movement. In addition, the man's wavelength is a lot of orders of magnitude smaller than even 1 atom diameter. The wave-function description is meaningless.

It's not invalid-it leads to a probability distribution that's peaked about the classical motion, i.e. the classical states, with a width that's so narrow, that non-classical configurations are not observable, that's all. So one recovers, this way, the classical description, as stressed above. But there's nothing conceptually wrong. One writes the action for a body of given mass and computes the corrections to the classical trajectory.  This is in no way different than when computing finite wavelength corrections to geometrical optics effects-something that had been noticed already by Hamilton, in the 19th century.

(Cf. also  Guillemin and Sternberg, ``Symplectic techniques in physics'', contents described here; http://tocs.ulb.tu-darmstadt.de/50951491.pdf )

I have repeatedly commented here on the issue of a smooth transition from quantum mechanics to classical mechanics without any response except evading statements like: I shall read your article(s) as soon as I have time.

I am firmly convinced that central questions like: "Where does quantum mechanical randomness come from?" or "How can one explain the collaps of the wave function?" cannot be answered without the willingness to abandon the obsolete vision of an empty vacuum. Clearly, this is what our senses insinuate because of their limited capabilities. But on a microscopical scale it is wrong and it lacks a sense of scientific openess when people cling stubbornly to the familiar classical image of an empty space and blame departure from classical mechanics dubiously on the influence of the observer or "the measurement". As if the stability of a hydrogen atom, for example, could be traced back to some observer performing measurements on it. The zero point motion of particles, the property of liquid helium to stay "molten" down to the very lowest temperatures, reflect directly the presence of vacuum fluctuations. As soon as one accounts for vacuum fluctuations the Schrödinger equation renders itself derivable from Newtonian mechanics, and practically all of quantum mechanics follows. We have shown this in two articles which I recommend for further reading. They merely require  some familiarity with vector calculus.

L. Fritsche and M. Haugk, “A new look at the derivation of the Schrödinger equation from Newtonian mechanics”, Ann. Phys. (Leipzig) 12, No.6, 371-403 (2003)

L. Fritsche and M. Haugk, “Stochastic Foundation of Quantum Mechanics and the Origin of Particle Spin”, arXiv:0912.3442v1, Dec. 17, 1-47 (2009)

Maybe someone of the contributors to this discussion can talk herself/himself into reading at least some of the essentials.

Stam, I accepted your criticism about the word "invalid", and replaced it by "meaningless". I invite you to calculate the wavelength of a man of, say 70kg mass, moving with, say, 1m/s. And then, to calculate those correction to the classical theory you speak of.

Now, I see that you indicate some reference. Hmm! You see, as we talk here about Feynman's path integral and passing from quantum to classical, I would be very glad if I could find some material showing how one passes from the quantum Lagrangian, to the classical one. The quantum Lagrangian depends on the field function and its gradient, the classical Lagrangian depends on positions and velocities. Could it be that you know some work that shows the transition from one to another? I didn't find such things in the book of Feynman and Hibbs.

   Kare, my idea is that a quantum observer would be able to give a classical (i.e., deterministic) description of quantum phenomena. He would probably be of the same size as quantum objects, and would not be subject to the Uncertainty Principle (UP). 

   However, I have heard that the UP is now customarily violated in laboratory conditions (!). See the following reference for more information:

Hofer, W. A.: "Elements of physics for the 21st century". arXiv: 1311.5470v1 [physics.gen-ph] 8 Nov 2013.  

The calculation of the quantum fluctuations of the trajectory of a massive object about the classical solution is straightforward, as mentioned: the fluctuations are proportional to Planck's constant and inversely proportional to the mass of the object-that's, either from the Schrödinger equation or from the path integral. 

A quantum observer provides a quantum description and is subject to the uncertainty principle, as a matter of course.

A straightforward calculation shows that it doesn't make sense to state that the uncertainty principle is ``violated''. For a generic system, that's not a harmonic oscillator, the variance isn't sufficient to characterize the fluctuations, that's all. The identities get correspondingly complicated. 

Stam,

You say

==> "The calculation of the quantum fluctuations of the trajectory of a massive object about the classical solution is straightforward, as mentioned: the fluctuations are proportional to Planck's constant and inversely proportional to the mass of the object"

The velocity of the object also has to appear in the calculus, why only the mass? Truely speaking I am interested in the opposite direction, as Coleman said, from quantum to classical. But, anyway, where did you see that calculus? It is probably very simple but I am too much busy. Can you give me the reference? 

I think the usual example of  a detector (macroscopic) which gives an answer +1 (macroscopically) when a spin enters in a state with s_z=1 and similarly gives a answer -1 when the spin state has  s_z=-1 in the z direction.

What happens when a spin poalrised in the x direction enters? Unless you make approximations, the apparatus must (Schroedinger equation is linear) wind up in a coherent superposition of two different macroscopic states. Of course, I am not unaware of the entire decoherence approach, and consistent histories, to get rid of this irritating fact. Yet we do see that this is the case: given the apparatus' behaviour for z-polarised spins, the behaviour for an x-polarised spin is determined. We have good arguments why we, as human beings, cannot actually see this happen. But we have no way to deny that it happens.

In such cases, it does not seem to me that we may readily say, that the fluctuations of macroscopic objects are small. Rather, they are macroscopic.

Sofia> It is probably very simple but I am too much busy.

There are some pretty nice formulas in the literature, one for the semiclassical time-dependent propagator (the one computed by the Feynman path integral) is attributed to a 1928 paper by van Vleck (later followed up by Dirac and Feynman, leading to the path integral), another for the energy dependent propagator derived in a 1967 paper by Martin Gutzwiller.  For explicit interpretation I would not say they are very simple, and their detailed derivations are not that simple. For practical purposes they are just some real prefactors, multiplying the phase factor determined solely by the action of the classical path.

Anyway, such more explicit formulas do not change anything to the (Schrödinger cat) problem mentioned by Francois: Consider a spin-1/2 particle moving in a magnetic field, rather heavy so that its position follows a classical path quite closely, but different classical paths for Sz = +1/2 and Sz = -1/2. Which means that the particle will follow both paths (each one mapping out a consistent history), with equal amplitudes, when they start out with say Sx = +1/2.  As described by quantum mechanics they go the left path and the right path. What leads to the conclusion that they go the left path or the right path? One answer is that the wave function do not describe individual systems, only ensembles of identical systems. Which triggers the question (best ignored) of how then we should describe the reality of individual systems.

==> (Kåre) "Consider a spin-1/2 particle moving in a magnetic field, rather heavy so that its position follows a classical path quite closely, . . ."

It doesn't smell good, "there is something rotten in Denmark". If your particle is so heavy that it follows an almost classical trajectory, there is a question whether QM is able to describe it properly. You see, QM and classical mechanics are extremal cases. One is for extremely tiny particles, the other for big objects. Stam wrote in his 1st comment on this page

==> (Stam) ". . . the quantum fluctuations of the trajectory of a massive object about the classical solution . . . the fluctuations are proportional to Planck's constant and inversely proportional to the mass of the object . . .".

So, there is a wide region "in between", not quantum, not yet classical, e.g. objects not so big and not so small. "Fluctuations" is a nice word, but in fact, how do such objects behave? Does the superposition of the wave-packets |z+> and |z-> still persist for them, or the object is located in one of the wave-packets?

==> (Kåre) "when they start out with say Sx = +1/2.  As described by quantum mechanics they go the left path and the right path. What leads to the conclusion that they go the left path or the right path? One answer is that the wave function do not describe individual systems, only ensembles of identical systems. . . ."

(Ahhh, Asher Peres with his unclear words!!!) In interference experiments, it couldn't happen that dark fringes are avoided, if it wouldn't be that each particle and particle avoids those fringes. On one single system whose preparation is not known to us, it is true that we can't guess the wave-function. However, when we prepare an experiment, we prepare each trial and trial, i.e. each individual system.

As to "and" vs. "or", the "or" wins. You already know from Sciarrino, who knows from Peres, that when we send a wave-packet on a beam-splitter, we get at the output:

|ψ> = |1>transmitted |0>reflected + i |0>transmitted |1>reflected 

It's an entanglement, and we either get the 1st coupling, |1>transmitted |0>reflected, or the 2nd coupling, but not both, because the energy conservation forbids here |1>transmitted|1>reflected.

So, when we get the coupling |1>transmitted |0>reflected, Peres' notation says that the detector on the reflected path sees nothing. The question remains what happens with the 2nd coupling. Does it disappear? 

``but in fact, how do such objects behave? Does the superposition of the wave-packets |z+> and |z-> still persist for them, or the object is located in one of the wave-packets?''

We know, from experiment, that very large objects indeed (molecules with molecular weight > 1000) can in fact interfere quantum mechanically. So it does not appear likely that there really is an upper cutoff where quantum mechanics actually fails. That is, however, the point of view of Ghirardi, Rimini and Weber, and it might still be true.

François,

if you have in mind the experiments with fullerenes, they are not an example of macroscopic objects interfering. If you read those experiments - you can find them in quant-ph eventually under the name Talbot-Lau interferometry - you see that they are brought to extremely low velocities, so as to ensure a wavelength grater than their linear dimensions.

About cutoff, it is not sharp, but to see better what I talk about, I invite you to calculate with which velocity a piece of chalk of, say, 50mg, should move in order to ensure the wavelength greater than its diameter, say 5mm. You''ll find that the required velocity is 2.5x10-25cm/s. So, for moving by 1cm, this piece of chalk has to spend 4x1024s ≈ 1017years.

Please remind me what is the age of the Universe.

However, the problem we talk about here, as well as in the thread about von Neumann's collapse postulate, is another one: consider an experiment in which we prepare single particle wave-packets, one after another and well separated in time. Each wave-packet passes a beam-splitter (BS). The wave-function at the output of the BS is of the form

|ψ> = |1>transmitted |0>reflected + i |0>transmitted |1>reflected.

Thus, if we place detectors on the way of both the transmitted and the reflected output, in half of the trials we get 1 transmitted particle and 0 reflected, and in the other half, vice-versa.

We have a SYMMETRY BREAKING between the two possibilities.The question is WHAT BREAKS THE SYMMETRY? What chooses in some trials the coupling  |1>transmitted |0>reflected  and in other trials the other coupling? You see, Coleman gave a beautiful speech, but it's not clear what was his conclusion. He said something that maybe we, the people, are quantum systems. He should better do the calculus with the piece of chalk.

I did not say fullerenes were macroscopic. In fact, I referred to later experiments, where they got up to 10'000 in molecular weights and higher (the number named in the previous post in in error). My claim is that these objects have a very fair claim to being in the mid-zone you speak about.

As to their having been brought to very low speeds, according to my source (arXiv:1310.83483v1 [quant-ph]) the molecules were going at 85 m/sec, about 300 km/hour: not really a snail's pace. As to the de Broglie wavelength, they make the following remark:

``All molecules of the library were evaporated at a temperature of about 600 K. We selected the velocity class around v = 85 m/s (vFWHM = 30 m/s) corresponding to a most probable de Broglie wavelength of approximately 500 fm. This is about four orders of magnitude smaller than the diameter of each individual molecule.''

So it appears to be possible to get particles to interfere even when their de Broglie wavelength is quite a bit smaller than the size of the object.

Of course, I agree that no chalk will probably ever be brought to interference. But the issue is, whether this is a mere issue of practicality, or whether there is a cutoff size where QM truly ceases to be valid. I believe the above experiments give a strong hint that it may be a bad idea to hope for an eventual breakdown of QM fundamentals.

``We have a SYMMETRY BREAKING between the two possibilities.''

Do we really? It altogether depends on whether you believe that you can, in principle, argue for measurement devices that really fail to obey quantum mechanical laws. Maybe it can be done. To my mind, however, all that has ever been said on the subject is, quite correctly, that it is easy to justify why we cannot see macroscopic devices in macroscopic superpositions. And the experiments I mention do indicate that the limit of validity of QM goes, at least, up to quite high masses. Can I store a measurement in a molecule of molecular weight 10'000? I guess it should not be impossible. But we have seen that such objects are still hopelessly quantum, at least when handled with due care. So a measurement apparatus can be susceptible of quantum behaviour. This, by the way, is very much related to the beautiful paper by N. Mott mentioned by Kare and yourself above.

This does not mean I have an opinion. I am mainly sharing my confusion.

François,

What you quote from that article, i.e. a velocity of 85 m/sec, is transversal, or longitudinal velocity? It seems longitudinal. As far as I remember, on longitudinal direction the particles in Talbot-Lau experiments behave classically. The interference is based mostly on their transversal behavior. Look at fig. 3a showing the fringes: the period is cca. 300nm, while, as far as I understand from the text, the diameter of the molecule is only a few nanometers. So, it's O.K.  

But, you can't have a meaninful conversation on Talbot-Lau iterferometry - in my modest opinion - without being familiar with its theory.

I'll continue later, it's almost morning in my country.

(continuation of my post on page 3)

==> (François) "the experiments I mention do indicate that the limit of validity of QM goes, at least, up to quite high masses."

You mix things. I repeat what happens at a beam-splitter:

(1) |ψ> = |1>transmitted |0>reflected + i |0>transmitted |1>reflected.

Both couplings appearing in the RHS impinge on detectors. There is nothing special about anyone of them. Neither the transmitted wave, nor the reflected wave, has some special advantages over the other. It's from this point of view that I say that there is SYMMETRY between them. And though, in each trial only one of these couplings is selected. This is what I call SYMMETRY BREAKING. Only one of the copulings is selected, and without any appearent reason. This selection has nothing to do with mass. In both the transmitted and reflected wave the mass of the particle is the same.

The situation is beautifully illustrated in Mott's article. A s-wave of α-particle exiting a radioactive nucleus has spherical symmetry. Though, the track of α-particle through an ionizable gas, is, in a given trial, a straight line, is not a spherical cloud. Again, symmetry breaking. Here too, mass is not an issue, the mass of the α-particle is the same whatever straight line is picked.

In quantum field theory we have all sort of cases of symmetry breaking. But in the non-relativistic QM the symmetry breaking occurs in the presence of macroscopic detectors, or macroscopic objects. Each point on the track through the ionizable gas is an agglomeration of ions.

Yes it is posible to go from QM to Classical Mechanics, but you

have to be carefull how you do it. For example taking h to zero in

Schrödinger equation Leeds to nonsense.

Generally if the action S is very much larger than h , then you can treat it Clasically. But vice versa you cannot deduce QM from Clasical theory, so Colemans remarks are quite true.

Well, but Mott's paper shows the point I make: as long as you stay in QM, the wave function is, and remains, spherically symmetric. It is a coherent, isotropic superposition of ionised tracks. Of course, if you then ask: which track actually materialised? you are asking a classical question, and you need to face the measurement issue. But if you do not ask that, if rather you ask, as Mott does, what is the quantum state of the quantum system containing both the alpha particle and the ionised gas, then you get an answer which, while indeed spherically symmetric, contains multiparticle correlations which express the existence of tracks.

As to the mass question: I did not make myself understood: it is not the mass of the test particle, undergoing quantum dynamics, which is at issue. Rather, it is the mass of the measuring apparatus. Could we make a measuring apparatus of molecular weight around 10'000? If so, could we confidently count on that measuring apparatus being classical? Of course, if one goes far enough, things are straightforward: when the measurement is carried out by a graduate student, we may feel moderately confident that we are in fact dealing with a classical system.

Now the issue is, in my mind, is it always meaningless to ask what a massive particle will do and apply quantum mechanics? The reason I do not think we can always simply say ``use classical mechanics for large objects and stop worrying'' is precisely this: the kind of experiments I mentioned suggests you cannot always disregard quantum effects, even in very large systems indeed: 10'000 molecular weight is getting in the order of magnitude of proteins.

SPONTANEOUS SYMMETRY BREAKING

https://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking#Dynamical_symmetry_breaking

For spontaneous symmetry breaking to occur, there must be a system in which there are several equally likely outcomes. The system as a whole is therefore symmetric with respect to these outcomes. However, if the system is sampled (i.e. if the system is actually used or interacted with in any way), a specific outcome must occur. Though the system as a whole is symmetric, it is never encountered with this symmetry, but only in one specific asymmetric state. Hence, the symmetry is said to be spontaneously broken in that theory. Nevertheless, the fact that each outcome is equally likely is a reflection of the underlying symmetry, which is thus often dubbed "hidden symmetry", and has crucial formal consequences.

``a specific outcome must occur. ''

That is *precisely* what does not happen until you perform a measurement, which is why, in Mott's paper, such a symmetry breaking does not occur. Similarly, I believe we have no symmetry breaking as long as the measuring devices are viewed as quantum objects

==> (François) "as long as you stay in QM, the wave function is, and remains, spherically symmetric."

I disagree with your approach. What happens with the α-particle is not up to us, i.e. if we stay or not, with the QM. Please see below.

The α-particle has indeed a spherically symmetric wave-packet inside the nucleus. A typical energy of α is 4MeV. That means a very short wavelength: mα = 6.64×10–24g, Eα = 4MeV = 6.64×10–6erg, implies a linear momentum pα = 0.939×10–14g cm/s. From this, one gets λα = 2πħ/p = 6.28×10–27/0.939×10–14 = 6.7×10–13cm. This is the order of magnitude of nuclear radius.

Now, out of the nucleus, as Mott explains, the Schrödinger equation has to be supplemented with an interaction term between α and the electron in the atom, probably something proportional with ~ 1/|rα - re|. This term is not spherically symmetrical in |rα|.

Though, I am not wise enough to say how is chosen a certain straight line in a specific trial.

==> (François) "if you then ask: which track actually materialised? you are asking a classical question, and you need to face the measurement issue."

Well, I don't use to disguise my ignorance under such words as "a classical question". In the recordings, we see a certain straight line. We know that the Hamiltonian has a symmetry breaking term, but that is not enough to help us in predicting which line is picked.

(To the rest of your comment I'll refer later.) 

If you read Mott's paper with care, you will find that he in fact calculates the wave function, in an appropriate approximation. It is not quite easy, but the point is the following: the resulting wave function *is* isotropic in the configuration space containing all the coordinates of the ions and those of the alpha particle. There is no symmetry breaking, nor could there be any mechanism for such an effect, since we are dealing with a very elementary system, a linear equation, in finitely  many degrees of freedom.

On the other hand, let us ask for the *conditional probability* that, if an atom at position R is ionised, then another atom should be ionised at position R'. This probability is indeed strongly peaked if R and R' lie on the same straight line, and negligibly small otherwise. Mott's main point, as I understand it, is that this is not in contradiction with isotropy of the wave function.

``This term is not spherically symmetrical in r_alpha''

Indeeed it is not. But it is *jointly* symmetrical in the atom and alpha particle position, which is the point. It is the global wave function, including all variables, that is symmetric. To quote Mott:

`` The difficulty that we have in picturing how it is that a spherical wave can produce a straight track arises from our tendency to picture the wave as existing in ordinary three dimensional space, whereas we are really dealing with wave functions in the multispace formed by the co-ordinates both of the alpha-particle and of every atom in the Wilson chamber.''

(quote at the end of the first page)

==> (François) "If you read Mott's paper with care"

I have the custom to read and check each equation. Do you? This is why it takes me time to read articles with a long mathematical treatment. So, I am far from finishing Mott's article.

But, there is more. You see, there is the case of lower dimensionality, with the two wave-packets at the output of the beam-splitter. Here, one doesn't have to explain how is chosen one straight direction from a continuum of 4π. Here the problem is how is chosen one of the two options, either |1>transmitted |0>reflected, or |0>transmitted |1>reflected. In addition, the two wave-packets, the transmitted and the reflected, are space-separated. So, I don't read Mott's article with big enthousiasm, because I am not convinced that it helps to understand the case of two space-separated wave-packets.

==> (François) "it is *jointly* symmetrical in the atom and alpha particle position"

Please don't foget that the Hamiltonian contains the kinetic terms in which the α and the electron's coordinates appear separately, not as |rα - re|.

But, I indeed have first to advance with Mott's article. If I had an article treating the output of the beam-splitter I were happier.

Well, it's late in my country, let's continue tomorrow.

OK. Just for rotational symmetry: if you rotate all positions of all atoms, as well as all the position of the alpha particle, as well as all corresponding momenta, then the Hamiltonian remains invariant.

==> (François) "if you rotate all positions of all atoms, as well as all the position of the alpha particle, as well as all corresponding momenta, then the Hamiltonian remains invariant "

No, it doesn't go like that. Spherical symmetry is something else, namely isotropy of the environment, for the α, For the α particle all the directions should be equivalent. Well, they are not, take the Hamiltonian and rotate rα by some angle, while keeping re fixed. You don't get the same Hamiltonian because |rα - re| changed.

Now, let's talk tomorrow!

I like the concept of spontaneous symmetry breaking as introduced by Sofia. But I think it should be considered as a metaphor, since there does not necessarily have to be any symmetry present to make a "spontaneous" selection of one from many options.

Within QFT there is also a precise mathematical way to phrase this: The two (or more) parts of an originally innocent superposition start interacting with increasingly large and complicated things.  Eventually each part of the superposition evolves into Hilbert spaces which lie in different superselection sectors. Then it makes no mathematical sense to consider the system as a superposition any more.

But, to me this is just a mathematical way to express the many-worlds scenario. What makes us experience the world as built over one specific superselection sector? Or rather, what happen to all the other sectors? Is there another version of me in all of them? What prevents me from choosing one where I can live a more successful and happy life? Or one where a different US president was elected?

In classical physics, one definite choice is selected during spontaneous symmetry breaking. For a dynamical description I think a non-linear evolution is required (how can one get around the superposition principle otherwise?). So, should we introduce nonlinearity in a beyond-Quantum Mechanics model? With entanglement and post-selection and all that, I don't this can be made to work without giving up locality.

in waves, locality seems not absolute. But it seems difficult to know what drives the original parts of superposition to interact with each other largely and largely and finally let the original symmetry to break into different symmetries.

   Dear Sofia,

  You have a very interesting general discussion but, sorry, without enough accuracy in the concepts. For instance, when you said:

   "For spontaneous symmetry breaking to occur, there must be a system in which there are several equally likely outcomes. The system as a whole is therefore symmetric with respect to these outcomes".

   This is not the condition for having spontaneous symmetry breaking:

1. You only need to have a symmetry of one action and no "there are several equally likely outcomes".

2. The vacuum must be degenerate and without following such symmetry.

   Dear Kare,

   Your sentence:

"Eventually each part of the superposition evolves into Hilbert spaces which lie in different superselection sectors. Then it makes no mathematical sense to consider the system as a superposition any more".

   I think that it needs much more explanation because the Hilbert spaces ( or other functional spaces) cannot depend of the "different superselection sectors". Behind the Hilbert equations are the motion equations of the system associated to the Hamiltonians which cannot be splitted arbitrarily. Perhaps you are speaking of the possibility of chosen different basis, but in such a case you are not avoiding the "superposition".

Dear Daniel,

I just quoted from Wikipedia, because it seemed to me correct. You say

==> "You only need to have a symmetry of one action and no "there are several equally likely outcomes" "

I do not understand you. I would be glad if you'd ellaborate a bit. Also, what you mean by "action"? I know only one concept of action in physics, namely the one appearing in Maupertuis' principle of least action. 

Of course that the problem is that we have in the wave-function, a couple of possible outcomes. This is our problem, we need to explain why only one of them appears, and which one of them.

Next, you say

==> "The vacuum must be degenerate and without following such symmetry."

Are you trying to say that for explaining why is chosen one of the components of the wave-function, we veed the vacuum? All this talk here is within the 1st quantization. Are you trying to say that the 1st quantization is not enough when we examine the measurement problem? I am open in front of any opinion, just please be clear!

Let me say this: there were trials in the past to put the picking of one component of the wave-function or the other, on the account of the vacuum fluctuations. Well, those trials failed totally. You see, sometimes the wave-function is not a fair superposition of the eigenstates of the measured operator. I mean, some of the eigenstates in the superposition appear with a bigger value of the absolute square amplitude, and other eigenstates with a lower value. Nobody explained how to tailor the vacuum fluctuation so as to fit those differences.

With kind regards!

  Dear Sofia,

 1. I do not understand you. I would be glad if you'd ellaborate a bit. Also, what you mean by "action"? I know only one concept of action in physics, namely the one appearing in Maupertuis' principle of least action.

    I'm sure that you and all the people of this forum knows what is the action, but let me to remember it. It's the integral (classically) of the Lagrangian between two instants of time. Physically its variation is zero and this provides the motion equations through the Euler-Lagrange equations. Besides its symmetries carry associated the Noether's conservation laws of the system. For instance, the energy conservation is due to the invariance under time translations of the action.

 2.Of course that the problem is that we have in the wave-function, a couple of possible outcomes. This is our problem, we need to explain why only one of them appears, and which one of them.

The wave-function is not obtained from an action variation and least taking into account the breaking of symmetry of the action. They are just eigenfunctions of a Hamiltonian. 

 3.Are you trying to say that for explaining why is chosen one of the components of the wave-function, we veed the vacuum?

   No, obviously not. Only tried to say that the spontaneous breaking of symmetry needs to fail as symmetry of the vacuum,i.e. the fundamental state.

 4.All this talk here is within the 1st quantization. Are you trying to say that the 1st quantization is not enough when we examine the measurement problem? I am open in front of any opinion, just please be clear!

  First quantization or second quantization are not related with the symmetries too. They only are related with the functional representation for the operators, usually Hilbert or Fock.

  5. Let me say this: there were trials in the past to put the picking of one component of the wave-function or the other, on the account of the vacuum fluctuations. Well, those trials failed totally. You see, sometimes the wave-function is not a fair superposition of the eigenstates of the measured operator. I mean, some of the eigenstates in the superposition appear with a bigger value of the absolute square amplitude, and other eigenstates with a lower value. Nobody explained how to tailor the vacuum fluctuation so as to fit those differences

     From my humble point of view this fail is logic. The collapse is due to the reduction of superposition states to one single one. This is made when the measurement happens, but this is not related with any symmetry and least with a spontaneous breaking of symmetry which is with a very accurate meaning in physics. This is an old problem of quantum mechanics that Heisenberg tried to solve using the concept of observable and Einstein put as one of the main criticisms to quantum mechanics as a complete theory (Bell identities and so on) ............

==> (Kåre) "Within QFT there is also a precise mathematical way to phrase this: The two (or more) parts of an originally innocent superposition start interacting with increasingly large and complicated things.  Eventually each part of the superposition evolves into Hilbert spaces which lie in different superselection sectors. Then it makes no mathematical sense to consider the system as a superposition any more."

I quite join Daniel's comment, though, with some additional objections (rather, lamentations). Why do we need QFT? Please see, people have sometimes the custom that, when a problem is hard to solve, to complicate it even more and blame the difficulty of the problem, on the nebulousness of the complications.

I don't say that somebody here does this, I just say that we have to be cautios and not fall into such procedures. So, I repeat, which reason we have to mix QFT here?

In Mott's problem with the α-particle and the ionizable gas, the α has a symmetrical Hamiltonian inside the nucleus, so an isotropic α-wave is emitted, exp(ikrα)/rα, where rα = |rα|. Withing the gas, an interaction term ~ 1/|rα - re| is added to the Hamiltonian. This fact doesn't go as far as to tell us which direction would be chosen for the α, just tells us that the isotropy is broken. On the other hand, with small bulks on ions gathered at each point of the α movement, thoses bulks are classical objects, not quantum - and Feynman told us what happens with them.

Though, a ring is missing in my chain of explanation. NOTHING in whatever I said, indicates which direction the α chooses. The fathers of the anti-hidden variables theorems, would say that we have no deterministic hidden variables.

==> (Kåre) "So, should we introduce nonlinearity in a beyond-Quantum Mechanics model?"

Ehhh! Ghirardi introduced non-linearity. It didn't help much.

==> (Kåre) "But, to me this is just a mathematical way to express the many-worlds scenario. What makes us experience the world as built over one specific superselection sector? Or rather, what happen to all the other sectors? Is there another version of me in all of them?" 

Ayyyy! You are a classical object, not quantum, you have a well-defined trajectory - see Feynman - and with Great God's help, stay in THIS world!!!! As to either many worlds, or true collapse with erasing of the other components in the quantum superposition, hmmm - please WAIT . . .

With kind regards!

Daniel, If one symmetry corresponds to a wave function described by the corresponding motion equation?

   Dear Jun,

  The standard symmetry associated to the wave functions (not a symmetry of the wave function, the action or the motion equation) is the U(1), because the wave function is only an amplitude of probability. Thus measurement doen´t work on the wave function but on its square directly related with the density of probability (Max Born mechanism). Notice that this symmetry belongs to the abelian group of gauge transformations to share with the gauge transformations of the electromagnetic potentials.

   One problem with quantum mechanics or quantum field theory is that the motion equation is not related with the action but with a partition function of the action. The partition function has a measure of the integral which could be associated to sums of trajectories and no to trajectories themselves. This means that topology is included in the equation of motion at difference of what happens in classical actions (chiral anomalies with instanton solutions, etc..calculated with a 2 Chern class or using Atiyah-Singer theorem). Thus the concept of locallity is lost and this was the main difficulty behind Einstein's criticism or the one or Bohm.

   Thus it is necessary to take care when we speak on the symmetries and wave function as if we could employ the intuition straightforwardly. This is not so simple, at least for me.

Daniel,

I wonder whether you read my comments, because you don't answer to them. 

I also wonder whether you ask yourself whether people understand your explanations. They are good, however, I am afraid that they require a level of knowledge that not everyone possesses. I asked you in another comment of mine to ellaborate a bit, but you didn't, and I renew my advice here. Please be more explicit in your last post. Can you?

==> (Daniel) "The standard symmetry associated to the wave functions (not a symmetry of the wave function, the action or the motion equation)"

All the discussions until now were about the symmetries in the Hamiltonian. Which symmetries in the wave-function, or action?

==> (Daniel) "Thus measurement doesn´t work on the wave function but on its square, directly related with the density of probability (Max Born mechanism)."

It's not square, it's absolute square. There is big truth in what you said, but do you know why? Does anybody know why? Because apparatuses are sensitive to the intensity of the waves illuminating them. While the mechanism producing such an intensity is based on summing the amplitudes contributions to the respective results, the apparatus is sensitive to the intensity.

==> (Daniel) Notice that this symmetry belongs to the abelian group of gauge transformations to share with the gauge transformations of the electromagnetic potentials."

I fail to understand what you talk about. Why do we need gauge transformations and electromagnetic potentials in this thread?

Sofia> Ayyyy! You are a classical object, not quantum

Surely not! There is absolutely not chance that life could exist without quantum mechanics. Actually, for reasons exactly opposite to the possibilities of making superpositions! I.e., because of the required stability and reproducibility furnished by quantisation. Kind of strange, isn't it? 

   Dear Sofia,

   I don't want to disturb you and you can follow. Please forget my posts!

Daniel,

I appreciate your posts. They don't disturb me! To the contrary.

You see, there come a couple of comments at a time, and I missed one of yours. I react to it now. I agree with you, just I want to clarify an issue: you say

==> "The collapse is due to the reduction of superposition states to one single one. This is made when the measurement happens, but this is not related with any symmetry and least with a spontaneous breaking of symmetry which is with a very accurate meaning in physics."

Let me explain: Coleman refered in his lecture to Mott's article which poses the problem why an α-particle leaving a nucleus as s-wave, is recorded as if moving on a straight line. Well, indeed the interaction with the ions in the chamber breaks the symmetry that characterized the Hamiltonian inside the nucleus.

It's Coleman who made a connection between this problem, and the collapse. I am not sure whether the connection is correct.

In what regards the so-called collapse of the wave-function, I still claim that it's a symmetry breaking, but in a different sense. Imagine the simplest case when the wave-function is a superposition of the eigenstates of some operator, each eigenstate with the same probability. Then, in a given trial of the experiment, all of them should be chosen. But it's not what happens, only one of them is chosen.

Dear Kåre,

==> "There is absolutely not chance that life could exist without quantum mechanics."

You asked the question whether there may be versions of you in other worlds. It's to this question that I answered telling that you are a macroscopic object. To be practical, please take a walk in your garden and calculate your wavelength. Then, please be so kind and tell me whether you can be in a superposition of states.

The atoms in our bodies are quantum systems in what regards their internal movements. Moreover, people that work many hours under sun without appropriate clothes get dark color - these are also quantum processes. But a copy of you in another world should be a man moving as a whole, not split into separated atoms. To spli him into atoms means to destroy him.

Dear Daniel,

Thank you for your teaching. I know little about quantum physics though I have interests.

I try to understand your post. There are Pauli matrices for spin 1/2. So there is topology. if right? Imagine to inverse spin, there maybe difference of actions on each axis. So the chiral current is not conserved and so the anomalies. If relevant? 

Sofia> But a copy of you in another world should be a man moving as a whole,

Yes, absolutely! That is the sort of consistent histories which in principle can be derived from quantum mechanics (as already outlined by Neville Mott), like one in principle can derive that the superposition of different histories have no observable consequences, so that the density matrix for an originally pure state can FAPP (for all practical purposes) be treated as a mixture.

And now I can look around to deduce that the history branch where I went outside in the garden to weed dandelions have not materialised, while the branch were I waste time on ResearchGate has. I am sure my neighbours wonder what happened to the first branch -- I will let KvZ explain that to them, in a lecture on Quantum Mechanics.

==> (Kåre) "And now I can look around to deduce that the history branch where I went outside in the garden to weed dandelions have not materialised, while the branch were I waste time on ResearchGate has."

I can tell people to calculate their wavelengths, but it's up to them whether they do it or not.

Also, I do not force people to join my research, s.t. I can't understand why do you throw me in face that you waste time on RG.

   Dear Jun,

   Sorry perhaps Sofia is right and I have explained me not well enough (it is difficult because the things of the subject in this thread are not very well defined). Sorry in advance!

1. Having spin is not a condition to have non trivial topology at all. Thus the Dirac matrices ( Majorana, Weyl,,,,,) are not showing any kind of non trivial topology.

2. What I tried to say is that the motion equation in quantum mechanics (i.e. Schrödinger equation) is not obtained trough an action (i.e. using a Lagrangian as in classical mechanics) but it employs a partition function where the action appears in one exponent.

3. In QM you have not defined trajectories in the space-time, thus the definition of the action needs to have been done through a partition function (with similarities of quantum statistics) where there are two integrals:

a- trajectories with time as the guiding parameter as the usual action (using a Lagrangian).

b- on the trajectories themselve. This needs to introduce a new measure of the this integral which doesn't needs to transform under the symmetries of the action. The spontaneous breaking of the chiral symmetry has its origin in this mathematical quantum behaviour.

c-The solution of the motion equation is through a Green function ( for instance the Feynman propagator) and boundary conditions which can induce non trivial topological solutions.

d- The wave functions relate each other, point by point, through these propagators, which can be also associated to non local information. For instance, you have can have different vacuums connect through wave functions using instantons. This is a pure topological solution ( i.e. non local and therefore non dependent of the point). Its calculation come using the two chern number associated to a curvature. Thus this is mainly used in QFT where the degrees of freedom are infinite instead in QM where they are finite.

e- thus it is a wasting of time to write states dependent of points (local) for knowing the wave function collapse in this case (and other more).

3. Dear Jun, let me tell you that my intention is not to teach anything here. My aim was just to try to help people that (from my humble point of view) were using as granted non well defined concepts as spontaneous breaking of symmetry, symmetry of a wave function or assuming the translation of the wave functions whitout taking into account more basic concepts as the possibility to have non local wave function transport, physical meaning of the measurement in QM or QFT, phase space associated to the motion of states and so on.

Kåre, can your cat bash classical mechanics a little? There are too many people that think it is god given. 

Dear Daniel,

Yes, somethimes you go too much quickly ( for me at least). For instance you say,

==> "e- thus it is a wasting of time to write states dependent of points (local) for knowing the wave function collapse in this case (and other more)."

Here I lost you. What you mean by states dependent of points? Which points, in which space? In the real 3D (or 4D) space?

Also, what you mean by "nonlocal wave function transport"? Maybe I know the phenomenon but not under these words. So, can you give an example?

With best regards!

   Dear Sofia,

   I am speaking in the case that you have topological solutions for the motion equation. For instan, see

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

Dear Daniel,

I am aware of that Wikipedia article. But I don't see what it has to do with the wave-function "collapse". The word "collapse" seems to hint that some part of the wave-function disappears (there is a question of mine on RG about what meant von Neumann by "collapse".)

What you think that instantons may have to do with the collapse? Or, do I read your words incorrectly? Can you be clearer?

With kind regards,

Sofia

I think that the study of such collapses or measurements or perhaps collisions is very important-

It leads to irreversibility and the arrow of time. Also to get a finite coherence length for quantum phenomena in a noisy environment.

   Dear Sofia,

   I'm not at all specialist in this issue and I'm only a theoretician that has curiosity on fundamental physics. This topic is obviously one of these subjects that it seems that we face every time that we start to teach every course. Please don't consider me with more knowledge than that and let me try to summarize my "humble opinion".

   It was von Neumann who reduced the wave function transport to two processes:

1. The wave function follows the Schrödinger equation in a deterministic form an even a local conservation of the "probalistic currents" (always assumed out of the interaction) can be stablished. This is a nice behaviour!

2. There is a non-deterministic, non-local and non-unitary behaviour when the "wave-function interacts with the enviroment". This is what happens in measurement and finally is the important physical output!

   The problem is how could be possible to explain seriously that Schrödinger equation did something out of its control? How is possible to say that in the functional space associated to the Hamiltonian all the possible superposition of basis can be reduced only to one in a single state?

   What is my humble solution (sorry, much better opinion because I didn't). Leave as correct the 1 assumption of von Newmann but to add that the interaction introduce new BOUNDARY conditions obligen to find non local solutions or topological solutions. One of them is the instantons that I have shown you for connecting different vacua but obviously it would be go one step further for concentrating in this concrete problem.

Dear Daniel,

Let me add one small ingredient to what you said. Please see this wave-function, simply to prepare by means of non-fair beam-splitters

(1) |ψ> = [1/(q2+1)][ |1>_t |0>_r + i q |0>_t |1>_r ]

where the subscript "t" means transmitted, and "r" means reflected. In this wave-function the reflected part is q2 times more intense than the transmitted part. Thus, we get q2 times more recordings at the output "r" than at "t". Assume also that we perform the experiment with different beams splitters, i.e. we change the value of q from one trial to another.

What I want to say with this, is that in "which way" experiments, no environment state, or vacuum fluctuations, or microscopic states of the detector, determine which wave-packet, "t" or "r", gives the response. The decision is taken within the wave-function. The detector microscopic states, or the vacuum fluctuations, etc. have a distribution of their own, not a distribution of 1 vs. q2 automatically adjusting itself to the value of q when we change the beam-splitter.

   Dear Sofia,

   Go ahead! Put one experiment with a good device to work and identify q as a physical parameter. If you can find this deterministic behaviour this would be a good achievement without any kind of doubt. But I am afraid that the things are not so simple. One of the problems is that when you make the experiment, part of your wave function is seat down here with me and as soon as you told me your expemental result, suddenly is going to disapper. Only for receiving the email and to pass the photons to my eyes and my brain could identify it. Miracle!

   This is a parallel physical process as the entropy in thermodynamics. This was an old problem that nobody knows at present how the second principle of thermodynamics follows. This is just a physical fact!

    What is important is that von Newmann gave also a definition of entropy using the concepts of quantum mechanics and he found that the information entropy of a system is the amount of "missing" information needed to determine a microstate, given the macrostate. This was more elaborated by Shannon for directly relate it with the information theory of the transmission of one message.

   In my humble opinion, there is a lack of definitions clear which allow to distinguish the distribution of the density of probability in the space-time and this must have one component independent of the coordinates chosen or the base taken for the states. This is purely topological field that is out of the assumptions of special relativity or general relativity where the finitness of the transmission of information is taken as physical axiom.

Dear Daniel,

Yes QM is complicated and difficult and is essentially different from classical physics. We must be careful when applying intuition. Thank you.

I think when we rotate spin, Dirac matrices describe how this spin state is achieved. So this means a topology between x,y,z.

Yes, there is not trajectory in QM like that in classical physics. The wave function in QM is probability wave function. Thank you for your explaination of line integral and partition function in QM.

I understand the second 'trajectory' or wave function you mentioned perhaps corresponds to trajectory described by spinor. And there is symmetry in spinor which does not changed with action.

Perhaps as the said topology exists, chiral currents are not conserved.

The wave vector can be generalized by vector potential and Dirac matrices, how the wave function move is difficult. 

Perhaps I have known some more of concepts listed. Thank you. But I am still in mess and confusion. 

 Dear Daniel,

==> " If you can find this deterministic behaviour this would be a good achievement without any kind of doubt."

No, no, I didn't say that the behavior is deterministic. To be specific, let's assume three values of q, i.e. q1, q2, q3, and collect separately the results in the trials with each q. In each such set, the results "r" and "t" would appear at random. However, in the set of q1 we are going to see that the result "r" appears q12 times more than the result "t", and correspondingly in the sets q2 and q3.

==> "when you make the experiment, part of your wave function . . . as soon as you told me your expemental result, suddenly is going to disapper."

DOES IT HAVE TO DISAPPEAR? Or the wave-function is just decohered and we can't get interference? Imagine that the detectors are not absorbing, i.e. a particle can be detected, though it exits the detector. Obviously, what exits the detector on "t" won't produce interference with what exits the detector on "r". But, is this testimony that something disappeared?

No doubt, quantum mechanics says that if we got a detection on "r", additional detectors that we may place on the paths "t" and "r", will still report "r". Does that mean that the coupling  |1>t |0>r was destroyed, or just disappeared? THIS IS WHAT I DOUBT.

I suggest to leave aside the concept of information. (Some people claim that by making a measurement we get new information about the studied object, information we didn't have before the measurement. Standard QM says no! We just decohere between the couplings in the wave-function, and catch one of them. Before the measurement, all the couplings existed, not only the one we caught.)

But, as I told you, I am reading an article now and maybe I'd be able to say more things.

    Dear Sofia,

    How can we decoherete? The conservation of the probability is implicit in Schrödinger equation while during measurement is lost, how is produced the decoherence and how is it measured physically

  Dear Jun,

  I'm sorry if I have help to produce confusion on your physical interpretation of Dirac equation. Rotation of an spinor is not related with topology or the change of topology. Perhaps you are thinking in the chiral anomaly where the conservation of the chiral current is not conserved due to a change of topology. This is due to the fact that the measure of the integral of the partition function doesn't follow the same symmetries as the action. The form to measure this change of topology is using the Atiyah-Singer theorem of the index and in this case corresponds to a second Chern number given by the gauge field of coupling. The second Chern number also is associated to the instantons that I have also discussed.

  It is absolutely normal that these concepts tourned out to be difficult to follow if you are not acquented with them. But let me to say that here in RG forum it is assumed, at least by me, that we are doing our opinions on a given issue but we are not trying to teach anything. My advice is that you go to basic literature on these questions better than trying to follow me.

Dear Daniel,

Do you ask how the wave-function is decohered within a trial of a measurement? Simply! Assume for simplicity that we place a detector only on the transmitted path. Let A be an atom in this detector, usually on the ground state, |A0>. After the interaction with the studied particle P, the atom A is ionized - state that I denote by |A+>|e- >, and P is scattered. So the joint state of the system consisting in the particle P and the atom A evolves as

(1) |ψ> = (q2+1)-½ |A0> [ |1>t |0>r + i q |0>t |1>r ]

--> (q2+1)-½ [ |A+>|e- > |1>t,scat |0>r + i q |A0> |0>t |1>r ]

You see? In absence of the detector, we could have brought the transmitted and the reflected wave-packets to cross one another, and would have obtained interference. But if the detector is in place, there won't be interference.

By the way, I sent you a message right now.

Best regards!

Dear Daniel,

Perhpas I did not expressed clearly. Sorry. I do not only mean spinor rotation. I mean change the state of spin, like for spin 1/2, change the spin state from +1 to -1, or vice versa. I think there is a topology of angular momentum to let this happen. 

   Dear Jun,

   I think that I understand what is the origin of your worry with the topology. You are, perhaps, thinking not in the spin associated to one particle but the spin (or helicity when mass zero) of excitations on the edge (surface) of a topological insulator ( T reversal time operator is associated to an inner symmetry) or a Wey semimetals.

   Well in such a case the helicity is locked to the linear momentum p of the electrons and its change of state is directly linked with the Z2 group and the topology of the whole manifold. One copy alone (e.g. graphene) drives to Haldane model while two corresponds to models as the Kane-Melle one. Are you speaking about that?

   It is true that in semimetals of Wey it is found that thanks to chiral anomaly associated to these helicities, there is negative magnetoresistance and other very appealling physical properties. In summary, the change of the state of the helicities or the chirality (with the locking linear momentum) is one consequence of the change of the general topology and no a change of topology due to this change of rotation. The rotation SO(3) or SO(2) doesn't changes topology, although a spheric symmetry is not topologically trivial an it needs to charts instead of one for covering it (in such aspect it is not trivial topologically but out of the scope of the spinors).

   By the way, my previous posts were in a very different direction, although I have employed the chiral anomaly as one example of how the topology enters in quantum physics. That is to say, how the there a non local information in the states which was one possibility to complement the collapse wave function problem.

   But what was really my interest in this thread is that I have found a confussion in the use of basic concepts as spontaneous symmetry breaking and others which could avoid to understand properly what was the physical phenomenon by it self. I hope that the people is not too angry.

   Dear Sofia,

   Thank you very much for your clear explanation. Thank you.

Let me now try to use it for fixing our language

   1. For me your model is deterministic because you have only one base for your Hilbert space and you can follow the evolution of the states.

   2. But here there is one problem (or contradiction). You only observe or measure one part of the wave function and by definition that is the real wave function.

   3. How could you claim that reflexion part is real and it exists if you cannot measure it by definition?

Dear Daniel,

You say

==> "2. . . . You only observe or measure one part of the wave function and by definition that is the real wave function.

   3. How could you claim that reflexion part is real and it exists if you cannot measure it by definition?"

Again for being clear I return to my example. In absence of detector we have at the output of a beam splitter

(1) |ψ> = [ T½|1>t |0>r + i R½ |0>t |1>r ],

where T and R are the transmission, respectively reflection coefficient, and T+R = 1.

I am not sure whether I understand your words "by definition that is the real wave function". Which definition? And what you mean by "real" wave-function. When you say that I cannot measure the reflection part, do you mean those trials in which I get a recording on the transmitted wave-packet?

Let me answer you in general: if a wave-packet produces an OBSERVABLE EFFECT we can assume that it EXISTS. But, again, for me to be able to continue, and for understanding each other, please clarify your words as I asked above.

With best regards,

Sofia

   Dear Sofia,

   Sorry if I explain not well enough. When I wrote:

2. But here there is one problem (or contradiction). You only observe or measure one part of the wave function and by definition that is the real wave function.

   I want to say that when you write transmision or reflexion as parts of the wave function, this is not realistic because you cannot know what is the part of the wave function which going to interact and what is the part which didn't do it, previous to the measurement or the interaction with the device.

   On the other hand, your wave function is assumed to have an only base of four terms { |0>t, |0>r, |1>t,|1>r} which is locally determined through the interaction with the measurement. But is it logic if you take into account that the actual wave function is distributed by all the space-time?

                                         =1

    How do you concentrate all your information in just a small region of the space-time to interact with the device of measurement?

   Dear Sofia,

  By the way, as a proof of how was the level of accuracy employed in you, if you read the most popular post that you find at upper part, you find:

Note: Classical mechanics is not the limiting case of v 0). For each case of this separation into classical and quantum, there is an orthogonal separation into non-relativistic and relativistic. Not that Nature care about such classification.

  Which doesn't coincides with the usual definition of what classical mechanics have as formal limits:

              v 0

where S is the action of the system, because relativistic mechanics is not considered as a part of classical mechanics in all that I know.

Dear Daniel,

I read your both last posts, but I want to focus on the 1st of them, (about the 2nd one I agree with you). You say

==> "When you write transmision or reflexion as parts of the wave function, this is not realistic because you cannot know what is the part of the wave function which going to interact and what is the part which didn't do it, previous to the measurement or the interaction with the device."

Let's take the things step by step. I return to the expression

(1) |ψ> = [ T½|1>t |0>r + i R½ |0>t |1>r ] .

Do you want to say that you disagree with it? Let me just remind you that both the transmitted and reflected wave-packet have an OBSERVABLE EFFECT.  Just bering them to cross one another. You'll get interference.

But, anyway, if you disagree with the expression (1), which expression would you put in its place?

(Thanks for the confirmation of my letter.)

   Dear Sofia,

  That expression could be correct, the problem is that is not complete, at least. The wave function have to depend of the coordinates as states of the Schrödinger equation and you can do it making depend the coefficints R and T of them. This is, I think the realistic case, but if you do it special relativity tells you that part of the information cannot reach the quantum collapse event. Thus there is here a problem.

   Obviously I have not thought on this issue enough as to write one solution, but I think that it is necessary to introduce topological information if the problem of the velocity is going to be overcome.

Dear Daniel,

Again it is not clear to me. You say

==> "The wave function have to depend of the coordinates as states of the Schrödinger equation and you can do it making depend the coefficints R and T of them. "

There is a combination of words here that I can't understand: "coordinates as states of the Schrödinger equation". R and T are just parameters.

I understood from your former posts that you place doubt on introducing in (1) both the transmitted and the reflected wave-packet. What does that have to do with the relativity?

==> "part of the information cannot reach the quantum collapse event."

I am at total loss. Collapse means that part of the wave-function seems to have disappeared. How can the collapse be "reached" by something? The combination of words "information cannot reach the collapse" is unclear to me. Also, I speak of the collapse of the WAVE-FUNCTION, I don't deal with INFORMATION, with entropy, etc. I strongly recommend not to widen the problem, it is enough complicated by itself. Please see my message.

    Dear Sofia,

    Of course R and T can be taken as parameters but the wave function has to depend of the points (space-time coordinates).

     You don't need to do anything with relativity. The problem is that the scalar field of the wave-function is reduced just to a point in one instant or in a very short time, thus this can be against the limit of c.

   Sorry, perhaps I have explained me very badly. Imagine that you have the wave function distributed in the suface of the Earth, if you think that it is one part in New York and you measures in Tel Aviv, the collapse means that

  1. You consider the information coming from New York

  2. The information contained in the wave function which much reach the collapse come also from New York. But if you do that, this propagation of the information can have a very high velocity.

Dear Daniel,

I hope to study topological insulator. I feel the description of helicity exitation on the surface of topological insulator is very interesting and imagine charge motion on the surface or not on the surface is complicated. 

   Perfect Jun.

   That is a very interesting subject which is nowadays in fashion and I am sure that sure that deserves worth. Good luck!

@Nicolis: "Concepts of quantum mechanics were developed, historically, from those of classical mechanics-but this is history, not logic."

Bohr would disagree. Of course, it is history, but it is also logic. According to Bohr's interpretation, to which most physicists nowadays at least pay lip service, you need classical notions to interpret quantum mechanics. Coleman says otherwise, but there is no need to be convinced by that. So it is a matter of interpretation.

"Of course it's possible to learn quantum mechanics first and deduce classical mechanics as a limiting case-there's no logical obstacle."

There is. How do you define the notionsof position or momentum without having a classical limit? The only way you could do quantum mechanics without classical mechanics would be to work entirely with undefinded notions first. Results of measurements would acquire meaning only in the limit. I doubt that you could call giving the mathematics of the theory without assigning meaning to the mathematical terms (there is no meaning to a wave function without a classical world behind...) teaching. So I repeat my claim that you cannot teach quantum mechanics before classical mechanics. You can babble about incomprehensible concepts at best. A theory is different from the mathematics behind.

On the other hand, you can define all the terms of relativity that require definition without having to know classical mechanics. You cannot do that with quantum mechanics.

Now, every theory including classical mechanics contains a number of undefined concepts of which you must have preconceived ideas. Force is such a concept in Newtonian mechanics that is roughly as undefined as points and straight lines are in Euclidean geometry. But it is visualizable and the theory can be understood without a precise definition of force. The concept becomes more precise as you go along with the theory. The only undefined concepts on which you can build in quantum mechanics are classical concepts. So you need classical mechanics before quantum mechanics.

"The interpretation of quantum mechanics doesn't depend on classical mechanics-it's the other way around."

That is a nice idea, but it seems unsubstantiated to me.