How Does Quantum Mechanics Combine Particle and Wave Descriptions?

[Loren Booda]

One of myriad descriptions of the wavepacket might be as a probabilistic representation of an entity's complementary measurements excluded from each other by the magnitude of Planck's constant.

 

[Demystifier]

1. Demystifier, you are a nice guy, but in that eprint you didn't know what you were doing.
... bla bla bla ...
To sumarize, the eprint has nothing in it.

As far as quantum mechanics is concerned, you actually disagree with almost everything said by almost everybody. I am glad to see that I am not an exception.

 

[zbyszek]

As far as quantum mechanics is concerned, you actually disagree with almost everything said by almost everybody. I am glad to see that I am not an exception.

Going with the herd? That's the scientific spirit!
Nice answer to a detailed argument, too.

Cheers!

 

[Demystifier]

Going with the herd? That's the scientific spirit!
Nice answer to a detailed argument, too.

The idea of a public forum is to write something that will be interesting to many people reading it, not just to one person. If anybody else here finds out that some of your arguments are viable, I will give a more scientific answer. If, one the other hand, you want to argue only with me, send me a private message.

 

[Anonym]

ueit:"In a double slit experiment it's the wall with the slits. An electron passing near such an object changes momentum. The mechanism behind this change is ignored so we shouldn't expect a good prediction of the individual detection event. So, besides the probable statistical character of the wavefunction itself, we have another approximation regarding the potential at the slits (which is assumed to be 0 although it's only 0 on average)."

You did not answer my question: Do you agree that lossless beamsplitter is real life realization of the "wall"?

 

[Anonym]

Zbyszek:” I don't think there has been much progress since Einstein. If anything it would be rather a regress. These days Bohr is perceived (unjustly again) as a winner of the duel with Einstein over the meaning of QM. So, not many guys are even aware that we are still missing a quantum theory of single objects and that QM is incomplete indeed.”

You are kidding. I am asking seriously. For example, investigations of R.J. Glauber and others established the connection between classical and quantum statistical mechanics. On the other side, the single particle approach also led to enormous progress in QT: QED, local gauge abelian and non-abelian interactions, electroweak unification, quarks, QCD, etc. However, in that game the role of “interpretations” is not clear. Looks like something stand alone.
May you present coherently what is the content of the “normal physicist” criticism of the standard approach to QM?

 

[Careful]

You are kidding. I am asking seriously. For example, investigations of R.J. Glauber and others established the connection between classical and quantum statistical mechanics. On the other side, the single particle approach also led to enormous progress in QT: QED, local gauge abelian and non-abelian interactions, electroweak unification, quarks, QCD, etc. However, in that game the role of “interpretations” is not clear. Looks like something stand alone.
May you present coherently what is the content of the “normal physicist” criticism of the standard approach to QM?

QFT has no realistic, local single event interpretation, period (and mind MWI is not realistic).

 

[Anonym]

Careful:"QFT has no realistic, local single event interpretation, period (and mind MWI is not realistic)."

I agree. I claim that non-relativistic QM is complete. I don't claim that QFT is complete. And this is not a question. We discuss the description of the statistical ensembles in terms of wave packets.

 

[Careful]

Careful:"QFT has no realistic, local single event interpretation, period (and mind MWI is not realistic)."

I agree. I claim that non-relativistic QM is complete. I don't claim that QFT is complete. And this is not a question. We discuss the description of the statistical ensembles in terms of wave packets.

You cannot discuss physics without taking into account special relativity, that is like going to restaurant and eat with bare hands (which might still be a habit in some parts of the world). I will reverse the question to you, what makes you think that nonrelativistic quantum mechanics has a sensible single event interpretation (there are some ``options'' so it is simply more efficient to ask you) ?

 

[vanesch]

(and mind MWI is not realistic).

... for sufficiently naive versions of "realistic"

(meaning: where events really, and uniquely, happen)

 

[Careful]

... for sufficiently naive versions of "realistic"

(meaning: where events really, and uniquely, happen)

Brilliant !

 

[ueit]

You did not answer my question: Do you agree that lossless beamsplitter is real life realization of the "wall"?

It may be, the electron interference experiments I know of were performed with copper gratings, magnetic fields, or crystallization planes.

 

[ueit]

Although, for non-relativistic QM, Bohmian mechanics is ontologically clearer, its "clearness" is sometimes overstated, because Bohmian mechanics needs TWO ontological parts:
- the particles, and that's what everybody stresses, and what looks like Newtonian mechanics with an added potential, so this seems at first sight to be very clear
- but there is ALSO, as an independent entity, the wavefunction, which does NOT live in spacetime, but which lives over configuration space all together. It is NOT a classical field, and it contains also all the "ghosts" of MWI.

The wavefunction describes the force acting on the particles. The particles exist in 3D space and the force acts upon them in 3d space as well. We need not to ascribe a fundamental character to the mathematical formalism needed to calculate that force. And what do you mean by ghosts?

You still have an interpretational problem, in the following sense: somehow, an "observer", although that observer has TWO ontological parts (its "particle" part, and its "wavefunction" part), can only be "aware" of its "particle part", and not of its "wavefunction" part, because if he were so, there wouldn't be any probabilistic part to it, and hence the predictions of BM wouldn't coincide with QM.

The wavefunction describes how the particles move. I don't understand your point about "awareness". Is this like saying that we are aware of the Moon but not of its gravitational force?

The other problem with BM of course is the fact that it is not compatible with a relativistic spacetime: there is no geometric formulation of BM possible (which would come down to being able to write BM in a lorentz-invariant way, which is not possible).

I think there is some debate about this issue.

Not at all. QM is defined in hilbert space, which is the functional space over configuration space. This only coincides with "normal space" in the case of a single point particle.

The Hilbert space is a mathematical construct. The derivation of the wavefunction involves the assumption of point particles existing in a 3d space + 1d time background.

There are a lot of misunderstandings about MWI. MWI is simply defined, as in "normal" QM, over Hilbert space. This Hilbert space has a basis which can be "indexed" using a configuration space of a classical system, and that classical system can be a field over spacetime, or a set of particles in Euclidean space, or... whatever.

Normal QM is defined on a Newtonian background, QFT on a Minkowski spacetime. The particles with their masses and various charges exist in this background. As MWI is only an interpretation it should be defined on the same background, isn't it?

A given "observer" in MWI corresponds to certain subspaces of Hilbert space which correspond to a certain "history of observations" (just like a given observer state in classical phase space corresponds to certain patches in phase space corresponding to a certain "record of observation"). Now, in classical phase space, we usually consider only ONE point which is wandering around (the "state of the universe") in phase space, following a Hamiltonian flow. This point will enter and leave certain "observer patches" and this will correspond to "experienced observations". Given that these patches, classically, are disjoint (you do not have a patch that corresponds at the same time to "the light bulb was on" and "the light bulb was off"), there is no ambiguity to any observation.

In Hilbert space, the patches of "states of observers" are subspaces of hilbert space. And the "state of the universe" is a vector in Hilbert space that follows the unitary evolution of the Hamiltonian flow. However, the difference is now that this state of the universe can have components in DIFFERENT observer subspaces at the same time.
In MWI, we simply say that these different and incompatible observations are then taking place in different "worlds", and that you, as an observer, are just experiencing one of these subspaces and not all of them, simply because we can only experience one subspace. The other subspaces then correspond to experiences of "copies". What matters, for a specific subjective observer, is, what is the probability that he will be one of the copies. It is the specific structure of the subspaces which makes us have the illusion of a "theatre that is like spacetime".
You could compare this situation with a classical phase space where there are different points corresponding to different "worlds" wandering around on the Hamiltonian flow. These different points can then be in different "observer patches" at the same time, but you will only experience "one of these patches".

Again, the hilbert space is a derived concept. What could be the meaning of the configuration space without first defining the background? The position of a particle depends on the number of dimensions in which the particle exists so it doesn't seem to me that the "theatre that is like spacetime" follows from quantum formalism but the other way arround.

This is not postulated a priori. It is because it follows naturally out of the Schroedinger evolution equation that this consideration is taken. The reason for postulating "many worlds" is not a crazy idea that is imported, it is because it follows from the formalism. One has introduced a specific EXTRA mechanism in quantum theory to GET RID OF IT, which is projection, but that extra mechanism is the core of all difficulties in QM: it is explicitly non-local and not Lorentz-invariant, irreversible, dynamically ill-defined (when exactly does it happen) etc... It is because of all these difficulties *introduced by the patch that is projection* that Everett first considered to get rid of it, and to keep the one and only dynamical law that is well-defined in QM: hamiltonian unitary evolution. But IF you keep that as a universal dynamics, well then you end up *naturally* with a state of the universe where observers occur in superpositions of "macroscopically different observations". It is just because of this natural appearance of different classical observation states in the "state of the universe" that the idea was then to see this as "parallel worlds". There's no more or no less to it. MWI is simply: let us take the unitary dynamics of quantum theory as fundamental and universal, without introducing a patch to make it fit "classical outcomes" which introduces a lot of difficulties.

So MWI is nothing else but: let us take the hilbert space formalism of QM, and its unitary dynamics, for real, and see what it tells, without wanting to force any specific a-priori of what "should" reasonably, come out.

I have a hard time understanding how can you write down a wavefunction without first assuming point particles in a 3d background. And if you assume that as real, then how can the wavefunction be real and the spacetime an illusion?

 

[karfiol]

there are no particles, there are only waves

When energy is spreading through space it has form of wave. That is default form of energy.

But when energy come in contact with something, it cease to be wave and become particle.

 

[zbyszek]

You are kidding. I am asking seriously. For example, investigations of R.J. Glauber and others established the connection between classical and quantum statistical mechanics. On the other side, the single particle approach also led to enormous progress in QT: QED, local gauge abelian and non-abelian interactions, electroweak unification, quarks, QCD, etc. However, in that game the role of “interpretations” is not clear. Looks like something stand alone.
May you present coherently what is the content of the “normal physicist” criticism of the standard approach to QM?

No, I'm not. We are talking QM here, right? No relativistic extensions.
The QM is incomplete in the sense that it is only statistical. No single events
theory at all. How many quantum physicist realize that today?
In the first half of 20th century, there was no MWI. Do you need a stronger
argument? :)
So, where is the progress on foundations of QM? What exactly did Glauber
do that qualifies as progress in QM?
You seem to distinguish between QM and quantum statistical mechanics.
What's the difference?

Cheers!

 

[zbyszek]

You cannot discuss physics without taking into account special relativity, that is like going to restaurant and eat with bare hands (which might still be a habit in some parts of the world). I will reverse the question to you, what makes you think that nonrelativistic quantum mechanics has a sensible single event interpretation (there are some ``options'' so it is simply more efficient to ask you) ?

Yes, you can. One can get relativistic equations out of nonrelativistic QM.
Here is an example.
Effective theories for low energy corner of nonrelativistic quantum liquids tend
to have Lorentz invariance, among other symmetries. Moreover, one can get something like Einstein's equations for propagation of excitations with, so called, acoustic metric.
c is replaced with the speed of sound in the liquid. See works of Volovik or
his book "The Universe in a helium droplet".

In short you have an absolute frame of reference (the one of the
center of mass of the liquid) and Lorenz invariance (for low energy excitations) in one system.

Cheers!

 

[zbyszek]

The idea of a public forum is to write something that will be interesting to many people reading it, not just to one person. If anybody else here finds out that some of your arguments are viable, I will give a more scientific answer. If, one the other hand, you want to argue only with me, send me a private message.

Thanks for the explanation!
I find my arguments viable and I am a part of the public. I hope. Or am I?

More, I've even read the manuscript publicly recommended by you with some
level of comprehension. Do I deserve an answer?

Forgive me my curiosity, but I cannot help wondering what "more scientific"
means in that case. Please, don't let me down!



Cheers!

 

[masudr]

Normal QM is defined on a Newtonian background

By that I imagine you mean 3 space + 1 time dimension. This is incorrect. It comes from Hamiltonian phase space.

Again, the hilbert space is a derived concept. What could be the meaning of the configuration space without first defining the background? The position of a particle depends on the number of dimensions in which the particle exists so it doesn't seem to me that the "theatre that is like spacetime" follows from quantum formalism but the other way arround.

You forget that there are elements of Q.M. that have absolutely no classical analogue. e.g. spin degrees of freedom. What are the "position co-ordinates" of that?

If you feel that the Q.M. is not defined on an abstract Hilbert space, then you are not talking about standard quantum mechanics here, you are talking about something else.

 

[Careful]

Yes, you can. One can get relativistic equations out of nonrelativistic QM.
Here is an example.
Effective theories for low energy corner of nonrelativistic quantum liquids tend
to have Lorentz invariance, among other symmetries. Moreover, one can get something like Einstein's equations for propagation of excitations with, so called, acoustic metric.
c is replaced with the speed of sound in the liquid. See works of Volovik or
his book "The Universe in a helium droplet".

In short you have an absolute frame of reference (the one of the
center of mass of the liquid) and Lorenz invariance (for low energy excitations) in one system.

Cheers!

Euh, I don't think anybody is worried about how to rewrite a non relativistic theory as an approximately relativistic one for the low energy modes (given for example the fact that Lorentz invariance is so well tested at high energies). I don't know about this reference but it sounds like saying that for sufficiently small x a Lorentz boost B(x) doesn't differ much from the corresponding rotation R(x). It would be much cooler to have a Galileian theory which allows for some coarse graining which *is* Lorentz invariant for *all* modes. This is achieved (but not entirely to my liking) by Holland amongst others in his paper about the hydrodynamic interpretation of Maxwell's theory (which has nothing to do with the old eather models I must add).

Careful

 

[vanesch]

The wavefunction describes the force acting on the particles. The particles exist in 3D space and the force acts upon them in 3d space as well. We need not to ascribe a fundamental character to the mathematical formalism needed to calculate that force.

If I only give you the positions (and the momenta, if you wish) of the particles, in BM, you are unable to calculate the force. This means that there is a dynamical content which is entirely contained in the wavefunction, and the wavefunction alone.

And what do you mean by ghosts?

As in BM, the wavefunction evolves strictly according to the Schroedinger equation, there is no projection, and hence there are exactly the same "branches" present as in MWI. That's why I sometimes say (to enerve Bohmians) that BM is MWI plus a tag

The wavefunction describes how the particles move. I don't understand your point about "awareness". Is this like saying that we are aware of the Moon but not of its gravitational force?

No, your brain is aware of its particle positions, but not of the wavefunction that goes with it.

I think there is some debate about this issue.

I don't believe those claims.

The Hilbert space is a mathematical construct. The derivation of the wavefunction involves the assumption of point particles existing in a 3d space + 1d time background.

But, 3d space (plus time) is also a mathematical construction...

Normal QM is defined on a Newtonian background

No, the degrees of freedom in normal QM are indexed using a Newtonian space (to enumerate the degrees of freedom as "x-position of particle 5"...).

Again, the hilbert space is a derived concept. What could be the meaning of the configuration space without first defining the background?

Well, you can simply have a totally different enumeration of the basis vectors in Hilbert space (for instance, N spin-1/2 systems, with no relation to any space !).

The position of a particle depends on the number of dimensions in which the particle exists so it doesn't seem to me that the "theatre that is like spacetime" follows from quantum formalism but the other way arround.

Yes, that's because it is put in by hand: you START by saying that you want a quantum system of dots in an Euclidean space. But you could just as well start from something totally different.

I have a hard time understanding how can you write down a wavefunction without first assuming point particles in a 3d background. And if you assume that as real, then how can the wavefunction be real and the spacetime an illusion?

As I said, you can consider a set of spin-1/2 systems, define a unitary dynamics over it, and go ahead.

 

[Anonym]

Zbyszek:” The QM is incomplete in the sense that it is only statistical. … So, where is the progress on foundations of QM? What exactly did Glauber do that qualifies as progress in QM? You seem to distinguish between QM and quantum statistical mechanics.
What's the difference?”

For me Born’s statistical interpretation is counter-intuitive. I don’t understand how single particle may do statistic with itself. Contrary, selfinterference is not only a mystery but self-evident as explained by P.A.M. Dirac and clearly demonstrated experimentally by A. Tonomura. The wave packet reduction (as explained by A. Einstein) is necessary in order to satisfy requirements of special relativity in the classical physics.
QM describes physics of the massive wave packets (not in the coherent basis). R.J. Glauber demonstrated that
<alpha|H|alpha>=omega*(alphasquare+1/2)
with alpha continuous. Glauber contribution is establishing the connection between quantum and classical statistical mechanics by using E.Schrödinger preliminary result. However, I don’t consider that story complete (therefore, I asking questions).
Foundations of non-relativistic QM were established by J.von Neumann through unification of the Heisenberg-Dirac theory of dynamical observables and Schrödinger theory of states. I consider that story complete. By quantum statistical mechanics I mean construction of tensor product states and description of many particle QM systems
( N>4 )using them. Indeed this is inherent part of QM. My distinction is between QM and the statistical interpretation of QM.

 

[Anonym]

Careful:"You cannot discuss physics without taking into account special relativity"

I agree with zbyszek answer. Special relativity as well as wave mechanics already resides in HJ formulation. Be careful, what peoples were doing before 1905? With respect to relativistic QM or QFT as you call it, don't worry. " raffinert ist der Herr Gott, aber boshaft ist Er nicht"

 

[Demystifier]

As in BM, the wavefunction evolves strictly according to the Schroedinger equation, there is no projection, and hence there are exactly the same "branches" present as in MWI. That's why I sometimes say (to enerve Bohmians) that BM is MWI plus a tag

I am a Bohmian and I also have a saying.
BM is MWI but with only one world. (Other worlds are tags.)

 

[Demystifier]

Yes, you can. One can get relativistic equations out of nonrelativistic QM.
Here is an example.
Effective theories for low energy corner of nonrelativistic quantum liquids tend
to have Lorentz invariance, among other symmetries. Moreover, one can get something like Einstein's equations for propagation of excitations with, so called, acoustic metric.
c is replaced with the speed of sound in the liquid. See works of Volovik or
his book "The Universe in a helium droplet".

In short you have an absolute frame of reference (the one of the
center of mass of the liquid) and Lorenz invariance (for low energy excitations) in one system.

This time I agree with you.
Maybe I will answer your questions, despite your rude qualifications such as that "I was not knowing what I was doing" and alike.

 

[Demystifier]

1. In the introduction you notice that the object that satisfies Klein-Gordon equation is not
a wave function but a field operator. However, in the third section you call it a wave
function anyway and even worse you introduce in eq. 3 a third quantized operator
on the right hand side. Do you realize that?
The left hand side of eq. 3, \psi, is already a "second quantized" operator and to get a wave function for a Fock state |n> you should have put \psi in place of \hat \phi in the RHS!

2. But even
there how do you pick initial conditions for the Bohmian trajectories (defined correctly
i.e. not as you did it)?
Don't you have to draw them from some probability density? If the answer is
afirmative then you have your "statistical transparency".

1. If you read it more carefully, you will notice that \psi and \hat{\phi} are not the same objects, despite the fact that they satisfy the same Klein-Gordon equation. In particlar, the former is a c-number function, whereas the latter is an operator. Eq. (3) is just a textbook relation between these two quantities. (Perhaps you was reading some other textbooks than I did.) In other words, this equation is a sort of bridge between first and second qutization and has nothing to do with third quantization.

2. If you read it more carefully, I admit that I do not always know the correct probability density, but the point is that it is not necessary to know it in a deterministic theory. For example, in classical mechanics you also do not know a priori the correct initial position of the particle nor the correct probability density, but classical mechanics still works very well.

 

[Anonym]

Ueit:” It may be, the electron interference experiments I know of were performed with copper gratings, magnetic fields, or crystallization planes.”

Ok,let us consider 50-50 for siplicity. The only thing I should know in order to proceed after “wall” is relative phase between the transmitted and reflected wave fields.It may be obtained from unitarity without knowledge of any details of underlined dynamics (V.Degiorgio, A.Zeilinger). Apparently, you may invent any complicated interaction you can imagine provided that outcome will be as predicted by them. I guess you cannot. I guess that only minimal coupling will do a job.

 

[Demystifier]

Zbyszek, as you clearly do not like the Bohmian interpretation (just as many others), I believe that you might like this anti-Bohmian interpretation of CLASSICAL mechanics:
http://xxx.lanl.gov/abs/quant-ph/0505143

 

[Careful]

Careful:"You cannot discuss physics without taking into account special relativity"

I agree with zbyszek answer. Special relativity as well as wave mechanics already resides in HJ formulation. Be careful, what peoples were doing before 1905? With respect to relativistic QM or QFT as you call it, don't worry. " raffinert ist der Herr Gott, aber boshaft ist Er nicht"

Euh, Zbyszek merely gave some examples of ``approximate Lorentz invariance'' in the low energy sector (which is unsurprising since any student knows how to switch between the relativistic and Galileian description in this case). So, what are you saying here, that one should not bother about Lorentz invariance in the high energy sector !? Then, I want to see how you can *naturally* embed Maxwell's theory in such framework without being in violation with Michelson Morely.

NOTE : I looked up the reference of Volovik http://ltl.tkk.fi/personnel/THEORY/volovik/book.pdf and indeed it confirms my supicion about giving up Lorentz, Gauge invariance etc... at sufficiently high energies. So relativity goes down the drain, but on the other hand QM isn't complete either (strange enough he doesn't appear to mention that) : embracing Galileian mechanics isn't sufficient to solve the entanglement ``paradox'' (if it needs to be solved in the first place). And of course, you should worry about those things ...

Careful

 

[vanesch]

I am a Bohmian and I also have a saying.
BM is MWI but with only one world.

Yes, the tag is given by the "particle positions", but you still need the wavefunction as a separate, physical and dynamical entity. And honestly, I have difficulties believing that you can formulate the particle dynamics in a relativistically invariant way. If you can do that, I'll become a bohmian too

 

[zbyszek]

Euh, Zbyszek merely gave some examples of ``approximate Lorentz invariance'' in the low energy sector (which is unsurprising since any student knows how to switch between the relativistic and Galileian description in this case). So, what are you saying here, that one should not bother about Lorentz invariance in the high energy sector !? Then, I want to see how you can *naturally* embed Maxwell's theory in such framework without being in violation with Michelson Morely.
Careful

Here is the Maxwell embeded himself :
http://xxx.lanl.gov/abs/gr-qc/0112041"

The idea is that for He3-A, close to the Fermi points the order parameter (atomic angular momentum for the liquid) satisfies Maxwell's equations.

Cheers!