How observation leads to wavefunction collapse?

[gato_]

about the question of waveform collapse, i found this usefull
http://arxiv.org/ftp/quant-ph/papers/0306/0306072.pdf
I think decoherence gives the right answer

 

[Demystifier]

about the question of waveform collapse, i found this usefull
http://arxiv.org/ftp/quant-ph/papers/0306/0306072.pdf
I think decoherence gives the right answer

See also
http://xxx.lanl.gov/abs/quant-ph/0312059
It seems that now most experts agree that decoherence does NOT solve the problem of wave-function collapse.

 

[meopemuk]

Meopemuk, I agree with you that QFT is actually a more sofisticated theory of particles. Yet, QFT does not answer the question why |\psi(x,t)|^2
represents the probability density of particle positions in the nonrelativistic limit.

What is \psi(x,t) in your formula? Is it quantum field or wave function? Quantum fields have nothing to do with probability densities. Particle wave functions can be defined in both nonrelativistic QM and in QFT.

 

[Demystifier]

What is \psi(x,t) in your formula? Is it quantum field or wave function? Quantum fields have nothing to do with probability densities. Particle wave functions can be defined in both nonrelativistic QM and in QFT.

\psi is a wave function. I know that it can be defined in QFT. But is it consistent to attribute a probabilistic interpretation to \psi in QFT?

 

[meopemuk]

\psi is a wave function. I know that it can be defined in QFT. But is it consistent to attribute a probabilistic interpretation to \psi in QFT?

Yes, it is possible to define particle wave functions in QFT with a probabilistic interpretation. The only diifculty is that in QFT the number of particles is not specified and is not conserved, so the most general wave function is a superposition of wave functions with different numbers of particles: 0-particle, 1-particle, 2-particle, ... etc.

 

[Demystifier]

Yes, it is possible to define particle wave functions in QFT with a probabilistic interpretation. The only diifculty is that in QFT the number of particles is not specified and is not conserved, so the most general wave function is a superposition of wave functions with different numbers of particles: 0-particle, 1-particle, 2-particle, ... etc.

So let us take the simplest possible case: massless uncharged scalar in a 1-particle state. For a given \psi, write the formula for calculating the probability density of particle positions!

 

[rewebster]

> To recognize the problem is sometimes not easier than to solve it.

I agree with this.

/Fredrik

Yes, and sometimes I think the problem is just too many parameters (some not recognised and/or adjusted for the specific situation)

 

[meopemuk]

So let us take the simplest possible case: massless uncharged scalar in a 1-particle state. For a given \psi, write the formula for calculating the probability density of particle positions!

This should be easy. I'll take $| \psi \rangle$ as a vector in the Fock space. As you said, it is an one-particle vector, so it lies entirely in the 1-particle sector of the Fock space. In this sector I can define the Newton-Wigner position operator $\mathbf{R}$ and its eigenvectors $| \mathbf{r} \rangle$. Then the position-space wave function corresponding to the state $| \psi \rangle$ is given by formula


$$\psi(\mathbf{r} ) = \langle \mathbf{r} | \psi \rangle$$

and the probability of finding the particle in a space region V is given by

$$\int \limits_{V} |\psi(\mathbf{r} )|^2 d^3r$$

With minor modifications, this construction can be repeated for multiparticle states as well.

 

[Demystifier]

OK, fine.
Of course, there is a problem of covariance, but there is no reason to repeat it.

Anyway, perhaps you might like my less radical (not Bohmian and still covariant) proposal for the solution of this problem:
http://xxx.lanl.gov/abs/quant-ph/0602024

 

[meopemuk]

OK, fine.
Anyway, perhaps you might like my less radical (not Bohmian and still covariant) proposal for the solution of this problem:
http://xxx.lanl.gov/abs/quant-ph/0602024

Thanks for the reference. I actually had this paper in my collection for a while. You can guess that I have a few objections. Many of them have been posted on this forum already. I am not sure if you want to start another round. Maybe we can agree to disagree, and heal our wounds for a while?

 

[gptejms]

With such a reasoning, we have TWO independent theories (nonrelativistic QM and relativistic QFT) that are mutually logically incompatible. ...

With what reasoning?

The only point I was trying to make was that \psi could not just be a mathematical entity as someone said.It's as much a wave as any other.You can't have a mathematical entity interfering with itself to give you an interference pattern.

 

[meopemuk]

The only point I was trying to make was that \psi could not just be a mathematical entity as someone said.It's as much a wave as any other.You can't have a mathematical entity interfering with itself to give you an interference pattern.

I think that wave function \psi is a purely mathematical entity. It would be incorrect to imagine that \psi is some kind of physical fluid or field that propagates in space, interferes with itself, collapses, etc. In fact, \psi is an abstract probability density amplitude, and nothing else.

We can contact physical systems in two situations: preparation and measurement. So, the goal of a good theory is to describe these two regimes and their interconnections. Nothing objective can be said about what happens to the physical system between the events of preparation of measurement, i.e., when we are not watching. Even if we say something about this intermediate regime, our statements cannot be experimentally verified (by definition), so they are not objective.

In order to predict results of measurements from known preparation conditions we build a theoretical model (quantum mechanics). This model involves Hilbert spaces, Hermitian operators, normalized wave functions, and other purely mathematical objects, which work according to some formal rules. These objects are not parts of the physical world, they are just mathematical symbols. All this math allows us (somewhat mysteriously) to predict results of physical measurements with great precision. However, this is not a reason to assume the existence of wave functions as some physical entities.

Eugene.

 

[gptejms]

These objects are not parts of the physical world, they are just mathematical symbols. All this math allows us (somewhat mysteriously) to predict results of physical measurements with great precision. However, this is not a reason to assume the existence of wave functions as some physical entities.

Eugene.

We end up agreeing to such statements because of our constant brain-washing.But it's like gulping down something we don't have in our mouth--btw how nice it would have been if we could gulp down mathematical entities rather than food and feel satiated!

I think physicists need to do more than carrying on with such mythological stories.

 

[meopemuk]

We end up agreeing to such statements because of our constant brain-washing.But it's like gulping down something we don't have in our mouth--btw how nice it would have been if we could gulp down mathematical entities rather than food and feel satiated!

I think physicists need to do more than carrying on with such mythological stories.

You may not agree with me and think that wavefunctions are some kinds of physical fluids. Fine. Then you will be pressed to answer a lot of unpleasant questions and to resolve a lot of strange paradoxes. One difficulty would be to explain how this diffuse fluid that is spread all over the world produces definite clicks of localized particle detectors. You will need to explain how this fluid interacts with detectors, why this interaction leads to the "collapse"? Even more interestingly, for consistency you will need to describe the detector in a quantum way, i.e., also by some kind of wavefunction fluid. Then you will have two fluids to worry about. How do they collapse? Do they interact with our brain?

You can spend all your life (and some people do) trying to answer these questions. I am saying that all these questions simply have no answer for a simple reason that they are not formulated in a manner that can be confirmed by experiment. Statements that cannot be verified by experiment are not about nature. They are either about our mathematical models or about philosophy. Wavefunctions are not "natural fluids". They are parts of our mathematical model of nature. Their collapse is also a part of our mathematical model. So, there is no need to worry that the collapse occurs instantaneously. There is nothing material that moves or changes during these "collapses".

Do you feel brainwashed yet?

 

[gptejms]

You may not agree with me and think that wavefunctions are some kinds of physical fluids. Fine. Then you will be pressed to answer a lot of unpleasant questions and to resolve a lot of strange paradoxes. One difficulty would be to explain how this diffuse fluid that is spread all over the world produces definite clicks of localized particle detectors. You will need to explain how this fluid interacts with detectors, why this interaction leads to the "collapse"? Even more interestingly, for consistency you will need to describe the detector in a quantum way, i.e., also by some kind of wavefunction fluid. Then you will have two fluids to worry about. How do they collapse? Do they interact with our brain?

No,no fluid is a very poor thing to imagine---it's nowhere near to a wave! If you read my posts carefully,you'll see that all I am saying is that psi represents a wave/field,not just a mathematical entity--I don't know how the fluid thing came.

Historically non-relativistic QM came before relativistic QM, and QFT.Suppose it was the other way round--QFT was discovered first and then the Schrodinger equation.Wht interpreeeetation would you then give to psi?You would then think it to be a (quantum ) field that satisfied the continuity equation---and so it could be given the probability interpretation---or one could interpret that, in the non-relativistic limit, the particle number is fixed.I think that's all there is to it.

They are parts of our mathematical model of nature. Their collapse is also a part of our mathematical model. So, there is no need to worry that the collapse occurs instantaneously. There is nothing material that moves or changes during these "collapses".

Oh really?Then why all the fuss about the measurement problem?

 

[Shahin]

In order to predict results of measurements from known preparation conditions we build a theoretical model (quantum mechanics). This model involves Hilbert spaces, Hermitian operators, normalized wave functions, and other purely mathematical objects, which work according to some formal rules. These objects are not parts of the physical world, they are just mathematical symbols. All this math allows us (somewhat mysteriously) to predict results of physical measurements with great precision. However, this is not a reason to assume the existence of wave functions as some physical entities.

Eugene.

I don't think there is something mysterious in all of that, simply, the maths had been developed in such way that "fit" with our world, because they are logically compatible with it.

Wavefunction is a mathematical entity, but just in the same way as the classical trayectory is. Is something mathematical which alow us to describe some things. However we have, for the wave function, a continuity equation that permit us to think in the modulus^2 (the probability) as a fluid, in just some aspects.

 

[lightarrow]

I don't think there is something mysterious in all of that, simply, the maths had been developed in such way that "fit" with our world, because they are logically compatible with it.

Wavefunction is a mathematical entity, but just in the same way as the classical trayectory is. Is something mathematical which alow us to describe some things. However we have, for the wave function, a continuity equation that permit us to think in the modulus^2 (the probability) as a fluid, in just some aspects.

I don't agree with it. Wavefunction is much more abstract. The wavefunction describing a system of 2 only particles is defined in a 6-dimension space. Is this space real?

 

[Shahin]

I don't agree with it. Wavefunction is much more abstract. The wavefunction describing a system of 2 only particles is defined in a 6-dimension space. Is this space real?

Well, what u mean with the word "real"? if you are asking me if the world we live in can be considered mathematically as a 6 dimensional euclidean space; of course it isn't true, but i don't see the relation between the number of variables you need for the wavefunction and the "level of asbtraction" that it has. Besides, you are talking of a system of two particles, and it is logical that you will need three spatial coordinates for each particle, in analogie with a classical trayectory. Anyway i am not telling that the wavefunction and the classical trayectory are the same, i just mean that both of them are mathematical entities, and probably de wavefunction is less intuitive, but nothing more.

 

[meopemuk]

No,no fluid is a very poor thing to imagine---it's nowhere near to a wave! If you read my posts carefully,you'll see that all I am saying is that psi represents a wave/field,not just a mathematical entity--I don't know how the fluid thing came.

Yes, I understand. Fluid, wave, or field... something material.

Historically non-relativistic QM came before relativistic QM, and QFT.Suppose it was the other way round--QFT was discovered first and then the Schrodinger equation.Wht interpreeeetation would you then give to psi?You would then think it to be a (quantum ) field that satisfied the continuity equation---and so it could be given the probability interpretation---or one could interpret that, in the non-relativistic limit, the particle number is fixed.I think that's all there is to it.

I am not sure we would do that. In my opinion, wave functions and quantum fields are completely different things that have nothing in common. Wave functions are probability amplitudes for measuring observables. Quantum fields are certain operators in the Fock space, which provide convenient building blocks for interaction Hamiltonians (or Lagrangians). Quantum fields are even farther from material world than wave functions are.

Oh really?Then why all the fuss about the measurement problem?

I don't know. I think it is just a lot of hot air.

 

[meopemuk]

Anyway i am not telling that the wavefunction and the classical trayectory are the same, i just mean that both of them are mathematical entities, and probably de wavefunction is less intuitive, but nothing more.

In my view, there is a significant difference between trajectories and wave functions. Trajectory is a collection of directly measurable values of an observable - position. Wave function is not directly observable. Trajectory is a record of facts. Wave function is a record of probabilities.

Eugene.

 

[masudr]

I don't agree with it. Wavefunction is much more abstract. The wavefunction describing a system of 2 only particles is defined in a 6-dimension space. Is this space real?

Well, the complete specification of just one particle in classical mechanics also requires a 6-dimensional space (3 for position and 3 for momentum: without the momentum values, you do not know the initial conditions and hence cannot determine the motion). This space is also no more real - it is specifically called phase space, not real space.

 

[gptejms]

I am not sure we would do that. In my opinion, wave functions and quantum fields are completely different things that have nothing in common. Wave functions are probability amplitudes for measuring observables. Quantum fields are certain operators in the Fock space, which provide convenient building blocks for interaction Hamiltonians (or Lagrangians). Quantum fields are even farther from material world than wave functions are.

Ok then let me know what you would do--you have QFT and then somebody comes up with the Schrodinger equation.What interpretation would you give to \psi?Let this question be open to all.

 

[lightarrow]

Well, the complete specification of just one particle in classical mechanics also requires a 6-dimensional space (3 for position and 3 for momentum: without the momentum values, you do not know the initial conditions and hence cannot determine the motion). This space is also no more real - it is specifically called phase space, not real space.

Of course, but those 6 parameters refers to the initial conditions, not to the location of the particle during its motion. With a vavefunction it's different: those 6 parameters refers just to the motion of the particle.

 

[lightarrow]

In my view, there is a significant difference between trajectories and wave functions. Trajectory is a collection of directly measurable values of an observable - position. Wave function is not directly observable. Trajectory is a record of facts. Wave function is a record of probabilities.
Eugene.

Exactly. I quote you.

 

[Demystifier]

Thanks for the reference. I actually had this paper in my collection for a while. You can guess that I have a few objections. Many of them have been posted on this forum already. I am not sure if you want to start another round. Maybe we can agree to disagree, and heal our wounds for a while?

In fact, I do not know what would be your main objections against THIS paper. Could you briefly indicate them?

 

[masudr]

Of course, but those 6 parameters refers to the initial conditions, not to the location of the particle during its motion. With a vavefunction it's different: those 6 parameters refers just to the motion of the particle.

No: the state space in single-particle classical mechanics is 6-dimensional. The state space in single-particle quantum mechanics is an infinite dimensional Hilbert space.

Specifically, the wavefunction is a representation of a vector in the latter space, and we can conveniently label the representation by 3 real numbers.

My point is that if you are going to compare CM with QM, then you must compare like for like. In this case, we are comparing state spaces, and I am saying how the state space in neither case should resemble anything to do with spacetime.

Here, you seem to have a problem that a 2-particle quantum system requires 6 parameters, and somehow this is problematic given that the space we know is only 3-dimensional. Do you still find it problematic given the above discussion on state space?

 

[Shahin]

In my view, there is a significant difference between trajectories and wave functions. Trajectory is a collection of directly measurable values of an observable - position. Wave function is not directly observable. Trajectory is a record of facts. Wave function is a record of probabilities.

Eugene.

Of course, as i have explained, the wavefunction and the classical trayectory are not the same. But in some way, they are similar, because they are mathematical entities that we use to describe some world we live in, but we use them in different range of scales.

 

[masudr]

Of course, as i have explained, the wavefunction and the classical trayectory are not the same. But in some way, they are similar, because they are mathematical entities that we use to describe some world we live in, but we use them in different range of scales.

As I have explained above, the wavefunction and trajectory aren't even qualitatively equivalent. The wavefunction corresponds to a point in state space in classical mechanics. This state space looks nothing like the actual space in which particles move (where they may trace out a trajectory).

 

[Shahin]

As I have explained above, the wavefunction and trajectory aren't even qualitatively equivalent. The wavefunction corresponds to a point in state space in classical mechanics. This state space looks nothing like the actual space in which particles move (where they may trace out a trajectory).

I disagree with you. It´s like if I say that the classical space in which a particle is moving is infinite-dimensional just because the fourier´s expansion (in some orthonormal system) of the three position´s functions is infinite. The point is that, to get all the information possible about a particle, you have to introduce in psi three numbers (x,y,z) for the position, and another for the time. Obviously, psi lives in a hilbert space of infinite dimensions and it is hardly different compared with a classical trayectory, but i just want to say that both of them are mathematical entities to describe the world, and in taht sense, they are the same.

PS: Are my posts too difficult for reading because my basic level of english?

 

[Schrodinger's Dog]

Your English is fine, in fact I wouldn't of guessed it wasn't your first language. I think physics is just one of those fields were there are a lot of different interpretations, and precision is important, whether your speaking in your native language or not, you'll no doubt get a lot of discussion on something as theoretically divisive as the wave function and the implications of the mathematics. A 12 page thread adequately demonstrates this

 

[meopemuk]

Ok then let me know what you would do--you have QFT and then somebody comes up with the Schrodinger equation.What interpretation would you give to \psi?Let this question be open to all.

I don't understand your question completely. I decided to describe my understanding of wave functions, quantum fields, Schroedinger equation and relativistic wave equations in quantum mechanics and in QFT. Hopefully, this will answer your question. This is going to be quite long. Please bear with me. Those who read S. Weinberg "The quantum theory of fields" vol. 1 may recognize that here I am trying to retell his book in few paragraphs.

First, there is no fundamental difference beween quantum mechanics and QFT. QFT is simply quantum mechanics applied to a class of systems in which the number of particles can change. For this reason, the Hilbert space of QFT cannot be a space with a fixed number of particles. The Hilbert space of QFT is constructed as a direct sum of n-particle spaces (or sectors), where n varies from 0 to infinity. This direct sum is called the Fock space. In the Fock space, there is a 1-dimensional no-particle subspace (n=0), which is called vacuum. There are also 1-particle subspaces (sectors n=1) for each type of particle in the theory: 1-electron sector, 1-photon sector. Then there are 2-particle sectors (n=2) with different combinations of particles: "2 electrons", or "2 photons", or "1 electron plus 1 photon", ... etc. Take a direct sum of all these sectors and you obtain the Fock space which is the Hilbert space of states in QFT.

All standard Rules of Quantum Mechanics work without change in this Fock space. Any state of the system is described by a state vector. Observables are described by Hermitian operators. You can build orthonormal bases there, define a Hamiltonian, etc. The only difference is that the number of particles is not fixed. If you have an arbitrary state vector $|\Psi \rangle$ in the Fock space, you can take projections of this vector on sectors with different numbers of particles n=0,1,2,3,... In each of these sectors you can have an orthonormal basis (e.g. a basis of position eigenvectors). So, in each of these sectors you can have an n-particle wave function. Then the total wave function corresponding to the state $\Psi \rangle$ is a suporposition of all these wave functions with complex coefficients. To write all this down would require a cumbersome notation. I will give you just a couple of simple examples.

Suppose that the state vector $|\Psi \rangle$ lies entirely in a 1-particle sector corresponding to a massive spinless particle. One can define a basis of position eigenvectors $| \mathbf{r} \rangle$ in this sector and find the wave function of $|\Psi \rangle$ in this representation $\psi (\mathbf{r}) = \langle \mathbf{r} | \Psi \rangle$.

The time evolution of any state vector in the Fock space is generally described by

$$-i \hbar \frac{\partial}{\partial t} | \Psi(t) \rangle = H | \Psi(t) \rangle$$ (1)

where $H = \sqrt{\mathbf{P}^2c^2 + M^2 c^4}$ is the Hamiltonian defined in the entire Fock space, $\mathbf{P}$ and $M$ are operators of the total momentum and total mass, respectively. Let us now consider the simple case in which there are no interaction terms in the Hamiltonian $H$. In particular, each n-particle sector remains invariant with respect to time evolution. In the 1-particle subspace described above, we can multiply eq. (1) by the bra vector $\langle \mathbf{r} |$ from the left, and take into account that $M$ is reduced to multiplication by a number $m$, which the mass of the particle. Then we get the Schroedinger equation for the wave function of one free particle in QFT

$$-i \hbar \frac{\partial}{\partial t} \psi(\mathbf{r}, t) = \sqrt{-\hbar^2 c^2 \nabla^2 + m^2 c^4} \psi(\mathbf{r}, t)$$

A similar construction can be repeated in the case of a two-particle state. The wavefunction may be written as $\psi (\mathbf{r}_1, \mathbf{r}_2)$, and the Schroedinger equation is

$$-i \hbar \frac{\partial}{\partial t} \psi (\mathbf{r}_1, \mathbf{r}_2)= \Bigl( \sqrt{-\hbar^2 c^2 \nabla_1^2 + m^2 c^4} + \sqrt{-\hbar^2 c^2 \nabla_2^2 + m^2 c^4}\Bigr)\psi (\mathbf{r}_1, \mathbf{r}_2)$$

In the case of multiparticle systems with interactions that can change the number of particles, it is almost impossible to write the Schroedinger equation in this notation, so it is more preferable to use the abstract form (1).


Now, you may ask, where are quantum fields? I didn't need this concept so far. In fact, quantum fields have no relationship to particle wave functions. Their role in QFT is completely different. They are needed in order to write down the interaction part $V$ of the full Hamiltonian $H = H_0 +V$. This is a non-trivial construction if we want to have a Hamiltonian that satisfies the principle of relativistic invariance. I can tell you the story of quantum fields and their wave equations (Dirac, Klein-Gordon, etc.) if we can agree that what I wrote so far makes sense.

Eugene.

 

[Mike2]

As I have explained above, the wavefunction and trajectory aren't even qualitatively equivalent. The wavefunction corresponds to a point in state space in classical mechanics. This state space looks nothing like the actual space in which particles move (where they may trace out a trajectory).

The path integral forumlation of quantum mechanics states that the wavefunction is every possible path (trajectory) from one state to the next. It reduces to the classical trajectory when the phases cancel almost everywhere except the classical path.

 

[masudr]

The path integral forumlation of quantum mechanics states that the wavefunction is every possible path (trajectory) from one state to the next. It reduces to the classical trajectory when the phases cancel almost everywhere except the classical path.

I'm not sure how this is related to anything I've said.

 

[Mike2]

I'm not sure how this is related to anything I've said.

A path is a trajectory through spacetime. The path integral speaks about the paths that a particle takes, and therefore takes into account trajectories. The path integral is also another definition of the wavefunction. Therefore, the wavefunction is related to trajectories, right?

 

[masudr]

I disagree with you. It´s like if I say that the classical space in which a particle is moving is infinite-dimensional just because the fourier´s expansion (in some orthonormal system) of the three position´s functions is infinite.

You can't actually say that. The particle in classical mechanics needs 6 quantities for a complete description. If you can describe it in less than 6, then you have done something quite remarkable.

3 quantities do not describe a particle in quantum mechanics. The value of the function at all possible co-ordinates (or all possible energy, or all possible momenta etc.) are required to fully describe just one single state.

The point is that, to get all the information possible about a particle, you have to introduce in psi three numbers (x,y,z) for the position, and another for the time.

Not only that, but you also need to have a complex-valued function of those 4 variables (i.e. an infinite set of complex numbers). However, in classical mechanics, all the information is contained in 6 numbers, and there is no need for a function. This is why a classical mechanical state space is 6-dimensional, whereas quantum mechanical state space is infinite dimensional.

Obviously, psi lives in a hilbert space of infinite dimensions and it is hardly different compared with a classical trayectory, but i just want to say that both of them are mathematical entities to describe the world, and in taht sense, they are the same.

I'm glad you agree that they are different, but you initially had a problem with QM because 2 particles require 6 variables, and I picked up on that. All I am saying is that, at any moment in time, in the single particle case, CM requires 6 numbers for a complete description, and QM requires an infinite set of numbers, but they can be indexed by 3 variables. Can you see why the two aren't even qualitatively equivalent?

PS: Are my posts too difficult for reading because my basic level of english?

Apart from the odd spelling mistake (e.g. trajectory, not trayectory), your English is remarkably good.