What Are the Implications of a New Relativistic Quantum Theory?

[meopemuk]

How wrong you are! Maybe you haven't noticed that because of your aversion to filed theoretic methods (which you denounce as mere mathematical tricks), but it is in every textbook where the LSZ formula is derived. For example, in Weinberg, this is handled in Section 10.2.

Note that on p.430, Omega_0 is the true vacuum, not the bare one, and the A_i are the renormalized Heisenberg fields with space-time arguments. The latter act on the physical Hilbert space spanned by the states of the form A_1 ... A_n Omega_0, though Weinberg doesn't emphasize this explicitly. But one can see it from the fact that he takes matrix elements between such states (_not_ between bare states!). This is _precisely_ the recipe that I had given in my explanation of the Wightman approach to QFT. Wightman didn't take his approach from nowhere, but only isolated the minimal stuff from the usual, nonrigorous treatment that one would have to make clear mathematical sense of in order to have a rigorous, nonperturbatively defined theory.

As far as I can tell, physical states and fields A_i are used only in abstract proofs, like in Weinberg's Chapter 10. This is because such objects have not been formulated in explicit forms necessary for real calculations. In actual QED calculations (see section 8.7 and Chapter 11) one still uses the bare particle picture, bare a/c operators and free fields.

This is foul play. If you criticize QED because it has no mathematically rigorous formulation so far, you must criticize your own theory for the same reason.

I am not criticizing QED for the lack of mathematical rigor. I am criticizing it, because it cannot offer a plausible solution for the time dynamics of the simplest 2-particle state. You said that in QED particles make sense in asymptotic states. Fine. Then let us prepare an asymptotic 2-particle state and let us follow its time evolution and see how its wave function changes in time. I guess that QED cannot do that. QED simply does not have an adequate Hamiltonian to do this job. This is my concern.

If you like this sort of advice, I have two pieces more:

1. The photon self-energy is infrared finite to first nontrivial order; see Weinberg (11.2.16) and (11.2.22).

This is a good one too! I guess that the corresponding dressed particle potential will be the same as Uehling potential in (11.2.38) and it will contribute a little bit to the Lamb shift (11.2.42)

2. Why don't you postulate that the photon has a tiny mass? This is experimentally indistinguishable from real QED, and has a number of advantages:
-- Massive photons have a position operator, and hence a fully adequate Schroedinger picture. This would make your philosophical position much better grounded.
-- Massive photons save you from all infrared problems. Without the infrared problems, Fock space is perturbatively fully adequate, and all my criticism regarding the IR problem and wrong asymptotics is no longer applicable.
-- With massive photons, you can calculate radiative corrections to Compton scattering and get a finite result for the Lamb shift.

I will think about this advice too. Thanks.

Eugene.

 

[A. Neumaier]

As far as I can tell, physical states and fields A_i are used only in abstract proofs, like in Weinberg's Chapter 10.

You can't tell far. All renormalized S-matrix calculations are done using the LSZ-formulas derived in Chapter 10 from this characterization of the physical states. Why else would Weinberg introduce the discussion of LSZ on p.429 by saying: ''will help us later in dealing with radiative corrections"? It is needed to define the renormalization recipes in a way that ensures correct scattering results between physical states!

From the LSZ formula (10.2.3), which is valid nonperturbatively (hence independent of any free fields), Weinberg gets on p.437-438 the usual Feynman rules (and he says so explicitly in the second sentence of p.438); the free field (or rather its propagator) appears at the bottom of p.437. But, as he emphasizes, for arbitrary spin, including bound states, and including the field renormalization factor 1/N (which was missing in the heuristic derivation in Chapter 6, Thereby the heuristic derivation is justified and improved. The notion of a bare versus renormalized field is introduced by Weinberg only on p.438/9, when he exploits LSZ to make the connection!

Weinberg urges the reader on p.292 that only the canonical approach gives reliable results in more complex situations, because the earlier heuristic derivation is no longer justified when there are rearrangement collisions, or when extra contact terms are needed in the Hamiltonian (see the footnotes on p.110 and p.145), and that ''when we come to non-Abelian gauge theories, in Volume II, this extra convenience will become a necessity''.

Of course, once one has the LSZ formula (10.2.3), one only needs to evaluate the right hand side using perturbation theory, and the latter simply uses the Wick rules for the free fields derived in Chapter 6 for scalar/spinor fields and in Chapter 8 for QED. (But this is no worse than your use of free fields in working out how your Hamiltonian acts on a given state.)

This is because such objects have not been formulated in explicit forms necessary for real calculations. In actual QED calculations (see section 8.7 and Chapter 11) one still uses the bare particle picture, bare a/c operators and free fields.

In Section 8.7 he only calculates Compton scattering without radiative corrections, which could already be done in 1930. At this order of accuracy, N=1. But once one includes radiative corrections, one needs the renormalized version derived in Chapter 10.
This is the reason why the 1-loop calculations are done in Chapter 11, with LSZ already proved!

I am not criticizing QED for the lack of mathematical rigor. I am criticizing it, because it cannot offer a plausible solution for the time dynamics of the simplest 2-particle state. You said that in QED particles make sense in asymptotic states. Fine. Then let us prepare an asymptotic 2-particle state and let us follow its time evolution and see how its wave function changes in time. I guess that QED cannot do that. QED simply does not have an adequate Hamiltonian to do this job. This is my concern.

Of course one can do it.It is just that nobody has been interested to work it out since one can never experimentally follow the time evolution of a real electron or photon.
Experimentally you prepare it somewhere, and you detect it somewhere, and from the statistics you can calculate cross sections. That's all.

If you believe that you can test the time evolution experimentally and therefore need the formulas to do so, it is _your_ job to derive then. QED tells you how to do it in principle, but the details (like with any calculation) must be done by those interested in the results. The way to do it is this: Given an arbitrary state Psi in the Wightman representation, where all its creation fields are taken at time t=0, you can compute its inner product with an asymptotic Fock state Phi by taking the inner product with the physical state Phi(t) where all its creation fields are at the same time t and letting t --> -inf. Now pick Psi such that to the desired accuracy only the 2-particle contributions survive, and you have your desired 2-particle in state at time t=0. You can propagate it in time simply by replacing the time t=0 by an arbitrary time.

It is not necessary to do this explicitly in order to see that it can be done. Actually doing it is worth only for someone who believes that such fictions as exact 2-particle in states exist and have a useful explicit time evolution. I am not among those, and hence will not do it.

 

[meopemuk]

Of course one can do it.It is just that nobody has been interested to work it out since one can never experimentally follow the time evolution of a real electron or photon.
Experimentally you prepare it somewhere, and you detect it somewhere, and from the statistics you can calculate cross sections. That's all.

In the low energy macroscopic world of our everyday experience it is possible to see time-dependent trajectories of charged particles (charged drops of oil, or specks of dust or whatever) interacting with each other. So, a complete theory should be able to calculate not only scattering cross-sections, but also time-dependent evolution of wave functions.

The way to do it is this: Given an arbitrary state Psi in the Wightman representation, where all its creation fields are taken at time t=0, you can compute its inner product with an asymptotic Fock state Phi by taking the inner product with the physical state Phi(t) where all its creation fields are at the same time t and letting t --> -inf. Now pick Psi such that to the desired accuracy only the 2-particle contributions survive, and you have your desired 2-particle in state at time t=0. You can propagate it in time simply by replacing the time t=0 by an arbitrary time.

Why don't you just write the Hamiltonian? In quantum mechanics the Hamiltonian contains all one needs to know to describe the time evolution. Every relativistic quantum theory must have 10 Hermitian generators of the Poincare group, one of which is the Hamiltonian. If Wightman approach is such a theory, then it must have an explicitly written Hamiltonian.

If you are not comfortable of doing this in electrodynamics, then please write the Hamiltonian of a 2-dimensional Phi^4 theory, where everything is understood, as you say. If I understand correctly, the Wightman et al. approach permits creation of two separated physical bosons moving toward each other. I presume that at each later time point one can measure (the probabilities of) how many physical particles there are in the system and what are their observables. This time evolution must be unitary, so there should exist a Hermitian Hamiltonian responsible for it.

It is not necessary to do this explicitly in order to see that it can be done. Actually doing it is worth only for someone who believes that such fictions as exact 2-particle in states exist and have a useful explicit time evolution. I am not among those, and hence will not do it.

I have no problem with the fact that initial asymptotic 2-particle state may acquire components in n-particle sectors as time evolution progresses and the two original particles move closer to each other and interact stronger. The full Hamiltonian should be able to describe the appearance of such n-particle contributions. However, it is known from experiment that for slowly moving projectiles these n-particle contributions are negligible. For example, one usually neglects radiation processes in the theory of electronic circuits, where electrons move very slowly.

Eugene.

 

[A. Neumaier]

In the low energy macroscopic world of our everyday experience it is possible to see time-dependent trajectories of charged particles (charged drops of oil, or specks of dust or whatever) interacting with each other. So, a complete theory should be able to calculate not only scattering cross-sections, but also time-dependent evolution of wave functions.

But a charged drop of oil is macroscopic, and well described by the classical Maxwell equations, as every engineer can tell you. Thus one only needs to look at the macroscopic limit of QED - and I had already described how to get that.

Why don't you just write the Hamiltonian?

H \psi(x_1,t_1,...x_N,t_N) = -i\sum_k d/dt_k \psi(x_1,t_1,...x_N,t_N)
I wonder why you couldn't derive that yourself from the prescription I gave.
The explicit form adds nothing to the description.

I have no problem with the fact that initial asymptotic 2-particle state may acquire components in n-particle sectors as time evolution progresses and the two original particles move closer to each other and interact stronger. The full Hamiltonian should be able to describe the appearance of such n-particle contributions.

It does, in the way I described.

 

[meopemuk]

H \psi(x_1,t_1,...x_N,t_N) = -i\sum_k d/dt_k \psi(x_1,t_1,...x_N,t_N)
I wonder why you couldn't derive that yourself from the prescription I gave.
The explicit form adds nothing to the description.

Few questions:

1. Can I interpret \psi as a wave function? I.e., is the square of \psi the probability density?

2. What is the meaning of N? Is it the number of particles?

3. Why there are N time labels?

4. What is the t-dependence of \psi? Without such explicit t-dependence the Hamiltonian remains undefined.

5. How this form can be used to calculate the interacting time evolution of an initial 2-particle state?

Thanks.
Eugene.

 

[A. Neumaier]

1. Can I interpret \psi as a wave function? I.e., is the square of \psi the probability density?
2. What is the meaning of N? Is it the number of particles?
3. Why there are N time labels?
4. What is the t-dependence of \psi? Without such explicit t-dependence the Hamiltonian remains undefined.
5. How this form can be used to calculate the interacting time evolution of an initial 2-particle state?

1. psi is a state in the physical Hilbert space. No further interpretation is necessary. Asymptotic states get a interpretation as superpositions of bound state tensor products through Haag-Ruelle theory.
2. No. It is just a convenient label.
3. Because one needs to choose some N to write down a state. But as in Fock space, one can have superpositions of states with different values of N.
4. The t-dependence is defined as usual form the t-independent initial condition at t=0 by the Schroedinger equation. Defining the Hamiltonian itself needs no t.
5. This is your task to figure out, not mine. Nobody else needs it in any application of QED. But I'll guide you into a simplified exercise related to your question in the thread
https://www.physicsforums.com/showthread.php?p=3174961 . Working this out should give you enough intuition about the more complicated cases that you are interested in.

 

[meopemuk]

5. This is your task to figure out, not mine. Nobody else needs it in any application of QED.

In my opinion, studying the time evolution of initial 2-particle states is rather interesting and important. The most common thing calculated in QFT is the scattering amplitude in a 2-particle initial state. I hope you would agree that scattering is a dynamical time-dependent process. So, it would be very educational to follow the time evolution of the colliding system from distant past to distant future and see how scattering amplitudes appear from this time-dependent treatment.

Moreover the time resolved 2-particle scattering could be even investigated experimentally if sufficiently precise instruments are used.

Eugene.

 

[A. Neumaier]

In my opinion, studying the time evolution of initial 2-particle states is rather interesting and important. The most common thing calculated in QFT is the scattering amplitude in a 2-particle initial state. I hope you would agree that scattering is a dynamical time-dependent process. So, it would be very educational to follow the time evolution of the colliding system from distant past to distant future and see how scattering amplitudes appear from this time-dependent treatment.

One wouldn't see more than one sees it from the usual QM way of deriving scattering amplitudes. Some formulas that are discarded after one has the scattering angle.

I'll follow the educational route in the other thread. You think it is very instructive, so I'll enable you to do it. But I won't do it myself.

Moreover the time resolved 2-particle scattering could be even investigated experimentally if sufficiently precise instruments are used.

How would you do it? You claimed in the photon thread that one cannot say anything about what is not observed. But if you try to observe, you won't get the desired scattering. So the situation here is as in the case of the two slits...

 

[meopemuk]

One wouldn't see more than one sees it from the usual QM way of deriving scattering amplitudes. Some formulas that are discarded after one has the scattering angle.

Scattering amplitudes only connect states in the remote past and remote future. I am interested at intermediate time points.

How would you do it? You claimed in the photon thread that one cannot say anything about what is not observed. But if you try to observe, you won't get the desired scattering. So the situation here is as in the case of the two slits...

I don't quite understand your objection. Quantum mechanics tells us that any isolated physical system is described by a time-dependent wave function, which satisfies the Schroedinger equation. The wave function is generally a collection of complex numbers - amplitudes, whose squares are exactly the probabilities of certain measurements. It is tacitly assumed that these measurements can be performed and experimental probabilities will match the theoretical ones. So, there is nothing fancy in my suggestion to perform observations in a time-dependent state of colliding particles.

In reality, such observations are difficult to perform, because collisions occur in a very small region of space in a short time interval. This is why most people are completely satisfied with the S-matrix description. However "difficult" does not mean "impossible". A complete theory of physical event must be able to describe the unitary time evolution of interacting states.

Eugene.

 

[A. Neumaier]

The wave function is generally a collection of complex numbers - amplitudes, whose squares are exactly the probabilities of certain measurements. It is tacitly assumed that these measurements can be performed and experimental probabilities will match the theoretical ones.

It doesn't need to be assumed. The claim of the generally accepted minimal interpretation is only that _if_ you can perform a discrete and perfect projective measurement then you get results with probabilities conforming to Born's rule. (This assumption is practically realizable only in measurements of spin or polarization degrees of freedom, or of the ''presence'' of a particle at a detector.)

However "difficult" does not mean "impossible".

It is you who are claiming beyond the minimal consensus that it is not impossible. Thus you'd be able to tell us how it is possible.

 

[meopemuk]

It is you who are claiming beyond the minimal consensus that it is not impossible. Thus you'd be able to tell us how it is possible.

My understanding is that when in quantum mechanics we write the wave function \psi(x,t) then we assume the following meaning: If we make a device that detects the presence of the particle in the volume V and turn on this device at time t, then the probability of this device actually clicking will be

\int_V |\psi(x,t)|^2

I don't know how exactly this device can be made or what are the problems with experimental errors associated with such measurements. This is the job of experimentalist to worry about such details.

Do you think that my understanding is incorrect?

Eugene.

 

[A. Neumaier]

My understanding is that when in quantum mechanics we write the wave function \psi(x,t) then we assume the following meaning: If we make a device that detects the presence of the particle in the volume V and turn on this device at time t, then the probability of this device actually clicking will be

\int_V |\psi(x,t)|^2
Do you think that my understanding is incorrect?.

Yes; this is only the kindergarden version - it attaches to the abstract wave function a plausible (and never checked) interpretation, so that the young kids are not afraid and go on learning the weird stuff they are told. Those who need to work with it on a real life level must unlearn all the kindergarden tales...

Your formula cannot be correct since it takes time till the detector responds to the interaction, and a click needs time to be produced and measured. But your probability p_kindergarden is independent of the time. For a particle in a monochromatic beam, where psi(x,t) =e^i(itp_0-ix dot \p)psi_0, you get
p_kindergarden=V^3|psi_0|^2,
as long as V is so small that the plane wave approximation is valid.

To see how the real detection probability looks like, compare your formula with that given by Mandel & Wolf in (14.8-16). For the detection probability p_expert of a single monochromatic photon in a momentum eigenstate, take m=n=1, and remember that the waiting time interval T is supposed to be small, since O(T^2) effects are neglected. One gets the formula
p_expert=alpha c S T/L^3,
where alpha is the detector efficiency, c is the speed of light, S is the area of the detector, and L^3 is the volume V within which the photon energy is supposed to be uniformly distributed (because of the plane wave approximation).

Now this was derived for photon detection, but I doubt that it is very different for an electron.

 

[meopemuk]

Yes; this is only the kindergarden version - it attaches to the abstract wave function a plausible (and never checked) interpretation, so that the young kids are not afraid and go on learning the weird stuff they are told. Those who need to work with it on a real life level must unlearn all the kindergarden tales...

This just confirms my suspicion that you and I understand the basic quantum mechanics very very differently.

I may agree that measurements in real life cannot exactly reproduce the probability given in my kindergarten formula. But this is a purely technical problem. In principle (though, perhaps, not in practice), one can improve the experimental equipment so as to achieve the exact QM result. I have many issues with Mandel & Wolf interpretation of quantum measurements, and I prefer to remain in my happy kindergarten for now.

Eugene.

 

[A. Neumaier]

I may agree that measurements in real life cannot exactly reproduce the probability given in my kindergarten formula. But this is a purely technical problem. In principle (though, perhaps, not in practice), one can improve the experimental equipment so as to achieve the exact QM result.

The exact QM result is in Mandel & Wolf, not in the kindergarden world of introductory QM texts. In the latter, everything is heavily idealized, reducing the complexity of the real world to something that can be reproduced in an exam, even by the average student.

I have many issues with Mandel & Wolf interpretation of quantum measurements, and I prefer to remain in my happy kindergarten for now.

Well, Mandel and Wolf are world-famous experts on quantum optics, which is the basis for understanding quantum measurements and for testing the foundations of quantum mechanics. Learning from them is not a bad idea.

But it takes time to grow up.

I am teaching tough, for those who are prepared to grow.

Knowing that it is difficult, I offer help - but not compromises.

 

[meopemuk]

Well, Mandel and Wolf are world-famous experts on quantum optics, which is the basis for understanding quantum measurements and for testing the foundations of quantum mechanics. Learning from them is not a bad idea.

Well, Feynman is not a lesser expert, but his interpretation of the double-slit experiment is rather different from Mandel & Wolf's.

Eugene.

P.S. Feynman is actually the principal in my kindergarten.

 

[A. Neumaier]

Well, Feynman is not a lesser expert, but his interpretation of the double-slit experiment is rather different from Mandel & Wolf's.

But probably the last measurement he took was during his undergraduate studies.
And he probably never analyzed one on the quantum mechanical level in his whole career.

P.S. Feynman is actually the principal in my kindergarten.

Yes; he created the ''Feynman Lectures on Physics'' to teach kindergarden kids at university.

Even fairy tales told by the Brothers Grimm http://en.wikipedia.org/wiki/Brothers_Grimm don't change the fact that the stories are fairy tales.

 

[A. Neumaier]

For a particle in a monochromatic beam, where psi(x,t) =e^i(itp_0-ix dot \p)psi_0, you get
p_kindergarden=V^3|psi_0|^2,
as long as V is so small that the plane wave approximation is valid.

compare your formula with that given by Mandel & Wolf in (14.8-16). For the detection probability p_expert of a single monochromatic photon in a momentum eigenstate, take m=n=1, and remember that the waiting time interval T is supposed to be small, since O(T^2) effects are neglected. One gets the formula
p_expert=alpha c S T/L^3,
where alpha is the detector efficiency, c is the speed of light, S is the area of the detector, and L^3 is the volume V within which the photon energy is supposed to be uniformly distributed (because of the plane wave approximation).

Note that experiment agrees with p_expert, and contradicts p_kindergarden !

 

[meopemuk]

p_kindergarden=V^3|psi_0|^2,

p_expert=alpha c S T/L^3,

Note that experiment agrees with p_expert, and contradicts p_kindergarden !

The kindergarten formula guarantees that the probability is always between 0 and 1, which is kind of nice. Does the expert formula offer the same guarantee? I don't see it, even in the case when the detector efficiency alpha=1.


In general, I think we are separated by a big philosophical divide. I prefer to think that if there was a sparkle on a luminescent screen, this simply means that an electron (which is a tiny particle) hit exactly at this location.

Your (and M&W) interpretation is different. (Please correct me if I misrepresent your views. I surely misrepresent them, because I don't understand them) You represent the electron as a continuous extended field, which somehow excites atoms in the entire screen. This excitation conspires to produce a sparkle at a single location. The location of the sparkle is sort of unpredictable, because of the chaotic behavior of the atoms in the screen.

These are two completely different views on quantum mechanics and on the origin of quantum uncertainties. They are as different as the corpuscular and wave pictures of the world. I don't know much about real experiments, but I can believe that both these pictures can explain observations. I am working entirely in the corpuscular picture. If you know an experiment where this picture fails completely, this would be a big shock to me. I would like to see the exact reference.

Eugene.

Eugene.

 

[meopemuk]

Yes; he created the ''Feynman Lectures on Physics'' to teach kindergarden kids at university.

There are many other quantum mechanics textbooks teaching the same stuff: Landau & Livgarbagez, Ballentine, to name a few. Are you suggesting to through them away and read only Mandel & Wolf from now on?

Eugene.

 

[A. Neumaier]

There are many other quantum mechanics textbooks teaching the same stuff: Landau & Livgarbagez, Ballentine, to name a few.

It is stated in the beginning as an interpretation aid without proof, and never taken up again in the context of real measurements where the claim would have to be justified. It is very common to make this sort of idealized assumption to get started; but once the formalism is established, this assumption is never used again.

For example, Landau & Lifgarbagez begin in Section 2 of their Vol. 3 with such a statement, but immediately replace it in (2.1) and (3.10) by the more correct version about the interpretation of the expectation value <K> = Psi^* K Psi, where K is an arbitrary observable (linear integral operator) depending on the form and values of the measurement. From then on, only the latter interpretation is used; never the fictitious, idealized introductory remark.

And it cannot be different, since quantum mechanics is used in many situations where the state vectors used in the formalism have no interpretation as a function of position - the whole of quantum information theory and the whole of quantum optics belonging to this category.

 

[A. Neumaier]

The kindergarten formula guarantees that the probability is always between 0 and 1, which is kind of nice. Does the expert formula offer the same guarantee? I don't see it, even in the case when the detector efficiency alpha=1.

The expert formula is derived under the assumption that alpha T is small, since higher order terms are neglected in the derivation. Thus p<<1.

The kindergarden formula is underspecified since you are not saying which volume is the relevant volume. So it says nothing about the actual experimental situation.

In general, I think we are separated by a big philosophical divide. I prefer to think that if there was a sparkle on a luminescent screen, this simply means that an electron (which is a tiny particle) hit exactly at this location.

Yes, we are at opposite sides of the particle -wave spectrum of interpretations.

You represent the electron as a continuous extended field, which somehow excites atoms in the entire screen. This excitation conspires to produce a sparkle at a single location.

Nothing needs to conspire; each excitable atom acts independently of the other, according to the incident intensity.

The location of the sparkle is sort of unpredictable, because of the chaotic behavior of the atoms in the screen.

Yes. This is what the usual semiclassical analysis reveals. Under ordinary circumstances, quantum corrections are tiny and can be neglected.

These are two completely different views on quantum mechanics and on the origin of quantum uncertainties. They are as different as the corpuscular and wave pictures of the world.

Yes, and the modern view is that Nature is governed by quantum field theory, not by Schroedinger equations for many-particle systems.

If you know an experiment where this picture fails completely, this would be a big shock to me.

We are going in circles. I had mentioned spontaneously broken theories, etc.. But you discounted as pure speculation what others found worthy of a Nobel prize.

 

[meopemuk]

The expert formula is derived under the assumption that alpha T is small, since higher order terms are neglected in the derivation. Thus p<<1.

But when I shine one photon on the photographic plate I get the detection probability of almost 1. Is your theory applicable in this case?

The kindergarden formula is underspecified since you are not saying which volume is the relevant volume. So it says nothing about the actual experimental situation.

V^3 is the volume of the particle detector. This could be the volume of the photographic plate, for example.

We are going in circles. I had mentioned spontaneously broken theories, etc.. But you discounted as pure speculation what others found worthy of a Nobel prize.

Well, I've asked you about *experimental* refutation of the probability interpretation of wave functions. You are talking about *theories*, which are not confirmed yet, by the way.

Eugene.

 

[meopemuk]

For example, Landau & Lifgarbagez begin in Section 2 of their Vol. 3 with such a statement, but immediately replace it in (2.1) and (3.10) by the more correct version about the interpretation of the expectation value <K> = Psi^* K Psi, where K is an arbitrary observable (linear integral operator) depending on the form and values of the measurement. From then on, only the latter interpretation is used; never the fictitious, idealized introductory remark.

The probability

\int_{V} |\psi(x)|^2 dx


can be also regarded as expectation value. In this case the Hermitian operator of observable is the projection on the subset V of the position operator spectrum. This observable can be characterized as a "yes-no experiment" or a question "is the particle present in the volume V?"

Eugene.

 

[meopemuk]

It [the probabilistic interpretation of wave function] is stated in the beginning as an interpretation aid without proof, and never taken up again in the context of real measurements where the claim would have to be justified. It is very common to make this sort of idealized assumption to get started; but once the formalism is established, this assumption is never used again.

The probabilistic interpretation is the centerpiece of the quantum logic approach to quantum mechanics. See chapter 1 in my book. In this approach, quantum mechanics is nothing but a modified (rather generalized) version of the probability theory. The probabilistic interpretation is used everywhere where QM is applied. The most basic things calculated in QM are probabilities of measurements. How are you going to get them without wave function and its probabilistic interpretation? I am very surprised to learn that other opinions exist on this matter.

Eugene.

 

[meopemuk]

Nothing needs to conspire; each excitable atom acts independently of the other, according to the incident intensity.

No, there is a huge conspiracy there. In your theory, before approaching the screen the electron is represented by a continuous field. This means that the electron charge is spread over large area. Nevertheless, the "click" occurs only in one place. I hope you wouldn't deny the fact that after the "click" the full electron charge resides in the neighborhood of the "clicked" atom. So, somehow this charge density cloud has condensed at one point. Not at two points, not at three points - always at one point. What is the explanation of this mysterious behavior?

The idea of independent excitable atoms does not seem to be a good explanation. It seems that all atoms in the screen agree to choose one of them as the "condensation" point and to send their portion of the incident electron's charge density exactly to this point. I call it a conspiracy.

Eugene.

 

[A. Neumaier]

The idea of independent excitable atoms does not seem to be a good explanation. It seems that all atoms in the screen agree to choose one of them as the "condensation" point and to send their portion of the incident electron's charge density exactly to this point. I call it a conspiracy.

If that is a conspiracy, then all the experiments done with Alice and Bob point to the same sort of conspiracy between far away particles in your favored interpretation.
This only proves that this sort of conspiracy is an unavoidable feature of QM.

 

[meopemuk]

If that is a conspiracy, then all the experiments done with Alice and Bob point to the same sort of conspiracy between far away particles in your favored interpretation.
This only proves that this sort of conspiracy is an unavoidable feature of QM.

Are you talking about entanglement?

I think that the conspiracy you are suggesting is much more troubling than entanglement. Your approach needs a real flow of charge density to one point from around the entire area of the detector. This is despite the fact that significant electrostatic repulsion needs to be overcome in order to concentrate the charge in one place. It seems that this flow of charge density can happen equally effectively independent on whether the detector is a conductor or an insulator.

On the other hand, nothing of that sort is needed in the particle interpretation. The "click" happens simply because a point-like electron hits the atom. End of story.

Eugene.

 

[strangerep]

[...]
So, somehow this charge density cloud has condensed at one point.
Not at two points, not at three points - always at one point.

It's misleading to say "always at one point". One must take account of the
time interval in the formula. There's a nonzero probability for more than
one click in a given time interval (assuming the incident beam remains "on").
And if one makes the time interval very small there's a very good
chance that nothing at all happens at any given point on the detector.

Such information involving time intervals is missing if one relies
only on the over-simplified picture that |\psi(x)|^2 is the
probability for finding the particle at x.

 

[Duderonimous]

This is an argument you don't get to witness first hand too often. Take notes kids.

 

[meopemuk]

It's not "always at one point". One must take account of the time interval
in the formula. There's a nonzero probability for more than one click in
a given time interval (assuming the incident beam remains "on").
And if one makes the time interval very small there's a very good
chance that nothing at all happens at any given point on the detector.

Such information involving time intervals is missing if one relies
only on the over-simplified picture that |\psi(x)|^2 is the
probability for finding the particle at x.

But I have arranged my experiment so that only one electron was emitted. So, if the detector's efficiency is 1, I must get one and only one click, provided that I've waited long enough to allow the electron to reach the detector.

Eugene.

 

[strangerep]

But I have arranged my experiment so that
only one electron was emitted.

Where did you "arrange" that? I scanned back through the earlier posts,
but didn't see the experimental arrangement described.

So, if the detector's efficiency is 1, I must get one and only one click,
provided that I've waited long enough to allow the electron to reach the detector.

Then you don't have an ensemble, so an interpretation involving probability
becomes problematic. If you do set up many repetitions of the experiment,
it's indistinguishable from a very low intensity beam striking a target.

 

[meopemuk]

Where did you "arrange" that? I scanned back through the earlier posts,
but didn't see the experimental arrangement described.

Sorry if it wasn't clear. In our discussions with Arnold we went back and forth between different threads, so this piece of info could be lost. In my posts I've always assumed emission of particles one-by-one. Here I've intentionally switched from photons (whose one-by-one emission can be problematic) to electrons, which can be easily emitted one-by-one, carry a unit charge and cannot be divided into smaller pieces.

Then you don't have an ensemble, so an interpretation involving probability
becomes problematic. If you do set up many repetitions of the experiment,
it's indistinguishable from a very low intensity beam striking a target.

I shoot one electron and get one dot on the luminescent screen or whatever detector was chosen. This is one member of the ensemble. Then I shoot the second electron and get the second dot in a different place. This is the second member of the ensemble. Then I repeat this procedure as many times as needed in order to form a representative ensemble of measurements. I am sure that each electron has produced a single dot for me. I also see that the distribution of dots on the screen forms a characteristic interference pattern. (I am doing the double-slit experiment here.)

From these observations I make a few conclusions:

1. Electrons are point-like particles, and luminescent sparkles are created by direct hits of incident electrons.

2. For each area of the detector I can measure the probability of it being hit by electrons. This is the number of electrons that have landed in this area divided by the total number of electron emitted.

3. For this double-slit setup I can calculate 1-electron quantum-mechanical wavefunction in the vicinity of the screen. Taking square of this wavefunction and integrating over area I get exactly the same probability as the one measured in 2.

So, standard 1-electron quantum mechanics gives a perfect description of the double-slit experiment in the corpuscular picture.

Eugene.

 

[A. Neumaier]

I think that the conspiracy you are suggesting is much more troubling than entanglement. Your approach needs a real flow of charge density to one point from around the entire area of the detector.

Unobservable fictions of the imagination need not be conserved.

The observable charge flow is the expectation of the current operator defined by the electron field. It satisfies the continuity equation rigorously and hence fully accounts for charge conservation.

 

[meopemuk]

The observable charge flow is the expectation of the current operator defined by the electron field. It satisfies the continuity equation rigorously and hence fully accounts for charge conservation.

I don't dispute the fact that the total charge is conserved in your model. But I don't understand what is the mechanism that forces this charge density to condense to one point against the force of Coulomb repulsion.

Eugene.

 

[A. Neumaier]

I don't dispute the fact that the total charge is conserved in your model. But I don't understand what is the mechanism that forces this charge density to condense to one point against the force of Coulomb repulsion.

The continuity equation not only tells that the total charge is conserved but also that at _every_ point in space and _every_ moment in time the inflow and outflow of charge balance exactly in the ensemble mean.

Nowhere in quantum mechanics one can have better conservation laws for energy and momentum - not even in the quantum mechanics of two nonrelativistic particles.

Thus requiring it of charge is unreasonable (and in fact untestable).

 

[meopemuk]

The continuity equation not only tells that the total charge is conserved but also that at _every_ point in space and _every_ moment in time the inflow and outflow of charge balance exactly in the ensemble mean.

Nowhere in quantum mechanics one can have better conservation laws for energy and momentum - not even in the quantum mechanics of two nonrelativistic particles.

Thus requiring it of charge is unreasonable (and in fact untestable).

Sorry, I don't understand your point here. I was not questioning the validity of continuity equation in your model. I can even close my eyes on non-conservation of energy and momentum. But what is really strange is the ability of the charge density to shrink to a point in some instances. This is reminescent of the QM wave function collapse, but much more troubling, because in your case the collapsing thing is not the imaginary probability density amplitude (as in QM), but real physical charge density.

Eugene.

 

[A. Neumaier]

Sorry, I don't understand your point here. I was not questioning the validity of continuity equation in your model.

You raised the objection that charge must flow to the firing point, and I replied that the continuity equation proves that nothing can be wrong with the flow of charge. (Tiny flows of charge happen all the time in a macroscopic body.)

I can even close my eyes on non-conservation of energy and momentum.

You have to close your eye on non-conservation of momentum already if you consider a particle picture of a double slit experiment!

But what is really strange is the ability of the charge density to shrink to a point in some instances. This is reminiscent of the QM wave function collapse, but much more troubling, because in your case the collapsing thing is not the imaginary probability density amplitude (as in QM), but real physical charge density.

The charge flows to the point where it is needed. It doesn't need to contract miraculously and instantly from everywhere. The fired electron removes charge locally, and this local deficiency is corrected for by inflow of a little bit of charge density from the neighborhood. (It is not really different from the flow of water in a bucket that has a little hole at the bottom from which drops leak out at random times, while the compensating inflow happens steadily at some other place.) The covariance and locality of quantum field theory ensures that nothing happens instantly.

 

[meopemuk]

The charge flows to the point where it is needed. It doesn't need to contract miraculously and instantly from everywhere. The fired electron removes charge locally, and this local deficiency is corrected for by inflow of a little bit of charge density from the neighborhood. (It is not really different from the flow of water in a bucket that has a little hole at the bottom from which drops leak out at random times, while the compensating inflow happens steadily at some other place.) The covariance and locality of quantum field theory ensures that nothing happens instantly.

I don't find your explanation convincing. I don't know of any physical force that would suck the whole widely distributed charge density to a single point. Charge densities normally have the tendency of self-repulsion. Here we have an example of a completely opposite effect. What kind of Maxwell equation can be written to demonstrate such an abnormal behavior?

Eugene.

 

[A. Neumaier]

I don't find your explanation convincing. I don't know of any physical force that would suck the whole widely distributed charge density to a single point. Charge densities normally have the tendency of self-repulsion. Here we have an example of a completely opposite effect. What kind of Maxwell equation can be written to demonstrate such an abnormal behavior?

There is a nontrivial and very rugged charge distribution already in the detector before anything arrives at it. Even in a single molecule like water you have a nontrivial charge distribution - one of the reasons water behaves as it does. This charge distribution changes and adapts continuously while the arriving wave reaches the body, and it keeps doing so during the time it takes to emit the electron.

Thus everything happens continuously, in conformance with the continuity equation. Nothing needs to be sucked into a point - the emitted electron is itself a moving charge distribution, radially expanding in its rest frame.

 

[meopemuk]

Nothing needs to be sucked into a point - the emitted electron is itself a moving charge distribution, radially expanding in its rest frame.

According to you, before the measurement the electron had the form of a charge density cloud extending over the range of several centimeters. After the measurement we see that one CCD pixel has its charge changed by -e, while all other pixels stay with the same (=0) charge. This looks like "sucked into a point" to me. How this behavior can be achieved by a "moving charge distribution, radially expanding in its rest frame"?

Eugene.

 

[A. Neumaier]

According to you, before the measurement the electron had the form of a charge density cloud extending over the range of several centimeters. After the measurement we see that one CCD pixel has its charge changed by -e, while all other pixels stay with the same (=0) charge. This looks like "sucked into a point" to me. How this behavior can be achieved by a "moving charge distribution, radially expanding in its rest frame"?

The charges are only moved a little, not concentrated to a point from everywhere:

A charge-coupled device (CCD) is a device for the movement of electrical charge, usually from within the device to an area where the charge can be manipulated, for example conversion into a digital value. This is achieved by "shifting" the signals between stages within the device one at a time. CCDs move charge between capacitive bins in the device, with the shift allowing for the transfer of charge between bins.
Often the device is integrated with an image sensor, such as a photoelectric device to produce the charge that is being read

(taken from http://en.wikipedia.org/wiki/Charge-coupled_device )

 

[meopemuk]

The charges are only moved a little, not concentrated to a point from everywhere:

Well, then we disagree on the meaning of the words "moved" and "concentrated".

Eugene.

 

[meopemuk]

Arnold,

I think we don't understand each other, because we are talking about different charges. The wikipedia article is talking about the charge, which moves from "an image sensor, such as a photoelectric device" to "an area where the charge can be manipulated, for example conversion into a digital value". All this charge movement occurs within one CCD array pixel in a well controlled fashion. This is a part of the registration process, i.e., the part of the process, which records the fact that the "image sensor" of that pixel has picked up some signal from an impinging photon or electron or some other external particle.

I was talking about the charge density wave, which you attribute to the single electron, which impinges on the CCD array from the outside. The integral of this charge density is equal to the electron's charge -e. When this wave approaches the array of CCD detectors, its density is distributed across a large area covering many pixels. However, only one pixel produces a signal. The whole initial charge (-e) of the wave gets concentrated within this one pixel. So, apparently, the initial distributed charge density has shrinked to the size of one pixel. This charge movement/migration/concentration/collapse happens across the whole area of the detector array. This abrupt shrinkage is different from the controlled charge migration in the circuitry attached to one particular pixel as described in the wikipedia article.

Eugene.

 

[A. Neumaier]

I think we don't understand each other, because we are talking about different charges.

Yes. You ignore all the charge that is already present in the CCD, and concentrate on the charge of the electron - of course if you do that, everything looks strange. But once you take into account the complete charge field, including that of the CCD, everything falls into place.

The wikipedia article is talking about the charge, which moves from "an image sensor, such as a photoelectric device" to "an area where the charge can be manipulated, for example conversion into a digital value". All this charge movement occurs within one CCD array pixel in a well controlled fashion. This is a part of the registration process, i.e., the part of the process, which records the fact that the "image sensor" of that pixel has picked up some signal from an impinging photon or electron or some other external particle.

I was talking about the charge density wave, which you attribute to the single electron, which impinges on the CCD array from the outside. The integral of this charge density is equal to the electron's charge -e. When this wave approaches the array of CCD detectors, its density is distributed across a large area covering many pixels. However, only one pixel produces a signal. The whole initial charge (-e) of the wave gets concentrated within this one pixel. So, apparently, the initial distributed charge density has shrinked to the size of one pixel. This charge movement/migration/concentration/collapse happens across the whole area of the detector array. This abrupt shrinkage is different from the controlled charge migration in the circuitry attached to one particular pixel as described in the wikipedia article.

Nothing shrinks abruptly. The new charge of total amount -e arrives continuously , thereby changing the charge distribution of the CCD continuously. This changing charge distribution undergoes a continuous evolution according to the laws of QM.
When a random pixels produces a signal, the local charge near that pixel is rearranged. Nowhere is there any shrinking or any abrupt change in the charge distribution.

 

[meopemuk]

Nothing shrinks abruptly. The new charge of total amount -e arrives continuously , thereby changing the charge distribution of the CCD continuously. This changing charge distribution undergoes a continuous evolution according to the laws of QM.
When a random pixels produces a signal, the local charge near that pixel is rearranged. Nowhere is there any shrinking or any abrupt change in the charge distribution.

Please let me know where is the gap in my logic: Suppose that we have a 1000X1000 CCD array with 1000000 pixels. Before the experiment the total charge of each pixel and attached circuitry is around zero. We send one electron in the form of a spread-out wave toward the array. The total charge of the electron wave is -e. The amount of charge projected onto each pixel is -e/1000000. So, just after the wave has touched the array the total charge of each pixel is -e/1000000. Then, according to you, some charge migration occurs between the pixels, so that the whole initial charge -e gets concentrated in one and only one pixel.

You say that this charge rearrangement occurs due to "a continuous evolution according to the laws of QM". I am wondering, what kind of physical interaction is responsible for such an unusual charge rearrangement? Are there any references discussing this interesting effect?

Eugene.

 

[A. Neumaier]

Please let me know where is the gap in my logic: Suppose that we have a 1000X1000 CCD array with 1000000 pixels. Before the experiment the total charge of each pixel and attached circuitry is around zero.

In this approximation, the charge arriving from the electron wave is also around zero.
So your argument amounts to redistributing around zero charge.

We send one electron in the form of a spread-out wave toward the array. The total charge of the electron wave is -e. The amount of charge projected onto each pixel is -e/1000000. So, just after the wave has touched the array the total charge of each pixel is -e/1000000. Then, according to you, some charge migration occurs between the pixels, so that the whole initial charge -e gets concentrated in one and only one pixel.

You say that this charge rearrangement occurs due to "a continuous evolution according to the laws of QM". I am wondering, what kind of physical interaction is responsible for such an unusual charge rearrangement? Are there any references discussing this interesting effect?

There is nothing interesting about this.

The charge distribution at a time t before the wave reaches the detector is Q(x,t). The integral over x is Q approx 0 (but can well be of the order of e), but locally Q(x,t) is nonzero. Q(x,t) changes stochastically with time, preserving the zero mean but fluctuating via small random movements of the local charge distribution. When the electron wave begins to touch the detector, Q(x,t) increases locally a tiny amount wherever the interference pattern allows it. The total charge increases, until after the whole wave packet reached the detector, the charge has changed by -e. The stochastic process governing the local redistribution of the charge is influenced by the external field and affects the way the charge field changes. At some time, one of the pixels fires, by transferring a whole electron to the register. This is again accompanied by a local change of charge density only.

Charge current is always locally conserved, and nowhere is a sign of the spooky process you claim must have happened.

 

[meopemuk]

The charge distribution at a time t before the wave reaches the detector is Q(x,t). The integral over x is Q approx 0 (but can well be of the order of e), but locally Q(x,t) is nonzero. Q(x,t) changes stochastically with time, preserving the zero mean but fluctuating via small random movements of the local charge distribution. When the electron wave begins to touch the detector, Q(x,t) increases locally a tiny amount wherever the interference pattern allows it. The total charge increases, until after the whole wave packet reached the detector, the charge has changed by -e. The stochastic process governing the local redistribution of the charge is influenced by the external field and affects the way the charge field changes. At some time, one of the pixels fires, by transferring a whole electron to the register. This is again accompanied by a local change of charge density only.

In this continuous charge model, is there an explanation for the fact that in observed systems (molecular ions or Millikan oil drops) we always see integer number of unit charges e?

Eugene.

 

[A. Neumaier]

In this continuous charge model, is there an explanation for the fact that in observed systems (molecular ions or Millikan oil drops) we always see integer number of unit charges e?

Of course. Total charge is conserved and quantized in QFT, and there is a corresponding superselection rule (while there is none for particle number). Thus at microscopically large distances one can separate only integer number of charges.

Things are different for charges embedded in matter, where fractional charges may appear in quantum wires.

 

[meopemuk]

Arnold,

I think we now agree that the field-based view (which you advocate) and the particle-based view (which I defend in my book) are two completely different ways to think about nature. I can possibly agree that both these ways have the right to exist and coexist. However, so far you wasn't able to convince me that the particle-based view has any fundamental flaws associated with it.

By the way, I should thank you again for the hint regarding IR infinities in radiative corrections. Using this idea I was able to reproduce the Uehling potential and the electron's anomalous magnetic moment in my approach. I will add this stuff to the next revision of the book. The Lamb shift is a tougher cookie. It would possibly require a full-blown Kulish-Faddeev-type infrared theory. I am working on it.

Eugene.

 

[A. Neumaier]

I think we now agree that the field-based view (which you advocate) and the particle-based view (which I defend in my book) are two completely different ways to think about nature. I can possibly agree that both these ways have the right to exist and coexist. However, so far you wasn't able to convince me that the particle-based view has any fundamental flaws associated with it.

It is _fundamentally_ flawed _only_ when there are massless fields, since then your Hamiltonian is not self-adjoint (else it would generate a finite perturbation series without IR divergences).

For massive QED, your approach is basically sound, and only the weird discussion about causality you associate with it is flawed. If you want to get insight into the latter, please respond to the thread https://www.physicsforums.com/showthread.php?t=474571