Exploring the Possibilities of a New Relativistic Quantum Theory

[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

 

[meopemuk]

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).

Do you have a proof that the dressed Hamiltonian I am using is not adequate?

... 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

Why do you think that my discussion of causality is "weird"? Perhaps we can discuss it here. I am not sure this discussion belongs to the thread "What is observable in a relativistic quantum field theory?" We already agreed that my approach is *not* a "quantum field theory" in your understanding. In your post you claim as something self-evident that "...relativity forbids the communication of information at a speed >c" and "...relativity forbids the propagation of influences at a speed >c." These claims are not evident to me. If by "relativity" you mean a theory based on two Einstein's postulates, then this theory cannot make such sweeping statements, because the second postulate (the invariance of the speed of light) refers only to one particular kind of particles - free massless photons - and therefore cannot be applied universally to all physical systems.

Perhaps you would like to add some other postulates to those used by Einstein? What are they?

Eugene.

 

[A. Neumaier]

Do you have a proof that the dressed Hamiltonian I am using is not adequate?

The occurrence of infrared divergences in the perturbative expansion is proof that your Hamiltonian is not self-adjoint, and hence inadequate.

Why do you think that my discussion of causality is "weird"? Perhaps we can discuss it here. I am not sure this discussion belongs to the thread "What is observable in a relativistic quantum field theory?" We already agreed that my approach is *not* a "quantum field theory" in your understanding.

OK, but let us discuss the classical case first. Do you think the conclusion at the end of Section I in the paper ''Relativistic Invariance and Hamiltonian Theories of Interacting Particles'' by Currie et al., Rev. Mod. Phys. 35, 350–375 (1963) is sound?

 

[meopemuk]

The occurrence of infrared divergences in the perturbative expansion is proof that your Hamiltonian is not self-adjoint, and hence inadequate.

In my approach the dressed particle interaction potential is simply fitted to renormalized scattering amplitudes in each perturbation order. Ideally, in traditional QED, such amplitudes should be available without any UV or IR divergences present. So, the dressed particle Hamiltonian must exist and must be divergence-free.

My only problem is that I haven't learned QED well enough to understand how to cancel IR divergences in the 4th order renormalized amplitude for electron-proton scattering. This is not a fundamental problem of the dressed particle approach, this is simply a result of my learning disability. As you correctly pointed out, IR divergences do not play any role in the Uehling potential and in the anomalous magnetic moment of the electron. Indeed, in my approach I got corresponding contributions to the dressed Hamiltonian in full agreement with established knowledge. The only remaining part is the dressed interaction responsible for the Lamb shift. I believe, this part of interaction can be obtained by using dressed particle approach in combination with Kulish-Faddeev scattering theory. I am working on it and I'll be happy to report the results to you when I'm done.

OK, but let us discuss the classical case first. Do you think the conclusion at the end of Section I in the paper ''Relativistic Invariance and Hamiltonian Theories of Interacting Particles'' by Currie et al., Rev. Mod. Phys. 35, 350–375 (1963) is sound?

Yes, I agree with this conclusion. This is actually the whole point of the paper: In interacting particle theories the relativistic invariance (=Poincare commutators) and the manifest covariance (=specific simple transformation rules for particle positions) cannot coexist. Since I am developing an interacting particle theory, this theorem has direct relevance to my approach. My solution to this "paradox" is to ignore the requirement of manifest covariance. I don't know where this requirement comes from, so I don't think it is important.

Eugene.

 

[A. Neumaier]

In my approach the dressed particle interaction potential is simply fitted to renormalized scattering amplitudes in each perturbation order. Ideally, in traditional QED, such amplitudes should be available without any UV or IR divergences present. So, the dressed particle Hamiltonian must exist and must be divergence-free.

That it must exist is your assumption (indeed only a disproved hope), not a theorem.

You ignore the infraparticle structure of electrons, and this invalidates your approach.

In contrast, the Wightman representation with the Wightman functions constructed perturbatively by Steinmann in http://archive.numdam.org/ARCHIVE/A...A_1995__63_4_399_0/AIHPA_1995__63_4_399_0.pdf is manifestly UV and IR finite. But its Hilbert space is not a Fock space.

My only problem is that I haven't learned QED well enough to understand how to cancel IR divergences in the 4th order renormalized amplitude for electron-proton scattering. This is not a fundamental problem of the dressed particle approach, this is simply a result of my learning disability.

You may believe that, but your lack of knowledge of more complex QED techniques also implies a lack of knowledge of structural results that make your dream impossible to carry out.

Yes, I agree with this conclusion. This is actually the whole point of the paper: In interacting particle theories the relativistic invariance (=Poincare commutators) and the manifest covariance (=specific simple transformation rules for particle positions) cannot coexist. Since I am developing an interacting particle theory, this theorem has direct relevance to my approach. My solution to this "paradox" is to ignore the requirement of manifest covariance. I don't know where this requirement comes from, so I don't think it is important.

But they do not assume _manifest_ covariance in the usual sense of the word. They assume only:
(i-iii) the three equations on p. 351;
(iv) that ''the generators P and J are assumed to have the standard form'';
(v) that ''the generators H, P, J , and K satisfy the Poisson bracket equations characteristic of the Lorentz group''.

So precisely which of their assumptions are you giving up?
And what do you have instead that makes the theory observer independent?

 

[meopemuk]

That it must exist is your assumption (indeed only a disproved hope), not a theorem. [\QUOTE]

In order to find the 4-th order contribution to the dressed interaction (on the energy shell) I need to solve equation (10.29). F_4^c on the right hand side is the 4-th order S-operator obtained in a traditional QFT calculation. This must be free of UV and IR infinities. V_2^2 is the finite 2nd order dressed particle interaction obtained in a previous step (10.27). For QED this interaction is described in section 10.3. The commutator term involves loop integrals, but these integrals are guaranteed to be convergent as explained in the last paragraph of subsection 10.2.7. So, it looks like a proven theorem to me.

You ignore the infraparticle structure of electrons, and this invalidates your approach. [\QUOTE]

What do you mean by "infraparticle structure of electrons" and how it can be observed?

In contrast, the Wightman representation with the Wightman functions constructed perturbatively by Steinmann in http://archive.numdam.org/ARCHIVE/A...A_1995__63_4_399_0/AIHPA_1995__63_4_399_0.pdf is manifestly UV and IR finite. But its Hilbert space is not a Fock space. [\QUOTE]

I thought we agreed not to mix field-based approach (Wightman & Steinmann) and particle-based approach (my book) in this thread.

But they do not assume _manifest_ covariance in the usual sense of the word. They assume only:
(i-iii) the three equations on p. 351;
(iv) that ''the generators P and J are assumed to have the standard form'';
(v) that ''the generators H, P, J , and K satisfy the Poisson bracket equations characteristic of the Lorentz group''.

So precisely which of their assumptions are you giving up?
And what do you have instead that makes the theory observer independent?

In the context of this paper, the assumption of _manifest_ covariance is equivalent to your assumption (iii). This is exactly the assumption, which I am willing to sacrifice. By doing that, I am not affecting the "observer independence". This property is guaranteed once we have assumption (v) in place.

Eugene.

 

[A. Neumaier]

In order to find the 4-th order contribution to the dressed interaction (on the energy shell) I need to solve equation (10.29). F_4^c on the right hand side is the 4-th order S-operator obtained in a traditional QFT calculation. This must be free of UV and IR infinities. V_2^2 is the finite 2nd order dressed particle interaction obtained in a previous step (10.27). For QED this interaction is described in section 10.3. The commutator term involves loop integrals, but these integrals are guaranteed to be convergent as explained in the last paragraph of subsection 10.2.7. So, it looks like a proven theorem to me.

Why then do you get IR problems? If things were IR finite, you could just evaluate them.

What do you mean by "infraparticle structure of electrons" and how it can be observed?

The intuition about it is discussed, e.g., in the introduction of http://arxiv.org/pdf/hep-th/9704212 . See also Section 6 of http://arxiv.org/pdf/math-ph/0509047

Annals of Physics 210 (1991), 112-136 discusses the relations to Kulish and Faddeev.

I thought we agreed not to mix field-based approach (Wightman & Steinmann) and particle-based approach (my book) in this thread.

We don't need to discuss it, but I still point out relationships and differences.

In the context of this paper, the assumption of _manifest_ covariance is equivalent to your assumption (iii). This is exactly the assumption, which I am willing to sacrifice. By doing that, I am not affecting the "observer independence". This property is guaranteed once we have assumption (v) in place.

What is your replacement for property (iii), i.e., what do you get for the commutator?

 

[meopemuk]

In order to find the 4-th order contribution to the dressed interaction (on the energy shell) I need to solve equation (10.29). F_4^c on the right hand side is the 4-th order S-operator obtained in a traditional QFT calculation. This must be free of UV and IR infinities. V_2^2 is the finite 2nd order dressed particle interaction obtained in a previous step (10.27). For QED this interaction is described in section 10.3. The commutator term involves loop integrals, but these integrals are guaranteed to be convergent as explained in the last paragraph of subsection 10.2.7. So, it looks like a proven theorem to me.

Why then do you get IR problems? If things were IR finite, you could just evaluate them.

First I would like to make a correction. You should read "these integrals are guaranteed to be *UV* convergent". In the case of the electron-proton interaction the UV convergence is not a problem. The problem is that this integral is IR-divergent. This integral is nothing but a next-to-the-leading-order contribution to the scattering amplitude of two charged particles. This contribution has been studied extensively beginning from the work

R. H. Dalitz, "On higher Born approximations in potential scattering", Proc. Roy. Soc. A206 (1951), 509

The source of the IR problem is also well-known: this is the persistent perturbation of asymptotic states due to the long-range Coulomb potential between two charges. The solution lies in Kulish-Faddeev scattering theory. So, I am positive that I'll be able to solve this riddle in the future.

Eugene.

 

[meopemuk]

What is your replacement for property (iii), i.e., what do you get for the commutator?

In the non-interacting case the commutator is the same as in CJS paper. In the interacting case both K and H generators contain interaction terms: H=H_0+V and K = K_0+Z, and one cannot guarantee that (iii) is satisfied. The actual commutator [Q,K] is a non-trivial expression, which depends on the chosen interaction terms V and Z. In somewhat abstract form this commutator is presented in (11.42).

Eugene.

 

[A. Neumaier]

In the non-interacting case the commutator is the same as in CJS paper. In the interacting case both K and H generators contain interaction terms: H=H_0+V and K = K_0+Z, and one cannot guarantee that (iii) is satisfied. The actual commutator [Q,K] is a non-trivial expression, which depends on the chosen interaction terms V and Z. In somewhat abstract form this commutator is presented in (11.42).

Writing H=H_0+g V +O(g^2) and K=K_0+g Z +O(g^2), could you please show how the W_{ij} term in
[Q_i,K_j]=Q_j[Q_i,H] + g W_{ij} + O(g^2) looks like?

 

[A. Neumaier]

I agree that the perturbative dressing transform accounts for all UV problems, and hence would be adequate for a perturbative Hamiltonian treatment of massive QED.

The problem is that this integral is IR-divergent.

This is proof that your Hamiltonian is not well-defined as a self-adjoint operator. If it were, the perturbation expansion would be well-defined at all orders.

The source of the IR problem is also well-known: this is the persistent perturbation of asymptotic states due to the long-range Coulomb potential between two charges. The solution lies in Kulish-Faddeev scattering theory.

But they change the Fock space to a much bigger inseparable Hilbert space, to accommodate the soft photon clouds.

 

[meopemuk]

Writing H=H_0+g V +O(g^2) and K=K_0+g Z +O(g^2), could you please show how the W_{ij} term in
[Q_i,K_j]=Q_j[Q_i,H] + g W_{ij} + O(g^2) looks like?

I can prove that the W_{ij} term is non-zero in the case of two classical charged particles.

First consider the Poisson bracket [Q,K]. In the case of (v/c)^2 approximation, K is the sum of (O.4) and (O.5) in Appendix O. For Q I take position r1 of the particle 1. Since interaction term (O.5) has zero bracket with r1, I conclude that [Q,K] does not depend on the coupling constant (q1q2 in my notation) in this approximation.

The Poisson bracket [Q,H] is calculated in (12.11) in the same approximation. This expression clearly depends on q1q2. So, it is impossible to satisfy [Q,K]=Q[Q,H]. There should be additional terms gW on the right hand side, which are not difficult to calculate explicitly.

Eugene.

 

[meopemuk]

This is proof that your Hamiltonian is not well-defined as a self-adjoint operator. If it were, the perturbation expansion would be well-defined at all orders.

I would rather say that traditional scattering theory (with asymptotic dynamics governed by H_0) is not adequate for charged particles, as was demonstrated by Kulish and Faddeev. My reliance on this (wrong) traditional theory is the reason for IR divergences in the Hamiltonian. If I switch to the (correct) Kulish-Faddeev scattering theory, the spurious IR divergences will disappear, I hope. Then I will be able to derive a fully consistent finite dressed particle Hamiltonian.

Eugene.

 

[A. Neumaier]

I can prove that the W_{ij} term is non-zero in the case of two classical charged particles.

First consider the Poisson bracket [Q,K]. In the case of (v/c)^2 approximation, K is the sum of (O.4) and (O.5) in Appendix O. For Q I take position r1 of the particle 1. Since interaction term (O.5) has zero bracket with r1, I conclude that [Q,K] does not depend on the coupling constant (q1q2 in my notation) in this approximation.

The Poisson bracket [Q,H] is calculated in (12.11) in the same approximation. This expression clearly depends on q1q2. So, it is impossible to satisfy [Q,K]=Q[Q,H]. There should be additional terms gW on the right hand side, which are not difficult to calculate explicitly.

Could you please get it explicitly for me? You may simplify the form of V and Z as much as is still consistent with your understanding of relativity - the simpler the better for later arguments, as long as W remains nonzero.

 

[A. Neumaier]

I would rather say that traditional scattering theory (with asymptotic dynamics governed by H_0) is not adequate for charged particles, as was demonstrated by Kulish and Faddeev. My reliance on this (wrong) traditional theory is the reason for IR divergences in the Hamiltonian. If I switch to the (correct) Kulish-Faddeev scattering theory, the spurious IR divergences will disappear, I hope. Then I will be able to derive a fully consistent finite dressed particle Hamiltonian.

The problem is that scattering theory defines three things:
A. the vacuum,
B. the 1-particle states, and
C. the multiparticle scattering.
You take A and B from scattering theory and hope to get by with a modified form of C. But as Kulish and Faddeev show, the IR photon cloud already changes B - as it turns single-electron states defined by a free Dirac equation into infraparticles.

This is impossible in your treatment since, by construction, 1-particle states are not affected at all by scattering. Thus you need to build in the infraparticle structure into the free states before the interaction is switched on - which requires the extension of your Fock space to the inseparable Hilbert space of Kibble, Kulish, and Faddeev.

 

[meopemuk]

The problem is that scattering theory defines three things:
A. the vacuum,
B. the 1-particle states, and
C. the multiparticle scattering.
You take A and B from scattering theory and hope to get by with a modified form of C. But as Kulish and Faddeev show, the IR photon cloud already changes B - as it turns single-electron states defined by a free Dirac equation into infraparticles.

This is impossible in your treatment since, by construction, 1-particle states are not affected at all by scattering. Thus you need to build in the infraparticle structure into the free states before the interaction is switched on - which requires the extension of your Fock space to the inseparable Hilbert space of Kibble, Kulish, and Faddeev.

I am not ready to discuss this right now. As I said already, I don't fully understand Kibble-Kulish-Faddeev theory. When I'm done with my homework I'll put this stuff in the book and we'll have a chance to discuss it.

Eugene.

 

[meopemuk]

Could you please get it explicitly for me? You may simplify the form of V and Z as much as is still consistent with your understanding of relativity - the simpler the better for later arguments, as long as W remains nonzero.

W can be obtained by multiplying two last terms in (12.11) by the first particle's position vector

$$gW_{ij} = \frac{q_1q_2}{8 \pi m_1m_2 c^2} \left( \frac{r_{1i}p_{2j}}{r} + \frac{(\mathbf{p}_2 \cdot \mathbf{r}) r_{1i}r_j}{r^3} \right)$$

Eugene.

 

[A. Neumaier]

W can be obtained by multiplying two last terms in (12.11) by the first particle's position vector

$$gW_{ij} = \frac{q_1q_2}{8 \pi m_1m_2 c^2} \left( \frac{r_{1i}p_{2j}}{r} + \frac{(\mathbf{p}_2 \cdot \mathbf{r}) r_{1i}r_j}{r^3} \right)$$

Is H_0 the part in (12.8) obtained by setting q_1q_2 to zero? Also, please tell me where I can find a formula for Z (to first nontrivial order). I didn't find in Section 11.2 or 12.1.

 

[meopemuk]

Is H_0 the part in (12.8) obtained by setting q_1q_2 to zero?

Yes it is.

Also, please tell me where I can find a formula for Z (to first nontrivial order). I didn't find in Section 11.2 or 12.1.

Eq. (O.5) in Appendix O.

Eugene.

 

[A. Neumaier]

Yes it is.
Eq. (O.5) in Appendix O.

Thanks. I'll attempt to start with your assumptions and demonstrate their weirdness.

But I need some further input, in order to be sure that I have the right starting point, and don't misunderstand your position:

Since you reject part of the common assumptions in classical relativity (limit speed c and relation (iii) of Currie et al.), could you please state in a compact way (or refer to the relevant statements in your book) what you consider to be the precise mathematical equivalent of
1. the relativity principle;
2. causality;
not phrased in terms of words but in terms of the formulas needed for checking whether a putative model satisfies these principles. (So, nothing about unidentified laws and laboratories, as in Postulate 2.1.) In particular, 1. and 2. should allow one to easily write down
3. formulas stating where and when an observer sees a particle in an interacting N-particle system change color when it is known that another observer in a different Lorentz frame sees the same particle change color at time t in position r.

 

[meopemuk]

Since you reject part of the common assumptions in classical relativity (limit speed c and relation (iii) of Currie et al.), could you please state in a compact way (or refer to the relevant statements in your book) what you consider to be the precise mathematical equivalent of
1. the relativity principle;

The mathematical equivalent of the relativity principle is the statement that an unitary representation of the Poincare group acts in the Hilbert space of each physical system. This statement is explained in Chapter 3, see especially subsection 3.2.4.

2. causality;

This basically means that the effect cannot precede the cause in any frame of reference. See subsections 11.1.4 and 11.4.3.

not phrased in terms of words but in terms of the formulas needed for checking whether a putative model satisfies these principles. (So, nothing about unidentified laws and laboratories, as in Postulate 2.1.) In particular, 1. and 2. should allow one to easily write down
3. formulas stating where and when an observer sees a particle in an interacting N-particle system change color when it is known that another observer in a different Lorentz frame sees the same particle change color at time t in position r.

To answer your question 3, I need to know what you mean by "color" and what is the Hermitian operator C corresponding to the observable "color". If the operator is C in one frame, then in the moving frame the corresponding operator is $$C' = \exp(iK \theta) C \exp(-iK \theta)$$, where K is the (interacting) boost generator. This relationship should be sufficient to tell everything about particle's "color" in the moving frame. The same formula $$r' = \exp(iK \theta) r \exp(-iK \theta)$$ connects particle positions in the frame at rest (r) and in the moving frame (r').

However, I do not apply this formula for transformations of "time" t, because time is not a usual observable in my approach. E.g., there is no "operator of time". See discussion in subsection 11.3.4. If you want to ask me how different observers perceive their respective times of the same event, you should tell me which event you are talking about. For example, the event can be associated with a collision of 2 particles. Then I can work out times t and t' based on formulas for r and r'.

Eugene.

Eugene.

 

[A. Neumaier]

The mathematical equivalent of the relativity principle is the statement that an unitary representation of the Poincare group acts in the Hilbert space of each physical system. This statement is explained in Chapter 3, see especially subsection 3.2.4.

This is a description in terms of quantum mechanics. But we are still discussing the classical version, as put forward in the Currie et al. paper (but without the law (iii)).
Since you insisted on a discussion independent of field aspects, and I believe that the particle aspects of the principles of relativity are completely independent of quantum mechanics, it is much simpler to discuss them in a classical setting, where the interpretational problems of quantum mechanics are absent.

This basically means that the effect cannot precede the cause in any frame of reference. See subsections 11.1.4 and 11.4.3.

Which formula there encodes this statement (defining causality in informal words) in a mathematically precise way?

To answer your question 3, I need to know what you mean by "color" and what is the Hermitian operator C corresponding to the observable "color".

This was just a metaphor to define a single point in space-time associated to some particle at a certain time. Since we are discussing the classical version, you may think of the color as a Poincare-invariant classical property that changes at random times, unrelated to the dynamics.

$$r' = \exp(iK \theta) r \exp(-iK \theta)$$ connects particle positions in the frame at rest (r) and in the moving frame (r').

Again, what is the classical version of this, suitable for the framework of Currie et al.?

However, I do not apply this formula for transformations of "time" t, because time is not a usual observable in my approach. E.g., there is no "operator of time". See discussion in subsection 11.3.4. If you want to ask me how different observers perceive their respective times of the same event, you should tell me which event you are talking about. For example, the event can be associated with a collision of 2 particles. Then I can work out times t and t' based on formulas for r and r'.

Suppose that the observer at rest sees the color change at time t in his rest frame. At which time t' does the moving observer see the change of color? In the standard discussions of relativity, the answer is well-determined. My question is whether you agree or differ with this standard setting in this point.

 

[meopemuk]

This is a description in terms of quantum mechanics. But we are still discussing the classical version, as put forward in the Currie et al. paper (but without the law (iii)).
Since you insisted on a discussion independent of field aspects, and I believe that the particle aspects of the principles of relativity are completely independent of quantum mechanics, it is much simpler to discuss them in a classical setting, where the interpretational problems of quantum mechanics are absent.

This is not difficult to translate to the language of classical mechanics if you replace "Hilbert space" -> "phase space" and "unitary transformation" -> "canonical transformation", i.e., continuous transformation of points in the phase phase space, which conserves the Poisson bracket.

Which formula there encodes this statement (defining causality in informal words) in a mathematically precise way?

I don't think there exists one formula that encodes the idea of causality. It depends on each particular physical process or series of events, where one can identify events-causes and events-effects. Then events-causes always happen before (or, at least, simultaneously with) the events-effects. That's what causality basically says. In the book I discuss causality on the simplest example of a two-particle system in Fig. 11.4

This was just a metaphor to define a single point in space-time associated to some particle at a certain time. Since we are discussing the classical version, you may think of the color as a Poincare-invariant classical property that changes at random times, unrelated to the dynamics.

Your example contains an internal contradiction, because a Poincare-invariant property cannot depend on time. If an observable has zero Poisson brackets with generators P, J, K, then it must have a zero bracket with H as well.

Again, what is the classical version of this, suitable for the framework of Currie et al.?

In the classical case, boost transformation of position (or any other dynamical variable) can be written as

$$r' = r + [K,r] \theta + [K,[K,r]] \theta^2/2 + \ldots$$

Here [..] means Poisson bracket. See second equation on page 402.

Suppose that the observer at rest sees the color change at time t in his rest frame. At which time t' does the moving observer see the change of color? In the standard discussions of relativity, the answer is well-determined. My question is whether you agree or differ with this standard setting in this point.

As I said before, I would need a more specific example of observable and its corresponding function on the phase space in order to answer your question. Since you assume that "color" changes with time, this means that it has a non-zero Poisson bracket with the Hamiltonian. This implies that the commutator with the boost generator is non-trivial too. In my approach the answer about the behavior of C under boosts would depend on the value of the Poisson bracket $$[K,C]$$.

Eugene.

 

[A. Neumaier]

This is not difficult to translate to the language of classical mechanics if you replace "Hilbert space" -> "phase space" and "unitary transformation" -> "canonical transformation", i.e., continuous transformation of points in the phase phase space, which conserves the Poisson bracket.

OK, thanks.

I don't think there exists one formula that encodes the idea of causality. It depends on each particular physical process or series of events, where one can identify events-causes and events-effects. Then events-causes always happen before (or, at least, simultaneously with) the events-effects. That's what causality basically says. In the book I discuss causality on the simplest example of a two-particle system in Fig. 11.4

But this means that the concept of causality is vague. How then can you maintain that causality is always preserved in your approach? If this is as theorem then causality (and with it cause and effect) must have a formal definition, and I ask you to reaveal to me this definition. But if it is not a theorem then how do you know that your claim is valid?

Your example contains an internal contradiction, because a Poincare-invariant property cannot depend on time. If an observable has zero Poisson brackets with generators P, J, K, then it must have a zero bracket with H as well.

No. This would hold for a deterministic process. But a stochastic process can be Poincare invariant without being constant. It only needs to be stationary in every timelike direction.

If you don't like that, then tell me a different way how you relate in your interpretation the space-time correspondence r(t) of a world line in the frame of an observer, and the space-time correspondence r(t) of the same world line in the frame of an observer in a different Lorentz frame, using whatever means you find appropriate.

In the classical case, boost transformation of position (or any other dynamical variable) can be written as
$$r' = r + [K,r] \theta + [K,[K,r]] \theta^2/2 + \ldots$$
Here [..] means Poisson bracket. See second equation on page 402.

OK, thanks. But a position is meaningless without an associated time. How does this extend to world lines r(t) transforming into r'(t')? Or is time a global concept in your theory?

As I said before, I would need a more specific example of observable and its corresponding function on the phase space in order to answer your question. Since you assume that "color" changes with time

It is enough to assume that the color changes exactly once, and abruptly from black to red, so that it can be used to label a particular point on the worldline. But you may also assume an arbitrary internal dynamics of the particle that assigns to it a changing color, and that let's one refer unambiguously to a particular point on the particle's worldline where the color had the value c.

The important thing is not the color but the correspondence of times. This correspondence is very easy in standard relativity, and I want to know whether it is the same in your interpretation of relativity, or by what it is replaced.

 

[meopemuk]

But this means that the concept of causality is vague. How then can you maintain that causality is always preserved in your approach? If this is as theorem then causality (and with it cause and effect) must have a formal definition, and I ask you to reaveal to me this definition. But if it is not a theorem then how do you know that your claim is valid?

I have a detailed discussion of causality in the case of 2-particle systems. See Fig. 11.4 and subsection 11.4.3. In this case, causality is a proven theorem.

No. This would hold for a deterministic process. But a stochastic process can be Poincare invariant without being constant. It only needs to be stationary in every timelike direction.

As I said elsewhere, I am interested only in few-particle systems. I am not sure what kind of *stochastic process* can occur in such systems.

If you don't like that, then tell me a different way how you relate in your interpretation the space-time correspondence r(t) of a world line in the frame of an observer, and the space-time correspondence r(t) of the same world line in the frame of an observer in a different Lorentz frame, using whatever means you find appropriate.

OK, thanks. But a position is meaningless without an associated time. How does this extend to world lines r(t) transforming into r'(t')?

Formulas for r'(t') in a moving reference frame can be found in subsection 11.4.3. See unmarked set of equations before (11.65). Similar formulas for r(t) in the frame at rest are obtained by setting \theta=0. In the classical case commutators should be replaced by Poisson brackets everywhere.

Or is time a global concept in your theory?

No, each observer measures its own time. This is emphasized by marking time labels associated with moving observer by the prime - t'.

It is enough to assume that the color changes exactly once, and abruptly from black to red, so that it can be used to label a particular point on the worldline. But you may also assume an arbitrary internal dynamics of the particle that assigns to it a changing color, and that let's one refer unambiguously to a particular point on the particle's worldline where the color had the value c.

The important thing is not the color but the correspondence of times. This correspondence is very easy in standard relativity, and I want to know whether it is the same in your interpretation of relativity, or by what it is replaced.

I cannot discuss the abstract observable of "color", because you haven't provided its commutators (Poisson brackets) with generators of the Poincare group, so the dynamics with respect to time translations and boosts remain unspecified.

Instead of abstract "color" I suggest to consider a more physical observable - the composition of an unstable particle. For example, one can call *black* the undecayed state of a muon and one can call *red* the decay products (= electron+neutrino+antineutrino). The probability of finding the *black* state and the time evolution of this probability (=decay law) has a precise definition in a quantum mechanical model of the decaying muon. The change of this decay law between different frames can be calculated unambiguously. See chapter 14. In first approximation one gets the usual Einstein's time dilation law. However, there are tiny corrections to this law, which depend on the strength of interaction that leads to the muon's decay.

Eugene.

 

[A. Neumaier]

I have a detailed discussion of causality in the case of 2-particle systems. See Fig. 11.4 and subsection 11.4.3. In this case, causality is a proven theorem.

In standard relativity, causality is defined by saying that a change in the dynamics of a system in a space-time region A (by adding there an external field) does not affect the values in any space-time region B such that all points x in A and y in B have a spacelike x-y.

Does this still hold, or what is your replacement of this? If your version of causality depends on the number of particles present, it would be very strange.

As I said elsewhere, I am interested only in few-particle systems. I am not sure what kind of *stochastic process* can occur in such systems.

If your theory is worth its salt it must apply also for many-particle systems, and must allow a reduced description where macroscopic bodies are treated as approximate point particles. These have internal degrees of freedom. So it should be possible to define a way of marking particular points on the world line - whether stochastic or not. It would be a bad feature of your theory if the time synchronization problem between world lines as seen by two different observers would depend on the details of the internal dynamics.

Formulas for r'(t') in a moving reference frame can be found in subsection 11.4.3. See unmarked set of equations before (11.65). Similar formulas for r(t) in the frame at rest are obtained by setting \theta=0. In the classical case commutators should be replaced by Poisson brackets everywhere.

May I take the formula on p.436 to be generally valid, or would it be different when more particles are present?

No, each observer measures its own time. This is emphasized by marking time labels associated with moving observer by the prime - t'.

OK.

I cannot discuss the abstract observable of "color", because you haven't provided its commutators (Poisson brackets) with generators of the Poincare group, so the dynamics with respect to time translations and boosts remain unspecified.

The correspondence of times should not depend on color or anything else being used to mark times. Choose whatever you want - and if the formulas on p. 436 are generally valid then we can completely dispense with the color or whatever problem.