What Are the Implications of a New Relativistic Quantum Theory?

[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.

 

[meopemuk]

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.

No, all these usual considerations about light cones and causality do not hold in my approach. In Hamiltonian theory particles interact via instantaneous direct potentials, so any perturbation at point A affects all surrounding points B immediately. (In your language A and B are separated by a space-like interval.) If this action-at-a-distance approach were used together with usual universal interaction-independent Lorentz transformations for boosts, then I would be in a big trouble, because the causality would be certainly violated: in some reference frame one would see that the effect at B occurs before the cause at A. However, I am *not* using standard Lorentz transformation formulas. Instead, I recognize that interaction in the Hamiltonian implies the presence of interaction terms in the boost operator. Therefore boost transformations of observables must have a non-trivial interaction-dependent form. As discussion in subsection 11.4.3 (page 436) shows, this is sufficient to guarantee causality. The arguments there are presented for a 2-particle system, but they should remain valid for any number of particles.

Actually, the causality of my approach can be understood very simply without much calculations. It is based on the relativistic invariance (=the equivalence of intertial reference frames). If interactions are instantaneous in one reference frame, then they must remain instantaneous in all other reference frames. So, it is not possible to find a frame in which the effect at point B occurs earlier than the causing event at point A.

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.

If there is an internal degree of freedom (like your "color"), then it must be present in the Hamiltonian. So, its time evolution and boost transformations should be obtainable from usual formulas, like

C(t) = \exp(iHt)C(0) \exp(-iHt)
C(\theta) = \exp(iK \theta)C(0) \exp(-iK \theta)

In this case I would be able to answer your questions about the timings of changes in C seen from different frames.

However, you suggest to change the particle's color in one frame "by hand" and ask what is the timing of this event in another frame? I cannot answer this question, because this artificial change of color is not governed by internal particle dynamics. In fact, the particle cannot be regarded as an isolated system anymore. So, the whole formalism of the Poincare group is not applicable.

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

These arguments do not depend on the number of particles. Perhaps I should re-write these formulas for the general case of N particles to avoid any confusion. Thanks for the idea.

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.

Yes, but as I've said above, this "whatever I want" thing must be a degree of freedom explicitly present in the Hamiltonian. Then I have no problem of discussing it. The closest thing that satisfies your (and my) criteria is the decay probability of unstable particle. See Chapter 14.

I am not sure that the correspondence of times between different frames should be independent on the type of physical process/property considered. This assumption may be true in the traditional special relativity, but it is no longer true in my approach, where boost transformations depend on interactions. Therefore, the "correspondence of times" can depend on what property of the interacting system we are looking at. As my example in Chapter 14 shows, the differences between special relativity predictions and my approach are extremely small. So, one can still get good accuracy by using SR formulas, like the Einstein's time dilation law.

Eugene.

 

[A. Neumaier]

No, all these usual considerations about light cones and causality do not hold in my approach. In Hamiltonian theory particles interact via instantaneous direct potentials, so any perturbation at point A affects all surrounding points B immediately.

How do you check experimentally whether two events are simultaneous?
You run into all the problems discussed when relativity is introduced, and have to answer them again from scratch.

As discussion in subsection 11.4.3 (page 436) shows, this is sufficient to guarantee causality. The arguments there are presented for a 2-particle system, but they should remain valid for any number of particles.

Then you should formulate it as such.

If there is an internal degree of freedom (like your "color"), then it must be present in the Hamiltonian. So, its time evolution and boost transformations should be obtainable from usual formulas, like

C(t) = \exp(iHt)C(0) \exp(-iHt)
C(\theta) = \exp(iK \theta)C(0) \exp(-iK \theta)

In this case I would be able to answer your questions about the timings of changes in C seen from different frames.

However, you suggest to change the particle's color in one frame "by hand" and ask what is the timing of this event in another frame? I cannot answer this question, because this artificial change of color is not governed by internal particle dynamics. In fact, the particle cannot be regarded as an isolated system anymore. So, the whole formalism of the Poincare group is not applicable.

No real system is isolated. Thus there must be ways to go from the exact theory to a more approximate theory that accounts for truncated descriptions, such as ignoring the details of the dynamics of internal degrees of freedom of a compound particle.

If you can't do that, your theory doesn't apply to reality but only to two particles alone in the universe.

I am not sure that the correspondence of times between different frames should be independent on the type of physical process/property considered. This assumption may be true in the traditional special relativity, but it is no longer true in my approach, where boost transformations depend on interactions.

Special relativity began (and is still mostly used) as a theory of macroscopic objects, which are therefore classically describable, without accounting for their microscopic degrees of freedom. There it satisfies the standard rules for Lorentz transformations.
You _must_ be able to recover this (experimentally well-confirmed) setting in an appropriate many-particle limit. If your theory doesn't reproduce this limit correctly, it is in disagreement with experiment.

 

[meopemuk]

How do you check experimentally whether two events are simultaneous?
You run into all the problems discussed when relativity is introduced, and have to answer them again from scratch.

I am not sure which problems you have in mind. I think it is reasonable to assume that each observer can measure accurately positions and times of events in his/her reference frame. Without this basic assumption we cannot even start doing physics. Each observer can determine unambiguously whether two events are simultaneous or not. I don't think special relativity ever questioned this statement. Events looking simultaneous for one observer could be not simultaneous for another observer. But this is a different matter.

No real system is isolated. Thus there must be ways to go from the exact theory to a more approximate theory that accounts for truncated descriptions, such as ignoring the details of the dynamics of internal degrees of freedom of a compound particle.

If you can't do that, your theory doesn't apply to reality but only to two particles alone in the universe.

I thought it is most interesting to understand how things look in the exact treatment. Making approximations should be an easy part.

Special relativity began (and is still mostly used) as a theory of macroscopic objects, which are therefore classically describable, without accounting for their microscopic degrees of freedom. There it satisfies the standard rules for Lorentz transformations.
You _must_ be able to recover this (experimentally well-confirmed) setting in an appropriate many-particle limit. If your theory doesn't reproduce this limit correctly, it is in disagreement with experiment.

I would be very much indebted if you can find a single example where my approach disagrees with existing experiments.

Eugene.

 

[A. Neumaier]

I think it is reasonable to assume that each observer can measure accurately positions and times of events in his/her reference frame. Without this basic assumption we cannot even start doing physics. Each observer can determine unambiguously whether two events are simultaneous or not. I don't think special relativity ever questioned this statement. Events looking simultaneous for one observer could be not simultaneous for another observer. But this is a different matter.

How would I tell whether the sun I see looks like it seems to look now (simultaneously) or 8 1/2 minutes ago (which standard relativity says)?

I thought it is most interesting to understand how things look in the exact treatment. Making approximations should be an easy part.

Not in your case, since the framework is so different than the traditional framework that it is nontrivial to recover the tradition in some limit. If it were indeed easy, why don't you derive from your form of the dynamics the standard macroscopic Maxwell equations with their standard relativistic transformation properties?

I would be very much indebted if you can find a single example where my approach disagrees with existing experiments.

If you tell me a clear mechanism how to apply your theory to problems where macroscopic particles (such as moving human observers) have internal clocks (without explicitly invoking quantum theory, i.e., particle decay) then I can look at standard examples. But as long as you have worked out only the behavior of two microscopic point particles, where nothing can be measured at two different times, very little can be said about how your theory would compare with traditional experiments.

 

[meopemuk]

How would I tell whether the sun I see looks like it seems to look now (simultaneously) or 8 1/2 minutes ago (which standard relativity says)?

We see the sun as it was 8 1/2 minutes ago. I don't think anybody can disagree about that.

Not in your case, since the framework is so different than the traditional framework that it is nontrivial to recover the tradition in some limit. If it were indeed easy, why don't you derive from your form of the dynamics the standard macroscopic Maxwell equations with their standard relativistic transformation properties?

I think we've discussed this point before. My position is that Maxwell equations and their "standard relativistic transformation properties" are *not* a rigorous way to describe classical electromagnetic phenomena. The rigorous way is given by a Hamiltonian Darwin-Breit theory (in a broader sense this theory includes also photon absorption/emission interactions discussed in section 14.2) with relativistic transformation properties provided by the interacting boost operator. This is discussed in chapter 12.

It is true that at this point I know only low-order terms in the full Darwin-Breit interaction Hamiltonian. However, (1) these terms are already sufficient to describe most (if not all) existing experiments; (2) higher order corrections can be derived in a controlled fashion after the infrared problem is solved.

If you tell me a clear mechanism how to apply your theory to problems where macroscopic particles (such as moving human observers) have internal clocks (without explicitly invoking quantum theory, i.e., particle decay) then I can look at standard examples. But as long as you have worked out only the behavior of two microscopic point particles, where nothing can be measured at two different times, very little can be said about how your theory would compare with traditional experiments.

Most experimental verifications of special relativity involve observations of microscopic particles, like photons. All these tests agree with my approach. See subsection 11.3.2.

Eugene.