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.

 

[A. Neumaier]

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

How do you know this in your version of relativity, where the Lorentz transformation law is different from the standard one?

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.

Yes, we discussed some of this before. But people use Maxwell's theory for many things, hence it must be derivable (using some approximation mechanism of your choice) from your theory if your theory is to describe macroscopic N-particle systems correctly.

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

Most experiments involve light, but not individual photons. Relativity developed in classical physics and light is usually treated classically in optics, except when looking for special quantum effects.

Moreover the interpretation of the standard experiments needs assumptions about the location of macroscopic objects emitting photons, and these follow the standard Lorentz transformation. Thus you need to be able to bridge this gap.


Since I couldn't get from you a recipe how to apply your relativity to classical macroscopic but pointlike objects like stars, I give up trying to interpret your theory.
Right or wrong, it is too difficult to use to be practical. The standard approach has all the advantages:
- a clear, simple, and context-independent transformation law;
- a well-studied and time-approved ontology, matching experiments;
- a working extension to field theory as practiced by engineers;
- a good quantum field version in which both UV and IR problems are under control;
- high order calculations are feasible, have been done, and agree with experiment.

If you can't tell how the standard view arises as a controlled and valid approximation to your theory, nobody is going to use your theory (even should it be consistent) - except perhaps to calculate some special effects involving only two particles. But already handling the IR correctly in your approach requires infinitely many photons...

 

[meopemuk]

How do you know this in your version of relativity, where the Lorentz transformation law is different from the standard one?

I don't need to know relativity in order to calculate that light travels from sun to Earth in 8.5 seconds. This is just distance/speed.

Yes, we discussed some of this before. But people use Maxwell's theory for many things, hence it must be derivable (using some approximation mechanism of your choice) from your theory if your theory is to describe macroscopic N-particle systems correctly.

The important thing is that all experimental electromagnetic effects are reproduced correctly in my approach. This is discussed in chapter 12

Since I couldn't get from you a recipe how to apply your relativity to classical macroscopic but pointlike objects like stars, I give up trying to interpret your theory.

The recipe for transformations to the moving frame is simple. If the star is not interacting with anything (i.e., is not a part of a double-star system) then boost transformations of star observables is the same as in traditional special relativity. If the star is a part of an interacting system, then boost transformations are derivable by formulas similar to those used in chapters 12 and 13 for time evolution. One just needs to replace the interacting Hamiltonian there with the interacting boost operator, whose explicit form is readily available.

- a clear, simple, and context-independent transformation law;

My point is that such an universal transformation law does not exists. If a particle/object/star is a part of a bigger interacting system then boost transformations are interaction-dependent and context-dependent.

The situation is the same as with time evolution. If a particle is isolated, then you can write universal formulas for its time evolution

x(t) = x(0) + p(0)t/m
p(t) = p(0)

However if the particle is a part of an interacting system, time evolution formulas become interaction-dependent and context-dependent.

I hope you agree that there is no universal context-independent description of time evolution. Why do you insist on universal context-independent evolution with respect to boosts?


Eugene.

 

[A. Neumaier]

The important thing is that all experimental electromagnetic effects are reproduced correctly in my approach. This is discussed in chapter 12

I consider the macroscopic Maxwell equations a very important experimental electromagnetic effect, but it is not reproduced in your Chapter 12.

The recipe for transformations to the moving frame is simple. If the star is not interacting with anything (i.e., is not a part of a double-star system) then boost transformations of star observables is the same as in traditional special relativity.

So you agree that in case of a star, one can neglect its internal degrees of freedom for the dynamics, and regard these internal properties as a clock. For example, if a star becomes a supernova, this is now a random event that can be used to label a particular time on the worldline of the star. Previously it sounded as if the internal structure must also be described by observables with a deterministic dynamics within your system.

My point is that such an universal transformation law does not exists. If a particle/object/star is a part of a bigger interacting system then boost transformations are interaction-dependent and context-dependent.

But everything is interacting with everything, though perhaps slightly.

The situation is the same as with time evolution. If a particle is isolated, then you can write universal formulas for its time evolution

x(t) = x(0) + p(0)t/m
p(t) = p(0)

However if the particle is a part of an interacting system, time evolution formulas become interaction-dependent and context-dependent.

What is different in traditional relativity and in your relativity is that space-time is traditionally (including all applications of Maxwell's equations) objective, while in your relativity space-time doesn't exist. This makes everything complicated.

 

[meopemuk]

I consider the macroscopic Maxwell equations a very important experimental electromagnetic effect, but it is not reproduced in your Chapter 12.

I consider Maxwell equations being *equations* not an *experimental effrect*. We have slight differences in our terminologies.

So you agree that in case of a star, one can neglect its internal degrees of freedom for the dynamics, and regard these internal properties as a clock. For example, if a star becomes a supernova, this is now a random event that can be used to label a particular time on the worldline of the star. Previously it sounded as if the internal structure must also be described by observables with a deterministic dynamics within your system.

My basic point is very simple. If you want to know the time evolution of a physical system (for example, the timing of the supernova explosion) you need to know its Hamiltonian. For an interacting system (supernova explosion occurs as a result of interactions in its interior) the Hamiltonian is interacting and non-trivial. So, the time evolution is non-trivial and interaction-dependent.

Similarly, if you want to know boost transformations of observables of the system (e.g., how the timing of the supernova explosion is seen by different moving observers) you need to know system's boost operator. For an interacting system (e.g., the supernova star) this operator is interacting and non-trivial. So, I cannot expect simple, universal and interaction-independent boost transformation rules.

However, as I've shown in section 14.3 the interaction-dependent corrections to the boost transformation law are very small. So, for all practical purposes it is OK to use traditional Lorentz transformations when discussing space-time coordinates of the supernova explosion.

But everything is interacting with everything, though perhaps slightly.

If interaction is weak then you can safely ignore my corrections to Lorentz transformations and use traditional special relativistic formulas.

What is different in traditional relativity and in your relativity is that space-time is traditionally (including all applications of Maxwell's equations) objective, while in your relativity space-time doesn't exist.

Yes, that's a good way to put it.

This makes everything complicated.

On the contrary, I think this makes everything very simple, especially when discussing gravitational effects and their quantum description. See chapter 13.

Eugene.

 

[A. Neumaier]

I consider Maxwell equations being *equations* not an *experimental effrect*. We have slight differences in our terminologies.

Only on a superficial level. In your terminology, I meant ''the experimental effects described and predicted by the macroscopic Maxwell equations''

However, as I've shown in section 14.3 the interaction-dependent corrections to the boost transformation law are very small. So, for all practical purposes it is OK to use traditional Lorentz transformations when discussing space-time coordinates of the supernova explosion.

If interaction is weak then you can safely ignore my corrections to Lorentz transformations and use traditional special relativistic formulas.

I think this should be emphasized - else nobody will want to use your formulas.

What is different in traditional relativity and in your relativity is that space-time is traditionally (including all applications of Maxwell's equations) objective, while in your relativity space-time doesn't exist.

Yes, that's a good way to put it.

But it means that you undo 100 years of progress since Minkowski. All the advantages of a space-time description are sacrificed on the altar of direct interactions. And with it, you sacrifice powerful nonperturbative metods, and everything becomes very cumbersome to do.

On the contrary, I think this makes everything very simple, especially when discussing gravitational effects and their quantum description. See chapter 13.

With all the standard relativity undone, your theory might indeed be consistent for massive particles (where there is no IR problem). But nevertheless, it remains (to my taste, and probably that of most physicists) weird, ugly, and hard to use beyond low order few-particle systems. I wonder how you'd do nonequilibrium statistical mechanics with it...

So I'll finish the discussion with this disagreement on very important fundamentals. You may have the final word if you wish.

 

[meopemuk]

But it means that you undo 100 years of progress since Minkowski.

I don't think you can call it unqualified progress. The incompatibility between Einstein & Minkowski relativity and quantum mechanics is still listed as the top problem in theoretical physics.

everything becomes very cumbersome to do... weird, ugly, and hard to use...

I don't think "dressed particle" theory is cumbersome etc. It is the same standard quantum mechanics applied to systems with variable number of particles. The major problem is that I still don't know high-order interaction terms in the Hamiltonian. But once these terms become known, the calculations and their interpretation are simple and transparent.

Eugene.

P.S. Arnold, thank you for your thoughtful comments.

 

[strangerep]

Eugene,

I have never properly understood why you think that direct-interaction is
necessary, and that interaction via an intermediate boson is "bad".

From past conversations, I got the impression it has something to do with
"no-go" theorems like Currie-Jordan-Sudarshan, but I never followed your
subsequent logic.

It seems to me that Currie-Jordan-Sudarshan just shows that, when position
is included in our dynamical framework, and accelerations are permitted,
then the Poincare group is not an adequate dynamical group.

But so what? If we have two (let's say classical) particles following mutually
accelerating worldlines under the influence of an interaction between them,
then a larger dynamical group is in play. An external inertial observer O1 might
try to describe the entire 2-particle system in terms of a Lorentz-invariant
interaction, but this must apply to the system as a whole. A different intrtial
observer O2 (in motion relative to O1) sees the system differently. Trump and
Schieve give a nice little spacetime diagram of this to show how the two inertial
observers cannot consistently agree on a center of mass for the system.
This illustrates the pitfalls of trying to analyze composite systems involving a larger
group in terms of only a much smaller group.

But if we analyze the system in terms of its larger dynamical algebra, things
become a bit clearer. E.g., in the Kepler problem, (and Hydrogen atom), a
larger group is involved (SO(4,2) - conformal group) and there's an extra
Casimir operator, related to the Laplace-Lenz-Runge vector. See, e.g.,

http://en.wikipedia.org/wiki/Laplace-Runge-Lenz_vector

which also contains a section about how this relates to the Hydrogen atom.
Maxwell's equations are known to be invariant under this larger group, not
merely the Poincare group, which I take to be a hint that physics is more than
Poincare.

If not CJS, then would you please try to make me understand why you insist
on direct interaction?

 

[meopemuk]

From past conversations, I got the impression it has something to do with
"no-go" theorems like Currie-Jordan-Sudarshan, but I never followed your
subsequent logic.

It seems to me that Currie-Jordan-Sudarshan just shows that, when position
is included in our dynamical framework, and accelerations are permitted,
then the Poincare group is not an adequate dynamical group.

The Poincare group structure has no relationship to dynamical properties of particular interacting or non-interacting *physical systems*. The Poincare group is a group of transformations between inertial *observers* or frames of reference or laboratories. In my book I make a sharp distinction between *physical systems* and *observers*. This terminology is explained in Introduction. So, the Poincare group can be defined even in a world populated by different inertial observers, but where there is not a single physical system. In a sense, the Poincare group can be regarded as a definition of what we call an inertial observer.

Since Galileo we know that there exists a certain class of observers (which we call inertial observers) or laboratories, which can be regarded as equivalent. If one such inertial observer O is given, then we can shift this observer in space or in time, rotate it or change its velocity and thus obtain a different observer O', who still belongs to the inertial observers class. Apparently, this set of "inertial transformations" must form some 10-parametric Lie group. There are not many 10-parametric Lie groups, which can play the role of the group of inertial transfrormations: the Galilei group, the de Sitter group, the Poincare group, ... As experience shows the Poincare group is the most realistic choice by far. Once again: this choice of the group is completely unrelated to physical systems that we are going to observe. CJS theorem is not related to this choice as well. This choice reflects only relationships between different inertial observers.


Once we established that different inertial observers are related to each by transformations forming the Poincare group and once we postulated that all these observers have equal rights (=the relativity principle), we are forced to admit that in the Hilbert space of states of *any* isolated physical system there must exist a unitary representation of the Poincare group (Chapter 3). Of course, this does not mean that all states of this system are related to each other by the group transformations. For example, there is no Poincare group element, which transforms the ground state of the hydrogen atom into an excited state of the same atom. This simply means that the representation of the Poincare group acting in the Hilbert space of the hydrogen atom is *reducible*. The ground state and the excited state belong to different irreducible components of this representation.

So, the Hilbert space of each physical system carries a representation of the Poincare group. There is an infinite number of inequivalent ways to construct a representation of the Poincare group even in the same Hilbert space. In the language of physics these different Poincare group representations are described as different types of interaction. For example, the reducible representation in the Hilbert space of the "electron+proton" system described above corresponds to the Coulomb potential acting between the two particles. If we decided that electron and proton interact by a different potential we would be forced to build a different representation of the Poncare group to reflect this fact.

The Poincare group remains in place even though there are non-zero accelerations in the interacting system. In fact, the Poincare group theory only guarantees that the center-of-mass of the system moves without acceleration, but it does not place any restriction on accelerations of individual subsystems.

Now, regarding CJS theorem: It says that if we have a multi-particle physical system, which is

(a) Poincare invariant in the sense described above
(b) interaction between particles is also Poincare-invariant

then particle world-lines in different reference frames are not related to each other by Lorentz formulas of special relativity.

The usual explanation of this "paradox" claims that particle-based description is not adequate and we need to switch to a field-based description, where particle trajectories become an ill-defined concept, so the contradiction disappears. I have never seen an attempt to explain the CJS "paradox" by claiming that the Poincare group itself is inadequate.

My explanation of the CJS "paradox" is that I don't see anything troubling in the fact that particle observables do not transform by Lorentz formulas. Such transformations have never been observed experimentally, so there is no problem if they are different from Lorentz formulas. This is why I place the word "paradox" in double-quotes.

I have never properly understood why you think that direct-interaction is
necessary, and that interaction via an intermediate boson is "bad".

[...]
If not CJS, then would you please try to make me understand why you insist
on direct interaction?

The direct interaction follows from my insistence on the point that only particles' degrees of freedom are observable, so only these degrees of freedom must be present in our theory. If this is so, then, when we build a Poincare-invariant interaction in the Hilbert space of a two-particle system electron+proton, we obtain interaction, which depends only on positions and momenta of these two particles. This interaction must be direct and instantaneous. This is required by the laws of conservation of the momentum and energy.

Suppose that the electron-proton interaction is retarded. The existence of interaction means that two particles exchange portions of energy-momentum between each other. The retarded character of this exchange means that there is a time lag between a change of the electron's energy-momentum and the corresponding change of the proton's energy-momentum. During this time the exchanged portion of energy-momentum must exist somewhere in the space between the two particles. This means that in addition to the particles' degrees of freedom we need to introduce additional degrees of freedom, which would be responsible for keeping the "traveling" energy-momentum. This should be degrees of freedom of the "field" or "virtual force carriers" or whatever you want to call them. These degrees of freedom have not been observed directly, which gives me the right to say that they simply do not exist and, therefore, inter-particle interactions must be direct and instantaneous.

Eugene.

 

[meopemuk]

Returning to our discussions with Prof. Neumaier about infrared divergences in the dressed particles approach... Recently I've found several papers whose goal is to fit "effective" particle interactions to calculated scattering amplitudes from quantum field theory, e.g., QED. The fitted amplitudes include renormalized loop integrals with properly eliminated infrared divergences. So, the resulting particle potentials are basically the same as I am trying to find in my approach. I am going to study these works and add new sections to the book.

B. R. Holstein, Effective interactions and the hydrogen atom, Am. J. Phys. 72 (2004), 333

A. Pineda, J. Soto, The Lamb shift in dimensional regularization, Phys. Lett. B, 420 (1998), 391

A. Pineda, J. Soto, Potential NRQED: The Positronium Case, http://www.arxiv.org/abs/hep-ph/9805424

S. N. Gupta, W. W. Repko, C. J. Suchyta, III, Muonium and positronium potentials, Phys. Rev. D, 40 (1989), 4100

S. N. Gupta, S.F. Radford, Quantum field-theoretical electromagnetic and gravitational two-particle potentials, Phys. Rev. D, 21 (1980), 2213

G. Feinberg, J. Sucher, Two-photon-exchange force between charged systems: Spinless particles, Phys. Rev. D, 38 (1988), 3763

Eugene

 

[A. Neumaier]

Recently I've found several papers whose goal is to fit "effective" particle interactions to calculated scattering amplitudes from quantum field theory, e.g., QED. The fitted amplitudes include renormalized loop integrals with properly eliminated infrared divergences. So, the resulting particle potentials are basically the same as I am trying to find in my approach. I am going to study these works and add new sections to the book.

Yes, there is a lot of work on NRQED, which is indeed the technique to get the most accurate calculations for the anomalous magnetic moment and the Lamb shift.

Like in your approach, they work in a Hamiltonian framework, but they expand in both
alpha and 1/c.

Working out how your approach relates to NRQED would be a valuable addition to your book.

 

[izh-21251]

I have no doubts that the new approach, recited in Eugene's book (along with the results of another "clothing people" - Shirokov, Shebeko et al.) deserves the name of a "new theory" - alternative to QFT.
And it definitely deserves all efforts invested in it so far and to be invested in future.
This is to be said contrary to some commentaries I saw in this thread.

Along with this, let me dare state my opinion, the comparison with the conventional approaches is an inevitable pay for a new theory to be brought into life...

 

[izh-21251]

I am coming up again with my misunderstanding of gauge equivalence... When do I eventually calm down? ))
(The original thread is -
https://www.physicsforums.com/showthread.php?p=3248968&posted=1#post3248968 )
Removing my appeals here can draw, as I'd like to hope, more attention to this problem...
Moreover, as my problem concerns the Clothing Approach, this could be so much the right place for it...

Let me briefly revise the question.
From the beginning, I was interested in the origin of gauge invariance in QFT. Weinberg showed, that the origin is in the transformation law of one-particle massless spin-1 states. C/A operators of photons cannot be the ingredients to build 4-vector causal fields. Since I am not concerned with fields at all - they are not (as believed here)))) the fundamental ingredients of the theory - this should not bother me a lot.
I avoid fields and start with Hamiltonian right away and use the creation/annihilation operators as basic ingredients.
What I see further is that the interaction in Hamiltonian (and the interaction in boost generator inavitably) looks different in different gauges. (CAN ANYONE SHOW HAMILTONIAN SAME IN DIFERENT GAUGES?)

(At this point I was given an advise to calm down and work in whatever gauge I like, say, in the Coulomb one.. ).
However, a lot of questions started to arise, which I could not pacify.

1. Having applied dressing (or clothing) procedure to different gauges, I obtained different representations of Hamiltonian (for clothing approach - different properties of clothed particles - mass shifts and interactions). Thus - the question arisen is - what is the relationship between Hamiltonians in different gauges. Do they lead to the same time evolution? Are they scattering equivalent? Since hamiltonian and Boost look differently in different gauges - do we have unique family of Poincare generators for every gauge?
Or am I wrong?

2. Weinberg gives a hint. One cannot compose a field out of massless spin-1 operators because of their specific transformation law under boost transformation in certain direction.
Thus I believe, that the gauge transformations are somehow connected with Lorentz-boosts.
What is this connection if it exists??

3. I believe, that Hamiltonians in different gauges lead to the same S-matrix, and this, as I understand it, usually is called gauge equivalence (compare to 'invariance' in QFT). Thus, in the dressing-clothing approach, I think one needs to establish the relationship between Hamiltonians in different gauges... At least I haven't seen it's been done anywhere.

4. If we agreed to abandon the concept of fields and establish approach dealing with particle operators... This might be a bit of a fantasy - but what if one eventually try to compose interaction term of QED Hamiltonian that looks equally in different gauges.
If this madness successful, one can speak of abandoning the idea of gauge invariance-equivalence principle in new theory.

 

[izh-21251]

By the way, the relationship between Dressing approach (where Hamiltonian is transformed) and Clothing approach after Shirokov, Shebeko (where particle operators are transformed) is mentioned in the book.
It is stressed, that mathematically these approaches are equal and resuls for observables obtained in both cases are the same.
Thus, why have you chosen particularly the Dressing formalism instead of Clothing one?

As seems to me, from the formal point of view, the approach, where one transform from from bare operators to clothed ones (and Hamiltonian stays intact) is more consistent, compared to the case, when Hamiltonian is transformed...
Though my understanding may be superficial...

 

[meopemuk]

... one can speak of abandoning the idea of gauge invariance-equivalence principle in new theory.

Yes, this is exactly what I suggest. My position is that field theory (with gauges etc.) is a wrong way to think about physics. The correct way to think about physics is in terms of particles and their direct interactions. The only obstacle to this new way of thinking is that we don't know exact interaction operators. Somehow, we should fit these interaction operators to observable experimental data. There is a lot of high-precision experimental information about particle scattering amplitudes contained in the theoretical S-matrix. So, the S-matrix is a good target for fitting interaction terms in the particle Hamiltonian.

Despite their bad theoretical foundation, quantum field theories (such as QED) are very good at one thing - they calculate the S-matrix in a very good agreement with observations. I have no idea how to explain this perfect match. This could be just a coincidence, or there could be a deeper explanation. I am not trying to find this explanation. I simply accept it as a postulate that the S-matrix calculated in QED is exact. So, the idea of the unitary dressing transformation is simply to fit inter-particle potentials of the clothed/dressed theory so that they yield exactly the same QED-calculated S-matrix.

I am not worrying about the gauge invariance issue in QED. As far as I know, the resulting S-matrix is gauge-independent. So, the fitting of the dressed/particle Hamiltonian to the QED S-matrix does not depend on the gauge chosen for S-matrix calculations in QED.


Eugene.

 

[meopemuk]

By the way, the relationship between Dressing approach (where Hamiltonian is transformed) and Clothing approach after Shirokov, Shebeko (where particle operators are transformed) is mentioned in the book.
It is stressed, that mathematically these approaches are equal and resuls for observables obtained in both cases are the same.
Thus, why have you chosen particularly the Dressing formalism instead of Clothing one?

As seems to me, from the formal point of view, the approach, where one transform from from bare operators to clothed ones (and Hamiltonian stays intact) is more consistent, compared to the case, when Hamiltonian is transformed...
Though my understanding may be superficial...

Yes, it is true that Shirokov-Shebeko approach is physically equivalent to my approach. So, the choice between these two approaches is a matter of personal preference. In the S-S approach I don't like the idea that we have two types of particles (and two types of particle c/a operators): the "bare" and "physical" ones. This makes things rather confusing to me. I prefer to think that there are only physical particles - those which we see in experiments, and that the definition of particles is independent on interactions that may act between them.

Then in my approach it becomes clear that the true problem with the traditional QED is a wrong choice of the Hamiltonian. This problem is solved by applying the unitary dressing transformation to this Hamiltonian. For me it is easier to comprehend the transition between two theories (between the field-based QED and the particle-based RQD) as a change in interaction potentials. The change in the particle content (as suggested by S-S) is more difficult to visualize and understand, at least for me pesonally.

Eugene.

 

[izh-21251]

So, the fitting of the dressed/particle Hamiltonian to the QED S-matrix does not depend on the gauge chosen for S-matrix calculations in QED.

Does your phrase mean that Dressed Hamiltonian is supposed to be gauge-independent??
This is so much important I think, because you are talking of time-evolution and obtaining the ONLY ONE CORRECT Hamiltonian for time evolution.

P.S. S-Matrix do not worry me indeed - I am sure it must be unique for different gauges - for different Hamiltonians in different gauges.

 

[izh-21251]

Yes, it is true that Shirokov-Shebeko approach is physically equivalent to my approach. I prefer to think that there are only physical particles - those which we see in experiments, and that the definition of particles is independent on interactions that may act between them.
Eugene.

By now it was more logical for me to think of transforming between different types of particle operators.
Though I saw it was a debate here about transforming between different Hilbert spaces of states... I am not an expert in such sort of things, but somehow I think there are not a reasons for concern.

Ok, thank you for clarifying.

 

[izh-21251]

Due to presence of interaction in Boost generator in the instant form of dynamics, you predict violation of (or corrections to) the laws of special relativity... such as time dilation law for relativistic unstable particle.

Is there a possibility to (indirectly) see this effect in experiment? At what scale these effects could emerge?
I am currently involved in processing data from ATLAS (LHC) detector, where unstable hadrons are produced in vast amounts... Could one potentially observe (measure) any of these effects there?

Thanks,
Ivan

 

[meopemuk]

Does your phrase mean that Dressed Hamiltonian is supposed to be gauge-independent??
This is so much important I think, because you are talking of time-evolution and obtaining the ONLY ONE CORRECT Hamiltonian for time evolution.

Yes, the dressed particle Hamiltonian is independent on gauges. The gauge is not a physical notion. There are no fields and gauges in the dressed particle theory. The true Hamiltonian is unique and can be, in principle, fully derived from experimental observations.

Eugene.

 

[meopemuk]

Due to presence of interaction in Boost generator in the instant form of dynamics, you predict violation of (or corrections to) the laws of special relativity... such as time dilation law for relativistic unstable particle.

Is there a possibility to (indirectly) see this effect in experiment? At what scale these effects could emerge?
I am currently involved in processing data from ATLAS (LHC) detector, where unstable hadrons are produced in vast amounts... Could one potentially observe (measure) any of these effects there?

Thanks,
Ivan

Yes, this can be done, in principle. However, the magnitude of the effect is so small that I do not expect any experimental results soon.

The most accurate measurement of the time dilation in particle decays was performed with muons on a circular orbit by Bailey et al. See references [305, 306] in the book. The experimental error was of the order 10^{-3}. In subsection 14.4.1 I estimated that the effect of interaction-dependent boosts has the order of 10^{-18}. I don't expect experimentalists to improve the precision of their measurements by 15 orders of magnitude any time soon.

Perhaps one can suggest a different experimental setup with modern particle accelerators, but I haven't got any good idea so far.

Eugene.