# Relativistic Quantum Dynamics

by meopemuk
Tags: dynamics, quantum, relativistic
P: 1,746
 Quote by dextercioby My 2cents (sorry to disrupt): try the other volume, written by Lifschitz, Pitayevskii and Berestetskii in which QED is discussed (volume 4 of the series, IIRC). I don't have this with me either, so I can't give you the section number. http://www.elsevierdirect.com/ISBN/9...lectrodynamics (the table of contents in on Amazon.com)
Yes, thanks. In sections 83 of this tome you'll find a derivation of the Darwin-Breit Hamiltonian from the 2nd order scattering matrix in QED. It is called "Breit equation" there; Darwin potential is just the spin-independent part of the full Darwin-Breit interaction potential. This derivation is basically the same as in section 10.3 of my book.

The dressed particle approach can be regarded as a generalization of the above derivation to higher orders of the perturbation theory.

Eugene.
P: 1,746
 Quote by A. Neumaier asymptotic bound states,
I don't think I have fundamental problems with those

 Quote by A. Neumaier states with infinitely many soft photons,
As I said already, this has not been done, but I think this is a technical issue

 Quote by A. Neumaier spontaneously broken symmetries, and topological effects involving solitons or instantons.
As far as I know these are purely theoretical exercises. Which experiments are you talking about?

Eugene.
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P: 11,952
 Quote by meopemuk Yes, thanks. In sections 83 of this tome you'll find a derivation of the Darwin-Breit Hamiltonian from the 2nd order scattering matrix in QED. It is called "Breit equation" there; Darwin potential is just the spin-independent part of the full Darwin-Breit interaction potential. This derivation is basically the same as in section 10.3 of my book. The dressed particle approach can be regarded as a generalization of the above derivation to higher orders of the perturbation theory. Eugene.
What you call <Darwin potential> I think it's a purely relativistic term (no spin involved) and can be derived purely from classical considerations. I remember seeing a derivation in a Romanian e-m book.
P: 1,746
 Quote by dextercioby What you call I think it's a purely relativistic term (no spin involved) and can be derived purely from classical considerations. I remember seeing a derivation in a Romanian e-m book.
Exactly. See my posts #13 and #18.

Eugene.
P: 1,943
 Quote by meopemuk I am not sure why you keep saying that infinite number of soft photons cannot fit into the Fock space? In the Fock space the number of particles can vary from 0 to infinity.
They can be any finite number. But this is not good enough.

The mean number of photons in the photon state accompanying a free electron is infinite.
Since one can show (and Bob_for_short was always eager to point this out, in his own language) that the overlap of this photon state with an arbitrary N-photon state is exactly zero for any N, the state of the photon cloud cannot be a Fock state.
P: 1,943
 Quote by meopemuk Could you please be more specific here. Why do you call these representations "fake"?
Fake = formal, ignoring the fact that distributions cannot be multiplied, and that the resulting operator is therefore not well-defined, let alone a generator of a 1-parameter group.

 Quote by meopemuk I am not sure that the "dressing" used in those "lower-dimensional incarnations" is the same dressing that I use in my approach. In my case the dressing transformation is applied to the Hamiltonian of QED, so the a/c operators of particles and the Fock space structure of the bare theory are not affected in any way.
Yes, but equivalently you could apply it to the a/c operators and leave the Hamiltonian untouched. This is essentially like choosing between the Schroedinger picture and the interaction picture. Then it is clear that all forms of dressing are of the same kind - transforming the ill-defined representation to one that works. But the dressing transform is only formally unitary - and if one adds the rigor (which is currently feasible only in dimensions <4) it turns out that it moves the Fock representation to an inequivalent one.

But it is impossible to discuss this with someone who ignores all field theoretical insight.
 Quote by meopemuk I do have a proof that the dressed QED Hamiltonian leads to the same S-matrix as the standard approach. This is done in section 10.2.
I presume you mean Theorem 10.2. The problem here is that your e^{i\Phi} does not exist as a unitary operator, so the argument given there and in Section 6.5.6 which you refer to is spurious.

But even if you argue that on the level of perturbation theory, this argument is adequate,
it doesn't change may main point: That for the _calculation_ of radiation corrections to S-matrix entries your formalism is utterly unsuitable, and you need to refer to Weinberg's calculations.

By the way, in 10.1 you say (p.351 top): ''The physical vacuum in QED is not just an
empty state without particles. It is more like a boiling “soup” of particles,
antiparticles, and photons.'' This is incorrect. It would be a boiling soup of bare particles, but these are eliminated by renormalization. The renormalized vacuum is completely empty and inert.
P: 1,943
 Quote by meopemuk The Darwin Hamiltonian can be found in section 65 "The Lagrangian to terms of second order"
But there it is unrelated to the Maxwell equations.

 Quote by meopemuk I have a different derivation logic than yours.
But you must convince the world of your logic. They want to see whether there is any advantage in going from standard QED to your version of it. Deriving the Maxwell equations from QED is very standard, while deriving them from your theory seems to impossible without first reconstructing Weinberg's version of it, since when done directly it produces all the strange effects you mention in your book.

 Quote by meopemuk I have no intention to derive Maxwell equations, because I don't believe in the idea of fields possessing energy and momentum.
A classical e/m field possesses energy and momentum. You won't convince anyone seriously interested in foundations of physics as practiced. maxwell equations are used a lot, thus they must be derivable from any foundation worth its salt. If you not even intend to do that, you give up the claim of having a foundation of electrodynamics. In that case, it is not interesting at all to discuss your theory further.

 Quote by meopemuk In chapter 12 I list a number of concrete examples, where this idea fails to describe even simplest configurations of charges. I would be most interested to hear your opinion about these examples.
I care about the bulk properties, not about the properties of point particles, which don't exist in nature.

 Quote by meopemuk Could you be more specific, which form factors you have in mind?
Weinberg, Section 10.6.

 Quote by meopemuk Form factors are derived from scattering experiments
They are the matrix elements of the electron current. They are defined independent of scattering, and contain among others information about the magnetic moment. Can you derive the anomalous magnetic moment of the electron without recourse to Weinberg? I couldn't find it in you index.

Of course, form factors may be probed by means of scattering experiments, but this is a different matter.
 Quote by meopemuk Non-trivial form factors and particle localizability are two different issues, in my opinion.
Yes. Point particles and particle localizability are two different issues, too.

But trivial form factors are synonymous with point particles. See the entry ''Are electrons pointlike/structureless?'' of Chapter B2 of my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/ph...html#pointlike
P: 1,943
 Quote by meopemuk Yes, thanks. In sections 83 of this tome you'll find a derivation of the Darwin-Breit Hamiltonian from the 2nd order scattering matrix in QED. It is called "Breit equation" there; Darwin potential is just the spin-independent part of the full Darwin-Breit interaction potential. This derivation is basically the same as in section 10.3 of my book. The dressed particle approach can be regarded as a generalization of the above derivation to higher orders of the perturbation theory.
Yes, but it is a one-way road: From the foundations to an approximately valid equation, from which Maxwell's equations cannot be recovered.
P: 1,943
Quote by meopemuk
 Quote by A. Neumaier spontaneously broken symmetries, and topological effects involving solitons or instantons.
As far as I know these are purely theoretical exercises.
It seems that either you don't know very far, or - since according to you the only things that exist are particles - you regard all quantum field theory as purely theoretical exercise.

 Quote by meopemuk Which experiments are you talking about?
One cannot go from quarks to hadrons without chiral symmetry breaking, http://en.wikipedia.org/wiki/Chiral_symmetry . And QCD would not be consistent with experiment without instantons, http://arxiv.org/pdf/hep-ph/9610451
 Sci Advisor P: 1,943 On top of p.133 of your book, I found the remark that ''the photon is not a true elementary particle as it is not described by an irreducible representation of the Poincar´e group. We will see in subsection 5.3.3 that a photon is described by a reducible representation of the Poincar´e group which is a direct sum of two irreducible representations with helicities +1 and -1.'' In your terminology (that only considers irreducible representations of the connected part of the Poincare group), the left-handed photon is elementary, and the right-handed photon is its antiparticle. Thus photons are elementary even according to this rule - just not all of them.
P: 1,746
 Quote by A. Neumaier They can be any finite number. But this is not good enough. The mean number of photons in the photon state accompanying a free electron is infinite. Since one can show (and Bob_for_short was always eager to point this out, in his own language) that the overlap of this photon state with an arbitrary N-photon state is exactly zero for any N, the state of the photon cloud cannot be a Fock state.
I think, this is just hairsplitting. This looks to me the same as saying that infinity does not belong to the real axis, because real axis is composed of finite numbers only. Well, formally this is true, but not in substance. Anyway, we can always incorporate infinite numbers in the real axis by means of the non-standard analysis. I believe, the same can be done with infinite numbers of photons in the Fock space.

Eugene.
P: 1,943
 Quote by meopemuk I think, this is just hairsplitting.
You'll discover that this sort of hair splitting causes real problems once you try to correctly handle soft photons in your approach. (I tried to find your blog from a few years ago where you had discussed your failure to handle the IR problem, but is seems to have disappeared.)
P: 1,746
 Quote by A. Neumaier I presume you mean Theorem 10.2. The problem here is that your e^{i\Phi} does not exist as a unitary operator, so the argument given there and in Section 6.5.6 which you refer to is spurious.
Yes, I work under the naive assumption that if operator $$\Phi$$ is Hermitian, meaning that $$\Phi^{\dag} = \Phi$$, then $$e^{i\Phi}$$ is unitary. This is Lemma F.4 in the Appendix. Could you provide an example, where this assumption is not correct?

 Quote by A. Neumaier But even if you argue that on the level of perturbation theory, this argument is adequate, it doesn't change may main point: That for the _calculation_ of radiation corrections to S-matrix entries your formalism is utterly unsuitable, and you need to refer to Weinberg's calculations.
The only reason that does not allow me to perform calculations of radiative corrections with dressed particles is the infrared divergence, i.e., the ill-defined limit $$\lambda \to 0$$ in chapter 9. As I said earlier, I don't know yet how to overcome this problem. So, your criticism is accepted. However, I also believe that this is a technical problem, which can be solved if some more brain power is applied. As in the standard QED, this problem may be solved by explicit consideration of large (infinite) number of soft photons, which is rather challenging mathematically.

 Quote by A. Neumaier By the way, in 10.1 you say (p.351 top): ''The physical vacuum in QED is not just an empty state without particles. It is more like a boiling “soup” of particles, antiparticles, and photons.'' This is incorrect. It would be a boiling soup of bare particles, but these are eliminated by renormalization. The renormalized vacuum is completely empty and inert.
Agreed. I've changed this piece to read "...boiling “soup” of *bare* particles..." Thank you.

Eugene.
P: 1,943
 Quote by meopemuk Yes, I work under the naive assumption that if operator $$\Phi$$ is Hermitian, meaning that $$\Phi^{\dag} = \Phi$$, then $$e^{i\Phi}$$ is unitary. This is Lemma F.4 in the Appendix. Could you provide an example, where this assumption is not correct?
The Hille-Yosida theorem says that e^{i\Phi) exists for a hermitian |phi if and only iff \Phi is self-adjoint.
See, e.g., (2.21) in http://arxiv.org/pdf/quant-ph/9907069 for a non-selfadjoint momentum operator; physically more relevant examples and the HY theorem itself are discussed in Vol.3 of the math physics treatise by Thirring.
 Quote by meopemuk The only reason that does not allow me to perform calculations of radiative corrections with dressed particles is the infrared divergence, i.e., the ill-defined limit $$\lambda \to 0$$ in chapter 9. As I said earlier, I don't know yet how to overcome this problem. So, your criticism is accepted. However, I also believe that this is a technical problem, which can be solved if some more brain power is applied.
Yes. The extra brain power is set free through getting rid of the restricting shackles of Fock space.
 Quote by meopemuk As in the standard QED, this problem may be solved by explicit consideration of large (infinite) number of soft photons, which is rather challenging mathematically.
It is challenging (since impossible) only when one insists on working in Fock space. Once one allows enough coherent states into the picture (which naturally arise in the field formulation but are orthogonal to all Fock states), the challenge disappears (on the level of rigor of theoretical physics - on the fully rigorous level, additional hurdles arise that are not yet fully overcome).
 Quote by meopemuk Agreed. I've changed this piece to read "...boiling “soup” of *bare* particles..."
But bare particles are no particles at all. They become that only if one tries to give the bare fields a particle interpretation - which they cannot have, since all these particles would be infinitely heavy after renormalization. Nobody except popularizers of physics thinks of bare particles as particles.
P: 1,746
 Quote by A. Neumaier But you must convince the world of your logic. They want to see whether there is any advantage in going from standard QED to your version of it. Deriving the Maxwell equations from QED is very standard, while deriving them from your theory seems to impossible without first reconstructing Weinberg's version of it, since when done directly it produces all the strange effects you mention in your book. A classical e/m field possesses energy and momentum. You won't convince anyone seriously interested in foundations of physics as practiced. maxwell equations are used a lot, thus they must be derivable from any foundation worth its salt. If you not even intend to do that, you give up the claim of having a foundation of electrodynamics. In that case, it is not interesting at all to discuss your theory further.
Let me be clear about it. I believe that Maxwell equations are fundamentally wrong. Yes, their solution can sometimes yield accurate results. But history of science knows many examples when incorrect theories were used successfully for centuries. I agree with your statement that Maxwell equations cannot be derived in my approach, but this doesn't worry me a bit.

Instead of Maxwell equations, I suggest to use the Darwin-Breit Hamiltonian derived in subsection 12.1.1. In chapter 12 I discuss a number of electromagnetic effects and how they can be explained/described using this Hamiltonian. If you know some other effect, where this Hamiltonian fails, then it would be a real problem for my approach. Do you know such an effect?

 Quote by A. Neumaier I care about the bulk properties, not about the properties of point particles, which don't exist in nature.
If Maxwell equation cannot describe the dynamics of two isolated electrons (I don't care whether you call them point particles or not), then how you can be sure that dynamics of billions of electrons in wires is described correctly?

 Quote by A. Neumaier Can you derive the anomalous magnetic moment of the electron without recourse to Weinberg? I couldn't find it in you index.
In my approach, the anomalous magnetic moment would appear as a 4th perturbation order correction to the Darwin-Breit potential. I have confessed already that the dressed particle transformation has been performed explicitly only in the 2nd and 3rd orders. The 4th order calculation is complicated by infrared divergences in loop integrals. For example, the relevant vertex renormalization integral in 9.2.6 contains expression $$\log(\lambda)$$, where $$\lambda \to 0$$. The resolution of this divergence is well understood in standard QED. It is related to consideration of soft photons. I hope that a similar solution can be found in the dressed particle approach as well. But I have not done that, and I am not ready to discuss it here.

 Quote by A. Neumaier But trivial form factors are synonymous with point particles.
I think, we simply used different terminologies. No disagreement on substance. I promise not to call electron and photon point particles anymore. I will call them "elementary particles".

Eugene.
P: 1,746
 Quote by A. Neumaier One cannot go from quarks to hadrons without chiral symmetry breaking, http://en.wikipedia.org/wiki/Chiral_symmetry . And QCD would not be consistent with experiment without instantons, http://arxiv.org/pdf/hep-ph/9610451
As I suspected, these are just theoretical speculations. The symmetry breaking and instantons/solitons have not been observed directly in experiment. Is it true?

Eugene.
P: 1,746
 Quote by A. Neumaier On top of p.133 of your book, I found the remark that ''the photon is not a true elementary particle as it is not described by an irreducible representation of the Poincar´e group. We will see in subsection 5.3.3 that a photon is described by a reducible representation of the Poincar´e group which is a direct sum of two irreducible representations with helicities +1 and -1.'' In your terminology (that only considers irreducible representations of the connected part of the Poincare group), the left-handed photon is elementary, and the right-handed photon is its antiparticle. Thus photons are elementary even according to this rule - just not all of them.
I think this is just a terminological issue. It depends on how we define "elementary particle". I decided to identify elementary particles with irreducible representations of the Poincare group. Since 1-photon space is not irreducible, then, according to my formal definition, photon is not elementary. Another point is that we cannot treat separately right-handed and left-handed photons, because one can always make a linear combination of them, which will defy such classification.

Eugene.
P: 1,746
 Quote by A. Neumaier You'll discover that this sort of hair splitting causes real problems once you try to correctly handle soft photons in your approach. (I tried to find your blog from a few years ago where you had discussed your failure to handle the IR problem, but is seems to have disappeared.)
Try http://meopemuk2.blogspot.com/ This blog was inactive for a few years, but still alive. You are welcome to post your thoughts there.

Eugene.

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