Exploring the Possibilities of a New Relativistic Quantum Theory

[meopemuk]

Dear moderator,

I would like to submit to the Independent Research forum my book "Relativistic Quantum Dynamics", which was published at the arXiv site as
http://www.arxiv.org/abs/physics/0504062

The book's abstract is

"This book is an attempt to build a consistent relativistic quantum theory of interacting particles. In the first part of the book "Quantum electrodynamics" we present traditional views on theoretical foundations of particle physics. Our discussion proceeds systematically from the principle of relativity and postulates of measurements to the renormalization in quantum electrodynamics. In the second part of the book "The quantum theory of particles" the traditional approach is reexamined. We find that formulas of special relativity should be modified to take into account interparticle interactions. We also suggest to reinterpret quantum field theory in the language of physical "dressed" particles. In this new formulation the fundamental objects are particles rather than fields. This approach eliminates the need for renormalization and opens up a new way for studying dynamical and bound state properties of quantum interacting systems. The developed theory is applied to realistic physical objects and processes including the hydrogen atom, the decay law of moving unstable particles, the dynamics of interacting charges, and boost transformations of observables. These results force us to take a fresh look at some core issues of modern particle theories, in particular, the Minkowski space-time unification, the role of quantum fields and renormalization, and the alleged impossibility of action-at-a-distance. A new perspective on these issues is suggested. It can help to solve the biggest problem of modern theoretical physics -- a consistent unification of relativity and quantum mechanics."

This book incorporates a number of peer-reviewed journal publications:

B. T. Shields, M. C. Morris, M. R. Ware, Q. Su, E. V. Stefanovich, R. Grobe, “Time dilation in relativistic two-particle interactions”, Physical Review A 82, No.5 (2010) 052116.

E.V. Stefanovich, "Is Minkowski Space-Time Compatible with Quantum Mechanics?" Foundations of Physics 32 (2002), 673-703.

E.V. Stefanovich, "Quantum Field Theory without Infinities", Annals of Physics 292 (2001), 139-156.

E.V. Stefanovich, "Quantum effects in relativistic decays". International Journal of Theoretical Physics 35 (1996), 2571-2586.

Regarding the experimental verification of the new theory, I have two suggestions. First, there is an indirect evidence for the proposed instantaneous interaction between charged particles in numerous existing experiments with "evanescent waves" quoted in subsection 11.4.4 of the book. Second, I also predict (small) deviations from Einstein's time dilation formula in decays of fast moving particles. This is explained in subsection 14.4.1 of the book.

Thank you very much.
Eugene Stefanovich.

 

[A. Neumaier]

The following piece of discussion is related to the approach taken in this book, and hence belongs here:

Now, you are saying that this picture is wrong or incomplete. If I understand correctly, you are saying that we need to introduce fields (the electric current and electromagnetic potentials) and solve Maxwell equations in both quantum and classical limits.

I was saying that every engineer uses the traditional Maxwell equations, and that any theory that deserves the name QED must be able to reproduce it in the appropriate approximation, no matter which starting point it takes. If you can derive it from your Hamiltonian particle theory, fine.

So please try to derive the classical Maxwell equations from the quantum ED proposed in your book!

 

[meopemuk]

So please try to derive the classical Maxwell equations from the quantum ED proposed in your book!

The classical limit is taken in chapter 12, where I argue that classical electrodynamics can be formulated without using the concepts of electric and magnetic fields. If radiation is not involved, then it is sufficient to use direct position- and velocity-dependent (Darwin-Breit) potentials between charged particles in a Hamiltonian approach. I discuss a number of concrete examples and show that experimental observations are reproduced pretty well. I also consider a few examples, where traditional field-based approach leads to yet unresolved paradoxes, but the new particle-based theory has no problems.

If radiation is involved, then we need to use (at least) the 3rd perturbation order of QED. The way to do that is outlined in section 14.2. Note that radiation is represented there as a collection of discrete photons rather than a continuous "electromagnetic field". This section discusses the emission of photons by accelerated charges and the rate formula for radiative transitions. I haven't put enough work into this section yet, so some numerical coefficients in formulas do not look right. But the whole idea should be transparent already.

Eugene.

 

[A. Neumaier]

The classical limit is taken in chapter 12, where I argue that classical electrodynamics can be formulated without using the concepts of electric and magnetic fields. If radiation is not involved, then it is sufficient to use direct position- and velocity-dependent (Darwin-Breit) potentials between charged particles in a Hamiltonian approach. I discuss a number of concrete examples and show that experimental observations are reproduced pretty well. I also consider a few examples, where traditional field-based approach leads to yet unresolved paradoxes, but the new particle-based theory has no problems.

But I don't see anywhere in Chapter 12 a derivation of the Maxwell equations, only a discussion about that you get something different.

So which equations for electrodynamics should engineers be using, based on your Hamiltonian theory?

How do you explain the fact that they have been happy with the standard Maxwell equations for more than a century?

If radiation is involved, then we need to use (at least) the 3rd perturbation order of QED. The way to do that is outlined in section 14.2. Note that radiation is represented there as a collection of discrete photons rather than a continuous "electromagnetic field". This section discusses the emission of photons by accelerated charges and the rate formula for radiative transitions. I haven't put enough work into this section yet, so some numerical coefficients in formulas do not look right. But the whole idea should be transparent already.

How do you explain that Hertz could discover electromagnetic radiation based on Maxwell's equation alone, without knowing about the existence of discrete photons?

 

[meopemuk]

But I don't see anywhere in Chapter 12 a derivation of the Maxwell equations, only a discussion about that you get something different.

So which equations for electrodynamics should engineers be using, based on your Hamiltonian theory?

How do you explain the fact that they have been happy with the standard Maxwell equations for more than a century?

Engineers can keep using Maxwell equations, which are resonable approximations for macroscopic cases, where one deals with huge numbers of charges forming continuous current densities. For example, in subsection 12.2.5 I show how one can derive a "kind of" Maxwell equation in my particle-based approach.

However, Maxwell equations are not very useful for describing systems of few point charges or magnetic moments. There are all kinds of paradoxes and violations of conservations laws in Maxwell's electrodynamics. Some of them are discussed in Chapter 12.

Another area, where Maxwell's electrodynamics seems to fail is related to experiments on superluminal propagation of evanescent waves, as discussed in subsection 11.4.4.

How do you explain that Hertz could discover electromagnetic radiation based on Maxwell's equation alone, without knowing about the existence of discrete photons?

History of science is full of funny ironies. Newton discovered "Newton rings", which is, actually, a quantum interference effect, though he never took a class on quantum mechanics.

Eugene.

 

[A. Neumaier]

Engineers can keep using Maxwell equations, which are resonable approximations for macroscopic cases, where one deals with huge numbers of charges forming continuous current densities. For example, in subsection 12.2.5 I show how one can derive a "kind of" Maxwell equation in my particle-based approach.

In 12.2.5, I only see one of the four Maxwell equations derived - after (12.34). But this one seems to be derived exactly - so I don't understand your talk about ''reasonable approximations''.

And what about the other three? Can you justify the engineering practice from your foundations?

However, Maxwell equations are not very useful for describing systems of few point charges or magnetic moments.

The important thing is that you recover the engineering practice, not something about hypothetical point charges, for which the Maxwell equations were never designed. There are no point charges in Nature.

History of science is full of funny ironies. Newton discovered "Newton rings", which is, actually, a quantum interference effect, though he never took a class on quantum mechanics.

But one doesn't need quantum mechanics to explain Newton rings.

On the other hand, you seem to get two kinds of electromagnetic radiation - the one coming from the engineer's Maxwell equations, and a second set coming from your photons. the Maxwell equations. This is very funny.

You need to explain why both are precisely the same thing.

 

[A. Neumaier]

Dynamics in relativistic QFT

I want to comment on your claim on p.347 of your book that

the “bare particle” Hamiltonian of QED is completely
unsuitable for calculations of time evolution.

Standard QED contains the full dynamical information about quantum fields in the Heisenberg picture.

The reason why the standard QFT textbooks only discuss scattering is that the usually introduce to applications in elementary particle physics, where all experimental raw information comes through scattering.

But books and papers on nonequilibrium thermodynamics show how to get the dynamical content of QED or other QFTs via the derivation of quantum kinetic equations (see, e.g., http://arxiv.org/pdf/1007.1099 ) or Kadanoff-Baym equations.

The reason why the dynamics is inherent in relativistic QFT even in the absence of an explicit Hamiltonian is that the Heisenberg dynamics in terms of the quantum fields is simply a time shift in the argument. Hence any knowledge about field expectation values (which is what Feynman rules are made for) translates directly into corresponding dynamical information. The only difference to the Feynman rules for scattering is that one needs to use the closed time path (CTP) = Schwinger-Keldysh formalism to get through the path-ordering (which replaces the usual time-ordering) all required field expectations.

 

[meopemuk]

In 12.2.5, I only see one of the four Maxwell equations derived - after (12.34). But this one seems to be derived exactly - so I don't understand your talk about ''reasonable approximations''.

This equation (the last one in subsection 12.2.5) resembles one of the Maxwell's equations only in form, but not in substance. First of all, in my approach there are no electric and magnetic fields, which in Maxwell theory are supposed to be kinds of forms of existence of matter - with their field energies, momenta, etc. In my approach there are only particles with their charges and magnetic moments and instantaneous forces acting between them.

In this equation, vector E_1 should be understood as "a vector quantity that is equal to the force exerted on a charged particle at rest divided by the charge value". Vector B_1 should be understood as "a vector quantity such that if we insert it in eq. (12.31) we will obtain the force acting on a moving particle".

And what about the other three? Can you justify the engineering practice from your foundations?

Perhaps I can derive other three "Maxwell equations" in a similar manner. I will think about that. However, even if this can be done, the similarity with the true field-based Maxwell theory will be only superficial.

My approach is basically equivalent to the Darwin Hamiltonian formulation of classical electrodynamics. This formulation is known to be pretty accurate. So, engineers can it them in place of Maxwell equations.

The important thing is that you recover the engineering practice, not something about hypothetical point charges, for which the Maxwell equations were never designed. There are no point charges in Nature.

What about electrons? They are described by an irreducible unitary representation of the Poincare group, and I can define electron states localized in the Newton-Wigner sense. This is my definition of a point particle.

But one doesn't need quantum mechanics to explain Newton rings.

On the other hand, you seem to get two kinds of electromagnetic radiation - the one coming from the engineer's Maxwell equations, and a second set coming from your photons. the Maxwell equations. This is very funny.

You need to explain why both are precisely the same thing.

I think that you need to do the explanation, not me.

In my book, photons and electrons are particles. The Newton rings, double-slit and other interference effects (including interference experiments with electrons, neutrons, etc.) are fully explained within quantum mechanics of point particles, e.g., as in Feynman famous discussions of the double-slit setup. There are no fields and no Maxwell equations.

On the other hand, it seems that you have a problem. You have the same double-slit interference effect explained quite differently by two different parts of your theory. There is a classical field explanation a la Young, Fresnel and Maxwell, and there is a quantum explanation a la Feynman. In my opinion, this is an ugly situation. In a consistent theory, one physical effect cannot have two different explanations.

Eugene.

 

[meopemuk]

The reason why the dynamics is inherent in relativistic QFT even in the absence of an explicit Hamiltonian is that the Heisenberg dynamics in terms of the quantum fields is simply a time shift in the argument.

I don't understand this at all. In quantum mechanics, the Hamiltonian is a generator of time translations. So, if you know the Hamiltonian then you know the time evolution of all states. Inversely, if you know the time evolution of all states, you should be able to recover your Hamiltonian. The time evolution is impossible to describe without a well-defined Hamiltonian operator. At least, if you are working within good old standard quantum mechanics. If you are talking about some other theory, then I don't know.

Eugene.

 

[A. Neumaier]

This equation (the last one in subsection 12.2.5) resembles one of the Maxwell's equations only in form, but not in substance. First of all, in my approach there are no electric and magnetic fields, [...]
Perhaps I can derive other three "Maxwell equations" in a similar manner. I will think about that. However, even if this can be done, the similarity with the true field-based Maxwell theory will be only superficial.

If your theory is to deserve the name QED it must be able to derive from it the Maxwell equations in the form actually used by engineers. For where else should these equations come from if not from the underlying microscopic theory?

Standard QED reduced to the classical Maxwell equations in a well-understood approximation. If you want to be taken serious with your replacement of QED by a pure particle theory you must be able to explain the classical Maxwell equations as well.
For the macroscopic matter engineers work with is composed of the same stuff of which your theory claims to be a complete description!

My approach is basically equivalent to the Darwin Hamiltonian formulation of classical electrodynamics. This formulation is known to be pretty accurate. So, engineers can it them in place of Maxwell equations.

Is it known to be equivalent to the Maxwell equations? In that case, you'd add to your book the proof of equivalence, and mention how you get the Darwin Hamiltonian formulation of classical electrodynamics as a limit of your quantum formulation.

What about electrons? They are described by an irreducible unitary representation of the Poincare group, and I can define electron states localized in the Newton-Wigner sense. This is my definition of a point particle.

The point-like nature of the bar electrons is destroyed by renormalization, through which the electron acquires a nontrivial form factor. For the details and for references, see the entry ''Are electrons pointlike/structureless?'' of Chapter B2 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#pointlike

I think that you need to do the explanation, not me.

No. You are proposing a theory that should replace standard QED. To be taken serious you need to convince people that they don't get spurious effects through adoption of the proposed alternative.

You have the same double-slit interference effect explained quite differently by two different parts of your theory. There is a classical field explanation a la Young, Fresnel and Maxwell, and there is a quantum explanation a la Feynman. In my opinion, this is an ugly situation. In a consistent theory, one physical effect cannot have two different explanations.

There is no problem since when the input states are coherent states the quantum explanation reduces exactly to the classical explanation.

 

[A. Neumaier]

I reply here to your comment from another thread since it better fits here:

I would like to stay at the same rigor level as in Weinberg's book. It seems that he is not particularly concerned about "dense subspaces", so I am not concerned either. For me it is sufficient if 10 Hermitian operators are found, which satisfy Poincare commutation relations. Then I would say that we have a relativistic quantum theory.

One can stay at the same rigor of Weinberg's book only if one also restricts one's claims to those justified by this level of rigor.

Weinberg nowhere claims that the Hilbert space of an interacting QFT is a Fock space - he knows better. He leaves room for all sorts of things that cannot be accommodated in Fock space but occur in physically relevant field theories - such as asymptotic bound states, states with infinitely many soft photons, spontaneously broken symmetries, and topological effects involving solitons or instantons.

For the purposes of scattering theory, which he develops in the book, it is enough to get the asymptotic structure right, and that is a Fock space if there are no massless particles. However, it is a Fock space in which the dynamics is free, not interacting. And it contains a separate field for _each_ bound state of the theory - something far from what one wants to start with. QED with massive photons is exceptional here in that it does not form bound states apart from the photon, electron, and positron.

But if, as in QED, there are massless particles, even the asymptotic space is no longer a Fock space because of infrared effects involving states containing infinitely many soft photons. There are no such states in a Fock space! You don't address the infrared problem at all (except, if I recall correctly, in a single footnote), hence miss these problems completely.

By ignoring what QFT has to say about the non-particle aspects of relativistic quantum mechanics, and what it has to say about theories in lower dimensions, where the structure is much better known), you deprive yourself of a lot of important insight, necessary to arrive at a valid view - and all the more so for constructing a valid conceptual alternative for standard QED.

 

[A. Neumaier]

I reply here to your comment from another thread since it better fits here:

There are two distinct ways to look at relativistic quantum theory of systems with variable number of particles (aka QFT).

Yes: The very successful mainstream way called quantum field theory, and a minority trickle of work promoted by people like Klink or Polyzou (who are compatible with the mainstream since they make no claim of equivalence to the standard approach) on the one hand and you and Vladimir Kalitvanski (on PF =Bob_for_short) on the other hand, both not being able to reproduce the successes of QED, but making claims of having something better.

In one approach (advocated by you) the Hilbert space has a non-Fock structure. [...] In the limit of widely separated particles the Fock-like structure of the Hilbert space gets restored, which allows you to define asymptotic n-particle states.

Only in the case where there are no massless particles. For QED, not even the asymptotic Hilbert space is a Fock space.

A different approach (that I read between the lines of Weinberg's textbook) is that particles are the primary objects. 2-particle Hilbert space is always a tensor product of 1-particle spaces, whether particles interact or not. Independent on interaction, one can always define and prepare pure n-particle states, which are always orthogonal to m-particle states. This leads to the Fock structure of the total Hilbert space, and this structure is independent on interactions. Interactions are introduced by specifying an unitary representation of the Poincare group in this Fock space.

Fake representations that cannot be turned into unitary representations of the groups in lower-dimensional incarnations of the same ideas -- since the dressing is in these cases provably equivalent to the non-Fock representations constructed by QFT.

As I understand, this approach is fully capable to explain scattering, decays, and the entire range of processes in which particles can be created or annihilated.

But not the extremely successful Maxwell equations used by engineers to create and maintain the modern world of technology.

Possibly, the best we can do is (1) recognize the differences

I always recognized (and emphasized) the differences: The first is able to explain everything one wants to explain, the other can just do some bound state calculations and already has lots of difficulties to incorporate radiative corrections. And for replicating the S-matrix results, you must rely on Weinberg's calculations, since you cannot do it on the version you built yourself.

(2) agree to coexist peacefully.

They coexist peacefully, with your approach being ignored by essentially anybody.

I am not trying to create a war, but to open your eyes to a large number of facts that you fail to see, which would change your work to become more respectable. Once you declare that you no longer want to listen to me, I'll stop trying to teach you the insights of the current state of the art.

 

[meopemuk]

Is it known to be equivalent to the Maxwell equations? In that case, you'd add to your book the proof of equivalence, and mention how you get the Darwin Hamiltonian formulation of classical electrodynamics as a limit of your quantum formulation.

The Darwin-Breit Hamiltonian is derived from dressed-particle QFT in the 2nd perturbation order in section 10.3. The whole chapter 12 is devoted to discussions of various electromagnetic experiments and their description within the Darwin-Breit approach. It is shown there how and where the results are similar/different to those obtained in classical electrodynamics. The equivalence of the Darwin Hamiltonian and Maxwell equations in the (v/c)^2 approximation has been proven in many textbooks. See, for example, section 12.6 in Jackson. I remember that Landau-Livgarbagez also has a similar discussion. Unfortunately, in engineering practice it is not possible to see effects in orders higher than (v/c)^2. So, Darwin Hamiltonian can replace Maxwell equations as an engineering tool. The only exception are radiation effects. They require 3-order approximation as described in chapter 14.

The point-like nature of the bar electrons is destroyed by renormalization, through which the electron acquires a nontrivial form factor. For the details and for references, see the entry ''Are electrons pointlike/structureless?'' of Chapter B2 of my theoretical physics FAQ at http://arnold-neumaier.at/physfaq/physics-faq.html#pointlike

In my approach, there is no difference between bare and physical electrons. Only physical electrons are discussed. Such electrons can be prepared in eigenstates of the Newton-Wigner position operator. The interaction potential between physical electrons is (roughly) 1/r, where the distance r can approach zero as close as one likes. These look like point-like particles to me.


Eugene.

 

[A. Neumaier]

The Darwin-Breit Hamiltonian is derived from dressed-particle QFT in the 2nd perturbation order in section 10.3. The whole chapter 12 is devoted to discussions of various electromagnetic experiments and their description within the Darwin-Breit approach. It is shown there how and where the results are similar/different to those obtained in classical electrodynamics. The equivalence of the Darwin Hamiltonian and Maxwell equations in the (v/c)^2 approximation has been proven in many textbooks. See, for example, section 12.6 in Jackson. I remember that Landau-Livgarbagez also has a similar discussion. Unfortunately, in engineering practice it is not possible to see effects in orders higher than (v/c)^2. So, Darwin Hamiltonian can replace Maxwell equations as an engineering tool.

But they use Maxwell. So you need to derive th Maxwell equations from the Darwin Hamiltonian to justify their successful practice.

The only exception are radiation effects. They require 3-order approximation as described in chapter 14.

I don't have Jackson. Darwin is not in the index of Vol.2 of my (German version of) Landau/Lifgarbagez on classical field theory; so please provide section numbers. In any case, it is very likely that any traditional derivation starts with Maxwell and ends up with Darwin, while you'd have to go the opposite way, since you need to derive Maxwell from your particle theory. Thus you cannot simply refer to their work but have to give your own derivation.

In my approach, there is no difference between bare and physical electrons. Only physical electrons are discussed. Such electrons can be prepared in eigenstates of the Newton-Wigner position operator. The interaction potential between physical electrons is (roughly) 1/r, where the distance r can approach zero as close as one likes. These look like point-like particles to me.

Then how do you recover the standard form factors of the electron, which prove its non-point-like nature?

 

[meopemuk]

But if, as in QED, there are massless particles, even the asymptotic space is no longer a Fock space because of infrared effects involving states containing infinitely many soft photons. There are no such states in a Fock space!

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. Isn't it?

Eugene.

 

[meopemuk]

Fake representations that cannot be turned into unitary representations of the groups in lower-dimensional incarnations of the same ideas -- since the dressing is in these cases provably equivalent to the non-Fock representations constructed by QFT.

Could you please be more specific here. Why do you call these representations "fake"? 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.

I always recognized (and emphasized) the differences: The first is able to explain everything one wants to explain, the other can just do some bound state calculations and already has lots of difficulties to incorporate radiative corrections. And for replicating the S-matrix results, you must rely on Weinberg's calculations, since you cannot do it on the version you built yourself.

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. You are right that this proof works only in the absence of infrared divergences. I agree that I wasn't able to resolve the infrared problem so far. But I see it as a technical problem rather than a showstopper. You are entitled to a different opinion, of course.

Eugene.

 

[dextercioby]

I don't have Jackson. Darwin is not in the index of Vol.2 of my (German version of) Landau/Lifgarbagez on classical field theory; so please provide section numbers.

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/9780750633710/Quantum-Electrodynamics (the table of contents in on Amazon.com)

 

[meopemuk]

I don't have Jackson. Darwin is not in the index of Vol.2 of my (German version of) Landau/Lifgarbagez on classical field theory; so please provide section numbers. In any case, it is very likely that any traditional derivation starts with Maxwell and ends up with Darwin, while you'd have to go the opposite way, since you need to derive Maxwell from your particle theory. Thus you cannot simply refer to their work but have to give your own derivation.

The Darwin Hamiltonian can be found in section 65 "The Lagrangian to terms of second order"
I have a different derivation logic than yours. I know that all experimental non-radiative electromagnetic effects are described very well by the Darwin Hamiltonian. So, I am satisfied with the fact that exactly this Hamiltonian can be derived from the dressed particle QED in the 2nd perturbation order and in the (v/c)^2 approximation. So, I am safe as far as comparison with experiment is concerned.

I have no intention to derive Maxwell equations, because I don't believe in the idea of fields possessing energy and momentum. 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.

Then how do you recover the standard form factors of the electron, which prove its non-point-like nature?

Could you be more specific, which form factors you have in mind? Form factors are derived from scattering experiments, and I claim that the scattering matrix in my approach is exactly the same as in standard QED and in experiment. So, I should be able to reproduce whatever form factors are there. Non-trivial form factors and particle localizability are two different issues, in my opinion.

Eugene.

 

[meopemuk]

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/9780750633710/Quantum-Electrodynamics (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.

 

[meopemuk]

asymptotic bound states,

I don't think I have fundamental problems with those

states with infinitely many soft photons,

As I said already, this has not been done, but I think this is a technical issue

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.

 

[dextercioby]

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.

 

[meopemuk]

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.

Exactly. See my posts #13 and #18.

Eugene.

 

[A. Neumaier]

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.

 

[A. Neumaier]

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.

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.

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.

 

[A. Neumaier]

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.

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.

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.

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.

Could you be more specific, which form factors you have in mind?

Weinberg, Section 10.6.

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.

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://arnold-neumaier.at/physfaq/physics-faq.html#pointlike

 

[A. Neumaier]

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.

 

[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.

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

 

[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.

 

[meopemuk]

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.

 

[A. Neumaier]

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.)

 

[meopemuk]

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?

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.

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.

 

[A. Neumaier]

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.

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.

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).

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.

 

[meopemuk]

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?

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?

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.

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.

 

[meopemuk]

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.

 

[meopemuk]

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.