Exploring the Possibilities of a New Relativistic Quantum Theory

[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.

 

[izh-21251]

Yes, the dressed particle Hamiltonian is independent on gauges. The gauge is not a physical notion.
Eugene.

This is a very important statement!
Can one explicitly derive this Hamiltonian?
When I apply dressing to Hamiltonians in different gauges - I obtain different resulting Hamiltinians? Am I mistaking somewhere??
--------------------------------------------
If one has at hand Hamiltonian, invariant with respect to gauge transformation, it will be a great result!
It will mean, that the principle of gauge invariance (taken almost in every textbook in QFT as fundamental principle of nature) is nothing but mathematical trick..

 

[meopemuk]

This is a very important statement!
Can one explicitly derive this Hamiltonian?
When I apply dressing to Hamiltonians in different gauges - I obtain different resulting Hamiltinians? Am I mistaking somewhere??
--------------------------------------------

I think here you are talking about field theory Hamiltonians. I can agree that these Hamiltonians are different in different gauges.

I was talking about the dressed particle Hamiltonian, which is unique and gauge-independent. See, for example, the Darwin-Breit Hamiltonian in section 10.3. I am now working on adding 4th order radiative corrections to this Hamiltonian.

If one has at hand Hamiltonian, invariant with respect to gauge transformation, it will be a great result! It will mean, that the principle of gauge invariance (taken almost in every textbook in QFT as fundamental principle of nature) is nothing but mathematical trick..

Yes, I believe that gauge invariance is *not* a fundamental principle of nature.

Eugene.

 

[izh-21251]

Thank you,
Ivan

 

[strangerep]

If one has at hand Hamiltonian, invariant with respect to gauge transformation, it will be a great result!

The Hamiltonian as a whole is already invariant with respect to gauge transformations.
The trouble is that the Hamiltonian is constructed from free fields (electron, photon)
which are not gauge-invariant separately.

People (starting with Dirac 1955) have tried to construct different basic fields. E.g.,
the following transformation

$$\psi(x) ~\to~ \Psi(x) ~:=~ \psi(x) \, e^{i\, C(x)} ~~,~~\mbox{where}~~ C(x) ~:=~ \int\! d^3z\, c_j(x-z) \, A^j(z)$$

with

$$- \; \frac{\partial c_k(x-z)}{\partial x_k} ~=~ e \, \delta^{(3)}(x-z)$$

ensures that $$\Psi(x)$$ is manifestly gauge invariant and (after the
equation for c(x) is solved explicitly) has the correct Coulomb field.

Applying this transformation to the usual QED Hamiltonian results in
something which contains no explicit gauge noninvariant quantities,
though at the price of a nonlocal integral over the EM field.

I think it would be interesting to find out whether there's any useful
relationship between this re-expressed Hamiltonian and Eugene's
dressed Hamiltonian(s).

It will mean, that the principle of gauge invariance (taken almost in every textbook
in QFT as fundamental principle of nature) is nothing but mathematical trick.

I think it's been known to be just a mathematical technique, or guiding method,
for a long time. But then something else is determining which interactions occur
in nature and, via their group theoretic details, the multiplet structure(s) of
elementary particles.

 

[meopemuk]

The Hamiltonian as a whole is already invariant with respect to gauge transformations.
The trouble is that the Hamiltonian is constructed from free fields (electron, photon)
which are not gauge-invariant separately.

Many years ago I wasted a lot of efforts trying to derive field Hamiltonians from the QED Lagrangian in different gauges and figuring out how they are related to each other and what is the meaning of their differences. Then I lost interest to these rather non-trivial exercises, because I understood that field degrees of freedom are not observable, and that only particle degrees of freedom have observable meaning. So, the true Hamiltonian of quantum theory must look like the dressed particle Hamiltonian, for which the idea of gauges is simply not applicable.

Eugene.

 

[meopemuk]

I think it's been known to be just a mathematical technique, or guiding method,
for a long time. But then something else is determining which interactions occur
in nature and, via their group theoretic details, the multiplet structure(s) of
elementary particles.

I also came to the conclusion that gauge invariance is simply a convenient heuristic trick for deriving interacting field Lagrangians. The physical meaning of the gauge invariance always remained obscure for me.

Eugene.

 

[strangerep]

Many years ago I wasted a lot of efforts trying to derive field Hamiltonians from the QED Lagrangian in different gauges and figuring out how they are related to each other and what is the meaning of their differences.

I agree. But the scheme I mentioned is different in that it seeks to find basic
fields that are themselves gauge invariant, instead of trying to pick this or that gauge
with attendant difficulties. What's not clear to me is whether the gauge degrees of
freedom will just pop up again somewhere else later.

 

[meopemuk]

I agree. But the scheme I mentioned is different in that it seeks to find basic
fields that are themselves gauge invariant, instead of trying to pick this or that gauge
with attendant difficulties. What's not clear to me is whether the gauge degrees of
freedom will just pop up again somewhere else later.

Your (and Dirac's) reformulation of fields looks like a completely different approach to QFT. Does it have the same agreement with experiment as the traditional version?

Eugene.

 

[izh-21251]

I still believe, that in particle theory gauge transformations is nothing but Lorentz-boost in certain direction.
Products of photonic C/A operators and their coefficient functions are not invariant under boosts of certain direction - namely these boosts do carry transformations from one gauge to another.
One can check - under boost, expression $$\epsilon^{\mu}(\vec{k},\lambda)a(\vec{k}},\lambda)$$ transfroms into $$(\epsilon^{\mu}(\vec{k},\lambda)+C(\theta)k^{\mu})a(\vec{k},\lambda)$$.
Here $$\epsilon$$ is the coefficient function, $$a$$ is the photon C/A operator, $$C$$ - complex number depending on the boost parameter $$\theta$$.
This $$C(\theta)k^{\mu}$$ term is not canceled, being put between spinors in the interaction operator, because the vertex is off the energy shell...
------------------------------------------------------------------------------------
You show the electron-proton ($$ep\rightarrow ep$$) interaction in the 2-nd order in the Coulomb gauge... If I work in different gauge (or make a boost, which I believe to be equal actions) - this potential will be different. Thus - the dressed Hamiltonian IS gauge dependent.
That is how I see the situation... OR I still don't understand something... ))

Exactly the same situation will occur with operators, say, of type $$e^{+}e^{-}\leftrightarrow\gamma\gamma$$ or $$e\gamma\rightarrow e\gamma$$ in the 2-nd perturbation order, and, e.g. $$ee\leftrightarrow ee\gamma$$ in the 3-rd.


In $$ee\leftrightarrow ee\gamma$$ after boost, an additional term, proportional to
$$k^{\mu}$$ will occur...
Maybe I missed something and you have at hand operators, that have different properties?

Thanks,
Ivan

 

[meopemuk]

You mentioned you have explicitly derived dressed QED Hamiltonian up to the third order in coupling..
Am I right?

Could you please say few words about potentials in the 2-nd order (like $$e^{+}e^{-}\leftrightarrow\gamma\gamma$$ or $$e\gamma\rightarrow e\gamma$$) and the 3-rd order ($$ee\leftrightarrow ee\gamma$$)..

I have only two kinds of potentials derived explicitly. One if them is the 2nd order electron-proton interaction $$p^{+}e^{-}\leftrightarrow p^{+}e^{-}$$ discussed in section 10.3. I also have some pieces of the 4th order radiative correction to this potential, but this work is not completed and not yet in the book. Calculation of the charge-charge elastic potentials from field theory is a well-studied subject. See, for example, section 83 in V. B. Berestetskii, E. M. Livgarbagez, L. P. Pitaevskii, "Quantum electrodynamics" and the list of references in post #189 of this thread.

The other calculated interaction is the 3rd order "bremsstrahlung" potential $$p^{+}e^- \leftrightarrow p^{+}e^- \gamma$$ in subsection 14.2.3.

Are they also independent of gauge (invariant with respect to gauge transformations)?
What are the differences between them and the alternative potentials, which one can obtain from the field-theoretical Hamiltonians in different gauges?

When deriving these particle potentials I actually not using the dressing transformation explicitly. So, I am not using the field-theoretical Hamiltonian, and, therefore, the gauge freedom is not present in calculations. In fact, I simply fit particle potentials to the known S-matrix elements in different QED perturbation orders. Basically, I am solving equations (10.27) - (10.30). QED S-matrix elements do not depend on the chosen gauge, so I don't need to worry about the gauge (in)dependence of my results.

As discussed in subsection 10.2.9, fitting to the S-matrix does not allow one to obtain a unique set of particle potentials. There are many scattering-equivalent sets. I have a feeling that this non-uniqueness is somehow related to the uncertainty, which one would get when trying to apply the unitary dressing transformation to field Hamiltonians in different gauges. But I haven't attempted to follow this idea through.

Eugene.