A paradox in electromagnetic theory

[meopemuk]

Well---it's not hidden anymore

are you looking at a specific facet, or the entire thing?

My starting point is the recognition that quantum electrodynamics (QED) is the most accurate theory of electromagnetic phenomena. So, the goal is to derive classical electrodynamics as a classical limit of QED. The major obstacle on this path is renormalization of QED, which makes its Hamiltonian infinite and ill-defined. I was able to overcome this obstacle by using the so-called "dressed particle" approach to quantum field theories:

E.V. Stefanovich "Quantum field theory without infinities", Ann. Phys. (NY) 292 (2001),139

E. V. Stefanovich, "Relativistic quantum dynamics", http://www.arxiv.org/abs/physics/0504062

The emerging classical limit is quite different from Maxwell's electrodynamics. For example, instead of the Lorentz force law it predicts that charged particles interact via instantaneous Darwin's potentials. I am now trying to figure out if this approach can lead to experimentally verifiable differences from Maxwell's theory. One interesting difference is that Darwin's potentials predict additional magnetic repulsive forces between electrons in current-carrying wires. It seems that there are experiments demonstrating exactly that

N. Graneau, T. Phipps Jr, D. Roscoe An experimental confirmation of longitudinal electrodynamic forces. EUROP. PHYS. J D, 15, (2001) 87 - 97

Eugene.

 

[meopemuk]

And Maxwell's theory is not actually internal inconsistent, because mathematically it does have a model. It's just that the theory contradicts certain common assumptions, such as the existence of point particles.

The existence of point particles, e.g., electrons, is a hard experimental fact. Moreover, if Maxwell's theory is fully consistent, then all paradoxes that I mentioned should have a successful resolution. I am not sure about that.

Eugene.

 

[rewebster]

I always have had trouble with the 'idea' of 'point particles'.

 

[meopemuk]

I always have had trouble with the 'idea' of 'point particles'.

What is wrong with this idea in your opinion?

Eugene.

 

[lugita15]

The existence of point particles, e.g., electrons, is a hard experimental fact. Moreover, if Maxwell's theory is fully consistent, then all paradoxes that I mentioned should have a successful resolution. I am not sure about that.

Eugene.

No. Maxwell's theory can be internally consistent without agreeing with experimental fact.

 

[meopemuk]

No. Maxwell's theory can be internally consistent without agreeing with experimental fact.

Possibly I was not clear enough. I am looking not only at internal mathematical consistency, but also at agreement with experiment. Both requirements are equally important. For example, if Maxwell's theory says that a charged magnet has a non-zero momentum, but experiment tells us that it is not moving, I see it as a sign of serious trouble.

Eugene.

 

[lugita15]

Possibly I was not clear enough. I am looking not only at internal mathematical consistency, but also at agreement with experiment. Both requirements are equally important. For example, if Maxwell's theory says that a charged magnet has a non-zero momentum, but experiment tells us that it is not moving, I see it as a sign of serious trouble.

Eugene.

As I said earlier, if you neglect quantum effects, all magnets are due to currents, which are just moving charged particles. Therefore, it should come to no surprise that a bunch of moving charged particles have a nonzero momentum.

 

[meopemuk]

As I said earlier, if you neglect quantum effects, all magnets are due to currents, which are just moving charged particles. Therefore, it should come to no surprise that a bunch of moving charged particles have a nonzero momentum.

But these charges move in a circle, so the total mechanical momentum of electrons in the wire is zero. I don't see how you can resolve this paradox without introducing some "hidden" non-observable momentum which would compensate the field momentum. In my opinion, if two non-observable things cancel each other, then we are allowed to forget about both of them.

Eugene.

 

[Hurkyl]

I don't see how you can resolve this paradox without introducing some "hidden" non-observable momentum which would compensate the field momentum.

How about qA? Both Wikipedia's article on momentum and Bo Thidé's online EMFT textbook indicate that the appropriate momentum for a particle is
p = mv + qA.
(Actually, Thidé states the relativistic version)

 

[rewebster]

What is wrong with this idea in your opinion?

Eugene.

The idea of 'point particles' and the present idea of 'fields' to me aren't logically consistent.

-----------------
Whether someone has put a name (hidden, local, inferred, Quasistatic, etc.) or not.

 

[meopemuk]

How about qA? Both Wikipedia's article on momentum and Bo Thidé's online EMFT textbook indicate that the appropriate momentum for a particle is
p = mv + qA.
(Actually, Thidé states the relativistic version)

I am not sure if using this definition of the particle momentum would solve the electromagnetic paradoxes with "hidden momentum".

The idea of 'point particles' and the present idea of 'fields' to me aren't logically consistent.

Same here. So, my proposal is to reject fields as physical entities and build electromagnetic theory in terms of interacting particles (e.g., electrons and photons) only.

Eugene.

 

[rewebster]

I haven't read your papers yet to get a better 'feel' for your work.

PS--you may want to 'edit' that last 'quote' to rewebster-- Hurkyl may want to 'sue'

 

[Hurkyl]

I am not sure if using this definition of the particle momentum would solve the electromagnetic paradoxes with "hidden momentum".

What is this "hidden momentum"?

(searches back)

I see you mentioned the momentum of the field as hidden, because you assert it cannot be verified experimentally. I don't see how you conclude that: the electric and magnetic field at any point can be measured with test charges.

 

[meopemuk]

PS--you may want to 'edit' that last 'quote' to rewebster-- Hurkyl may want to 'sue'

Fixed. Thanks.

What is this "hidden momentum"?

You can find a few references to "hidden momentum" paradoxes in my post #31.

I see you mentioned the momentum of the field as hidden, because you assert it cannot be verified experimentally. I don't see how you conclude that: the electric and magnetic field at any point can be measured with test charges.

Then we are mesuring momenta of test charges, but not the field momentum. I think it is important to note that electric and magnetic fields are not observable by themselves. We can only measure properties of particles (position, velocity, momentum, spin, acceleration, etc.). My idea is to get rid of this non-observable redundant notion of fields and formulate the theory in terms of particles directly interacting with each other.

Eugene.

 

[Hurkyl]

Then we are mesuring momenta of test charges, but not the field momentum ... My idea is to get rid of this non-observable redundant notion of fields

By measuring the forces on the test charges, we obtain the values for the electric and magnetic field. (And thus obtain the momentum density)

 

[meopemuk]

By measuring the forces on the test charges, we obtain the values for the electric and magnetic field. (And thus obtain the momentum density)

This would make sense if you believe that fields are some "material entities" which possesses momentum and energy independent on particles. So that in order to find the full momentum of a system of charged particles it is necessary to add the momentum of all particles and the field momentum.

I don't believe in this idea. I think that a theory of directly interacting charged particles and photons can be formulated without any involvement of electric and magnetic fields (as independent material entities having their own momentum and energy). I understand that this is a complete departure from classical electrodynamics as it was known for 150 years. I also understand that my project is very far from completion. However, it is also clear to me that Maxwell's electrodynamics has a number of nasty paradoxes, and that some new approaches would be needed if we want to solve these paradoxes.

Eugene.

 

[Hurkyl]

This would make sense if you believe that fields are some "material entities" which possesses momentum and energy independent on particles.

All I need for it to make sense is to believe there is something called a "field" that has values at points and a quantity called "momentum" that can be computed from the values of the field.

If that quantity called "momentum" is involved in a generalization of the law of conservation of momentum, then I would even argue the label "momentum" is an appropriate name for that quantity.

If everything I just mentioned can be computed by physical experiment, and their computed values behave according to some theory, then I would even argue that that theory is a good description of "reality".

However, it is also clear to me that Maxwell's electrodynamics has a number of nasty paradoxes,

Pseudoparadox. A paradox is a logical contradiction; merely being counterintuitive is not enough. And, of course, 'counterintuitive' is a highly subjective notion.

I think that a theory of directly interacting charged particles and photons can be formulated

I think you're right. And I expect that if you can manage this, then you'll wind up with something closely resembling a sea of photons whose stress-energy obeys Maxwell's equations, and that interacts with charged particles via the Lorentz force.

 

[cesiumfrog]

Just to expand on this "hidden momentum" business:

a basic example might be a (neutral) square solenoid standing in an (upward) electric field. Say you close a switch, allowing a battery to produce a clockwise electron current in the solenoid. The external field accelerates charges on one side, and decelerates them on the other side, with the consequence that the electrons at the bottom have a higher average velocity than on the top. This gives rise to a net sideways momentum.

It's called hidden because you wouldn't normally consider it, but nonetheless (since it is predicted by such a trivial exercise in relativistic mechanics) it hardly makes sense to deny it exists. You can certainly perform experiments to verify the assumptions of the derivation.

But does that mean conservation of momentum as a whole is violated when you close the switch? Not if you believe the produced fields also carry an exactly opposite momentum, as per classical electrodynamics.

 

[meopemuk]

All I need for it to make sense is to believe there is something called a "field" that has values at points and a quantity called "momentum" that can be computed from the values of the field.

If that quantity called "momentum" is involved in a generalization of the law of conservation of momentum, then I would even argue the label "momentum" is an appropriate name for that quantity.

If everything I just mentioned can be computed by physical experiment, and their computed values behave according to some theory, then I would even argue that that theory is a good description of "reality".

My problem with this is that electric and magnetic fields cannot be measured directly. Their momentum and energy are also non-measurable quantities. Experiments measure properties of particles. So, nothing will be lost if fields are excluded from the theory and only particles are left.

Note that measurements of the momentum, angular momentum, and energy of electromagnetic radiation do not contradict my above statements. It is more realistic to consider EM radiation as a collection of particles - photons, rather than continious electromagnetic fields. The inadequacy of the continuum field picture becomes obvious when one considers radiation of very low intensity, where individual photons can be distinguished.

Eugene.

 

[Hurkyl]

My problem with this is that electric and magnetic fields cannot be measured directly. Their momentum and energy are also non-measurable quantities. Experiments measure properties of particles.

It is measurable. I can perform an experiment that determines the value of the electric field at a point. Therefore, the value of the electric field at that point is a measurable quantity, that was measured by experiment.


Could you describe an example of an experiment that directly measures anything at all, by your definition? Every physical property, even of particles, would appear to be indirect by such a strict measure. For example...

How do I measure the position of something? I fire electromagnetic radiation at it, which the thing scatters or absorbs in some fashion. The scattered radiation interacts electromagnetically with the cones and rods in my eye, and so forth.

How do I measure the weight of something? I construct a device in elastic equilibrium (which is moderated by electromagnetic forces), and measure its position. I position the object so that its only substantial interactions are gravitation attraction to the Earth and electromagnetic repulsion with my device. I then measure the new position of my device. I repeat this experiment with a standard object, whose weight I've defined to be a predetermined value, and I can calculate the weight of my object.

How do I measure the mass of something? I measure the weight as above, I do another experiment to measure the acceleration due to gravity, and combine the results.

So, nothing will be lost if fields are excluded from the theory and only particles are left.

Actually, something is lost; knowledge of the charge and current distributions does not determine the electromagnetic field. If you knew the mass, charge, position, velocity, and acceleration of every particle in the universe simultaneously, that is not sufficient to predict the future motion of particles.

This information needs to be reclaimed if you are going to attempt to reformulate classical EM.

It is more realistic to consider EM radiation as a collection of particles - photons, rather than continious electromagnetic fields.

You are forgetting diffraction. :tongue:

The inadequacy of the continuum field picture becomes obvious when one considers radiation of very low intensity, where individual photons can be distinguished.

No, this yielded an inadequacy of the classical picture of mechanics. And to the best of my knowledge, the electromagnetic field strength tensor makes the passage to quantum electrodynamics essentially unchanged.

 

[meopemuk]

It is measurable. I can perform an experiment that determines the value of the electric field at a point. Therefore, the value of the electric field at that point is a measurable quantity, that was measured by experiment.


Could you describe an example of an experiment that directly measures anything at all, by your definition? Every physical property, even of particles, would appear to be indirect by such a strict measure. For example...

How do I measure the position of something? I fire electromagnetic radiation at it, which the thing scatters or absorbs in some fashion. The scattered radiation interacts electromagnetically with the cones and rods in my eye, and so forth.

How do I measure the weight of something? I construct a device in elastic equilibrium (which is moderated by electromagnetic forces), and measure its position. I position the object so that its only substantial interactions are gravitation attraction to the Earth and electromagnetic repulsion with my device. I then measure the new position of my device. I repeat this experiment with a standard object, whose weight I've defined to be a predetermined value, and I can calculate the weight of my object.

How do I measure the mass of something? I measure the weight as above, I do another experiment to measure the acceleration due to gravity, and combine the results.

I can agree with your argument. Both points of view are possible. Currently, there is no experimental way to prove or disprove the existence of fields. One can say that particles are bundles of fields, or one can say that there are no fields, simply particles interact with each other without intermediaries at a distance. At this point, the choice between these two alternatives is a matter of personal belief and philosophy rather than hard-core physics. However, I hope that it will be possible to conduct an experiment that will clearly distinguish these two possibilities.

Actually, something is lost; knowledge of the charge and current distributions does not determine the electromagnetic field. If you knew the mass, charge, position, velocity, and acceleration of every particle in the universe simultaneously, that is not sufficient to predict the future motion of particles.

Why not? In the classical (non-quantum) world it is sufficient. Note also that photons must be included in the list of particles.

You are forgetting diffraction.

No, I am not. Photons are massless, so quantum effects with photons are visible at macroscopic scale. Diffraction, interference, and other wave properties of light are simply manifestations of quantum properties of individual photons. It may sound strange that Grimaldi's diffraction, Newton's rings, and Young's double slit were first quantum experiments. But I think it is true.

Eugene.

 

[meopemuk]

I recently found a wonderful website with hundreds of articles that cover all kinds of problems and paradoxes in classical electromagnetism.

http://puhep1.princeton.edu/~mcdonald/examples/EM/

Eugene.