Struggles With the Continuum - Part 3 - Comments

[john baez]

john baez submitted a new PF Insights post

Struggles With the Continuum - Part 3

Continue reading the Original PF Insights Post.

 

[ShayanJ]

Thanks John, really exciting posts!

 

[Greg Bernhardt]

Great series John!

 

[vanhees71]

Great overview of the "radiation-reaction problem", but isn't the conclusion rather that point charges in the literal sense are still strangers to classical field theory? For "particles" of "finite extent", there seem not to be any problems. See, e.g., the paper

Medina, Rodrigo. "Radiation reaction of a classical quasi-rigid extended particle." Journal of Physics A: Mathematical and General 39.14 (2006): 3801.
http://stacks.iop.org/JPhysA/39/3801
http://arxiv.org/pdf/physics/0508031

 

[atyy]

But is a discrete spacetime a credible alternative? Can we formulate the known laws of physics in a discrete spacetime? Currently, the answer seems to be "no", as there is no consensus lattice standard model because of the problem of chiral fermions interacting with non-Abelian gauge fields.

 

[bcrowell]

The summary of radiation reaction and point charges is very helpful -- I hadn't heard of a lot of the more recent work. A question that's been bothering me for a long time is whether there is any sense in which one can prove that a system of charged particles can't be stable. This seems to be something that all physicists are convinced must be true, but that nobody knows how to prove, or even how to state rigorously. Acceleration is a necessary but not sufficient condition for radiation, so one can't just declare that such a system automatically radiates away its energy to infinity. It's conceivable that for a set of initial conditions of measure zero, radiation vanishes exactly and the system is stable. One would have to prove that this doesn't happen. Unfortunately, it doesn't seem possible even to get started on such a proof, because there just isn't any satisfactory, canonical, self-consistent theory of interacting point charges.

 

[PAllen]

But is a discrete spacetime a credible alternative? Can we formulate the known laws of physics in a discrete spacetime? Currently, the answer seems to be "no", as there is no consensus lattice standard model because of the problem of chiral fermions interacting with non-Abelian gauge fields.

I don't see what Medina's paper (linked by Vanhees) has to do with discrete spacetime. It just posits that a classical field theory can accommodate matter bodies (meeting criteria determined by the field theory), but not point particles.

 

[atyy]

I don't see what Medina's paper (linked by Vanhees) has to do with discrete spacetime. It just posits that a classical field theory can accommodate matter bodies (meeting criteria determined by the field theory), but not point particles.

I was responding to the initial thoughts in this Insight: "In these posts, we’re seeing how our favorite theories of physics deal with the idea that space and time are a continuum, with points described as lists of real numbers. We’re not asking if this idea is true: there’s no clinching evidence to answer that question, so it’s too easy to let ones philosophical prejudices choose the answer. Instead, we’re looking to see what problems this idea causes, and how physicisists have struggled to solve them."

 

[klotza]

Very interesting article. For those interested, my friend James Hedberg, now at the City University of New York, wrote a historical/theoretical overview of fieldless E&M.

http://www.jameshedberg.com/docs/directactionelectrodynamics.pdf

 

[Jano L.]

john baez submitted a new PF Insights post

Struggles With the Continuum - Part 3

Continue reading the Original PF Insights Post.

In short, the electric field is ‘infinite’, or undefined, at the particle’s location. So, it is unclear how to define the ‘self-force’ exerted by the particle’s own electric field on itself.

Indeed! It also means the Poynting theorem is invalid at the points where the particles are (it still holds in between them).

This makes its interpretation as a work-energy theorem invalid. It is thus impossible to infer anything about energy from the Poynting theorem.

In relativistic electrodynamics, the electric field has energy density equal to $\frac{\epsilon_0}{2}|E|^2$.

That is true, but only for regular source distributions, i.e. in macroscopic theory. For point particles, there is no reason to think EM energy is given by this formula. If we use it anyway, we can get anything. The examples you gave and other well-known problems are a result of ignoring this basic problem.

Theory of point particles is not necessarily limit of theory of regular continuous charge distributions (ironically the latter is even more difficult than the former). Point particles have consistent relativistic theory free of the above-mentioned 'self-force' and the original Poynting theorem does not play any role, as it is not valid where the particles are. There is a different theorem of similar significance, which however expresses energy as multi-linear function of individual fields of particles, instead of quadratic function of one field.

Cf.

J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534. (in German, the first paper I know that gives consistent theory of point particles)

http://dx.doi.org/10.1007/BF01331692R. C. Stabler, A Possible Modification of Classical Electrodynamics, Physics Letters, 8, 3, (1964), p. 185-187. (in English, shorter but presents the same core idea)

http://dx.doi.org/10.1016/S0031-9163(64)91989-4

 

[vanhees71]

I still don't understand these attempts to modify classical electrodynamics, because despite the words at the end of the paper, it is clearly not true that, e.g., single electrons don't radiate when accelerated by electromagnetic fields. The accelerator physicists would be very glad, if this was true since then you'd not have a problem with synchrotron radiation. I cannot follow the final conclusion of the paper. How can a theory, which contradicts the Klein-Nishina formula of the Compton effect be right, which is a non-radiative process, be considered right? Usual classical electrodynamics is not in contradiction with this formula. At least it is compatible with the Thompson cross section in the low-energy limit, i.e., for $E_{\gamma} \ll m_{\text{electron}} c^2$.

I still think the most convincing conclusion of the failure of a completely self-consistent classical theory of electromagnetically interacting charged point particles is that the very notion of a point particle in classical electrodynamics is flawed.

Even in QED the problem is not really solved, because it's not known, whether QED does really exist as a non-perturbative QFT. Of course in the sense of perturbation theory the radiation-reaction problem is solved by the usual renormalization procedure to get rid of the UV divergences and soft-photon resummations (or equivalently the use of the correct asymptotic states) to get rid of the infrared and collinear divergences due to the masslessness of the photon.

 

[Jano L.]

I still don't understand these attempts to modify classical electrodynamics, because despite the words at the end of the paper, it is clearly not true that, e.g., single electrons don't radiate when accelerated by electromagnetic fields. The accelerator physicists would be very glad, if this was true since then you'd not have a problem with synchrotron radiation.

Stabler explains this himself in his paper:The predictions (i) - (iii) above are not as dire as they appear at first glance because radiation
observed in classical electromagnetic phenomena (ƛ $\gg \hbar/mc$) usually arises coherently from a large
number of equivalent radiators
...
Since few measurements of classical electromagnetic phenomena involve very small numbers of coherently radiating particles, it is possible that the above theory is not contradicted by experimental data. Most important it may not yet have been demonstrated that there is any radiated energy for the case N=1.Back to your example, in synchrotron charged particles move in so-called bunches, one of which contains billions of charged particles. Many charged particles are thus moving and accelerating similarly - in a highly coherent/correlated way. In such a situation, the energy radiated is practically the same in both theories.

Also as far as I know, Stabler's doubt in the last sentence above is still valid; there is currently no experimental proof that single isolated electron loses energy when accelerating in external field.

I cannot follow the final conclusion of the paper. How can a theory, which contradicts the Klein-Nishina formula of the Compton effect be right, which is a non-radiative process, be considered right? Usual classical electrodynamics is not in contradiction with this formula. At least it is compatible with the Thompson cross section in the low-energy limit, i.e., for $E_{\gamma} \ll m_{\text{electron}} c^2$.

That is a good remark. The goal for the new theory is not to reproduce the ideas of the older theory, but to be consistent with experiments and to provide new views on them. The standard calculation of scattering cross-section in EM theory considers one particle in external field, which is not at all how the experiment is done. On the other hand, for many coherently moving particles (which seem to be present in the scattering experiments that measured the cross sections in the past), the Frenkel-Stabler kind of theory gives result close to Thomson's formula.

I still think the most convincing conclusion of the failure of a completely self-consistent classical theory of electromagnetically interacting charged point particles is that the very notion of a point particle in classical electrodynamics is flawed.

But the point of both papers and of mine here is precisely that the notion of point particle is not flawed at all, at least not from the point of view of internal consistency of the theory. Experimentally, there may be difficulties but that's to be expected from any model.

 

[PAllen]

I would add that perhaps the greatest champion of classical field theories (Einstein) would agree with the spirit of the papers Vanhees linked in #4 - that point particles are conceptually at odds with field theories; that ideally, a complete field theory models matter as a field concentration, thus inherently finite size with minimum size related to the nature of the field theory. I find the resolution to the self energy issues in these (linked in #4) papers ('particles' must be of finite minimum size, all derived with classical EM) far more satisfactory than adding a notion of point particles that needs additional rules.

Of course, this is aesthetic preference. Experiment is not likely to be relevant because neither classical point particles nor classical small charged bodies exist, and the regimes that might distinguish are going to dominated by QED differences from classical EM.

 

[haushofer]

Just want tot say that I'm really enjoying this series! Looking forward for more!

 

[vanhees71]

Ad #12: This is of course true. I don't think that one has been able to accelerate just one elementary particle and measure whether there is radiation emitted from it or not.

 

[Demystifier]

I believe the review paper by Eric Poisson is also relevant here:
http://lanl.arxiv.org/abs/gr-qc/0306052
See also the update
http://lanl.arxiv.org/abs/1102.0529

 

[vanhees71]

That's addressing the even more complicated question of motion in gravitational fields (or relativistically spoken in curved background spacetime) and the associated radiation. Anyway, also these authors come to the conclusion that a single accelerated charge radiates off radiation energy, i.e., that there is a radiation-reaction force on the particle ("self energy").

 

[marmoset]

I think this paper, 'A rigorous derivation of electromagnetic self force' by Gralla, Hart and Wald, is relevant here (cited 76 times).

http://lanl.arxiv.org/abs/0905.2391

 

[ndokos]

The link to the Feynman lectures is slightly broken: the backslash in it should not be there.

 

[Jim Stuttard]

I can't imagine more understandable explanations even if I don't understand a lot of them :)
There's a full version of the janssen paper at:

https://netfiles.umn.edu/users/janss011/home page/electron.pdf

HNY

 

[john baez]

The summary of radiation reaction and point charges is very helpful -- I hadn't heard of a lot of the more recent work.

Thanks! I learned a lot while writing this article; the new work by Kijowski is very nice, but it takes some work to understand it.

A question that's been bothering me for a long time is whether there is any sense in which one can prove that a system of charged particles can't be stable. This seems to be something that all physicists are convinced must be true, but that nobody knows how to prove, or even how to state rigorously.

I think it's possible now to state it rigorously, but nobody has proved it.

Unfortunately, it doesn't seem possible even to get started on such a proof, because there just isn't any satisfactory, canonical, self-consistent theory of interacting point charges.

You might not consider Kijowski's theory satisfactory - it has annoying features - but it's rigorously well-defined, it has a lot of good properties, and I'd say it's the best theory we have of interacting point charges, so it makes some sense to investigate how it behaves. It might even be the best possible theory of interacting point charges where the fields obey Maxwell equations, energy is locally conserved, and energy is given by a formula that physicists would consider reasonable for electrodynamics.