When do Maxwell's Equations break down?

[thegreenlaser]

I'm trying to get a sense of how widely applicable Maxwell's equations really are. I've read that electrodynamics becomes non-linear in the Schwinger limit where electric field strengths get high enough, but are there other situations where Maxwell's equations are insufficiently accurate? What about in the low electric field limit or the high frequency limit? What about something like nano-engineering where the length scales are small and quantum effects have to be taken into account? Would a nano-engineer ever have to use quantum electrodynamics, or would they normally just use some sort of mixture between Maxwell's equations and Schrödinger (e.g. use Maxwell to find the hamiltonian and then solve the Schrödinger equation)?

 

[jedishrfu]

I found this arxiv article on it:

http://arxiv.org/abs/1012.1068

 

[thegreenlaser]

I found this arxiv article on it:

http://arxiv.org/abs/1012.1068

Correct me if I'm wrong, but isn't that article pretty much just talking about low-frequency approximations to Maxwell's equations, rather than cases where Maxwell's equations don't apply?

 

[jedishrfu]

Correct me if I'm wrong, but isn't that article pretty much just talking about low-frequency approximations to Maxwell's equations, rather than cases where Maxwell's equations don't apply?

You may be right as I read it, I saw they were looking at relativistic limits that they found. I think EM is limited on one side by Relativity and on the other by QM. But I don't know enough to say but I did find a historical reference that may work:

http://www.jstor.org/stable/229644?seq=2

and Wikipedia talks about other limits (quoted below):

http://en.wikipedia.org/wiki/Maxwell's_equations#Limitations_for_a_theory_of_electromagnetism

Limitations for a theory of electromagnetism

While Maxwell's equations (along with the rest of classical electromagnetism) are extraordinarily successful at explaining and predicting a variety of phenomena, they are not exact laws of the universe, but merely approximations. In some special situations, they can be noticeably inaccurate. Examples include extremely strong fields (see Euler–Heisenberg Lagrangian) and extremely short distances (see vacuum polarization). Moreover, various phenomena occur in the world even though Maxwell's equations predicts them to be impossible, such as "nonclassical light" and quantum entanglement of electromagnetic fields (see quantum optics). Finally, any phenomenon involving individual photons, such as the photoelectric effect, Planck's law, the Duane–Hunt law, single-photon light detectors, etc., would be difficult or impossible to explain if Maxwell's equations were exactly true, as Maxwell's equations do not involve photons. For the most accurate predictions in all situations, Maxwell's equations have been superseded by quantum electrodynamics.

Variations

Popular variations on the Maxwell equations as a classical theory of electromagnetic fields are relatively scarce because the standard equations have stood the test of time remarkably well.

Magnetic monopoles

Maxwell's equations posit that there is electric charge, but no magnetic charge (also called magnetic monopoles), in the universe. Indeed, magnetic charge has never been observed (despite extensive searches)[note 4] and may not exist. If they did exist, both Gauss's law for magnetism and Faraday's law would need to be modified, and the resulting four equations would be fully symmetric under the interchange of electric and magnetic fields.[24][25]

 

[WannabeNewton]

I think EM is limited on one side by Relativity and on the other by QM.

In what way(s) would EM be limited by relativity? It is already a fully relativistic theory.

 

[Vanadium 50]

The equations in materials assume that the media are continuous - i.e. not composed of discrete charges. That approximation works well for long times and large systems.

 

[Andy Resnick]

I'm trying to get a sense of how widely applicable Maxwell's equations really are. <snip>

Maxwell's equations relating the four fields E, D, B, and H; and charge Q and current I can be quantized but are not Lorentz invariant. By using vector and scalar potentials instead of the fields, the equations become Lorentz invariant and can still be quantized.

Constitutive relations relating E and D, B and H can be done in a fully Lorentz-invariant manner, and there are also several compatible transformation laws for the fields E, D, B, and H: Minkowski, Chu, Boffi, and Amperian.

My favorite reference for the electrodynamics of moving media is Penfield and Haus, "Electrodynamics of Moving Media". The bottom line is that Maxwell's equations are a fundamentally correct microscopic description of electrodynamics.

 

[thegreenlaser]

Maxwell's equations relating the four fields E, D, B, and H; and charge Q and current I can be quantized but are not Lorentz invariant. By using vector and scalar potentials instead of the fields, the equations become Lorentz invariant and can still be quantized.

Constitutive relations relating E and D, B and H can be done in a fully Lorentz-invariant manner, and there are also several compatible transformation laws for the fields E, D, B, and H: Minkowski, Chu, Boffi, and Amperian.

My favorite reference for the electrodynamics of moving media is Penfield and Haus, "Electrodynamics of Moving Media". The bottom line is that Maxwell's equations are a fundamentally correct microscopic description of electrodynamics.

But aren't there still situations where Maxwell's equations are insufficient, and quantum electrodynamics must be used? What I'm really trying to understand is when exactly those situations arise. Is it just whenever quantum mechanics comes into play, or are there still a lot of quantum mechanical situations where Maxwell's equations provide a sufficiently accurate answer?

 

[BruceW]

stuff like spontaneous emission of radiation is not explained very well by classical (i.e. non-quantum) electrodynamics. Once you introduce quantum fields, then spontaneous emission is naturally explained by the theory. There are probably a lot of other examples, but this is the simplest one I can think of right now.

 

[WannabeNewton]

The canonical Maxwell equations have to be modified in the presence of gravitational fields but I would definitely not call this a "breakdown" of the equations but rather a necessary modification imposed upon by general covariance.

 

[Andy Resnick]

But aren't there still situations where Maxwell's equations are insufficient, and quantum electrodynamics must be used? What I'm really trying to understand is when exactly those situations arise. Is it just whenever quantum mechanics comes into play, or are there still a lot of quantum mechanical situations where Maxwell's equations provide a sufficiently accurate answer?

QED relates to the interaction of matter and the electromagnetic field; Maxwell's equations only refer to the electromagnetic field. To be sure, constitutive equations are used along with Maxwell's equations in the presence of ponderable matter.

Are you really asking about constitutive relations?

 

[Andy Resnick]

The canonical Maxwell equations have to be modified in the presence of gravitational fields but I would definitely not call this a "breakdown" of the equations but rather a necessary modification imposed upon by general covariance.

Not really; once the field equations are written in terms of an electromagnetic field tensor, the only change is "commas to semicolons", to use a common convention for derivatives. See, for example, MTW pg. 568.

 

[Jano L.]

But aren't there still situations where Maxwell's equations are insufficient, and quantum electrodynamics must be used?

Yes, Maxwell's equations by themselves are often insufficient to describe and explain what is going on. As Andy Resnick pointed out, important part of the physical description is some model for the matter, which is much more difficult in general and thus people tend to use non-general simplified models, different for different situation. In some cases, consideration of such model can be reduced into adoption of so-called "constitutive relations", or better said, functions describing behaviour of material in the presence of the EM field, like permittivity $\epsilon(\omega)$ for dielectrics and conductivity $\sigma(\omega)$ for metals.

So Maxwell's equations are not $sufficient$ even in the classical theory. They need to be amended by additional assumptions - model of matter, and also some boundary conditions to provide satisfactory description of what is going on. That could be answer to your question, but since you mentioned quantum electrodynamics, let's go further and rephrase the question into

what are the situations where the Maxwell equations for classical fields $\mathbf E,\mathbf B$ are not applicable ?

I suspect that this is the actual question most participants are trying to answer. I do not know well of any situation which would require us to drop Maxwell equations. The fact is that they have been very robust and are present even in quantum electrodynamics, although with somewhat different meaning and mathematical properties - they are "quantized".

There are optics experiments which some physicists regard as proof of necessity of quantization, but the borderline has been shifting many decades to more and more obscure phenomena accompanied by constant increase of the amount of phenomena that can actually be traced back to behaviour of matter instead of quantization of the EM field. Planck's spectrum, spontaneous emission, Lamb shift are all phenomena that were first explained on the basis of energy quantization, but later were shown to be present even if Maxwell's equations are not quantized. Check neo-classical theory by E.T Jaynes and coworkers, and stochastic electrodynamics by Marshall, Boyer and others.

...are there still a lot of quantum mechanical situations where Maxwell's equations provide a sufficiently accurate answer?

Yes, the phenomena like absorption and dispersion of light are usually being described by the Schroedinger equation + classical Maxwell equations quite well. The spontaneous emission mentioned above is not a problem of the Maxwell equations and has nothing to do with quantization - there is spontaneous emission in the classical theory as well, only the model of the matter has to allow for it - either allow for the presence of bath or radiation reaction forces - Jaynes explained this in his papers

http://bayes.wustl.edu/etj/articles/is.qed.necessary.pdf
http://bayes.wustl.edu/etj/articles/radiative.effects.pdf
http://bayes.wustl.edu/etj/articles/electrodynamics.today.pdf

 

[Rena Cray]

The canonical Maxwell equations have to be modified in the presence of gravitational fields[...]

How is that? Maxwell's equations can be put in covariant form. However, I'm not sure if there is a difference between the original and the covariant form.

 

[Rena Cray]

stuff like spontaneous emission of radiation is not explained very well by classical (i.e. non-quantum) electrodynamics. Once you introduce quantum fields, then spontaneous emission is naturally explained by the theory. There are probably a lot of other examples, but this is the simplest one I can think of right now.

Sorry, I don't buy this. How do we define the randomness of spontaneous emission? In response we mumble something about monte carlo casinos and roulette wheels. How do we define this classical randomness? The only way I can think of that isn't pretentious is to refer to the measurement of quantum states.

This too is circular reasoning. If you thing there is a way out of this circuitous stuff, I'm all ears.

 

[BruceW]

I totally agree. The semi-classical explanation of spontaneous emission is no good. We have to go to full quantum field theory to get a proper description.

 

[vanhees71]

Indeed, it is not so simple to find examples where the classical picture of the electromagnetic field breaks down. Even in the quantumtheoretical realm a lot of phenomena can be explained by a semiclassical description, where the electromagnetic field is treated still as a classical field (including the photoelectric effect!).

The places, where the full quantum theory of the radiation field is needed are whenever it comes to genuine quantum-field theoretical issues like vacuum polarization and other radiative corrections (i.e., in Feynman-diagram language of QED perturbation theory graphs with loops), of which the historically important pillars are the anomalous magnetic moment of the electron (deviation of the gyrofactor from the semiclassical Dirac-equation value of 2) and the Lambshift of the hydrogen-atomic spectral lines.

More modern and somewhat simpler to understand phenomena are typical quantum optical phenomena like quantum beats, for which already the Wikipedia article

http://en.wikipedia.org/wiki/Quantum_beats

or the famous experiments with entangled photon pairs a la Aspect, Zeilinger, et al.

 

[WannabeNewton]

How is that? Maxwell's equations can be put in covariant form. However, I'm not sure if there is a difference between the original and the covariant form.

Well there clearly is a modification because the original form assumes a flat derivative operator and ignores the fact that derivative operators couple to the gravitational field; replacing the flat derivative operator uniquely associated with the Minkowski metric with a general curved derivative operator uniquely associated with an arbitrary metric is a difference.

 

[thegreenlaser]

So Maxwell's equations are not $sufficient$ even in the classical theory. They need to be amended by additional assumptions - model of matter, and also some boundary conditions to provide satisfactory description of what is going on. That could be answer to your question, but since you mentioned quantum electrodynamics, let's go further and rephrase the question into

Just to clarify what I'm asking, this kind of answer is something I'm not really looking for (the rest of your post was quite helpful though).

In my view, once you introduce material models of permittivity, conductivity and permeability (constitutive relations), you're really working with an approximation of Maxwell's equations. In principle, Maxwell's equations should give a perfect description of the electromagnetic behaviour of materials if you could somehow account for each individual electron and proton in the material. We just use the continuum approximation and generate material models because it's impossible to keep track of that many charges at once. If it's possible to eliminate all error by considering every single electron and proton individually rather than using permittivity/conductivity/permeability models, then I would say the error lies with the material approximation rather than with Maxwell's equations themselves. That error is still important, but it's not really what I'm curious about. I'm asking specifically about the breakdown of the "microscopic" Maxwell equations, where every charge/current is considered explicitly.

I guess it's kind of just a matter of semantics, but hopefully that clarifies what I'm asking.

 

[Jano L.]

In my view, once you introduce material models of permittivity, conductivity and permeability (constitutive relations), you're really working with an approximation of Maxwell's equations.

In a sense yes, if by "Maxwell's equation" here you mean microscopic Maxwell-Lorentz equations, since the macroscopic Maxwell equations can be viewed as averaged microscopic Maxwell-Lorentz equations.

In principle, Maxwell's equations should give a perfect description of the electromagnetic behaviour of materials if you could somehow account for each individual electron and proton in the material.

(bold mine)

Maxwell's equations by themselves cannot give full description; the bold part means we need to introduce additional microscopic model of matter, in addition to the microscopic Maxwell equations. That is what the Newton equations, the Schroedinger or the Dirac equation are for.

It is hard to prove that Maxwell's equations are wrong, when they are just one grain in the calculational scheme and most of the experiments involve interacting matter described by a much more complicated mathematical model.

 

[BruceW]

I'm asking specifically about the breakdown of the "microscopic" Maxwell equations, where every charge/current is considered explicitly.

I'm pretty sure that the microscopic Maxwell's equations are 'correct' except for when we get to the quantum regime. i.e. there are some situations where the microscopic Maxwell equations give incorrect predictions, but quantum electrodynamics gives correct predictions. But then, it's maybe not so clear-cut to say exactly when we need to use quantum, and when we can use classical to get an approximately correct answer. In fact, in essence, this is the problem of when can we say that a quantum field acts semi-classically. And as far as I know, there is no general way of knowing this for a general experiment.

 

[voko]

If one works with electrons and protons directly, there are still things like vacuum polarization and pair production, which effectively give vacuum material properties. That could be regarded as a break-up of classical equations.