Maxwell's Equations and Quantum Electrodynamics: A Comprehensive Overview

[idea2000]

Hi,

I'm just someone trying to learn a bit more about quantum. I was wondering
what the relationship between maxwell's equations and quantum electrodynamics is? Are they compatible? Are maxwell's equations for solving macroscopic problems only? How
do you determine the E field for an electron when you don't know where it is? Any help would be greatly appreciated. Thanks!

 

[meopemuk]

Hi,

I'm just someone trying to learn a bit more about quantum. I was wondering
what the relationship between maxwell's equations and quantum electrodynamics is? Are they compatible? Are maxwell's equations for solving macroscopic problems only? How
do you determine the E field for an electron when you don't know where it is? Any help would be greatly appreciated. Thanks!

Photon quantum fields satisfy equations that formally resemble Maxwell's equations. However, I haven't seen a rigorous approach in which a classical limit of quantum electrodynamics would be taken so that trajectories of interacting charged particles could be calculated. I suspect that such an approach is not possible for two (related) reasons. First, the QED Hamiltonian (or Lagrangian) is formulated in terms of "bare" particles rather than "physical" particles observed in experiments. Second, this Hamiltonian contains infinite counterterms, which make it useless for any time evolution calculations.

The charge and mass renormalization counterterms in QED are designed to provide finite and accurate results for the S-matrix. So, within QED one can calculate such things as scattering cross-sections or energies of bound states. However, I haven't seen applications of QED to interacting time-dependent processes. I would be interested to learn if such applications exist and if they are comparable with experiment.

Eugene.

 

[Haelfix]

"However, I haven't seen applications of QED to interacting time-dependent processes."

They exist and have been studied, as a simple google search shows. Of the top of my head, QED with finite temperature time dependence might be a good place to start, I also seem to get a lot of hits for sonoluminescence.
Obviously some of the approximations that are used are phenomenological in nature and in general i'd probably first tackle the problem with Dirac theory.

I am not aware of the specifics, I know more about time dependant QCD (a much harder problem).

 

[meopemuk]

"However, I haven't seen applications of QED to interacting time-dependent processes."

They exist and have been studied, as a simple google search shows. Of the top of my head, QED with finite temperature time dependence might be a good place to start, I also seem to get a lot of hits for sonoluminescence.
Obviously some of the approximations that are used are phenomenological in nature and in general i'd probably first tackle the problem with Dirac theory.

I am not aware of the specifics, I know more about time dependant QCD (a much harder problem).

I am interested in simple time-dependent problems (such as interacting trajectories of two charged particles) solved in renormalized QFT (QED or QCD) without "phenomenological approximations". It seems that "finite temperature" and sonoluminescence are complicated many-body phenomena for which direct application of the QFT Hamiltonian is not possible. Are there any time-dependent problems that can be solved within QCD and compared with nuclear physics experiments? I would appreciate references.

Eugene.

 

[idea2000]

Hi, I have to admit that I'm not very familiar with some of the topics that have been mentioned.

Perhaps I can understand a little more by first asking a simpler question:

Does the E field of a single electron (whose postion and momentum are unknown) satisfy Gauss' Law if the surface is drawn large enough to sufficiently cover the electron cloud?

 

[meopemuk]

Does the E field of a single electron (whose postion and momentum are unknown) satisfy Gauss' Law if the surface is drawn large enough to sufficiently cover the electron cloud?

Quantum electrodynamics doesn't bother about such things as wavefunctions of particles and electric fields produced by them.

The only physical property that can be reliably calculated within QED is the so-called S-matrix. The S-matrix is a simplified description of scattering processes (i.e., special kinds of interacting time evolutions in which particles are separated and non-interacting in the distant past and in the distant future, and are interacting only during short time interval of collision). For such processes, the S-matrix allows one to find the final state (in the distant future) from the initial state (in the distant past). Moreover, the S-matrix contains information about energies of bound states of particles. These kinds of properties (scattering cross-sections and bound state energies) are measured in high-energy physics experiments, and predictions of QED agree with data very well.

The thing you are talking about - the electric field of a single electron at rest - doesn't have a proper definition in the traditional QED.

Eugene.

 

[reilly]

I suggest you look at Mandel and Wolf's book on quantum optics -- 700 or 800 pages of non-relativistic QED -- plenty of wave functions, and few S-matrices. Further, the electric field for a single particle at rest in QED is quite well defined with appropriate use of wave packets. See, for example, Zee's book on field theory. I'm sorry to say that the main points in your post are incorrect.
Regards
Reilly Atkinson

Quantum electrodynamics doesn't bother about such things as wavefunctions of particles and electric fields produced by them.

The only physical property that can be reliably calculated within QED is the so-called S-matrix. The S-matrix is a simplified description of scattering processes (i.e., special kinds of interacting time evolutions in which particles are separated and non-interacting in the distant past and in the distant future, and are interacting only during short time interval of collision). For such processes, the S-matrix allows one to find the final state (in the distant future) from the initial state (in the distant past). Moreover, the S-matrix contains information about energies of bound states of particles. These kinds of properties (scattering cross-sections and bound state energies) are measured in high-energy physics experiments, and predictions of QED agree with data very well.

The thing you are talking about - the electric field of a single electron at rest - doesn't have a proper definition in the traditional QED.

Eugene.

 

[meopemuk]

Further, the electric field for a single particle at rest in QED is quite well defined with appropriate use of wave packets. See, for example, Zee's book on field theory. I'm sorry to say that the main points in your post are incorrect.
Regards
Reilly Atkinson

Unfortunately, I don't have Zee's book. Could you please help me to understand how he defines one-electron states? Apparently, the straightforward definition (where $a^{\dag}_p$ is the electron creation operator)

$$| \mathbf{p} \rangle = a^{\dag}_p |0 \rangle$$

doesn't work, because this state is not an eigenstate of the interacting Hamiltonian H of QED, and in the course of time evolution

$$| \mathbf{p}(t) \rangle = \exp(\frac{i}{\hbar}Ht ) | \mathbf{p} \rangle$$

this state acquires components in many-particle sectors of the Fock space. For example, trilinear terms of the type $a^{\dag}c^{\dag}a$ (where $c^{\dag}$ is the photon creation operator) in H lead to the emission of photons from the free electron. This seems rather unphysical to me.

I am not even talking about the fact that the interacting Hamiltonian H contains infinite counterterms, so the product $H | \mathbf{p} \rangle$ is ill-defined.

Eugene.

 

[Anonym]

I suggest you look at Mandel and Wolf's book on quantum optics -- 700 or 800 pages of non-relativistic QED -- plenty of wave functions, and few S-matrices. Further, the electric field for a single particle at rest in QED is quite well defined with appropriate use of wave packets. See, for example, Zee's book on field theory. I'm sorry to say that the main points in your post are incorrect.

Indeed I agree with you completely, but in what sense you use words non-relativistic QED?

Regards, Dany.

 

[idea2000]

Hi,

Thanks for all the replies. Is it safe to say that there haven't been many attempts to relate QED to Maxwell's?

 

[f95toli]

Hi,

Thanks for all the replies. Is it safe to say that there haven't been many attempts to relate QED to Maxwell's?

It depends on what you mean by QED. If by QED you simply mean models using equations where the EM field is quantized (as in all quantum optics) I would say you can quite easily relate the two. There is of course never an exact equivalence since Maxwells equations are classical. However, you can "derive" second quantization of the EM field by starting from Maxwell's equations and then (non-rigourosly) introduce quantization by writing the field in terms of harmonic oscillators.
Also, all QED equations must of course agree with Maxwell's equation in the "classical" regime (i.e. high temperatures or large drive powers), simply because we know that Maxwell's equation are correct in this limit.


If you want to read more I suggest you look in a good book on quantum optics. e.g. Gerry&Knights "Introduction to Quantum Optics".

 

[quetzalcoatl9]

However, you can "derive" second quantization of the EM field by starting from Maxwell's equations and then (non-rigourosly) introduce quantization by writing the field in terms of harmonic oscillators.

i don't know much about QED, so please excuse my ignorance..but why is this non-rigorous? the connection between a classical field and a collection of harmonic oscillators can always rigorously be made.

do rather mean how we postulate the functional form of the lagrangian density? i would agree that this part is non-rigorous

Q

 

[f95toli]

i don't know much about QED, so please excuse my ignorance..but why is this non-rigorous? the connection between a classical field and a collection of harmonic oscillators can always rigorously be made.

do rather mean how we postulate the functional form of the lagrangian density? i would agree that this part is non-rigorous

Q

Well, the problem is that at some point you have to go from a classical field with continuous variables to a quantum description with discrete excitations of energy $$\hbar\omega$$ so it is hardly surprising that you can't derive quantized equations (i.e. equations which include annhilation and creation operators) of the field from Maxwell's equation.

There is always at least one "leap of faith" involved, where it occurs depends on the derivation. One example: one usually starts by assuming that the field is confined to a volume L^3 with periodic boundary conditions, in this case the field is of course quantized even in classical physics (just plain old standing waves) and can therefore be described in terms of oscillators. The problem is that at the end of the derivation one then let's L to infinity, but one nevertheless assumes that the field is STILL quantized in quantas of $$\hbar\omega$$. Moreover, the excitation (photon) is also assumed to be spatially localized (which of course isn't the case for a classical field in an infinite volume).
It is a bit like trying to derive Planck's formula for blackbody radiation using classical physics, at some point you have to ASSUME that the field is quantized.

There is a rather long discussion about this is Gardiners's "Quantum Noise", but derivations of the kind I described above can be found in any book on quantum optics.

 

[quetzalcoatl9]

The problem is that at the end of the derivation one then let's L to infinity, but one nevertheless assumes that the field is STILL quantized in quantas of $$\hbar\omega$$. Moreover, the excitation (photon) is also assumed to be spatially localized (which of course isn't the case for a classical field in an infinite volume).
It is a bit like trying to derive Planck's formula for blackbody radiation using classical physics, at some point you have to ASSUME that the field is quantized.

yes, i am familiar with the derivation of Rayleigh-Jeans/Planck distributions, but is quantization of the field really that hard to swallow? I thought maybe you were balking at having to introduce the gauge potential or something.

 

[f95toli]

yes, i am familiar with the derivation of Rayleigh-Jeans/Planck distributions, but is quantization of the field really that hard to swallow? I thought maybe you were balking at having to introduce the gauge potential or something.

Not at all, I was merely pointing out that it is not possible to go from a classical description (using Maxwell's equations) to a quantized form without making some assumptions. Hence, there is no rigorous way to derive quantization of the field which I guess was the original question. However, there is still a definite connection between the classical- and quantum picture.
Note, however, that the whole concept of a photon is very hard to understand in the framwork of classical electrodynamics EVEN if you accept that the energy is quantized; you still have to explain how it can have an exact energy (i.e. frequency); classicaly this can only happen for a wave that exists for an in infinite time; which is of course is not the case since we know that photons are quite well localized in space-time.