Does classical vacuum permittivity have a QM analog?

[LarryS]

According to Maxwell's Equations, the classical vacuum permittivity and permeability have a very important role: They determine the speed of light. But it seems like these two important concepts are not as precisely defined in the quantum world. Are there rigorous analogs of these two constants in QM and do those quantum analogs also determine the speed of light?

 

[Avodyne]

Yes, they are defined in the same way. You get quantum electrodynamics from canonical quantization of classical electrodynamics.

 

[LarryS]

What if the medium is not a vacuum? Are they defined like other QM observables, as linear operators on a Hilbert Space?

 

[Avodyne]

At the simplest level of description, the permittivity and permeability would be parameters in the hamiltonian (just like, in the hamiltonian for the harmonic oscillator, the particle mass and oscillation frequency are parameters). If you want a more refined level of description of the medium, where you make it out of dynamical atoms, then the permittivity and permeability would correspond to some complicated linear operators.

 

[DrDu]

The permittivity of the vacuum is nothing else than $\epsilon=1/c_0^2$, i.e. it can be expressed in terms of the speed of light in vacuum. The speed of light, besides being the speed of light, is simply the conversion factor to convert units of time into units of length. We know from special relativity that space and time coordinates can be combined into a 4-vector and can be interchanged to some extent when going to another inertial system.
So $c_0$ is basically fixed by the symmetry of spacetime and thus also permittivity.
The second part of the question is why light moves at the speed of light.
Well, light is made up of massless photons, which means that their energy is purely kinetic as mass is the energy in the rest frame. Would it be possible to switch to an inertial frame where the photon were at rest, then the photon would not exist at all, as there would be no photon in this rest frame as it's energy vanishes there. Now from special relativity we know that only the speed of light is the same in all inertial frames. Particularly, we can't stop a particle moving at the speed of light by switching to another reference frame.
So you see that the speed of light and thus also epsilon is rather a geometric property of space-time than a quantum mechanical observable.

 

[vanhees71]

The permittivity of the vacuum is just to accommodate a pretty unnatural choice of units for electromagnetic phenomena, the SI units, to a natural choice of units like the Heaviside-Lorentz units. The only universal physical constant entering electromagnetism as compared to Newtonian mechanics is the socalled "speed of light", which should rather be called the "universal limiting speed" of relativistic spacetime.

https://en.wikipedia.org/wiki/International_System_of_Units

The most natural choice of units in relativity is, of course, to measure time intervals in the same units as space intervals, using the universal limiting speed. Indeed that's realized in the SI since 1983, because in the SI one defines the unit of time by a certain hyperfinestructure transition in Cesium (since 1967):

The duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

Then the metre as the basic unit for length is defined by fixing the value of the limiting speed:

The distance traveled by light in vacuum in 1/299792458 second.

This strange number, of course, only comes about to keep the metre at the same length as its original definition as $10^{-7}$ of the length of the meridian going through Paris. Nature doesn't care much about this distance, and thus it's not a natural unit and not reproducible easily (the story behind defining the metre by measuring the length of the meridian is however a very exciting adventure story, quite dangerous for the brave scientists doing right this).

The permittivity of the vacuum, $\epsilon_0$, in the SI units for electromagnetism is totally artificial. While to keep the speed of light $c$ in usual units like m/s makes some sense, because we are used to measure times in seconds and distances in metres, the introduction of a fourth unit in electromagnetism (in the SI the Ampere for electric currents) is totally artificial, because it makes us measure the electric components of the electromagnetic field in a different unit than the magnetic components, but relativity tells us that there is only one electromagnetic field, described by an antisymmetric 2nd-rank Minkowski tensor, the Faraday tensor $F_{\mu \nu}$. This is analogous to the usual system of units used in the US, giving distances in miles and height's in feet and inches. Of course the SI units are just chosen to have convenient numbers when dealing with everyday applications in electrical engineering and not for fundamental theory of electromagnetism. In theory thus the Gaussian system of units or sometimes its rationalized version, the Heaviside-Lorentz units (the latter used particularly in high-energy particle theory since in these units QED uses the same conventions as the more modern siblings QCD and QFD, making together the Standard Model of elementary particles).

The same concerning the units is true for quantum theory. Then another fundamental constant enters the game compared to classical physics (classical usually are called all physical theories not taking quantum theory into account, no matter whether you use a Galilei-Newton or a relativistic space-time model) is Planck's unit of action $h$ or more modern the modified Planck constant $\hbar=h/(2 \pi)$. In a natural system of units you set $\hbar=1$. Together with $c=1$ you've only one unit left, for which in high-energy particle theory one usually chooses GeV (giga electron volts) for masses, energies and momenta. In this system length and time intervals are measured in 1/GeV. You can as well measure lengths and times in some unit (usually fm, i.e., $10^{-15} \;\mathrm{m}$). To switch back and forth between these choices, you only have to remember that in usual units $\hbar c \simeq 0.197 \; \text{MeV} \, \text{fm}$.

To get finally rid of this remaining arbitrariness in the choice of units, you can take also general relativity and gravitation into account. There the fundamental constant is Newton's Gravitational constant $G$, which you may choose to set to 1. Then all quantities are given as pure numbers. However, this (modified) Planck system of units is not in common use in physics (except sometimes in the General Relativity community), because the numbers become pretty inconvenient since the Planck length is so tiny and the Planck energy scale so huge:

https://en.wikipedia.org/wiki/Planck_units

 

[bhobba]

The permittivity of the vacuum is nothing else than $\epsilon=1/c_0^2$, i.e. it can be expressed in terms of the speed of light in vacuum.

Exactly.

Its physical meaning is rather trivial in a QM sense - its a reflection of space-time geometry.

Thanks
Bill

 

[LarryS]

I understand the above arguments that the EM permittivity and permeability as well as the speed of light are just conversion parameters, especially if you START your argument assuming with SR as a GIVEN. Now I am not really questioning the validity of SR, but I think it is interesting to, at least hypothetically, to question the PHYSICAL origin of the two EM constants. For example, during the time of Faraday and Maxwell how was the value of the vacuum permittivity determined? Was it determined by EXTRAPOLATING from the permittivity values determined by measuring it from NON-vacuum mediums?

 

[DrDu]

I don't know whether this is useful. The Fields D and H are really mixtures of the electric and magnetic fields E and B with corresponding polarizations and magnetizations P and M, e.g. D=E+P. The polarisation and magnetization are in turn determined from the microscopic charge distributions. On the other hand this means that you can live without them and consider directly the microscopic Maxwell equations which contain only E and B and the microscopic charges and currents.
I think the real core of your question is how to appropriately chose units for the Maxwell equations.
SI is clearly a bad choice for theoreticians.

 

[vanhees71]

I understand the above arguments that the EM permittivity and permeability as well as the speed of light are just conversion parameters, especially if you START your argument assuming with SR as a GIVEN. Now I am not really questioning the validity of SR, but I think it is interesting to, at least hypothetically, to question the PHYSICAL origin of the two EM constants. For example, during the time of Faraday and Maxwell how was the value of the vacuum permittivity determined? Was it determined by EXTRAPOLATING from the permittivity values determined by measuring it from NON-vacuum mediums?

In Maxwell's and Faraday's time of course nobody had the idea to introduce artificial units as the SI. They used, however, also a very confusing system of different units. The natural one is due to Gauss, with just one fundamental constant, the "speed of light", which is, as we now know, rather to be identified with the general limiting speed of relativistic space-time (and as such also just a conversion factor adjusted to the used units for space and time, and this is also the way as it is defined in the SI since 1983). The most natural units are due to Heaviside and Lorentz, who rationalized the Gaussian units by getting rid of the nasty $4 \pi$ factors in Maxwell's equations. Then they appear, where they belong, namely in Coulomb's Law for the electric field of a static point charge (and the so derived Green's function of the vacuum Maxwell equations).

The SI introduces (artificially) another unit, i.e., switching from the MKS (metre, kilogram, second) (or originally CGS=centimeter, gram, second) of the Gaussian or Heaviside-Lorentz units to the MKSA system, introducing the Ampere as another base unit of the system of units, which goes back to Georgi. The advantage is that the units for electric charge and magnetization do not contain non-integer powers of other base units as in the CGS systems. Also the numbers for everyday applications in electrical engineering become more convenient.

To understand the beauty of electromagnetic theory, which is after all the paradigmatic example of a Poincare covariant local field theory with massless vector field, which thus necessarily must be formulated as a gauge theory, the SI units are a pest, and it's really sad that the theory-textbook literature seems to almost always switch to the SI. Even Jackson did this in the 3rd edition of his famous book, but then switching to Gaussian units at places where the SI is too ugly to write down, namely when it comes to the formulation of electromagnetism in a manifestly Poincare covariant way, which in fact is the way the subject should be taught in my opinion.

 

[PhilDSP]

The concepts of permittivity and permeability were quite essential in the classical field of Optics. The standard text on the subject is Born and Wolf's Principles of Optics. But even recently there has been interest in Quantum Optics in studying the potential equivalence of QFT models with Photon Wave Mechanics (as Smith and Raymer call it).

In this paper Dragoman formulates a preliminary development of single and multi-photon quantum wave functions and Hamiltonians which are solutions to the Maxwell Equations given certain stipulations:
http://arxiv.org/ftp/quant-ph/papers/0606/0606096.pdf

Smith and Raymer go further to show the equivalence or complement of similar single and multi-photon quantum wave functions (which incorporate the classical permittivity and permeability values) with QFT solutions:
http://arxiv.org/pdf/quant-ph/0605149.pdf

 

[Avodyne]

I understand the above arguments that the EM permittivity and permeability as well as the speed of light are just conversion parameters, especially if you START your argument assuming with SR as a GIVEN. Now I am not really questioning the validity of SR, but I think it is interesting to, at least hypothetically, to question the PHYSICAL origin of the two EM constants. For example, during the time of Faraday and Maxwell how was the value of the vacuum permittivity determined? Was it determined by EXTRAPOLATING from the permittivity values determined by measuring it from NON-vacuum mediums?

Coulomb measured it with a torsion balance. I'm pretty sure this was done in air, not in an (approximate) vacuum. But the permittivity of air is within 0.1% of the permittivity of vacuum. I don't know when measurements became accurate enough for that to matter. Presumably, at some point, the torsion balance was used in an (approximate) vacuum to get the vacuum permittivity.

 

[LarryS]

Thank you all!