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When Maxwell developed the electromagnetic wave equation, he took the wave velocity to be dictated by the permittivity and permeability of the aether. But the Michelson-Morley experiment demonstrated that there was no aether. This being the case, are the permittivity and permeability of free space (i.e. vacuum), which determine the speed of electromagnetic waves, dictated by the quantum mechanical zero point energy field?

John Petrovic
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No, the vacuum permittivity and permeability have nothing to do with quantum mechanics. These dimensionful parameters only exist because of historical reasons: we chose an arbitrary system of units, and the result is the presence of some redundant parameters in the equations.

A wiser choice of units is the Gaussian system, where neither $\epsilon_0$ nor $\mu_0$ appears in any formula: these constants are reabsorbed into the electric and magnetic fields. The mere fact that one can formulate a complete theory of electric and magnetic phenomena with no reference to $\epsilon_0$ and $\mu_0$ implies that the actual value these constants has no intrinsic meaning.

In gaussian units, Maxwell's equations read \begin{align} \nabla\cdot\boldsymbol E&=4\pi\rho\\ \nabla\cdot\boldsymbol B&=0\\ \nabla\times\boldsymbol E&=-\frac{1}{c}\frac{\partial\boldsymbol B}{\partial t}\\ \nabla\times\boldsymbol B&=\frac{4\pi}{c}\boldsymbol J+\frac{1}{c}\frac{\partial\boldsymbol E}{\partial t} \end{align} and the Lorentz force is $$ \boldsymbol F=q\left(\boldsymbol E+\frac{1}{c}\boldsymbol v\times \boldsymbol B\right) $$

As you can see, these equations encapsulate the whole theory of electrodynamics, and they make no reference to the vacuum permittivity and permeability of the vacuum. These parameters only have historical relevance, not physical.

AccidentalFourierTransform
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    This is true, but the product $\epsilon_0\mu_0$ shows up in your equation in the form of $c$. – Jahan Claes Jan 19 '17 at 18:47
  • Then explain to me how Maxwell calculated the velocity of electromagnetic waves, which was unknown to him at the time. – John Petrovic Jan 19 '17 at 18:47
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    @JahanClaes No, $c$ is much more fundamental than $\epsilon_0$ and $\mu_0$. You chose to read $c$ as $1/\epsilon_0\mu_0$. In gaussian units there is no fundamental need to identify $c$ as some function of $\epsilon_0,\mu_0$. – AccidentalFourierTransform Jan 19 '17 at 18:49
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    @JohnPetrovic he did use the SI value of $\epsilon_0$ and $\mu_0$. But, again, this only emphasises the historical relevance of $\epsilon_0,\mu_0$; physically, there is no need to ever introduce such constants. Instead of measuring $\epsilon_0,\mu_0$, one measures $c$ directly. – AccidentalFourierTransform Jan 19 '17 at 18:51
  • @AccidentalFourierTransform it's undeniable, whatever units you choose, that the electric field has a certain strength. One could ask to explain why the strength is what it is. That amounts to asking "why is $\epsilon_0$ what it is?". That doesn't go away by cleverly picking units. Then I'm just asking, "why did Nature pick those units?" – Jahan Claes Jan 19 '17 at 18:59
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    I am not sure I adhere to the thesis that vacuum permittivity and permeability are not valid physical quantities/concepts but only mere historical accidents. The only thing that gaussian units show is that their values is an historical accident not necessarily the corresponding concepts. For example, which charges do you put in your right hand side of the Maxwell-Gauss equation? Bare charges of the corresponding QFT or renormalised charges due to vacuum polarisation? – gatsu Jan 19 '17 at 19:06
  • @JahanClaes sorry but that makes little sense: to ask about the strength of the electric field, you must specify some (a priori arbitrary) units, and this reflects the fact that the actual value of $E$ is arbitrary: in some units you may have $E=1 \mathrm u_1$ and in other units you may have $E=10^{200}\ \mathrm u_2$. You can only ask about the magnitude of dimensionless parameters, such as the charge of the electron. But, in that case, I fear that physics has very little to say: as of today, there is no working theory that explains the value of fundamental parameters such as $m_e$ or $e$. – AccidentalFourierTransform Jan 19 '17 at 19:09
  • @gatsu 1) If you can take a model and recast it into a form that one less free parameter than before, then it means that such a parameter was irrelevant/redundant to begin with. 2) I dont want to sound condescending but your comment seems to reflect some misunderstanding of bare vs renormalised parameters. Moreover, if you are using QFT then you have to use photons, not Maxwell's equations. It is just meaningless to ask whether $\rho,J$ should be bare or renormalised. You're mixing the sheep and the goats. – AccidentalFourierTransform Jan 19 '17 at 19:16
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    @AccidentalFourierTransform: "1) If you can take a model and recast it into a form that one less free parameter than before, then it means that such a parameter was irrelevant/redundant to begin with". No, that's not what it means, it just means that you set one of the constants to 1. The same thing happens in QFT when working in "natural units". Now, you may want to prefer a unit system in which electric charge is not fundamental but that's not my case. Regarding point 2) yes you are condescending but you may be right about my lack of understanding. – gatsu Jan 19 '17 at 19:56
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    @gatsu 1) you cannot take an arbitrary parameter and set it to 1: you can only do that if the parameter is redundant to begin with, thus my previous comment. The electric charge is fundamental in any system of units. The units have nothing to do with a parameter being fundamental. 2) sorry, I tried not to sound condescending, I sincerely apologise if I offended you. – AccidentalFourierTransform Jan 19 '17 at 20:00
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    @AccidentalFourierTransform, I think that Maxwell would agree with me that the speed of electromagnetic waves depends on something that is physical in nature. This speed is not a magic number. – John Petrovic Jan 19 '17 at 20:16
  • @AccidentalFourierTransform: If you have a dimensional parameter in a system of equations, you can always decide to work in a system of units where this parameter has value 1. Moreover, who decides which "redundant" parameter should be gotten rid off ? One could keep the charge based on the Ampere and express all masses in these terms. My remark on the non-fundamental aspect of the charge in CGS was more on the fact that it becomes a derived quantity. Regarding point 2 I was referring to this. – gatsu Jan 19 '17 at 20:19
  • @JohnPetrovic - It is entirely possible to explain Maxwell's insight without referring to ε0 or μ0: When you charge and discharge a capacitor, you can do a measurement on the electric force (from the charge), and on the magnetic forces from the corresponding discharge current. If you combine these various measurements in the right way, you'll calculate a universal value with units of velocity. People did this in the 1850s and found a constant which was awfully similar to the speed of light (measured independently). Maxwell showed that it wasn't a coincidence ... if light was electromagnetic! – Steve Byrnes Jan 26 '17 at 13:56
  • But why it is nonzero? – WunderNatur Aug 07 '17 at 16:56
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Vacuum permittivity and permeability are just constants of proportionality to relate electromagnetic units (such as fields) to kinematic units (like forces). In SI, they arise from the definition of the Ampere which is defined such that currents (and voltages) have decent units (in SI) to work with in the day-to-day life. You can always change your unit system (like switching from SI to CGS), this will cause a change in the value of those constants ($\epsilon_0$ becomes $\frac{1}{4\pi}$ and $\mu_0$ becomes $\frac{4\pi}{c^2}$). It is even possible to chose a system of units such that one of them is not necessary (e.g. $\epsilon_0=1$).

Often in physics, we like to put fundamental quantities (like the speed of light) to 1. This simplifies equations at the cost of changing the units (if $c=1$ mass and energy have the same units).

As you can see, there is absolutely no links between the value of some constants and QFT. Those constants are defined within a units system and thus can change from one system to another. There is even units systems where a bunch of constants can be put to 1. Those systems are called "natural units" and a few examples are given here.

fgoudra
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  • Fgoudra, the speed of light is not a fundamental, it has its value for a physical reason. The essence of my question is to inquire into the nature of that reason. That is how physics progresses. It seems to me that the zero point energy field is the logical place to look. Is anyone looking? – John Petrovic Jan 19 '17 at 23:27
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    As long as I know, the values of fundamental constants (speed of light, elementary charge, Boltzmann Constant, electron mass, Planck constant, etc...) have no definite origin for now (they cannot be expressed as a combination of other fundamental constants or math constants). Of course it depends of the theory you are using but, if you take General Relativity which is one of the most fundamental theory of the universe as of now, the speed of light has no origins it just pops out in equations because of the basic postulates. – fgoudra Jan 20 '17 at 14:39
  • Ok I think I understand your question. In fact, what we call fundamental constants are values which appears in physic equations but have no other specific origin. For example, the planck constant comes from a guess (from Planck) that photons have energy proportional to the frequency. Some constants can, technically, be computed from QFT but they all happen to diverge to infinity and the theory must be renormalized to give those constant a finite value (it is chosen to give the accepted measured value). The point is that fundamental constants cannot be (for now) deducted from other things. – fgoudra Jan 20 '17 at 18:18
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I have seen Maxwell's Equations written in at least 4 different ways. I'm not sure that excluding the showing explicitly of vacuum permittivity and permeability proves anything about those items. You would then need to show that you don't need them to construct capacitors and inductors -- thinking of course in simple terms: C = epsilonA/d and L = mu(N^2*[4*pi*r^2])/l, N = turns, r = radius, and l = length of a coil.

More interestingly a German scientist named Heim used no more than Planck's Constant, the Gravitational Constant, and epsilon_0 and mu_0 above to derive the masses of the three neutrinos of current solar physics interest. I plugged them into a basic Cosmology Model (of my own construction). Result was the predicted value of the radius of a neutron = 1.27E-15 meters. Generally researchers [first] assume a radius for the neutron (commonly 1.25E-15 m) and find a Cosmology to derive from it. I went the other direction. But, it was Cosmology after all. His particle values were for (today) -- not 10 Billion years ago. I can scale all his values rather easily. But I dread the thought of what must happen to epsilon and mu, though still epsilon*mu = 1/c^2. Some poor scientist must reintroduce these into Schrodinger's Equations. Then see how they effect electron orbits and space gas dynamics. Type I Supernovas noticed "Dark Energy" which might merely come from changes in epsilon effecting changes in electron cloud sizes -- and nothing more.

Community
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William C. McKee
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To change your system of units so that some constants, like $\epsilon_{0}$ and $\mu_{0}$, disappear (are set to 1) changes the VALUE of the constants but not their RELEVANCE. These two constants determine the constants of proportionality of the electric field and the magnetic field and their relationship to the speed of light, in ANY system of units, will always exist — be physical.

Buzz
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LarryS
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