In this post, I look to share my thoughts on the theoretical possibility of the EM drive. If possible, it seems to me that the correct answer will only be described with Quantum Field Theory, and is inherently a non-classical process. Unfortunately, not many people know much about quantum field theory; even the people who do would not admit to understand the notion of “what is the quantum vacuum”. In this post, I will try to elucidate a bit of what I have learned about the quantum vacuum over the years, and describe how such an object could lead to such wildly crazy experimental results, such as the EM drive, which seems to break momentum conservation.

For those who don’t know, the EM drive is a RF resonant cavity thruster device proposed by Robert Shawyer. Essentially, it is a closed cone-shaped hunk of metal which has microwave radiation (light, or photons) bouncing around it. I like to picture it as a photon gas enclosed to the shape of a cone. The device is therefore powered by something similar to a microwave oven, and as a source of energy.

The source of energy is not the problem, but rather the source of momentum. Any theory of electromagnetism, whether it is classical or quantum mechanical, will have the notion of radiation pressure. The photons carry momentum, and therefore can cause a pressure, which is the average force per area. In the classical equation for radiation pressure, there is the speed of light, which is typically considered to be constant.

Perhaps you have read one of my posts on the Scharnhorst effect. It states that boundary conditions, such as the shape of a cone, act as dispersive medium. Essentially, if you are in empty space and are inside a cone, it is as if you are not in a vacuum. This is a very confusing point in quantum field theory, and I will attempt to describe it in layman’s terms.

Roughly speaking, in quantum electrodynamics (the QFT of electromagnetism), you state that classical mechanics is a good approximation, but not the exact solution. Whenever you calculate anything, you can find the “tree level amplitude”, which gives back the exact solution in classical electromagnetism. The power of QFT allows you to calculate “one loop amplitudes”, which are essentially quantum corrections. These quantum corrections represent scattering of the classical particles with the quantum vacuum.

To say the least, if you grew up your whole life learning classical mechanics, you will be very confused by these “virtual particles”, which represent interactions with the quantum vacuum and are only in loop amplitudes, not tree level. Basically, a photon can “briefly turn into” a virtual electron/positron pair and then turn back into a photon later. The one loop Feynman diagram for this process is shown below.

The wiggly line on the left represents a photon coming in, while the circle represents the electron/positron pair. We can see that the circle closes, and the photon is reemitted on the other side.

If we think of Feynman diagrams in position space, we instantly realize how a cone might be able to affect 1-loop calculations. The way I envision it is that on the larger side of the cone, there are more 1-loop diagrams contributing to the process, since you could fit bigger circles in that area without being cut off. However, if you are near the pinch of the cone, the vacuum diagram contributions become less and less.

This analogy is where many physicists may start to disagree with me (including the professor who taught me QFT at MIT), but this was my first analogy for why 1-loop diagrams affect the speed of light. Essentially, the photon always moves at some very fast speed, perhaps even at an infinite speed, or maybe larger that the speed of light that we classical view. It is the virtual state that slows down the particles, since they are now massive electron/positron pairs. This analogy has to be wrong in some sense, since the virtual particles are off-shell and not on-shell, but I don’t want to open this can of worms. In short, it seems conceivable that if I have more 1-loop diagrams, perhaps they slow the actual particles down.

I remember a time where I tried to explain this to my professor, and he utterly disagreed with me. I then told a friend, and he showed me the work of Scharnhorst, which has the mathematics to back up my ideas. Essentially, the conceptual picture is right. If you have two metal plates really close together, light can move slightly faster than c, since you cut out vacuum modes. I have gotten a lot of ridicule whenever I try to describe this to anyone, because people instantly say it “violates special relativity”, which is hilarious, since this result of from QFT, which is inherently a theory of special relativity. No causality is violated.

So now I have hopefully convinced you that the speed of light is varying as I go up and down the height of the cone. It is very slight, yet varying. If the speed of light varies, than the radiation pressure varies, and this could cause an imbalance in forces, causing propulsion! Not so fast… Scharnhorst himself said Casimir plates would have a change in the speed of light by of order 10^(-40), so how could this possibly result in something we could measure in a laboratory today? I still have this confusion, which will be elucidated by the calculation I did and will briefly describe next.

If anyone who is familiar with statistical mechanics is still confused about what the quantum vacuum is, think of it as a quantum mechanical action reservoir. There was a moment where I thought I was about to discover this on my own, but then I remembered that I had skimmed through this paper a few months earlier: http://arxiv.org/abs/physics/0605068

If anyone is still hung up on how a conservation law such as momentum conservation could be broken, you must realize that it comes from the ambiguity in your definition of momentum in the first place. The simplest resolution is to say that the quantum vacuum has momentum, and therefore we are pushing off of the quantum vacuum. This statement is oversimplistic, however, and very subtle. Let’s also make the point that it is not as if this is the first conservation law that has been broken by a QFT. Anomalous behavior is one of the most interesting topics in QFT. Also, scale invariance is broken in QFT through renormalization.

My first attempt to describe the phenomenon was to first assume that the virtual particles must go on-shell. This means that p^2 = m^2, which is a thing in special relativity. In QFT, the virtual particles can take on any crazy value, including ones where p^2 is negative, which seems to be like an imaginary mass particle. My adviser at UCLA has developed methods that calculate 1-loop amplitudes from purely on-shell kinematics, suggesting that you should be able to find the solution from the on-shell case. As a rough guess, it seemed reasonable to assume that the only way such a crazy EM drive result could occur is if the photon energy density is great enough to create an electron positron pair.

So I started my first calculation, can you get a high enough photon energy density to excite this on-shell electron/positron pair over the volume of the whole cone? I got into some deep discussions with experimentalists about this. Both of them concluded that we did not have enough power, but would in a couple decades. I actually disagreed with these professors on a technical detail, which gave me a lower value of the power. Okay, so I found a way to argue that maybe there is a high enough energy density to create an on-shell pair, but how much force would be generated?

The next thing I did was attempt to calculate the speed of light. Assuming microwave frequencies and reasonable laboratory sizes, I realized that you would need approximately 10^11 photons all interacting at once. This is quite a difficult calculation! I have calculated 2 photon scattering, and maybe I have written code to calculate 5 gluon scattering, but never in my life have I even attempted 10^11 particle scattering. I came up with a very very crude approximation for the speed of light. Roughly speaking, the photons move at c when they are in the “classical state”, or are not scattering with the quantum vacuum. The way I looked at this was as a two state system. There exists the classical state (speed c), and the 1-loop quantum state (speed 0) The quantum state had a speed of 0, since we will assume that the electron/positron pair will come to rest in the cavity for a brief moment of time. The final speed is therefore the speeds times the probabilities of each state occurring, so we could see that if the quantum state has a larger probability, then the photon’s effective speed of light will slow down.

Okay, so I have this crude approximation for the effective speed of the photon, but how do I calculate the probabilities? Well… I don’t… This is harder than anything I have ever imagined. I can say this, however. Since there are 10^11 photons interacting, there will be 10^11 vertices on the 1-loop diagram. With each vertex, you have to multiply by a factor of 1/137, which is known as the fine structure constant. So, I know that the probability of the quantum state is equal to something times (1/137)^10^11. This is a verrrrrrrrry small number. To top it off, the radiation pressure equation has a 1/c^3 dependence. The difference in pressure would be even smaller!

So I concluded YES! There does exist an EM drive! However, the pressure gradient generated is so small, it would be impossible to ever measure it in a lab.

I also want to stressed that at multiple points I made many assumptions TRYING to make the EM drive make sense. I assumed we had enough energy to put the particles on-shell. I even assumed that the speed of the electrons is zero, which is unlikely. I tried my hardest to get as large of a result as possible, and it still ended out tiny!

There are only two ways in which I could reconcile such an experimental result with my theoretical understanding:

1) Off-shell states happen frequently and affect things

2) The group velocity of light is somehow altered, but not by the Scharnhorst effect.

I encourage any questions, comments, concerns, refutations, corrections, etc! I want to get to the bottom of understanding if and how such a device is possible.

Hey, I’ve been following your posts sometimes, interesting thoughts ! I was at UCLA 3 years ago as an exchange student…

As a condensed matter phd, I would say you could use dielectrics to modulate the speed of light. You create pairs (real I think) of electrons and holes. The relative permittivity of the material can be 6 orders of magnitude higher than vacuum as in certain polymers for example, then the light is really slow ! You can then maybe stack different materials to get a gradient.

Haven’t made any calculations, just responded to your thoughts…

Sometime complexity allows much more powerful results than simple and abstract ones, I think.

I also really like the article you cited about QM vacuum

Can I ask a question – Is the photon energy density, or the photons per unit volume, the same at the wide end of the cavity, as it is at the narrow end? I know there is more pressure overall on the wide end, but it has a larger area. But is there also a higher energy density? Thank you.

My conventional knowledge of QFT would make me assume that the energy density would be constant throughout, why would it be different? But we never really learned anything about boundary conditions… Now, it seems clear to me that this effect may be possible due to some sort of resonance phenomena due to the irregular shape of the device. Gaining intuition for how any boundary effects work quantum mechanically is notoriously difficult. For example, Casimir plates attract, but if you have square wave shaped plates, like square teeth, then they repel! You just have to do the bitchy calculations to figure it out. And that would have to be some ridiculous research project involving weird numerical simulations that I have no idea how to start with.