Summing the natural numbers

For my first entry, I would like to do a calculation that has always resulted in a debate among my physics friends. A while back we came across a formula where you sum up all the natural numbers and get the answer -1/12. This clearly makes no sense, and yet it is used in actual physics calculations. Are physicists crazy, or is there at least some truth to this formula? Let us find out.

This formula shows up in a number of different areas of physics. I first saw it appear in a calculation of the Casimir effect, and so I will reproduce it here and try to convince the skeptics that physicists do actually know what they're doing. I will set .

The Casimir effect is due to the “vacuum energy” of a system. Let's remind ourselves what that is. Remember that in a quantum field theory, we have creation operators and annihilation operators . The vacuum of a quantum system is defined to be the state annihilated by all annihilation operators: for all . If we have a free scalar field theory, and we calculate the Hamiltonian, or total energy of the system, we get the following result:

where . It would be nice if the vacuum had zero energy. After all, the vacuum is an absence of anything, including energy. We know that gives nothing when acting on the vacuum, so let's commute the second of the two terms in to put it on the right. We use the commutation relation :

This doesn't look quite like what we want. If we act on the vacuum state with this operator, the first term vanishes, but the second term remains. The second term is the vacuum energy, or zero-point energy. Let us focus on that:

We notice immediately that it is infinite, and in fact infinite in two ways. The first infinity comes from the -function, evaluated at 0, where the infinite spike sits. The second infinity comes from the fact that we are integrating over , which goes roughly like , and so clearly diverges.

This situation looks hopeless. How can we ever deal with a theory if we always get infinity when we try to calculate the energy? The usual way to proceed is to not worry. We just ignore the infinite constant. Does that make sense? Well, in experiments, we can only ever measure a difference in energy. We use this argument in classical mechanics all the time, when we choose where we want the zero of potential energy to be. So if we can only measure differences in the energy, then clearly the infinite constants cancel. No big deal.

If you're not satisfied by this argument, let's dig a little deeper. Suppose that the universe in which our theory lives is a box with side length , where we let to get an infinite universe. Notice that

We thus see that the -function diverges arises because , the volume of our theory's universe, is infinitely large. This divergence is called an infra-red divergence, because it comes from the exponent when , giving a zero frequency. Using this interpretation, it is clear why we are getting this infinity — we are trying to calculate the vacuum energy of the entire infinite universe. The problem goes away when we work only with the energy density:

That's one of the infinities taken care of. It seems we have managed to calculate the energy density of empty space (and it's still infinite.) Could this infinite energy density possibly have anything to do with the real world? To save you the suspense, I'll tell you that the answer is yes. This is where the Casimir effect comes into it.

Let us consider two parallel infinite conducting plates a distance apart. To avoid the infra-red divergences described above, we will put them in a universe with the -direction periodic, and of length . In other words, for a field , we set . To take care of the and directions we will divide by the area and calculate the vacuum energy per unit area.

We really want to look at electric fields, but a scalar field is easier to deal with, so let's pretend it represents an electric field. This means that it vanishes on the conducting plates; when is on a plate. Taking the Fourier transform of , we can look at its Fourier modes:

Because the field is trapped between two plates, this of course means that the -momentum modes are quantized, and so . Now that we have what looks like, we can write down the vacuum energy per unit area:

Here we used the fact that our “electric” field is massless, and we are also summing over all possible instead of integrating, because takes on discrete (quantized) values. Note that this expression is for the vacuum energy in between the plates. To get the energy outside the plates, we remember that we are considering a universe that is periodic in the -direction, and so the space outside the plates is really the same as the space inside the plates, except now the distance is instead of . We thus have that the energy outside is , and the total energy is .

The integral above can be done, but it's messy. Since I only want to illustrate the concepts and not get bogged down in technicalities, I'm going to simplify the problem. Let's get rid of the and directions and consider only the 1 dimensional problem. This means that the energy between the plates becomes

This is much nicer, but what exactly is the point? The expression is still infinite! The problem comes from the , which can get arbitrarily large. This corresponds to arbitrarily high momentum modes. But remember that we are dealing with a physical system. It's safe to say that no actual physical conducting plates are able to withstand electric fields with arbitrarily high momentum. This means that we can neglect the very high momentum modes, where we define “very high” to mean for some arbitrary distance scale . To achieve this, insert a factor of into the sum above:

For small , exponential factor is close to 1 and gives terms that are approximately correct. The terms of large vanish. We now have a finite expression, so let's start manipulating it, using the fact that is large and so .

Notice that this is still infinite in the limit where , which would give back our original expression.

Now we can calculate the total vacuum energy in the system, which is given by

This is still infinite. However, remember that total energy is something that we cannot measure. We can only measure energy differences. We can also measure forces; let's see what sort of force this expression gives us. If we treat this as a potential energy, then we can look at (minus) the slope to determine the force:

We can now send and and still get a finite answer. The result is the Casimir force, .

If we had done the integral in 3 dimensions instead of simplifying to 1 dimension, the result would have been

The real Casimir force is then twice as large as this, because photons have 2 polarization states (they are not scalars, as we assumed they were for simplicity.) It has been measured experimentally, which is of course the most important fact about this whole process.

Notice that we did not make any bogus claims, such as saying that . It would certainly have given the same result if we had simply used this instead of introducing an exponential cut-off, as you can check for yourself.

Now that the mathematical details are out of the way, what does it all mean? We have found that the energy of the vacuum, of empty space, is in fact relevant physically, and can give rise to a force. If you take two conducting plates and put them parallel to each other, they will experience an attractive force. This force is due to vacuum fluctuations — virtual photons popping in and out of existence in empty space are influenced by the plates, in such a way that an net force is induced. That is pretty cool.

For more information, I recommend the quantum field theory lecture notes by Dr David Tong, which is where I got this calculation.