The Casimir Effect: Ramanujan Revisited

There's been a small flurry of hits (a few dozen actually) on one of my posts from almost three   four  years ago, when I talked about how you can use one of Ramanujan's weird infinite sums to calculate the Casimir effect. There seem to be two unrelated reasons for this new activity: a math video by a guy named Brady Haran that got reposted on Slate Magazine and got an awful lot of hits...and an ongoing discussion on mymathforum.com on the same topic, where my new correspondent Balarka Sen mentioned my post. It's quite a coincidence because I do almost no math on this blog, and Balarka doesn't do any physics...and the only two real math topics I've ever done...the quintic equation and that Ramanujan series....turn out to be huge topics of interest for Balarka.

The topic of the internet video is the series

1 + 2 + 3 + 4 + 5 ....

And I have to say I've never understood how that's supposed to work. My interest was in the alternating series:

1 - 2 + 3 - 4 + 5....

and that's the one I used to calculate the Casimir Effect. Or at least that's what I thought. When I look it over again, (especially after watching that guy's video) it's almost arguable that I really did use the first series.

Except there's a difference the way I do it. I actually have a physically motivated reason for being able to shuffle those series around the way I do. And I'm not sure I made it perfectly clear in my original post. So I think I want to go over it again.

The idea is based on a fact of quantum field theory that I have to admit I don't really understand: that a standing-wave mode of the electromagnetic field cannot be totally at rest: it must have a minimum energy of one-half quantum. It's a very small amount of energy, but the problem is there is an infinite number of modes. So does that mean space is filled with an infinite amount of energy? Hard to say. Because in the case of an infinite universe, each of those mode enegies is spread over an infinite volume. What's the local energy density? Hard to say.

But we can calculate the case of a finite box. We're going to take our "box" to be one-dimensional...in other words, a parallel-plate capacitor...because that will simplify counting of the modes. Le'ts say the plates are one millimeter apart. We have modes with one standing wave, two, three, etc...with corresponding frequencies of 1, 2, 3 etc. The mode energies are proportional to the mode frequencies, so we get a total energy of

1 + 2 + 3 + 4 + 5.....

which looks a lot like infinity. (I'm basically letting Plank's Constant = 1, in case you were wondering).

Now let's do something funny. Let's compress the capacitor and see how much the energy increases. The funny thing here is that each of the modes deforms continuously....so by the time the plates are twice as close together, the energy is:

2 + 4 + 6 + 8 + 10....

which looks like twice as much energy. But wait....

The whole problem with these infinite sums is that we assume the presence of infinitely high frequencies. That's a little crazy. Are there actually gamma rays between those capacitor plates...and not just gamma rays, but super-ultra high frequency gamma rays beyond anything imaginable? It doesn't make sense.

Maybe we should only be counting up to a specific frequency cutoff. Lets see how our sums work then. For the capacitor at 1 mm, we have:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8.....; and,counting only up to the same frequency we get

     2    +     4     +   6     +   8..... for the compressed capacitor at 0.5 mm!

If you count it this way, the compressed capacitor has actually less energy that the one you started with. How can that be? Well... maybe it means that unlike a cylinder full of air, which resists being compressed, the parallel plate capacitor has a negative pressure...the plates attract. That would be the Casimir Effect.

But can we calculate it? We're still looking at those monotonically increasing infinite series, which I find intractable. But there's a trick. I can handle the alternating infinite series. Because I assume in physics that the stuff at ultra-high frequencies which is wildly fluctuating between positive and negative must logically just cancel out. It's not hard to verify this numerically. Take that series

1 - 2 + 3 - 4 + 5....

and put it into Excel, and then put a very gradual Gaussian envelope over it, to gently supress the ultra-high frequencies. It's not hard to verify that the value tends to 0.25, once your Gaussian is wide enough, and it stabilizes there as your Gaussian tends to infinite width. So how can we apply this to our capacitor?

I figured out I can make it work by analyzing pressures instead of energies. The mode pressures in the capacitor are proportional to the energies:

1 + 2 + 3 + 4 + 5....

but when you bring the plates twice as close together, the mode pressures don't double: they quadruple. Because pressure is energy per unit volume, and you now have twice the energy in half the volume. So comparing the pressures:

1.0 mm capacitor:   1 + 2  +  3  +  4  +  5  +  6  +  7  +  8....
0.5 mm capacitor:         4       +     8        +    12     +    16....

(Notice I've lined them up so the frequencies agree.) We still don't know what the pressure is between the plates, but oddly enough, we can now calculate the change in pressure when we went from d=1 to d=0.5  .  It's the difference between the two series, which is just the alternating series we've been talking about:

dP = 1 - 2 + 3 - 4 + 5....= 0.25

Now...the people who want the monotonic series to add up to -1/12, they say: look what you've done: you've just said S - 4S = 0.25, and they solve for S and get -1/12. I don't know about that. I can't justify it physically. But it turns out I'm going to get the same result, with my own brand of physical reasoning.

If the change in pressure of the capacitor was 1/4 in compressing from 1 to 0.5, then what do you think it would have been expanding going from 1 to 2 mm? It's not hard to verify by dimensional analysis that it would have been exactly a quarter as much, or 1/16. And in going from 2 to 4?...1/64. You can see that in expanding the plates from 1 millimeter to infinity, the pressure increased by:

1/16  +  1/64   +   1/256.....   = 1/12

But the there can't be any pressure between the plates when they're infinitely far apart, can there? So we...wait for it...renormalize. We say that the pressure at infinity is zero, and the pressure at 1 millimeter separation is...-1/12.

Now as for the people who say I could have got that result from the get-go just by using Ramanujan's result for divergent series...I disagree. Yes, superficially you could write as I did, from the mode pressures, the sum:

1 + 2 + 3 + 4 + 5....= -1/12

But that's wrong. There is infinite pressure between the plates. This calculation neglects the ever-so-slightly-larger infinite pressure which is also found outside the plates. It's the difference between those infinities that adds up to -1/12. And that's why the plates experience a force drawing them together.