Fractal Dimensions Should Modify The Casimir Effect

Back in the 1920s, Theodor Kaluza and Oskar Klein developed an idea that unified Maxwell’s theory of electromagnetism with Einstein’s theory of relativity.

That was an impressive feat but it had one small drawback. In the Kaluza-Klein model, the universe has 5 dimensions.

Kaluza and Klein were unfazed, however. They suggested that the fifth spatial dimension existed only on the Planck scale at distances of the order of 10^-35 m.

In fact, they imagined it as curled up on itself. So if it were possible to traverse space at this scale, any travellers in the fifth dimension would constantly end up where they started.

Since then, numerous physicists have used a similar approach to explain why we don’t see the extra dimensions their theories predict.

A more recent idea is that the extra dimension may flit in and out of existence, like other quantum objects on this scale. And if this happens, the extra dimension would only partially exist. Physicists describe these dimensions as fractal–half way between one integer and the next.

Being so small, little can be done to confirm or refute the idea of fractal dimensions. At least, that’s what physicists thought until now.

Today, however, Hongbo Cheng at the East China University of Science and Technology in Shanghai says it may be possible to spot the difference.

He has calculated how extra fractal dimensions would influence the Casimir effect. This is the mysterious force that pushes two parallel conducting plates together when they are only a tiny distance apart.

The effect is caused by the maelstrom of particles flitting in and out of existence at the Planck scale. These particles all have an associated wavelength. If the gap between the plates is smaller than this wavelength, then the particle cannot fit in the gap and so is excluded. When this happens, the excess of particles outside the plates tend to push them together.

Cheng says that if the distance between the plates is about the same as the scale of any extra dimension, then this must effect the Casimir force too. In fact, he says that this force will be stronger if the extra dimension is integral than if it is fractal but that the exact nature of the difference is sensitive to the fractal degree.

The question of course is whether the difference can be measured unambiguously. If it can, we may have a rather interesting test of the nature of space-time.

But don’t hold your breath. Measurements of the Casimir effect are famously difficult. The Casimir force is tiny; so small, that it wasn’t measured until 1997.

And even now, physicists often do not agree on the direction this force will take in a given experimental set up, let alone its magnitude. Even Cheng doesn’t stick his neck out to quantify the size of the new force.

An interesting experiment but clearly one that will be fraught with difficulties, when and if it ever becomes feasible, .

Ref: arxiv.org/abs/1106.4610: The Casimir Effect For Parallel Plates In The Spacetime With A Fractal Extra Compactified Dimension