We don't really know yet.
When you are asking what something looks like, you are asking to describe it in relation to its surrounding space. The problem with asking this question of a fundamental particle in the quantum worldis that we do not have a working understanding of what space[time] is like at these scales. Without THAT, we cannot stand even begin to answer this question. There are some indirect clues that could help us constrain the answer in a meaningful way.
Take the simplest particle we know about, the electron. Experiments show that it is just a knot in the electromagnetic field with no solid surface or internal structure. As you scatter more and more energetic particles off of it, all you see is a region of increasingly higher electric field strength. One thing is for sure, it cannot be a tiny sphrical ball with its electric charge on its surface. Such a system would explode instantly from the enormous electrostatic repulsive forces. Also, because they obey special relativity, they cannot have an extended shape because as they travel through space, their shape would change and no two observers would agree who was observing the correct shape for an elementry particle. Any physical extension would violate
A theory called quantum electrodynamics has been tested to a phenomenal number of decimal places, 10 or more, and still agrees with experimental data on the electron and how it behaves quantum mechanically. The theory stipulates that the electron is a pure point particle with absolutely no internal structure. If you added internal structure, the theory would violate special relativity.
Recently, experimenters have found that in certain kinds of experiments, there MAY be a weak departure from the predictions of QED and experiment, but the scale at which this happens is about 10^-20 centimeters or so, and at energies above 100 GeV. Future experiments will check this.
Also, some recent results at Fermilab seem to indicate that quarks may have some internal structure making them less than fundamental, however, these experiments are far less conclusive and under considerable controversy given the statistical significance of the results.
On the theoretical side, it has been widely expected for decades now that at a scale of 10^-33 centimeters, the structure of space-time will cease to be the implacable, smooth 'surface' we use in modern quantum theory. It will become something quite bizarre like the figure at the top of this page. Perhaps an unimaginable froth of mini-worm holes, quantum loops or strings wiggling about in some strange kind of hyperspace with a dozen or more dimensions. At these scales, all particles loose their point line character. All quantum fields reduce to some more complex topological structure as the superstring theorists advocate.
Conceptually, I have not the slightest idea how to interpret the mathematics, however, if the mathematics lead to testable and verifiable predictions, how we think of the mathematics becomes a moot issue. Just like we cannot accept quantum indeterminacy, duality, and special relativity as 'common sense' ideas, yet this is the way nature seems to work.
Return to Dr. Odenwald's Gravity page at the Astronomy Cafe Blog.