(with apologies)

When I left things off last week, we were stuck with the somewhat schizophrenic viewpoint that we have two kinds of entities in the world, particles- pointlike masses – that interact with each other through an intermediate agent, the field. Each particle is capable of distorting the field configuration, creating then a potential for other particles to feel its presence through that distortion.

The particle concept starts falling apart when considering the issue of self-interaction: does the particle itself feel the distortion in the field configuration it creates? If so, how does that influence the particle motion? It seems impossible to keep track which particle created which distortion of the field, if particles respond to the fields it seems clear they ought to respond also to the distortions they create. However, that leads to all kinds of nonsense…

There are two sets of problems that arise once trying to deal with the issue of self interaction. First, that force the particle exerts on itself tends to be very large, infinite in fact if we believe the particle is exactly a mathematical point. Even if we consider the particle to be a very small distribution of matter, but not exactly pointlike, that force is enormous, and tends to tear that matter distribution apart. Just think about an electron as a spherical shell of negatively charged matter – what holds that shell together? recall that charges of the same type repel each other.

Second set of problems arises if we include the self-force in the equations that determine the motion of the particle. The equation then become higher order in derivatives. Newton’s laws give rise to second order differential equations, which means that two pieces of initial data, say the initial position and velocity, specify a solution. Not so for higher order equations, those have more solutions, and need more data to select a specific solution. Moreover, some of those extra solutions are really bizarre, they have non-local and non-causal effects, clearly there is something wrong…

So, small particle do not make sense in classical mechanics. This should not very surprising, we now know that the world of the small is governed by quantum mechanics, so we probably need to quantize something to get small distribution of matter such as particles, but what do we quantize exactly?