  We investigate whether quantum theory can be understood as the continuum
limit of a mechanical theory, in which there is a huge, but finite, number of
classical 'worlds', and quantum effects arise solely from a universal
interaction between these worlds, without reference to any wave function. Here
a `world' means an entire universe with well-defined properties, determined by
the classical configuration of its particles and fields. In our approach each
world evolves deterministically; probabilities arise due to ignorance as to
which world a given observer occupies; and we argue that in the limit of
infinitely many worlds the wave function can be recovered (as a secondary
object) from the motion of these worlds. We introduce a simple model of such a
'many interacting worlds' approach and show that it can reproduce some generic
quantum phenomena---such as Ehrenfest's theorem, wavepacket spreading, barrier
tunneling and zero point energy---as a direct consequence of mutual repulsion
between worlds. Finally, we perform numerical simulations using our approach.
We demonstrate, first, that it can be used to calculate quantum ground states,
and second, that it is capable of reproducing, at least qualitatively, the
double-slit interference phenomenon.
