  Motivated by describing time-evolutions of noncommutative worlds, I discuss
symmetry-preserving evolutions of noncommutative worlds of finite dimensional
representation spaces. An interesting issue in such evolutions is that there
can be transitions of representation spaces of symmetry, and therefore the
evolutions are generally non-unitary or symmetry-violating. The central idea of
this paper is that a main world evolves by emitting baby worlds which
compensate the violations of the symmetry of the main world. Tracing out the
states of the baby worlds, the symmetry-preserving evolutions of the density
matrices of the main world are obtained. I give a simple example with SU(2)
symmetry, which can be regarded as an evolving quantum two-sphere. This simple
model has some attractive features resembling our universe: it gets born from a
vacuum and its entropy of geometric origin grows. I also discuss the evolutions
in Heisenberg picture.

High Energy Physics - Theory;General Relativity and Quantum Cosmology