Non-unitary evolutions of noncommutative worlds with symmetry

Motivated by describing time-evolutions of noncommutative worlds, I discusssymmetry-preserving evolutions of noncommutative worlds of finite dimensionalrepresentation spaces. An interesting issue in such evolutions is that therecan be transitions of representation spaces of symmetry, and therefore theevolutions are generally non-unitary or symmetry-violating. The central idea ofthis paper is that a main world evolves by emitting baby worlds whichcompensate the violations of the symmetry of the main world. Tracing out thestates of the baby worlds, the symmetry-preserving evolutions of the densitymatrices 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 simplemodel has some attractive features resembling our universe: it gets born from avacuum and its entropy of geometric origin grows. I also discuss the evolutionsin Heisenberg picture.Comment: 10 pages, 2 figures, LaTe