Quantized scalar fields in a closed anisotropic universe

We formulate the quantum theory of a neutral scalar field, either minimally or conformally coupled to gravity, in nonrotating Bianchi type IX (mixmaster) universes. Creation and annihilation operators in the Heisenberg picture develop in time under a general Bogolubov transformation, which we calculate for a particular difinition of those operators. That transformation expresses mathematically the physical processes of particle creation and mode mixing, the latter being a new effect, absent in simpler models. Our results should be useful for investigating the possible damping of anisotropy in the expansion of closed, homogeneous universes as a result of particle creation and mode mixing.