ua nt - p h / 98 03 08 2 v 1 2 9 M ar 1 99 8 Geometry of the Hilbert space and the Quantum Zeno Effect

We show that the quadratic short time behaviour of transition probability is a natural consequence of the inner product of the Hilbert space of the quantum system. We provide a relation between the survival probability and the underlying geometric structure such as the Fubini-Study metric defined on the projective Hilbert space of the quantum system. This predicts the quantum Zeno effect even for systems described by non-linear and non-unitary evolution equations, within the collapse mechanism of the wavefunction during measurement process.