REALITY AND GEOMETRY OF STATES AND OBSERVABLES IN QUANTUM THEORY

The determination of the quantum state of a single system by protective observation is used to justify operationally a formulation of quantum theory on the quantum state space (projective Hilbert space) $\cal P$. Protective observation is extended to a more general quantum theory in which the Schrodinger evolution is generalized so that it preserves the symplectic structure but not necessarily the metric in $\cal P$. The relevance of this more general evolution to the apparant collapse of the state vector during the usual measurement, and its possible connection to gravity is suggested. Some criticisms of protective observation are answered. A comparison is made between the determination of quantum states using the geometry of $\cal P$ by protective measurements, via a reconstruction theorem, and the determination of space-time points by means of the space-time geometry, via Einstein's hole argument. It is argued that a protective measurement may not determine a time average.