Institute for Mathematical Physics Geometrical Formulation of Quantum Mechanics Geometrical Formulation of Quantum Mechanics

States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a KK ahler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics which, although equivalent to the standard algebraic formulation , has a very diierent appearance. In particular, states are now represented by points of a symplectic manifold (which happens to have in addition a compatible Riemannian metric), observables are represented by certain real-valued functions on this space and the Schrr odinger evolution is captured by the sym-plectic ow generated by a Hamiltonian function. There is thus a remarkable similarity with the standard symplectic formulation of classical mechanics. Features|such as uncertainties and state vector reductions|which are spe-ciic to quantum mechanics can also be formulated geometrically but now refer to the Riemannian metric|a structure which is absent in classical mechanics. The geometrical formulation sheds considerable light on a number of issues such as the second quantization procedure, the role of coherent states in semi-classical considerations and the WKB approximation. More importantly, it suggests generalizations of quantum mechanics. The simplest among these are equivalent to the dynamical generalizations that have appeared in the literature. The geometrical reformulation provides a uniied framework to discuss these and to correct a misconception. Finally, it also suggests directions in which more radical generalizations may be found.