Geometrical Formulation of Quantum Mechanics
Abhay Ashtekar, Troy A. Schilling
TL;DR
The paper presents a geometric reformulation of quantum mechanics in which states are points of a Kähler quantum phase space, the projective Hilbert space, and dynamics are described as Hamiltonian flows generated by expectation-value functions. Observables become real-valued functions whose Hamiltonian flows preserve the Kähler structure, with the Poisson bracket reproducing the operator commutator and the Riemann bracket encoding quantum uncertainties. It then develops a gauge-reduction framework to obtain the reduced state space and shows how measurements and probabilities acquire intrinsic geometric meaning via geodesic distances on the phase space. Finally, the authors outline a unified approach to generalizations of quantum mechanics, relate nonlinear dynamics to Weinberg-type frameworks, and discuss semiclassical links through coherent states and WKB, highlighting open problems and potential paths toward deeper generalizations.
Abstract
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 Kähler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics which, although equivalent to the standard algebraic formulation, has a very different 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 Schrödinger evolution is captured by the symplectic flow 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 specific 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 unified framework to discuss these and to correct a misconception. Finally, it also suggests directions in which more radical generalizations may be found.