Background Independent Quantum Mechanics, Gravity, and Physics at Short Distance: Some Insights

In the present discussion Background Independent framework of Quantum Mechanics and its possible implications in the studies of gravity and Physics at short distance are addressed. The expression of the metric of quantum state space $g_{\mu \nu }$ which is intrinsically a quantized quantity, is identified in terms of Compton wavelength as: $[<\partial_\mu\psi\mid\partial_\nu\psi> - <\partial_\mu\psi\mid\psi> <\psi\mid\partial_\nu\psi>] = 1 / \lambdabar_C^2 (=m_0^2c^2 / \hbar^2})$. The discussion also sheds light on the notion of neighborhood in quantum evolution, and various constraints on the possibilities of smallness of distance in the quantum mechanical limits.