Quantum Spacetime Geometry and Quantum Relativity

We present a Lie symmetry theoretical formulation of `relativistic' quantum mechanics with a Minkowski metric operator defining the inner product between state vectors, and a noncommutative geometric picture of spacetime naturally comes out of it.

The latter is taken to give a new formulation of quantum mechanics in curved spacetime with very nice properties the conventional formulation based on Schrödinger wavefunction representation missed. Quantum reference frame is a natural notion of reference frame for the noncommutative quantum spacetime. The Quantum Relativity Principle under the quantum reference frame transformations would give a new approach to quantum gravity. The talk gives a sketch of our major results on the subject matter with emphasis on the background perspectives.