Quantum Mechanics and General Relativity

This manuscript presents a comprehensive theoretical analysis aimed at unifying quantum mechanics and general relativity to advance the study of gravitational waves. We explore the quantum treatment of spacetime, employing advanced mathematical frameworks to address phenomena observable at Planckian and larger scales. Initial sections detail the quantum foundations of spacetime, focusing on vacuum fluctuations and the Hamiltonian framework for gravitational interactions. Subsequent discussions include the quantization of the gravitational field through perturbative approaches, defining the metrics of spacetime perturbations, and elaborating on the quantum state perturbations modeled as gravitons.

The interaction between gravitational waves and dark matter is modeled using quantum field theory, with implications for dark matter astrophysics and cosmological observations detailed. Theoretical extensions into Loop Quantum Gravity (LQG) and string theory are proposed to investigate their effects on gravitational wave propagation and potential detectability via gravitational interferometry.

We also introduce new quantum mechanical formulations to describe interactions between subatomic particles within gravitational fields, proposing experimental setups to validate these theories. Overall, this paper aims to bridge gaps in current theoretical physics and suggest methodologies for observing quantum gravitational effects in cosmological contexts.