Quantization in noninertial reference frames and curved spacetime

In quantum mechanics, the reference frames of the observers are often not explicitly mentioned and quantization for a physical system is usually performed in inertial reference systems. Quantization in noninertial reference systems is an important topic and should be studied systematically. In the framework of nonrelativistic quantum mechanics, the quantum mechanical equations in noninertial reference frames could be derived from those in inertial reference frames using unitary transformation operators. Here, we use canonical quantization scheme to derive quantum mechanics wave equations and Hamiltonian operator in general noninertial coordinate systems in nonrelativistic and relativistic spacetime directly from the classical Hamiltonian formulation based on the basic physical principles. As a special case, the Schrödinger equation in rotating coordinate system of nonrelativistic spacetime, and the KG equation and the first-order wave equation in rotating coordinate systems of Minkowski spacetime were derived from the classical Hamiltonian using canonical quantization scheme. In curved spacetime, the matrix-valued vector field γ^a in the first-order wave equation determines the spacetime metric g_ab and spacetime geometry and has great gauge freedom. In the gauge defined by the gauge constraint γ^b∇_bγ^a = 0, the Dirac equation in curved spacetime is reduced to our first-order wave equation. A Hamiltonian operator in general spacetime is derived from the first-order wave equation, which is Hermitian only in a special class of spacetimes.