Quantum Gravity Framework 3: Relative Time Formulation and Simple Applications to derive conventional Hamiltonians.
In this paper I discuss the further update of the quantum gravity framework. The last versions are Quantum Gravity Framework 1 , Quantum Gravity Framework 2. When I first wrote the paper , I had only hunch about how to go about deal with problem of time. I did not even know how apply the time constraint to understand the dynamics of path integral formulation of quantum gravity. But I have developed a better understanding of in it. This I have presented in this paper. Basically, there is quite a bit of flexibility with respect to which we can study the evolution of quantum state of fields both in space-time and in the internal space of particles. Basically, the choice of the dynamic frame more and more seems to depend on the observer: That is conscious itself. So, in the next update quantum gravity framework 4.0, conscious is included.
In quantum gravity framework 2.0 I discussed four parts: self-time evolution, self-time decoherence, global quantum decoherence and a fourth proposal related to scale invariance, determinism, and continuum limit. In this update 3.0, I generalize the concepts to make them more relative. In this paper I discuss the relative nature of the four proposals in quantum gravity formulation 2.0 with respect to choices of observation paths in configuration space and foliation of the space-time, and I discuss the relative formulations as the quantum evolutions.
I introduce the relative-time evolution which is the more general form of self-time evolution and its impact on the other proposals. I also introduce the concept of rest frame foliation which I believe is physically significant in understanding global quantum reduction. This is more important in understanding conscious which will be included the framework 4.0
In this paper first I discuss the applications of relative-time formulation. I have done very detailed and thorough calculation of these. This is quite unusual give that currently we are discussing the ideas in only in heuristic form.
First, I discuss how to get the conventional quantum mechanics from a relativistic Hamiltonian constraint using the relative-time constraint. This is quite straightforward. Then I discuss how the time constraint formulation can be used to derive non-relativistic quantum mechanics in the context of particles. This calculation requires a thorough understanding of various terms of Hamiltonian of general relativity as applied to asymptotically flat space times.
This starts with various basic assumptions, regarding collection of particles interacting with each other by gravitational and gauge fields. We start with Hamiltonian constraint and apply the relative-time evolution formulation. By systematic deductions, simplifying using flat space approximations, I derive the conventional Hamiltonian including the Newtonian gravity terms. I also briefly discuss deriving the conventional Hamiltonian formulation using the relative-time evolution in the context of field theory.