Hope for Quantum Gravity: The Hamiltonian=Stress Energy Tensor
With the struggle for quantum gravity has gone on for a while, it's worth looking at if there even is quantum gravity. Does it exist? The following should hopefully provide some encouragement. I also devise what I believe to be a new technique, setting the Hamiltonian from quantum mechanics equal to the stress-energy tensor where both subscore thingies equal 0, which gives us energy density. I came up with this idea independently, and the most I could find on this topic was, "...in texts on quantum field theory, field Hamiltonians that correspond to the time component of the canonical stress-energy tensor...are widely used, without even mentioning the relevant problems, while in general relativity, the concept of energy is often considered, for just those reasons, to be badly defined, and in some texts, the discussion is avoided completely." (https://arxiv.org/pdf/gr-qc/0510044.pdf) However, we will use that equivalence as the guide for the rest of this paper. The following is what I believe to be physics' first attempt using this equivalence, using Einstein's general relativity equation. We will assume k=0 (that we have a flat, Euclidean universe, consistent with current measurements) and we will disregard dark energy and the cosmological constant. We will use standard cosmological geometry. a in these equations represents a(t)=average supercluster spacing/400000000 light years. So, a(present)=1. The makes our metric tensor, in the 00, equal to -1 right now, but I didn't factor that into the equations,
Well, I'm not an expert mathematician, but I can interpret some things from the very bottom equation. The left side is the fabric of spacetime times the front factor (c^4/8piG), and we can see how that tells the energy of the quantum mechanical wavefunction how to move throughout spacetime, as if this was any regular mass-energy. Likewise, we see how the quantum mechanical wavefunction can tell the spacetime of general relativity how to curve. In conclusion, there is hope for quantum gravity, and hopefully, we can continue to dive more into equations like mine (one on the very bottom) to solve physics' greatest problem.
NOTE: I neglected in my math that T00 gives us energy density, which is J/(m^3), while the Hamiltonian just gives us energy in Joules (J). We assume that the volume of the electron or any other Standard Model particle is 0 a lot, but it really does have a mass, just super small, so we have to use that. We know that protons have a volume that's super small, so some calculations and assumptions for each particle can help us with this. Where we inserted the Hamiltonian into the general relativity equation, we divide that by the particle's volume in m^3 (yes, it's small making that part of the equation very large). Now our units should work out and this equation might be somewhat useful.
NOTE: Not all my math may be correct (I'm only in high school), but the general principles should apply: there is some basic, basic foundation for quantum gravity, and we might find use from setting the Hamiltonian equal to the stress-energy tensor in both dimension of time (00). We also can still describe a quantum physical object with a wave function's effect on spacetime, and spacetime's effect on a quantum physical object.