Quantum State Geometry - Towards Quantum Gravity

Abstract

We present a minimal operational route towards quantum gravity from three axioms: (1) time = Fisher distinguishability $dt = \hbar\, ds / \Delta E$, (2) bounded spectrum $E \le E_{\max}=c^5/G$, (3) $r_q \ge r_s$. From this single constant $E_{\max}$ we derive:$G = \hbar c^5/E_{\max}^2$,$L_{\min}=\hbar c/E_{\max}$,modified dispersion $v/c = [1-(E/E_{\max})^2]^{3/2}\approx 1-\tfrac32(E/E_{\max})^2$ (quadratic, GRB-safe, $k_2\sim10^{-38}$, $\Delta t\sim5\times10^{-19}$\,s for $D=10^{26}$\,m, $E=10$\,GeV),GUP $[x,p]=i\hbar(1+p^2c^2/E_{\max}^2)$,bounce $H^2=\tfrac{8\pi G}{3}\rho(1-\rho/\rho_{\max})$,regular black hole core $f(r)=1-2GMr^2/(r^3+r_0^3)$, Fisher vs.\ Einstein locality and CHSH Bell $S_q=2\sqrt2>2$ proof. No linear LIV: linear $n=1$ is ruled out by GRB090510 ($E_{QG}>9.3\times10^{19}$\,GeV), quadratic $n=2$ survives ($M_2>7.13\times10^{11}$\,GeV). We gravitize the bound of QM, not quantize GR. Towards, not completed.