Probing the Planck scale: The modification of the time evolution operator due to the quantum structure of spacetime

The propagator which evolves the wave-function in NRQM, can be expressed as a matrix element of a time evolution operator: i.e $G_{\rm NR}(x)= \langle{\mathbf{x}_2}|{U_{\rm NR}(t)}|{\mathbf{x}_1}\rangle$ in terms of the orthonormal eigenkets $|{\mathbf{x}}\rangle$ of the position operator. In QFT, it is not possible to define a conceptually useful single-particle position operator or its eigenkets. It is also not possible to interpret the relativistic (Feynman) propagator $G_R(x)$ as evolving any kind of single-particle wave-functions. In spite of all these, it is indeed possible to express the propagator of a free spinless particle, in QFT, as a matrix element $\langle{\mathbf{x}_2}|{U_{\rm R}(t)}|{\mathbf{x}_1}\rangle$ for a suitably defined time evolution operator and (non-orthonormal) kets $|{\mathbf{x}}\rangle$ labeled by spatial coordinates. At mesoscopic scales, which are close but not too close to Planck scale, one can incorporate quantum gravitational corrections to the propagator by introducing a zero-point-length. It turns out that even this QG corrected propagator can be expressed as a matrix element $\langle{\mathbf{x}_2}|{U_{\rm QG}(t)}|{\mathbf{x}_1}\rangle$. I describe these results and explore several consequences. It turns out that the evolution operator $U_{\rm QG}(t)$ becomes non-unitary for sub-Planckian time intervals while remaining unitary for time interval is larger than Planck time. The results can be generalised to any ultrastatic curved spacetime.