In field theory, the analog of the data set is the couple , where is a 3d surface bounding a finite spacetime region, and is a field configuration on . …The data from a local experiment (measurements, preparation, or just assumptions) must in fact refer to the state of the system on the entire boundary of a finite spacetime region. The field theoretical space is therefore the space of surfaces and field configurations on . Quantum dynamics can be expressed in terms of an [probability] amplitude . Following Feynman’s intuition, we can formally define in terms of a sum over bulk field configurations that take the value on the boundary . …Notice that the dependence of on the geometry of codes the spacetime position of the measuring apparatus. In fact, the relative position of the components of the apparatus is determined by their physical distance and the physical time elapsed between measurements, and these data are contained in the metric of . …What is happening is that in background-dependent QFT we have two kinds of measurements: those that determine the distances of the parts of the apparatus and the time elapsed between measurements, and the actual measurements of the fields’ dynamical variables. In quantum gravity, instead, distances and time separations are on an equal footing with the dynamical fields. This is the core of the general relativistic revolution, and the key for background-independent QFT (Rovelli, 2004, p. 23).
In this sense, Einstein’s hole, as a symbol of process, has reasserted its physical primacy over Hilbert’s Cauchy surface, as a symbol of instantaneous state (see Section 2.7).