## 7 Conclusion: The Hole Argument Redivivus

The problem becomes even more delicate for quantum systems, in which the existence of the quantum of action is taken into account. The finite value of precludes the measurement of a complete set of classical observables by a single compound procedure. It becomes important to show that a complete set of quantum observables, as defined by the theory, can indeed be so measured in principle. Non-relativistic quantum mechanics and quantum electrodynamics, have been show to meet this criterion; and it has been employed as a test of proposals for what should be the fundamental physical quantities defined in quantum gravity (see Bergmann and Smith, 1982; Amelino-Camelia and Stachel, 2009). Rovelli (2004) and Oeckl (2008, 2013) have shown how to define such measurements on the hypersurface bounding a four-dimensional region of space-time, even in a background-independent theory.

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).