## Final Report Summary - PHILOQUANTUMGRAVITY (Philosophy of Canonical Quantum Gravity)

The main objective of this research project in philosophy of physics was to address from a philosophical perspective one of the most important open problems in theoretical physics, namely the formulation of a quantum theory of Einstein’s theory of gravity (general relativity). From a methodological standpoint, we have proceeded by developing a conceptual analysis of the mathematical foundations of the physical theories at stake. In order to pursue its main objective, this project relied on separate analysis of the foundations of general relativity and gauge theories on the one hand and quantum mechanics on the other. The common conceptual thread that runs through both subprojects was provided by the onto-logical notion of identity such as it is reconceptualized in the framework of the “homotopical paradigm” in contemporary mathematics. Very briefly, this paradigm substitutes the trivial (in the sense that it requires no demonstration) and entity-independent principle of identity a = a by a non-trivial and entity-dependent principle based on a notion of identity qua identification: each identification between an entity a and itself (i.e. each automorphism of a) can be understood as a proof of the fact that the proposition a = a is true. By using the formal and conceptual framework provided by the theory of Cartan connections, we have argued that a generally curved spacetime can be constructed by “connecting” in a curved manner elementary building blocks given by Klein geometries (such as Minkowski spacetime). The internal symmetries (or equivalently the non-trivial identities) of these Klein geometries entail both the local Lorentz invariance and the diffeomorphism invariance of general relativity. Since identical copies of a symmetric object (in this case the Klein geometries) cannot be canonically identified, the assemblage of the symmetric building blocks requires introducing an additional geometric structure, a (Cartan) connection describing the gravitational field. In the framework provided by this gauge-theoretical comprehension of general relativity, we have argued that this theory relies on two independent principles, the principle of background independence and the principle of local affine symmetry. Regarding quantum mechanics, we have provided an ontological interpretation of the Heisenberg indeterminacy relations by arguing that these relations can be understood in as a consequence of the fact that quantum states have non-trivial identities. Moreover, the non-trivial identity of a quantum state is given by a linear group of automorphisms classified by the quantum numbers that define the state. It follows that the non-trivial and entity-dependent identity of a quantum state conveys all the information required to identify (or individuate) the state. We have described these group-theoretical interpretations of general relativity and quantum mechanics in terms of a localization and a linearization of Klein’s Erlangen program respectively. We have finally conjectured that the notion of spin network used in loop quantum gravity can be understood as a straightforward synthesis of the relativistic localization and the quantum linearization of Klein’s Erlangen program.