Quantum gravity corrections to quantum field theory: Born-Oppenheimer approach to the canonical formalism

The role of time is intrinsically different between Quantum Mechanics and General Relativity: while the former associates time with an external observer, the latter unifies time and space, making them indistinguishable in a covariant framework. The absence of a clear time variable in GR stems from its symmetry and parametrized nature, resulting in the so-called frozen formalism. For this reason, the search for a theory of Quantum Gravity must face the challenge of time absence in the Wheeler-de Witt equation. Efforts to quantize gravity have led to various approaches to define time, categorized into pre-quantization, post-quantization, and timeless proposals. This thesis focuses on post-quantization time constructions, particularly within the Wentzel-Kramers-Brillouin approach, which perturbatively expands the wave function to derive dynamical equations. Previous attempts have shown that the introduction of an internal clock from gravitational variables yields non-unitary dynamical effects on the matter sector at the next order. This thesis implements a Born-Oppenheimer-like scheme that separates the matter and gravitational sectors, leveraging their distinct energy scales: the matter's faster evolution is contrasted with the slower gravitational field, both properly quantum. Two novel time constructions are proposed, making use of a fast component derived from introducing the kinematical action or (reparametrized) Gaussian frame fixing respectively; the discussion of their geometrical and physical meaning proves that both are essentially tied to the concept of a reference system. These clocks for the matter subsystem overcome previous non-unitarity concerns, resulting in an Hermitian dynamics at the first order where quantum-gravitational corrections emerge. A direct equivalence between the two implementations is proved in the homogeneous minisuperspace setting. The present investigation also faces the challenge posed by the dependence of the matter wave functional on intrinsically quantum gravitational components, particularly evident in the cosmological context. To address this, a more rigorous Born-Oppenheimer separation of dynamics is proposed, distinguishing the classical gravitational background from its small quantum fluctuations (i.e. gravitons) and then proper quantum matter contributions. By introducing an appropriate gauge choice for the gravitons' sector, the zero-th order of this model allows to recover the standard Quantum Field Theory dynamics. We show how this refined scheme can be combined with the concept of a reference fluid time (or equivalently the kinematical action one), offering a unitary evolution for the quantum matter subsystem with quantum gravity corrections, free of previously mentioned concerns. Such unified approach clarifies the quantum nature of gravitational components and shows how gauge requirements address the emergence of quantum gravity effects in subsequent orders of the expansion. The central achievement of the present thesis is the development of a suitable Born-Oppenheimer scheme for the quantum gravity-matter system, in which the matter's evolution modified by quantum gravitational effects has a unitary character. This framework offers insights into how quantum gravity influences our understanding of the universe and contributes to a deeper comprehension of gravitational phenomena.