Quantum Gravity and Inflation as an Open System: A New Paradigm

As part of our program to develop a general theory of relativity for open systems, we introduce a covariant theory that incorporates the effects of classical and quantum spacetime alterations in a new metric tensor that effectively includes these alterations, thereby generating a Riemannian manifold from a new varied action without boundary terms. We illustrate the theory by studying an inflationary model which incorporates the quantum feedback effects of spacetime on the dynamics of the inflaton field $\hatφ$, enabling the simultaneous quantization of $\hatφ$ and the fluctuating gravitational field $\hatΩ$ without relying on perturbative theory. We obtain an exact solution for the modes of both fields, $\hatφ$ and $\hatΩ$, which comply with different quantum algebras. The normalization of the inflaton field modes that is obtained is intrinsically related to the geometric field modes, in such a way that the quantization of the fields results from an expression that links the geometric fields with the physical fields. Finally, the quadratic fluctuations of spacetime are calculated, and its spectrum is analyzed.