Quantum Geometry Effects in Quantum Field Theory: Hamiltonian constraint Generates Gravity-Matter Entanglement

Abstract

In this paper, we address a foundational challenge in quantum field theory on curved spacetime by developing a consistent framework within loop quantum gravity. We introduce a methodology for defining meaningful superpositions of quantum geometry and matter states. This is achieved by identifying a restricted subspace of the gravitational phase space, which ensures unitary equivalence among Fock representations of a scalar field across different quantum geometries. Within the resulting well-defined state space, we derive weak solutions to the quantum Hamiltonian constraint of general relativity. Furthermore, we generalize the Hartle-Hawking vacuum state to this quantum geometric framework. The resulting state exhibits the inherent entanglement between geometry and matter, which arises from the quantum Hamiltonian constraint of general relativity. This work establishes a principled framework for studying geometry-matter entanglement and offers new insights into the quantum foundations of the black hole information paradox.

Figures (1)

• Figure 1: The Fock states $|\vec{k}_{\vec{n}}(\mathfrak{g})\rangle$ and $|\vec{k}'_{\vec{n}}(\mathfrak{g}')\rangle$ (illustrated by red wavy lines) constructed on the classical geometries $\mathfrak{g}$ and $\mathfrak{g}'$ (illustrated by grey surfaces) respectively. The superposition state $|\vec{k}_{\vec{n}}(\mathfrak{g})\rangle+|\vec{k}'_{\vec{n}}(\mathfrak{g}')\rangle$ is ill-defined because of the following two reasons; First, $|\vec{k}_{\vec{n}}(\mathfrak{g})\rangle$ and $|\vec{k}'_{\vec{n}}(\mathfrak{g}')\rangle$ are constructed on different classical geometries $\mathfrak{g}$ and $\mathfrak{g}'$ so that they belong to different Hilbert spaces $\mathcal{H}^\phi_{\mathfrak{g}}$ and $\mathcal{H}^\phi_{\mathfrak{g}'}$, while one cannot guarantee a unitary equivalence $\mathcal{H}_{\mathfrak{g}}^\phi\cong \mathcal{H}_{\mathfrak{g}'}^\phi$ for arbitrary $\mathfrak{g}\neq \mathfrak{g}'$; Second, $|\vec{k}_{\vec{n}}(\mathfrak{g})\rangle$ and $|\vec{k}'_{\vec{n}}(\mathfrak{g}')\rangle$ must be associated to the classical geometries $\mathfrak{g}$ and $\mathfrak{g}'$ respectively by the Hamiltonian constraint, while the classical geometries can not be superposed.