QSD V : Quantum Gravity as the Natural Regulator of Matter Quantum Field Theories
Thomas Thiemann
TL;DR
The paper provides a rigorous, non-perturbative canonical framework in which gravity acts as the natural regulator for matter quantum field theories in four dimensions. By employing real connection variables and a volume operator-based regularization, it constructs densely defined, finite Hamiltonians for gauge, fermionic, and Higgs sectors that remain well-defined in the continuum limit. A general Structure Theorem is presented for converting a broad class of weight-one Hamiltonian densities into anomaly-free, diffeomorphism-covariant operators. The work also demonstrates cylindrical consistency, partial positivity/self-adjointness for key pieces, and outlines the full solution space of the diffeomorphism and Hamiltonian constraints, arguing the theory is nontrivially rich and physically meaningful.
Abstract
It is an old speculation in physics that, once the gravitational field is successfully quantized, it should serve as the natural regulator of infrared and ultraviolet singularities that plague quantum field theories in a background metric. We demonstrate that this idea is implemented in a precise sense within the framework of four-dimensional canonical Lorentzian quantum gravity in the continuum. Specifically, we show that the Hamiltonian of the standard model supports a representation in which finite linear combinations of Wilson loop functionals around closed loops, as well as along open lines with fermionic and Higgs field insertions at the end points are densely defined operators. This Hamiltonian, surprisingly, does not suffer from any singularities, it is completely finite without renormalization. This property is shared by string theory. In contrast to string theory, however, we are dealing with a particular phase of the standard model coupled to gravity which is entirely non-perturbatively defined and second quantized.