What sort of laws shape the universe? The answer is given by understanding two
things: 1) what is space-time, 2) what is the dynamics of space-time.
The first question has no unique answer. In fact, there could be many answers to it, among some of which are: metric fields, frame fields and connections, causal sets, causal sites, strings, topological fields, spin networks, etc. Usually any reasonable assumption for simplifying a physical system such as space-time is acceptable as long as its dynamics satisfies some physical facts. What makes a theory successful is mostly its dynamics.
So far there have been many different models for studying space-time dynamics in quantum scales; from perturbative to non-perturbative and also from continuum to discretized ones.
The traditional quantum gravity started with fixing a background metric and defining quantum effects as the perturbations of background metric. The diffeomorphism constraint is linearized and it appears to be solvable. This theory in this format becomes similar to a local field theory with two degrees of freedom defined at each point. The Feynman rules for constructing the diagrams are obtained from the Einstein-Hilbert Lagrangian coupled to matter using standard procedures of quantum field theory.
Comparing to other field theories, in this localized field theory of gravity the perturbative part of metric turns out to interact with itself on the fixed background by some additional derivatives. This suggests the interaction coefficient to be of length dimension.
Quantum field theory of gravity has an infinite number of complicated interaction vertices. Where the momentum of the internal loop propagator becomes arbitrary large the loop diagrams diverge. In other theories such as chromodynamics the similar UV divergences appear; however those theories are renormalizable. Renormalizability means that the divergences from high energy scales can be absorbed into the redefinition of original parameters appearing in the theory. However, in the quantum field theory of gravity, we cannot eliminate UV divergences in the original Lagrangian because the gravity coupling G carries dimension of length.
To be continued ...
