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3.7 Relational dynamics

The typical form of Hamiltonian constraint operators in loop quantum gravity following [259292Jump To The Next Citation Point] prevents single-spin network states from being physical, i.e., annihilated by the constraint operator. As a consequence of the creation of new edges and vertices by the constraint, physical states cannot be based on a single graph but must be superpositions of different graph states. Such superpositions can be complicated, but generally they can be understood as encoding the relational dynamics of gravity in the absence of an absolute time. Rather than using an external time coordinate one chooses an internal time variable from the dynamical fields. In general situations, no global time exists but locally evolution can be described in suitable variables. Especially in cosmology it is often convenient to use the spatial volume as internal time and thus, at least formally, expand a state in volume eigenstates. Since each action of the Hamiltonian constraint changes spins and the graph, and thus the volume, its basic action can be seen to provide elementary moves of a dynamically changing lattice (see [73Jump To The Next Citation Point] for more details). Typically, larger volumes require finer graphs and thus the underlying lattice is refined as the universe expands. New degrees of freedom emerge while the universe grows, which is a characteristic feature of quantum gravity.

In the full theory, finding solutions and decomposing them in volume eigenstates is currently out of reach. But qualitative aspects of this process can be incorporated in models as described in the following. The models, in turn, can often be analyzed much more easily and thus allow crucial tests of the general framework. Several non-trivial consistency checks have by now been performed and given valuable insights into the dynamics of loop quantum gravity.


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