The Continuum Limit of Loop Quantum Gravity: A Framework for Solving the Theory

Abstract

The construction of a continuum limit for the dynamics of loop quantum gravity is unavoidable to complete the theory. We explain that such a construction is equivalent to obtaining the continuum physical Hilbert space, which encodes the solutions of the theory. We discuss iterative coarse graining methods to construct physical states in a truncation scheme and explain in which sense this scheme constructs a renormalization flow. We comment on the role of diffeomorphism symmetry as an indicator for the continuum limit. Solving the dynamics of loop quantum gravityLoop quantum gravity led to a rigorous non-perturbative framework, in which to formulate the dynamics of quantum gravity. It allowed fascinating insights into quantum geometry and a possible structure of quantum space time. To get a complete picture of the dynamics of the theory -in the form of constructing the so-called physical Hilbert space of wave functions satisfying the Wheeler deWitt equation -we need to construct the continuum limit. In the framework presented here physical states, i.e. solutions of the Wheeler deWitt equations which give the equations of motions of the theory, are constructed by taking the refinement limit via a coarse graining procedure.The conceptual underpinnings of this framework rely on the inductive limit Hilbert space construction used in loop quantum gravity to define the continuum (so far kinematical) Hilbert space. We point out the powerful concept of this inductive limit construction if one allows for a generalization of the refinement maps that define the inductive limit Hilbert spaces. It leads to a framework in which physical states are computed in a truncation scheme, where the type of truncation is determined by the dynamics itself.This procedure allows for an understanding of the dynamics of quantum gravity on all scaleswhere a notion of scale is given by the coarseness or fineness of configurations. The different scales of the theory are connected via the cylindrical consistency condition inherent in the inductive limit construction. This replaces the notion of renormalization flow in theories with a background scale.We start our considerations with a short explanation of the inductive limit construction in section 2 and discuss the difference between kinematical and dynamical understanding of the continuum limit. In section 3 we start with the task to construct the physical Hilbert space of the theory and explain that it necessitates the construction of the refinement limit for the dynamics of the theory. This results in an iterative coarse graining scheme, in which physical states -or amplitude maps -are constructed in a certain truncation, labelled by the coarseness or fineness of the discrete structures involved. The relation of this scheme with a renormalization flow is clarified in section 4. Concrete realizations of this scheme in the form of (decorated) tensor network methods are shortly explained in section 5. We then point out the powerful notion of diffeomorphism symmetry for discrete systems ...

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

“…Setting all labels to be trivial for the spin network basis we obtain a (quantum deformation of the) Ashtekar-Lewandowski vacuum, expressed on a fixed triangulation or dual graph. Note however that a refinement of these states (in order to define the continuum limit) would require different embedding maps [57,58]. The entire construction rather assumed a quantum deformed BF vacuum and we can therefore expect that the operators we consider here are cylindrically consistent with respect to this vacuum.…”

“…Alexander moves, which will lead to another (larger) Hilbert space. The question is to specify so-called embedding or refining maps, that map states from a 'coarser' Hilbert space to states in a 'finer' Hilbert space [57,58]. Such embeddings impose a certain vacuum state [9] and allow the construction of a continuum Hilbert space via an inductive limit (if certain consistency conditions are satisfied) [15][16][17].…”

“…To restore this symmetry one would have to consider a continuum limit [19,20,[104][105][106], which can be constructed via an auxiliary coarse graining flow [58]. For coarse graining schemes along the lines of [27][28][29][30][31][32][33] the finiteness of the state spaces constructed here facilitates a numerical implementation.…”

Fusion basis for lattice gauge theory and loop quantum gravity

“…Firstly, there are holonomy operators along root-based closed cycles of Γ. These 9 We use the word 'holonomy' for the group-valued path-ordered exponential of a connection along a path between two points on the manifold. It transforms covariantly upon gauge transformations at its starting and ending points, and it is invariant upon any other gauge transformation.…”

“…Therefore, if we want the gluing to be mirrored at the level of the state spaces, this face must be associated with a trivial holonomy. This suggests to define first the gluing operation on basis states based on graphs via 9) and then to extend this by linearity to arbitrary states. The before the last term in the diagram above signals that equivalence relations (section 2.2) are generally used at this point.…”