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 present iterative coarse graining methods to construct physical states in a truncation scheme and explain in which sense this scheme represents a renormalization flow. We comment on the role of diffeomorphism symmetry as an indicator for the continuum limit. 1 Solving the dynamics of loop quantum gravity Loop 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 theory – in the form of constructing the so–called physical Hilbert space – we need to construct the continuum limit. In the framework presented here physical states, i.e. solutions of 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 scales – which we here argue is to understand in terms of 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 in section 6. The realization of this diffeomorphism symmetry is necessary for the definition of physical states, however also indicates that a continuum limit is reached. In this sense physical states can only be defined in the continuum limit. We end with a discussion and outlook of future developments in section 7. 2 Continuum limit in canonical loop quantum gravity Loop quantum gravity is formulated as a continuum theory, we therefore should clarify the need for a continuum limit in canonical loop quantum gravity. To this end we will shortly discuss how