Abstract
A hallmark of non-perturbative theories of quantum gravity is the absence of a fixed background geometry, and therefore the absence in a Planckian regime of any notion of length or scale that is defined a priori. This has potentially far-reaching consequences for the application of renormalization group methods a la Wilson, which rely on these notions in a crucial way. We review the status quo of attempts in the Causal Dynamical Triangulations (CDT) approach to quantum gravity to find an ultraviolet fixed point associated with the second-order phase transitions observed in the lattice theory. Measurements of the only invariant correlator currently accessible, that of the total spatial three-volume, has not produced any evidence of such a fixed point. A possible explanation for this result is our incomplete and perhaps naive understanding of what constitutes an appropriate notion of (quantum) length near the Planck scale.
Highlights
The Wilsonian concept of renormalization has been of immense importance for our understanding of quantum field theory and its relation to critical phenomena in statistical mechanics and condensed matter physics
The lattice field theories address the question of existence of certain quantum field theories using the Wilsonian picture: if the continuum quantum field theory exists as a limit of the lattice field theory when the cut-off is removed, there exists a UV fixed point of the renormalization group
It is identical to a (Euclidean) de Sitter universe volume profile and the configurations created by the Monte Carlo (MC) simulations can be viewed as quantum geometries that fluctuate around this background
Summary
The Wilsonian concept of renormalization has been of immense importance for our understanding of quantum field theory and its relation to critical phenomena in statistical mechanics and condensed matter physics. It allows one to address in a non-perturbative way the question of whether or not a given quantum field theory exists, the simplest example being a φ4-theory in four dimensions This is a perturbatively renormalizable quantum field theory, so one can fix the physical mass and the physical coupling constant of the theory, and to any finite order in the coupling constant calculate the correlation functions. The lattice field theories address the question of existence of certain quantum field theories using the Wilsonian picture: if the continuum quantum field theory exists as a limit of the lattice field theory when the cut-off is removed (the lattice spacing goes to zero), there exists a UV fixed point of the renormalization group One can approach such a fixed point in the following way: choose observables which define the physical coupling constants of the theory and measure them for a certain choice of the bare coupling constants used to define the lattice theory.
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