Julian Rennert , Applied Mathematics, University of Waterloo
Loop Quantum Gravity with a cosmological constant
Loop Quantum Gravity is a proposal for a non-perturbative and diffeomorphism invariant canonical quantization of general relativity which leads to remarkable conclusions about the structure of space and time at the Planck scale. Similarly to the discreteness obtained in standard quantum theory from quantization of classical systems the loop quantization of general relativity leads to a discrete structure of space itself. Taking into account that we observe in our universe a non-vanishing positive cosmological constant one certainly wants to implement it in our quantum theory of gravity and then ask how it affects the quantum structure of space and time.
The objective of my research is to understand how quantum groups can arise in Loop Quantum Gravity and how exactly they capture the presence of a cosmological constant. This entails already at the classical level many interesting mathematical questions concerning generalized (quasi-) Poisson manifolds and their subsequent quantization. In the quantum theory we aim to utilize so called tensor operators to construct observables and probe the quantum geometry of constantly curved `atoms of space'.