A study of quantum sequential growth dynamics, observables in causal sets and renormalization.

Abstract

Causal Set theory is a discrete framework for Quantum Gravity that assumes that space-time is discrete and causal structure is the property of space-time that needs to be quantized. The causal structure is a partial order and thus, causal sets are locally finite posets. In this thesis, we study the fundamental dynamics or causal sets known as Sequential Growth Dynamics (SGD) where we start with one element and keep on growing causal sets by adding one element at a time and assigning transition probability to these transitions. We study the quantum SGD where rather than assigning probability to transition, we assign vectors lying in some Hilbert space to causal sets and linear operators over that Hilbert space to transition amplitudes. We show that under the generalized assumptions of SGD the algebra of transition amplitudes is commutative if the transition amplitudes are invertible. We also study the observables in SGD by which we mean the set of questions we can ask, like probability of occurrence of certain causal structure. We construct measures of certain observables. Towards the end, we study the dynamics in case where the universe collapses and re-forms again and again in the SGD. We find the limit points of dynamics for more general type of collapse than those studied before.