Causality and the arrow of time in process theories

The objective of this thesis is to study the concept of causality and its interplay with the arrow of time in quantum process theories and beyond. The framework of process theories is particularly intuitive since it is accompanied by a diagrammatic formalism. Its formal underpinnings lie in Categorical Quantum Mechanics, a field which has recently seen enormous success both in the foundations of physics and in quantum information theory. Taking a process-theoretic viewpoint we examine possible ways to time-symmetrize quantum theory. We identify the causality condition as the root of the asymmetry and we proceed by eliminating it in a variety of ways. We furthermore view that process theories can be faithfully captured by operad algebras, which is an alternative to their identification with symmetric monoidal categories. The landscape of operad algebras allows us to introduce novel notions of process theories, namely time neutral process theories. To provide more intuition about them, we establish a link with compact closed categories. Finally, we present an algebraic framework to describe the evolution of fields, perceived as causal process theories, in discretised spacetimes. We form connections with algebraic quantum field theory (AQFT) as well as quantum cellular automata (QCA).