Best,

Joe

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Abstract: It is argued that only relational quantities, such as the distance between two objects, are physically meaningful. Consequently, unobservable quantities, such as the global position of a particle, should be considered as artifacts of our mathematical description. In this work, we review current approaches, based on coherent and incoherent group averaging, aimed at identifying and describing relational quantities, and discuss some difficulties arising within these approaches. In particular, we study subsystems defined w.r.t. reference frames and show that unexpected phenomena occur once we consider "unsharp" reference frames instead of ideal ones. Furthermore, we introduce a rigorous definition of relational subsystems based on group averaged operator algebras. However, we prove several theorems which suggest that group averaged operators cannot be used in quantum field theory to define a relational subsystem structure that is compatible with the causal structure of spacetime. We argue that such difficulties arise because approaches involving group averaging can only capture a subset of all relational quantities contained within a theory. Subsequently, we propose a general framework, based on probability theory, that allows for a rigorous description of all relational quantities. We demonstrate that within this framework many difficulties encountered when using group averaging can be overcome.