Formalism Locality in Quantum Theory and Quantum Gravity

Standard physical theories, such as Quantum Theory and General Relativity, rely heavily on the assumption of a definite causal structure. In Quantum Theory, this is manifested in the fixed background time used for state evolution, while in General Relativity, the causal structure is determined by the metric. However, a fundamental theory of Quantum Gravity is expected to be both probabilistic and possess indefinite causal structure, where it may not be a matter of fact whether two events are space-like or time-like separated. This creates a conceptual crisis for traditional formulations that depend on evolving states across space-like hypersurfaces.

To address this, the author proposes the causaloid framework. This approach shifts the focus from time-evolving states to the correlation of recorded data. The framework is defined by its F-locality: for any arbitrary spacetime region, predictions are made using only mathematical objects pertaining to that specific region. By avoiding the need to refer to a global causal structure, the causaloid framework provides a neutral mathematical language capable of describing systems where causal relationships are not fixed.

Hardy demonstrates that Quantum Theory, specifically the interaction of pairwise qubits, can be reformulated within the causaloid framework. By using a 'causaloid product' and a two-step prediction process—first determining if a probability is well-defined and then calculating it—the framework unifies various quantum operations (like tensor products and sequential compositions) into a single, consistent structure. This shows that the causaloid framework is not merely a theoretical abstraction but a robust alternative to standard quantum formalisms.