Speaker
Description
In generally covariant theories, physical configurations are defined modulo diffeomorphisms, and the induced equivalence relation is highly singular. In particular, recent results show that complete sets of observables need not be Borel-definable, and their existence may fail within standard measurable frameworks. This obstructs any direct parametrisation of spacetime geometry in terms of well-behaved invariants.
We adopt an operational formulation in which spacetime is specified by a measurable space of events, a probability measure, and a causal accessibility relation encoded by a measurable kernel. Physical equivalence is restricted to measure-preserving transformations. In this setting, the relevant invariant is the probability law obtained by sampling the kernel. By the reconstruction results of Gromov and Vershik, such matrix distributions provide complete invariants of measurable structures of this type, up to measure-preserving isomorphism. They therefore separate equivalence classes at the level of observable data, modulo null sets.
This leads to a state space of operational spacetimes realised as a subset of probability measures on infinite matrices. Within appropriate regularity classes, this space admits a natural structure as an infinite-dimensional manifold; in particular, it can be modelled as a Fréchet manifold when described in terms of smooth densities or moment coordinates. This provides a well-defined differential framework compatible with its statistical interpretation.
We propose to quantise causal structure on this space. The differential structure allows the construction of a cotangent bundle and, under suitable conditions, a canonical symplectic form. This yields a phase space for operational causal configurations. One can then construct a representation into a projective Hilbert space, thereby associating quantum states to classical causal data. In this representation, amplitudes encode invariant statistical features of causal relations.
This framework defines quantum causal states directly from operationally accessible structures. It bypasses the non-Borel-definability of observables in the fully diffeomorphism-invariant setting, while retaining the essential causal and measure-theoretic content of spacetime.
References:
Panagiotopoulos, A., Sparling, G. A. J., & Christodoulou, M. (2023). Incompleteness Theorems for Observables in General Relativity. PRL 131, 171402.
Vershik, A. M. (2004). Random metric spaces and universality. Russian Mathematical Surveys 59(2), 259–295.
Kuchař, K. V. (1992). Time and interpretations of quantum gravity. In Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics.