Incompleteness Theorems for Observables in General Relativity

Abstract: One of the biggest open problems in mathematical physics has been the problem of formulating a complete and consistent theory of quantum gravity. Some of the core technical and epistemological difficulties come from the fact that General Relativity (GR) is, fundamentally, a geometric theory and, as such, it oughts to be `generally covariant', i.e. invariant under change of coordinates by the arbitrary diffeomorphism of the ambient manifold. The Problem of Observables is a famous instance of the difficulties that general covariance brings into quantization: no non-trivial diffeomorphism-invariant quantity has ever been reported on the collection of all spacetimes.

It turns out that there is a good reason for this! In a recent joint work with Marios Christodoulou and George Sparling we employ methods from Descriptive Set Theory (DST) in order to show that no complete observables for full GR can be Borel definable. In other words, the problem of observables is to `analysis' what the Delian problem was to `straightedge and compass'. This talk will provide a gentle introduction to the main concepts of DST, where they come from, and how they connect to the problem of observables. We will then discuss the aforementioned Incompleteness Theorem, as well as the next steps in a research program that aims to further uncover this new connection between theoretical physics and DST.

general relativity and quantum cosmologymathematical physicslogicquantum physics

Audience: advanced learners

( paper | video )

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Tricontinental Quantum Fundamentals Seminar

Series comments: The Tricontinental Quantum Fundamentals Seminar (3C-QFS) is an online seminar series connecting researchers from Australia, Europe, and America. Our goal is to facilitate discussion between the three continents and bring together those working on foundational topics in relativistic quantum physics, covering but not limited to:

• Measurement, localisation, and causality in relativistic quantum theory
• Energy conditions and quantum inequalities
• Non-locality and observer-dependence of entanglement
• Quantum reference frames and symmetries in quantum theory
• Holography, AdS/CFT, and analogue gravity

For 3C-QFS, we wish to foster an inclusive community and promote open discussion by encouraging a casual atmosphere during the seminars.

The seminars are held online via Zoom, consisting of 40-60 minute talks followed by additional time for questions. We meet every three weeks on Tuesday, alternating between two times in order to connect research communities across two of the three continents.

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The seminar series is primarily for students and academics active in the foundations of physics. To sign up for the mailing list and join the seminars live, please complete the registration form on the 3C-QFS website.

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