Nitica Sakharwade - Possible futures: How to construct quantum space-time with indefiniteness

The speaker begins by expressing gratitude to the organizers and introduces the topic of quantum gravity, which is likened to a formidable mountain to climb. The focus shifts to the field of indefinite causal structures, referencing a 2006 paper by Alison Hardy that discusses the integration of general relativity's deterministic yet non-fixed causal structure with quantum theory's probabilistic but fixed causal structure. The speaker aims to build a framework for probabilistic theories with non-fixed causal structures, using concepts such as operationalism, finite dimensions, and the Choi-Jamiołkowski isomorphism. The talk outlines the development of the 'Causaloid' framework and other theoretical advancements in quantum networks and process matrices.

The speaker proceeds to discuss the utility of toy models in understanding quantum foundations, citing the famous model by Robert B. Griffiths. The goal is to construct a toy model that reproduces quantum phenomena like superposition, entanglement, and teleportation within a classical local theory, while omitting contextuality and nonlocality. The model aims to provide insights into the interpretation of quantum theory and serve as a simplified playground for examining phenomena. The speaker introduces 'Mightdoor Theory,' a model designed to represent a locally definite but globally indefinite causal structure, emphasizing the use of local laboratories and joining rules.

The speaker elaborates on adding space-time elements to the toy model, setting it in 1+1 dimensions for simplicity. Each local laboratory is associated with a perception of flat space-time and communicates using photons, constrained by null lines representing future communication paths. The laboratories are depicted as triangles, with inputs and outputs, and an operational choice for the direction of photon transmission. The speaker introduces a coloring law as a local physical law that governs the interaction between neighboring triangles, ensuring stable causality and preventing closed timelike curves.

The speaker delves into the implications of the coloring law and operational choices within the toy model, highlighting the absence of closed timelike curves and the model's ability to represent relativistic phenomena through coarse-graining. The model allows for a Gaussian hypersurface analogue, where initial conditions and operational choices lead to an uncertain future causal structure. The speaker discusses the concept of a 'switch' in the literature, which involves superpositions of causal structures and the potential for local laboratories to perceive a definite causal order within a globally indefinite structure.

The speaker addresses the tension between local and global causal orders, emphasizing that the model allows for operational choices that enable a future of possibilities, governed by a local physical law through the coloring law. The model does not provide analogues for all dynamic causal structures found in relativity and is not fully operational, but rather presents a clash between local and global perspectives. The speaker wraps up by suggesting that the model could be generalized to higher dimensions if motivated, and invites questions about potential conflicts between local and global causal orders.

In the final paragraph, the speaker contemplates future directions for the model, including generalizing it to higher dimensions and addressing the challenges of geometry and complexity that arise. They discuss the possibility of using higher-dimensional tessellations and the difficulties faced in generalizing such models, referencing the work on Finman checkerboards. The speaker also touches on the need to define curvature for the indefinite matrix and concludes with an open discussion about the local-to-global conflict in the context of space-times with unusual causal structures.