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When twistors met loops

26 November 2019
Roger Penrose
Twistor and shout Roger Penrose’s talk was the highlight of the Twistors and Loops conference. Credit: F Viadotto

Loop Quantum Gravity and Twistor Theory have a lot in common. They both have quantum gravity as a main objective, they both discard conventional spacetime as the cornerstone of physics, and they have both taken major inspiration from renowned British mathematician Roger Penrose. Interaction between the two communities has been minimal so far, however, due to their distinct research styles: mathematically oriented in Twistor Theory, but focused on empirical support in Loop Gravity. This separation was addressed in the first week of September at a conference held at the Centre for Mathematical Researches (CIRM) at the Campus of Luminy in Marseille, where about a hundred researchers converged for lively debates designed to encouraged cross-fertilisation between the two research lines.

Both Twistor Theory and Loop Gravity regard conventional smooth general-relativistic spacetime as an approximate and emerging notion. Twistor theory was proposed by Roger Penrose as a general geometric framework for physics, with the long-term aim of unifying general relativity and quantum mechanics. The main idea of the theory is to work on the null rays, namely the space of the possible path that a light ray can follow in spacetime, instead of the manifold of the points of physical spacetime. Spacetime points, or events, are then seen as derived objects: they are given by compact holomorphic curves in a complex three-fold: twistor space. It is remarkable how much the main equations of fundamental physics simplify when formulated in these terms. The mathematics of twistors has roots in the 19th century Klein correspondence in projective geometry, and modern Twistor Theory has had a strong impact on pure mathematics, from differential geometry and representation theory to gauge theories and integrable systems.

Could allying twistors and loops be dangerous?

Loop gravity, on the other hand, is a background-independent theory of quantum gravity. That is, it does not treat spacetime as the background on which physics happens, but rather as a dynamical entity itself satisfying quantum theory. The conventional smooth general relativistic spacetime emerges in the classical (ℏ→0) limit, in the same manner as a smooth electromagnetic field satisfying the Maxwell equations emerges from the Fock space of the photons in the classical limit of quantum electrodynamics. Similarly, the full dynamics of classical general relativity is recovered from the quantum dynamics of Loop gravity in the suitable limit. The transitions amplitudes of the theory are finite in the ultraviolet and are expressed as multiple integrals over non compact groups. The theory provides a compelling picture of quantum spacetime. A basis in the Hilbert space of the theory is described by the mathematics of the spin networks: graphs with links labelled by SU(2) irreducible representations, independently introduced by Roger Penrose in the early 1970s in an attempt to a fully discrete combinatorial picture of quantum physical space. Current applications of Loop Gravity include early cosmology, where the possibility of a bounce replacing the Big Bang has been extensively studied using Loop gravity methods, and black holes, where the theory’s amplitudes can be used to study the non-perturbative transition at the end of the Hawking evaporation.

The communities working in Twistors and Loops share technical tools and conceptual pillars, but have evolved independently for many years, with different methods and different intermediate goals.  But recent developments discussed at the Marseille conference saw twistors appearing in formulations of the loop gravity amplitudes, confirming the fertility and the versatility of the twistor idea, and raising intriguing questions about possible deeper relations between the two theories.

The conference was a remarkable success. It is not easy to communicate across research programs in contemporary fundamental physics, because a good part of the field is stalled in communities blocked by conflicting assumptions, ingrained prejudices and seldom questioned judgments, making understanding one another difficult. The vibrant atmosphere of the Marseille conference cut through this.

The best moment came during Roger Penrose’s talk. Towards the end of a long and dense presentation of new ideas towards understanding the full space of the solutions of Einstein’s theory using twistors, Roger said rather dramatically that now he was going to present a new big idea that might lead to the twistor version of the full Einstein equations  – but at that precise moment the slide projector exploded in a cloud of smoke, with sparks flying. We all thought for a moment that a secret power of the Universe, worried about being unmasked, had interfered. Could allying twistors and loops be dangerous?

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