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New Mathematics Bridges Holography and Twistor Theory



A diagram representing a twistor ̵
1; an extended entity in space and time that can be thought of as a two-dimensional beam of light. Credit: Penrose, R. & Rindler, W. (1986). Spins and Space-Time (Cambridge monographs on mathematical physics). Cambridge: Cambridge University Press.

The modern theoretical physicist faces a strenuous rise. "As we learn more, the reality becomes more subtle, the absolute becomes relative, the fix becomes dynamic, the definition is laden with uncertainty," writes physicist Yasha Neiman.

A professor and director of the Quantum Gravity Unit at the Okinawa Institute for Science and Technology Graduate University (OIST), he deals daily with this puzzle. Quantum gravity, Neiman's branch of physics, aims to unite quantum mechanics, which describes the scale of atoms and subatomic particles, with Einstein's theory of general relativity – the modern theory of gravitation as the curvature of space and time. How, he asks, can physicists write equations if the geometry of space itself is subject to quantum uncertainty? Quantum gravity, the current limit in fundamental theory, proved to be more difficult to alienate than earlier concepts, according to Neiman.

"With the notion of space slipping between our fingers, we are looking for alternative bases on which to base our description on the world," he writes.

This search for alternative stops is essentially a search for a new language to describe the reality – and it is the subject of his most recent work, which was published in the Journal of High Energy Physics . In the work, Neiman proposes a new perspective on the geometry of space and time based on established approaches in physics, such as holography and twistor theory.

Holography is an offshoot of string theory The theory that the universe consists of one-dimensional objects called strings was developed in the late 1990s. Holography imagines the ends of the universe as the surface of an infinitely large sphere that forms the boundary of the space. Even if geometry fluctuates within this sphere, this "limit at infinity" can remain fixed on the sphere surface.

Yasha Neiman deals daily with complex puzzles in quantum gravity. Credit: OIST

For the past 20 years, holography has been an invaluable tool for experiments in quantum gravity. However, astronomical observations have shown that this approach does not really apply to our world. "The accelerated expansion of our universe and the finite speed of light are constricting to limit all possible observations, present or future, to a finite – albeit very large – region of space," Neiman writes.

In such a world, the limit at infinity, where the holographic image of the universe is based, is no longer physically significant. A new frame of reference might be necessary – one that does not seek to find a solid surface in space, but leaves the space entirely behind.

In the 1960s, physicist Roger Penrose proposed such an attempt to understand quantum gravity as a radical alternative. In Penrose's Twistor Theory, geometric points are replaced by twisters that are most similar to the stretched, beamlike shapes. In this twistor room, Penrose discovered a highly efficient way to represent fields moving at the speed of light, such as electromagnetic and gravitational fields. However, the reality is more than fields – each theory must also consider the interactions between fields, such as the electric force between charges, or, in the more complicated case of general relativity, the gravitational attraction that results from the energy of matter field itself. The inclusion of the interactions of general relativity in this picture, however, has proven to be a daunting task.

So we can express a full quantum gravity theory in the Twistor language, perhaps simpler than the General Theory of Relativity, but with both fields and interactions fully accounted for? Yes, according to Neiman.

Neiman's model builds on higher spin gravity, a model developed by Mikhail Vasiliev in the 1980s and 1990s. Higher spin gravity can be thought of as the "smaller cousin" of string theory, "too simple to reproduce general relativity, but very instructive as a playground of ideas," as Neiman puts it. In particular, it is perfectly suited to explore possible bridges between holography and twistor theory.

On the one hand, as discovered by Igor Klebanov and Alexander Polyakov in 2001, higher spin gravitation, as well as string theory, can be described holographically. His behavior in space can be completely captured as a limit at infinity. On the other hand, his equations contain twistor-like variables, even though they are still bound to specific points in ordinary space.

Starting from these starting points, Neiman & # 39; s paper takes an extra step and constructs a mathematical dictionary that combines the languages ​​of holography and twistor theory.

"The underlying mathematics that sets this story apart are square roots," Neiman writes. "It's about identifying subtle ways in which a geometric operation such as rotation or reflection can be performed halfway A sly square root is like a crack in a solid wall that opens it in two and reveals a new world.

The use of square roots has a long history in mathematics and physics. In fact, the intrinsic shape of all matter particles – such as electrons and quarks – as well as rotators are described by a square root of ordinary directions in space. In a subtle technical sense, Nieman's method of connecting space, its limit at infinity, and torsion space, goes back to such a square root.

Neiman hopes his conceptual proof can pave the way to a quantum theory Gravity not based on a limit at infinity

"It takes a lot of creativity to uncover the code of the world," says Neiman. "And it's a pleasure to fiddle with it."


Further research:
Russian scientists find flaws in popular theories of gravity

Further information:
Yasha Neiman, The Holographic Dual of the Penrose Transformation, Journal of High Energy Physics (2018). DOI: 10.1007 / JHEP01 (2018) 100

Provided by:
Okinawa Institute of Science and Technology


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