A two-way mathematical ‘dictionary’ could connect quantum physics to number theory

Several areas of mathematics have developed in complete isolation, using their own “undecipherable” coded languages. In a new study published in Proceedings of the National Academy of SciencesTamás Hausel, professor of mathematics at the Austrian Institute of Science and Technology (ISTA), presents “great algebras”, a two-way mathematical “dictionary” between symmetry, algebra and geometry, which could strengthen the connection between the distant worlds of quantum physics and number theory.

Mathematics, the most exact of scientific disciplines, can be considered the ultimate quest for absolute truth. However, mathematical paths to truth often require overcoming enormous obstacles, such as conquering unimaginable mountain peaks or building giant bridges between isolated continents.

The world of mathematics is full of mysteries and several mathematical disciplines have developed in a tortuous manner, in total isolation from each other. Thus, establishing an irrefutable truth about complex phenomena of the physical world calls upon intuition and a good dose of abstraction.

Even fundamental aspects of physics push mathematics to new heights of complexity. This is especially true for symmetries, through which physicists have theorized and discovered a whole zoo of subatomic particles that make up our universe.

In an exceptionally ambitious project, Hausel, a professor at the Austrian Institute of Science and Technology (ISTA), has not only conjectured but also proven a new mathematical tool called “large algebras.”

This new theorem is comparable to a “dictionary” that deciphers the most abstract aspects of mathematical symmetry using algebraic geometry. By operating at the intersection of symmetry, abstract algebra, and geometry, large algebras use more tangible geometric information to summarize sophisticated mathematical information about symmetries.

“Thanks to large algebras, information from the ‘tip of the mathematical iceberg’ can give us unprecedented insights into the hidden depths of the mysterious world of symmetry groups,” Hausel says.

With this mathematical breakthrough, Hausel seeks to consolidate the link between two distant fields of mathematics.

“Imagine, on one side, a world of mathematical representations of quantum physics, and on the other, far, far away, the purely mathematical world of number theory. With the present work, I hope to have taken another step toward establishing a stable connection between these two worlds.”

Don’t lose the translation anymore

The 17th-century philosopher and mathematician René Descartes showed us that we could understand the geometry of objects using algebraic equations. He was thus the first to “translate” mathematical information between these previously separate domains.

“I like to think of the relationships between different mathematical domains as dictionaries that translate information between often non-mutually intelligible mathematical languages,” Hausel says.

So far, several such mathematical “dictionaries” have been developed, but some translate information in only one direction, leaving the information on the way back completely encrypted. Moreover, the term “algebra” today encompasses both classical algebra, as in Descartes’ time, and abstract algebra, that is, the study of mathematical structures that cannot necessarily be expressed by numerical values. This adds an additional level of complexity. Hausel now uses abstract algebra and algebraic geometry as a two-way “dictionary.”

A skeleton and nerves

In mathematics, symmetry is defined as a form of “invariance”. The group of transformations that keep a mathematical object unchanged is called a “symmetry group”. These are classified as “continuous” (e.g., rotating a circle or sphere) or “discrete” (e.g., mirroring an object). Continuous symmetry groups are represented mathematically by matrices, rectangular arrays of numbers.

From a matrix representation of a continuous symmetry group, Hausel can calculate the large algebra and represent its essential properties geometrically by drawing its “skeleton” and “nerves” on a mathematical surface.

The skeleton and nerves of the big algebra give rise to interesting, 3D-printable shapes that recapitulate sophisticated aspects of the original mathematical information, thus closing the circle of translation.

“I am particularly excited about this work because it offers us a completely new approach to the study of representations of continuous symmetry groups. With large algebras, mathematical “translation” works not only in one direction, but in both.”

Connecting Isolated Continents in a Vast World of Mathematics

How might large algebras strengthen the connection between quantum physics and number theory, two fields of mathematics that are seemingly light years apart? First, the mathematics underlying quantum physics relies heavily on matrices, which are rectangular arrays of numbers.

However, these matrices are usually “noncommutative,” meaning that multiplying the first matrix by the second does not give the same result as multiplying the second by the first. This poses a problem in algebra and algebraic geometry, because noncommutative algebra is not yet well understood.

Large algebras now solve this problem: once computed, a large algebra is a commutative “mathematical translation” of a noncommutative matrix algebra. Thus, the information initially contained in noncommutative matrices can be decoded and geometrically represented to reveal their hidden properties.

Second, Hausel shows that large algebras reveal not only the relationships between related symmetry groups, but also when their “Langlands duals” are related. These duals are a central concept in the purely mathematical world of number theory. In the Langlands program, a very complex and large-scale dictionary that seeks to connect isolated mathematical “continents,” Langlands duality is a concept or tool that allows mathematical information to be “mapped” between different categories.

“In my work, large algebras appear to relate different symmetry groups precisely when their Langlands duals are related, a rather surprising result with possible applications in number theory,” Hausel says.

“Ideally, large algebras would allow me to connect Langlands duality in number theory to quantum physics,” Hausel says.

So far, he has been able to demonstrate that large algebras solve problems on both continents. The fog has begun to clear, and the continents of quantum physics and number theory have glimpsed their respective mountains and shores on the horizon. Soon, rather than connecting the continents only by boat, a bridge of large algebras could make it easier to cross the mathematical strait that separates them.