## Revealing Light

**1 December 2015**

**Light is critical to our existence. It played a role in how organic molecules - and therefore living cells - developed on Earth in the first place, but now we use it for medicine, communications, entertainment and culture. Even something as everyday as supermarket checkouts scan product barcodes using light!**

2015 is the International Year of Light, so when I stumbled upon a recent publication from one of our pure mathematicians, Dr Oded Yacobi, from the School of Mathematics and Statistics here at the University of Sydney, titled 'Categorification and Heisenberg Doubles arising from Towers of Algebras', I just had to share how the two relate! He published this in January 2015 with his colleague Associate Professor Alistair Savage from the University of Ottawa, Canada.

We might ask: What is light? Physicists define visible light as electromagnetic radiation of a wavelength between 400 and 700 nanometres. So light is radiant energy. The kick to our reasoning is that to study this energy we have had to come to terms with the fact that it behaves both as a particle and a wave at the same time. A wave: as in a continuous flow of energy. A particle: as in discrete, or separate, packets of energy (called photons) explained through quantum mechanics. We need two completely separate explanations to understand how light works.

Einstein wrote: "It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do."

A key player in the understanding of the wave-particle duality was Werner Heisenberg, a German physicist who published a rather intriguing paper in 1927 on what he called the uncertainty principle. He explained the paradox of the duality as a consequence of the act of observing something, that he proved actually had an impact at quantum level. In other words, we were having trouble studying light at the quantum level, as the very act of trying to observe it was changing its behaviour. His discovery hinged on his unique insight in how to mathematically encode and deal with the symmetries he was seeing in the workings of photons.

You remember symmetries from primary school: reflections, rotations, translations and glides. A symmetry is something that remains invariant under a transformation. These were your first steps into quantum mechanics!

Werner Heisenberg's work was so groundbreaking it directly led to the fundamental mathematical objects to look for when seeking out symmetries across all of nature. We called these Heisenberg Algebras. But could there be further hidden symmetries? I mean how can you know? When do you give up the search?

An exciting moment came in the 1990s. This new mathematical field called Categorification began opening up ways to uncover physical symmetries that had been invisible to our older methods. It's already built many bridges between different parts of mathematics which were not seen as connected before. And our past experience shows that through finding symmetries we often get led to resolutions of many physical mysteries. Could there possibly be a better way to understand light that avoids any dualities? Can we resolve the dichotomy between quantum mechanics and general relativity? Dr Oded Yacobi and his collaborator are helping to build this very theory. Even looking for symmetries in the theory itself!

From light, to Heisenberg's search for its fundamentals, to the generalisation of Heisenberg's mathematics, to new complementary mathematics where Yacobi studied 'Categorification and Heisenberg Doubles arising from Towers of Algebras' - these are the wonders of the interconnected web of thinking that happens at the forefront of research. It might make our heads hurt, but it's fun to gain the perspective!

Full reference:

- Savage, A., Yacobi, O. (2015). Categorification and Heisenberg doubles arising from towers of algebras.
*Journal of Combinatorial Theory, Series A*, 129, 19-56.