Saturday, March 28, 2009

Majorana Representations and Platonic Solids

A few years ago, I was asked to contribute a paper to a book celebrating the centenary of the birth of Ettore Majorana (1906-1938), a gifted Italian theoretical physicist who died during World War II in mysterious circumstances. I got this call because in my graduate work under Sir Roger Penrose (see here for example), I had used Majorana's representation of quantum spin states to investigate the conceptual foundations of quantum theory. (Roger had effectively rediscovered the Majorana representation just prior to my arrival at Oxford.)

I agreed to contribute a paper to Majorana centenary, but I didn't have any suitable projects in the pipeline. So, while driving down to New York one night along the Taconic - the setting for more than one productive daydream - I started thinking about spin states. Reflecting on a particular variety of spin states called "coherent" spin states, I wondered "how far from coherent" a state could get. It immediately occurred to me to look at (inverse) Majorana representations of Platonic solids. These states would, in a sense, be the opposite of the familiar coherent spin states of quantum theory, and so I gave them the name "anticoherent" spin states.

Following this chain of reasoning leads to some geometric curiosities, including a basis for complex five-dimensional space consisting of five states whose Majorana representations form a dodecahedron's worth of interlocking tetrahedra! The resulting paper is available online here (see especially the figures at the end). It was my honor to receive the 2006 Majorana Prize for it, as part of the centennial celebration of Majorana's legacy in contemporary theoretical physics.

These anticoherent states were such beautiful little objects that I felt sure they must have some physical importance. After all, coherent states are often said to be "as classical as possible," so perhaps the anticoherent states would be useful for exhibiting exotic quantum phenomena such as one encounters in quantum information theory. In the paper, I alluded to a couple of potential physical implications, but I'm not an expert on quantum information theory so I couldn't say much. But I was happy to see recently that some physicists who do know a thing or two have found some interesting physical properties of the anticoherent states! See Kolenderski and Demkowicz-Dobrzanski, "Optimal state for keeping reference frames aligned and the platonic solids," Phys. Rev. A 78, 052333 (2008), available here (subscription required).