NON-ABELIAN OPERATIONS

AN OPERATION IS COMMUTATIVE (OR ABELIAN) if the order of its application is irrelevant to the result: A • B = B • A. It is non-commutative (or non-abelian) if the result depends on the order of application.

By the way, to many, mathematician Niels Henrik Abel is familiarized by lower-case abelian just as Sir Isaac Newton is with newtonian. 

Niels Henrik Abel, 1802-1829, Norwegian mathematician, died at 26 of tuberculosis, leaving “mathematicians enough to keep them busy for five hundred years,” a contemporary said.

Examples. In arithmetic, addition is abelian: A + B = B + A. So is multiplication: A x B = B x A. But neither subtraction nor division is abelian: 6-2= 4, but 2-6 =-4. And 6÷2 =3, whereas 2÷6=2/6=1/3.

Also, quaternions have non-abelian multiplication.

Page turning is abelian: Flipping forward two pages then three pages is the same as flipping three pages then two. 

All this is preface to “Non-Abelian Physics in Light and Sound,” by Yi Yang et al. in AAAS Science, February 23, 2024. 

Controlling Light and Sound. “Research in non-Abelian phenomena,” Science reports (choosing upper-case Abelian), “has recently undergone rapid developments in photonics and acoustics, introducing a new perspective on controlling light and sound at the fundamental level.” 

By the way, the article offers another non-abelian operation: book rotation: “If you rotate it 90° sequentially along two perpendicular axes, you will find that switching the orders of the two rotations will result in different final orientations of the book. This is known as the non-commutativity of the three-dimensional rotation of a classical rigid body. In modern physics, there are many examples in a similar vein.”

Science offers an hourglass example of abelian versus non-abelian operation. At left, different entry order into the single apex has no effect on flow of sphere-like sand. At right, three apices give a braiding pathway of brick-like sands.

Abstract. “Non-Abelian phenomena arise,” the researchers note in their Abstract, “when the sequence of operations on physical systems influences their behaviors. By possessing internal degrees of freedom such as polarization, light and sound can be subjected to various manipulations, including constituent materials, structured environments, and tailored source conditions…. Recent developments have constituted a versatile testbed for exploring non-Abelian physics at the intersection of atomic, molecular, and optical physics; condensed matter physics; and mathematical physics.” 

Non-Abelian Physics. “In this review,” the researchers write, “we discuss the theoretical foundations and experimental advances of non-Abelian physics in light and sound, epitomized by non-Abelian topological charges, non-Abelian gaugefields, non-Abelian mode dynamics, and non-Hermitian non-Abelian phenomena…. In these cases, the underlying non-Abelian topological invariants are no longer integers but are represented by matrices like quaternions.” 

The added dimensionality of quaternions (of the form a+bi+cj+dk) gives a benefit in mathematical modeling of “the rich number of available control knobs….” As cited in Wikipedia, “Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis.”

The researchers cite, “Recent examples include the braiding, pumping, and Bloch oscillations of light and sound, ranging from microwave waveguides to integrated chips…. To this end, opportunities are emerging in photonics and acoustics because of the rich numbers of available control knobs, such as duality, polarization, angular momentum, Bloch bands, particle numbers, symmetric-protected subspace, and gauge field–induced degeneracy.” 

Image by W. Symons from Wikipedia. 

Heady stuff, this. Yet it’s heartening when seemingly theoretical mathematics has practical applications. Like a composer finding his sea chanty benefits the work load under sail. ds

© Dennis Simanaitis, SimanaitisSays.com, 2024