In his talk “Stranger Things (In Math)”, Professor Ivan Contreras sets out to explore the ‘Nether’ of mathematics: Non-commutative Algebra.
In our daily lives, many of our basic operations are commutative, which means the order of the components does not affect the result. For example, 2 + 3 is the same as 3 + 2, and scissors wins no matter if it’s paper vs. scissors or scissors vs. paper.
Now imagine a world of operations where order does matter, so 2 + 3 does not equal 3 + 2. What could we say about such a (strange) world? In our commutative world, if we want to solve x^2 = 0 there is only one solution, and x^2 = 1 has two solutions. In a non-commutative world x^2 = 0 and x^2 = 1 can have infinitely many solutions! Similarly, many other basic facts we take for granted go out the window.
In this world, Professor Contreras introduces us to a host of non-commutative “strange things”. One such thing is an operator known as the commutator, which measures how ‘non-commutative’ another operator is. More precisely, the commutator [,] is defined as [a, b] = a * b – b * a, where * is the given operator. What could this possibly be used for?
Rather surprisingly, the famous Heisenberg’s Uncertainty Principle in quantum mechanics is explained by the commutator. This principle states that the position and momentum of an object cannot be measured with absolute certainty; in other words, the more precise one measurement is, the less precise the other measurement will be. We can express this as [x̂, p̂], where x̂ and p̂ are the position and momentum vectors respectively. This non-commutativity comes from the fact that if we measure the position after the momentum, we would get a different result than if we measured the momentum after the position! Spooky, but also very real.
In fact, our daily lives are surrounded by events in which the order matters, and these non-commutative “strange things” provides us with tools to understand them.
Math, like many other fields in STEM, contains such unexplored worlds, that might look spooky or strange from a distance. But through the exploration process (which includes defining new mathematical objects and studying their properties and applications), we realize that such worlds are not that scary, and indeed offer important insights about our world.