What happens when you swap the order of two matrices A and B? The difference, AB − BA, is called a commutator. It measures how much two operators fail to commute.
In finite dimensions, the rule is simple: a matrix is a commutator if and only if its trace is zero. But when we move to operators on infinite-dimensional space, this rule breaks down. The trace is no longer defined for all operators, and finite-dimensional criteria cannot be extended.
This talk explores how mathematicians had to reinvent the theory of commutators as they moved from matrices to operators on infinite-dimensional Hilbert spaces.
