Duality is one of the most – if not the most – fundamental principles in our universe. It’s a principle that, once in your gut, transforms the way you approach equations and algebraic concepts; putting a perimeter around each problem rather than letting the edges disappear into the obscure distance. However, unless you’ve taken specific […]

There are many hard things to grasp in quantum mechanics: the algebra of commutators Hamiltonians & Lagrangians Hilbert-Space Sturm-Liouville theory the inexplicable profusion of complex numbers, etc. One of these things is the ‘quantum mode’. We’re told that such modes live in an infinite-dimensional space, and that in order to identify them analytically we need […]