The structure of the quantum harmonic oscillator by an adjunction of categories

A categorical framework for the quantum harmonic oscillator

We describe how the structure of the state space of the quantumharmonic oscillator can be described by an adjunction of categories, that encodes the raising and lowering operators into a commutative comonoid.Generalised coherent states arise through the hom-set isomorphismsdefining the adjunction, and we prove that they are eigenstates of the lowering operators.Surprisingly, generalised exponentials also emerge naturally in this setting, and we demonstrate that coherent states are produced by the expansiononential of a raising morphism acting on the zero-particle state.Finally, we examine all of these constructions in a suitable category of hilbert spaces,and find that they reproduce the conventional mathematical structures.