A categorical framework for the quantum harmonic oscillator

TL;DR

This paper introduces a categorical framework for the quantum harmonic oscillator that generalizes traditional structures without relying on Hilbert spaces, revealing new insights into coherent states and exponential operators.

Contribution

It develops a general categorical formulation of the quantum harmonic oscillator, unifying raising/lowering operators and coherent states without Hilbert space dependence.

Findings

Reproduces conventional quantum harmonic oscillator structures in Hilbert space categories.

Shows coherent states as exponentials of raising morphisms on the zero-particle state.

Uncovers natural emergence of generalized exponentials in the categorical setting.

Abstract

This paper describes how the structure of the state space of the quantum harmonic oscillator can be described by an adjunction of categories, that encodes the raising and lowering operators into a commutative comonoid. The formulation is an entirely general one in which Hilbert spaces play no special role. Generalised coherent states arise through the hom-set isomorphisms defining 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 exponential 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.