International Journal of Theoretical Physics

, Volume 47, Issue 12, pp 3408–3447 | Cite as

A Categorical Framework for the Quantum Harmonic Oscillator

  • Jamie VicaryEmail author


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. 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.


Quantum Category Fock space Canonical commutation relations Harmonic oscillator 


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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Imperial College LondonLondonUK

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