A Categorical Framework for the Quantum Harmonic Oscillator
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 homset 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 zeroparticle state. Finally, we examine all of these constructions in a suitable category of Hilbert spaces, and find that they reproduce the conventional mathematical structures.
 Publication:

International Journal of Theoretical Physics
 Pub Date:
 December 2008
 DOI:
 10.1007/s1077300897724
 arXiv:
 arXiv:0706.0711
 Bibcode:
 2008IJTP...47.3408V
 Keywords:

 Quantum;
 Category;
 Fock space;
 Canonical commutation relations;
 Harmonic oscillator;
 Quantum Physics;
 Mathematical Physics;
 Mathematics  Category Theory
 EPrint:
 44 pages, many figures