A Categorical Framework for the Quantum Harmonic Oscillator
- 82 Downloads
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.
KeywordsQuantum Category Fock space Canonical commutation relations Harmonic oscillator
Unable to display preview. Download preview PDF.
- 2.Blute, R., Panangaden, P., Seely, R.A.G.: Fock Space: A Model of Linear Exponential Types (1994) Google Scholar
- 7.Goldstern, M.: Completion of Semirings (1985) Google Scholar
- 8.Isham, C.: Some reflections on the status of conventional quantum theory when applied to quantum gravity. Presented at Stephen Hawking’s 60th Birthday Symposium, University of Cambridge (2002) Google Scholar
- 10.Mac Lane, S.: Categories for the Working Mathematician. Springer, Berlin (1997) Google Scholar
- 11.Melliès, P.-A.: Categorical Semantics of Linear Logic: a Survey (2008) Google Scholar
- 12.Penrose, R.: Applications of negative-dimensional tensors. In: Welsh, D.J.A. (ed.) Combinatorial Mathematics and Its Applications. Academic Press, New York (1971) Google Scholar
- 13.Selinger, P.: Idempotents in dagger categories. In: Proceedings of the 4th International Workshop on Quantum Programming Languages, July 2006 Google Scholar