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A Categorification of the Boson Oscillator

In this paper, we investigate the categorical description of the boson oscillator. Based on the categories constructed by Khovanov, we introduce a categorification of the Fock states and the corresponding inner product of these states. We find that there are two different categorical definitions of the inner product of the Fock states. These two definitions are consistent with each other, and the decategorification results also coincide with those in conventional quantum mechanics. We also find that there are some interesting properties of the 2-morphisms which relate to the inner product of the states.