Minimal length, maximal momentum, and Hilbert space representation of quantum mechanics

Kempf et al. in Ref. [A. Kempf, G. Mangano, and R. B. Mann, Phys. Rev. D 52, 1108 (1995).] have formulated a Hilbert space representation of quantum mechanics with a minimal measurable length. Recently it has been revealed, in the context of doubly special relativity, that a test particles’ momentum cannot be arbitrarily imprecise and therefore there is an upper bound for momentum fluctuations. Taking this achievement into account, we generalize the seminal work of Kempf et al. to the case that there is also a maximal particles’ momentum. Existence of an upper bound for the test particles’ momentum provides several novel and interesting features, some of which are studied in this paper.

Figure 1

Variation of $⟨{\phi }_{{\zeta }^{\prime }}|{\phi }_{\zeta }⟩$ versus $\zeta -{\zeta }^{\prime }$.

Figure 2

Variation of $⟨{\phi }_{{\epsilon }^{\prime }}^{\mathrm{ml}}|{\phi }_{\epsilon }^{\mathrm{ml}}⟩$ versus $\epsilon -{\epsilon }^{\prime }$.