Hilbert space representation for quasi-Hermitian position-deformed Heisenberg algebra and Path integral formulation

Position deformation of a Heisenberg algebra and Hilbert space representation of both maximal length and minimal momentum uncertainties may lead to loss of Hermiticity of some operators that generate this algebra. Consequently, the Hamiltonian operator constructed from these operators are also not Hermitian. In the present paper, with an appropriate positive-definite Dyson map, we establish the Hermiticity of these operators by means of a quasi-similarity transformation. We then construct Hilbert space representations associated with these quasi-Hermitian operators that generate a quasi-Hermitian Heisenberg algebra. With the help of these representations we establish the path integral formulation of any systems in this quasi-Hermitian algebra. Finally, using the path integral of a free particle as an example, we demonstrate that the Euclidean propagator, action, and kinetic energy of this system are constrained by the standard classical mechanics limits.