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Casimir effect in polymer scalar field theory

In this paper, we study the Casimir effect in the classical geometry of two parallel conducting plates, separated by a distance $L$, due to the presence of a minimal length $λ$ arising from a background independent (polymer) quantization scheme. To this end, we polymer-quantize the classical Klein-Gordon Hamiltonian for a massive scalar field confined between the plates and obtain the energy spectrum. The minimal length scale of the theory introduces a natural cutoff for the momenta in the plane parallel to the plates and a maximum number of discrete modes between the plates. The zero-point energy is calculated by summing over the modes, and by assuming $λ\ll L$, we expressed it as an expansion in powers of $1/N$, being $N=L/ λ$ the number of points between the plates. Closed analytical expressions are obtained for the Casimir energy in the cases of small and large scalar mass limits.