In this work we study a completely degenerate Fermi gas at zero temperature by a semiclassical approximation for a Hamiltonian that arises in polymer quantum mechanics. Polymer quantum systems are quantum mechanical models quantized in a similar way as in loop quantum gravity, allowing the study of the discreteness of space and other features of the loop quantization in a simplified way. We obtain the polymer modified thermodynamical properties for this system by noticing that the corresponding Fermi energy is exactly the same as if one directly polymerizes the momentum pF. We also obtain the expansion of the corresponding thermodynamical variables in terms of small values of the polymer length scale λ. We apply these results to study a simple model of a compact one-dimensional star where the gravitational collapse is supported by electron degeneracy pressure. As a consequence, polymer corrections to the mass of the object are found. By using bounds for the polymer length found in Bose–Einstein condensates experiments we compute the modification in the mass of the compact object due to polymer effects of order ~ 10-8. This result is similar to the other order found by different approaches such as generalized uncertainty principle (GUP), and that certainly is within the error reported in typical measurements of white dwarf masses.
Effective equation for quasi-one dimensional tube-shaped Bose–Einstein condensates
