  As the loop quantum gravity is based on polymer quantization, we will argue
that the polymer length (like string length) can be several orders larger than
the Planck length, and this can have low energy consequences. We will
demonstrate that a short distance modification of a quantum system by polymer
quantization and by string theoretical considerations can produce similar
behavior. Moreover, it will be demonstrated that a family of different deformed
Heisenberg algebras can produce similar low energy effects. We will analyze
such polymer correction to a degenerate Fermi gases in a harmonic trap, and its
polymer corrected thermodynamics.
