What are the mechanical degrees of freedom of the Dirac field?

The study of the behavior of quantum fields at very high energies, possibly at the Planck scale, is an open problem today. Recent attempts by Hossain et al [1] explore a polymer quantized scalar field in which the canonical algebra of the tower of oscillators making up the field is replaced by the polymer one, inspired in loop quantum gravity [2]. A smoking gun of such a quantization appears in the form of deformed dispersion relations at very high energies and hence in the corresponding propagator. In this work we provide some steps towards the generalization of these results to a Dirac field. In particular we use a Fourier decomposition to look for the analogue of the oscillators of the scalar field. It turns out the corresponding energy spectrum can be intepreted as containing for each mode the contribution of four Fermi oscillators. [ABSTRACT FROM AUTHOR]