  Linear Canonical Transformations (LCTs) are known in signal processing and
optics as the generalization of certain useful integral transforms. In quantum
theory, they can be identified as the linear transformations which keep
invariant the canonical commutation relations characterizing the coordinates
and momenta operators. In this work, the possibility of considering LCTs to be
the elements of a symmetry group for relativistic quantum physics is studied
using the principle of covariance. It is established that Lorentz
transformations and multidimensional Fourier transforms are particular cases of
LCTs and some of the main symmetry groups currently considered in relativistic
theories can be obtained from the contractions of LCTs groups. It is also shown
that a link can be established between a spinorial representation of LCTs and
some properties of elementary fermions. This link leads to a classification
which suggests the existence of sterile neutrinos and the possibility of
describing a generation of fermions with a single field. Some possible
applications of the obtained results are discussed. These results may, in
particular, help in the establishment of a unified theory of fundamental
interactions. Intuitively, LCTs correspond to linear combinations of
energy-momentum and spacetime compatible with the principle of covariance.
