  The main purpose of this work is to identify invariant quadratic operators
associated with Linear Canonical Transformations (LCTs) which could play
important roles in physics. LCTs are considered in many fields. In quantum
theory, they can be identified with linear transformations which keep invariant
the Canonical Commutation Relations (CCRs). In this work, LCTs corresponding to
a general pseudo-Euclidian space are considered and related to a phase space
representation of quantum theory. Explicit calculations are firstly performed
for the monodimensional case to identify the corresponding LCT-invariant
quadratic operators then multidimensional generalizations of the obtained
results are deduced. The eigenstates of these operators are also identified. A
first kind of LCT-invariant operator is a second order polynomial of the
coordinates and momenta operators and is a generalization of reduced momentum
dispersion operator. The coefficients of this polynomial depend on the mean
values and the statistical variances-covariances of the coordinates and momenta
operators themselves. It is shown that these statistical variances-covariances
can be related with thermodynamic variables. Two other LCT-invariant quadratic
operators, which can be considered as the number operators of some
quasipartciles, are also identified: the first one is a number operator of
bosonic type quasiparticles and the second one corresponds to fermionic type.
This fermionic LCT-invariant quadratic operator is directly related to a spin
representation of LCTs. It is shown explicitly, in the case of a
pentadimensional theory, that the eigenstates of this operator can be
considered as basic quantum states of elementary fermions. A classification of
the fundamental fermions, compatible with the Standard model of particle
physics, is established from a classification of these states.
