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The main purpose of this work is to identify invariant quadratic operatorsassociated with Linear Canonical Transformations (LCTs) which could playimportant roles in physics. LCTs are considered in many fields. In quantumtheory, they can be identified with linear transformations which keep invariantthe Canonical Commutation Relations (CCRs). In this work, LCTs corresponding toa general pseudo-Euclidian space are considered and related to a phase spacerepresentation of quantum theory. Explicit calculations are firstly performedfor the monodimensional case to identify the corresponding LCT-invariantquadratic operators then multidimensional generalizations of the obtainedresults are deduced. The eigenstates of these operators are also identified. Afirst kind of LCT-invariant operator is a second order polynomial of thecoordinates and momenta operators and is a generalization of reduced momentumdispersion operator. The coefficients of this polynomial depend on the meanvalues and the statistical variances-covariances of the coordinates and momentaoperators themselves. It is shown that these statistical variances-covariancescan be related with thermodynamic variables. Two other LCT-invariant quadraticoperators, which can be considered as the number operators of somequasipartciles, are also identified: the first one is a number operator ofbosonic type quasiparticles and the second one corresponds to fermionic type.This fermionic LCT-invariant quadratic operator is directly related to a spinrepresentation of LCTs. It is shown explicitly, in the case of apentadimensional theory, that the eigenstates of this operator can beconsidered as basic quantum states of elementary fermions. A classification ofthe fundamental fermions, compatible with the Standard model of particlephysics, is established from a classification of these states.

Invariant quadratic operators associated with Linear Canonical Transformations and their eigenstates