Invariant quadratic operators associated with Linear Canonical Transformations and their eigenstates

TL;DR

This paper identifies invariant quadratic operators linked to Linear Canonical Transformations in quantum physics, explores their eigenstates, and relates them to fundamental fermions and the Standard Model.

Contribution

It introduces new LCT-invariant quadratic operators, including generalizations of momentum dispersion and fermionic number operators, with explicit eigenstate characterizations and physical interpretations.

Findings

Identified LCT-invariant quadratic operators in phase space.

Connected statistical variances with thermodynamic variables.

Classified fundamental fermions via eigenstates of fermionic operators.

Abstract

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.