Invariant quadratic operators associated with linear canonical transformations and their eigenstates

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. In quantum physics, LCTs are the 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. 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. In this context, new quantum corrections to the ideal gas state equation are deduced from correction to the Hamiltonian operator of non-relativistic free quantum particles that is suggested by LCT-covariance. Two other LCT-invariant quadratic operators, which can be considered as the number operators of some quasiparticles, 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 relativistic 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.

Quantum and Relativistic corrections to Maxwell-Boltzmann ideal gas model from a Quantum Phase Space approach

“…This formulation has been introduced and developed firstly in the references [1,[33][34][35].This approach deals with a formulation of quantum mechanics but the phase space which is considered is a classical one. Another approach has been considered and developed through the works [22][23][24][25] in which the concept of "quantum phase space" is introduced. For a monodimensional case, this quantum phase space was defined as the set {(〈𝑥〉, 〈𝑝〉)} of the possible mean values 〈𝑥〉 and 〈𝑝〉 of the coordinate operator𝒙and momentum operator𝒑corresponding to quantum states denoted |〈𝑧〉⟩ which are themselves eigenstates of the operator 𝒛defined by the relation [22][23][24][25] 𝒛 = 𝒑 − 2𝑖 ℏ ℬ𝒙 (7) in which ℬ is the statistical variance of the momentum operator corresponding to the state |〈𝑧〉⟩ itself.…”

“…It was shown in [24] that the expression of the Hamiltonian operator 𝑯 𝑄 of a nonrelativistic free particle, for a monodimensional motion, compatible with the uncertainty principle and the concept of quantum phase space is…”

“…Using results from the reference [24], it may be deduced that the density operator 𝝆 of a nonrelativistic particle at equilibrium with a heat bath of temperature 𝑇 is given by the relation (quantum canonical distribution)…”

“…In this work, we consider the study on new kind of corrections for the Maxwell-Boltzmann ideal gas model which can be considered as related with the quantum nature of phase space. The approach uses results from the references [22][23][24][25].…”

“…Relativistic corrections are studied too. Our approach is based on the concepts of quantum phase space and phase space representation of quantum mechanics that were considered in the references [22][23][24][25]. These concepts are described briefly in the section 2.…”