Properties of Phase Space Wavefunctions and Eigenvalue Equation of Momentum Dispersion Operator

Invariant quadratic operator associated with Linear Canonical Transformations

“…Our main purpose in this work is to identify the general operator which is invariant under the action of any LCT. The results that are established generalize and bring more clarification to the formulation that we developed in our previous works [9][10][11][12][13][14][15][16][17][18].…”

“…Through our previous works [9][10][11][12][13][14][15][16][17][18], we have implicitly use the eigenstates of the operators associated with the particular dispersion operator (1) to construct what we have called a phase space representation of quantum theory. Taking into account the results that we have established through this paper, it is clear that we have to replace these states by their generalization (84) which are the eigenstates of the general LCT -invariant operator (82).…”

“…In our previous works concerning a phase space representation of quantum theory and Linear Canonical Transformations [9][10][11][12][13][14][15][16][17][18], we have introduced operators called reduced dispersioncodispersion operators, denoted ℶ 𝜇𝜈 + , and the corresponding eigenstates to define the called phase space representation. It was shown in [17] that the operator ℶ + = 𝜂 𝜇𝜈 ℶ 𝜇𝜈 + , the 𝜂 𝜇𝜈 being the contravariant components of the metric tensor associated with the considered pseudoeuclidian space, is invariant under the action of some particular case of LCTs called Isodispersion LCTs.…”

“…{ 𝒜 00 ℬ 00 = 1 4 𝒶 0 0 𝒷 0 0 = 1 2 (𝒶 0 0 ) 2 = 𝒜 00 (𝒷 0 0 ) 2 = ℬ 00 (14) 𝒛 0 = (𝒑 𝟎 + 2𝑖ℬ 00 𝒙 𝟎 ) (…”

“…The multidimensional generalization of the obtained results will also be studied. Like in our previous work [9][10][11][12][13][14][15][16][17][18], we use a system of notation based on the approach considered in [20]. We use also boldface for the operators to distinguish them.…”