Properties of Elementary Fermions of the Standard Model deduced from Linear Canonical Transformations

Ranaivoson, Ravo Tokiniaina; Andriambololona, Raoelina; Rakotoson, Hanitriarivo

doi:10.48550/arxiv.1806.07228

Cited by 2 publications

(17 citation statements)

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“…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].…”

Section: -Introductionsupporting

confidence: 61%

“…It can be verified that the operator ℶ 00 + in (13) are invariant under the action of a set of particular LCT called Isodispersion LCT in our previous work [17]. They are explicitly defined by the relation (36) below.…”

Section: -Introductionmentioning

confidence: 85%

“…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. Some particular case of the Isodispersion LCTs were identified as the Lorentz-like and Fractional Fourier-like transforms.…”

Section: -Introductionmentioning

confidence: 99%

“…Let us consider a multidimensional theory corresponding to a pseudo-euclidian space with dimension 𝑁 (𝜇, 𝜈 = 0, 1, … 𝑁 − 1). We have established in our previous works [9,15,17] that some equivalent expressions of a reduced dispersion-codispersion operator ℶ 𝜇𝜈 + are (we work in the natural unit with the reduced Planck's constant ℏ = 1 and speed of light 𝑐 = 1)…”

Section: -Introductionmentioning

confidence: 99%

“…Through the next section, we will identify the generalization of this operators which is invariant under the action of any LCT. We will also establish the main relations, analog of ( 14), ( 15), ( 16), (17) which are associated with this general operator and are covariant under the action of any LCT. The multidimensional generalization of the obtained results will also be studied.…”

Section: -Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Invariant quadratic operator associated with Linear Canonical Transformations

Ranaivoson¹,

Andriambololona²,

Hanitriarivo

2021

Preprint

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The main purpose of this work is to identify the general quadratic operator which is invariant under the action of Linear Canonical Transformations (LCTs). LCTs are known in signal processing and optics as the transformations which generalize certain useful integral transforms. In quantum theory, they can be identified as the linear transformations which keep invariant the canonical commutation relations characterizing the coordinates and momenta operators. In this paper, LCTs corresponding to a general pseudo-Euclidian space are considered. Explicit calculations are performed for the monodimensional case to identify the corresponding LCT invariant operator then multidimensional generalizations of the obtained results are deduced. It was noticed that the introduction of a variance-covariance matrix, of coordinate and momenta operators, and a pseudo-orthogonal representation of LCTs facilitate the identification of the invariant quadratic operator. According to the calculations carried out, the 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 these coordinates and momenta operators themselves. The eigenstates of the LCT invariant operator are also identified with it and some of the main possible applications of the obtained results are discussed.

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Section: -Introductionsupporting

confidence: 61%

Section: -Introductionmentioning

confidence: 85%

Section: -Introductionmentioning

confidence: 99%

Section: -Introductionmentioning

confidence: 99%

Section: -Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Invariant quadratic operator associated with Linear Canonical Transformations

Ranaivoson¹,

Andriambololona²,

Hanitriarivo

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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Linear Canonical Transformations in relativistic quantum physics

Ranaivoson¹,

Andriambololona²,

Rakotoson

et al. 2021

Phys. Scr.

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Linear Canonical Transformations (LCTs) are known in signal processing and optics as the generalization of certain useful integral transforms. In quantum theory, they can be identified as the linear transformations which keep invariant the canonical commutation relations characterizing the coordinates and momenta operators. In this work, the possibility of considering LCTs to be the elements of a symmetry group for relativistic quantum physics is studied using the principle of covariance. It is established that Lorentz transformations and multidimensional Fourier transforms are particular cases of LCTs and some of the main symmetry groups currently considered in relativistic theories can be obtained from the contractions of LCTs groups. It is also shown that a link can be established between a spinorial representation of LCTs and some properties of elementary fermions. This link leads to a classification which suggests the existence of sterile neutrinos and the possibility of describing a generation of fermions with a single field. Some possible applications of the obtained results are discussed. These results may, in particular, help in the establishment of a unified theory of fundamental interactions. Intuitively, LCTs correspond to linear combinations of energy-momentum and spacetime compatible with the principle of covariance.

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Properties of Elementary Fermions of the Standard Model deduced from Linear Canonical Transformations

Cited by 2 publications

References 0 publications

Invariant quadratic operator associated with Linear Canonical Transformations

Invariant quadratic operator associated with Linear Canonical Transformations

Linear Canonical Transformations in relativistic quantum physics

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