| Preface |
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xi | |
| Acknowledgments |
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xiii | |
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1 | (4) |
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1.1 The Concept of a Spinor Structure of Physical Space |
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1 | (1) |
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1.2 What should We Expect from the Spinor Structure of the Physical Space |
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2 | (3) |
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2 On the Geometry of Spaces and Spinor Structure |
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5 | (30) |
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2.1 Cartan's Classification for 2-Spinors |
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5 | (2) |
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2.2 Pseudovector Space Π3 and Spatial Spinor ξ |
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7 | (2) |
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2.3 The Vector Space E3 and the Spinor Model E3 |
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9 | (4) |
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2.4 Spatial Spinor ξa3 (a1 + ia2) and the Cauchy-Riemann Condition |
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13 | (2) |
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2.5 Calculating δξ and δnξ |
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15 | (1) |
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2.6 Features of the Spinor Field ξ |
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16 | (3) |
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2.7 Spinor η(b3, b1 + ib2) and Cauchy-Riemann non-Analyticity |
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19 | (2) |
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2.8 Continuity Properties of the Spinor η |
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21 | (1) |
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2.9 Comparing the Models ξ and η |
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22 | (1) |
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2.10 Spatial Spinors in Cylindric Parabolic Coordinates |
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23 | (6) |
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2.11 The Spinors ξ and η in Parabolic Coordinates |
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29 | (2) |
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2.12 Spatial Spinors in Spherical Coordinates |
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31 | (4) |
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3 Spinor Structure. Kustaanheimo--Stiefel and Hopf Bundles |
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35 | (8) |
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35 | (1) |
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3.2 The Spinor Extension of a Pseudo Vector Space Model |
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36 | (1) |
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3.3 The Spinor Extension of a Vector Space Model |
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37 | (2) |
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3.4 Parabolic Coordinates and Spatial Spinors |
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39 | (1) |
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3.5 The Connection between the Variables Ua and Va |
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40 | (3) |
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4 The Spin Covering for the Full Lorentz Group and the Concept of Fermion Parity |
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43 | (10) |
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4.1 Lorentz Group Theory: History of the Subject |
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43 | (3) |
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4.2 The Spinor Covering of the Total Lorentz Group |
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46 | (2) |
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4.3 Representations of Extended Spinor Groups |
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48 | (2) |
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4.4 Representations of the Coverings for L↑+- and L↑↓+ |
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50 | (1) |
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4.5 On Reducing Spinor Groups to a Real Form |
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51 | (1) |
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52 | (1) |
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5 Spinor Space Structure and Solutions of Klein--Fock--Gordon Equation |
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53 | (14) |
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5.1 Parabolic Cylindrical Coordinates |
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53 | (2) |
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5.2 Solutions of the Klein-Fock-Gordon Equation |
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55 | (3) |
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5.3 The Basis Wave Functions and Spinor Space Structure |
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58 | (3) |
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5.4 The Explicit Form of a Diagonalized Operator A |
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61 | (1) |
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5.5 Orthogonality and Completeness of the Basis in Spinor Space |
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61 | (2) |
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5.6 On Matrix Elements in Vector and in Spinor Models |
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63 | (1) |
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5.7 On Schrodinger Particle in Spinor Space |
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64 | (1) |
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65 | (2) |
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6 Fermion in Riemannian Space-Time |
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67 | (32) |
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6.1 The Tetrad Method by Tetrode--Weyl--Fock--Ivanenko |
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67 | (3) |
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6.2 Spinor Gauge Transformations and the (3+1)-Splitting |
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70 | (2) |
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6.3 Spinor Transformations, and the (2 + 2)-Splitting |
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72 | (1) |
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6.4 Examples of Spinor Gauge Transformations |
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73 | (2) |
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6.5 On Bispinor Transformations at Arbitrary Basis |
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75 | (2) |
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6.6 Nonrelativistic Approximation in the Dirac Equation |
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77 | (3) |
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6.7 On Gauge Invariance of the Pauli Equation |
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80 | (2) |
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6.8 Newman--Penrose Coefficients in Spinor Approach |
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82 | (3) |
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6.9 Gauge Transformations |
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85 | (2) |
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6.10 Spin Coefficients in Spherical Tetrad |
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87 | (1) |
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6.11 Dirac Equation in Orthogonal Coordinates and Tetrads |
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88 | (2) |
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6.12 Dirac Equation and Ricci Coefficients |
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90 | (2) |
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6.13 Gauge Properties of the Vectors Ba and Ca |
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92 | (1) |
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6.14 Connection with the Newman--Penrose Formalism |
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93 | (3) |
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6.15 Majorana 4-Spinor Fields in Riemannian Space-Time |
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96 | (1) |
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6.16 Gauge Transformations and Spinor Space Structure |
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97 | (2) |
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7 Polarization Optics and 2-Spinors |
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99 | (8) |
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7.1 Polarization of the Light: General References |
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99 | (1) |
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100 | (1) |
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7.3 The Polarization of Light. The Stokes--Mueller Formalism |
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100 | (1) |
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7.4 Jones Complex 2-Dimensional Formalisms |
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101 | (6) |
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8 Polarization Optics and 4-Spinors |
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107 | (24) |
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8.1 4-Spinors and Completely Polarized Light |
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107 | (5) |
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8.2 Stokes 4-Vectors and 4-Tensors. Additional Constraints |
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112 | (4) |
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8.3 4-Spinors and Kustaanheimo-Stiefel Approach |
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116 | (1) |
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8.4 4-spinor Representation for Space-Time Vectors |
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117 | (6) |
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8.5 On Possible Jones 4-Spinors for Partially Polarized Light |
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123 | (8) |
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9 Transitivity for the Lorentz Group and Polarization Optics |
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131 | (20) |
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9.1 On the Lorentz Little Group in Optics |
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131 | (2) |
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9.2 The Transitivity Problem for the Lorentz Group |
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133 | (1) |
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9.3 3-Dimensional Mueller Matrices |
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134 | (2) |
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9.4 On the Construction of Mueller 3-Matrices from Polarization Measurements |
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136 | (2) |
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9.5 Mueller 4-Matrices Relating Two Stokes 4-Vectors |
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138 | (4) |
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9.6 On Determining Mueller Matrices from Polarization Measurements |
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142 | (2) |
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9.7 Transitivity and Diagonalization of Quadratic Forms |
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144 | (7) |
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10 Parameters of Lorentz Matrices and Transitivity in Polarization Optics |
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151 | (12) |
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10.1 On Establishing the Parameters of a Lorentz Matrix |
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151 | (4) |
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10.2 On Identifying Lorentz--Mueller Matrices within the Total Linear Group GL(4,R) |
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155 | (1) |
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10.3 On the Expansion of the Lorentz Matrices in Dirac Basis |
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156 | (3) |
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10.4 On Parameters of Lorentz Matrices and Transitivity |
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159 | (4) |
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11 Factorizations for 3-Rotations and the Polarization of the Light |
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163 | (16) |
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163 | (1) |
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11.2 One Special Case for 2-Element Factorization |
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164 | (4) |
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11.3 All the Six 2-Element Factorizations |
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168 | (2) |
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11.4 A Special Case of the 3-Element Factorization |
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170 | (6) |
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11.5 The Six 3-Element Factorizations |
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176 | (1) |
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177 | (2) |
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12 On the Determining of Mueller Matrices from Polarization Measurements |
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179 | (8) |
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179 | (2) |
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12.2 3-Dimensional Mueller Matrices of Rotational Type |
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181 | (1) |
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12.3 4-Dimensional Mueller Matrices of Lorentzian Type |
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182 | (5) |
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13 Elementary Constituents of the Group SL(4, R) and Mueller Matrices |
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187 | (30) |
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13.1 Elementary Constituents of the Group GL(4, R) |
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187 | (4) |
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13.2 On (Pseudo-) Euclidean Rotations of Stokes 4-Vectors |
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191 | (4) |
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13.3 Pseudo-Euclidean Rotations and Partially Polarized Light |
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195 | (3) |
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13.4 Lorentzian Transformations and Completely Polarized Light |
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198 | (1) |
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13.5 On Deformations of the Stokes 4-Vectors |
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199 | (2) |
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13.6 On Other Subgroups of SL(4, R) |
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201 | (1) |
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13.7 The Sixteen 1-Parametric Subgroups in SL(4, R) |
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201 | (11) |
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13.8 Varying the Degree of Polarization of the Light |
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212 | (5) |
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14 Degenerate 4-Dimensional Matrices with Semi-Group Structure |
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217 | (48) |
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14.1 One Independent Vector: Variant I(k) |
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218 | (6) |
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14.2 One Independent Vector: Variant I(m) |
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224 | (5) |
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14.3 One Independent Vector: Variant I(n) |
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229 | (3) |
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14.4 One Independent Vector: Variant I(l) |
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232 | (3) |
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14.5 Two Independent Vectors: Variant II(k,m) |
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235 | (9) |
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14.6 Two Independent Vectors: Variant II(l,n) |
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244 | (4) |
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14.7 Two Independent Vectors: Variant II(k,l) |
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248 | (4) |
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14.8 Two Independent Vectors: Variant I(n,m) |
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252 | (1) |
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14.9 Two Independent Vectors: Variant II(k,n) |
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252 | (4) |
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14.10 Two Independent Vectors: Variant II(m,l) |
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256 | (1) |
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14.11 Three Independent Vectors: Variants I(k,m,n) and I(m,k,l) |
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256 | (2) |
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14.12 Three Independent Vectors: Variants I(n,l,k) and I(n,l,m) |
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258 | (3) |
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14.13 Degenerate Matrices of Rank 3 |
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261 | (3) |
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264 | (1) |
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15 Degenerate Mueller Matrices, Semigroups and Projective Geometry |
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265 | (64) |
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265 | (2) |
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15.2 One Independent Vector (κ0, k) |
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267 | (15) |
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15.3 One Independent Vector, Variant I(m) |
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282 | (7) |
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15.4 One Independent Vector (n0, n) |
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289 | (6) |
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15.5 One Independent Vector (l0, l) |
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295 | (3) |
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15.6 Two Independent Vectors, Variants II(k,m) |
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298 | (7) |
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15.7 Two Independent Vectors, Variants II(l,n) |
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305 | (3) |
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15.8 Two Independent Vectors II(k,l) |
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308 | (2) |
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15.9 Two Independent Vectors, Variants I(n,m) |
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310 | (2) |
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15.10 Two Independent Vectors, Variants I(k,n) |
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312 | (2) |
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15.11 Two Independent Vectors, Variants I(m,l) |
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314 | (2) |
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15.12 Three Independent Vectors, I(k,m,n) |
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316 | (2) |
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15.13 Three Independent Vectors I(k,m,l) |
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318 | (1) |
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15.14 Three Independent Vectors I(n,l,k) |
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319 | (1) |
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15.15 Three Independent Vectors I(n,l,m) |
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320 | (1) |
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15.16 Degenerate Mueller matrices of rank 3 |
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321 | (7) |
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328 | (1) |
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16 Diagonalization of Quadratic Forms and the Mueller Formalism |
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329 | (28) |
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16.1 Diagonalizing the Mueller Quadratic Forms |
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329 | (9) |
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16.2 Particular Variants for the Vanishing Determinant |
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338 | (14) |
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16.3 Some Surfaces of Third Order |
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352 | (3) |
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355 | (2) |
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17 Finsler-Type Structures and Det-Based Classification of Mueller Manifolds |
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357 | (22) |
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357 | (1) |
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17.2 Mueller Matrices in Spinorial Representation |
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358 | (1) |
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17.3 Finsler Metrics and KCC Stability |
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359 | (4) |
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17.4 The (h, ν)-model. Einstein Equations |
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363 | (2) |
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17.5 The Vertical Kern RlM Model |
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365 | (2) |
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17.6 The Horizontal Kern RlM Model |
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367 | (2) |
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17.7 Finslerian Det-classification for the Mueller Set |
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369 | (4) |
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17.8 Riemannian KCC-invariants for SO3(R) |
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373 | (6) |
| References |
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379 | (34) |
| Index |
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413 | |
