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Symmetries of Equations of Quantum Mechanics SYMMETRIES OF EQUATIONS OF QUANTUM MECHANICS TABLE OF CONTENTS Chapter I. LOCAL SYMMETRY OF BASIS EQUATIONS OF RELATIVISTIC QUANTUM THEORY 1. Local Symmetry of the Klein-Gordon-Fock Equation 1.1.Introduction ............................................ 1 1.2.TheIAoftheKGFEquation ............................... 3 1.3. Symmetry of the d’Alembert Equation . 5 1.4.LorentzTransformations................................... 6 1.5.ThePoincaréGroup ...................................... 9 1.6.TheConformalTransformations............................ 12 1.7. The Discrete Symmetry Transformations . 14 2. Local Symmetry of the Dirac Equation 2.1.TheDiracEquation...................................... 16 2.2. Various Formulations of the Dirac equation . 17 2.3.AlgebraoftheDiracMatrices ............................. 19 2.4.SOsandIAs ........................................... 19 2.5. The IA of the Dirac Equation in the Class M1 ................. 20 2.6.TheOperatorsofMassandSpin ........................... 22 2.7. Manifestly Hermitian Form of Poincaré Group Generators . 23 2.8. Symmetries of the Massless Dirac Equation . 24 2.9. Lorentz and Conformal Transformations of Solutions of the Dirac Equation............................................... 25 2.10.P-, T-, and C-Transformations............................. 27 3. Maxwell’s Equations 3.1.Introduction .......................................... 28 3.2. Various Formulations of Maxwell’s Equations . 30 i 3.3. The Equation for the Vector-Potential . 32 3.4. The IA of Maxwell’s Equations in the Class M1 ............... 33 3.5. Lorentz and Conformal Transformations . 34 3.6. Symmetry Under the P-, T-, and C-Transformations . 38 3.7. Representations of the Conformal Algebra Corresponding to a Field withArbitraryDiscreteSpin ............................... 39 3.8. Covariant Representations of the Algebras AP(1,3) and AC(1,3) .. 40 3.9. Conformal Transformations for Any Spin . 43 Chapter II. REPRESENTATIONS OF THE POINCARÉ ALGEBRA AND WAVE EQUATIONS FOR ARBITRARY SPIN 4. IR of the Poincaré Algebra 4.1.Introduction ........................................... 44 4.2.CasimirOperators....................................... 45 4.3.BasisofanIR ......................................... 46 4.4. The Explicit Form of the Lubanski-Pauli Vector . 48 4.5. IR of the Algebra A(c1,n) ................................. 50 4.6. Explicit Realizations of the Poincaré Algebra . 53 4.7. Connections with the Canonical Realizations of Shirokov-Foldy-Lomont-Moses............................. 55 4.8.CovariantRepresentations ................................ 58 5. Representations of the Discrete Symmetry Transformations 5.1.Introduction ........................................... 60 5.2. Nonequivalent Multiplicators of the Group G8 ................ 62 5.3. The General Form of the Discrete Symmetry Operators . 64 5.4. The Operators P, T, and C for Representation of Class I ......... 67 5.5. Representations of Class II ................................ 70 5.6. Representations of Classes III-IV .......................... 71 5.7. Representations of Class V ............................... 73 5.8.ConcludingRemarks .................................... 75 6. Poincaré-Invariant Equations of First Order 6.1.Introduction ........................................... 75 6.2. The Poincaré Invariance Condition . 77 6.3. The Explicit Form of the Matrices ßµ ....................... 78 β 6.4. Additional Restristions for the Matrices µ ................... 79 ii 6.5. The Kemmer-Duffin-Petiau (KDP) Equation . 81 6.6. The Dirac-Fierz-Pauli Equation for a Particle of Spin 3/2 . 82 6.7.TransitiontotheSchrödingerForm ......................... 85 7. Poincaré-Invariant Equations without Redundant Components 7.1.PreliminaryDiscussion................................. .88 7.2.FormulationoftheProblem ............................... 89 I II 7.3. The Explicit Form of Hamiltonians Hs and Hs ................. 92 7.4. Differential Equations of Motion for Spinning Particles . 96 7.5. Connection with the Shirokov-Foldy Representation . 98 8. Equations in Dirac’s Form for Arbitrary Spin Particles 8.1. Covariant Equations with Coefficients Forming the Clifford Algebra............................................... 100 8.2. Equations with the Minimal Number of Components . 101 8.3. Connection with Equations without Superfluous Components . 103 8.4.LagrangianFormulation................................ 104 8.5. Dirac-Like Wave Equations as a Universal Model of a Particle withArbitrarySpin...................................... 105 9. Equations for Massless Particles 9.1.BasicDefinitions ...................................... 108 9.2. A Group-Theoretical Derivation of Maxwell’s Equations . 109 9.3. Conformal-Invariant Equations for Fields of Arbitrary Spin . 110 9.4.EquationsofWeyl’sType................................ 112 9.5. Equations of Other Types for Massless Particles . 115 10. Relativistic Particle of Arbitrary Spin in an External Electromagnetic Field 10.1. The Principle of Minimal Interaction . 116 10.2. Introduction of Minimal Interaction into First Order Wave Equations............................................. 117 10.3. Introduction of Interaction into Equations in Dirac’s Form . 119 10.4. A Four-Component Equation for Spinless Particles . 121 10.5. Equations for Systems with Variable Spin . 122 10.6. Introduction of Minimal Interaction into Equations without Superfluous Components........................................... 123 iii 10.7. Expansion in Power Series in 1/m ........................ 124 10.8. Causality Principle and Wave Equations for Particles of Arbitrary Spin ................................................. 127 10.9. The Causal Equation for Spin-One Particles with Positive Energies.............................................. 128 Chapter III. REPRESENTATIONS OF THE GALILEI ALGEBRA AND GALILEI-INVARIANT WAVE EQUATIONS 11. Symmetries of the Schrödinger Equation 11.1.TheSchrödingerEquation .............................. 131 11.2. Invariance Algebra of the Schrödinger Equation . 132 11.3. The Galilei and Generalized Galilei Algebras . 134 11.4. The Schrödinger Equation Group . 136 11.5.TheGalileiGroup..................................... 138 11.6. The Transformations P and T ............................ 139 12. Representations of the Lie Algebra of the Galilei Group 12.1. The Galilei Relativity Principle and Equations of Quantum Mechanics ........................................... 140 12.2.ClassificationofIRs................................... 141 12.3. The Explicit Form of Basis Elements of the Algebra AG(1,3) ... 142 12.4. Connections with Other Realizations . 144 12.5.CovariantRepresentations .............................. 146 12.6. Representations of the Lie Algebra of the Homogeneous Galilei Group................................................ 148 13. Galilei-Invariant Wave Equations 13.1.Introduction ......................................... 154 13.2. Galilei-Invariance Conditions . 154 13.3. Additional Restrictions for Matrices ßµ .................... 155 λ 13.4. General Form of Matrices ßµ in the Basis ;l,m> ............ 157 13.5. Equations of Minimal Dimension . 159 13.6. Equations for Representations with Arbitrary Nilpotency Indices 163 iv 14. Galilei-Invariant Equations of the Schrödinger Type 14.1. Uniqueness of the Schrödinger equation . 165 14.2. The Explicit Form of Hamiltonians of Arbitrary Spin Particles . 167 14.3.LagrangianFormulation................................ 170 15. Galilean Particle of Arbitrary Spin in an External Electromagnetic Field 15.1. Introduction of Minimal Interaction into First-Order Equations . 171 15.2. Magnetic Moment of a Galilei Particle of Arbitrary Spin . 173 15.3. Interaction with the Electric Field . 175 15.4. Equations for a (2s+1)-Component Wave Function . 176 15.5. Introduction of the Minimal Interaction into Schrödinger-Type Equations ............................................ 179 15.6.AnomalousInteraction................................ 180 Chapter IV. NONGEOMETRIC SYMMETRY 16. Higher Order SOs of the KGF and Schrödinger Equations 16.1. The Generalized Approach to Studying of Symmetries of Partial DifferentialEquations................................... 183 16.2.SOsoftheKGFEquation............................... 185 16.3. Hidden Symmetries of the KGF Equation . 188 16.4. Higher Order SOs of the d’Alembert Equation . 190 16.5.SOsoftheSchrödingerEquation......................... 191 16.6. Hidden Symmetries of the Schrödinger Equation . 193 16.7. Symmetries of the Quasi-Relativistic Evolution Equation . 196 17. Nongeometric Symmetries of the Dirac Equation 17.1. The IA of the Dirac Equation in the Class M1 ............... 198 17.2. Symmetries of the Dirac Equation in the Class of Integro- DifferentialOperators ................................... 202 17.3. Symmetries of the Eight-Component Dirac Equation . 203 17.4. Symmetry Under Linear and Antilinear Transformations . 206 17.5. Hidden Symmetries of the Massless Dirac Equation . 209 v 18. The Complete Set of SOs of the Dirac Equation 18.1.IntroductionandDefinitions............................. 211 18.2. The General Form of SOs of Order n .................... 212 18.3. Algebraic Properties of the First-Order SOs . 213 18.4. The Complete Set of SOs of Arbitrary Order . 216 18.5.ExamplesandDiscussion .............................. 220 18.6. SOs of the Massless Dirac Equation . 221 19. Symmetries of Equations for Arbitrary Spin Particles 19.1.SymmetriesoftheKDPEquation ........................ 223 19.2. Arbitrary Order SOs of the KDP Equation
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