Lagrangian Quantum Field Theory in Momentum Picture, Free Fields

LAGRANGIAN QUANTUM FIELD THEORYIN MOMENTUM PICTURE.FREE FIELDS......Page 3
NOTICE TO THE READER......Page 6
Contents......Page 7
List of Conventions......Page 13
Preface......Page 15
Acknowledgments......Page 17
Lagrangian Formalism andthe Momentum Picture......Page 19
2. Lagrangian Formalism......Page 20
3. Derivatives Relative to Operatorsand Lagrangian Formalism......Page 22
4. Heisenberg Relations......Page 25
5. The Momentum Picture ofMotion......Page 26
6. General Aspects of Lagrangian Formalismin Momentum Picture......Page 30
7. On the Momentum Representation andParticle Interpretation......Page 35
8. The Momentum Picture as 4-Dimensional Analogue ofthe Schr¨odinger One......Page 37
9. Conclusion......Page 40
Free Scalar Fields......Page 41
1. Introduction......Page 42
2. Description of Free Neutral Scalar Fieldin Momentum Picture......Page 43
3. Analysis of the Klein-Gordon Equation......Page 47
4. Frequency Decompositions and Their PhysicalMeaning......Page 50
5. The Klein-Gordon Equations in Terms ofCreation and Annihilation Operators......Page 51
6. Commutation Relations......Page 54
7. Vacuum and Normal Ordering......Page 60
8. State Vectors and Transitions between Them......Page 62
9. Description of Free Scalar Field......Page 65
10. Analysis of the Field Equations......Page 72
11. Frequency Decompositions and Their PhysicalMeaning......Page 74
12. The Field Equations in Terms of Creation andAnnihilation Operators......Page 75
13. The Charge and Orbital Angular MomentumOperators......Page 79
14. Commutation Relations......Page 84
15. Vacuum and Normal Ordering......Page 90
16. State Vectors......Page 94
17. Conclusion......Page 96
Free Spinor Fields......Page 99
2. Description of Free Spinor Field in Momentum Picture......Page 100
3. Analysis of the Dirac Equation(s)......Page 105
4. Frequency Decompositions......Page 111
5. The Dynamical Variables in Terms ofCreation and Annihilation Operators......Page 118
6. The Field Equations in Terms of Creation andAnnihilation Operators......Page 128
7. Anticommutation Relations......Page 133
8. Vacuum and Normal Ordering......Page 141
9. State Vectors......Page 145
10. On the Choice of Lagrangian and Its Consequences......Page 149
11. Conclusion......Page 160
Free Vector Fields......Page 163
1. Introduction......Page 164
2. Description of Free Vector Field in Momentum Picture......Page 165
3. Analysis of the Field Equations......Page 171
4. Frequency Decompositions andCreation and Annihilation Operators......Page 176
5. The Dynamical Variables inTerms of Creation and Annihilation Operators......Page 181
6. The Field Equations in Terms of Creation andAnnihilation Operators......Page 190
7. Commutation Relations......Page 194
8. Vacuum and Normal Ordering......Page 200
9. State Vectors and Particle Interpretation......Page 204
10.1. Problems in the Massless Case......Page 208
Conjecture 10.2.......Page 210
Conjecture 10.3.......Page 214
11. On the Choice of Lagrangian......Page 216
12.1. Description of Free Massless Vector Fieldswithout the Lorenz Condition......Page 222
12.2. Analysis of the Euler-Lagrange Equations......Page 224
12.3. Dynamical Variables......Page 226
12.4. The Field Equations......Page 227
12.5. Discussion......Page 228
13. Conclusion......Page 229
Commutation Relationsfor Free Fields......Page 231
1. Introduction......Page 232
2. Lagrangians, Euler-Lagrange Equationsand Dynamical Variables......Page 234
3. On the Uniqueness of the Dynamical Variables......Page 240
4. Heisenberg Relations......Page 246
5. Types of Possible Commutation Relations......Page 252
5.1. Restrictions Related to the Momentum Operator......Page 253
5.2. Restrictions Related to the Charge Operator......Page 258
5.3. Restrictions Related to the Angular Momentum Operator(s)......Page 261
6. Inferences......Page 266
7. State Vectors, Vacuum and Mean Values......Page 273
8. Commutation Relations forSeveral Coexisting Different Free Fields......Page 280
8.1. Commutation Relations Connected with the MomentumOperator. Problems and Their Possible Solutions......Page 282
8.2. Commutation Relations Connected with the Charge andAngularMmomentum Operators......Page 287
8.3. Commutation Relations between the Dynamical Variables......Page 288
8.4. Commutation Relations under the Uniqueness Conditions......Page 291
9. Conclusion......Page 293
References......Page 295
Index......Page 301