Mathematical Aspects of The Quantum Theory of FieldsPDF电子书下载

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Introduction 1

Remarks about Functional and Spectral Representation 4

Part Ⅰ．Field Operators 11

1．Simultaneous Spectral Representation of Infinitely Many Operators 11

2．Commutation Rules and Improper Operators 13

3．The Differential Equations 17

4．The Energy Integral 20

5．Motivation of the Configuration Space Representation 25

Part Ⅱ．Particle Re presentation 28

6．Biquantization 28

7．Remark on the Occupation Number Representation 35

8．Annihilation and Creation Operators 37

9．Time Variation of Annihilation and Creation Operators and Representation of Field Operators 45

10．Trace Operators 46

11．Oscillators 51

12．Hermite Functionals and Integration over the Hilbert Space 52

Bibliography to PartsⅠand Ⅱ 63

Part Ⅲ．Boson Field in Interaction with a Given Source Distribution 65

13．Expectation Values of the Energy and the Number of Bosons 65

Representations 67

Operators and Differential Equations 68

Asymptotic Expectation Values 72

Modified Particles 77

14．Particle Representation of the States of the Boson Field Modified by a Source Distribution 79

First Form of the Operator T 83

Second Form of the Operator T 86

Modified Vacuum State 88

Transformation to the Modified Particle Representation 89

Probabilities in General 91

Derivation 92

Probabilities in Special Cases 93

15．Transition Probabilities 95

Transition Probabilities for the Vacuum State 97

General Transition Probabilities 99

Perturbation 102

Method of Spectral Transformation 103

16．Boson Fields under the Influence of a Source Distribution Which Varies in Time 104

Infinitely Slow Switch-on 107

Removing Sinks from Sources 109

Switching Off 109

Lorentz Invariant Formulation 110

17．Modified Vacuum States 113

Influence of a Source Distribution 115

Probability Distribution of the Energy 118

Blbliography to Part Ⅲ 120

Supplementary References to Parts Ⅰ and Ⅱ 120

part Ⅳ．Occupation Number Representation and Fields Different Kinds 121

18．Occupation Number Representation 121

Particle and Occupation Representation 123

Occupation Functionals 125

Correspondence between Occupation Functionals and Particle Representers 128

Representation by Occupation Functionals 129

Different Forms of the Occupation Representation 130

Occupation Functions of a Discrete Variable 130

Annihilation and Creation Operators 132

Formal Operations 134

Biquantized Operators 136

Modified Vacuum State 137

Expectation Values 138

Equidistribution State 139

19．Myriotic and Amyriotic Fields 139

Functionals of the Second Type 142

Annihilation and Creation Operators 145

The Functionals f(v) 147

Representation by Functionals φ(v) 148

Proof that Myriotic Fields Possess No Vacuum States 149

Equidistribution State 151

Expectation Values 152

Infrared Catastrophe 153

Occupation Functions of a Discrete Variable 153

Myriotic Field in a Box 155

20．Probabilities and Expectation Values for the Equidistribution State 156

Evaluation of Iw(τ) for Polynomials τ(ν) 157

Evaluation of the Expression Iw(τ) by Complex Integration 160

Conditions on λ(s),w(？),and h(z) 162

Evaluation of Probabilities 164

Saddle Point Method 166

Conditioned Equidistribution State 166

21．Occupation Number Representation for Fermion Fields 168

Representation of the First Type 169

Representation of the Second Type 173

Equidistribution State 174

Definition of the Functional F(v) 175

Evaluations for the F-Equidistribution State 176

Properties of the Projector F 179

Partly Myriotic Fields 181

Bibliography to Part Ⅳ 183

Part Ⅴ．Fields Modified by Linear Homogeneous Forces 185

22．Boson Fields under the Influence of Spring Forces 185

Adjoint and Conjugate Operators 186

Various Types of Problems 187

Modified and Unmodified Particle Representation 188

Modified Energy Operator for Single Particles 189

Modified Creation and Annihilation Operators 190

Remarks about Energy Operators 191

Modified Particle Representation 192

Canonical Transformations 193

23．General Homogeneous Linear Transformation of Creation and Annihilation Operators 194

Pseudo-biquantized Operators 197

First Commutator Identity 199

Exponential Function of Pseudo-biquantized Operators 200

First Similarity Rule,First Form of the Operator T 201

24．E-ordering of the Canonical Transformation 201

Second Commutator Identity 202

Composition Rule 203

Second Form of the Canonical Transformation 204

Modified Vacuum State 207

First Decomposition 209

Relations between E,F,G,and Y 210

Trace Relations 211

Conditions for the Existence of the Canonical Transformation 212

25．Third and Fourth Form of the Canonical Transformation 214

Third and fourth Form of the Transformation T 214

Second Decomposition 215

Final Form of the Transformation T 217

Identification of the Third and Fourth Form of T 217

Second and Third Similarity Rule 219

Composition Rule for Biquantized Operators 220

Identity of the Fourth and the Second Form of the Operator T 221

26．Application to Boson Fields 222

Reduction of the Quantized to the Unquantized Field Problem 223

Conditions for the Existence of the Canonical Transformation 224

Special Cases 224

27．Transition Operator．Scattering Operator 227

Method of Spectral Transformation 227

Transition Operator 228

Direct Method 229

Properties of the Transition Operator 230

Time Variation of the Vacuum State 231

Asymptotic Transition Probabilities 232

Scattering Operator 233

Scattering Operator According to Yang and Feldman 234

Asymptotic Field for Scattering Operator 235

Justification 235

Interpretation 237

28．A Modified Electron-Positron Field 239

Dirac Electron 240

Transformation of the Quantum Variables 241

Electron-Positron Field 245

A Modified Electron-Positron Field 247

Linear Transformation of Operators A into Operators B 247

Modified Vacuum State 249

Vacuum Transition Probability 249

Perturbation Approximation 251

Bibliography to Part Ⅴ 251

Appendix:Lorentz Invariant Treatment of Boson Fields 257

29．Unquantized Field 257

Momentum Representation 258

Equations for Wave Amplitudes 261

Inverse Operators 262

Inner Products 263

30．Boson Field Subject to Homogeneous Forces 265

Comments and Corrections 269

Supplementary Bibliography 272

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