《Mathematical Aspects of The Quantum Theory of Fields》求取 ⇩

Introduction1

Remarks about Functional and Spectral Representation4

Part Ⅰ．Field Operators11

1．Simultaneous Spectral Representation of Infinitely Many Operators11

2．Commutation Rules and Improper Operators13

3．The Differential Equations17

4．The Energy Integral20

5．Motivation of the Configuration Space Representation25

Part Ⅱ．Particle Re presentation28

6．Biquantization28

7．Remark on the Occupation Number Representation35

8．Annihilation and Creation Operators37

9．Time Variation of Annihilation and Creation Operators and Representation of Field Operators45

10．Trace Operators46

11．Oscillators51

12．Hermite Functionals and Integration over the Hilbert Space52

Bibliography to PartsⅠand Ⅱ63

Part Ⅲ．Boson Field in Interaction with a Given Source Distribution65

13．Expectation Values of the Energy and the Number of Bosons65

Representations67

Operators and Differential Equations68

Asymptotic Expectation Values72

Modified Particles77

14．Particle Representation of the States of the Boson Field Modified by a Source Distribution79

First Form of the Operator T83

Second Form of the Operator T86

Modified Vacuum State88

Transformation to the Modified Particle Representation89

Probabilities in General91

Derivation92

Probabilities in Special Cases93

15．Transition Probabilities95

Transition Probabilities for the Vacuum State97

General Transition Probabilities99

Perturbation102

Method of Spectral Transformation103

16．Boson Fields under the Influence of a Source Distribution Which Varies in Time104

Infinitely Slow Switch-on107

Removing Sinks from Sources109

Switching Off109

Lorentz Invariant Formulation110

17．Modified Vacuum States113

Influence of a Source Distribution115

Probability Distribution of the Energy118

Blbliography to Part Ⅲ120

Supplementary References to Parts Ⅰ and Ⅱ120

part Ⅳ．Occupation Number Representation and Fields Different Kinds121

18．Occupation Number Representation121

Particle and Occupation Representation123

Occupation Functionals125

Correspondence between Occupation Functionals and Particle Representers128

Representation by Occupation Functionals129

Different Forms of the Occupation Representation130

Occupation Functions of a Discrete Variable130

Annihilation and Creation Operators132

Formal Operations134

Biquantized Operators136

Modified Vacuum State137

Expectation Values138

Equidistribution State139

19．Myriotic and Amyriotic Fields139

Functionals of the Second Type142

Annihilation and Creation Operators145

The Functionals f(v)147

Representation by Functionals φ(v)148

Proof that Myriotic Fields Possess No Vacuum States149

Equidistribution State151

Expectation Values152

Infrared Catastrophe153

Occupation Functions of a Discrete Variable153

Myriotic Field in a Box155

20．Probabilities and Expectation Values for the Equidistribution State156

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

Evaluation of the Expression Iw(τ) by Complex Integration160

Conditions on λ(s),w(？),and h(z)162

Evaluation of Probabilities164

Saddle Point Method166

Conditioned Equidistribution State166

21．Occupation Number Representation for Fermion Fields168

Representation of the First Type169

Representation of the Second Type173

Equidistribution State174

Definition of the Functional F(v)175

Evaluations for the F-Equidistribution State176

Properties of the Projector F179

Partly Myriotic Fields181

Bibliography to Part Ⅳ183

Part Ⅴ．Fields Modified by Linear Homogeneous Forces185

22．Boson Fields under the Influence of Spring Forces185

Adjoint and Conjugate Operators186

Various Types of Problems187

Modified and Unmodified Particle Representation188

Modified Energy Operator for Single Particles189

Modified Creation and Annihilation Operators190

Remarks about Energy Operators191

Modified Particle Representation192

Canonical Transformations193

23．General Homogeneous Linear Transformation of Creation and Annihilation Operators194

Pseudo-biquantized Operators197

First Commutator Identity199

Exponential Function of Pseudo-biquantized Operators200

First Similarity Rule,First Form of the Operator T201

24．E-ordering of the Canonical Transformation201

Second Commutator Identity202

Composition Rule203

Second Form of the Canonical Transformation204

Modified Vacuum State207

First Decomposition209

Relations between E,F,G,and Y210

Trace Relations211

Conditions for the Existence of the Canonical Transformation212

25．Third and Fourth Form of the Canonical Transformation214

Third and fourth Form of the Transformation T214

Second Decomposition215

Final Form of the Transformation T217

Identification of the Third and Fourth Form of T217

Second and Third Similarity Rule219

Composition Rule for Biquantized Operators220

Identity of the Fourth and the Second Form of the Operator T221

26．Application to Boson Fields222

Reduction of the Quantized to the Unquantized Field Problem223

Conditions for the Existence of the Canonical Transformation224

Special Cases224

27．Transition Operator．Scattering Operator227

Method of Spectral Transformation227

Transition Operator228

Direct Method229

Properties of the Transition Operator230

Time Variation of the Vacuum State231

Asymptotic Transition Probabilities232

Scattering Operator233

Scattering Operator According to Yang and Feldman234

Asymptotic Field for Scattering Operator235

Justification235

Interpretation237

28．A Modified Electron-Positron Field239

Dirac Electron240

Transformation of the Quantum Variables241

Electron-Positron Field245

A Modified Electron-Positron Field247

Linear Transformation of Operators A into Operators B247

Modified Vacuum State249

Vacuum Transition Probability249

Perturbation Approximation251

Bibliography to Part Ⅴ251

Appendix:Lorentz Invariant Treatment of Boson Fields257

29．Unquantized Field257

Momentum Representation258

Equations for Wave Amplitudes261

Inverse Operators262

Inner Products263

30．Boson Field Subject to Homogeneous Forces265

Comments and Corrections269

Supplementary Bibliography272