Verfügbarkeit: Noncovariant Gauges in Canonical Formalism

Noncovariant Gauges in Canonical Formalism:

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1. Verfasser:	Burnel, André (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Berlin [u.a.] Springer 2009
Schriftenreihe:	Lecture Notes in Physics 761
Schlagworte:	Gauge fields (Physics) Quantum field theory Renormalization (Physics) Eichtheorie
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XV, 236 S.
ISBN:	9783540699200 9783540699217

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adam_text	Contents Canonical Quantization for Constrained Systems .................. 1 1.1 Canonical Quantization of Mechanical Unconstrained Systems .... 1 1.1.1 Lagrangian Formalism ................................ 1 1.1.2 Hamiltonian ........................................ 1 1.1.3 Poisson Brackets ..................................... 2 1.1.4 Quantization ........................................ 2 1.2 Field Theory ............................................... 3 1.2.1 Lagrangian and Lagrangian Density ..................... 3 1.2.2 Hamiltonian ........................................ 3 1.2.3 Poisson Brackets ..................................... 4 1.2.4 Quantization ........................................ 4 1.2.5 The Free Scalar Field ................................. 5 1.2.6 Solution of the Klein-Gordon Equation .................. 5 1.3 The Electromagnetic Field ................................... 6 1.4 Primary and Secondary Constraints ........................... 8 1.4.1 Primary Constraints .................................. 8 1.4.2 Secondary Constraints ................................ 8 1.4.3 First and Second Class Constraints ...................... 9 1.4.4 Quantization in the Presence of Second-Class Constraints .. 10 1.4.5 Quantization in Presence of First-Class Constraints ........ 11 1.4.6 BRST Quantization .................................. 13 1.4.7 Summary ........................................... 14 1.5 Back to the Free Massless Vector Field ........................ 14 1.5.1 Lagrangian and Constraints ............................ 14 1.5.2 Gauge Fixing ....................................... 16 1.6 Covariant Gauge for the Free Massless Vector Field .............. 16 1.6.1 Commutation Relations for Any Time ................... 18 1.6.2 Creation and Annihilation Operators .................... 21 1.6.3 The Gupta-Bleuler Formalism ......................... 22 References ..................................................... 23 Contents Quantization of the Free Electromagnetic Field in General Class III Linear Gauges ................................................. 25 2.1 Introduction ............................................... 25 2.2 Lagrangian and Field Equations .............................. 26 2.2.1 The Lagrangian ...................................... 26 2.2.2 Euler-Lagrange Equations ............................. 26 2.2.3 Derived Field Equations .............................. 28 2.2.4 Sketching the Solution of the Cauchy Problem ............ 28 2.2.5 Particular Cases of Interest ............................ 30 2.2.6 Planar-Type Gauges .................................. 31 2.3 Constraint Analysis ......................................... 32 2.3.1 Canonical Momenta and Primary Constraints ............. 32 2.3.2 The Hamiltonian ..................................... 33 2.3.3 Constraint Chains .................................... 33 2.4 Singular Frames ............................................ 33 2.5 Commutation Relations for Any Time ......................... 34 2.5.1 Canonical Commutation Relations ...................... 34 2.5.2 Commutation Relations Involving the S Field ............. 34 2.5.3 Commutation Relations Involving the S Field ............ 36 2.5.4 Commutation Relations Involving B=d -A .............. 38 2.5.5 Commutation Relations Between Ад ѕ ................... 42 2.5.6 Summary of the Commutation Relations for Any Time ..... 44 2.6 Creation and Annihilation Operators ........................... 45 2.6.1 Momentum Space Expansion of the Fields ............... 45 2.6.2 Commutation Relations Between Creation and Annihilation Operators ............................ 50 2.6.3 Summary of the Algebra of Creation and Annihilation Operators ............................ 55 2.7 The Gupta-Bleuler Formalism ................................ 55 2.8 Covariance Problems ....................................... 56 2.8.1 Translation Invariance ................................ 56 2.8.2 Lorentz Transformations .............................. 58 References ..................................................... 60 Quantization of the Free Electromagnetic Field in Class II Axial Gauges ........................................................ 61 3.1 Introduction ............................................... 61 3.2 Lagrangian and Field Equations .............................. 61 3.2.1 The Lagrangian ...................................... 61 3.2.2 Euler-Lagrange Equations ............................. 62 3.2.3 Derived Field Equations .............................. 63 3.2.4 Summary of the Field Equations ........................ 63 3.3 Constraint Analysis and Effective Hamiltonian .................. 63 3.4 Solution of Field Equations .................................. 65 3.4.1 Solution of η dS = Q ................................. 65 Contents xi 3.4.2 Solutíonof (п-д+к)Ѕ = 0 ........................... 66 3.4.3 Solution of и · dB = (η2 -a)S......................... 67 3.4.4 Elementary Solution of и · дОА = О ..................... 67 3.4.5 Solution of the Cauchy Problem for DA(x~z) = LxDn(x-z) 70 3.4.6 Elementary Solution of (n · d)2DA =0.................. 70 3.5 Commutation Relations ..................................... 71 3.5.1 Equal Time Commutators ............................. 71 3.5.2 Commutation Relations Involving the i-Field ............ 72 3.5.3 Commutation Relations Involving the S -Field ............ 72 3.5.4 Commutation Relations Involving д ■ A .................. 73 3.5.5 Commutation Relations [Αμ(χ), Αν(ζ) ................... 74 3.6 Association of a Feynman Propagator with the Operator η · д ...................................... 76 3.7 Associating Creation and Annihilation Operators with Fields ...... 77 3.7.1 TheS-Field ......................................... 78 3.7.2 Other Fields ......................................... 78 3.8 Correct Momentum Space Expansion of the Fields ............... 80 3.8.1 The S-Field ......................................... 80 3.8.2 The B-Field ......................................... 80 3.8.3 The S -Field ......................................... 82 3.8.4 The Afield ........................................ 83 3.9 A Toy Model for the Unphysical 5 and В Fields ................. 84 3.10 Conclusions ............................................... 85 3.11 Interpolating Between Axial and Relativistic Gauges ............. 85 3.12 Summary ................................................. 86 References ..................................................... 86 4 Gauge Fields in Interaction ..................................... 87 4.1 Introduction ............................................... 87 4.2 General Formalism of Gauge Invariance ....................... 88 4.2.1 General Transformations .............................. 88 4.2.2 Infinitesimal Transformations .......................... 89 4.2.3 Remarks ............................................ 90 4.3 Class III Gauges in Yang-Mills Theory ........................ 91 4.3.1 Building the Lagrangian .............................. 91 4.3.2 Field Equations ...................................... 94 4.3.3 Canonical Momenta and Constraints .................... 94 4.3.4 The Hamiltonian ..................................... 95 4.3.5 TheBRSTCharge ................................... 95 4.3.6 Ghost Number ....................................... 97 4.3.7 Global Internal Symmetry ............................. 97 4.3.8 Physical States ...................................... 98 4.3.9 Problems of Covariance ............................... 98 References ..................................................... 99 Contents Perturbation Theory: RenormaUzatìon and AU That ...............101 5.1 Introduction ...............................................101 5.2 The S-Matrix ..............................................101 5.2.1 Definition and Properties ..............................101 5.3 Perturbative Expansion ......................................103 5.3.1 Expansion and Consequences of the Unitarity Condition ... 103 5.3.2 Consequences of the Causality Condition ................104 5.3.3 The First Term 7Ί ....................................106 5.3.4 Nonunicity of the Tn ..................................108 5.3.5 The Fixed Part of Tn ..................................108 5.3.6 Wick s Theorem .....................................109 5.3.7 Feynman Rules ...................................... Ill 5.3.8 Feynman Rules in Momentum Space ....................113 5.4 Divergences, Power Counting and Renormalizability .............118 5.4.1 Example ............................................118 5.4.2 Power Counting and Superficial Degree of Divergence .....118 5.4.3 Renormalizability by Power Counting ...................120 5.5 Dimensional Regularization of Covariant Divergent Integrals ......120 5.5.1 A General One-Loop Integral ..........................120 5.5.2 Euclidean Space .....................................121 5.5.3 Use of the Feynman Formula ..........................121 5.5.4 Elimination of the Denominators .......................122 5.5.5 Complex Dimension .................................122 5.5.6 Tensor Integrals .....................................123 5.6 Extension to General Covariant or Noncovariant Integrals in a Preferred Frame ........................................124 5.6.1 The General One-Loop Integral ........................124 5.6.2 Euclidean Space .....................................125 5.6.3 Elimination of the Denominators .......................126 5.6.4 Calculation of the Derivatives ..........................127 5.6.5 Introduction of the Feynman Variables ..................128 5.6.6 Integration Over A ...................................129 5.6.7 Regularization of Ultraviolet Divergences ................129 5.6.8 Consequences of the Nonsingularity of the B~~ λ Matrix .....130 5.7 Computation of the Ghost Loop With Two Legs .................131 5.8 Renormalization and Counter-Terms ...........................132 5.8.1 Various Renormalization Schemes ......................133 5.8.2 Multiplicative Renormalization ........................134 5.9 Summary .................................................135 References .....................................................136 Contents Slavnov-Taylor Identities for Yang-Mills Theory ...................137 6.1 Introduction ...............................................137 6.2 The Reduction Formula .....................................137 6.3 One-Particle Irreducible Vertex Functions ......................138 6.4 Yang-Mills Theory in a General Class III Gauge .................139 6.4.1 The Lagrangian and Superficially Divergent Processes .....139 6.4.2 BRST Symmetry, Field Equations and Canonical Commutation Relations ...............................141 6.4.3 Commuting Derivatives and Time-Ordered Products .......143 6.5 The Ward-Takahashi-Slavnov-Taylor Identity for the Gluon Self-Energy ...............................................143 6.5.1 Covariant Gauges ....................................143 6.5.2 General Gauges .....................................146 6.5.3 Renormalization .....................................148 6.6 Identity for the Three-Gluon Vertex Function ...................149 6.6.1 Derivation of the Identity ..............................150 6.6.2 Renormalization .....................................150 6.7 Ghost Propagator ...........................................151 6.8 Ghost-Ghost-Gluon Vertex ...................................153 6.8.1 Identity in Coordinate Space ...........................153 6.8.2 Momentum Space ....................................153 6.8.3 Remark ............................................154 6.9 Multiplicative Renormalization ...............................154 6.10 Summary .................................................156 References .....................................................156 Field Theory Without Infinities ..................................157 7.1 Introduction ...............................................157 7.2 Iterative Construction of the ¿ -Matrix Without Time-Ordering .....158 7.3 Splitting of Causal Distributions into Advanced and Retarded Parts . 159 7.3.1 Distribution and Fourier Transform .....................159 7.3.2 Order of Singularity of a Distribution ...................160 7.3.3 Splitting of Distribution with Negative Order of Singularity . 161 7.3.4 Nonnegative Singularity Order .........................162 7.4 Application to Yang-Mills Theory .............................164 7.4.1 First Order ..........................................164 7.4.2 Example of a One-Loop Process: The Gluon Self-energy ... 179 7.4.3 Calculation of 1 {ξ; κ)................................ 186 7.4.4 Calculation of /2(A; Jfj , κ·,)............................. 187 7.4.5 Calculation of the Tensor Distributions ..................188 7.4.6 The Final Result for the Gluon Self-energy ...............190 7.5 Summary .................................................190 References .....................................................191 xiv Contents 8 Gauges with a Singular С Matrix ................................193 8.1 Introduction ...............................................193 8.2 The Ghost Loop Contribution ................................194 8.2.1 The Leibbrandt Gauges ...............................194 8.2.2 Interpolating Between Leibbrandt and Relativistic Gauges .. 197 8.3 Singularities Generated by the Lack of Power Counting in Ultra-Violet Divergent Perturbative Theory ...................199 8.3.1 The Loop Integration in a General Gauge ................200 8.3.2 Restriction to Loops with Two External Particles ..........201 8.3.3 Further Restriction for m < 6..........................201 8.3.4 Interpolating Between Leibbrandt and Relativistic Gauges .. 205 8.4 Cancellation of Divergences at a = 0 in the Self-Energy ..........207 References .....................................................208 9 Conclusion ....................................................209 A Notations ......................................................211 В A Useful Fourier Transform .....................................213 С Generalized Functions ..........................................215 C.I Elementary Solutions of the Klein-Gordon Equation .............215 C.I.I The A Function ......................................215 С 1.2 The AF Function ....................................216 C.2 The E Function ............................................216 C.2.1 Definition ..........................................216 C.2.2 Derivation of the Elementary Solution ...................216 C.2.3 Zero-Time Properties .................................217 C.2.4 Integration Over ko ...................................218 C.2.5 The Cauchy Problem Associated with D2 ................218 C.3 Generalized Functions Associated with the Dc Operator ..........219 C.3.1 Elementary Solution D^ .............................220 C.3.2 Positive Frequency Part ...............................221 C.3.3 Negative Frequency Part ..............................222 C.3 .4 Elementary Solution with Causal Support ................222 C.3. 5 Generalization with a Mass Parameter ...................223 C.3.6 Analogous of the Feynman Propagator ..................224 C.3.7 Solution of the Cauchy Problem ........................225 C.3.8 Remarks ............................................226 C.4 Generalized Functions Associated with D£ .....................226 C.4. 1 The Elementary Function with Causal Support ............226 C.4.2 Zero Time Properties .................................227 C.4.3 Integration Over ko ...................................227 C.4.4 The Cauchy Problem .................................228 C.5 Generalized Functions Associated with DOc ...................228 Contents xv C.S.I Elementary Solution with Causal Support ................228 C.5.2 Zero-Time Properties of Fc and its Time Derivatives .......230 C.5.3 Preferred Frame .....................................231 C.5.4 The Cauchy Problem .................................231 C.6 The G-Functions ...........................................232 Index .............................................................235
adam_txt	Contents Canonical Quantization for Constrained Systems . 1 1.1 Canonical Quantization of Mechanical Unconstrained Systems . 1 1.1.1 Lagrangian Formalism . 1 1.1.2 Hamiltonian . 1 1.1.3 Poisson Brackets . 2 1.1.4 Quantization . 2 1.2 Field Theory . 3 1.2.1 Lagrangian and Lagrangian Density . 3 1.2.2 Hamiltonian . 3 1.2.3 Poisson Brackets . 4 1.2.4 Quantization . 4 1.2.5 The Free Scalar Field . 5 1.2.6 Solution of the Klein-Gordon Equation . 5 1.3 The Electromagnetic Field . 6 1.4 Primary and Secondary Constraints . 8 1.4.1 Primary Constraints . 8 1.4.2 Secondary Constraints . 8 1.4.3 First and Second Class Constraints . 9 1.4.4 Quantization in the Presence of Second-Class Constraints . 10 1.4.5 Quantization in Presence of First-Class Constraints . 11 1.4.6 BRST Quantization . 13 1.4.7 Summary . 14 1.5 Back to the Free Massless Vector Field . 14 1.5.1 Lagrangian and Constraints . 14 1.5.2 Gauge Fixing . 16 1.6 Covariant Gauge for the Free Massless Vector Field . 16 1.6.1 Commutation Relations for Any Time . 18 1.6.2 Creation and Annihilation Operators . 21 1.6.3 The Gupta-Bleuler Formalism . 22 References . 23 Contents Quantization of the Free Electromagnetic Field in General Class III Linear Gauges . 25 2.1 Introduction . 25 2.2 Lagrangian and Field Equations . 26 2.2.1 The Lagrangian . 26 2.2.2 Euler-Lagrange Equations . 26 2.2.3 Derived Field Equations . 28 2.2.4 Sketching the Solution of the Cauchy Problem . 28 2.2.5 Particular Cases of Interest . 30 2.2.6 Planar-Type Gauges . 31 2.3 Constraint Analysis . 32 2.3.1 Canonical Momenta and Primary Constraints . 32 2.3.2 The Hamiltonian . 33 2.3.3 Constraint Chains . 33 2.4 Singular Frames . 33 2.5 Commutation Relations for Any Time . 34 2.5.1 Canonical Commutation Relations . 34 2.5.2 Commutation Relations Involving the S Field . 34 2.5.3 Commutation Relations Involving the S' Field . 36 2.5.4 Commutation Relations Involving B=d -A . 38 2.5.5 Commutation Relations Between Ад'ѕ . 42 2.5.6 Summary of the Commutation Relations for Any Time . 44 2.6 Creation and Annihilation Operators . 45 2.6.1 Momentum Space Expansion of the Fields . 45 2.6.2 Commutation Relations Between Creation and Annihilation Operators . 50 2.6.3 Summary of the Algebra of Creation and Annihilation Operators . 55 2.7 The Gupta-Bleuler Formalism . 55 2.8 Covariance Problems . 56 2.8.1 Translation Invariance . 56 2.8.2 Lorentz Transformations . 58 References . 60 Quantization of the Free Electromagnetic Field in Class II Axial Gauges . 61 3.1 Introduction . 61 3.2 Lagrangian and Field Equations . 61 3.2.1 The Lagrangian . 61 3.2.2 Euler-Lagrange Equations . 62 3.2.3 Derived Field Equations . 63 3.2.4 Summary of the Field Equations . 63 3.3 Constraint Analysis and Effective Hamiltonian . 63 3.4 Solution of Field Equations . 65 3.4.1 Solution of η dS = Q . 65 Contents xi 3.4.2 Solutíonof (п-д+к)Ѕ = 0 . 66 3.4.3 Solution of и · dB = (η2 -a)S. 67 3.4.4 Elementary Solution of и · дОА = О . 67 3.4.5 Solution of the Cauchy Problem for DA(x~z) = LxDn(x-z) 70 3.4.6 Elementary Solution of (n · d)2DA =0. 70 3.5 Commutation Relations . 71 3.5.1 Equal Time Commutators . 71 3.5.2 Commutation Relations Involving the i-Field . 72 3.5.3 Commutation Relations Involving the S'-Field . 72 3.5.4 Commutation Relations Involving д ■ A . 73 3.5.5 Commutation Relations [Αμ(χ), Αν(ζ) . 74 3.6 Association of a Feynman Propagator with the Operator η · д . 76 3.7 Associating Creation and Annihilation Operators with Fields . 77 3.7.1 TheS-Field . 78 3.7.2 Other Fields . 78 3.8 Correct Momentum Space Expansion of the Fields . 80 3.8.1 The S-Field . 80 3.8.2 The B-Field . 80 3.8.3 The S'-Field . 82 3.8.4 The Afield . 83 3.9 A Toy Model for the Unphysical 5 and В Fields . 84 3.10 Conclusions . 85 3.11 Interpolating Between Axial and Relativistic Gauges . 85 3.12 Summary . 86 References . 86 4 Gauge Fields in Interaction . 87 4.1 Introduction . 87 4.2 General Formalism of Gauge Invariance . 88 4.2.1 General Transformations . 88 4.2.2 Infinitesimal Transformations . 89 4.2.3 Remarks . 90 4.3 Class III Gauges in Yang-Mills Theory . 91 4.3.1 Building the Lagrangian . 91 4.3.2 Field Equations . 94 4.3.3 Canonical Momenta and Constraints . 94 4.3.4 The Hamiltonian . 95 4.3.5 TheBRSTCharge . 95 4.3.6 Ghost Number . 97 4.3.7 Global Internal Symmetry . 97 4.3.8 Physical States . 98 4.3.9 Problems of Covariance . 98 References . 99 Contents Perturbation Theory: RenormaUzatìon and AU That .101 5.1 Introduction .101 5.2 The S-Matrix .101 5.2.1 Definition and Properties .101 5.3 Perturbative Expansion .103 5.3.1 Expansion and Consequences of the Unitarity Condition . 103 5.3.2 Consequences of the Causality Condition .104 5.3.3 The First Term 7Ί .106 5.3.4 Nonunicity of the Tn .108 5.3.5 The Fixed Part of Tn .108 5.3.6 Wick's Theorem .109 5.3.7 Feynman Rules . Ill 5.3.8 Feynman Rules in Momentum Space .113 5.4 Divergences, Power Counting and Renormalizability .118 5.4.1 Example .118 5.4.2 Power Counting and Superficial Degree of Divergence .118 5.4.3 Renormalizability by Power Counting .120 5.5 Dimensional Regularization of Covariant Divergent Integrals .120 5.5.1 A General One-Loop Integral .120 5.5.2 Euclidean Space .121 5.5.3 Use of the Feynman Formula .121 5.5.4 Elimination of the Denominators .122 5.5.5 Complex Dimension .122 5.5.6 Tensor Integrals .123 5.6 Extension to General Covariant or Noncovariant Integrals in a Preferred Frame .124 5.6.1 The General One-Loop Integral .124 5.6.2 Euclidean Space .125 5.6.3 Elimination of the Denominators .126 5.6.4 Calculation of the Derivatives .127 5.6.5 Introduction of the Feynman Variables .128 5.6.6 Integration Over A .129 5.6.7 Regularization of Ultraviolet Divergences .129 5.6.8 Consequences of the Nonsingularity of the B~~ λ Matrix .130 5.7 Computation of the Ghost Loop With Two Legs .131 5.8 Renormalization and Counter-Terms .132 5.8.1 Various Renormalization Schemes .133 5.8.2 Multiplicative Renormalization .134 5.9 Summary .135 References .136 Contents Slavnov-Taylor Identities for Yang-Mills Theory .137 6.1 Introduction .137 6.2 The Reduction Formula .137 6.3 One-Particle Irreducible Vertex Functions .138 6.4 Yang-Mills Theory in a General Class III Gauge .139 6.4.1 The Lagrangian and Superficially Divergent Processes .139 6.4.2 BRST Symmetry, Field Equations and Canonical Commutation Relations .141 6.4.3 Commuting Derivatives and Time-Ordered Products .143 6.5 The Ward-Takahashi-Slavnov-Taylor Identity for the Gluon Self-Energy .143 6.5.1 Covariant Gauges .143 6.5.2 General Gauges .146 6.5.3 Renormalization .148 6.6 Identity for the Three-Gluon Vertex Function .149 6.6.1 Derivation of the Identity .150 6.6.2 Renormalization .150 6.7 Ghost Propagator .151 6.8 Ghost-Ghost-Gluon Vertex .153 6.8.1 Identity in Coordinate Space .153 6.8.2 Momentum Space .153 6.8.3 Remark .154 6.9 Multiplicative Renormalization .154 6.10 Summary .156 References .156 Field Theory Without Infinities .157 7.1 Introduction .157 7.2 Iterative Construction of the ¿'-Matrix Without Time-Ordering .158 7.3 Splitting of Causal Distributions into Advanced and Retarded Parts . 159 7.3.1 Distribution and Fourier Transform .159 7.3.2 Order of Singularity of a Distribution .160 7.3.3 Splitting of Distribution with Negative Order of Singularity . 161 7.3.4 Nonnegative Singularity Order .162 7.4 Application to Yang-Mills Theory .164 7.4.1 First Order .164 7.4.2 Example of a One-Loop Process: The Gluon Self-energy . 179 7.4.3 Calculation of 1\{ξ; κ). 186 7.4.4 Calculation of /2(A; Jfj , κ·,). 187 7.4.5 Calculation of the Tensor Distributions .188 7.4.6 The Final Result for the Gluon Self-energy .190 7.5 Summary .190 References .191 xiv Contents 8 Gauges with a Singular С Matrix .193 8.1 Introduction .193 8.2 The Ghost Loop Contribution .194 8.2.1 The Leibbrandt Gauges .194 8.2.2 Interpolating Between Leibbrandt and Relativistic Gauges . 197 8.3 Singularities Generated by the Lack of Power Counting in Ultra-Violet Divergent Perturbative Theory .199 8.3.1 The Loop Integration in a General Gauge .200 8.3.2 Restriction to Loops with Two External Particles .201 8.3.3 Further Restriction for m < 6.201 8.3.4 Interpolating Between Leibbrandt and Relativistic Gauges . 205 8.4 Cancellation of Divergences at a = 0 in the Self-Energy .207 References .208 9 Conclusion .209 A Notations .211 В A Useful Fourier Transform .213 С Generalized Functions .215 C.I Elementary Solutions of the Klein-Gordon Equation .215 C.I.I The A Function .215 С 1.2 The AF Function .216 C.2 The E Function .216 C.2.1 Definition .216 C.2.2 Derivation of the Elementary Solution .216 C.2.3 Zero-Time Properties .217 C.2.4 Integration Over ko .218 C.2.5 The Cauchy Problem Associated with D2 .218 C.3 Generalized Functions Associated with the Dc Operator .219 C.3.1 Elementary Solution D^ .220 C.3.2 Positive Frequency Part .221 C.3.3 Negative Frequency Part .222 C.3 .4 Elementary Solution with Causal Support .222 C.3. 5 Generalization with a Mass Parameter .223 C.3.6 Analogous of the Feynman Propagator .224 C.3.7 Solution of the Cauchy Problem .225 C.3.8 Remarks .226 C.4 Generalized Functions Associated with D£ .226 C.4. 1 The Elementary Function with Causal Support .226 C.4.2 Zero Time Properties .227 C.4.3 Integration Over ko .227 C.4.4 The Cauchy Problem .228 C.5 Generalized Functions Associated with DOc .228 Contents xv C.S.I Elementary Solution with Causal Support .228 C.5.2 Zero-Time Properties of Fc and its Time Derivatives .230 C.5.3 Preferred Frame .231 C.5.4 The Cauchy Problem .231 C.6 The G-Functions .232 Index .235
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id	DE-604.BV035147655
illustrated	Not Illustrated
index_date	2024-07-02T22:29:26Z
indexdate	2024-07-09T21:23:23Z
institution	BVB
isbn	9783540699200 9783540699217
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-016814941
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physical	XV, 236 S.
publishDate	2009
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publisher	Springer
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series	Lecture Notes in Physics
series2	Lecture Notes in Physics
spelling	Burnel, André Verfasser (DE-588)136454798 aut Noncovariant Gauges in Canonical Formalism André Burnel Berlin [u.a.] Springer 2009 XV, 236 S. txt rdacontent n rdamedia nc rdacarrier Lecture Notes in Physics 761 Gauge fields (Physics) Quantum field theory Renormalization (Physics) Eichtheorie (DE-588)4122125-4 gnd rswk-swf Eichtheorie (DE-588)4122125-4 s DE-604 Lecture Notes in Physics 761 (DE-604)BV000003166 761 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016814941&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Burnel, André Noncovariant Gauges in Canonical Formalism Lecture Notes in Physics Gauge fields (Physics) Quantum field theory Renormalization (Physics) Eichtheorie (DE-588)4122125-4 gnd
subject_GND	(DE-588)4122125-4
title	Noncovariant Gauges in Canonical Formalism
title_auth	Noncovariant Gauges in Canonical Formalism
title_exact_search	Noncovariant Gauges in Canonical Formalism
title_exact_search_txtP	Noncovariant Gauges in Canonical Formalism
title_full	Noncovariant Gauges in Canonical Formalism André Burnel
title_fullStr	Noncovariant Gauges in Canonical Formalism André Burnel
title_full_unstemmed	Noncovariant Gauges in Canonical Formalism André Burnel
title_short	Noncovariant Gauges in Canonical Formalism
title_sort	noncovariant gauges in canonical formalism
topic	Gauge fields (Physics) Quantum field theory Renormalization (Physics) Eichtheorie (DE-588)4122125-4 gnd
topic_facet	Gauge fields (Physics) Quantum field theory Renormalization (Physics) Eichtheorie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016814941&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000003166
work_keys_str_mv	AT burnelandre noncovariantgaugesincanonicalformalism

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