Methods of Quantization: Lectures Held at the 39. Universitätswochen für Kern-und Teilchenphysik, Schladming, Austria

234 Pages·2001·1.705 MB·English

by  Bernard L.G. Bakker (auth.), H. Latal, W. Schweiger (eds.)

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H. Latal W. Schweiger (Eds.)
Methods
of Quantization
Lectures Held at the 39. Universita¨tswochen fu¨r
Kern- und Teilchenphysik, Schladming, Austria
1 3
Editors
H.Latal
W.Schweiger
Institutfu¨rTheoretischePhysik
Universita¨tGraz
Universita¨tsplatz5
8010Graz,Austria
SupportedbytheO¨sterreichischeBundesministeriumfu¨rWirtschaft,
VerkehrundKunst,Vienna,Austria
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Methodsofquantization:lecturesheldatthe39.Universita¨tswochenfu¨r
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(Lecturenotesinphysics;572)
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Preface
This volume contains the written versions of invited lectures presented at
the“39.InternationaleUniversita¨tswochenfu¨rKern-undTeilchenphysik”in
Schladming, Austria, which took place from February 26th to March 4th,
2000. The title of the school was “Methods of Quantization”. This is, of
course, a very broad field, so only some of the new and interesting develop-
ments could be covered within the scope of the school.
About 75 years ago Schro¨dinger presented his famous wave equation and
Heisenberg came up with his algebraic approach to the quantum-theoretical
treatment of atoms. Aiming mainly at an appropriate description of atomic
systems, these original developments did not take into consideration Ein-
stein’s theory of special relativity. With the work of Dirac, Heisenberg, and
Pauliitsoonbecameobviousthataunifiedtreatmentofrelativisticandquan-
tumeffectsisachievedbymeansoflocalquantumfieldtheory,i.e.anintrinsic
many-particle theory. Most of our present understanding of the elementary
building blocks of matter and the forces between them is based on the quan-
tizedversionoffieldtheorieswhicharelocallysymmetricundergaugetrans-
formations. Nowadays, the prevailing tools for quantum-field theoretical cal-
culations are covariant perturbation theory and functional-integral methods.
Being not manifestly covariant, the Hamiltonian approach to quantum-field
theories lags somewhat behind, although it resembles very much the familiar
nonrelativisticquantummechanicsofpointparticles.Aparticularlyinterest-
ing Hamiltonian formulation of quantum-field theories is obtained by quan-
tizing the fields on hypersurfaces of the Minkowsi space which are tangential
to the light cone. The “time evolution” of the system is then considered in
“light-cone time” x+ = t+z/c. The appealing features of “light-cone quan-
tization”, which are the reasons for the renewed interest in this formulation
ofquantumfieldtheories,werehighlightedinthelecturesofBernardBakker
and Thomas Heinzl. One of the open problems of light-cone quantization is
theissueofspontaneoussymmetrybreaking.Thiscanbetracedbacktozero
modes which, in general, are subject to complicated constraint equations. A
general formalism for the quantization of physical systems with constraints
waspresentedbyJohnKlauder.Theperturbativedefinitionofquantumfield
theories is in general afflicted by singularities which are overcome by a regu-
larizationandrenormalizationprocedure.Structuralaspectsoftherenormal-
VI Preface
ization problem in the case of gauge invariant field theories were discussed
in the lecture of Klaus Sibold. A review of the mathematics underlying the
functional-integral quantization was given by Ludwig Streit.
Apart from the topics included in this volume there were also lectures
on the Kaluza–Klein program for supergravity (P. van Nieuwenhuizen), on
dynamical r-matrices and quantization (A. Alekseev), and on the quantum
Liouville model as an instructive example of quantum integrable models (L.
Faddeev). In addition, the school was complemented by many excellent sem-
inars. The list of seminar speakers and the topics addressed by them can be
foundattheendofthisvolume.Theinterestedreaderisrequestedtocontact
the speakers directly for detailed information or pertinent material.
Finally,wewouldliketoexpressourgratitudetothelecturersforalltheir
effortsandtothemainsponsorsoftheschool,theAustrianMinistryofEdu-
cation,Science,andCultureandtheGovernmentofStyria,forprovidinggen-
erous support. We also appreciate the valuable organizational and technical
assistanceofthetownofSchladming,theSteyr-Daimler-PuchFahrzeugtech-
nik, Ricoh Austria, Styria Online, and the Hornig company. Furthermore,
we thank our secretaries, S. Fuchs and E. Monschein, a number of gradu-
ate students from our institute, and, last but not least, our colleagues from
the organizing committee for their assistance in preparing and running the
school.
Graz, Heimo Latal
March 2001 Wolfgang Schweiger
Contents
Forms of Relativistic Dynamics
Bernard L.G. Bakker ............................................ 1
1 Introduction ................................................ 1
2 The Poincar´e Group ......................................... 3
3 Forms of Relativistic Dynamics ............................... 4
3.1 Comparison of Instant Form, Front Form, and Point Form ... 6
4 Light-Front Dynamics........................................ 9
4.1 Relative Momentum, Invariant Mass ...................... 9
4.2 The Box Diagram....................................... 14
5 Poincar´e Generators in Field Theory........................... 19
5.1 Fermions Interacting with a Scalar Field................... 20
5.2 Instant Form........................................... 20
5.3 Front Form (LF)........................................ 21
5.4 Interacting and Non-interacting Generators on an Instant
and on the Light Front .................................. 22
6 Light-Front Perturbation Theory .............................. 23
6.1 Connection of Covariant Amplitudes
to Light-Front Amplitudes ............................... 24
6.2 Regularization.......................................... 26
6.3 Minus Regularization.................................... 26
7 Triangle Diagram in Yukawa Theory........................... 27
7.1 Covariant Calculation .................................. 28
7.2 Construction of the Current in LFD....................... 30
7.3 Numerical Results ...................................... 37
8 Four Variations on a Theme in φ3 Theory ...................... 37
8.1 Covariant Calculation ................................... 39
8.2 Instant-Form Calculation ................................ 42
8.3 Calculation in Light-Front Coordinates .................... 47
8.4 Front-Form Calculation.................................. 49
9 Dimensional Regularization: Basic Formulae .................... 51
10 Four-Dimensional Integration ................................. 52
11 Some Useful Integrals........................................ 53
References ..................................................... 53
VIII Contents
Light-Cone Quantization: Foundations and Applications
Thomas Heinzl ................................................. 55
1 Introduction ................................................ 55
2 Relativistic Particle Dynamics ................................ 58
2.1 The Free Relativistic Point Particle ....................... 58
2.2 Dirac’s Forms of Relativistic Dynamics .................... 64
2.3 The Front Form ........................................ 68
3 Light-Cone Quantization of Fields ............................. 74
3.1 Construction of the Poincar´e Generators................... 74
3.2 Schwinger’s (Quantum) Action Principle................... 76
3.3 Quantization as an Initial- and/or Boundary-Value Problem.. 78
3.4 DLCQ – Basics......................................... 84
3.5 DLCQ – Causality ...................................... 88
3.6 The Functional Schro¨dinger Picture ....................... 94
3.7 The Light-Cone Vacuum................................. 96
4 Light-Cone Wave Functions................................... 98
4.1 Kinematics............................................. 99
4.2 Definition of Light-Cone Wave Functions .................. 101
4.3 Properties of Light-Cone Wave Functions .................. 104
4.4 Examples of Light-Cone Wave Functions................... 105
5 The Pion Wave Function in the NJL Model..................... 113
5.1 A Primer on Spontaneous Chiral Symmetry Breaking ....... 114
5.2 NJL Folklore........................................... 117
5.3 Schwinger–Dyson Approach .............................. 121
5.4 Observables ............................................ 127
6 Conclusions................................................. 136
References ..................................................... 138
Quantization of Constrained Systems
John R. Klauder................................................ 143
1 Introduction ................................................ 143
1.1 Initial Comments ....................................... 143
1.2 Classical Background.................................... 144
1.3 Quantization First: Standard Operator Quantization ........ 145
1.4 Reduction First: Standard Path Integral Quantization ....... 146
1.5 Quantization First (cid:1)≡ Reduction First ..................... 147
1.6 Outline of the Remaining Sections ........................ 148
2 Overview of the Projection Operator Approach
to Constrained System Quantization........................... 148
2.1 Coherent States ........................................ 148
2.2 Constraints ............................................ 149
2.3 Dynamics for First-Class Systems......................... 150
2.4 Zero in the Continuous Spectrum ......................... 151
2.5 Alternative View of Continuous Zeros ..................... 152
Contents IX
3 Coherent State Path Integrals
Without Gauge Fixing ....................................... 152
3.1 Enforcing the Quantum Constraints....................... 153
3.2 Reproducing Kernel Hilbert Spaces ....................... 154
3.3 Reduction of the Reproducing Kernel ..................... 155
3.4 Single Regularized Constraints ........................... 156
3.5 Basic First-Class Constraint Example ..................... 157
4 Application to General Constraints ............................ 158
4.1 Classical Considerations ................................. 158
4.2 Quantum Considerations ................................ 160
4.3 Universal Procedure
to Generate Single Regularized Constraints ................ 162
4.4 Basic Second-Class Constraint Example ................... 164
4.5 Conversion Method ..................................... 165
4.6 Equivalent Representations .............................. 166
4.7 Equivalence of Criteria for Second-Class Constraints ........ 167
5 Selected Examples of First-Class Constraints.................... 168
5.1 General Configuration Space Geometry.................... 168
5.2 Finite-Dimensional Hilbert Space Examples ................ 170
5.3 Helix Model............................................ 172
5.4 Reparameterization Invariant Dynamics ................... 173
5.5 Elevating the Lagrange Multiplier
to an Additional Dynamical Variable...................... 175
6 Special Applications ......................................... 176
6.1 Algebraically Inequivalent Constraints..................... 176
6.2 Irregular Constraints .................................... 178
7 Some Other Applications
of the Projection Operator Approach .......................... 180
References ..................................................... 181
Algebraic Methods of Renormalization
Klaus Sibold.................................................... 183
1 Generalities................................................. 183
1.1 Renormalization Schemes ................................ 183
1.2 The Action Principle.................................... 186
1.3 Green Functions and Operators........................... 189
2 The Quantization of Gauge Theories........................... 190
2.1 The Abelian Case....................................... 190
2.2 BRS Transformations ................................... 192
2.3 The Slavnov–Taylor Identity ............................. 194
3 Applications ................................................ 197
3.1 The Electroweak Standard Model......................... 197
3.2 Supersymmetry in Non-linear Realization .................. 201
3.3 SUSY Gauge Theories................................... 202
References ..................................................... 205
X Contents
Functional Integrals for Quantum Theory
Ludwig Streit................................................... 207
1 Introduction ................................................ 207
2 White Noise Analysis ........................................ 208
2.1 Smooth and Generalized Functionals ...................... 210
2.2 Characterization of Generalized Functionals Φ∈(S)∗........ 210
2.3 Calculus............................................... 212
3 Quantum Field Theory....................................... 213
3.1 The Vacuum Density.................................... 213
3.2 Dynamics in Terms of the Vacuum........................ 214
4 Feynman Integrals........................................... 216
4.1 The Interactions........................................ 218
4.2 The Morse Potential .................................... 220
References ..................................................... 221
Seminars ..................................................... 223