Table Of ContentLecture Notes in Physics
Chueng-Ryong Ji
Relativistic
Quantum
Invariance
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Chueng-Ryong Ji
Relativistic Quantum
Invariance
Chueng-RyongJi
DepartmentofPhysics
NorthCarolinaStateUniversity
Raleigh,NC,USA
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-981-19-7948-4 ISBN978-981-19-7949-1 (eBook)
https://doi.org/10.1007/978-981-19-7949-1
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vii
About This Book
This book describes the invariant nature of the relativistic quantum field theories
utilizing the idea of interpolating the instant form dynamics and the light-front
dynamics.Whilethelight-frontdynamics(LFD)basedonthelight-fronttimewas
proposedbyDiracin1949,there hasnotyetbeenasalientreview ontheconnec-
tion between the LFD and the instant form dynamics (IFD) based on the ordinary
time. Reviewing the connection between LFD and IFD using the idea of inter-
polating the two different forms of the relativistic dynamics, one can learn the
distinguished features of each form and how one may utilize those distinguished
features in solving the complicate relativistic quantum field theoretic problems
more effectively. With the ongoing 12-GeV Jefferson Lab experiments and the
forthcoming electron-ion collider (EIC) facilities, the internal structures of the
nucleon and nuclei are vigorously investigated in particular using the physical
observables defined in the LFD rather than in the IFD. This book offers a clear
demonstrationonwhyandhowtheLFDismoreadvantageousthantheIFDforthe
studyofhadronphysics,illustratingthedifferencesandsimilaritiesbetweenthese
twodistinguishedformsofthedynamics.Itaimsatpresentingthebasicfirst-hand
knowledge of the relativistic quantum field theories, describing why and how the
different forms of dynamics (e.g. IFD and LFD) can emerge in them, connecting
theIFDandtheLFDusingtheideaoftheinterpolation,anddemonstratingexplicit
examplesoftheinterpolationinquantumelectrodynamicsandotherfieldtheories.
While the level of presentation is planned mainly for the advanced undergradu-
ate students and the beginning graduate students, the topics of the interpolation
between the IFD and the LFD are innovative enough for even the experts in the
field to appreciate its usefulness.
ix
Contents
1 Introduction ......................................................... 1
1.1 Special Relativity ............................................... 1
1.2 Four-Vectors and Relativistic Collisions .......................... 3
1.3 Symmetry Properties of Special Relativity ....................... 8
1.4 Poincaré Group ................................................. 11
References ........................................................... 16
2 InterpolationBetweenInstantFormDynamicsandLight-Front
Dynamics ............................................................ 17
2.1 Dirac’s Proposition .............................................. 17
2.2 Interpolating Scattering Amplitudes .............................. 20
2.3 Kinematic Transformations of Particle Momenta ................. 26
2.4 Application of Transformations on Interpolating Scattering
Amplitudes ...................................................... 38
2.5 Interpolating Scattering Amplitudes in Infinite Momentum
Frame ........................................................... 41
2.6 Conclusions ..................................................... 45
References ........................................................... 46
3 InterpolationofQuantumElectrodynamics .......................... 49
3.1 Introduction ..................................................... 49
3.2 Formal Derivation of the Interpolation of QED ................... 54
3.2.1 Scattering Theory ........................................ 54
3.2.2 Canonical Field Theory .................................. 61
+ −
3.3 Toy Calculation of e e Annihilation Producing Two Scalar
Particles ......................................................... 70
3.3.1 Collinear Scattering/Annihilation, θ =π .................. 75
3.3.2 Summary of e+e− → Two Scalar Particles ............... 81
3.4 Interpolating Helicity Scattering Probabilities .................... 84
+ −
3.4.1 e e Pair Annihilation into Two Photons ................. 84
3.4.2 Compton Scattering ...................................... 97
3.5 Summary and Conclusion ....................................... 107
References ........................................................... 108
xi
xii Contents
4 InterpolationofQuantumChromodynamicsin1+1Dimension ...... 111
4.1 Introduction ..................................................... 112
4.2 The Mass Gap Equation ......................................... 115
4.2.1 The Hamiltonian Method ................................. 116
4.2.2 The Feynman Diagram Method ........................... 122
4.2.3 Behavior of the Gap Equation When Approaching
the Light-Front ........................................... 125
4.3 The Mass Gap Solution ......................................... 128
4.4 Chiral Condensate and Constituent Quark Mass .................. 134
4.4.1 The Chiral Condensate ................................... 134
4.4.2 The Fermion Propagator and Constituent Mass ............ 137
4.5 The Bound-State Equation ....................................... 146
4.6 The Bound-State Solution ....................................... 151
4.6.1 Spectroscopy ............................................. 152
4.6.2 Wavefunctions ........................................... 158
4.6.3 Quasi-PDFs .............................................. 162
4.7 Conclusion and Outlook ......................................... 168
References ........................................................... 170
QEDAppendixA ........................................................ 173
QCD1+1AppendixB .................................................... 221
