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Gauge Theories in Particle Physics From Relativistic Quantum Mechanics to QED by Ian J.R. Aitchison and Anthony J.G. Hey



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Gauge Theories in Particle Physics From Relativistic Quantum Mechanics to QED written by Ian J.R. Aitchison and Anthony J.G. Hey . This new fourth edition retains the two-volume format, which has been generally well received, with broadly the same allocation of content as in the third edition. The principal new additions are, once again, dictated by substantial new experimental results – namely, in the areas of CP violation and neutrino oscillations, where great progress was made in the first decade of this century. Volume 2 now includes a new chapter devoted to CP violation and oscillations in mesonic and neutrino systems. Partly by way of preparation for this, volume 1 also contains a new chapter, on Lorentz transformations and discrete symmetries. We give a simple do-it-yourself treatment of Lorentz transformations of Dirac spinors, which the reader can connect to the group theory approach in appendix M of volume 2; the transformation properties of bilinear covariants are easily managed. We also introduce Majorana fermions at an early stage. This material is suitable for first courses on relativistic quantum mechanics, and perhaps should have been included in earlier editions (we thank a referee for urging its inclusion now).
To make room for the new chapter in volume 1, the two introductory chapters of the third edition have been condensed into a single one, in the knowledge that excellent introductions to the basic facts of particle physics are available elsewhere. Otherwise, apart from correcting the known minor errors and misprints, the only other changes in volume 1 are some minor improvements in presentation, and appropriate updates on experimental numbers. Volume 2 contains significantly more in the way of updates and additions, as will be detailed in the Preface to that volume. But we have continued to omit discussion of speculations going beyond the Standard Model; after all, the crucial symmetry-breaking (Higgs) sector has only now become experimentally accessible.


Gauge Theories in Particle Physics From Relativistic Quantum Mechanics to QED written by Ian J.R. Aitchison and Anthony J.G. Hey cover the following topics.


  • I Introductory Survey, Electromagnetism as a Gauge Theory, and Relativistic Quantum Mechanics

  • 1 The Particles and Forces of the Standard Model
    1.1 Introduction: the Standard Model
    1.2 The fermions of the Standard Model
    1.2.1 Leptons
    1.2.2 Quarks
    1.3 Particle interactions in the Standard Model
    1.3.1 Classical and quantum fields
    1.3.2 The Yukawa theory of force as virtual quantum exchange
    1.3.3 The one-quantum exchange amplitude
    1.3.4 Electromagnetic interactions
    1.3.5 Weak interactions
    1.3.6 Strong interactions
    1.3.7 The gauge bosons of the Standard Model
    1.4 Renormalization and the Higgs sector of the Standard Model
    1.4.1 Renormalization
    1.4.2 The Higgs boson of the Standard Model
    1.5 Summary
    Problems

  • 2 Electromagnetism as a Gauge Theory
    2.1 Introduction
    2.2 The Maxwell equations: current conservation
    2.3 The Maxwell equations: Lorentz covariance and gauge invariance
    2.4 Gauge invariance (and covariance) in quantum mechanics
    2.5 The argument reversed: the gauge principle
    2.6 Comments on the gauge principle in electromagnetism
    Problems

  • 3 Relativistic Quantum Mechanics
    3.1 The Klein–Gordon equation
    3.1.1 Solutions in coordinate space
    3.1.2 Probability current for the KG equation
    3.2 The Dirac equation
    3.2.1 Free-particle solutions
    3.2.2 Probability current for the Dirac equation
    3.3 Spin
    3.4 The negative-energy solutions
    3.4.1 Positive-energy spinors
    3.4.2 Negative-energy spinors
    3.4.3 Dirac’s interpretation of the negative-energy solutions of the Dirac equation
    3.4.4 Feynman’s interpretation of the negative-energy solutions of the KG and Dirac equations
    3.5 Inclusion of electromagnetic interactions via the gauge principle: the Dirac prediction of g = 2 for the electron
    Problems

  • 4 Lorentz Transformations and Discrete Symmetries
    4.1 Lorentz transformations
    4.1.1 The KG equation
    4.1.2 The Dirac equation
    4.2 Discrete transformations: P, C and T
    4.2.1 Parity
    4.2.2 Charge conjugation
    4.2.3 CP
    4.2.4 Time reversal
    4.2.5 CPT
    Problems

  • II Introduction to Quantum Field Theory

  • 5 Quantum Field Theory I: The Free Scalar Field
    5.1 The quantum field: (i) descriptive
    5.2 The quantum field: (ii) Lagrange–Hamilton formulation
    5.2.1 The action principle: Lagrangian particle mechanics
    5.2.2 Quantum particle mechanics `a la Heisenberg–Lagrange–Hamilton
    5.2.3 Interlude: the quantum oscillator
    5.2.4 Lagrange–Hamilton classical field mechanics
    5.2.5 Heisenberg–Lagrange–Hamilton quantum field mechanics
    5.3 Generalizations: four dimensions, relativity and mass
    Problems

  • 6 Quantum Field Theory II: Interacting Scalar Fields
    6.1 Interactions in quantum field theory: qualitative introduction
    6.2 Perturbation theory for interacting fields: the Dyson expansion of the S-matrix
    6.2.1 The interaction picture
    6.2.2 The S-matrix and the Dyson expansion
    6.3 Applications to the ‘ABC’ theory
    6.3.1 The decay C ? A + B
    6.3.2 A + B ? A + B scattering: the amplitudes
    6.3.3 A + B ? A + B scattering: the Yukawa exchange mechanism, s and u channel processes
    6.3.4 A + B ? A + B scattering: the differential cross section
    6.3.5 A + B ? A + B scattering: loose ends
    Problems

  • 7 Quantum Field Theory III: Complex Scalar Fields, Dirac and Maxwell Fields; Introduction of Electromagnetic Interactions
    7.1 The complex scalar field: global U(1) phase invariance, particles and antiparticles
    7.2 The Dirac field and the spin-statistics connection
    7.3 The Maxwell field Aµ(x)
    7.3.1 The classical field case
    7.3.2 Quantizing Aµ(x)
    7.4 Introduction of electromagnetic interactions
    7.5 P, C and T in quantum field theory
    7.5.1 Parity
    7.5.2 Charge conjugation
    7.5.3 Time reversal
    Problems

  • III Tree-Level Applications in QED

  • 8 Elementary Processes in Scalar and Spinor Electrodynamics
    8.1 Coulomb scattering of charged spin-0 particles
    8.1.1 Coulomb scattering of s+ (wavefunction approach)
    8.1.2 Coulomb scattering of s+ (field-theoretic approach)
    8.1.3 Coulomb scattering of s-
    8.2 Coulomb scattering of charged spin- 12 particles
    8.2.1 Coulomb scattering of e- (wavefunction approach)
    8.2.2 Coulomb scattering of e-(field-theoretic approach)
    8.2.3 Trace techniques for spin summations
    8.2.4 Coulomb scattering of e+
    8.3 e-s+ scattering
    8.3.1 The amplitude for e-s+ ? e-s+
    8.3.2 The cross section for e-s+ ? e-s+
    8.4 Scattering from a non-point-like object: the pion form factor in e-p+ ? e-p+
    8.4.1 e- scattering from a charge distribution
    8.4.2 Lorentz invariance
    8.4.3 Current conservation
    8.5 The form factor in the time-like region: e+e- ? p+p- and crossing symmetry
    8.6 Electron Compton scattering
    8.6.1 The lowest-order amplitudes
    8.6.2 Gauge invariance
    8.6.3 The Compton cross section
    8.7 Electron muon elastic scattering
    8.8 Electron–proton elastic scattering and nucleon form factors
    8.8.1 Lorentz invariance
    8.8.2 Current conservation
    Problems

  • 9 Deep Inelastic Electron–Nucleon Scattering and the Parton Model
    9.1 Inelastic electron–proton scattering: kinematics and structure functions
    9.2 Bjorken scaling and the parton model
    9.3 Partons as quarks and gluons
    9.4 The Drell–Yan process
    9.5 e+e- annihilation into hadrons
    Problems

  • IV Loops and Renormalization

  • 10 Loops and Renormalization I: The ABC Theory
    10.1 The propagator correction in ABC theory
    10.1.1 The O(g2) self-energy ?[2] C (q2)
    10.1.2 Mass shift
    10.1.3 Field strength renormalization
    10.2 The vertex correction
    10.3 Dealing with the bad news: a simple example
    10.3.1 Evaluating ?[2]C (q2)
    10.3.2 Regularization and renormalization
    10.4 Bare and renormalized perturbation theory
    10.4.1 Reorganizing perturbation theory
    10.4.2 The O(g2 ph) renormalized self-energy revisited: how counter terms are determined by renormalization conditions
    10.5 Renormalizability
    Problems

  • 11 Loops and Renormalization II: QED
    11.1 Counter terms
    11.2 The O(e2) fermion self-energy
    11.3 The O(e2) photon self-energy
    11.4 The O(e2) renormalized photon self-energy
    11.5 The physics of ?¯ [2] ? (q2)
    11.5.1 Modified Coulomb’s law
    11.5.2 Radiatively induced charge form factor
    11.5.3 The running coupling constant
    11.5.4 ?¯ [2] ? in the s-channel
    11.6 The O(e2) vertex correction, and Z1 = Z2
    11.7 The anomalous magnetic moment and tests of QED
    11.8 Which theories are renormalizable – and does it matter? Problems

  • Appendix
    A Non-relativistic Quantum Mechanics 361
    B Natural Units 365
    C Maxwell’s Equations: Choice of Units 369
    D Special Relativity: Invariance and Covariance 371
    E Dirac d-Function 377
    F Contour Integration 387
    G Green Functions 393
    H Elements of Non-relativistic Scattering Theory
    H.1 Time-independent formulation and differential cross section
    H.2 Expression for the scattering amplitude: Born approximation
    H.3 Time-dependent approach
    I The Schr¨odinger and Heisenberg Pictures
    J Dirac Algebra and Trace Identities
    J.1 Dirac algebra
    J.1.1 ? matrices
    J.1.2 ?5 identities
    J.1.3 Hermitian conjugate of spinor matrix elements
    J.1.4 Spin sums and projection operators
    J.2 Trace theorems
    K Example of a Cross Section Calculation
    K.1 The spin-averaged squared matrix element
    K.2 Evaluation of two-body Lorentz-invariant phase space in ‘laboratory’ variables
    L Feynman Rules for Tree Graphs in QED
    L.1 External particles
    L.2 Propagators
    L.3 Vertices

  • References

  • Index

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