Special topic course Fall 2001-Spring 2002 Dr. V. Shubov
Mathematical Methods of Quantum
Field Theory
I. Introduction
1. Lagrangian and Hamiltonian formalisms in Classical Mechanics.
2. Review of basic principles of Quantum Mechanics.
3. Group symmetry in Classical and Quantum Mechanics.
3.1 Main transformation groups (Lie groups) and their generators (Lie Algebras): SO(3), Lorentz, and Poincare groups.
3.2 Classical case: Hamiltonian actions of symmetry groups on a state space.
3.3 Quantum case: projective representations of symmetry groups in a Hilbert space. Relation between projective and unitary representations. Spin groups: SU(2), SL(2,C).
3.4 Wigner’s philosophy: relativistic quantum particles as unitary irreducible representations of the Poincare group in a Hilbert space.
II Classical Field Theory
1. Lagrangian formalism:
Stationary action principle, field equations, variational derivatives.
2. Hamiltonian formalism.
3. Group symmetry of the action functional and conservation laws.
3.1 Group transformations of field functions.
3.2 Noether’s theorem.
3.3 The energy-momentum tensor.
3.4 The angular-momentum tensor and the spin tensor.
3.5 The charge and the current vector.
4. The free scalar field.
4.1 Lagrangian. The Klein-Gordon equation.
4.2 Momentum representation. Positive and negative frequency components.
5. The electromagnetic field.
5.1 4-vector potential and gauge transformations. Lorentz gauge.
5.2 Lagrangian. Maxwell equations.
5.3 Energy-momentum tensor and spin.
5.4 Momentum representation.
6. The Dirac field.
6.1 Dirac equation.
6.2 Clifford algebra generated by Dirac matrices.
6.3 Action of the Poincare group on spinor fields. Relativistic invariance of the Dirac operator.
6.4 Lagrangian formalism for Dirac field.
6.5 Energy-momentum, spin, current, and charge.
6.6 Momentum representation for Dirac field.
6.7 Fundamental observables in momentum representation.
7. Introduction to Yang-Mills theory.
7.1 General relativity principle in the inner space.
7.2 Yang-Mills (gauge) fields as compensating fields.
7.3 Electromagnetic field as an example of a Yang-Mills field with an abelian gauge group SU(1).
7.4 Examples of field theories describing interacion of gauge fields with fields of matter.
(i) Quantum Electrodynamics (QED).
(ii) Quantum Chromodynamics (QCD).
7.5 Geometric interpretation of Yang-Mills fields: connections in vector bundles.
III. Quantum Theory of Free Fields
1.1 Creation and annihilation operators.
1.2 Fock’s space.
1.3 Normal ordering (Wick’s quantization).
2. Quantization of the free scalar field.
1.1 Creation and annihilation operators as quantum counterparts of the field’s
positive and negative frequency components.
1.2 Commutation relations for creation and annihilation operators (derivation
from the Heisenberg’s commutation relations).
1.3 Commutation relations for quantum field operators. Pauli-Jordan
commutation function.
1.4 Green’s functions for the Klein-Gordon equation.
(i) Advanced and retarded Green’s functions.
(ii) Feynman’s propagator (casual Green’s function).
1.5 Construction of the Fock’s space.
1.6 Fundamental observables as operators in Fock’s space.
(i) Expression in terms of creation and annihilation operators.
(ii) Normal ordering and elimination of an infinite energy of the vacuum.
1.7 Relativistic invariance of the quantized field.
3. Quantization of the electromagnetic field.
3.1 Photon creation and annihilation operators.
Problem of indefinite metrics in Fock’s space.
3.2 Gupta-Bleuler’s formalism. Quantum analog of the Lorentz gauge.
3.3 Green’s functions of the electromagnetic field. Photon propagator.
4. Quantization of the Dirac field.
4.1 Creation and annihilation operators for electrons and positrons. Anticommutation relations.
4.2 Anticommutation relations for field operators in coordinate representation.
4.3 Construction of the Fock’s space. Particles and antiparticles.
4.4 Pauli principle and Fermi-Dirac statistics.
4.5 Fundamental observables as operators in the Fock’s space (energy-momentum, charge, projection of spin, number of particles and antiparticles).
4.6 Positive definiteness of the quantum energy operator.
4.7 Green’s functions of the Dirac field. Feynman’s propagator for electrons and positrons.
IV. Quantum Theory of Interacting Fields.
1. S-matrix (scattering operator) in quantum mechanics.
1.1 Interaction picture of the dynamics of a quantum system.
1.2 Definition of the S-matrix.
1.3 Representation of the S-matrix in the form of a chronological exponential.
2. Quantum theory of the self-interacting scalar field.
2.1 Interaction picture of the dynamics. S-matrix as a chronological exponential.
2.2 Reduction of the S-matrix to the normal form
(i) Expansion of chronological products. Wick’s theorem.
(ii) Feynman diagrams for normal products in the perturbation expansion of the S-matrix.
2.3 Feynman diagrams for the matrix elements of the S-matrix.
3. Interacting system of electromagnetic and Dirac field: quantum electrodynamics.
3.1 Interaction Lagrangian of the electromagnetic and Dirac fields.
3.2 Expansion of the S-matrix into power series in powers of the interaction.
3.3 Evaluation of chronological products in quantum electrodynamics.
3.4 Feynman diagrams.
3.5 Divergent diagrams. Needs for renormalization.
3.6 Examples of calculation of second-order processes.
V. Feynman Path Integral in Quantum Mechanics and Quantum Field Theory
1. Path Integral in Quantum Mechanics.
1.1 Definition of the path integral for a quantum system with one degree of freedom: state space (Hamiltonian) form of the path integral formula for the matrix element of the quantum evolution operator.
1.2 Configuration space (Lagrangian) form of the path integral.
1.3 Derivation of the configuration space version of the path integral. The Trotter-Lie formula.
1.4 Derivation of the state space version of the path integral using Weyl’s quantization.
1.5 Generalization to systems with n degrees of freedom.
1.6 Path integral for a free particle.
1.7 Path integral for harmonic oscillator.
(i) Functional Taylor’s formula for the action.
(ii) Evaluation of the functional Gaussian integral.
(iii) Determinant of the Sturm-Liouville operator.
1.8 Path integral for forced harmonic oscillator.
1.9 Path integral method for perturbed harmonic oscillator. Perturbation expansion in terms of the generating functional.
2. Path integral in field theory.
2.1 Path integral formula for self-interacting scalar field theory in Hamiltonian and Lagrangian form.
2.2 Free scalar field.
(i) Formal calculation of the Gaussian functional integral.
(ii) Spatial Fourier transformation and factorization into oscillator path integrals.
2.3 Free scalar field with an exteral source.
(i) Formal calculation of the path intergal.
(ii) Factorization into oscillator integrals.
(iii) Passing to the infinite time interval and recovery of the Feynman’s propagator.
2.4 Self-interacting scalar field.
(i) Perturbation series for Green’s functions in terms of the generating functional.
(ii) Path integral version of the Wick’s theorem.
(iii) Feynman diagrams for Green’s functions in the coordinate and momentum representations.
2.5 The Lehman, Symanzik, and Zimmermann (LSZ) reduction formula.
(i) Derivation for perturbed harmonic oscillator.
(ii) Statement and derivation for self-interacting scalar field.
(iii) Feynman diagrams for S-matrix.
2.6 Alternative approach: path integral in the holomorphic (coherent state) representation.
(i) Holomorphic representation for free oscillator.
(ii) Oscillator path integral in holomorphic representation.
(iii) Generating functional for S-matrix in scalar field theory and Feynman diagrams.
2.7 Path integral method in Quantum Electrodynamics.
(i) Free electromagnetic field.
(ii) Free Dirac field.
(iii) Interacting electromagnetic and Dirac fields. Derivation of Feynman rules.
VI. Renormalization Theory
1. Renormalization in self-interacting scalar field theory.
1.1 Divergent diagrams.
1.2 Renormalization procedure. Counterterms.
1.3 Renormalization group.
2. Renormalization in QED.
1.1 Divergent diagrams.
1.2 Furry’s theorem.
1.3 Counterterms.
1.4 Mass and charge renormalization.