Quantum Field Theory II (undergraduate school) / Quantum Field Theory
(graduate school)
Objectives
The main goal of the course is to introduce tools in quantum field theory,
with an emphasis on perturbative approach.
Path-integral formulation, Symmetry and Ward identity,
quantization of gauge theory, perturbative expansion,
LSZ reduction formula,
1PI effective action, regularization and renormalization,
renormalization group, QED amplidutdes, etc, will be discussed.
Special Announcement:
Hours and Rooms:
Mondays 14:55-16:40 at Room 285 of Rigakubu-1-Go-Kan
Office Hours: Thursdays 9am-10am, Online
(zoom url given in UTAS/ITC-LMS).
Instructor:
Kentaro Hori
Kavli IPMU, Kashiwa Campus
kentaro.hori _at_ ipmu.jp
Course Plan:
Path integrals
Symmetry and Ward identity
Quantization of gauge theory
Perturbative expansion
LSZ reduction formula
Effective action
Regularization and renormalization
Renormalization group
QED
References:
Peskin and Schroeder,
An Introduction to Quantum Field Theory
Weinberg,
The Quantum Theory of Fields, I & II
Coleman,
Quantum Field Theory: Lectures of Sidney Coleman
(book),
Notes from Sidney Coleman's Physics 253a
(arXiv)
Kugo,
Geijiba no Ryoushiron, I & II (J = in Japanese)
Abers and Lee,
Gauge Theories
...
Path-integrals:
Feynman-Hibbs,
Faddeev in
Les Houches 1975,
Abers-Lee Section 11-12,
Sakita-Kikkawa (J),
Ohnuki-Suzuki-Kashiwa (J),
Kawarabayashi (J), ...
Quantization of gauge theories:
Faddeev,
Jackiw in Les Houches 1983,
Kugo Ch5 (J), ...
(For symplectic geometry:
Marsden-Weinstein (for symplectic quotients),
McDuff-Salamon,
Guillemin-Sternberg, ...)
LSZ reduction formula: Lecture is based on the exposition in Coleman
(14 & 15 in arXiv; 14 in book).
Grading Scheme:
Reports.
Report problem.
Submit your report via ITC-LMS. Deadline: January 14, 2024.
Progress:
Lecture 1 (October 16):
From operator formalism to path-integral,
Symmetry and Ward identity,
from path-integral to operator formalism (not finished).
Lecture 2 (October 23):
Remaining part of "PI to Op".
Fermioninc systems: algebra and calculus of anticommuting variables.
Lecture 3 (October 30):
Fermionic mechanics,
commutation relation and its representation, path-integral
representation of transition amplitudes and partition function.
Introduction to gauge theories.
Lecture 4 (November 6):
Introduction to gauge theory, path-integral quantization of gauge theories
(not finished):
gauge fixing condition, Faddeev-Popov determinant and ghosts,
the gauge fixed system.
Lecture 5 (November 13)
(correction):
Path-integral quantization of gauge
theories continued, BRST symmetry, physical observables;
Hamiltonian formulation and canonical quantization.
Lecture 6 (November 20):
Correlation functions vs VEV of
time ordered products; Free fields: finite system, real scalar, a part of
gauge fixed Maxwell theory.
Lecture 7 (November 27): Particle interpretation;
Free fermions: finite system, Dirac fermion, ghost system;
The gauge fixed Maxwell theory (full), computation of BRST cohomology.
Lecture 8 (December 4): Introduction to perturbation
theory: Feynman diagrams, connected vs disconnected.
Lecture 9 (December 11):
particle spectrum and interactions from correlation functions:
spectral decomposition of two point functions, scattering process
and asymptotic states.
Lecture 10 (December 18): LSZ reduction formula,
S-matrix in perturbation theory; 1PI effective action.
Lecture 11 (December 25):
Regularization and renormalization.
Lecture 12 (January 15): Renormalization revisited,
computation of the effective potential, the physical meaning of
the potential; Digression: Wilsonian renormalization group.
Lecture 13 (January 22): Renormalization group,
computation of beta function in 4d phi4 theory, RG fixed points,
RG improvement of effective potential.
Lecture 14 (January 29):
Renormalization of QED: Ward identity,
power counting, gauge invariant regularization, computation at one-loop.
Lecture 15 (February 5):
Renormalization of QED:
one-loop computation continued, Slavnov-Taylor identity,
renormalized perturbation theory,
RG flow of the coupling.
Additional notes/exercises:
For lecture 1:
Examples of path-integrals; free particle in a line, harmonic oscillator,
free particle in a circle.
For lecture 2: (a) Symmetry-twisted
partition function,
(b) Ward identity in d-dimensions.
For lecture 3: (a) Computation of
transition amplitude and partition functions,
(b) Multiple pairs of fermions,
(c) Ghost system.
For lecture 4: some math exercises.
For lecture 5: (a) Hermiticity of
the gauge fixed system,
(b) symplectic view on constrained systems.
For lecture 6: (a) free complex scalar,
(b) canonical quantization of
Maxwell theory,
(c) massive vector field,
(d) scalar propagator.
For lecture 7: BRST cohomology
of Maxwell theory.
For lecture 8: Decomposition to connected parts.
For lecture 10: (a) LSZ reduction formula
for m to n scattering,
(b) Proof of the properties of 1PI
effective action.
For lecture 11: Excercise.
For lecture 12: Detail of
computation of the effective potential.
For lecture 15: (a) Fermion self-energy
for a general gauge parameter.
(b) Computation in dimensional regularization.
Plan:
October 16:
From operator formalism to path-integral,
Symmetry and Ward identity,
from path-integral to operator formalism.
October 23:
Remaining part of "PI to Op".
Fermioninc systems: algebra and calculus, fermionic mechanics,
commutation relation and its representation, path-integral
representation of transition amplitudes and partition function.
October 30 (Online): Fermionic mechanics,
commutation relation and its representation, path-integral
representation of transition amplitudes and partition function.
November 6: Quantization of gauge theory.
November 13: Path-integral quantization of
gauge theories continued; Hamiltonian formulation and operator quantization.
November 20: Correlation functions vs VEV of
time ordered products; Free fields: finite system, real scalar, a part of
gauge fixed Maxwell theory.
November 27: Dirac fermion, ghost system;
Full system of gauge fixed Maxwell theory, BRST cohomology.
December 4: Introduction to perturbation
theory: Feynman diagrams, connected vs disconnected.
December 11:
particle spectrum and interactions from correlation functions
(spectral decomposition and LSZ reduction formula).
December 18: LSZ reduction formula,
S-matrix in perturbation theory; 1PI effective action.
December 25: Regularization and renormalization.
January 15: Renormalization revisited,
computation of the effective potential, the physical meaning of the potential.
January 22: Renormalization group,
computation of beta function in 4d phi4 theory, RG fixed points,
RG improvement of effective potential.
January 29: Renormalization of QED:
Ward identity, power counting, computation at one-loop.
February 5 (additional, online):
Renormalization of QED:
one-loop computation continued, Slavnov-Taylor identity,
renormalized perturbation theory,
RG flow of the coupling.