Theory of Elementary Particles
Objectives
The main theme of the course is non-Abelian gauge theory in 3+1 dimensions.
Yang-Mills action, BRST quantization, renormalization,
renormalization group (beta-function),
anomalies, theta angle, instantons, solitons, will be discussed.
Special Announcement:
There will be an additional lecture on July 29.
Hours and Rooms:
Mondays 13:00-14:45 at Room 285 of Rigakubu-1-Go-Kan
Instructor:
Kentaro Hori
Kavli IPMU, Kashiwa Campus
kentaro.hori _at_ ipmu.jp
Course Plan:
Non-Abelian gauge theory: the classical theory
Quantization
Renormalization and Beta function
Anomalies
Theta Vacua
Yang-Mills instantons
Solitons
References:
Homepage of
QFTII 2023-2024
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)
Descent: Zumino's lecture "Chiral anomalies and differential geometry," in Les Houches 1983 (also in Current algebra and anomalies)
Instantons: Coleman's lecture "The uses of instantons," in
Aspects of symmetry
Jackiw's lecture "Topological investigations of quantized gauge theories," in Les Houches 1983 (also in Current algebra and anomalies).
Yang-Mills instantons: Michael F. Atiyah, Geometry of Yang-Mills fields (a copy is recorded in
Atiyah's Collected works, Volume 5: Gauge Theories).
Grading Scheme: Reports.
Reprt Problem.
Submit your report via UTOL. Deadline: 23:59, August 2, 2024.
Progress:
Lecture 1 (April 8): Conventions; Yang-Mills theory,
Coupling to matter field: scalars, fermions.
Lecture 2 (April 15): Symmetry and Ward identity;
Quantization of gauge theories.
Lecture 3 (April 22):
Introduction to perturabation theory: Feynman diagrams,
connected vs disconnected.
Lecture 4 (May 7):
More on diagrams, 1PI effectiove action, Ward identity
for 1PI effective action, Ward identities in gauge theories.
Lecture 5 (May 20):
Regularization and renormalization.
Lecture 6 (revised)
(May 27):
Renormalizability of gauge theories.
Lecture 7 (revised)
(June 3): Renormalization group;
Renormalization group flow of 4d non-Abelian gauge theories (preparation).
Lecture 8 (June 10): Renormalization group flow
of 4d non-Abelian gauge theories: computation of the one-loop diagrams.
Lecture 9 (June 17): Renormalization group flow
of 4d non-Abelian gauge theories:
the one-loop beta function of the gauge coupling constant.
Lecture 10 (June 24): Anomaly: statement and
computation. (A correction to the lecture is included.)
Lecture 11 (July 1): Anomaly: computation
continued, Wess-Zumino consistency condition, Fujikawa's method.
Lecture 12 (July 8): Anomaly: Chern-Simons form,
Anomaly descent, relationship between
d=2n+2 axial anomaly and d=2n chiral anomaly.
Lecture 13 (July 22): Instantons in quantum mechanics:
instantons and anti-instantons, dilute gas approximation, periodic potential.
Lecture 14 (July 29): Instantons in 4d gauge theories,
theta vacua.
Additional notes:
For Lecture 1: Some math exercises
For Lecture 2: Path integrals
((a) examples,
(b) symmetry-twisted partition functions,
(c) Ward identities in a general dimensions)
Fermionic path-integrals
((a) amplitudes and partition functions,
(b) multiple pairs of fermions,
(c) ghost systems)
Right and left actions;
Hermiticity of gauge fixed system
Quantization of gauge theories in operator formalism
(symplectic view)
For Lecture 3: (a) Free theories,
(b) Perturbative expansion is an asymptotic expansion,
(c) Decomposition to connected parts.
For Lecture 4: (a)
Proof of the properties of the 1PI effective action.
For Lecture 5: (a)
Computation of one loop integrals.
For Lecture 6: (a) Possible form of divergent
variation (some math),
(b)
New counter terms at the N-th loop.
For Lecture 7: (a) Constraints on the
coefficients of 1PI effective action.
For Lecture 8: (a) Evaluation of the one-loop
diagrams.
For Lecture 10: (a) Axial anomaly
as a chiral anomaly.
For Lecture 11: (a) Details of chiral anomaly
computation,
(b) More on Fujikawa's method:
(b-1) chiral anomaly,
(b-2)
axial anomaly in a general gauge and gravitational background,
(c) Axial anomaly and
the index of Dirac operator,
Witten index in supersymmetric quantum mechanics.
For Lecture 12: (a) More on integrality,
(b) Descent and its derivation,
(c) Gravitational anomalies,
(d) More on descent.
For Lecture 13: (a) Ratio of determinants.
For Lecture 14: (a) Quantization of gauge theory
(a sequel to the additonal notes symplectic view/quantization of .. for Lecture 2),
(b) Construction of instantons (one instanton
in SU(2), ADHM construction).
Plan:
April 8: Conventions; Yang-Mills theory,
Coupling to matter field: scalars, fermions; Quantization of gauge
theories.
April 15: Symmetry and Ward identity; Quantization of gauge theories.
April 22: Introduction to perturabation theory.
May 7 (Tuesday!):
1PI effectiove action, Zinn-Justin equation.
May 13: No lecture due to an event of the School of Science.
May 20: Regularization and renormalization.
May 27: Renormalizability of gauge theories.
May 31 (Friday): No lecture.
June 3: Renormalization group.
June 10: Renormalization group of 4d non-Abelian
gauge theories: computation of the one-loop diagrams.
June 17: Renormalization group of 4d non-Abelian
gauge theories: the one-loop beta function of the gauge coupling constant.
June 24: Anomaly: statement and computation.
July 1: Anomaly: computation and general structure
July 8: Anomaly: Chern-Simons form,
Anomaly descent, relationship between
d=2n+2 axial anomaly and d=2n chiral anomaly
July 22: Instantons in quantum mechanics:
instantons and anti-instantons, dilute gas approximation, periodic potential
July 29: Instantons in 4d gauge theories,
theta vacua