Quantum Field Theory 2
A.Y. 2025/2026
Learning objectives
Expand the core ideas of relativistic quantum field theory which have been introduced in Quantum Field Theory 1, specifically in what
concerns analiticity, symmetry and invariance
concerns analiticity, symmetry and invariance
Expected learning outcomes
At the end of this course the student:
1.Will be able to use unitarity and the optical theorem to understand the analytic properties of amplitudes;
2.Derive the Ward identities for symmetres realized in Wigner-Weyl form;
3.Prove Glodstone's theorem for spontaneously broken symmetries, both at the classical and quantum level;
4.Construct and compute the effective potential;
5.Quantize a gauge theory and derive its Feynman rules with various gauge choices
6.Construct a gauge theory with massive field via the Higgs mechanism;
7.Renormalize quantum electrodymanics perturbatively;
8.Understand the quantum breaking of classical symmetries related to scale invariance (including chiral anomalies);
9.Write donw and solve the Callan-Symanzik equation (renormalization group equation);
10.Compute the operator-product (Wilson) expansion and the anomaloud dimensions of operators entering it.
1.Will be able to use unitarity and the optical theorem to understand the analytic properties of amplitudes;
2.Derive the Ward identities for symmetres realized in Wigner-Weyl form;
3.Prove Glodstone's theorem for spontaneously broken symmetries, both at the classical and quantum level;
4.Construct and compute the effective potential;
5.Quantize a gauge theory and derive its Feynman rules with various gauge choices
6.Construct a gauge theory with massive field via the Higgs mechanism;
7.Renormalize quantum electrodymanics perturbatively;
8.Understand the quantum breaking of classical symmetries related to scale invariance (including chiral anomalies);
9.Write donw and solve the Callan-Symanzik equation (renormalization group equation);
10.Compute the operator-product (Wilson) expansion and the anomaloud dimensions of operators entering it.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
A. Unitarity and analyticity
1. The optical theorem
2. Feynman diagrams and Cutkosky rules
3. Decay amplitudes
B. Ward identities
1. Symmetries and current algebra
2. The Ward identity for the two-point Green function
3. Ward identities from the path integral
4. Examples: QED and Φ^4 theory
C. Spontaneous symmetry breaking
1. Goldstone's theorem in classical field theory
2. Goldstone's theorem and Ward identities
3. The effective potential
D. Gauge invariance
1. Geometric interpretation
2. Nonabelian gauge theories
3 Quantization of constrained systems and Faddeev formula
4. Quantization of gauge theories
5. The Higgs mechanism
E. Renormalization
1. Renormalization of QED
2. Scale invariance
3. Running coupling
4. The CallanSymanzik equation and the renormalization group
5. The operatorproduct expansion (Wilson expansion)
F. The chiral anomaly
1. Conservation of the axial current
2. The theta vacuum
1. The optical theorem
2. Feynman diagrams and Cutkosky rules
3. Decay amplitudes
B. Ward identities
1. Symmetries and current algebra
2. The Ward identity for the two-point Green function
3. Ward identities from the path integral
4. Examples: QED and Φ^4 theory
C. Spontaneous symmetry breaking
1. Goldstone's theorem in classical field theory
2. Goldstone's theorem and Ward identities
3. The effective potential
D. Gauge invariance
1. Geometric interpretation
2. Nonabelian gauge theories
3 Quantization of constrained systems and Faddeev formula
4. Quantization of gauge theories
5. The Higgs mechanism
E. Renormalization
1. Renormalization of QED
2. Scale invariance
3. Running coupling
4. The CallanSymanzik equation and the renormalization group
5. The operatorproduct expansion (Wilson expansion)
F. The chiral anomaly
1. Conservation of the axial current
2. The theta vacuum
Prerequisites for admission
Knowledge of the basics of relativistic quantum field theory, special relativity, and path integral methods as covered in the Quantum Field Theory I course.
Teaching methods
The course consists of blackboard lectures in which the individual topics included in the syllabus are presented, first introducing the basics and then discussing the main conceptual points and computational techniques. Interaction with the students in class is very much encouraged through questions and discussions.
Teaching Resources
M.E. Peskin, D.V. Schroeder: An introduction to Quantum Field Theory; Addison-Wesley, 1995 (for specific topics and reference textbook)
T.P. Cheng, L.F. Li, Gauge Theory of Elementary Particle Physics; Oxford University Press, 1985 (for special topics)
S. Coleman: Aspects of Symmetry; Cambridge University Press, 1985 (for special topics)
R. Jackiw: Topological Investigations of Quantized Gauge Theories: in Current Algebra and Anomalies; Princeton University Press, 1985 (for special topics)
A. Zee, Quantum Field Theory in a Nutshell; Princeton University Press, 2010 (for extra insight, especially at a conceptual and qualitative level)
S. Weinberg: The Quantum Theory of Fields: Vol. I (foundations); Cambridge University Press, 1995 (for extra insight, especially at a more advanced and formal level)
T.P. Cheng, L.F. Li, Gauge Theory of Elementary Particle Physics; Oxford University Press, 1985 (for special topics)
S. Coleman: Aspects of Symmetry; Cambridge University Press, 1985 (for special topics)
R. Jackiw: Topological Investigations of Quantized Gauge Theories: in Current Algebra and Anomalies; Princeton University Press, 1985 (for special topics)
A. Zee, Quantum Field Theory in a Nutshell; Princeton University Press, 2010 (for extra insight, especially at a conceptual and qualitative level)
S. Weinberg: The Quantum Theory of Fields: Vol. I (foundations); Cambridge University Press, 1995 (for extra insight, especially at a more advanced and formal level)
Assessment methods and Criteria
The exam is an oral examination of about 45 minutes, during which the student is asked to discuss one topic selected among those included in the
syllabus of the course. During the exam, the student is asked a number of questions of variable complexity, which aim at ascertaining his basic understanding of the various topics covered in class, his ability to place them in the more general context of quantum field theory, and his ability to think critically and autonomously using these methods. Given the very extensive number of topics covered in the course, the students can ask in the beginning of the exam to exclude questions in one of the sets of topic included in the syllabus.
syllabus of the course. During the exam, the student is asked a number of questions of variable complexity, which aim at ascertaining his basic understanding of the various topics covered in class, his ability to place them in the more general context of quantum field theory, and his ability to think critically and autonomously using these methods. Given the very extensive number of topics covered in the course, the students can ask in the beginning of the exam to exclude questions in one of the sets of topic included in the syllabus.
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 6
Lessons: 42 hours
Professor:
Forte Stefano
Professor(s)