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Quantum field theory 2

A.Y. 2019/2020

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.

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

Course syllabus and organization

### Single session

Responsible

Lesson period

First semester

**Course syllabus**

Syllabus:

1. Unitarity ed analiticity

1.1 The opitcal theorem

1.2 Feynman diagrams and Cutkosky rules

1.3 Decay amplitudes

2. The Ward identities

2.1 Symmetries and current algebra

2.2 Ward identities for the two-point Green function

2.3 The Ward identities from the path integral

2.4 Examples: QED and Φ^4

3. Spontaneous symmetry breaking

3.1Goldstone's theorem in classical field theory

3.2 Goldstone's theorem and Ward identities

3.3 The effective potential

4. Gauge invariance

4.1 Geometrical interpretation

4.2 Non-abelian gauge theories

4.3Quantization of constrained systems and Faddeev's formula

4.4 Quantization of gauge theories

4.4. The Higgs mechanism

5 Renormalization

5.1 Renormalization of QED

5.2 Scale invariance

5.3 The running coupling constant

5.4 The Callan-Symanzik equaton and the renormalization group

5.5 The operator-product (or Wilson) expansion

6. The chiral anomaly

6.1Non-conservation of the axial current

6.1 The theta vacuum

1. Unitarity ed analiticity

1.1 The opitcal theorem

1.2 Feynman diagrams and Cutkosky rules

1.3 Decay amplitudes

2. The Ward identities

2.1 Symmetries and current algebra

2.2 Ward identities for the two-point Green function

2.3 The Ward identities from the path integral

2.4 Examples: QED and Φ^4

3. Spontaneous symmetry breaking

3.1Goldstone's theorem in classical field theory

3.2 Goldstone's theorem and Ward identities

3.3 The effective potential

4. Gauge invariance

4.1 Geometrical interpretation

4.2 Non-abelian gauge theories

4.3Quantization of constrained systems and Faddeev's formula

4.4 Quantization of gauge theories

4.4. The Higgs mechanism

5 Renormalization

5.1 Renormalization of QED

5.2 Scale invariance

5.3 The running coupling constant

5.4 The Callan-Symanzik equaton and the renormalization group

5.5 The operator-product (or Wilson) expansion

6. The chiral anomaly

6.1Non-conservation of the axial current

6.1 The theta vacuum

**Prerequisites for admission**

Knowledge of the basics of relativistic quantum field theory, special relativity and path integral methods

**Teaching methods**

The course consists of blackboard lectures in which the individual topic included in the syllabus are presented, fist introducing the basics and then discussing the main conceptual points and computational technique. Interaction with the students in class is very much encouraged, through questions and discussions.

**Teaching Resources**

Recommended references:

M.E. Peskin, D.V. Schroeder: An introduction to Quantum Field Theory; Addison-Wesley, 1995 (reference textboos)

T.P. Cheng, L.F. Li, Gauge Theory of Elementary Particle Physics; Oxford University Press, 1985 (for specific topics)

S. Coleman: Aspects of Symmetry; Cambridge University Press, 1985 (for specific topics)

R. Jackiw: Topological Investigations of Quantized Gauge Theories: in Current Algebra and Anomalies; Princeton University Press, 1985 (for specific topics)

A. Zee, Quantum Field Theory in a Nutshell; Princeton University Press, 2010 (for deepening of specific topics at a more qualitative level)

S. Weinberg: The Quantum Theory of Fields: Vol. I (foundations); Cambridge University Press, 1995 (for more formal developments)

M.E. Peskin, D.V. Schroeder: An introduction to Quantum Field Theory; Addison-Wesley, 1995 (reference textboos)

T.P. Cheng, L.F. Li, Gauge Theory of Elementary Particle Physics; Oxford University Press, 1985 (for specific topics)

S. Coleman: Aspects of Symmetry; Cambridge University Press, 1985 (for specific topics)

R. Jackiw: Topological Investigations of Quantized Gauge Theories: in Current Algebra and Anomalies; Princeton University Press, 1985 (for specific topics)

A. Zee, Quantum Field Theory in a Nutshell; Princeton University Press, 2010 (for deepening of specific topics at a more qualitative level)

S. Weinberg: The Quantum Theory of Fields: Vol. I (foundations); Cambridge University Press, 1995 (for more formal developments)

**Assessment methods and Criteria**

The exam is an oral test of about one hour, 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 increasing 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.

FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 6

Lessons: 42 hours

Professor:
Forte Stefano

Professor(s)