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Quantum field theory 1

A.Y. 2020/2021

Learning objectives

The course provides an introduction to relativistic quantum

field theory, its theoretical foundations, and its application to the

perturbative computation of scattering processes.

field theory, its theoretical foundations, and its application to the

perturbative computation of scattering processes.

Expected learning outcomes

The course provides an introduction to relativistic quantum

field theory, its theoretical foundations, and its application to the

perturbative computation of scattering processes.

Risultati di apprendimento attesi (inglese )

At the end of this course the student will know how to

Decouple the dynamics of coupled finite-and infinite-dimensional

system in terms of normal coordinates

Obtain a classical field as the continuum limit of a system of coupled

harmonic oscillators

Construct a relativistic classical field theory for scalar, vector and

spin 1/2 fields

Determine the conserved currents in the presence of both internal and

space-time symmetry, specifically the enrrgy-momentum tensor

Quantize a free scalar field and construct its Fock space

Quantize a Fermi field

Obtain the time evolution of a quantum field theory from its path

integral

Compute the path integral and propagator for a free field theory of

Bosons or Fermions

Write down the path integral for an interacting field theory and use

it to calculate Green functions

Relkate aplitudes to Green functions through the reduction formula

Determine the Feynman rules for a given theory from the path integral

Compute amplitudes and cross-sections for simple processes

Understand the origin of divergences in perturbative computations, and

how to tame them through regularization and renormalization

Determine the Feynman rules for a renormalized field theory

Determinare le regole di Feynman per una teoria rinormalizzata

Understand under which conditions a theory is renormalizable or not,

and what it means

field theory, its theoretical foundations, and its application to the

perturbative computation of scattering processes.

Risultati di apprendimento attesi (inglese )

At the end of this course the student will know how to

Decouple the dynamics of coupled finite-and infinite-dimensional

system in terms of normal coordinates

Obtain a classical field as the continuum limit of a system of coupled

harmonic oscillators

Construct a relativistic classical field theory for scalar, vector and

spin 1/2 fields

Determine the conserved currents in the presence of both internal and

space-time symmetry, specifically the enrrgy-momentum tensor

Quantize a free scalar field and construct its Fock space

Quantize a Fermi field

Obtain the time evolution of a quantum field theory from its path

integral

Compute the path integral and propagator for a free field theory of

Bosons or Fermions

Write down the path integral for an interacting field theory and use

it to calculate Green functions

Relkate aplitudes to Green functions through the reduction formula

Determine the Feynman rules for a given theory from the path integral

Compute amplitudes and cross-sections for simple processes

Understand the origin of divergences in perturbative computations, and

how to tame them through regularization and renormalization

Determine the Feynman rules for a renormalized field theory

Determinare le regole di Feynman per una teoria rinormalizzata

Understand under which conditions a theory is renormalizable or not,

and what it means

**Lesson period:** Second semester
(In case of multiple editions, please check the period, as it may vary)

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

Second semester

**Course syllabus**

--Classical field theory

+ normal coordinates

+ the continuum limit and classical fields

+ equations of motion

+ Noether's theorem

--Field quantization: free fields

+quantisation of the scalar field and Fock space

+several degrees of freedom: the charged field and the spin-one field

+fermionic fields: the Dirac field

--Interacting fields

+interactions and time evolution

+the path integral

+the propagator

+the path integral for fermions

--Amplitudes

+the interaction vertex

+the reduction formula

+Feynman rules

--Leading order computation of physical processes:

+computation of the amplitude

+kinematics and reference frames

+the cross-section

--Renormalisation

+divergences and their meaning

+renormalised perturbation theory

+renormalisability

+ normal coordinates

+ the continuum limit and classical fields

+ equations of motion

+ Noether's theorem

--Field quantization: free fields

+quantisation of the scalar field and Fock space

+several degrees of freedom: the charged field and the spin-one field

+fermionic fields: the Dirac field

--Interacting fields

+interactions and time evolution

+the path integral

+the propagator

+the path integral for fermions

--Amplitudes

+the interaction vertex

+the reduction formula

+Feynman rules

--Leading order computation of physical processes:

+computation of the amplitude

+kinematics and reference frames

+the cross-section

--Renormalisation

+divergences and their meaning

+renormalised perturbation theory

+renormalisability

**Prerequisites for admission**

Nonrelativistic quantum mechanics. Special relativity. The Lagrangian formulation of classical mechanics.

**Teaching methods**

The course consists of 42 hours of blackboard lectures, during which the basic theory and techniques of quantum field theory are presented, and some applications are worked out. A teaching assistant is available for helping the students with questions and for discussing problems.

**Teaching Resources**

Reference textbook

M. Maggiore: A Modern Introduction to Quantum Field Theory; Oxford

University Press, 2005

Further reading

M.E. Peskin, D.V. Schroeder: An Introduction to Quantum Field Theory; Addison-Wesley, 1995

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

A. Zee, Quantum Field Theory in a Nutshell; Princeton University Press, 2010

V. Radovanovic: Problem Book in Quantum Field Theory; Springer, 2007

M. Maggiore: A Modern Introduction to Quantum Field Theory; Oxford

University Press, 2005

Further reading

M.E. Peskin, D.V. Schroeder: An Introduction to Quantum Field Theory; Addison-Wesley, 1995

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

A. Zee, Quantum Field Theory in a Nutshell; Princeton University Press, 2010

V. Radovanovic: Problem Book in Quantum Field Theory; Springer, 2007

**Assessment methods and Criteria**

The final exam consists of a written two-hour-long test and a half-hour oral exam. In the written test the student is asked to work out standard applications, such as deriving the Feynman rules for a given theory and use them to calculate amplitudes. In the oral exam, the student is asked to discuss one of the topics from the course syllabus, chosen on the spot. All previous written exams are available (with solutions) from the instructor's website.

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

Lessons: 42 hours

Professor:
Forte Stefano

Professor(s)