OVERVIEW

The course will explain in which way  Quantum Mechanics and Special Relativity are integrated in a coherent conceptual framework --- which is the one of Quantum Field Theory (QFT) --- whose fundamental physical principles are causality and locality. The course will analyze the physical principles at the basis of QFT, will describe some of his most important physical predictions such as the existence of anti-particles, the spin-statistics connection, and particle production. It will cover invariant perturbation theory, Feynman diagrams, quantization of massless fields of higher helicity and the BRS formalsims for the quantization of gauge theories. 

AIMS AND CONTENT

LEARNING OUTCOMES

The course will discuss how Quantum Mechanics and Special Relativity can be integrated into a coherent theoretical framework whose fundamental physical principles are those of causality and locality. The major achievements and mathematical methods of this theoretical synthesis, such as the spin-statistics theorem, the invariant theory of perturbations, the quantization of non-Abelian gauge theories, BRS symmetry, and anomalies, will be systematically illustrated.

AIMS AND LEARNING OUTCOMES

At the end of this course,

1. The student will understand why it is not possible to formulate quantum mechanics in a way compatible with special relativity in the context of a theory with a fixed number of particles;
2. The student will understand why causality and locality are the  physical principles which are the basis of relativistic quantum mechanics;
3. The student will understand why causality and locality imply the spin-statistic theorem and the existence of anti-particles;
4. The student will master the basic methods of the  mathematical theory of linear and projective representations of the symmetry groups and Lie algebras relevant to relativistic physics;
5. The student will understand how spatial parity, charge conjugation and time inversion symmetries are implemented in relativistic quantum field theory and why they can be violated in a relativistic invariant theory;
6. The student will be able to compute, at the lowest order in perturbation theory, by means of  Feynman diagrams, cross section and decay rates of interacting relativistic particles;
7. The student will be able to compute propagators of generic relativistic quantum field theories;
8. The student will understand why the relativistic description of massless particles of helicity 1 requires the introduction of gauge invariant quantum field theories;
9. The student will be able to compute, by means of the BRST method, Feynman rules for a gauge theory with an arbitrary gauge-fixing choice;
10. The student will be able to determine the restrictions that  both Lorentz invariance and invariance under the discrete symmetries P, C, and T, impose on quantum amplitudes of relativistic processes. 

PREREQUISITES

Non-relativistic quantum mechanics.

TEACHING METHODS

Traditional lectures and problem solving sessions in class, aimed to exemplify the theoretical methods and concepts discussed in the course and to develop the ability of the student to master the mathematical tools necessary to solve concrete physical problems.

SYLLABUS/CONTENT

1. Relativistic symmetry in Quantum Mechanics. 

   1.1 Elements of representation theory: linear and projective representations.
   1.2 Unitary and irreducible representations. Conjugates of linear complex representations.
   1.3 The method of induced representation.    
   1.4 Irreducible and unitary representations of the group of non-homogeneous Lorentz transformations. 
   1.5 The action of space-time inversions on particle representations.
   1.6 Irreducible finite dimensional representations of the algebra of homogenous Lorentz transformations.
   1.7 Relativistic fields.

2. Free relativistic invariant field equations
  
    2.1 Klein-Gordon equation
    2.2 Noether theorem and the bilinear invariant form on the space of solution of relativistic invariant field equations.
    2.3 Relativistic second quantization
    2.4 Particles and  antiparticles
    2.5 Causal relativistic fields
    2.6 Spin-statistic theorem
    2.7 Proca equation for massive vector fields
    2.8 Weyl and Dirac equations for spinorial fields 
    2.9 The action of discrete symmetries P, C, T on relativistic fields

3.  Interacting relativistic quantum fields

     3.1 Invariant perturbation theory
     3.2 In and out states and the scattering matrix 
     3.3 Feynman rules for the elements of the scattering matrix
     3.4 The causal Feynman prescription for the propagators
     3.5 S matrix, decay times and cross sections

4. Massless fields
     
    4.1 The connection between relativistic invariance and gauge invariance for the photon field. 
    4.2 Quantum electrodynamics in Landau gauge
    4.3 Gupta-Bleuler quantization for the electrodynamics in covariant gauges 
    4.4 BRST symmetry and the quantization of electrodynamics in generic gauges 
    4.5 Non-abelian gauge theories and their quantization in the BRST framework

5. The functional formulation of relativistic quantum field theory
 
    5.1 LSZ theorem
    5.2 Schwinger-Dyson equation for the functional generator of Green functions
    5.3 Solving Schwinger-Dyson equations with Feynman functional integral
    5.4 Computing Feynman functional integral for free theories

    5.5 Ward identities and Anomalies.

RECOMMENDED READING/BIBLIOGRAPHY

- L. D. Landau, E. M. Lifsits, Meccanica Quantistica, Teoria Relativistica, Editori Riuniti Edizioni Mir, Roma (1976);

- S. Weinberg, The Quantum Theory of Fields, Vol 1, Cambridge University Press, Cambridge, (1996);

- Lectures notes and a collection of exercises and problems with solutions will be available on-line. 

TEACHERS AND EXAM BOARD

Exam Board

CAMILLO IMBIMBO (President)

NICOLA MAGGIORE

STEFANO GIUSTO (President Substitute)

SIMONE MARZANI (Substitute)

LESSONS

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam is divided into two parts, written and oral.

The written test can be done at home. The written test consists of a problems articulated in several questions, regarding topics covered during the course: to each question, a score is assigned and explicitly specified on the exam sheet. The sum of the scores of all the questions is 32/30. To have access to the oral exam a minimum total score of 18/30 is required.

The details of the exam modalities are discussed in class with the students at the beginning of the course.

ASSESSMENT METHODS

The questions of the written exams are of variable difficulty, in order to achieve an accurate evaluation of the competence achieved by the student. The student must show, by solving a concrete physical problem, to understand the fundamental concepts of relativistic quantum field theory and to master the computational tools illustrated in the course.

The oral exam is lead by the professor responsible for the course and by another expert, who is usually a professor of the department of physics. The oral exam lasts about 30 minutes. The oral exam is divided into two parts: the first part is a discussion of the written test, in particular of the questions or the points which have not been correctly or completely answered by the student. The second part consists of a question on a topic which is different from the ones of the written test. The student is asked to present a topic covered in the course and lecture about it on the blackboard in his own personal way,  in order to evaluate his abilities of synthesis and of personal elaboration of the subject matter. 

Exam schedule

FURTHER INFORMATION

Students who have valid certification of physical or learning disabilities on file with the University and who wish to discuss possible accommodations or other circumstances regarding lectures, coursework and exams, should speak both with the instructor and with Professor Sergio Di Domizio (sergio.didomizio@unige.it), the Department’s disability liaison.