Physical Applications of Group Theory
A.Y. 2022/2023
Learning objectives
The course aspires to teaching students the complex of group theory basics:
especially finite groups and Lie groups.
At the end of the course, students will know group theory methods, necessary to deal with classic problems of unified models of fundamental interactions.
especially finite groups and Lie groups.
At the end of the course, students will know group theory methods, necessary to deal with classic problems of unified models of fundamental interactions.
Expected learning outcomes
The student will acquire the following skills, after attending the course:
1) He will be able to write a Lagrangian for any field theory that is
invariant under whichever gauge symmetry.
2) He will be capable to study the breaking pattern of the gauge symmetry for whatever model of unified forces.
3) He will attain the expertise to write gauge and Yukawa interactions,
including all factor coefficients for each component of all involved fields.
4) He will become skillful to build whatever representation for any Lie group.
1) He will be able to write a Lagrangian for any field theory that is
invariant under whichever gauge symmetry.
2) He will be capable to study the breaking pattern of the gauge symmetry for whatever model of unified forces.
3) He will attain the expertise to write gauge and Yukawa interactions,
including all factor coefficients for each component of all involved fields.
4) He will become skillful to build whatever representation for any Lie group.
Lesson period: Second 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
L'insegnamento sara' erogato nell'anno accademico 2023/24
Lesson period
Second semester
Course syllabus
PHYSICAL APPLICATIONS OF GROUP THEORY
Axioms of group theory.
Order of a group: finite and infinite group.
The credibility criterion of a physical theory.
Definition of permutation of the elements of a set: the symmetric group Sn.
Isomorphisms.
The Cayley's theorem.
The direct product of two groups.
Lagrange's theorem.
The partition of a set and the equivalence relations.
Cauchy's Theorem.
Normal subgroups.
The commutator of two group elements.
Definition of homomorphism.
First isomorphism theorem.
Groups of unitary matrix and the Lie algebra.
The structure constants of a Lie group.
The Cartan subalgebra.
The roots of a Lie algebra.
Outline of grand unification of forces and gauge theories.
Matrix representations of groups.
Young's tableaux.
Tensors, tensor products, invariant tensors.
List of all simple Lie algebras. Exceptional groups.
Building Lie algebra representations with the dynkin-cartan method.
Branching Rules.
Unification groups: SU(5) and SO(10).
Automorphisms and semidirect product.
Axioms of group theory.
Order of a group: finite and infinite group.
The credibility criterion of a physical theory.
Definition of permutation of the elements of a set: the symmetric group Sn.
Isomorphisms.
The Cayley's theorem.
The direct product of two groups.
Lagrange's theorem.
The partition of a set and the equivalence relations.
Cauchy's Theorem.
Normal subgroups.
The commutator of two group elements.
Definition of homomorphism.
First isomorphism theorem.
Groups of unitary matrix and the Lie algebra.
The structure constants of a Lie group.
The Cartan subalgebra.
The roots of a Lie algebra.
Outline of grand unification of forces and gauge theories.
Matrix representations of groups.
Young's tableaux.
Tensors, tensor products, invariant tensors.
List of all simple Lie algebras. Exceptional groups.
Building Lie algebra representations with the dynkin-cartan method.
Branching Rules.
Unification groups: SU(5) and SO(10).
Automorphisms and semidirect product.
Prerequisites for admission
Geometry 1 and Linear Algebra:
product of matrices , determinant , inverse matrix . System of Cartesian axes in n dimensions, rotations, translations, equations of the plane and straight lines, eigenvalues, eigenvectors of a matrix etc ...
product of matrices , determinant , inverse matrix . System of Cartesian axes in n dimensions, rotations, translations, equations of the plane and straight lines, eigenvalues, eigenvectors of a matrix etc ...
Teaching methods
Face-to-face learning, exercises.
Teaching Resources
M.A. Armstrong
Groups and symmetry
R. Slansky
Group theory for unified model building
PHYSICS REPORTS (Review Section of Physics Letters) 79, No. 1 (1981) 1—128.
Groups and symmetry
R. Slansky
Group theory for unified model building
PHYSICS REPORTS (Review Section of Physics Letters) 79, No. 1 (1981) 1—128.
Assessment methods and Criteria
Written and oral exam. Evaluation criteria: good understanding of the topics covered in the course, ability to apply the concepts learned.
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 6
Lessons: 42 hours