Physics

I.INTRODUCTION TO SYMMETRIES IN PHYSICS II. GROUPS- FUNDAMENTAL CONCEPTS 1. Definitions and examples: groups consisting of numbers, functions, linear operators. The general linear group GL(V). The group of permutations of a set S. New groups from old ones. 2. Homomorphisms of groups. Definition, examples, elementary properties. Kernel and image are always subgroups. Isomorphisms; automorphisms. 3. Normal subgroups. Quotients of groups (cosets). The quotient set G/H is a group if H is a normal subgroup [with proof]. III. GROUPS - FINITE GROUPS 1. Definition and examples. The rearrangement theorem [with proof]. Theorem: the order of a subgroup divides the order of a group [with proof]. 2. Cyclic subgroups. Examples of groups by order (up to order 6, and all primes). IV. GROUPS - BROTHERHOOD 1. The groups SU(2) and SO(3) and their topological properties. Description of SU(2) as a 3-sphere. 2. Brotherhood: SU(2) is a 2-covering of SO(3). Explicit construction of the covering map [proof with all details]. V. MANIFOLDS AND DIFFERENTIAL FORMS 1. Manifolds. The concept of manifold: extrinsic and intrinsic viewpoint. Definitions: local charts, atlases, maximal atlases. Why do we need maximality in the definition? Differentiable maps. Diffeomorphisms. 2. Tangent and cotangent spaces. Tangent vectors: geometric viewpoint,, linear structure, coordinate basis. Algebraic viewpoint: tangent vectors as directional derivatives. Transformation of cordinate bases when changing coordinates (local charts). Ricci calculus: components of vectors are controvariant, bases are covariant. The Ricci golden rule (mnemonic trick). Equivalence of the three viewpoints on tangent vectors. 3. Differential of a smooth map. Agreement with the differential of a real-valued function. Covectors and cotangent space. Ricci calculus: components of covectors are covariant, dual bases are controvariant. 4. Differential forms on a manifold. Linear k-forms in a linear space: definition, wedge product, geometric interpretation. Differential forms, wedge product. The exterior differential (or Cartan's differential). Examples of differentiation of forms. Relation with vector analysis. Poincaré lemma, converse of the Poincaré lemma (without proof). Pull back of differential forms. 5. Reformulation of electromagnetism via differential forms. Lorentz force and Faraday tensor. Maxwell equations rewritten in invariant form. Existence of tetrapotential. VI. LIE GROUPS 1. Definition of Lie group. The Lie algebra of a Lie group (geometric viewpoint). Lie algebras (in general). Examples: GL(n, R), GL(n, C), SL(n, R), O(n), SO(n). 2. Lie subgroups. Fundamental criterion to be a Lie subgroup (Cartan's criterion). 3. More examples: - generalized orthogonal groups O(p,q) - symplectic groups Sp(n, R) - Euclidean groups and (generalized) Poincaré groups P(1,n) - special linear groups SL(n, C) - unitary groups: U(n), SU(n). 4. Homomorphisms of Lie groups. 5. Matrix Lie groups. Is every Lie group a matrix Lie group? A counterexample. 6. Topological properties of Lie groups: analysis of compactness and connectedness of classical Lie groups. 7. Matrix exponential and logarithm. Definition and continuity of exp. Basic properties of the exponential. Computing the exponential in practice. Matrix logarithm: definition, continuity and local injectivity. Linear approx. Lie product formula, magic formula det(exp(A)) = exp(tr(A)), one-parameter subgroups. 8. Lie algebra of a matrix Lie group Lie algebras of classical Lie groups: gl(n, C), gl(n,R), sl(n,C), sl(n,R), u(n), su(n), o(n)=so(n). Lie algebras of generalized orthogonal groups (including the Lorentz group). 9. Induced Lie algebra homorphisms 10. The exponential map of a Lie algebra. Definition. Examples of lack of surjectivity and lack of injectivity. Fundamental Theorem (without proof). Exp is a local diffeomerfism from the Lie algebra to the Lie group. Identification of the matrix Lie algebra with the geometric Lie algebra, for every matrix Lie group. 11. The universal covering group ("Brothers are ubiquitous") 12. A closer look to the Lorentz and Poincaré groups VII. REPRESENTATIONS OF LIE GROUPS 1. Symmetries in Quantum Mechanics and (projective) representations 2. Basic concepts of representation theory 3. Representations of sl(2,C). Complete classification of irreducible representations. 4. Relation between the irreducible representations of a matrix Lie group and the irreducible representations of its Lie algebra. Two paradigmatic cases: SU(2) and SO(3). 4. Representations of sl(3,C). Basis and main commutators. Weights and roots: the fundamental tools. The highest weight theorem. A coordinate-free approach. The Cartan subalgebra and the Weyl group. Symmetries of the weighs and the roots in the intrinsic metric. Serendipity: analogy between roots diagrams and elementary particles diagrams. VIII. EPILOGUE A glimpse to the mathematical structure of Yang-Mills theories. Epilogue: "The Unreasonable Effective-ness of Mathematics in the Natural Sciences" (E. Wigner)

Good knowledge of the fundamental concepts of linear algebra, differential and integral calculus in several variables, functional analysis in Hilbert and Banach spaces. The students are also supposed to have an elementary knowledge of Analytic Mechanics (both Lagrangian and Hamiltonian formulation) and of Quantum Mechanics (Hilbert space formulation). The aforementioned contents correspond, for example, to the following courses in the Bachelor programme in Physics at "La Sapienza" University of Rome: - Analisi, Geometria, Analisi Vettoriale - Modelli e Metodi Matematici della Fisica - Meccanica Analitica e Relativistica - Meccanica Quantistica

B.C. Hall: Lie Groups, Lie Algebras, and Representations. An Elementary Introduction. Springer, 2003. M.A. Naĭmark: Linear representations of the Lorentz group. Pergamon Press, 1964. M.P. Do Carmo: Differential Forms and Applications. Springer, 2000. C. von Westenholz: Differential Forms in Mathematical Physics. North-Holland, 1978.

Lectures on the theoretical concepts with classroom exercises

Attendance to the lectures is not mandatory but strongly recommended.

The exam consists of an interview on the most relevant topics of the course. To be admitted to the interview the student is asked to present the solution to 5 selected problems ("starred problems"), chosen among those proposed by the lecturer during the course. The interview will initially be focused on the solution to one of these problems. The successful student will be able to illustrate theoretical concepts and mathematical proofs discussed and explained during the course. The student will be asked to apply the methods learned during the course to examples and situations similar to those that were discussed during the course. The evaluation will take into account: - correctness and completeness of the concepts illustrated by the student; - clarity and rigor of presentation; - analytical development of the theory; - problem-solving skills. To achieve a score of 30/30 cum laude, the student must demonstrate that she/he has acquired excellent knowledge of all the topics covered during the course, being able to link them in a logical and coherent way, and she/he is moreover able to apply such methods to problems (slightly) different than those presented in the course.

Lectures on the theoretical concepts with classroom exercises

I.INTRODUCTION TO SYMMETRIES IN PHYSICS II. GROUPS- FUNDAMENTAL CONCEPTS 1. Definitions and examples: groups consisting of numbers, functions, linear operators. The general linear group GL(V). The group of permutations of a set S. New groups from old ones. 2. Homomorphisms of groups. Definition, examples, elementary properties. Kernel and image are always subgroups. Isomorphisms; automorphisms. 3. Normal subgroups. Quotients of groups (cosets). The quotient set G/H is a group if H is a normal subgroup [with proof]. III. GROUPS - FINITE GROUPS 1. Definition and examples. The rearrangement theorem [with proof]. Theorem: the order of a subgroup divides the order of a group [with proof]. 2. Cyclic subgroups. Examples of groups by order (up to order 6, and all primes). IV. GROUPS - BROTHERHOOD 1. The groups SU(2) and SO(3) and their topological properties. Description of SU(2) as a 3-sphere. 2. Brotherhood: SU(2) is a 2-covering of SO(3). Explicit construction of the covering map [proof with all details]. V. MANIFOLDS AND DIFFERENTIAL FORMS 1. Manifolds. The concept of manifold: extrinsic and intrinsic viewpoint. Definitions: local charts, atlases, maximal atlases. Why do we need maximality in the definition? Differentiable maps. Diffeomorphisms. 2. Tangent and cotangent spaces. Tangent vectors: geometric viewpoint,, linear structure, coordinate basis. Algebraic viewpoint: tangent vectors as directional derivatives. Transformation of cordinate bases when changing coordinates (local charts). Ricci calculus: components of vectors are controvariant, bases are covariant. The Ricci golden rule (mnemonic trick). Equivalence of the three viewpoints on tangent vectors. 3. Differential of a smooth map. Agreement with the differential of a real-valued function. Covectors and cotangent space. Ricci calculus: components of covectors are covariant, dual bases are controvariant. 4. Differential forms on a manifold. Linear k-forms in a linear space: definition, wedge product, geometric interpretation. Differential forms, wedge product. The exterior differential (or Cartan's differential). Examples of differentiation of forms. Relation with vector analysis. Poincaré lemma, converse of the Poincaré lemma (without proof). Pull back of differential forms. 5. Reformulation of electromagnetism via differential forms. Lorentz force and Faraday tensor. Maxwell equations rewritten in invariant form. Existence of tetrapotential. VI. LIE GROUPS 1. Definition of Lie group. The Lie algebra of a Lie group (geometric viewpoint). Lie algebras (in general). Examples: GL(n, R), GL(n, C), SL(n, R), O(n), SO(n). 2. Lie subgroups. Fundamental criterion to be a Lie subgroup (Cartan's criterion). 3. More examples: - generalized orthogonal groups O(p,q) - symplectic groups Sp(n, R) - Euclidean groups and (generalized) Poincaré groups P(1,n) - special linear groups SL(n, C) - unitary groups: U(n), SU(n). 4. Homomorphisms of Lie groups. 5. Matrix Lie groups. Is every Lie group a matrix Lie group? A counterexample. 6. Topological properties of Lie groups: analysis of compactness and connectedness of classical Lie groups. 7. Matrix exponential and logarithm. Definition and continuity of exp. Basic properties of the exponential. Computing the exponential in practice. Matrix logarithm: definition, continuity and local injectivity. Linear approx. Lie product formula, magic formula det(exp(A)) = exp(tr(A)), one-parameter subgroups. 8. Lie algebra of a matrix Lie group Lie algebras of classical Lie groups: gl(n, C), gl(n,R), sl(n,C), sl(n,R), u(n), su(n), o(n)=so(n). Lie algebras of generalized orthogonal groups (including the Lorentz group). 9. Induced Lie algebra homorphisms 10. The exponential map of a Lie algebra. Definition. Examples of lack of surjectivity and lack of injectivity. Fundamental Theorem (without proof). Exp is a local diffeomerfism from the Lie algebra to the Lie group. Identification of the matrix Lie algebra with the geometric Lie algebra, for every matrix Lie group. 11. The universal covering group ("Brothers are ubiquitous") 12. A closer look to the Lorentz and Poincaré groups VII. REPRESENTATIONS OF LIE GROUPS 1. Symmetries in Quantum Mechanics and (projective) representations 2. Basic concepts of representation theory 3. Representations of sl(2,C). Complete classification of irreducible representations. 4. Relation between the irreducible representations of a matrix Lie group and the irreducible representations of its Lie algebra. Two paradigmatic cases: SU(2) and SO(3). 4. Representations of sl(3,C). Basis and main commutators. Weights and roots: the fundamental tools. The highest weight theorem. A coordinate-free approach. The Cartan subalgebra and the Weyl group. Symmetries of the weighs and the roots in the intrinsic metric. Serendipity: analogy between roots diagrams and elementary particles diagrams. VIII. EPILOGUE A glimpse to the mathematical structure of Yang-Mills theories. Epilogue: "The Unreasonable Effective-ness of Mathematics in the Natural Sciences" (E. Wigner)

Good knowledge of the fundamental concepts of linear algebra, differential and integral calculus in several variables, functional analysis in Hilbert and Banach spaces. The students are also supposed to have an elementary knowledge of Analytic Mechanics (both Lagrangian and Hamiltonian formulation) and of Quantum Mechanics (Hilbert space formulation). The aforementioned contents correspond, for example, to the following courses in the Bachelor programme in Physics at "La Sapienza" University of Rome: - Analisi, Geometria, Analisi Vettoriale - Modelli e Metodi Matematici della Fisica - Meccanica Analitica e Relativistica - Meccanica Quantistica

B.C. Hall: Lie Groups, Lie Algebras, and Representations. An Elementary Introduction. Springer, 2003. M.A. Naĭmark: Linear representations of the Lorentz group. Pergamon Press, 1964. M.P. Do Carmo: Differential Forms and Applications. Springer, 2000. C. von Westenholz: Differential Forms in Mathematical Physics. North-Holland, 1978.

Lectures on the theoretical concepts with classroom exercises

Attendance to the lectures is not mandatory but strongly recommended.

The exam consists of an interview on the most relevant topics of the course. To be admitted to the interview the student is asked to present the solution to 5 selected problems ("starred problems"), chosen among those proposed by the lecturer during the course. The interview will initially be focused on the solution to one of these problems. The successful student will be able to illustrate theoretical concepts and mathematical proofs discussed and explained during the course. The student will be asked to apply the methods learned during the course to examples and situations similar to those that were discussed during the course. The evaluation will take into account: - correctness and completeness of the concepts illustrated by the student; - clarity and rigor of presentation; - analytical development of the theory; - problem-solving skills. To achieve a score of 30/30 cum laude, the student must demonstrate that she/he has acquired excellent knowledge of all the topics covered during the course, being able to link them in a logical and coherent way, and she/he is moreover able to apply such methods to problems (slightly) different than those presented in the course.

Lectures on the theoretical concepts with classroom exercises