Physics

GIANLUCA PANATI
GIANLUCA PANATI

GENERAL OBJECTIVES:

The main goal of the course is to introduce students to the mathematical theory of groups (mainly: discrete groups and compact Lie groups) by a Mathematical Physics approach which emphasizes the role of representations of symmetries in terms of states or observables of the corresponding theory. Such an approach allows an immediate comparison between classical theories (Poisson brackets) and quantum theories (commutators).


SPECIFIC OBJECTIVES:

A - Knowledge and understanding
OF1) To know the fundamental concepts in the theory of finite groups and matrix Lie groups, and in the theory of their linear, unitary or projective representations.
OF2) To know the mathematical structure of the Lie groups which more often appear in physical theories, and to understand the relation between such groups and the symmetries of the physical theory.
OF3) To understand the role of symmetries and Lie groups in (relativistic) field theories.
OF4) To understand the mathematical language of differential forms, and the reformulation of electromagnetism in such a language.

B – Application skills
OF 5) To be able to compute the commutation relations among the generators of the Lie algebra of a given (matrix) Lie group; to be able to explicitly compute such commutation relations in the most relevant cases: the rotation group, the Poincaré group, and the group SU(3).
OF 6) To be able to compute the tensor product of two representations of the rotation group, by using the Wigner Eckart theorem; to be able to interpret the result of such a computation in the application to compound systems (e.g. molecules).
OF7) To be able to determine whether a given differential form is closed and/or exact; to be able to translate the concepts concerning differential forms in the analogous concepts of vector analysis (gradient, curl or rotational, divergence) and vice versa.

C - Autonomy of judgment
OF 8) To be able to critically read an advanced book on symmetries in physics.
OF 9) To be able to integrate the knowledge acquired within the course, in order to apply them in the context of different physical theories, in connection e.g. with high energy physics or with condensed matter physics.

D – Communication skills
OF 10) Ability to discuss the symmetries of a physical system by appropriately using the language of differential forms and Lie groups.

E - Ability to learn
OF 11) Ability to read advanced monographies and research papers, which usually use the mathematical language of Lie groups and differential forms.
OF 12) Ability to "construct" a physical theory, by implementing in the theory the symmetries of the physical system under investigation, using Lie algebras and Lie groups as a fundamental tool.

📘 **Group Theory in Mathematical Physics – Learning Objectives**

**General Aims**

The course provides a comprehensive introduction to the mathematical structures underlying symmetry in physics.
It develops the theory of groups and Lie groups, Lie algebras, and their representations, emphasizing both the algebraic and geometric viewpoints and their applications to classical and quantum physics.
Examples include SU(2), SO(3), the Lorentz and Poincaré groups, and a quick outlook to the mathematical foundations of gauge theories.

**Knowledge and Understanding**

Upon completion of the course, students will have:

* a solid understanding of the fundamental concepts of group theory, including subgroups, quotient groups, and homomorphisms;
* a rigorous comprehension of Lie groups and Lie algebras, their structure, topology, and relation through the exponential map;
* familiarity with the representation theory of Lie algebras and Lie groups, including the complete classification of irreducible representations of ( \mathfrak{sl}(2,\mathbb{C}) ) and of ( \mathfrak{sl}(3,\mathbb{C}) );
* an understanding of how group-theoretical and geometric tools describe physical symmetries and conservation laws.

**Applying Knowledge and Understanding**

Students will be able to:

* identify the symmetry group associated with a given physical system and exploit it to simplify, classify, or solve the system;
* compute and interpret the Lie algebra of classical matrix Lie groups such as SU(n), SO(n), and the Lorentz group;
* apply the exponential map, commutation relations, and representation theory to describe rotations, spin, and symmetries in quantum mechanics and field theory;
* use the language of manifolds and differential forms to reformulate physical laws in invariant form (e.g. electromagnetism via differential forms);
* connect the mathematical formalism to the structure of gauge theories and other modern formulations of fundamental physics.

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**Making Judgements**

Students will develop the ability to:

* critically assess the role of symmetry and group structure in the formulation of physical laws;
* evaluate the coherence and effectiveness of group-theoretical models when applied to physical phenomena;
* recognise deep analogies between mathematical structures (e.g. root systems) and physical patterns (e.g. particle multiplets and conservation symmetries).

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**Communication Skills**

Students will be able to:

* present concepts, proofs, and physical interpretations of group-theoretical results with mathematical rigor and clarity;
* communicate effectively using precise algebraic and geometric language, both in written and oral scientific contexts;
* engage in interdisciplinary discussions linking mathematics, physics, and geometry.

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**Learning Skills**

The course will equip students with the ability to:

* pursue independent study and research in advanced topics such as representation theory, differential geometry, and quantum field theory;
* integrate algebraic, analytical, and geometric methods in the development of new theoretical models;
* approach current research literature in mathematical physics with confidence and conceptual depth.

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

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)

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.