LECTURES ON SYMMETRY-ASSISTED COMPUTATION

Scientific problems have an internal 'beauty', called, referred to, precisely speaking, as their 'symmetry'. The symmetry arises, often, from the fact that the scientific problem refers to an object (a molecule, a crystal) and the object itself has some 'symmetry' elements, but in more abstract situations, such as those arising in particle physics and quantum technologies, symmetry is often the only known (and relevant!) fact about the problem. The scope of these Lecture Notes is to educate how to recognize the symmetry of a scientific problem and how to use symmetry to understand, manipulate and, finally, solve it. The principle guiding these Lecture Notes is that 'learning by doing' is the only way that young students can later become productive in science, business and industry. The lecture Notes have, essentially, two components. The first one reports the content of a set of lectures, held at ETH Zurich at the master and PhD level, frequented mainly by students from the department of Physics, Chemistry and Material Science. The lectures were accompanied by a set of student projects on various scientific subjects related to symmetry. These projects ended with a manuscript, worked out by the students themselves and edited into the second component of these Lecture Notes.

Contents:

• Lectures:
  • Geometrical, Algebraic, and Analytic Aspects of Symmetry
  • Representation Theory of Groups with Average
  • SO₃ and the General Method of Infinitesimal Transformations
  • The Symmetry Group of the Operator and Some Practical Applications
  • Point Groups and Space Groups of Crystallography
  • Applications to Solid-State Physics
  • The Relativistic Electronic Structure of Atoms and Solids
  • Young Diagrams and Particle Physics
  • The Permutation Group and Its Applications to Many-Body Problems
  • Group Theory and Phase Transitions
• Exercises:
  • Exercises to Lecture 1
  • Exercises to Lecture 2
  • Exercises to Lecture 3
  • Exercises to Lecture 4
  • Exercises to Lecture 5
• Projects:
  • Project for Lecture 2: Functional Analytic and Numerical Aspects of Eigenvalue Problems
  • Project for Lecture 4: Symmetry Arguments in Classical Mechanics
  • Project for Lecture 4: Crystal Field Splitting and the Jahn–Teller Effect
  • Project for Lecture 4: Vibrational Modes of the NH₃ Molecule
  • Project for Lecture 5: Frieze Patterns
  • Project for Lecture 5: An Algebraic Proof of the Hessel Theorem
  • Project for Lecture 6: Empty-Lattice Band Structure
  • Project for Lecture 6: Symmetry Analysis of Graphene Band Structure and Dirac Cones
  • Project for Lecture 6: Topological Protection by Non-symmorphic Degeneracy
  • Project for Lecture 7: Topological Aspects of Continuous Groups and the Universal Covering Groups
  • Project for Lecture 7: Optical Spin Orientation in Atoms and Solids
  • Project for Lecture 9: Quantum Chemistry

Readership: Intended for advanced undergraduates, graduates and PhD students in the fields of physics in general and condensed matter physics and quantum physics in particular.

Key Features:

• Symmetry arguments are key elements in modern research and, accordingly, the number of textbooks dedicated to illustrating group theory and its application in science is very large. Some books have very high mathematical content, others are more for practitioners. This set of lectures aims at being precise but, in the same time, practical enough to allow the reader to understand the significance and the outreach of the various mathematical tools introduced. These lectures are aimed at "educating" students — in recognizing the symmetry elements of a problem and incorporating them into the solution. In the exposition, every content is spelled out in detail, step by step — situations terminated by sentences such as "it is obvious that ..." or "it follows straightforwardly that ..." are explicitly avoided. The topics are, accordingly, designed to be self-explanatory (except for the most advanced mathematical content, which is referred to a more suitable literature), as the mathematical tools needed are introduced ad-hoc and in great details. The students have also the opportunity of recognizing the relevance of symmetry arguments in practical situations, described by a set of student projects that are chaperoned by the lectures and build a second, conspicuous component of these Lecture Notes. By working out these projects, future students should be encouraged to construct, with the guidance of their teachers, their own project on a novel subject involving symmetry, so that they can compare themselves with their "peers" at ETH Zurich. These Lecture Notes are not intended to be used as a comprehensive and extensive treatise of group theory, their representations and their applications. There is a long list of textbooks that better fulfil this task. They are primarily designed 1. to provide a platform, based on meticulous work, for initiating and solving novel, symmetry related scientific problems and 2. to learn how to "go beyond", without accepting anything for granted