Lectures on Symmetry-Assisted Computation

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【簡介】 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. 【目錄】 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

General Yang–Mills Symmetry From Quark Confinement to an Antimatter Half-Universe

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This monograph expounds on general Yang–Mills symmetry, a new symmetry based on arbitrary vector gauge functions and Hamilton\'s characteristic phase functions in the gauge transformations of Abelian and non-Abelian groups. General Yang–Mills symmetry includes the conventional gauge symmetries as special cases and is useful for describing phenomena at scales ranging from the super-macroscopic such as dark matter, to the ultra-microscopic such as the quantum 3-body problem of baryons. Moreover, this symmetry supports the Broader Particle-Cosmology framework based on particle physics and quantum Yang–Mills gravity in flat space-time, which can explain why the gravitational force is always attractive. This volume also discusses how CPT invariance in particle physics suggests a "Big Jets" model for the birth of the universe, proposing one explanation for the dearth of anti-matter in our universe. Finally, we discuss a simplified quantum shell model for N baryons with a quark Hamiltonian and a Sonine–Laguerre equation that gives reasonable eigenvalues for the energies of the 29 N baryons. Sample Chapter(s) Preface Chapter 1: Underpinnings Contents: Preface Overview About the Authors Broader Particle-Cosmology: New Perspectives On Dark Matter, Dark Energy, And The Missing Anti-Matter: Underpinnings A Model for Dark Matter A Model for Dark Energy The Question of the Missing Antimatter: The Big Jets Model The Evolution of the Universe Symmetry-Unified Quark-Cosmic Model Based On General Yang-Mills Symmetry: A Universal Principle of Interactions for Quarks and Leptons Finite Fermion Self-Masses and a Non-Propagating Phase Field Quark Confinement and the Accelerated Cosmic Expansion A Simple Harmonic Oscillator Model for 3-Quark Confinement Total Symmetry-Unified Model and Violations of All Internal Gauge Symmetries by Yang-Mills Gravity Appendices: Quantum Yang-Mills Gravity vs. Classical Einstein Gravity A New Gauge Invariant Phase Equation for Bound Fermions in Superconductors Quantum Shell Model for 3-Quark Bound States Epilogue Author Index Subject Index Readership: Graduate students, researchers interested in theoretical physics, astrophysics and cosmology.