Introduction to Continuous Symmetries

From Space-Time to Quantum Mechanics

Franck Laloë, Nicole Ostrowsky, Daniel Ostrowsky

• Publisher: Wiley-VCH
• Year: 2023
• Print ISBN: 9783527414161
• eBook ISBN: 9783527840540
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Applied Mathematics

Introduction to Continuous Symmetries

Powerful and practical symmetry-based approaches to quantum phenomena

In Introduction to Continuous Symmetries, distinguished researcher Franck Laloë delivers an insightful and thought-provoking work demonstrating that the underlying equations of quantum mechanics emerge from very general symmetry considerations without the need to resort to artificial or ambiguous quantization rules. Starting at an elementary level, this book explains the computational techniques such as rotation invariance, irreducible tensor operators, the Wigner—Eckart theorem, and Lie groups that are necessary to understand nuclear physics, quantum optics, and advanced solid-state physics.

The author offers complementary resources that expand and elaborate on the fundamental concepts discussed in the book's ten accessible chapters. Extensively explained examples and discussions accompany the step-by-step physical and mathematical reasoning. Readers will also find:

• A thorough introduction to symmetry transformations, including fundamental symmetries, symmetries in classical mechanics, and symmetries in quantum mechanics
• Comprehensive explorations of group theory, including the general properties and linear representations of groups
• Practical discussions of continuous groups and Lie groups, in particular SU(2) and SU(3)
• In-depth treatments of representations induced in the state space, including discussions of Wigner's Theorem and the transformation of observables

Perfect for students of physics, mathematics, and theoretical chemistry, Introduction to Continuous Symmetries will also benefit theoretical physicists and applied mathematicians.

Table of contents

• Cover
• Title Page
• Copyright
• Contents
• Preface
• Introduction
• I Symmetry transformations
• AI Eulerian and Lagrangian points of view in classical mechanics
• BI Noether’s theorem for a classical field
• II Some ideas about group theory
• AII Left coset of a subgroup; quotient group
• III Introduction to continuous groups and Lie groups
• AIII Adjoint representation, Killing form, Casimir operator
• IV Induced representations in the state space
• AIV Unitary projective representations, with finite dimension, of connected Lie groups. Bargmann's theorem
• BIV Uhlhorn-Wigner theorem
• V Representations of Galilean and Poincaré groups: mass, spin, and energy
• AV Proper Lorentz group and SL(2C) group
• BV Commutation relations of spin components, Pauli–Lubanski four-vector
• CV Group of geometric displacements
• DV Space reflection (parity)
• VI Construction of state spaces and wave equations
• AVI Relativistic invariance of Dirac equation and non-relativistic limit
• BVI Finite Poincaré transformations and Dirac state space
• CVI Lagrangians and conservation laws for wave equations
• VII Rotation group, angular momenta, spinors
• AVII Rotation of a spin 1/2 and SU(2) matrices
• BVII Addition of more than two angular momenta
• VIII Transformation of observables under rotation
• AVIII Short review of classical tensors
• BVIII Second-order tensor operators
• CVIII Multipole moments
• DVIII Density matrix expansion on tensor operators
• IX Internal symmetries, SU(2) and SU(3) groups
• AIX The nature of a particle is equivalent to an internal quantum number
• BIX Operators changing the symmetry of a state vector by permutation
• X Symmetry breaking
• APPENDIX
• Bibliography
• Index
• EULA