Introduction to Continuous Symmetries

In dem neuen Werk von Franck Laloe wird ein symmetriebasierter Ansatz zum grundlegenden Verständnis der Quantenmechanik vorgestellt ? zusammen mit den entsprechenden Rechentechniken, die Studierende höherer Semester in den Bereichen Nuklearphysik, Quantenopik und Festkörperphysik benötigen.

Autorentext
Franck Laloë is a researcher at the Kastler-Brossel laboratory of the Ecole Normale Supérieure in Paris. His first assignment was with the University of Paris VI before he was appointed to the CNRS, the French National Research Center. His research is focused on optical pumping, statistical mechanics of quantum gases, musical acoustics and the foundations of quantum mechanics.

Klappentext

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.

  Inhalt
  I Symmetry Transformations
  A Fundamental Symmetries
  B Symmetries in Classical Mechanics C Symmetries in Quantum Mechanics
  A_I Euler's and Lagrange's Views in Classical Mechanics
  1 Euler's Point of View
  2 Lagrange's Point of View

  II Notions on Group Theory
  A General Properties of Groups
  B Linear Representations of a Group
  A_II Residual Classes of a Subgroup; Quotient Group
  1 Residual Classes on the Left
  2 Quotient Group

  III Introduction to Continuous Groups and Lie Groups
  A General Properties B Examples
  C Galileo and Poincaré Groups
  A_III Adjoint Representation, Killing Form, Casimir Operator
  1 Representation Adjoint to the Lie Algebra
  2 Killing Form; Scalar Product and Change of Basis in L
  3 Totally Antisymmetric Structure Constants
  4 Casimir Operator

  IV Representations Induced in the State Space
  A Conditions Imposed on Transformations in the State Space
  B Wigner's Theorem
  C Transformations of Observables
  D Linear Representations in the State Space
  E Phase Factors and Projective Representations
  AIV Finite-Dimensional Unitary Projective Representations of Related Lie Groups
  1 Case Where G is Simply Connected
  2 Case Where G is P-Connected
  BIV Uhlhorn-Wigner Theorem
  1 Real Space
  2 Complex Space

  V Representations of the Galileo and Poincaré Groups: Mass, Spin and Energy
  A Galileo Group
  B Poincaré Group
  AV Some Properties of the Operators S and W2
  1 Operator S
  2 Eigenvalues of the Operator W2
  BV Geometric Displacement Group
  1 Reminders: Classical Properties of Displacements
  2 Associated Operators in the State Space
  CV Clean Lorentz Group
  1 Link with the Group SL(2,C)
  2 Small Group Associated with a Four-Vector
  3 Operator W2
  D_V Space Reflections (Parity)
  1 Action in Real Space
  2 Associated Operator in the State Space
  3 Retention of Parity

  VI Construction of State Spaces and Wave Equations
  A Galileo Group, Schrödinger Equation
  B Poincaré Group, Klein-Gordon and Dirac Equations
  A_VI Lagrangians of Wave Equations
  1 Lagrangian of a Field
  2 Schrödinger's Equation
  3 Klein-Gordon Equation
  4 Dirac's Equation

  VII Irreducible Representations of the Group of Rotations, Spinors
  A Irreducible Unitary Representations of the Group of Rotations
  B Spin 1/2 Particles; Spinors
  C Composition of the Kinetic Moments
  A_VII Homorphism Between SU(2) and Rotation Matrices
  1 Transformation of a Vector P Induced by an SU(2) Matrix
  2 The Transformation is a Rotation
  3 Homomorphism
  4 Link to the Reasoning of Chapter VII
  5 Link with Bivalent Representations

  VIII Transformation of Observables by Rotation
  A Vector Operators B Tensor Operators
  C Wigner-Eckart Theorem
  D Decomposition of the Density Matrix on Tensor Operators
  AVIII Basic Reminders on Classical Tensors
  1 Vectors
  2 Tensors
  3 Properties
  4 Tensoriality Criterion
  5 Symmetric and Antisymmetric Tensors
  6 Special Tensors
  7 Irreducible Tensors
  BVIII Second Order Tensor Operators
  1 Tensor Product of Two Vector Operators
  2 Cartesian Components of the Tensor in the General Case
  C_VIII Multipolar Moments
  1 Electrical Multipole Moments
  2 Magnetic Multipole Moments
  3 Multipole Moments of a Quantum System for a Given Kinetic Moment Multiplicity J

  IX Groups SU(2) and SU(3)
  A System of Discernible but Equivalent Particles
  B SU(2) Group and Isospin Symmetry
  C Symmetry SU(3)
  AIX the Nature of a Particle Is Equivalent to an Internal Quantum Number
  1 Partial or Total Antisymmetrization of a State Vector
  2 Correspondence Between the States of Two Physical Systems
  3 Physical Consequences
  BIX Operators Changing the Symmetry of a State Vector by Permutation
  1 Fermions
  2 Bosons

  X Symmetry Breaking
  A Magnetism, Breaking of the Rotation Symmetry
  B Some Other Examples

  APPENDIX
  I The Reversal of Time
  1 Time Reversal in Classical Mechanics
  2 Antilinear and Antiunitary Operators in Quantum Mechanics
  3 Time Reversal and Antilinearity
  4 Explicit Form of the Time Reversal Operator
  5 Applications