Franck Laloë

Introduction to Continuous Symmetries (eBook, PDF)

From Space-Time to Quantum Mechanics
Übersetzer: Ostrowsky, Nicole; Ostrowsky, Daniel

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Wiley-VCH
• Erscheinungstermin: 29. Juni 2023
• Englisch
• ISBN-13: 9783527840540
• Artikelnr.: 68385478

Autorenporträt

Franck Laloë is a researcher at the Kastler-Brossel Laboratory of the Ecole Normale Supérieure in Paris, France. His research is focused on optical pumping, the statistical mechanics of quantum gases, musical acoustics, and the foundations of quantum mechanics.

Inhaltsangabe

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
A_IV Finite-Dimensional Unitary Projective Representations of Related Lie Groups
1 Case Where G is Simply Connected
2 Case Where G is P-Connected
B_IV 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
A_V Some Properties of the Operators S and W_2
1 Operator S
2 Eigenvalues of the Operator W_2
B_V Geometric Displacement Group
1 Reminders: Classical Properties of Displacements
2 Associated Operators in the State Space
C_V Clean Lorentz Group
1 Link with the Group SL(2,C)
2 Small Group Associated with a Four-Vector
3 Operator W_2
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
A_VIII 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
B_VIII 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)
A_IX 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
B_IX Operators Changing the Symmetry of a State Vector by Permutation
1 Fermions
2 Bosons

X Symmetry Breaking
A Magnetism, Breaking of the Rotation