1Introduction to Quantum Theory
7110.2.1. Derivation of the Klein-Gordon Equation
21.1. Historical Background
7210.2.2. Interpretation and Properties: 10.2.3. Applications and Significance
31.2. The need for a new theory
7310.3. Antimatter and the Positron
41.3. Wave-particle Duality
7410.3.1. The Positron: The First Antiparticle
51.4. The Uncertainty Principle
7510.3.2. Production and Applications of Antimatter: 10.3.3. The Matter-Antimatter Asymmetry Puzzle
61.5. The Quantum World
7610.4. Relativistic Quantum Fields
71.6. Mathematical Foundations
7710.4.1. The Need for Quantum Fields
8Linear Algebra and Hilbert Spaces
7810.4.2. The Quantum Field: A New Perspective
92.1 Vector Spaces
7910.4.3. Particle Creation and Annihilation
102.2 Linear Operators
8010.4.4. Quantization of Fields: 10.4.5. Interactions and Feynman Diagrams
112.3. Hilbert Spaces
8110.5. Lorentz Group and Representations
122.4. Dirac Notation
8210.5.1. The Lorentz Group
132.5. Tensor Products
8310.5.2. Representations of the Lorentz Group
142.6. Spectral Theory
8410.5.3. Lorentz Invariance and Relativistic Wave Equations: 10.5.4. Applications and Significance
15The Schrödinger Equation
8510.6. Mathematical Techniques and Approximations
163.1. The wave function
8610.6.1. Perturbation Theory
173.2. The time-dependent Schrödinger equation
8710.6.2. Renormalization
183.3. The time-independent Schrödinger equation
8810.6.3. Non-Perturbative Methods: 10.6.4. Mathematical Tools and Techniques
193.4. Boundary conditions
89Quantum Field Theory
203.5. Normalization
9011.1. Canonical Quantization of Fields
213.6. Mathematical properties
9111.1.1. Classical Field Theory
22Quantum Mechanics in Hilbert Space
9211.1.2. The Quantization Procedure
234.1. Observables and Hermitian operators
9311.2. The Klein-Gordon Field
244.2. Measurement and the Born rule
9411.2.1. The Classical Klein-Gordon Field: 11.2.2. Quantization of the Klein-Gordon Field
254.3. The uncertainty principle (mathematical formulation)
9511.3. The Dirac Field
264.4. Symmetries and conservation laws
9611.3.1. The Classical Dirac Field: 11.3.2. Quantization of the Dirac Field
274.5. Commutation relations
9711.4. Gauge Theories and Quantum Electrodynamics
284.6. The harmonic oscillator
9811.4.1. Gauge Invariance and Local Gauge Symmetry
29Angular Momentum and Spin
9911.4.2. Quantum Electrodynamics (QED)
305.1. The concept of spin
10011.4.3. Feynman Diagrams and Perturbation Theory: 11.4.4. Applications and Extensions of Gauge Theories
315.2. Angular momentum operators
10111.5. Renormalization
325.3. Rotation matrices and representations
10211.5.1. Regularization and Renormalization
335.4. The addition of angular momenta
10311.5.2. Renormalization Group and Running Couplings: 11.5.3. Renormalizability and Effective Field Theories
345.5. Spin wavefunctions
10411.6. Mathematical Foundations and Challenges
355.6. The spin-orbit interaction
10511.6.1. Functional Integrals and Path Integrals
36Perturbation Theory and Approximation Methods
10611.6.2. Operator Algebras and Axiomatic Field Theory
376.1. The time-independent perturbation theory
10711.6.3. Gauge Theories and Geometric Quantization
386.2. Non-degenerate perturbation theory
10811.6.4. Renormalization and Constructive Field Theory
396.3. Degenerate perturbation theory
10911.6.5. Ongoing Challenges and Future Directions
406.4. The variational method
110Path Integrals and Functional Methods
416.5. The WKB approximation
11112.1 The Path Integral Formulation
426.6. Mathematical techniques and convergence
11212.2 Feynman’s Sum Over Histories
43Identical Particles and Symmetries
11312.3 Functional Integrals and Measures
447.1. The principle of indistinguishability
11412.4 Perturbative Expansions
457.2. Bosons and fermions
11512.5 Applications and Examples
467.3. Symmetric and antisymmetric wavefunctions
11612.6 Mathematical Foundations and Techniques
47Symmetric wavefunctions:
117Condensed Matter Physics
48Antisymmetric wave functions:: Consequences and applications:
11813.1 Crystalline Solids and Lattices
497.4. The Pauli exclusion principle
11913.2 Band Theory
507.5. Permutation groups and representations
12013.3 Semiconductors
517.6. Applications and examples
12113.4 Superconductivity
52Quantum Statistics
12213.5 Topological Phases of Matter
538.1. The Maxwell-Boltzmann distribution
12313.6 Mathematical Techniques and Models
548.2. The Bose-Einstein distribution
124Quantum Gravity
558.3. The Fermi-Dirac distribution
12514.1 The Problem of Quantum Gravity
568.4. Partition functions and thermodynamic quantities
12614.2 Canonical Quantum Gravity
578.5. Blackbody radiation
12714.3 Loop Quantum Gravity
588.6. Mathematical techniques and approximations
12814.4 String Theory
59Quantum Information and Computation
12914.5 Noncommutative Geometry
609.1. Qubits and quantum circuits
13014.6 Mathematical Challenges and Open Questions
619.2. Quantum algorithms
131Mathematical Tools and Techniques
629.3. Quantum Cryptography
13215.1. Functional Analysis
639.4. Quantum error correction
13315.2. Differential Geometry
649.5. Quantum entanglement
13415.3. Group Theory and Representations
659.6. Mathematical foundations and complexity
13515.4. Topology and Algebraic Geometry
66Relativistic Quantum Mechanics
13615.5. Stochastic Processes
6710.1. The Dirac Equation
13715.6. Numerical Methods and Simulations
6810.1.1. Derivation of the Dirac Equation
138Glossary
6910.1.2. Interpretation of the Dirac Equation: 10.1.3. Mathematical Formalism and Applications
139Index
7010.2. The Klein-Gordon Equation
