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Quantum Mechanics and Quantum Field Theory

Book Description

Explaining the concepts of quantum mechanics and quantum field theory in a precise mathematical language, this textbook is an ideal introduction for graduate students in mathematics, helping to prepare them for further studies in quantum physics. The textbook covers topics that are central to quantum physics: non-relativistic quantum mechanics, quantum statistical mechanics, relativistic quantum mechanics and quantum field theory. There is also background material on analysis, classical mechanics, relativity and probability. Each topic is explored through a statement of basic principles followed by simple examples. Around 100 problems throughout the textbook help readers develop their understanding.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Dedication
  5. Contents
  6. Preface
  7. Introduction
  8. Part I Non-relativistic
    1. 1 Mathematical prelude
      1. 1.1 Bounded operators
      2. 1.2 Unbounded operators
      3. 1.3 Self-adjoint operators
      4. 1.4 Compact operators
    2. 2 Classical mechanics
      1. 2.1 Hamiltonian mechanics
      2. 2.2 Examples
      3. 2.3 Canonical transformations
      4. 2.4 Symmetries
    3. 3 Quantum mechanics
      1. 3.1 Principles of quantum mechanics
      2. 3.2 Canonical quantization
      3. 3.3 Symmetries
      4. 3.4 Perspectives and problems
    4. 4 Single particle
      1. 4.1 Free particle
      2. 4.2 Particle in a potential
      3. 4.3 Spectrum
      4. 4.4 The harmonic oscillator
      5. 4.5 Scattering
      6. 4.6 Spin
    5. 5 Many particles
      1. 5.1 Two particles
      2. 5.2 Identical particles
      3. 5.3 n-particles
      4. 5.4 Fock space
    6. 6 Statistical mechanics
      1. 6.1 Mixed states
      2. 6.2 Equilibrium states
      3. 6.3 Free boson gas
      4. 6.4 Free fermion gas
      5. 6.5 Interacting bosons
      6. 6.6 Further developments
  9. Part II Relativistic
    1. 7 Relativity
      1. 7.1 Principles of relativity
      2. 7.2 Minkowski space
      3. 7.3 Classical free fields
      4. 7.4 Interacting classical fields
      5. 7.5 Fundamental solutions
    2. 8 Scalar particles and fields
      1. 8.1 Scalar particles
      2. 8.2 Scalar fields
      3. 8.3 Charged scalar field
    3. 9 Electrons and photons
      1. 9.1 Spinors
      2. 9.2 Electrons
      3. 9.3 Dirac fields
      4. 9.4 Photons
      5. 9.5 Electromagnetic field
    4. 10 Field theory on a manifold
      1. 10.1 Lorentzian manifolds
      2. 10.2 Classical fields on a manifold
      3. 10.3 Quantum fields on a manifold
  10. Part III Probabilistic methods
    1. 11 Path integrals
      1. 11.1 Probability
      2. 11.2 Gaussian processes
      3. 11.3 Brownian motion
      4. 11.4 The Feynman–Kac formula
      5. 11.5 Oscillator process
      6. 11.6 Application: ground states
    2. 12 Fields as random variables
      1. 12.1 More on Gaussian processes
      2. 12.2 The Schrödinger representation
      3. 12.3 Path integrals – free fields
      4. 12.4 Vacuum correlation functions
      5. 12.5 Thermal correlation functions
    3. 13 A nonlinear field theory
      1. 13.1 The model
      2. 13.2 Regularization
      3. 13.3 Infinite volume
      4. 13.4 Path integrals – interacting fields
      5. 13.5 A reformulation
  11. Appendix A Normed spaces
  12. Appendix B Tensor product
  13. Appendix C Distributions
  14. References
  15. Index