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Introduction to Quantum Effects in Gravity

Book Description

This is the first introductory textbook on quantum field theory in gravitational backgrounds intended for undergraduate and beginning graduate students in the fields of theoretical astrophysics, cosmology, particle physics, and string theory. The book covers the basic (but essential) material of quantization of fields in an expanding universe and quantum fluctuations in inflationary spacetime. It also contains a detailed explanation of the Casimir, Unruh, and Hawking effects, and introduces the method of effective action used for calculating the back-reaction of quantum systems on a classical external gravitational field. The broad scope of the material covered will provide the reader with a thorough perspective of the subject. Every major result is derived from first principles and thoroughly explained. The book is self-contained and assumes only a basic knowledge of general relativity. Exercises with detailed solutions are provided throughout the book.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface
  6. Part I Canonical quantization and particle production
    1. 1 Overview: a taste of quantum fields
      1. 1.1 Classical field
      2. 1.2 Quantum field and its vacuum state
      3. 1.3 The vacuum energy
      4. 1.4 Quantum vacuum fluctuations
      5. 1.5 Particle interpretation of quantum fields
      6. 1.6 Quantum field theory in classical backgrounds
      7. 1.7 Examples of particle creation
    2. 2 Reminder: classical and quantum theory
      1. 2.1 Lagrangian formalism
        1. 2.1.1 Functional derivatives
      2. 2.2 Hamiltonian formalism
      3. 2.3 Quantization of Hamiltonian systems
      4. 2.4 Hilbert spaces and Dirac notation
      5. 2.5 Operators, eigenvalue problem and basis in a Hilbert space
      6. 2.6 Generalized eigenvectors and basic matrix elements
      7. 2.7 Evolution in quantum theory
    3. 3 Driven harmonic oscillator
      1. 3.1 Quantizing an oscillator
      2. 3.2 The “in” and “out” states
      3. 3.3 Matrix elements and Green’s functions
    4. 4 From harmonic oscillators to fields
      1. 4.1 Quantum harmonic oscillators
      2. 4.2 From oscillators to fields
      3. 4.3 Quantizing fields in a flat spacetime
      4. 4.4 The mode expansion
      5. 4.5 Vacuum energy and vacuum fluctuations
      6. 4.6 The Schrödinger equation for a quantum field
    5. 5 Reminder: classical fields
      1. 5.1 The action functional
      2. 5.2 Real scalar field and its coupling to the gravity
      3. 5.3 Gauge invariance and coupling to the electromagnetic field
      4. 5.4 Action for the gravitational and gauge fields
      5. 5.5 Energy-momentum tensor
    6. 6 Quantum fields in expanding universe
      1. 6.1 Classical scalar field in expanding background
        1. 6.1.1 Mode expansion
      2. 6.2 Quantization
      3. 6.3 Bogolyubov transformations
      4. 6.4 Hilbert space; “a- and b-particles”
      5. 6.5 Choice of the physical vacuum
        1. 6.5.1 The instantaneous lowest-energy state
        2. 6.5.2 Ambiguity of the vacuum state
      6. 6.6 Amplitude of quantum fluctuations
        1. 6.6.1 Comparing fluctuations in the vacuum and excited states
      7. 6.7 An example of particle production
    7. 7 Quantum fields in the de Sitter universe
      1. 7.1 De Sitter universe
      2. 7.2 Quantization
        1. 7.2.1 Bunch–Davies vacuum
      3. 7.3 Fluctuations in inflationary universe
    8. 8 Unruh effect
      1. 8.1 Accelerated motion
      2. 8.2 Comoving frame of accelerated observer
      3. 8.3 Quantum fields in inertial and accelerated frames
      4. 8.4 Bogolyubov transformations
      5. 8.5 Occupation numbers and Unruh temperature
    9. 9 Hawking effect. Thermodynamics of black holes
      1. 9.1 Hawking radiation
        1. 9.1.1 Schwarzschild solution
        2. 9.1.2 Kruskal–Szekeres coordinates
        3. 9.1.3 Field quantization and Hawking radiation
        4. 9.1.4 Hawking effect in 3 + 1 dimensions
      2. 9.2 Thermodynamics of black holes
        1. 9.2.1 Laws of black hole thermodynamics
    10. 10 The Casimir effect
      1. 10.1 Vacuum energy between plates
      2. 10.2 Regularization and renormalization
  7. Part II Path integrals and vacuum polarization
    1. 11 Path integrals
      1. 11.1 Evolution operator. Propagator
      2. 11.2 Propagator as a path integral
      3. 11.3 Lagrangian path integrals
      4. 11.4 Propagators for free particle and harmonic oscillator
        1. 11.4.1 Free particle
        2. 11.4.2 Quadratic potential
        3. 11.4.3 Euclidean path integral
        4. 11.4.4 Ground state as a path integral
    2. 12 Effective action
      1. 12.1 Driven harmonic oscillator (continuation)
        1. 12.1.1 Green’s functions and matrix elements
        2. 12.1.2 Euclidean Green’s function
        3. 12.1.3 Introducing effective action
        4. 12.1.4 Calculating effective action for a driven oscillator
        5. 12.1.5 Matrix elements
        6. 12.1.6 The effective action “recipe”
        7. 12.1.7 Backreaction
      2. 12.2 Effective action in external gravitational field
        1. 12.2.1 Euclidean action for scalar field
      3. 12.3 Effective action as a functional determinant
        1. 12.3.1 Reformulation of the eigenvalue problem
        2. 12.3.2 Zeta function
        3. 12.3.3 Heat kernel
    3. 13 Calculation of heat kernel
      1. 13.1 Perturbative expansion for the heat kernel
        1. 13.1.1 Matrix elements
      2. 13.2 Trace of the heat kernel
      3. 13.3 The Seeley–DeWitt expansion
    4. 14 Results from effective action
      1. 14.1 Renormalization of the effective action
      2. 14.2 Finite terms in the effective action
        1. 14.2.1 EMT from the Polyakov action
      3. 14.3 Conformal anomaly
  8. Appendix 1 Mathematical supplement
    1. A1.1 Functionals and distributions (generalized functions)
    2. A1.2 Green’s functions, boundary conditions, and contours
    3. A1.3 Euler’s gamma function and analytic continuations
  9. Appendix 2 Backreaction derived from effective action
  10. Appendix 3 Mode expansions cheat sheet
  11. Appendix 4 Solutions to exercises
  12. Index