| Preface | p. ix |
| Canonical quantization and particle production | p. 1 |
| Overview: a taste of quantum fields | p. 3 |
| Classical field | p. 3 |
| Quantum field and its vacuum state | p. 4 |
| The vacuum energy | p. 7 |
| Quantum vacuum fluctuations | p. 8 |
| Particle interpretation of quantum fields | p. 9 |
| Quantum field theory in classical backgrounds | p. 9 |
| Examples of particle creation | p. 10 |
| Reminder: classical and quantum theory | p. 13 |
| Lagrangian formalism | p. 13 |
| Functional derivatives | p. 15 |
| Hamiltonian formalism | p. 17 |
| Quantization of Hamiltonian systems | p. 19 |
| Hilbert spaces and Dirac notation | p. 20 |
| Operators, eigenvalue problem and basis in a Hilbert space | p. 22 |
| Generalized eigenvectors and basic matrix elements | p. 26 |
| Evolution in quantum theory | p. 29 |
| Driven harmonic oscillator | p. 33 |
| Quantizing an oscillator | p. 33 |
| The "in" and "out" states | p. 35 |
| Matrix elements and Green's functions | p. 38 |
| From harmonic oscillators to fields | p. 42 |
| Quantum harmonic oscillators | p. 42 |
| From oscillators to fields | p. 43 |
| Quantizing fields in a flat spacetime | p. 45 |
| The mode expansion | p. 48 |
| Vacuum energy and vacuum fluctuations | p. 50 |
| The Schrodinger equation for a quantum field | p. 51 |
| Reminder: classical fields | p. 54 |
| The action functional | p. 54 |
| Real scalar field and its coupling to the gravity | p. 56 |
| Gauge invariance and coupling to the electromagnetic field | p. 58 |
| Action for the gravitational and gauge fields | p. 59 |
| Energy-momentum tensor | p. 61 |
| Quantum fields in expanding universe | p. 64 |
| Classical scalar field in expanding background | p. 64 |
| Mode expansion | p. 66 |
| Quantization | p. 67 |
| Bogolyubov transformations | p. 68 |
| Hilbert space; "a- and b-particles" | p. 69 |
| Choice of the physical vacuum | p. 71 |
| The instantaneous lowest-energy state | p. 71 |
| Ambiguity of the vacuum state | p. 74 |
| Amplitude of quantum fluctuations | p. 78 |
| Comparing fluctuations in the vacuum and excited states | p. 80 |
| An example of particle production | p. 81 |
| Quantum fields in the de Sitter universe | p. 85 |
| De Sitter universe | p. 85 |
| Quantization | p. 88 |
| Bunch-Davies vacuum | p. 90 |
| Fluctuations in inflationary universe | p. 92 |
| Unruh effect | p. 97 |
| Accelerated motion | p. 97 |
| Comoving frame of accelerated observer | p. 100 |
| Quantum fields in inertial and accelerated frames | p. 103 |
| Bogolyubov transformations | p. 106 |
| Occupation numbers and Unruh temperature | p. 107 |
| Hawking effect. Thermodynamics of black holes | p. 109 |
| Hawking radiation | p. 109 |
| Schwarzschild solution | p. 110 |
| Kruskal-Szekeres coordinates | p. 111 |
| Field quantization and Hawking radiation | p. 115 |
| Hawking effect in 3 + 1 dimensions | p. 118 |
| Thermodynamics of black holes | p. 120 |
| Laws of black hole thermodynamics | p. 121 |
| The Casimir effect | p. 124 |
| Vacuum energy between plates | p. 124 |
| Regularization and renormalization | p. 125 |
| Path integrals and vacuum polarization | p. 129 |
| Path integrals | p. 131 |
| Evolution operator. Propagator | p. 131 |
| Propagator as a path integral | p. 132 |
| Lagrangian path integrals | p. 137 |
| Propagators for free particle and harmonic oscillator | p. 138 |
| Free particle | p. 139 |
| Quadratic potential | p. 139 |
| Euclidean path integral | p. 142 |
| Ground state as a path integral | p. 144 |
| Effective action | p. 146 |
| Driven harmonic oscillator (continuation) | p. 146 |
| Green's functions and matrix elements | p. 146 |
| Euclidean Green's function | p. 148 |
| Introducing effective action | p. 150 |
| Calculating effective action for a driven oscillator | p. 152 |
| Matrix elements | p. 154 |
| The effective action "recipe" | p. 157 |
| Backreaction | p. 158 |
| Effective action in external gravitational field | p. 159 |
| Euclidean action for scalar field | p. 161 |
| Effective action as a functional determinant | p. 163 |
| Reformulation of the eigenvalue problem | p. 164 |
| Zeta function | p. 166 |
| Heat kernel | p. 167 |
| Calculation of heat kernel | p. 170 |
| Perturbative expansion for the heat kernel | p. 171 |
| Matrix elements | p. 172 |
| Trace of the heat kernel | p. 176 |
| The Seeley-DeWitt expansion | p. 178 |
| Results from effective action | p. 180 |
| Renormalization of the effective action | p. 180 |
| Finite terms in the effective action | p. 183 |
| EMT from the Polyakov action | p. 185 |
| Conformal anomaly | p. 187 |
| Mathematical supplement | p. 193 |
| Functionals and distributions (generalized functions) | p. 193 |
| Green's functions, boundary conditions, and contours | p. 202 |
| Euler's gamma function and analytic continuations | p. 206 |
| Backreaction derived from effective action | p. 212 |
| Mode expansions cheat sheet | p. 216 |
| Solutions to exercises | p. 218 |
| Index | p. 272 |
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