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