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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