Master-Pflichtseminar : QFT in curved spacetimes

SCHEDULE & NOTES             Basic bibliography      

What all this is about

Particle physics experiments are performed in a background that can be very well approximated by a Minkowski spacetime. This approximation is however not applicable to extreme astrophysical and cosmological environments. QFT in CST is expected to provide an accurate description of quantum phenomena in a regime in which the spacetime cannot be longer described by a Minkowski metric but quantum gravity itself can be neglected.

In this seminar, we will extend the canonical quantization in Minkowski spacetime to curved spacetimes (Lorentzian manifolds). This extension will allow us to disentangle the important structures and concepts of QFT from the simplifications emerging in the flat spacetime approximation. In particular, we will discover that familiar concepts like the vacuum state or the number of particles become ambiguous and observer-dependent on spaces without any particular symmetry. The material considered in this seminar is not only of academical interest. The theory makes spectacular predictions such as the generation of primordial perturbations during inflation and the Hawking radiation of black holes.


The seminar is intended to be self-consistent regarding basic QFT concepts. Although a previous exposure to this topic in certainly positive, it is not necessary to have taken a course on Quantum Field Theory. The relevant aspects of canonical quantization (mode expansion, creation and annihilation operators, vacuum definition, Fock space construction, etc…) will be covered during the course. A basic knowledge of General Relativity (fields in arbitrary coordinate systems, concepts of proper distance and proper time, Schwarzschild and FRW metrics,… ) is highly recommended, specially in the last part. Familiarity with some concepts from quantum optics (coherent states, squeezed states,…) could be also helpful but not necessary.


The emphasis of the seminar will be on physical concepts rather than on an axiomatic formulation of QFT in CST. Most of the topics and examples will be simplified to the barest possible minimum that still contains the relevant physical information (free scalar fields only, conformally flat metrics, reduced dimensions, etc…). The course will be divided in 3+1 parts:

1. Introduction to QFT in non-trivial backgrounds

We will start by discussing the quantization of  a scalar field with a time dependent mass.

1.1. Quantization of a scalar field with a time dependent mass.
1.2. Ambiguity of the vacuum state and correlation functions.
1.3 The Schwinger effect.

2. Inflationary cosmology

Inflation is nowadays a well established paradigm fully consistent with observations and able to explain the generation of an almost scale invariant spectrum of primordial perturbations giving rise to structure formation. We will apply the techniques developed in the first part of the course to simplified models capturing the main aspects of the inflationary dynamic. The depletion of the inflaton condensate after inflation will be also considered.

2.1 Introduction to inflation.
2.2 Quantum field theory in de Sitter spacetime.
2.3 Quantum fluctuations during inflation.
2.4 Inflationary observables.
2.5 Decoherence without decoherence.
2.6 (P)reheating after inflation.
2.7 Preheating in the lattice.

3. Accelerated observers and black holes

Before Hawking’s seminal papers, spherically-symmetric static black holes were assumed to be completely inert. Now we believe that black holes radiate, quantum mechanically and thermally, at a temperature proportional to their surface gravity. In this part of the course we will discuss the so-called Unruh effect and the classical and semiclassical theory of black holes.

3.1. The Unruh effect.
3.2  Hawking radiation.
3.3  KMS condition and Green functions.
3.4  Euclidean field theory: revisiting Unruh, Hawking and de Sitter.
3.5. The black hole information paradox.
3.6  AdS/CFT and the information paradox.
3.7. Analog gravity.

+1. Miscellanea

4.1. Effective action for time-dependent harmonic oscillator in quantum mechanics.
4.2. Effective action in a external gravitational field.
4.3. The heat kernel computation.
4.4. The conformal anomaly.
4.5.  Why Quantum Gravity? What is a graviton?
4.6 The ADM formalism.
4.7 The Wheeler-deWitt equation.



– Introduction to Quantum Effects in Gravity, Viatcheslav Mukhanov and Sergei Winitzki, Cambridge University Press, 2007. Also online
– Quantum Fields in Curved Space, N.D. Birrell, P.C.W. Davies, Cambridge University Press, 1982.
– Quantum Field Theory in Curved Spacetime, L.E. Parker and D.J. Toms, Cambridge University Press, 2009.
– Vacuum Quantum effects in Strong Fields, A.A. Grib, S.G. Mamayev and V.M. Mostepanenko, Friedmann Laboratory publishing, 1994.
– Physical Foundations of Cosmology, Viatcheslav Mukhanov, Cambridge University Press, 2005.
– The Global Approach to Quantum Field Theory, B. S. Dewitt, Oxford University Press, 2003.
– Aspects of quantum field theory in curved space-time, SA Fulling, Cambridge University Press, 1989.
– Quantum Gravity, Claus Kiefer, Oxford University Press 2007.
– General Relativity, Robert M. Wald, University of Chicago Press,1984.

Online lecture notes

– Quantum Field Theory in Curved Spacetime, L. H. Ford, arXiv:gr-qc/9707062.
– Topics in Quantum Field Theory in Curved Space, J. Haro, arXiv:1011.4772.
– Quantum Field Theory in Curved Spacetime, J. Preskill, Physics 236c.
– Quantum field theory in curved spacetime, Christopher J. Fewster. Lectures.
– Introduction to Quantum Fields in Curved Spacetime and the Hawking Effect, T. Jacobson, arXiv:gr-qc/0308048.
– Beyond flat-space quantum field theory, S. S. Seahra, Lectures.
– TASI Lectures on Inflation, Daniel Baumann, arXiv:0907.5424.
– Lectures on Inflation, Leonardo Senatore, arXiv:1609.00716.
– Inflationary Perturbations: the Cosmological Schwinger Effect, Jerome Martin, arXiv:0704.3540.
– Inflation and Reheating, Juan Garcia-Bellido, Lectures.
– Black Holes, P.K. Townsend, arXiv:gr-qc/9707012.
– A Primer for Black Hole Quantum Physics, R. Brout et al., arXiv:0710.4345.
– Black holes and Hawking radiation in spacetime and its analogues, T. Jacobson, arXiv:1212.6821.
– Quantum fields near Black Holes, A. Wipf, arXiv:hep-th/9801025.
– Introduction to the Theory of Black Holes, G. ’t Hooft. Lectures
– Quantum Field Theory, Black Holes and Holography, C. Krishnan, arXiv:1011.5875.
– Introduction to Black Hole Evaporation, P.-H. Lambert, arXiv:1310.8312.
– The information paradox: A pedagogical introduction, Samir D. Mathur, arXiv:0909.1038.
– The Black Hole information problem, Joseph Polchinski,  arXiv:1609.04036


 Selected articles and reviews

–  Semiclassicality and Decoherence of Cosmological Perturbations, David Polarski and Alexei A Starobinsky, gr-qc/9504030.
– Towards the theory of reheating after Inflation, Lev Kofman, Andrei Linde, Alexei Starobinsky, arXiv:hep-ph/9704452.
– The Unruh effect and its applications, Luis C. B. Crispino, Atsushi Higuchi, George E. A. Matsas, arXiv:0710.5373.
– Particle creation by black holes, S. W. Hawking, Commun.Math. Phys. (1975) 43: 199.
– Essential and inessential features of Hawking radiation, Matt Visser, arXiv:hep-th/0106111.
–  Temperature, periodicity and horizons, S.A. Fulling,  S. N. M. Ruijsenaars.
– What Exactly is the Information Paradox?, Samir D. Mathur, arXiv:0803.2030.
– The Dynamics of General Relativity, Arnowitt, Deser and Misner, arXiv:gr-qc/0405109.
– Ahmed Almheiri, Donald Marolf, Joseph Polchinski, James Sully, Black Holes: Complementarity or Firewalls?
– Ahmed Almheiri, Donald Marolf, Joseph Polchinski, Douglas Stanford, James Sully, An Apologia for Firewalls
– Analog gravity, Carlos Barcelo, Stefano Liberati, Matt Visser, arXiv:gr-qc/0505065 and references therein.