Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
Fax: 519-746-4319
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This course introduces quantum field theory from scratch and then develops the theory of the quantum fluctuations of fields and particles. We will focus, in particular, on how quantum fields are affected by curvature and by spacetime horizons. This will lead us to the Unruh effect, Hawking radiation and to inflationary cosmology. Inflationary cosmology, which we will study in detail, is part of the current standard model of cosmology which holds that all structure in the universe - such as the distribution of galaxies - originated in tiny quantum fluctuations of a scalar field and of space-time itself. For intuition, consider that quantum field fluctuations of significant amplitude normally occur only at very small length scales. Close to the big bang, during a brief initial period of nearly exponentially fast expansion (inflation), such small-wavelength but large-amplitude quantum fluctuations were stretched out to cosmological wavelengths. In this way, quantum fluctuations are thought to have seeded the observed inhomogeneities in the cosmic microwave background - which in turn seeded the condensation of hydrogen into galaxies and stars, all closely matching the increasingly accurate astronomical observations over recent years. The prerequisites for this course are a solid understanding of quantum theory and some basic knowledge of general relativity, such as FRW spacetimes.
Announcement (25 Jan.): Preliminary indications from Perimeter Institute are that we will be able to move to in-class teaching from after reading week. I was also told that there is a chance that PI might open to us earlier.
Jan. 5 (Wed), Lecture 1: Notes, Video
Historical introduction. The role of QFT in the standard models of particle physics and cosmology.
Jan 10 (Mon), Lecture 2: Notes, Video
Quantum fluctuations. Klein Gordon equation. Mode decomposition. Second quantization.
Jan 12 (Wed), Lecture 3: Notes, Video
Mode decomposition. Infrared regularization. Mode oscillators. Probability distribution for fields.
Jan 17 (Mon), Lecture 4: Notes, Video
Field eigenstates. Wave functionals. Schroedinger equation of the 2nd quantized Klein Gordon field.
Jan 19 (Wed), Lecture 5: Notes, Video
Particles as excitations of mode oscillators. External versus parametric particle creation.
Jan 24 (Mon), Lecture 6: Notes, Video
In and out operators. Fock bases. Resonance. Driving creates coherent states. Classicality.
Jan 26 (Wed), Lecture 7: Notes, Video
Bogolubov transformation. Quantum field driven by a classical current, then by a quantum current.
Jan 31 (Mon), Lecture 8: Notes, Video
Light-matter interaction. Absorption and emission by Unruh DeWitt detectors. Unruh effect.
Feb 2 (Wed), Lecture 9: Notes, Video
Functional differentiation. Legendre transform to Lagrangians. Quantization as a Fourier transform.
Feb 7 (Mon), Lecture 10: Notes, Video
Functional derivative of differentiated functions. Action functional. Covariance. Curvature.
Feb 9 (Wed), Lecture 11: Notes, Video
Einstein action and equation. D'Alembert operator. Generally covariant Klein Gordon Hamiltonian.
Feb 14 (Mon), Lecture 12: Notes, Video
Mode functions. Darboux theorem. Solving free QFT on any globally hyperbolic curved spacetime.
Feb 16 (Wed), Lecture 13: Notes, Video
Conservation and covariance of the CCRs. Stone von Neumann theorem. General Bogolubov maps.
Reading Week.
From this time onward, our lectures will be in person in the Bob room.
Feb 28 (Mon), Lecture 14: Notes, Video, 1pm Bob room (Special time!)
K.G. field in FRW Spacetimes. Conformal time. Chi field. Hamiltonians. Energy momentum tensor.
Mar 2 (Wed), Lecture 15: Notes, Video, 4pm Bob room
Quantization of K.G. field in FRW spacetimes. Bogolubov transformations. Pair creation of particles.
Mar 7 (Mon), Lecture 16: Notes, Video, 4pm Bob room
Particle production through expansion. Lowest energy state is not the vacuum. Adiabatic vacuum.
Mar 9 (Wed), Lecture 17: Notes, Video, 4pm Bob room
Quantum field fluctuation spectra in terms of box variances and correlators. Ultraviolet divergence.
Mar 14 (Mon), Lecture 18: Notes, Video, 4pm Bob room
Amplifications of quantum field fluctuations vs. particle creation. De Sitter horizon and inflation.
Mar 16 (Wed), Lecture 19: Notes, Video, 4pm Bob room
Calculation of the field fluctuation spectrum of a scalar field during a de Sitter inflationary period.
Mar 21 (Mon), Lecture 20: Notes, Video, 4pm Bob room
Origin of inflation, slow roll and re-heating. Quantum fluctuations of the inflaton and of the metric.
Mar 23 (Wed), Lecture 21: Notes, Video, 4pm Bob room
Decomposition of metric fluctuations. Dynamics of Mukhanov variable and tensor polarizations.
*** Special time: Mar 29 (Tue) ***, Lecture 22: Notes, Video, 1:30pm Bob room
Standard model of early universe cosmology. Example power law inflation. Experimental status.
Mar 30 (Wed), Lecture 23: Notes, Video, 4pm Bob room
Unruh effect for uniform acceleration from Bogolubov transformations. Stress energy.
Apr 4 (Mon), Lecture 24: Notes, Video, 4pm Bob room
Schwarzschild spacetime and its coordinates. Boulware and Kruskal vacua. Hawking radiation.
Remark to those who are not enrolled: I invite anybody to download these lecture notes for study purposes and to view the recordings, even without being enrolled in the course. If you do, please send me an email though, I'd just like to know. Thanks!
To some extent, we will follow this textbook: V. Mukhanov, Sergei Winitzki, Introduction to Quantum Effects in Gravity, Cambridge University Press, June 2007. It has plenty of homework problems including solutions. I strongly recommend making use of them.
See an early version of it: Introduction to Quantum Effects in Gravity (PDF).
Department of Applied Mathematics
University of Waterloo
Waterloo, Ontario
Canada N2L 3G1
PDF files require Adobe Acrobat Reader
The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is co-ordinated within our Office of Indigenous Relations.