Department of Applied Mathematics

University of Waterloo

Waterloo, Ontario

Canada N2L 3G1

Fax: 519-746-4319

PDF files require Adobe Acrobat Reader

**Term:**Winter 2024.**Course codes:**AMATH872 / PHYS785**Instructor:**Achim Kempf**Prerequisite:**AMATH673 or PHYS702 or consent of instructor. Some knowledge of general relativity.**Time/venue:**Tuesdays and Thursdays 4-5:20pm in the Alice room at Perimeter Institute. First lecture: Tue, Jan 9.**Reading week:**no lectures from Sat, Feb 17, 2024 to Sun Feb 25, 2024**Discussions/tutorial with prof: TBA****Office hours:**by arrangement.

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.

- The grades are currently planned to be based on a project, and essay and a presentation. Details TBA.

*If you have cold/flu/covid symptoms, do not come to class. Do the right thing, which is taking care of yourself and getting healthy first.*

- For Lecture
*V*ideos (of my previous teaching), scroll down to the bottom of this page. - All important updates will always be posted here.

- For Lecture

In the list below, the videos are from my pre-pandemic teaching of this course. New recordings are planned for Winter 2024.

**Jan 9 (Tue), Lecture 1: Notes, Video. Note: exceptionally, this lecture will be in the Bob room.**

Historical introduction. The role of QFT in the standard models of particle physics and cosmology.

**Jan 11 (Thu), Lecture 2: Notes, Video. Note: exceptionally, this lecture will be in the Bob room.**

Quantum fluctuations. Klein Gordon equation. Mode decomposition. Second quantization.

**Jan 16 (Tue), Lecture 3: Notes, Video. Note: from here, the lectures will be in the Alice room.**

Mode decomposition. Infrared regularization. Mode oscillators. Probability distribution for fields.

**Jan 18 (Thu), Lecture 4: Notes, Video**

Field eigenstates. Wave functionals. Schroedinger equation of the 2nd quantized Klein Gordon field.

**Jan 23 (Tue), Lecture 5: Notes, Video**

Particles as excitations of mode oscillators. External versus parametric particle creation.

**Jan 25 (Thu), Lecture 6: Notes, Video**

In and out operators. Fock bases. Resonance. Driving creates coherent states. Classicality.

**Jan 30 (Tue), Lecture 7: Notes, Video**

Bogolubov transformation. Quantum field driven by a classical current, then by a quantum current.

**Feb 1 (Thu), Lecture 8: Notes, Video**

Light-matter interaction. Absorption and emission by Unruh DeWitt detectors. Unruh effect.

**Feb 6 (Tue), Lecture 9: Notes, Video**

Functional differentiation. Legendre transform to Lagrangians. Quantization as a Fourier transform.

**Feb 8 (Thu), Lecture 10: Notes, Video**

Functional derivative of differentiated functions. Action functional. Covariance. Curvature.

**Feb 13 (Tue), Lecture 11: Notes, Video**

Einstein action and equation. D'Alembert operator. Generally covariant Klein Gordon Hamiltonian.

**Feb 15 (Thu), Lecture 12: Notes, Video**

Mode functions. Darboux theorem. Solving free QFT on any globally hyperbolic curved spacetime.

**Reading Week.
Feb 27 (Tue), Lecture 13: Notes, Video**

Conservation and covariance of the CCRs. Stone von Neumann theorem. General Bogolubov maps.

K.G. field in FRW Spacetimes. Conformal time. Chi field. Hamiltonians. Energy momentum tensor.

**Mar 5 (Tue), Lecture 15: Notes, Video**

Quantization of K.G. field in FRW spacetimes. Bogolubov transformations. Pair creation of particles.

**Mar 7 (Thu), Lecture 16: Notes, Video**

Particle production through expansion. Lowest energy state is not the vacuum. Adiabatic vacuum.

**Mar 12 (Tue), Lecture 17: Notes, Video**

Quantum field fluctuation spectra in terms of box variances and correlators. Ultraviolet divergence.

**Mar 14 (Thu), Lecture 18: Notes, Video**

Amplifications of quantum field fluctuations vs. particle creation. De Sitter horizon and inflation.

**Mar 19 (Tue), Lecture 19: Notes, Video**

Calculation of the field fluctuation spectrum of a scalar field during a de Sitter inflationary period.

**Mar 21 (Thu), Lecture 20: Notes, Video**

Origin of inflation, slow roll and re-heating. Quantum fluctuations of the inflaton and of the metric.

**Mar 26 (Tue), Lecture 21: Notes, Video**

Decomposition of metric fluctuations. Dynamics of Mukhanov variable and tensor polarizations.

**Mar 28 (Thu), Lecture 22: Notes, Video**

Standard model of early universe cosmology. Example power law inflation. Experimental status.

**Apr 2 (Tue), Lecture 23: Notes, Video**

Unruh effect for uniform acceleration from Bogolubov transformations. Stress energy.

**Apr 4 (Thu), Lecture 24: Notes, Video**

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).

- N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space, CUP, 1984.
- S.A. Fulling, Aspects of Quantum Field Theory in Curve Space-Time, CUP, 1989.
- A.R. Liddle, D. H. Lyth, Cosmological Inflation and Large-Scale Structure, CUP, 2000.
- T. Jacobson, Introduction to Quantum Fields in Curved Spacetime and the Hawking Effect, http://arxiv.org/abs/gr-qc/0308048
- L.H. Ford, Quantum Field Theory in Curved Spacetime, http://arxiv.org/abs/gr-qc/9707062

Department of Applied Mathematics

University of Waterloo

Waterloo, Ontario

Canada N2L 3G1

Fax: 519-746-4319

PDF files require Adobe Acrobat Reader

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

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 the Office of Indigenous Relations.