Department of Applied Mathematics

University of Waterloo

Waterloo, Ontario

Canada N2L 3G1

Fax: 519-746-4319

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**Term:**Winter 2022.**Course codes:**AMATH872 / PHYS785**Instructor:**Achim Kempf**Prerequisite:**AMATH673 or PHYS702 or consent of instructor. Some knowledge of general relativity.**Time/venue:**Mondays and Wednesdays (initially online) but now the lectures are in person, Mon+Wed, 4-5:20pm in the Bob room at Perimeter Institute.**Reading week:**no lectures Feb.19-27, 2022**Discussions/tutorial with prof:**Fridays 4-5pm in PI's Bob room, except: (via Zoom on March 25th).**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 will be based on a project.

- For the description of the project, click here. Part A is explained in some detail to get you started. In Part B you can give the project your own focus.
- Deadline (15-20 pages max, in pdf format):
**Friday, 29 April 2022, 6:00 pm**

**Also:**Brief summaries of a week's lectures (1/2 page per week) are to be submitted as a pdf file via email to the instructor:

- The summaries should be in full sentences, in your own words, concisely explaining what were the key points of the lectures.
- The homework will be graded pass/fail. The homework does not enter the calculation of the course grade.
- However, at least 8 of the 11 homework summaries need to be passes to pass the course.
- In your email's subject line, use the word "summary".
- Submissions are due Fridays 10pm to akempf at the usual uwaterloo.ca.:

14 Jan for Lectures 2+3

21 Jan for Lectures 4+5

28 Jan for Lectures 6+7

4 Feb for Lectures 8+9

11 Feb for Lectures 10+11

18 Feb for Lectures 12+13

*** reading week here ***

4 March for Lectures 14+15

11 March for Lectures 16+17

18 March for Lectures 18+19

25 March for Lectures 20 +21

1 April for Lectures 22+23

No summaries are to be submitted for lectures 1 and 24

**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. **

**The recordings will stay online here.****Do not come if you don't wish to come, have symptoms or have tested positive.****Arrive early enough because you will probably need to do a rapid test at the reception.**

**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).

- 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.