**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.**Also accessible on Zoom:**here**Reading week:**no lectures from Sat, Feb 17, 2024 to Sun Feb 25, 2024**Discussions/tutorial with prof:**after class**Office hours:**by arrangement.**Grades:**The grades are based on multiple components:

- An essay/project:

- Topic: Study of quantum fluctuations of fields, with visualizations
- Length: 10 to 14 pages
- A guide to get started on the project is here.
- The project will count 50% toward the grade
- Deadline for submitting the projects (via Crowdmark): Monday, April 15, 2pm

- A short presentation, counting 50%. Day: TBA
- Lecture summaries:

- Each Monday, submit a one page summary of the main messages of the lectures of the past week, in pdf format via Crowdmark.
- Since there are 24 lectures in this course, there will be 12 summaries.
- The summaries won't be graded
- To pass the course, submit at least 9 acceptable summaries (out of the 12)

- An essay/project:

## Content

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 quantuhttps://pirsa.org/c24003m theory and some basic knowledge of general relativity, such as FRW spacetimes.

## Grades

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

**Health precautions (permanent)**

*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

## Lectures, lecture notes and videos:

In the list below, the videos are from my pre-pandemic teaching of this course. The new lecture video recordings in Winter 2024 are collected here.

**Jan 9 (Tue), Lecture 1: Notes, Old Video. **

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

**Jan 11 (Thu), Lecture 2: Notes, Old Video. **

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

**Jan 16 (Tue), Lecture 3: Notes, Old Video. **

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

**Feb 29 (Thu), Lecture 14: Notes, OLd Video**

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

**Apr 4 (Thu), Lecture 24: Notes, Old 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!

## Textbook

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).

## Additional literature

- 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