Department of Applied Mathematics

University of Waterloo

Waterloo, Ontario

Canada N2L 3G1

Fax: 519-746-4319

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**Term:**Winter 2020.**Course codes:**AMATH872 / PHYS785**Instructor:**Achim Kempf**Prerequisite:**AMATH 673 or PHYS 702 or consent of instructor. Some knowledge of general relativity.**Time:**Tuesdays and Fridays, 4-5:20pm**First lecture:**4pm, Tuesday, January 7th, 2020, Alice Room, PI**Venue:**Alice room, Perimeter Institute**Reading week:**no lectures Feb.18-21**Office hours:**by appointment

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. The topic is quantum field fluctuations in flat and curved space-time.
**For the description of the project, click here.**- Deadline (to email in a max of 20 pages pdf):
**Sunday, 19 April 2020, 6:00 pm**

Lecture 1: Historical introduction. The role of QFT in the standard models of particle physics and cosmology.

Lecture 2: Quantum fluctuations. Klein Gordon equation. Mode decomposition. Second quantization.

Lecture 3: Mode decomposition. Infrared regularization. Mode oscillators. Probability distribution for fields.

Lecture 4: Field eigenstates. Wave functionals. Schroedinger equation of the 2nd quantized Klein Gordon field.

Lecture 5: Particles as excitations of mode oscillators. External versus parametric particle creation.

Lecture 6: In and out operators. Fock bases. Resonance. Driving creates coherent states. Classicality.

Lecture 7: Bogolubov transformation. Quantum field driven by a classical current, then by a quantum current.

Lecture 8: Light-matter interaction. Absorption and emission by Unruh DeWitt detectors. Unruh effect.

Lecture 9: Functional differentiation. Legendre transform to Lagrangians. Quantization as a Fourier transform.

Lecture 10: Functional derivative of differentiated functions. Action functional. Covariance. Curvature.

Lecture 11: Einstein action and equation. D'Alembert operator. Generally covariant Klein Gordon Hamiltonian.

Lecture 12: Mode functions. Darboux theorem. Solving free QFT on any globally hyperbolic curved spacetime.

Lecture 13: Conservation and covariance of the CCRs. Stone von Neumann theorem. General Bogolubov maps.

Lecture 14: K.G. field in FRW Spacetimes. Conformal time. Chi field. Hamiltonians. Energy momentum tensor.

Lecture 15: Quantization of K.G. field in FRW spacetimes. Bogolubov transformations. Pair creation of particles.

Lecture 16: Particle production through expansion. Lowest energy state is not the vacuum. Adiabatic vacuum.

Lecture 17: Quantum field fluctuation spectra in terms of box variances and correlators. Ultraviolet divergence.

Lecture 18: Amplifications of quantum field fluctuations vs. particle creation. De Sitter horizon and inflation. The video is here.

Lecture 19: Calculation of the field fluctuation spectrum of a scalar field during a de Sitter inflationary period. The video is here.

Lecture 20: Origin of inflation, slow roll and re-heating. Quantum fluctuations of the inflaton and of the metric. The video is here.

Lecture 21: Decomposition of metric fluctuations. Dynamics of Mukhanov variable and tensor polarizations. The video is here.

Lecture 22: Standard model of early universe cosmology. Example power law inflation. Experimental status. The video is here.

Lecture 23: Unruh effect for uniform acceleration from Bogolubov transformations. Stress energy. The video is here.

Lecture 24: Schwarzschild spacetime and its coordinates. Boulware and Kruskal vacua. Hawking radiation. The video is here.

Lecture 25: Interacting QFT through the path integral. Feynman graphs. The video is here.

The video recordings of the lectures 1-17 of 2020 (recorded before the closure due to the Covid19 outbreak) are here: PIRSA

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.

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

Lecture 1: Historical introduction. The role of QFT in the standard models of particle physics and cosmology.

Lecture 2: Quantum fluctuations. Klein Gordon equation. Mode decomposition. Second quantization.

Lecture 3: Mode decomposition. Infrared regularization. Mode oscillators. Probability distribution for fields.

Lecture 4: Field eigenstates. Wave functionals. Schroedinger equation of the 2nd quantized Klein Gordon field.

Lecture 5: Particles as excitations of mode oscillators. External versus parametric particle creation.

Lecture 6: In and out operators. Fock bases. Resonance. Driving creates coherent states. Classicality.

Lecture 7: Bogolubov transformation. Quantum field driven by a classical current, then by a quantum current.

Lecture 8: Light-matter interaction. Absorption and emission by Unruh DeWitt detectors. Unruh effect.

Lecture 9: Functional differentiation. Legendre transform to Lagrangians. Quantization as a Fourier transform.

Lecture 10: Functional derivative of differentiated functions. Action functional. Covariance. Curvature.

Lecture 11: Einstein action and equation. D'Alembert operator. Generally covariant Klein Gordon Hamiltonian.

Lecture 12: Mode functions. Darboux theorem. Solving free QFT on any globally hyperbolic curved spacetime.

Lecture 13: Conservation and covariance of the CCRs. Stone von Neumann theorem. General Bogolubov maps.

Lecture 14: K.G. field in FRW Spacetimes. Conformal time. Chi field. Hamiltonians. Energy momentum tensor.

Lecture 15: Quantization of K.G. field in FRW spacetimes. Bogolubov transformations. Pair creation of particles.

Lecture 16: Particle production through expansion. Lowest energy state is not the vacuum. Adiabatic vacuum.

Lecture 17: Quantum field fluctuation spectra in terms of box variances and correlators. Ultraviolet divergence.

Lecture 18: Amplifications of quantum field fluctuations vs. particle creation. De Sitter horizon and inflation.

Lecture 19: Calculation of the field fluctuation spectrum of a scalar field during a de Sitter inflationary period.

Lecture 20: Origin of inflation, slow roll and re-heating. Quantum fluctuations of the inflaton and of the metric.

Lecture 21: Decomposition of metric fluctuations. Dynamics of Mukhanov variable and tensor polarizations.

Lecture 22: Standard model of early universe cosmology. Example power law inflation. Experimental status.

Lecture 23: Unruh effect for uniform acceleration from Bogolubov transformations. Stress energy.

Lecture 24: Schwarzschild spacetime and its coordinates. Boulware and Kruskal vacua. Hawking radiation.

Lecture 25: (assigned reading) Interacting QFT through the path integral. Feynman graphs. Video here.

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 promised 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 centralized within our Indigenous Initiatives Office.