
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, 45:20pm in the Bob room at Perimeter Institute.
 Reading week: no lectures Feb.1927, 2022
 Discussions/tutorial with prof: Fridays 45pm in PI's Bob room, except: (via Zoom on March 25th).
 Office hours: by arrangement.
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 spacetime 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 smallwavelength but largeamplitude 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.
Grades

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 (1520 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
Lectures (initially online) and lecture notes:
Announcement (25 Jan.): Preliminary indications from Perimeter Institute are that we will be able to move to inclass 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
Lightmatter
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
reheating.
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!
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 SpaceTime, CUP, 1989.
 A.R. Liddle, D. H. Lyth, Cosmological Inflation and LargeScale Structure, CUP, 2000.
 T. Jacobson, Introduction to Quantum Fields in Curved Spacetime and the Hawking Effect, http://arxiv.org/abs/grqc/0308048
 L.H. Ford, Quantum Field Theory in Curved Spacetime, http://arxiv.org/abs/grqc/9707062