-
Term:
Winter
2022.
- Course codes:ÌýAMATH872 / PHYS785
- Instructor: Achim Kempf
- Prerequisite: AMATH673 orÌýPHYS702Ìýor consent of instructor. Some knowledge of general relativity.
- Time/venue:ÌýMondays and WednesdaysÌý(initially online) but now the lectures areÌýin person,ÌýMon+Wed, 4-5:20pm in the Bob room at Perimeter Institute.
- Reading week: no lecturesÌýFeb.19-27, 2022
- Discussions/tutorial with prof: Fridays 4-5pm in PI's Bob room, except:Ìý.
- Office hours: by arrangement.Ìý
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 theÌýUnruh 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.
Grades
-
The
grades
will
be
based
on
aÌýproject.Ìý
Ìý- 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.ÌýÌý
- Deadline (15-20 pages max, inÌýpdfÌýformat): Friday, 29ÌýApril 2022, 6:00Ìýpm
-
Also:
Brief
summaries
of
a
week's
lectures
(1/2
page
per
week)
are
to
be
submitted
as
a
pdf
file
viaÌýemail
to
the
instructor:
Ìý- 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.ÌýÌý
- In your email's subject line, use the word "summary".
-
Submissions
are
dueÌýFridays
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 FebÌýfor Lectures 8+9
11 FebÌýfor Lectures 10+11
18 FebÌýfor Lectures 12+13
*** reading week here ***
4 MarchÌýfor Lectures 14+15
11 MarchÌýfor Lectures 16+17
18 MarchÌýfor Lectures 18+19
25 MarchÌýfor Lectures 20 +21
1 AprilÌýfor Lectures 22+23
No summaries are to be submitted for lectures 1 and 24
Lectures (initiallyÌýonline) and lecture notes:Ìý
Announcement (25 Jan.): Preliminary indications from Perimeter Institute are that we will be able toÌýmove 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),ÌýLecture
1:
Notes,ÌýÌýÌý
Historical
introduction.
The
role
of
QFT
in
the
standard
models
of
particle
physics
and
cosmology.
Jan
10Ìý(Mon),
Lecture
2:
Notes,Ìý
Quantum
fluctuations.
Klein
Gordon
equation.
Mode
decomposition.
Second
quantization.
Ìý
Jan
12Ìý(Wed),
Lecture
3:ÌýNotes,Ìý
Mode
decomposition.
Infrared
regularization.
Mode
oscillators.
Probability
distribution
for
fields.
Jan
17Ìý(Mon),
Lecture
4:ÌýNotes,Ìý
Field
eigenstates.
Wave
functionals.
Schroedinger
equation
of
the
2nd
quantized
Klein
Gordon
field.
Jan
19Ìý(Wed),
Lecture
5:ÌýNotes,Ìý
Particles
as
excitations
of
mode
oscillators.
External
versus
parametric
particle
creation.
Jan
24Ìý(Mon),
Lecture
6:ÌýNotes,Ìý
In
and
out
operators.
Fock
bases.
Resonance.
Driving
creates
coherent
states.
Classicality.
Jan
26Ìý(Wed),
Lecture
7:ÌýNotes,
Bogolubov
transformation.
Quantum
field
driven
by
a
classical
current,
then
by
a
quantum
current.
Jan
31
(Mon),
Lecture
8:ÌýNotes,
Light-matter
interaction.ÌýAbsorption
and
emission
by
Unruh
DeWitt
detectors.
UnruhÌýeffect.
Feb
2Ìý(Wed),
Lecture
9:ÌýNotes,
Functional
differentiation.ÌýLegendre
transform
to
Lagrangians.
Quantization
as
a
Fourier
transform.
Feb
7
(Mon),
Lecture
10:ÌýNotes,
Functional
derivative
of
differentiated
functions.
Action
functional.
Covariance.
Curvature.
Feb
9
(Wed),
Lecture
11:ÌýNotes,
ÌýÌý
Einstein
action
and
equation.
D'Alembert
operator.ÌýGenerally
covariant
Klein
Gordon
Hamiltonian.
Feb
14
(Mon),
Lecture
12:ÌýNotes,
Ìý
Mode
functions.
Darboux
theorem.
Solving
free
QFT
on
any
globally
hyperbolic
curved
spacetime.
Feb
16
(Wed),
Lecture
13:ÌýNotes,
Ìý
Conservation
and
covariance
of
the
CCRs.
Stone
von
Neumann
theorem.
General
Bogolubov
maps.
Reading
Week.Ìý
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.Ìý
- Arrive early enough because you will probably need to do a rapid test at the reception.ÌýÌý
Feb
28
(Mon),
Lecture
14:ÌýNotes,
,
1pm
Bob
room
(Special
time!)Ìý
K.G.
field
in
FRW
Spacetimes.
Conformal
time.
Chi
field.
Hamiltonians.
Energy
momentum
tensor.
Mar
2
(Wed),
Lecture
15:ÌýNotes,
,
4pm
Bob
roomÌý
Quantization
of
K.G.
field
in
FRW
spacetimes.
Bogolubov
transformations.
Pair
creation
of
particles.
Mar
7
(Mon),
Lecture
16:ÌýNotes,
,Ìý4pm
Bob
room
Particle
production
through
expansion.
Lowest
energy
state
is
not
the
vacuum.
Adiabatic
vacuum.
Mar
9
(Wed),
Lecture
17:ÌýNotes,
,Ìý4pm
Bob
room
Quantum
field
fluctuation
spectra
in
terms
of
box
variances
and
correlators.
Ultraviolet
divergence.
Mar
14Ìý(Mon),
Lecture
18:ÌýNotes,
,Ìý4pm
Bob
roomÌý
Amplifications
of
quantum
field
fluctuations
vs.
particle
creation.
De
Sitter
horizon
and
inflation.Ìý
Mar
16
(Wed),
Lecture
19:ÌýNotes,
,Ìý4pm
Bob
roomÌý
Calculation
of
the
field
fluctuation
spectrum
of
a
scalar
field
during
a
de
Sitter
inflationary
period.ÌýÌý
Mar
21
(Mon),
Lecture
20:ÌýNotes,
,Ìý4pm
Bob
roomÌý
Origin
of
inflation,
slow
roll
and
re-heating.
Quantum
fluctuations
of
the
inflaton
and
of
the
metric.Ìý
Mar
23
(Wed),
Lecture
21:ÌýNotes,
,Ìý4pm
Bob
roomÌý
Decomposition
of
metric
fluctuations.
Dynamics
of
Mukhanov
variable
and
tensor
polarizations.Ìý
***
Special
time:
Mar
29Ìý(Tue)
***,
Lecture
22:ÌýNotes,
,
1:30pm
Bob
roomÌý
Standard
model
of
early
universe
cosmology.
Example
power
law
inflation.
Experimental
status.Ìý
Mar
30
(Wed),
Lecture
23:ÌýNotes,
,Ìý4pm
Bob
roomÌý
Unruh
effect
for
uniform
acceleration
from
Bogolubov
transformations.
Stress
energy.
Apr
4
(Mon),
Lecture
24:ÌýNotes,
,Ìý4pm
Bob
roomÌý
Schwarzschild
spacetime
and
its
coordinates.
Boulware
and
Kruskal
vacua.
Hawking
radiation.Ìý
Remark
to
those
who
are
not
enrolled:
I
invite
anybodyÌýto
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!
Ìý
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 makingÌýuse of them.
See
an
early
version
of
it:
Introduction
to
Quantum
Effects
in
Gravity
(PDF).
Ìý
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,
- L.H. Ford, Quantum Field Theory in Curved Spacetime,
.