In statistical mechanics one dispenses with the idea of set
of numbers that describe a system perfectly and instead
one must work with the beautiful and mysterious entity
know as the 'partition function'. The partition function is
just the sum, over all the possible states of the system,
of the probabilities of existing in that particular state. If
the state of a system is a continuous variable, you do an
integral instead of a sum, and if there is more than one degree of
freedom you do more than one integral.
The partition function may seem somewhat arbitrary but its
power lies in the fact that the value will depend on the
external thermodynamic properties (hence 'function' in the
name). For example in a gas it will depend on the volume,
temperature, and pressure. The logarithm of the partition
function is the free energy which is the energy of the
system after the thermal energy has been subtracted. This
quantity represents the available energy to do work, and
the thermal energy is removed because you're not getting
that energy back into anything useful. As an example of this,
the pressure of a system is actually just
the change in free energy with the system's volume. If you attached the
gas to a spring, the system would do work on the spring by
A duality is just a pair of statistical ensembles that have
the same partition function. They are said to be dual
because the two systems behave the same way under external
This is where things get weird. Replace statistical mechanics,
which has a large but finite number of degrees of freedom, with
statistical field theory, where the field value at each point
in space-time is a degree of freedom. Say that the field theory
interacts with electromagnetism and set up an
experiment where we are modify the electric and magnetic fields,
observing the thermodynamic response of the field theory.
This is like changing the volume to measure pressure except
that was just one variable whereas now we have six variables
(three for E-field and three for B-field) at every point in
space-time. The partition function is now calculated by doing
a path integral (a fancy way of saying an integral at every point in
space-time). It is now a 'functional' which means that it
takes functions (in this case the field value as a function of
space-time) as input rather than any finite number of parameters.
In statistical field theory a duality is a pair of different field
theories that have the same partition-function functional with
respect to an external, probing, field.
Particle-vortex duality was something discovered in the 70's by
people working on condensed matter systems like superconductors. On
the one hand you have a regular old scalar bosonic field theory
(scalar meaning that that the field has a number at each point in
space-time, and bosonic means that the number is just a normal
number). This theory is referred to as 'particle' in the duality
because the quantum excitations of a field theory are
exactly what we interpret as particles. On the other hand we have
and Abelian-Higgs field theory which, for the experts, is a U(1)
gauge field that interacts strongly with a scalar Higgs field.
I won't go into how this works but its interesting to note that
a classical field of this type has vortex solutions which are
effectively a result of the field getting itself tangled up
These knots behave like particles in the sense that
they are local excitations of the field with integer number, but
their existence is purely a consequence of classical physics and
topology rather than quantum effects. The marvelous thing is that
these two theories, when coupled to an appropriate external field,
can be shown to be dual to one another in the sense of partition
functions. There is no observable difference, using the external
field to probe, between the quantum-like particle excitations
and vortex-like particle excitations, and that is the sort of thing
that keeps me up at night.
My final report wasn't actually on the particle-vortex duality,
I decided to focus on another, more recent duality which is
called 3D bosonizaiton. If you remember from earlier that boson
field means that the field values are 'just normal numbers'. The
reason I make that distinction is because there are fermionic
fields which use Grassman numbers instead. The base unit of a Grassman number, usually
represented by theta, is like an imaginary number in that it can
be multiplied and divided by real numbers except that instead of
i x i = -1 we have theta x theta = 0. In general
fermionic particles have spin which is quantized magnetic flux.
It turns out that for a certian type of theory (Chern-Simons theory)
which is purely classical, magnetic flux is quantized in a way that
is very similar to the field-tangling of the vortexes. 3D
bosonization duality is between fermionic particles on the one
hand and normal, scalar, particles with magnetic flux attached
through its interaction with a Chern-Simons field on the other.
Most big projects have ups and downs, and some are littered
with summits and abyssal canyons. This thesis was one of the
latter. The idea was relatively simple: create a machine learning
algorithm capable of searching images for line-like features.
Since cosmic strings are as-of-yet unobserved,
the goal was to establish a method that was better than the existing
algorithms at discovering these strings.
I was working with my friend Chris Waddell, who was in the same
program with me and with whom I shared most of my classes. As it
turned out, cosmology is hard and machine learning is hard, and
both are notorious for their concepts being not-that-well-defined.
The project was separated into two halves which roughly coincided
with the two semesters over which we worked. The first was
understanding and modelling the actual physics behind the
hypothetical objects known as cosmic strings, and the second
was building a machine learning algorithm that would be able to
look at a collection of images and tell us if there are string-like
objects in the images.
Cosmic strings are a type of topological defect. These
are common in materials that undergo phase transitions, for example in
ferromagnets where the topological defects are domain walls
that mark the boundary between collections of atoms whose magnetic
moments all point in the same direction. Cosmic strings are similar
except that the defect in question is a line rather than a wall, and
the phase transition is the big bang instead of cooling magma.
Ok, lets back up. The Big Bang is not really something that happened.
Shocking, I know. It's actually just the point in time beyond which we stop
trusting our friend The Standard Model. It's hard to make predictions beyond
that point but there are a lot of compelling arguments that support
the idea of a period of extremely rapid inflation before the big
bang. Every physicist and their grandmother have a favourite
notion of what might have caused inflation, but a common idea is that
it was caused by some sort of field with very high energy density. In
this model, inflation occurs until the field undergoes a phase transition
at the Big Bang, and it is this phase transition that predicts the
existence of cosmic strings: regions of extremely concentrated energy
where the inflationary field winds around itself.
Pretty cool right? You can see now why people are interested in discovering
them, as I'm sure it would constitute a Nobel Prize for whoever does. But where
to look? The field that drives inflation had better be pretty dense with energy
for it to cause the universe to expand so rapidly, and this means that the
cosmic strings have a massive gravitational field. This is good. Gravity bends
light so you just have to search for parts of the sky that look like a fractured
mirror and there you go, Nobel Prize. In reality, it turns out that these
strings are separated by so much distance that chances are they've long since
left our observable universe. All is not lost, however, because you just have to look
back in time, back, back, and further back to the first light that could freely
propagate in the universe.
This light is what makes up the Cosmic Microwave Background (CMB), coming from the epoch
of recombination, about four hundred thousand years after the big bang, when matter
turned from an opaque soup of electrons and protons to a gas of neutral Hydrogen.
There are several projects that measure the CMB, shown above is the WMAP image of the full
sky, but since then the Plank telescope has retrieved higher resolution images. There
is also the South Pole Telescope (SPT) which covers a smaller
portion of the sky but has a much higher resolution. This is best
for looking at cosmic strings because they are such thin strictures
in the image.
Machine learning algorithms need data so we simulated a large number
of images of the CMB that were faithful to the South Pole Telescope's
specifications. We then superposed onto the image artifacts which
represented the cosmic strings. The algorithm was rewarded for outputting a signal
in places where a string exits and penalized for a signal in regions
where there were no strings at all. The diagram below shows the progression
of a preprocessed image as it passes through the machine learning algorithm. The resolution
of the image decreases in each step because we don't care so much where the
string is as we do that there is a string at all.
Part of the difficulty with looking for cosmic strings is that cosmologists
are not certain about exactly how dense the strings are. This is because
inflation is not that well understood and predictions can range across many orders
of magnitude. It is easier to see the strings if they are heavier, and the
gravity distorts more light. So when I said before that cosmic strings havent been
discovered, it's not because we haven't looked in the right part of the sky.
In reality physicists are fairly certain that strings above a certain mass don't exist, and the goal is
to push the boundary of that mass limit. Our supervisor had recently come back from
sabatical in Switzerland where he worked on an algorithm that could hypothetically
establish a new bound, and at the end of a year we were able to cut that almost in half.
QFT in de Sitter Space-Time
I often aim high with the projects that I tackle. There is little
fun in knowing that you are capable of doing something, and then
proceeding to do it. I like a little danger, to promise results I
don't believe I am capable of delivering and then finding a way to
make it happen. Sometimes this gets me into some hot water, but
nothing that a few all-nighters can't fix. This project, however,
is one that I'm still surprised I was able to pull off.
De Sitter space-time is an idealized version of the universe where there is
nothing but dark energy, allowing it to expand exponentially. It is often considered
in the context of inflation where the exponential expansion of the universe
vastly dwarfs any effects from matter or dark matter which produce polynomial-time
expansion. To a mathematician de Sitter spacetime is very interesting
because it is maximally symmetric, meaning that it has the same symmetries
(e.g. rotation and translation) as normal Minkowski space. There is a theorem by one of my
favourite mathematicians, Emmy Noether, which states that every symmetry leads to
a conserved quantity. For example, momentum is conserved because the universe is
symmetric under translational symmetry, energy is conserved by time-translation symmetry,
and angular momentum is conserved due to rotational symmetry. In de Sitter space,
the symmetries are different and lead to a different set of conservation laws.
In quantum mechanics the conserved quantities play a special role as generators
of the symmetry, which means that they act on states by applying the symmetry
transformation. For example the ubiquitous Hamiltonian operator, which in flat space
tells you how your quantum system evolves in time, is actually just the
energy operator which is conserved by time-translation symmetry. In de Sitter
space-time there is no time-translation symmetry because the universe is expanding.
Instead there is what is called dilation symmetry which is a combination of
advancing in time and scaling the space. It is a symmetry by virtue of the
exponential nature of the expansion. It is still possible to construct the
Hamiltonian operator in the de Sitter case, which tells you how the fields evolve in
time, but it is not particularly useful because what you find is that
all sorts of particles are being created in pairs, drowning out any
measurement that you might be trying to make.
Instead you work with the dilation operator, which is associated with the Dilation
symmetry that I described. It describes a form of time evolution, like the Hamiltonian, with
the added benefit that it is conserved, allowing you to carry forwards a lot of
language from flat quantum field theory (QFT) like the notion of particle states. For most of the project
I was tasked with figuring out exactly what this dilation operator looked like. In flat space
the basic modes of particles are sinusoidal functions. Calculating things
like the Hamiltonian operator are relatively straight-forward because you can
use Fourier analysis and lots of ugly stuff cancels out. In fact Fourier analysis is
so fundamental to QFT that it becomes second nature to many theoretical physicists.
In de Sitter space-time there is no such structure. The basic particle modes are these
strange Henkel functions that diverge in places, and have very few useful identities.
So when you start off trying to calculate something like the Dilation operator, it
quickly devolves into pages and pages of algebra.
By the end of the project I was able to come up with an expression
for the Dilation operator, shown above. I had spent the winter break learning
general relativity, the first half of the semester learning quantum field thoery,
and the second half of the semester working on that equation, often staying up until four in the morning
doing calculations. When your expressions take a whole page to write out, even a single
line of algebra can take hours to ensure that everything is correct. And in a few
instances, when I'd end up with a result that didn't pass the various consistency
checks, it could take days of going through all the steps to find a little sign error,
or a transcription mistake buried somewhere pages back. In those cases all those pages would
have to be redone.
My supervisor, I think, was surprised that I was able to produce anything at all,
but in a lot of ways the result was unsatisfying. For one thing the dilation operator
is supposed to be conserved under the dilation symmetry, but you'll notice
a lot of places where the greek letter eta appears (eta parametrises dilations and is
some function of time), which seems to indicate that the result will be time-dependent.
Another thing is that there are terms in that expression that don't conserve particle
number, and so you have the same problem as I described before when I was talking about
finding the Hamiltonian in de Sitter space-time. By this point I had developed an obsession
with this project, and it wasn't hard to convince my supervisor to keep working on the
project through the summer. So while I was working on the Super Kamiokande project at
the University of Toronto, I spent nights and weekends doing battle with these unwieldly
equations. It was in this time that I had several breakthroughs, moments where things would
fall into place in such a way that elicited profound euphoria. And finally, when all the
dust settled, the equation above settled in this simple and beautiful form
Both problems that I mentioned had been solved, all of the eta-dependence disappeared through a
host of cancellations and so did all the terms that violated particle number conservation.
There was only one problem left which I worked out before I left for CERN. Particles
with zero momentum (colloquially called zero modes) were not included in the above
calculation and had to be added in by hand. This was only necessary because of the
strange properties of Henkel functions, and it spelled disaster because they contributed an infinite amount
to the dilation operator from integrating over all of space. It is common in
physics to pretend that the universe has finite volume to fix this problem, but in that case,
although the answer would be finite, the operator would no longer be conserved. We say
that evergy is leaking through the boundary of the universe, and the problem is not
solved by moving the boundary further away. These were the things that I had to interpret
in the context of quantum field theory in order to have a complete and satisfactory
conclusion to the epic adventure.