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PROFESSOR: As you know,
Professor Guth is away.
I'm substituting for
today, he didn't leave me
with a particularly
coherent game plan,
so I'm going to begin with
where he thinks we should start.
Please jump in if I am
just repeating something
that he has already
described to you guys,
or if there's anything you
like me go over a little bit
more detail, I will
do my best here.
So, I'm working off of
a fairly rough plan.
But let me just
quickly describe what--
based on what Alan has explained
to me --what we're planning
to talk about today,
and if there's
any adjustments
you think I should
be making that would be great.
So, the game plan for today.
What I want to do
very quickly is
hit on a couple of
the key points which
I believe you talked
about last week,
which is a quick review
of the essential features
of symmetries of
the gauge fields
the make up the standard model.
Now, I believe you
guys did in fact talk
about this last week,
at least briefly.
And you talked about how
you can take these things
and embed them in a larger
gauge group, the group SU(5).
I'm not going to talk
about that too much,
but I want to just quickly
hit on a few elements related
to this before we get into that.
From this we'll then talk about
the Higgs mechanism-- really
I'm going to talk
about the Higgs field,
I'm not going to talk
about the Higgs mechanism
quite so much as motivate
why it is necessary--
and then talk about how
the Higgs field behaves
and why it's important for
the next problem, which
is what is called the
cosmological monopole problem.
To be more specific
magnetic monopole problem.
I confess I feel a little
bit awkward talking
about this problem
on behalf of Alan.
This would be kind of
like if you were planning
on studying Hamlet and there
was this guy W. Shakespeare who
was listed as the
instructor and you walk in
and discover there's this
guy Warren Shackspeare, who's
actually going to be teaching
or something like that.
I kind of feel like Warren here.
This stuff really is Allen's
thing, so it's sort of,
I'm probably going to leave this
at the denouement of all this
when you actually get
into inflation to him.
I may have a little bit of time
at the end to just motivate
it a little bit, but the grand
summary will come from him.
OK, so, as discussed by
Alan the standard model
describes all the
fundamental interactions
between particles
via gauge theories.
OK, and these gauge theories
all have a combined symmetry
group that is traditionally
written in a somewhat
awkward form, SU(3)
cross SU(2) cross U(1).
U(1) could be an
SU(1) for reasons
which I'll elucidate
a little bit more
clearly in just a moment.
There's really no point in
putting the S on that one.
So each of these
things essentially
labels the particular
symmetry group.
So, the "S" an
element of SU(n) is
a matrix that is n x n, that
is unitary-- that's the U.
Unitary just means that the
inverse and the transpose
of the matrix at the same,
actually the Hermitian
conjugate because they
can be complex, in fact,
they generally are.
And it has determinant of 1.
That's what the special refers
to, special, the S in SU(n)
stands for special unitary n.
So, the S means
that the determinant
is one-- that's what's
special about it-- unitary
is this idea that the inverse
Hermitian conjugate are
the same, and then n
refers to all these things.
So, that tells us that the
gauge degrees of freedom
are related by a symmetry
that looks like a 3
by 3 matrix with
these properties,
as listed there for the
SU(3) piece of the symmetry.
SU(2) means it's
a 2 by 2 matrix.
And U(1) means it's a one by one
matrix, what's a 1 by 1 matrix?
It's a number, its
a complex number.
And that's why we
don't really need
to put an "s" in front of it.
If it's a complex
number its determinant
is 1 if it's just a complex
number whose modules is one.
That's why we don't bother
with the S on the U(1).
So, I think you've already
hit on some of this
but this is sort of useful
to review because it's
going to set up why we
need to introduce a Higgs
mechanism in a little bit.
Let me just quickly
hit on what the details
structure of this looks
like for you want to think
is the easiest one understand,
So, as I just said,
a one by one matrix is
just a complex number.
So that means that any
element of this group
is a complex number, which
we can write in the form z
equals ei theta, where
theta is a real number.
Now, the thing which
is I want to hit on
in this, the reason I want to
describe this a little bit is,
this may not smell
like the gauge symmetry
that you're used to if you
study classical E&M. Some of you
here are in 807
with me right now,
and we've gone over this
quite a bit recently.
How is this akin to the gauge
group that we are normally
used to when we
talk about the gauge
freedom of electricity
and magnetism?
Well, it turns out
there's actually
a very simple relationship
between one and the other,
rather between this view
of it and the way we learn
about it when we study
classical E&M. It's simply
that we use a somewhat different
language, because when we talk
about it in this group
theoretic picture we're doing it
in the way that is sort of
tuned to a quantum field theory.
So, the way we have learned
about electromagnetic gauge
symmetry in terms of the
fields sort of goes as follows.
We actually work with the
potentials, and so what we do
is we note that the potentials
Amu, which you can write
as a four vector, whose
time-like component is
the negative of the
scalar potential,
and whose spatial
components are just
the three components of
the vector potential.
So, this potential and
this potential-- --okay
this is possibly module of
factor of c somewhere in here
but I'm going to imagine
the speed of light
has been set equal to 1.
Both of those potentials
generate the same E&B fields.
OK, again you still
should be looking at this
and thinking to yourself
what the hell does
this have to do with the
U(1) as we presented it here.
I've given you a
bunch of operations
that involve some kind of
a scale or function of time
and space.
And I've added particular
components of this four vector
in this way, what does that
to do with this multiplication
by a complex number?
Well, where it comes
from is that when
we study E&M, not as a
classical field theory
but as a quantum
field theory, we
have a field that
describes the electron.
So, where it comes
from is that when
you examine the Dirac field,
which is the quantum field
theory that governs the
electron, when you change gauge
the electron field acquires
a local phase change.
So in particular,
what we find is
that if we have a field
5x, which those of you who
have taken a little bit
of quantum field theory
should know this is
actually a spinner field,
but for now, just think of it
as some kind of a field that
under the field
equations of quantum
electrodynamics-- the Dirac
equation or high order
ones that have been developed
by Feynman, Schwinger,
and others-- under
a change of gauge
this goes over to
si prime of x, which
equals e to the--
terrible notation
I realized-- 1e is obviously
the root of natural logs,
e sub 0 is the fundamental
electric charge.
OK, can everyone read that?
I didn't block it too badly here
I'm not used to this classroom.
So, here's the thing to note,
is that this field lambda, which
we learned about
in classical E&M
directly connects to the phase
function of the Dirac field
in quantum electrodynamics.
So, our gauge symmetry
is simply expressed
in the quantum version
of electrodynamics
by a function of the form e
to the i real number, where
that real number is the
fundamental electric charge
times the classical
gauge generator.
So, this is what is meant when
people say that electrodynamics
is a U(1) gauge theory.
Now, I'm not going to go
into this level of detail
for the other two
gauge symmetry that
are built into the
standard model.
But, what I want
you to understand
is that the root idea
is very, very similar.
It's just now, instead of my
gauge functions looking like e
to the i, some kind of a local
gauge phase of x multiplying
my functions, my
quantities which
generate the gauge
transformation are going
to become complex
value matrices.
So that makes them a
lot more complicated,
and it's responsible
for the fact
that the weak and the
strong interactions
are non-abelian paid which
order you perform the gauge
transformation in matters.
Question.
AUDIENCE: What's the
physical significance
of them being non-abelian?
PROFESSOR: Yes.
So, what is a
physical significance
of them being non-abelian?
I'm trying think of a really
simple way to put this,
it's-- Alan would have an
answer to this right off the top
of his head, so I
apologize for this--
this isn't the kind of thing
that I work on every day so I
don't have an answer right
at the very top of my head,
unfortunately.
Let me get back to
you on that one, OK,
that's something I can't
give you a quick answer to.
It's an excellent question and
it's an important question.
Any other questions?
OK, so, here's a basic
picture that we have.
So, we find is that the
strong interactions have
a similar structure where
my need to e to the i factor
goes over to a 3 by 3 matrix,
and the weak interactions
in a similar structure with my
e to the i factor going over
to a 2 by 2 complex matrix.
OK, what does this have
to do with cosmology?
In fact, as an enormous
amount to do with cosmology,
as we'll see over the course
of the rest of course.
Part of the thing which is
interesting about all this
is that we have strong
experimental reasons,
and theoretical
reasons to believe,
that the different symmetries
that these interactions
participate in, the
different symmetries that we
see them having, that isn't the
way things have always been.
So, in particular when the
universe was a lot hotter
and denser these different
symmetries actually
all began to look the same.
In particular the one which
is particularly important,
and you guys have surely heard
of this, is that the SU(2)--
if we just focus on electric
and the weak piece of this--
SU(2) cross U(1).
So, this is associated
with the gauge boson that
carry the weak force, OK, the
z boson, the w plus, and the w
minus.
And your U(1) ends up being
associated with the photon.
In many ways, when
you actually look
at the equations that
govern these things,
they seem very,
very similar to one
another except that
the-- here's partly
an answer to your
question I just
realized-- the gauge
generators of these things
have a mass
associated with them.
That mass ends up
being connected
to the non-abelian
nature of these things.
That's not the whole answer,
but it has a connection to that.
That's one thing
which I do remember,
like I said I feel
this is really
Alan's perfect framework
here and I'm just
a posture in bad shoes.
So if we look at this
thing, what we see
is that these symmetry groups,
what's particularly interesting
is that U(1) can be regarded
as a piece of SU(2).
And we would expect
that in a perfect world
they would actually
be SU(2) governing
both the electric and
the weak interactions.
Whereby perfect
I mean everything
is a nice balmy 10 to the 16th
GeV throughout all of space
time, and all the different
vector bosons happily
exchange with one another,
not caring with who is who.
It's actually not
very perfect if you
want to teach a physics class
and have a nice conversation,
but if you are interested in
perfect symmetry among gauge
interactions it's
very, very nice.
So, the fact that
these are separate
is now-- I was about to use the
word believed but it's stronger
and that, we now know
this for sure thanks
to all the exciting work
that happened at the LHC
over the past year or two-- the
fact that these symmetries are
separate is due to what is
called spontaneous symmetry
breaking.
So, let's talk very
briefly about what
goes into this spontaneous
symmetry breaking.
So SU(2) turns
out to actually be
isomorphic to the group
of rotations on a sphere.
So, when you think
about something
that has perfect
SU(2) symmetry it's
as though you have perfect
symmetry when you move around
through a whole host
of different angles.
OK, so you move through all
of your different angles
and everyone looks exactly
identical to all the others.
If you break that
symmetry it may
mean you're picking out
one angle as being special,
and then you only retain
a symmetry with respect
to the other angle.
And essentially, that is what
happens when SU(2) breaks off
in a U(1) piece of it.
Something has occurred
that picked out
one of these directions.
And by the way, you have to
think very abstractly here.
This is not necessarily a
direction in physical space
we're talking
about here but it's
a direction in the
space of gauge fields.
So, if we imagine that all of
these, my gauge fields in some
sense the different
components of them defined
in some abstract
space direction,
initially these
things are completely
symmetric with
respect to rotations
in some kind of an abstract
notion of a sphere.
And then something
happens to freeze
one of the directions
and only symmetries
with respect to one of the
angles remains the same.
Let's just write that
out, when SU(2)'s symmetry
is broken so one of the
directions in the space
of gauge fields is
picked out as special.
That direction
then ends up being
associated with
your U(1) symmetry.
So, what is the mechanism that
actually breaks the symmetry
and causes this to happen.
Well, this is what the
Higgs field is all about.
The idea is there is some field
that fills all of space time.
It has the property that
at very high energies
it is extremely symmetric,
with to respect all these gauge
fields, all directions and
sort of gauge field space
look exactly the same.
And then as things cool, as
the energy density goes down
by the temperature of the
expanding universe, cooling
everything off, the Higgs field
moves to a particular place
that picks out some direction
in the space of gauge fields
as being special.
So let's make this a
little bit more concrete.
OK.
You guys have probably heard
quite a lot about the Higgs
field over the
past couple years,
months-- what actually is it?
Well, the field itself is
described by a complex doublet.
So, if you actually see someone
write down a Higgs field what
they will actually
write down is h,
being a two components
spinner, whose components are
h1 of x, h2 of
x-- where x really
stands for space time
coordinates, so that's time
and all of your
spatial coordinates--
and both h1 and h2
are complex fields.
The thing which is particularly
key to understanding
the importance of this
thing is that h transforms,
under gauge transformations,
with elements of SU(2).
So, if you want to
change gauge the way
you're going to do it is you're
going to have some new Higgs
field.
So remember, if U(2)
is an element of SU(2)
we call it the
two by two matrix.
This is what they look like
in a new gauge OK-- pardon me
a second I don't see
a clock in this room,
I just want to make sure I
know the time, thank you.
OK, so, what are we
going to do with this?
Well, there's a couple
features which it must have,
so the Higgs field
fills all of space time
and it has an energy
density associated
with it, which we will call
just the potential energy.
It's really an energy
density, but, whatever.
The energy density that is
associated with this thing
must be gauge invariant.
OK, even when you're
working with strong fields
and weak fields, the lesson
of gauge invariance from E&M
still holds.
OK, one of the key points was
that the gauge fields affect
potentials, they allow us
to manipulate our equations
to put things into a form where
the calculation may be easier.
But at the end of the day, there
are certain things it actually
exert forces that
cause things to happen,
those must be invariant to
the gauge transformation.
Energy density is
of those things.
If you were to get
into your spaceship
and go back to
the early universe
and actually take a little
scoop of early universe out
and measure the energy density,
A, that would be cool, but B
it would be something
that couldn't actually
depend on what
gauge you were using
to make your measurements.
That is something that is
a complete artifice of how
you want to set
up the convenience
of your calculation.
So, in order for the energy
density to be gauge invariant
we have to find a gauge
invariant quantity that
is constructed from this, which
is the only thing the energy
density can depend on.
This means, let's call
our energy density V,
it's the potential
energy density.
So, it can only depend on
the following combination
of the fundamental fields Pretty
much just what you'd expect.
This is sort of the
equivalent to saying
that if you're working
in spherical symmetry
the electrostatic
potential can only
depend on the distance
from a point charge.
This is a very similar
kind of construct
here, where I'm taking
the only quantity that
follows in a fully
symmetric way, of calling
the fact that this is a
special unitary matrix that I
can construct from these things.
So then, where all
the magic comes
in is in how the Higgs field
potential energy density
varies as a function of
this h, this magnitude of h.
So, as I plot v as
a function of h,
in order to get your
spontaneous symmetry breaking
to happen what you want
is for the minimum of V,
the minimum potential
energy, to occur somewhere
out at a non-zero value
of the Higgs field H.
Now, why is that so special?
The thing that is so
special about that
is that when I constructed
this magnitude of h,
I actually lost a
lot of information
about the Higgs field.
OK, let's just say for
the sake of argument
that this minimum
occurs at a place where
the Higgs field in some system
of units has a value of 1.
So, all I need to do
is as my universe cools
what I'm going to
want is energetically,
my potential is going to want
to go down to its minimum.
So, that just means that as
the universe is cooling, maybe
at very, very early times
when everything is extremely
hot and dense, I'm up here
where the potential energy is
very high.
As the universe expands,
as everything cools,
it moves over to
here, it just moves
to someplace where the Higgs
field takes on a value of 1.
And that's exactly correct,
that is what ends up happening.
But remember, the minimum
occurs at some value
in which the magnitude
of this field
does not equal zero, but given
that value-- where again let's
just say for this for
sake of specificity
that we set it equal to
the magnitude of this thing
equal to 1 in some
units-- there's actually
an infinite number of
configurations that correspond
to that because this
is a complex number,
this is a complex number.
I could put it all
into little h1,
and I could set into the value
where that thing is completely
real, or I could put it all
into little h2 being completely
imaginary or all on to h1 being
all imaginary, halfway into h1,
halfway into h2.
There are literally an
infinite number of combinations
that I can choose
which are consistent
with this value of the
magnitude of H. So, yeah--
AUDIENCE: So, I
don't know if I'm
putting too much physical
significance on the gauge,
but with the other cases
of spontaneous symmetry,
briefly, that we discussed
you can always measure.
OK, I've broken my symmetry,
and now it's lined up this way,
or there's something measurable.
Now, the field
has to be physical
because the fact that
you have gauge symmetry
gives you some concerned
quantity, right?
But, how can I measure what
direction in gauge space
that I picked out?
PROFESSOR: So, that
is, let me talk
about this just a
little bit more.
I think answering your
question completely
is not really
possible, but there
is a residue of that is
in fact very interesting,
and let me just lay out a couple
more facts about what actually
happens with this
gauge symmetry,
and it's not going to
answer your question
but it's going to give you
something to think about.
OK, so that's an excellent
and very deep question,
and there are really
interesting consequences.
And this is a case
where my failure
to answer the previous
one is because there's
details I can't
remember, in this case,
I think it's because
there's details we actually
don't understand fully.
Research into the mechanism of
electroweak symmetry breaking,
which is what this
is all about, is
one of the hot topics in
particle physics right now.
AUDIENCE: I was just
wondering if gravity
has any gauge symmetry
associated with it.
PROFESSOR: It does, but it fits
in a very, very different way,
and with the exception of the
fairly speculative framework
of string theory-- which I
think is very, very promising,
but it's just
sufficiently removed
from experimental
verification that I'm
going to have to label it
speculative-- it doesn't quite
tie in in the same way.
And that's the best
I can say right now.
The gauge symmetries
of general relativity
are, at the classical
level, they correspond
to coordinate transformations,
at a quantum level,
there's not such a
simple way to put it.
All right, where
was I, OK, sorry
I didn't get to your question.
So, the point we made here is
that we have spontaneously,
when we actually choose which
one of these infinite number
of values we're going
to have, we just
randomly break the symmetry.
OK, and you guys
apparently have already
talked a little bit about
spontaneous symmetry breaking.
The analogy that
people often make
is to the freezing
of water, OK, prior
to the water entering its
solid phase its completely
rotationally symmetric,
then at a certain point
crystalline planes
start to form,
the water forms,
all the molecules
get set into a
particular orientation,
you lose that
rotational symmetry.
In this case, we started
out with a theory,
with a set of interactions
that were completely symmetric
in sort of gauge field space.
And now by settling
down and picking
a particular special
value of h1 and h2
we have at least nailed
down one direction.
It's like we've defined
a crystalline plane,
and so now things, suddenly,
aren't as symmetric.
And we start to pick
out preferred directions
in our gauge fields.
What we can do
with this is really
a topic for a
whole other course,
and that course is called
quantum field theory,
but I will sketch a
couple of the consequences
and this gets directly to
the answer your questions.
So, one of the
consequences of this
is that once we have picked
out a particular direction,
electrons and neutrinos
are different.
When the Higgs field
is equal to zero
there is no difference between
an electron and a neutrino.
They obey exactly
the same equation,
there's literally no
difference between them.
Once we have actually
settled on an h1 and an h2
some combination of the
fundamental underlying fields
comes together, acquires a mass,
acquires an electric charge,
and we say A-HA thou
beist an electron.
It wasn't like that in the
original unbroken symmetry.
AUDIENCE: Also, [INAUDIBLE]?
PROFESSOR: Presumably,
but I'm going
to stick with just
these for now,
but I've I'm pretty sure
that's the case, yeah.
That gets into even more
complications of course
because the additional
generations are actually
consequence presumably of some
broken higher level symmetry,
which is even poorly,
more poorly understood.
But you raise a good point.
So, that's one partial
answer your question.
How one can actually
walk that backwards
to understand this thing
about the initial state?
That's hard to say.
I actually think
this particular one
is one of the profound
and interesting aspects
of this, in part because we now
know the neutrino has a mass.
We have no idea what that is,
and in fact we only really
have bounds on the mass, such
that we know it is non-zero,
and we have upper
limits that are
set by very indirect
measurements.
But the actual
values of the mass
are very, very
poorly constrained.
Within the standard
model you just
take the electroweak
interaction,
introduce a Higgs coupling
and allow the symmetry
to be spontaneously broken,
the neutrino mass is zero.
Full stop zero.
So something's not
right, we're actually
missing something here.
People have kind of jury
rigged the standard model
to put in the masses by
hand, and it works OK,
but it's not
completely satisfying.
And a lot of experiments
going on right now
to explore the neutrino
sector are hopefully
going to open us up to a
deeper understanding of this
and may say a lot about all this
physics, which is at present,
pretty poorly understood.
The consequence, which has
received the most popular
press, and what you
guys have certainly
seen about in newspapers,
given the results that came out
from the LHC over the past year
is that quarks and leptons have
mass, or put more
specifically, rest mass.
To understand what
this actually means
I think you really need to
ask yourself what is mass
meant to be.
Well, the idea is you calculate
the spectrum of oscillations
associated with the
fields of your theory,
and then if your theory
predicts a discrete spectrum
of oscillations, it doesn't
even have to be discrete
but predict some
spectrum of oscillations,
then for every oscillation
frequency omega
there's an associated
mass that is just
H bar omega over c squared.
If your omega has
some lower bound that
is greater than zero,
then your theory
has particles with
nonzero rest mass.
Without going into the
details-- and this again
is something which
those of you who
are going to go on to study
this in more detail in a higher
level course, which is
fairly standard stuff is done
in probably the first or maybe
late in the first or early
in the second semester of a
typical quantum field theory
course-- what you'll find is
that when the Higgs field is
zero then quarks
and leptons have,
the field that describes quarks
and leptons-- and yes including
mu and tau, so including
all the leptons, this one
I'm very confident
on-- the spectrum
goes all the way to zero
if the Higgs field is zero.
But when the Higgs field becomes
non-zero, roughly speaking,
it shifts the spectrum
over for these particles.
There's an interaction between
the things like the electron
field in the Higgs field or the
up quark field and the Higgs
field, which shifts the
spectrum over just enough
so that the frequency
is never allowed
to go below some minimum.
AUDIENCE: Going back
a bit, I'm confused
about how picking a specific
value to the Higgs field
is breaking SU(2)
symmetry and not U(1),
because it seems like we're
fixed on a circle, right?
PROFESSOR: That's right what
U(1) is a symmetry on a circle,
SU(2) is kind of like symmetry
on a sphere, essentially.
AUDIENCE: Right, so how are we
not picking a specific value
[INAUDIBLE] circle [INAUDIBLE]?
PROFESSOR: Well, what we're
doing is, think of it this way,
imagine SU(2) is a
symmetry on a sphere,
and then when we break
the SU(2) symmetry
it's like we're picking
some circle on that sphere.
So, we've broken one circle,
we've picked one circle,
but now we're allowed
to go anywhere
on that remaining circle,
which is a U(1) symmetry.
Does that help?
Yeah, OK good.
And it comes down to the
fact if you sort of count up
your degrees of freedom,
it has to do with the fact
you you've got four, you
have two complex numbers,
so there's four real parameters
associated with this thing,
and they are isomorphic to sort
of rotations in a three space
and you're adding
one constraint.
OK, so let me just finish
making this point here again.
So, when h does not
equal zero, spectrum
get shifted for the
quarks and leptons,
so everything picks up
a little bit of a mass.
And the final one,
final consequence
which we're going
to talk about today,
is that the universe is filled
with magnetic monopoles.
We all remember studying
Maxwell's equations
learning that del dot
b is equal to 4 pi
times the density
of magnetic charge--
this all makes
perfect sense, right?
Well, this is actually something
that when it first sort of came
out and people begin to
appreciate this thing with sort
of a "Um, well everything
else works so well,
maybe we're just not
looking hard enough. " So,
it was a bit of a surprise.
So, where do these magnetic
monopoles come from?
And essentially, the
magnetic monopoles
are going to turn out to be
a consequence of the fact
that when spontaneous
symmetry breaking happens
it doesn't happen
everywhere simultaneously.
So, think again about-- yeah?
AUDIENCE: Doesn't that
bring up possibility
that the symmetry could
break in different ways
in different places?
PROFESSOR: That is in
fact exactly what this
is going to be.
Magnetic monopoles are in
fact exactly a consequence
of this, yes.
Give me a few
moments to step ahead
to fill in a couple of the
gaps, but you're basically
already there.
So, think about crystalline
crystal formation again.
Imagine you have,
we could do ice
if you like or choose something
that's got a little bit more
of an interesting
crystalline structure.
Imagine you have a big bucket
full of molten quarts, OK.
So, if you have a
big thing of quartz
that you want to sort of freeze
into a single gigantic crystal,
what you typically do if
you'd like to do this is
you actually seed
it with a little bit
of a starter crystal.
So, you put a little bit
of crystal into this thing,
and what that does is it sort of
defines a preferred orientation
of the crystal axes,
so that as things
start to cool in the
vicinity of that they have
a preferred orientation
to grab on to.
And that seed then
gradually gets
bigger and bigger and bigger,
and all the little crystals
as they form near
it tend to latch
onto the preexisting
crystalline structure,
and that allows you
to grow actually
extremely large crystals.
I don't know if anyone here is
doing a year off with the LIGO
project but these guys have to
make these sort of 100 kilogram
mirrors of very pure either
Sapphire or silicon dioxide,
and when you make 100
kilograms of crystal
you need to build it
really, really carefully.
It's extremely important
for the optical purposes
that all the axes
associated with the crystal
will be pointing in
the right direction.
Otherwise you spend
$100,000 on this thing
and it ends up being the
world's prettiest paperweight.
So, similar things happen
when the Higgs field cools.
Let's imagine that
we've got our universe,
time going forward like this,
and at some point over here
the universe cools enough
that's the Higgs field condenses
into some particular direction.
And symmetry is spontaneously
broken right at this one point
over here.
So, I'm going to draw
my diagram over there
and put some words over here.
I shouldn't say Higgs
field cools enough,
the universe cools
enough so that the Higgs
field breaks the symmetry.
So, just to be concrete, let's
imagine that at 0.1 over here
it takes on a field of the
value one for h1 and I for h2.
So just for concreteness
imagine it looks something
like this at this point.
And so what happens is as
the university continues
to expand other areas
are going to cool off.
The bits that are
closest to it are
going to see that
there is already
a preferred orientation
defined by the Higgs field.
And so it's energetically
favorable for those regions
of the universe to fall
into the same alignment
and so there'll be a
region in space times
that grows here as the universe
cools, in which the Higgs
field all falls into
this configuration, which
I will call h1.
But suppose somewhere
over here at 0.2,
and the key thing is
that initially 0.2
is going to be so far away
from 0.1 that these points are
out of causal contact
with one another.
I can not send a message
from event one to event two.
The Higgs field
also reaches a point
that the universe cools enough
that at 0.2, just you know,
it's a system that's not
in thermal equilibrium.
So, some places are
going to be a little bit
hotter than others,
some are going
to be a little bit cooler.
And so, at these two
points it just so happened
that the Higgs field
got to the point
where it could spontaneously
break the symmetry.
So at 0.2 the Higgs field
also got to the point
where it could spontaneously
break its symmetry.
And the only thing
that's got to happen
is, remember the only
constraint we have is
that the magnitude
of the Higgs field
be equals to some value-- I
should normalize that to root 2
in units I want to
use but whatever.
Let's say on this one
my h1 is equal to y,
and h2 is equal to minus 1.
So, it's basically
the same thing
but all the fields
are multiplied by i.
It's the same
magnitude, so it's going
to have the same
potential energy.
So that's cool.
Clearly this is allowed,
and now all the regions
in the universe that
are close to the this
are going to sort of smell
this particular arrangement
of the Higgs field
and say OK, that's
preferred arrangement
I want to go into.
So, we have two separate
values of the Higgs field
that are happily swooping
out space time here.
This gets to the
excellent question
I was just asked a moment ago--
what happens when they collide?
As the universe expands
and gets cooler, all of it
is going to end
up getting swooped
into either the field that
was seeded at event one,
or the field that was
seeded in to event two,
but at a certain
point we're going
to get the bits where they're
smashing into one another.
So what happens when these
different domains come
into contact with one another?
The absolutely full
and probably correct
answer is we don't know.
The reason is that we don't
really, to be perfectly blunt,
fully understand every little
detail about the symmetry
breaking, or about the
structure of whatever
grand unified theory brings
all these things together
at the temperatures at
which this is happening.
Because this is happening
when the universe has
a temperature of like
10 to the 16th GeV.
And so it's way
beyond the domain
of where we can push things.
But we can, as physicists
are fond of doing,
we can paramaterize
our ignorance,
and we can ask
ourselves, well what
happens if these various
parameters that characterize
my grand unified theory take on
the following plausible kinds
of parameters.
And what we find is
that generically,
when you have two different
domains where the Higgs
field takes on different
values like this, when
these domains come
into contact you
get what are called
topological defects.
The topological defects come
in three different flavors.
To understand something
about those flavors
you have to know a
little bit about what
happens in general when
you have phase transitions,
and different regions
of your medium
go through a phase transition
with different values
of the parameters.
So, it's a general
case that whenever
you have some kind
of a phase transition
and you have domains
of different phase
that come into contact
with one another,
your field will attempt
to smoothly match itself
across the boundary.
But that can be very difficult.
So if you imagine these
particular two cases
that I have here,
that's essentially
saying that when
these two domains
coming to contact
with one another
there's going to be sort
of a transition zone
where the field is
attempting to rotate
from one value of the
Higgs to the other.
And it's going to pick some
value that is in some sense
intermediate to
those two things.
So that, let's say we
continue these up here,
so that the collision is
occurring right in this place
here, in this little locus
of events in space time.
I have Higgs field 2 over
here, Higgs field 1 over here,
and I've got some crazy
intermediate field that
goes between the
two of them, which
is trying to sort of force
itself to smoothly transition
from one to the other.
In so doing, I might end
up pushing my field away
from the minimum, in
which case there will then
be some energy
trapped in that layer.
And there's a reason we
do this level the class
in a bit of a hand
wavy way, I mean
it's very, very complicated
to get the details right.
But the key thing we see is
that in doing this match,
the field has to do some
pretty silly shenanigans order
to make everything
kind of match up
and we can be left with
odd observable consequences
from the energy associated
with the Higgs field getting
pinned down at
that boundary here.
Now, the details of the
forms of this boundary
vary a lot depending upon
to the specific assumptions
you make about your underlying
grand unified theory.
OK, so I should
back up for a bit.
I'm sort of assuming here when
I discuss all this that there
is some underlying SU(5)
theory which describes
the strong weak and
electromagnetic interactions
are very, very high temperatures
as one gigantic thing.
And we're getting
to the point now
where all the different
interactions are beginning
to just sort of
crystallize out of it.
There's a lot of different ways
you can pack your underlying,
fundamental, what we now think
of as our standard model,
into SU(5) grand
unified theories.
And so the ways in
which we can get
different topological
defects depend
upon how we choose to do that.
So defect flavor one is you get
something called a domain wall.
When we do this
the fields attempts
to make itself smoothly match
from one region of Higgs field,
say from Higgs 1 to Higgs 2.
It succeeds, but
you end up with kind
of a two dimensional structure--
a wall-- in which there's
some kind of anomalous field
that is just pinned down there.
And so we end up
with a big sheet.
So in a theory
like this, it would
predict that somewhere out
in the universe if there were
regions in which the
Higgs field had taken
on a different
value than the one
that we encounter
around us right now,
it could be somewhere out
gigaparsecs away, essentially
a giant sheet of some kind.
And there would be
weird, anomalous behavior
associated with it.
People have really
looked long and hard
to try to find things
like this and in fact it
would be expected to leave
interesting residuals
in the cause of
microwave background.
My understanding
of the literature
is that there are
actually now very
strong bounds on the
possibility of having
a grand unified theory
that leads to domain walls.
And so this kind of
a topological defect
is observationally disfavored.
So this, I should
mention, only occurs
in some grand unified theories.
Basically, As we move on to
the other flavors of defects
we end up just going down a step
in dimensionality associated
with the little kinks
that are left over when
the different domains come
into contact with one another.
Flavor two, we would get
what's called a cosmic string.
Some of you may
have heard of this.
This is essentially, at its
core, just a one dimensional,
it could be gigparsecs long,
but one dimensional, truly
one dimensional--
essentially just
a point in the other two
dimensions-- string of mismatch
Higgs field with some kind of
an energy density associated
with it when the different
domains get in contact.
AUDIENCE: Do we
have any estimate
of how close in actual space
these different regions would
have started?
PROFESSOR: We do and I'm
actually going to get to that.
So, let me give you
two answers to that.
One of them is you are going
to estimate that apparently
on PSET 10, according to the
notes that Alan left for me.
But I'm going to spell out
for you the arguments that
go into it in the last
10 minutes of a class.
But yeah, so let me just quickly
finish up this one because this
again-- so a cosmic string is
sort of like a one dimensional
analog of a domain wall.
And because it would be
this sort of long one
dimensional structure, that
has actually up a lot of energy
sort of pinned down to it
by the fact it has a Higgs
anomaly associated with it, it
would be strongly gravitating
and so it would leave really
interesting signatures.
It was thought for a while
that cosmic strings might
have been the sort of original
gravitational anomalies
that seeded some of
the structures we see
in the universe today.
Again, it's now pretty
highly disfavored.
If cosmic strings
exist, they don't
appear to contribute very
much to the budget of mass
in our universe.
I should also mention
that this is only
predicted by some grand
unifying theories.
If you guys are
curious about this
I suggest when Alice
back you ask him
what the difference
between these sums,
why some predict a
domain wall, some
predict the cosmic strings.
Flavor three is where
you end up with the Higgs
field essentially being
able to smoothly transition
without leaving any defect
anywhere except at a zero
dimensional point.
So you end up with just a
little knot in the Higgs field.
And for reasons that I
will outline very soon,
it turns out that this little
not must carry magnetic charge,
and so it must be a
magnetic monopole.
The domain walls and
the cosmic strings
are, as I've emphasized,
only predicted
by certain specific
grand unified theories.
Magnetic monopoles are actually
predicted by all of them.
Question.
AUDIENCE: What does it mean to
have a one dimensional domain
wall, because there's no
different region separated
by one [INAUDIBLE].
PROFESSOR: That's right.
So what ends up
happening, and this
is where I think
you're going to have
to ask Alan to sort of follow
up on this a little bit.
So, as the domains come into
contact with one another.
The fields do their best to
smoothly transition from one
to the other.
And grand unified theories
that predict a cosmic string,
they succeed pretty
much everywhere.
They're able to actually
smoothly make it all go away
so you don't end up with
feel being pinned down
anywhere, except in a little
one dimensional singularity that
is somewhere along where the two
dimensional services originally
met.
And that is--
there's details there
that I'm not even
pretending to explain.
And as I say, those
are only predicted
by certain kinds of
grand unified theories.
All of them will then predict
that even if you don't have
that, that cosmic string
will then shrink itself down
and it'll just be left with
a little knot of Higgs field,
where there's a little
bit of residual mismatch
between the two regions.
AUDIENCE: Do all three
types of defects carry
a magnetic charge,
or only the knots?
PROFESSOR: I think
only the knots.
They do carry other
kinds of fields,
though, in particular the
other ones gravitate, in fact
all them gravitate,
and so that's
one of the ways in
which people have
tried to set observational
limits on these things.
In particular
there have recently
been a fair amount of
work of people trying
to set limits on cosmic strings
from gravitational lensing,
and there was really
a lot of excitement
because people thought they
discovered want a couple
years ago.
And they saw
basically two quasars
that looked absolutely
identical, that were separated
or scale that was just
right to be a cosmic string.
And then people actually looked
at with better telescopes,
and saw they had absolutely
nothing to do with one another.
They were not cosmic, they were
not lenses, it's just every now
and then God is
screwing with you.
OK, so without going into some
of the details what you have,
these little point like
defects-- and I'm short on time
so I'm going to kind of go
through this a little bit
in a sketchy way enough so
that I can pay for you how
to do some calculations
you're going to need to do.
So the point like defects
end up being regions,
where at that point the
Higgs field actually takes
the value zero.
So remember I was describing how
when you have two regions where
the Higgs fields are both
taking on values such as there
at the minimum of the
Higgs potential energy,
and they come in to
match one another,
and what we have a
boundary condition
that very far away the
Higgs field has values
such as the energy is minimized.
And there is a theorem, which
in his notes Alan-- the way
he describes it is
he gives you a figure
and outlines the
various things that
are necessary for the
theorem to be true,
and invites you to think
deeply for a moment
and until insight
comes to you, I guess.
And when you put this ingredient
that the Higgs field has
this asymptotic, very far
away value that drives you
to the minimum of the field,
and yet it must change value
somewhere in the
middle, the theorem
requires that there be one
point at which H equals 0.
And apparently, this
is a consequence
in all grand unified theories.
So, recall, H equals 0.
This is a point at which
the potential energy
density can be huge.
So, when you have a little
point like defect like this,
it looks like a massive nugget,
little massive particle.
You can in fact calculate
the total amount
of energy associated
with this particle.
If you do so just including the
influence of the Higgs field,
the calculation
basically goes like this.
It's very similar to the way we
calculate the energy associated
with electric and magnetic
fields in electrodynamics.
Ask yourself, how much
energy is contained
in a sphere of
radius, capital R,
centered on this little
knot of Higgs field.
Well, it's going to
look like 4pi times
an integral of the
gradient of the Higgs field
squared r squared
dr It turns out,
when you calculate the
[INAUDIBLE] of the Higgs field
around one of these
little defects,
it's actually very complicated
close to the defect,
but as you get far away
it has a very simple form.
The gradient goes
as 1 over r, it
tells you the field
itself actually
goes something like log.
That means your energy looks
something like, R squared,
1 over R squared dr which goes
as R, which diverges as you
make the sphere bigger
and bigger and bigger.
So, what's the mistake we made?
Well, the Higgs
field doesn't always
just sit there and
operate on itself.
The Higgs field actually
couples pretty strongly
to all of our vector bosons.
Particularly, it
couples pretty strongly
to electric and magnetic fields.
So, we have to repeat
this calculation including
the interaction of the Higgs
field with the E&D field.
And in Alan's notes he
gives you some references
on this because this is not
the kind of calculation you
can really sketch
out very easily
in an undergraduate class.
To make this
integral convergent,
the only way it can be done
is if that little nugget
of Higgs field is endowed
with magnetic charge.
You need to have a monopolar
magnetic field that ends up
putting in interaction
terms, that
make the divergence of
this integral go away.
So, I at last get to the
punchline of all this,
we are left inevitably, if we
accept the whole foundation
story of particle physics that
the different interactions were
unified in some high energy
scale and then froze out.
We are driven
inevitably to the story
that defects in the Higgs field
create magnetic monopoles.
Now, I realize I'm out of time,
so let me just quickly sketch
a few interesting
facts about this
and there's a few exercises
that you guys are apparently
going to look at in your
homework assignment.
When we do this
calculation, one which
is I believe just
referenced in the notes
that Alan has for the
class, we learn a couple
of things about this
magnetic charge.
One of them is that if you
work in the fundamental unit,
say CGS units, the value
of the magnetic charge,
we'll call that g,
is exactly 1 over
2 alpha where alpha
is a fine structure
constant, times the
electric charge.
So if you have two
magnetic monopoles
they attract each
other with a force
that is-- so 1 over 2 alpha is
approximately 68.5 I think--
and so it would be 68.5
squared times the force of two
electric charges at
that same distance.
We also end up
learning the mass.
It turns out to be 1 over alpha
times the scale of GUT symmetry
breaking.
Anyone recall what the scale
of GUT symmetry breaking is?
10 to the 16 GeV.
So, this is a particle, 1
over alpha is approximately 10
to the two, so this
is a particle that
has a mass of about
10 to the 18th GeV,
in other words it's a single
particle with a mass of 10
to the 18th protons.
This is approximately
one microgram.
If you put one of these things
on a scale it could measure it,
that's bloody big.
So getting to the last bit
of the class, which I am just
going to very basically
quote the answer.
The question becomes how often
do these things get created
and here I'm going to
refer to Alan's notes.
What you'll find
is that, remember
when we sketched our original
picture of this thing
we looked at regions
of the universe
where the Higgs
field was initially
seeded with different values.
In order for the Higgs field
to take on different values,
initially, these regions had
to be out of causal contact
with one another.
So we are going to require
that the initial seed areas be
separated by a distance,
which is the correlation
length, which has to
be less than or of
order the horizon distance.
You can get a lower bound on
this thing by imagining that
it's-- sorry let me
say one other thing.
If you do that, then you
can estimate that the number
density associated
with these things,
the number density
of these monopoles
will be 1 over the
correlation length cubed.
To get a lower
bound on the number
density of these things, set
the correlation length exactly
to the horizon distance, and
then do the following exercise.
So first, let's set
up the correlation
length equal to the
horizon distance.
Set the density in monopoles
equal to the mass of a monopole
over rH cubed, normalized
to the critical density.
If you do this,
you will find that
just due to magnetic
monopoles alone,
the density of the universe.
PROFESSOR 2: Excuse
me, professor
PROFESSOR: Yes, I'm wrapping
up right this second.
PROFESSOR 2: It's seven minutes.
You were supposed
to end at 10:55.
PROFESSOR: I'm substitute
teaching, I'm sorry.
OK, so this tells
us that we are at 10
to the 20 of the
critical density.
And a consequence
is that the universe
is approximately two years old.
I will let Alan pick
it up from there.
