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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.