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22. The Higgs Field and the Cosmological Magnetic Monopole Problem

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visit MIT OpenCourseWare at ocw.mit.edu 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.

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8 thoughts on “22. The Higgs Field and the Cosmological Magnetic Monopole Problem”

  1. apburner1 says:

    If you're going to write everything you say on the board, word for word, why not just hand out copies and call it day?

  2. Mark Wolstencroft says:

    Higgs Boson as a potential point in space-time where Magnetic Fields coalesce into a singular place or Monopole before 'flipping' and becoming Dark Energy???

  3. Yashasvi Vashishtha says:

    I was researching about Higgs field and the origin of mass, I developed a research paper proposing that the mass of any particle is not its own property and hence it states that the particle is gaining its mass from external interactions. I want to know if should continue working on this research paper or my ideas and interpretations are not realistic?

    Here are my views in a detail,

    Abstract
    The standard model of particles describes the different particles based on their properties, in which Higgs boson was not yet discovered. On 4 July 2012, the Large Hadron Collider found the remaining particle “the Higgs Boson.” Following the existence of Higgs boson, the proof for the existence of Higgs field was also found. The theory of Higgs field defines mass as the result of the interaction of fundamental particles with the Higgs field. The interaction resists the fundamental particles from traveling at the speed of light. The process of encountering resistance due to Higgs field is when an object gains mass. The mass due to Higgs field is not its own mass. All the particles of the universe are massless by themselves. The mass known currently is due to Higgs field as Higgs field is present universally.

    Introduction
    Today in the 21st century, we measure something in two forms “Mass and Weight.” Mass is the amount of matter in the substance. Whereas Weight is the effect of gravitation force on the mass of an object. We have studied that anything that has mass and occupies space is matter. According to our current understanding of particle physics, a proton is made up of 2 up quarks, and 1 down quark and these quarks are bound together by gluons. The mass when combined with 2 up quarks and 1 down quarks should be equal to the mass of the proton, as gluons are massless. But this combined mass is only 1 % of the total mass of the proton. The remaining 99% mass of the proton is due to the energy of the gluons known as quantum chromodynamics binding energy. Through the mass composition, we can conclude that the energy is equal to mass as mass is due to the energy of gluons. Because the mass is due to the quantum chromodynamics binding energy of gluons they should be considered the matter. Contrast to these mathematical calculations, the current theory of particle physics describes gluons as massless. Due to the inconsistency of data, I conclude that the definition of matter is wrong. Thus, proving the currently known mass of proton wrong according to E=Mc2 as the energy and mass are interchangeable in certain conditions but not equal. As the definition of mater was crafted when the Higgs field was not yet discovered, the definition becomes invalid after the L.H.C discovered the Higgs boson on 4 July 2012. Following the discovery of Higgs boson the Higgs field was proven to have existed, as Higgs boson is an excitation in the Higgs field. After the Higgs field was confirmed to exist, its properties hold true after that. The most important property of the Higgs field is that it gives mass to fundamental particles when interacting with the Higgs field. For example, proton’s mass was measured when the Higgs field was not yet then discovered, so the mass of proton measured was in the presence of Higgs field. This implies that the mass measured was due to the interaction with the Higgs field and not the own respective mass of the proton. This theory holds true for everything which has mass according to our current understanding of measurement of mass.
    As all particles are made up of fundamental particles, so when the particle for example proton interacts with the Higgs field, the fundamental particles that make up the proton also interact as a proton is a name given to the composition of 2 up quarks and 1 down quark held together by gluons. The properties of Higgs field states that any fundamental particle when interacts with the Higgs field, it generates mass. Thus, concluding that the mass of a proton is due to Higgs field.

  4. Carl Fabian says:

    Is this course on graduate level (phd) ?

  5. tomatocan says:

    OH,….CHAAAAALLLK. more chalk.

  6. John Michael Stock says:

    This is a strange lecture style writing everything you say

  7. B BIll says:

    After listening to Alan's lectures for hours this kind of lecture style is, sorry, but just horrible. Why this rushing? Don't try to fill the blackboard with text, just write the minimum and try to explain what is hard to understand and what you are thinking about all that. Make space for interesting questions and ideas…

  8. Kaushal Timilsina says:

    Can symmetry breaking under the gauge transformation be understood, well not exactly but kind of as:
    There were two linearly independent bases for the spinor field before the symmetry breaking and under the transformation the two independent spinor state collapses to give a spinor field with two linearly dependant or just one linearly independent component-direction. And because of the singularity associated with the transformation, the symmetry breaking is irreversible unless, the single component spinor breaks down to a combination of two independent states. And so, avoiding singularity for some special reason-to describe the steady state at equilibrium during transition as described by the Higgs mechanism, the spontaneous symmetry breaking produces two different interactions from what used to be a single combined form. Does that make any sense?
    The symmetry breaking happened on a sphere( infinite solution-from non zero minimum field). Contrast to that if it had happened on a point(single solution-from a zero minimum) the lower Omega would be bounded at zero. Does the former imply that even if we keep on finding more fundamental particle in the course of scientific advancement, we will never hit the Omega = zero ; setting the bound that real particles could not be made up of virtual particles and the two are distinct. This idea is particularly really exciting to me because I recently wrote an ameteur discussion that "all real particles are emergent from the localization of gravitational waves(also quantization).
    Is it possible that the smooth transition can happen two ways, like you could connect two points with an concave or convex paths. But since this is a single point of defect, the two states by quantum transformation are superposed into like a superposition of the two states, in the same point-giving the dipole?
    Also, Is two structures with different orientation special or could structures with more orientations could come together. What would happen when a third orientation hits the boundary of the two? Could it be that the orientation would weigh in towards either one because of the amplitude of the real components of the third guy aligns more towards either of the two, and so the transition influences the whole structure to acquire the same orientation? The Higgs field being zero at the boundary allows only one of the two poles to be real, and hence creating the magnetic monopole in the theory. This would mean that magnetic monopoles could exist together with this other imaginary state.
    In condensed matter systems, the quasi particles when described as point like defects, can propagate. Could the defect in the Higgs field propagate, as the energy densities on either sides of the boundary (which is now almost just the point) evolve differently? Could there be an oscillatory behavior? If you could have different pairs of defects form groups and smoothen out, and the fact that the defect particles could propagate as in condensed matter system, and oscillate as described by the differently evolving, energy densities on different sides; could this possibly allow the same particle in such a field to have mass that changes as the system evolves, and could it have anything to do with Majorana mass?

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