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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.

Design & Developed By ThemeShopy

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?

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???

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.

Is this course on graduate level (phd) ?

OH,….CHAAAAALLLK. more chalk.

This is a strange lecture style writing everything you say

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…

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?