We did not take the class last semester of course they don't want to clash the semester, okay?
Okay, so everyone more or less knows what the deal is there's not too much different
We the way the course is going to be run this semester from last semester
My name is still Alex Scott's name is still Scott he's still the TA
We have slightly different office hours, which are posted here
The classroom is at this classroom, which you figured out
Remarkably enough nobody went to the one that was listed on
Let's see yeah as with last semester if you're not registered for the course
But you are attending the lectures
send me an email so I can add you to the email list cuz like if I have to cancel the course or
Reschedule something that way you can be notified
Grading it's just gonna be problem set to the semester no file or anything like that although the final was fun
Did you guys enjoy the final was it exciting?
Pretty good. It's pretty pretty fun right. It's fun to grade, so let's see
Do we need to go over this I mean it's all the same stuff. I said last semester problem sets
You have to turn in your own work as opposed to someone else's work
You need if you're gonna be late
You'll be penalized unless you get an extension from me or Scott the day before or before. It's due
The textbook, I'm gonna keep using Schwartz as a textbook. I think because
It's actually quite good for the stuff later on in the semester when we get to the standard model
So I think we're gonna stick with it
But I am gonna sort of lean a little bit more heavily on strid Niki as an alternate text
I think it's very it's very good
It's pretty different from shorts, and it's also available online for free so like if you go to the course web page
Will have a link to it so at least the giraffes version of Street Nikki's book is
publicly available for free and it's practically the same as the published version except it has a
few more exciting typos
If you did if you did check out
Just a word about supplementary reading you might want to take a look at a lot of the sort of standard quantum field theory
Books that I like like writer are great
But don't have quite as much of the more advanced stuff that we're going to be studying this semester
So I think Dave tongs lecture notes are great writer is great Z is great
But you may find that after the first few weeks month of this course. They're not actually covering what you need them to cover
That's gonna shredder is great
but
There are a couple
Problems with the way that you presenter normalization but aside from that I think it's really great, so that's what I really recommend
you
know Tom bankses book on normalizations really great and kind of quite different
So you should take a look at it if you want a good resource on our normalization?
It's quite advanced so you should be prepared for that, but it's great
And then the other thing is that I am going to try and do a little bit more
Of applications of condensed matter physics this semester as in Justin's book is kind of the classic
It's also like a telephone book and it's like
1,500 pages long or something so it's not the sort of thing you're just going to want to like
Browse through I don't son day afternoon, but it is really good so you should consider taking a look at that
You know this course is going to be more or less a continuation of last semesters for the administrative details are really not any different
I
Did give a syllabus a brief syllabus at the end of sort of an outline of what we're going to study
Note that
The word were normalization is used one two three four five times
In the first two lines of the course outline
That's really kind of the heart of what we're going to be studying this semester so from my point of view there are two
Important things that we're going to study this semester first we're gonna study
We're normalization, and then we're going to study non abelian gauge theory that is to say I'm those theory
Normalization is kind of the core
observation
And tool that we use to understand quantum field theory not just in particle physics
But also in applications to dense matter physics to cosmology and so forth so
We'll be starting with a kind of
I'll say old-fashioned approach to row normalization a kind of sort of standard
Textbook approach to the normalization as you might have learned it in a class
20 years ago
so in this case
So we'll be doing things like doing a detailed study of the normalization of scalar field theories and upon electrodynamics
And we'll be along the way sort of using what we learn about Rijn or Malaysia to
understand how perturbative correct Corrections are computed sort of
precisely in quantum field theory using Fineman diagrams
before moving on to a kind of more general approach to normalization using what's known as wilsonian or a normalization of the
And this will really this is really I think where quantum field theory shines
it's where you can kind of understand the quantum field theory as a
General tool for studying the dynamics of all sorts of different systems so effective field theory is kind of the buzzword here
It's the strategy that we use to if I hand you a physical system
You identify
The symmetries of that system you make a guess for what the basic dynamical degrees of freedom are for that system and then effective field
Theory gives you a recipe for studying the dynamics of that system
Systematically at low energies so for example effective field theory is of course used in particle physics all the time
They're effective so the standard model is believed to be an effective field theory that this is an approximation to so-called
Fundamental theory, but if you study nuclear physics there's effective theories of nuclear physics
But in addition in condensed matter physics when we talk about the quantum field theory descriptions of various condensed matter systems
We're always talking about effective field theory and even in the last several years effective field theory has been used to study
cosmologies when we stood in the large-scale structure of the universe
There's an effective field theory that describes that it's an effective field theory involving broken symmetries
Just like the effective field theories that describe nuclear physics, so it's a very very powerful
Common language that we can use to study physics and all sorts of different regimes
And that's kind of I think the heart of this second semester of field theory
So kind of the first I would say half of this course is going to be spent this
semester on normalization
I'll try and be generous about throwing in applications to things aside from particle physics
I'm gonna try and talk a little bit about
the theory of critical exponents and
Universality classes, it's matter physics
We'll see how that goes to some extent that depends on on how quickly I can get through some of the earlier material
And then in the second half of this semester
We're really going to be diving into the quantum field theories that are relevant for the standard model of particle physics
So we'll talk about yang-mills theory otherwise known as non abelian gauge theory. This is the generalization of electromagnetism
To describe the nuclear forces the weak nuclear force on the strong nuclear force
Will talk about quantum chromodynamics, which is the theory of the strong nuclear force describing the interactions of quarks and gluons?
before moving on to the electroweak force
The electroweak force is interesting because it will be the first example that we'll encounter this this semester of a theory with
Broken symmetries with spontaneously broken symmetries so to understand that we'll need to understand the Higgs mechanism
Which of course is very exciting because we just observed the Higgs boson which is the particle associated with?
The Higgs mechanism, and we'll probably also talk about some other
examples of spontaneously broken symmetry so for example superconductivity
Is a theory of spontaneously broken symmetry in the bcs model?
And then finally at the end. We'll sum it all up by
describing the standard model
We'll see how far we get I mean one can teach a whole course on the standard model
And there is a whole course on the standard model taught here every other year
So of course we won't get into a huge amount of detail
but we will talk a bit about the standard model some of its features and
Are there any questions about the organization of this course about structure about what we really will not cover any thoughts yes
This is the 500 level string theory class yeah, yeah, I
Mean it's fine with me. It's probably the problem so if we wanted to change the time to the course that
Guess is fine with me, but it's a matter of
Seeing that everyone's available whose teacher is yo.hannes teacher. Yeah, how many people would like to would be taking that course
if it were I
mean
Okay
Yeah, I mean it's a pretty basic like if you actually want to learn string theory I guess there's no harm in taking that course
But it is pretty Elementary
I mean that course is kind of geared for people who don't know quantum field theory if you know quantum field theory you should do
A real string theory course now of course we don't have a real string theory course the semester
Despite what Minerva says yeah Nerva is like
Because like it's always showing your classes that don't actually exist in some classes do exist, but don't show up on there
That's so you know. It's very exciting
Yeah, I mean, I'm not I don't discourage anyone from taking that class, but it's not like the most
Awesome introduction to string theory cuz it's pretty elementary although, maybe yo.hannes is you know inclined textbook easy?
Yeah if he's using Sui buckets pretty elementary I mean I would recommend
You know set aside a weekend and read-throughs a wee vodka and you'll be fine
But we could talk about any we can change the time of the course
There's only like 10 people you know there's not that many people in this course, so I'd be happy to change it if that's more
Easier I mean what time is a on us of course that's exactly this I
Mean let me think we could go earlier on Tuesday Thursday 1:00 to 2:30
Is there anyone who would object?
You would have to see what you guys have course you okay you three I mean
I don't want to go later because it's a little
It was like 4:00 to 5:30 was a little late the day and it's kind of hard. You know I'd rather not go later
I mean we could go like
Monday Wednesday would go to morning's how do you guys like the mornings?
Not everyone likes the mornings I mean if we had it like what
There's the gr
How many of you are in gr a couple of you how many of you here a couple of you from the achiara?
One is gr
10:30 to 12:00 okay, um I think on nine
Nine is difficult for me. You know nine is difficult for me. I
Could do it, but like you would have to pay me more
I would have to be paid more than I'm being paid now how to teach your classes
Mornings are what yeah, what about?
Yeah, we I mean we can do Monday Wednesdays morning
They're only four or five people already here
Yeah
You can't you can't you can't you can never officially do anything basically?
You can all like I can't officially change the location of this room
But I can hook this room and tell people to come here. That's fine, so it's like
I can't officially change the time of the course. I could not officially do it. I mean what about Monday Wednesday mornings was that?
Okay, and then we have 12 Monday Wednesday afternoons you guys are busy
This is the problem. This is Monday Wednesday afternoon. You said you were busy
Does anyone know are you guys is that a yes that you're busy, okay all afternoon?
More or less yeah, see this is the problem when you try and reschedule classes it never actually works
Asked you Hannes again. It's he's a nice guy. I mean usually
Yeah
Yeah unless anyone else has any bright ideas
What's that
Wednesday Friday Monday Friday, what's your problem your problem is Wednesday's your Monday Monday?
I'm running out of days
We could do 11:30 on Tuesday Thursday, I'm trying to think do we have because usually there's a lot of seminars at lunch times
So so you guys want to do 32
You can't make it okay
Yeah
It's like this is the thing getting 15 people in a room like nobody's ever gonna be able to
Okay, well, I guess we're just gonna have the class when it's scheduled and that I mean we could do 4 to 5:30
It was sort of it's sort of not an ideal time at the end of the day how many people did people?
Mind having it we could do 4 to 5:30 if we need to it's not ideal for me, but we can do it
Do people want to do that how many people would rather
Have it from 2:30 to 4:00 versus 4 to 5:30 raise your hand for 2:30 to 4:00, okay
And you're okay
To reschedule
It's not worth if it's not
Yeah, tell you Honus that I want him to move the class
and then maybe
It's so much more fun if I have other if I communicate through the medium of the through the medium of
It's very little bit right, but it's a more fun way
Okay, good, okay
so let's just
Start out with some physics so
Ya know finals the finals I
Mean you're all adults like it's time to leave behind childish things
At some point you grow up you you leave behind your teddy bears and your stuffed animals
Well, you can keep your teddy bears and your stuffed animals, but finals. I think you complete so no files
I mean unless someone wants a final you want a final
Yeah, you want to find them you enjoyed it you can ever find, okay?
What if we had a final problem set that's worth as much as a problem set, but it's called a final
Hope you like to say would you like that?
Sometimes people do this actually they have a fuckwit unquote final problem, so
It's that we yeah there yeah, yeah, do you want it we can do that gargling you could have a final
Yeah
Do you wanna how many people read you and if you'd like a final? It's okay. You can raise your hand it's fine
It's fine, how many people would rather have a final than just problems
Okay, can we just?
Let's just you problem since then okay. I'll call the last one a fun
but
Yeah, I'll call it the finals. Yeah. Good that way everyone will be happy, okay good
so
Last semester we developed a systematic way of understanding perturbation theory in quantum field theory
Using the Fineman diagram expansion but
We really only focused on the leading terms in
the Fineman diagram expansion of say scattering amplitudes or
correlation functions in quantum field theory and we really did very little to
discuss the computation of
perturbative Corrections involving loops in Fineman diagrams and
One reason is that of course in a perturbative expansion the sub leading perturbative Corrections are small
So loop corrections in quantum field theory will be relatively small
But we also saw
That these loop corruptions in
Fineman diagrams in the Fineman diagram expansion often are naively divergent
And we saw this in a couple different
contexts and
Which one of which I'll remind you of in just a few minutes
But the important point that one always has to keep in mind when thinking about these divergences in
Quantum field theory is that these divergences are always unphysical
Well I shouldn't say always unphysical these divergences
Unless they have a physical meaning are in physical. I suppose. That's a bit of a tautology, but that doesn't mean it's not true
The ultraviolet divergences let us say the short distance
divergences that arise in quantum field theory are always unphysical and
These divergences in the computation of loops and quantum field theory arise
When we compute
The wrong quantity that is to say when we compute on physical quantities
And
What we always want to keep in mind when we encounter a divergence in quantum field theory is that physical quantities?
That is to say physical observables
in quantum field theory
are finite
But it's very easy to get confused and to compute something that you think is physical
but is secretly not observable and hence has some sort of divergence and
So what I'd like to do is I would like to start by
recalling an example
That we discussed on the last day of class in December
So I don't remember exactly how much a detail I discussed this example
and so I'm just going to remind you of this example just so that we have just because it's a great example to
Understand the origin of some of these divergences as well as they are cure
before we try and implement
more systematically the procedure by which
Infinities are removed from the computation of physical quantities in quantum field theory
So the example that I want to consider is massless light of the fourth theory
So, this is a theory of a
scalar field
With a kinetic term where I've set the mass
equal to zero and
I wish to consider the scattering of
two particles into two other particles
So
the amplitude for that scattering, I'll call n m and
At leading order that's just given by this tree level diagram
Which is finite, and it's just given by lambda okay the value of the coupling constant appearing in the Lagrangian
However
At
Next order in perturbation Theory there start to be loop Corrections
So for example here's one loop correction
You can see that it's proportional to lambda squared because it has two interaction vertices
And it also has one loop integral
remember that
The way that we proceed when we compute Fineman diagrams
is
that we
enforce momentum conservation
at each one of the vertices in a Fineman diagram and
Then we integrate over all of the undetermined momenta so for example if I let P 1 plus P 2
Equal to P
Then there'll be one undetermined momentum, so this would be
That I've called Kay here and
One can always so one can always do a change of variables
to let you know K plus twelve or K plus P or K minus P or something like that be the undetermined momentum, but
the answer at the end of the day will not
Affect whether or not you do a change of variables like that so here for example. I've called the lower. You know the lower
Leg of this loop in the Fineman diagram the undetermined fermentable hey
I could have called it the upper one if I wanted to
That would just be some change of variables in the integral over the undetermined momentum at the end of the day
And it wouldn't actually affect the final answer
Now actually this is not the only
Contribution to this to-to-to scattering amplitude at this order in perturbation theory
This is the S channel contribution there are also T
And u channels and just for the purposes of illustration for this example. I'm just going to consider the S channel for today
Okay, if you wanted to actually do the computation correctly, then you would need to consider the other channels as well
But let's just talk about the S channel because really I'm not trying to
write down a precise answer but more to make a point so
Do you remember this I did this the last class of last semester right okay?
So I'm gonna just do it in a little bit more detail now just to highlight
Some features that we're gonna be using later on in this class and for the next couple of weeks, so
What is the value this contribution to the scattering amplitude?
well you have the coupling constant squared you have a symmetry factor of two and
Then you have the integral over
this undetermined
momentum and
Then you have two propagators a 1 over K squared propagator and a 1 over P minus K squared propagator
and
The important point is that if you stare at this integral for a second
It's an integral and you think about how it behaves when K is very large
Then you see that it's an integral d4k
But it's got four powers of K in the denominator so if K is very large compared to P
Then this looks like an integral D for K over K to the fourth
Which is logarithmically divergent?
So the question, then is how we should deal with this law of divergence
So the simplest thing you could imagine doing
Is just cutting off this integral at very large values of the absolute value of K and
Then you could go ahead and perform this integral we integrate d4k naught over all values
But where we've cut it off at some very large value of K
Let's say lambda
so if we cut off the integral
Probably misspoke cut off there
At values of K. Which are less than some scale lambda
Then you know that
This should be you should get a divergence in this integral which looks like the log of lambda
so in particular
just by
Lawrence invariance
you know so lambda is a dimensional quantity it has dimensions of energy and
by Lawrence invariance
You know that that integral right there has to be a function of P squared that is to say s
So s is the mandolin variable P squared
So you know just by the dimensional scaling of this integral that it's long divergent and the fact that it has to bend on s
That the leading term in this
In lambda has to go like log s over lambda square
Here, I put in a lambda squared here because s has dimensions of energy squared
And this is really by Lawrence invariance
And
Just the scaling properties of this integral as you rescale k dimensional analysis essentially
so later on
Probably next class. We're going to develop a whole set of technologies for computing integrals like this. They're really not that
Hard to compute in fact when we compute integrals like this in this course
We're probably we're generally not going to be introducing a hard cutoff. That is to say
Cutting off the value of the K integral at some finite value of lambda we'll be introducing some more sophisticated
kinds of ways of
regularizing or cutting off these integrals
but for now
let's just talk about this hard cutoff method where we just cut off the integral at some absolute value of lambda and
Then if you do that it's a sort of straightforward calculation
To go ahead and compute this integral
Because we're going to be spending a lot of time later on this semester doing this calculation
I'm not going to bother
Computing that cutoff integral for you right now. I'll just tell you the answer
So it goes like log s over lambda squared
Top where lambda is the cut off times the coupling constant lambda squared along with some numerical coefficient
Okay, so that means that if you go ahead and compute the
two to two scattering amplitude
Okay, bye-bye there
You get the leading contribution
which was lambda and
The sub leading contribution coming from the one loop correction
which is
small in the sense that
It's order lambda squared rather than lambda
but it also naively has a
logarithmic divergence
so it's a little confusing because
Ostensibly, it's second-order in perturbation theory because that coupling constant little lambda is small
But naively you might think it's divergent because it has the log of
Capital lambda here and capital lambda is some very small number
It's the place where we cut off the integral so as you took capital lambda to zero this
Second term would appear to be logarithmic lee divergence
So this naively appears to be a disaster for quantum field theory because you have an infinity
You don't know what to do with it well
You don't know yet what to do with it, and so it appears that quantum field theory is really
failing
But here is where we really have to remember that physical quantities
Are finite and that we really need to
Phrase the answer here
In terms of physically observable quantities
So I've written the answer for the two to two scattering amplitude M
Here in terms of the coupling constant lambda
Which typically goes by the name of a bear coupling constant?
And lambda is a parameter that
Appears in the Lagrangian, but it is not something that is directly observable
You don't observe a Lagrangian what you observe is things like scattering amplitudes or
correlation functions
the four-point function
Or two - two scattering matrix element
Is something that is observable
So the question
Then is how do we rephrase our answer for the scattering amplitude M in terms of something that's physically observable
and the answer really
Arises if you think carefully about what it is you mean by this coupling constant lambda, okay?
Lambda is supposed to measure a strength of the interactions between
Two of these five particles and so a
perfectly reasonable definition
For the coupling constant of the theory is simply that it's equal to minus
the two - two scattering matrix element at some particular value of energy
That I'll call s not here
So really what we want to do is we want to think about the physically observable coupling constant
Which measures the strength of the interactions and theory as being defined as?
The to-to-to scattering amplitude at some particular value of s that. I've called s naught
So indeed you could see that if you just looked at tree level then
The scattering amplitude M is
Just equal to minus lambda the coupling constant in the Lagrangian and so that a tree level
This is a perfectly consistent definition. This is the correct definition and so lambda sub R is
equal to
lambda at tree level
But then once you start including loop Corrections
You have to include this second term here
That involves the logarithmically divergent term and
So lambda sub R is often referred to as a burn or alized coupling
Hence the subscript R or sometimes it's just referred to as the physical coupling
So then we could take this formula for lambda of r the r enormous coupling in terms of the bare coupling lambda
And you could then go ahead and solve
for lambda in terms of lambda sub r
That's just some trivial
algebraic manipulations
So I've just written down the answer here all I've done is taken this formula here for lambda sub R in
Terms of lambda and I've inverted it working only two quadratic order in the coupling constant lambda
which is assumed to be small and
Then you could go ahead and plug that back in to our formula for M up above
So we could go ahead and plug that into our formula for M
And what you're left with
So let me write this up more carefully is an
expression for the two to two scattering amplitude
written in terms of
The physical coupling lambda sub R
So
Let's just take a second to stare at that expression here
so
You can see what?
We've done is we've written the two two two scattering amplitude m at any value of the center of mass energy s in
terms of its value at a particular
fiducial
value of the center of mass energy s pellet and
so of course
When s is equal to s naught that sub leading term vanishes, and the scattering amplitude is just lambda and so what really?
The quantum the scalar field theory is doing for us is
it's
Allowing us to relate the two to two scattering and the values of the two to do scattering amplitude at different values of the energy
Because you know the original coupling lambda and Lagrangian is not something you've ever observed all you ever observe are scattering
Amplitudes and so all the quantum field theory can do for us is relate different values of those scattering amplitudes to one another
And so once you think about quantum field theory in that sense not as computing
scattering amplitudes in terms of some constants appearing in both gironjin
But rather describing a set of relationships between different scattering amplitudes, then everything is going to be nice and physically finite
and
You can see you could think about this too due to scattering amplitude really as defining a coupling constant lambda
That is a function of the energy scale s and so you see that what's happening here in this
Expression is that the value of that coupling constant is depending logarithmically on mass
so as you probe the theory of different energy scales the theory might appear to be either more weakly or more strong decoupled and
That's something that is going to come back again and again in this course. That's a sort of universal feature of
quantum field theory
the fact that it has a logarithmic dependence on energy scale as
Opposed to say a power law dependence on energy scale is a special feature of five to the fourth theory
It'll be shared by some other theories that we study
But not all theories other theories you might have a polynomial dependence on energy rather than logarithmic
Okay so the essential point is that in quantum field theory?
We really only relates different observables to one another
And if we do so you
Know if you write down expressions for scattering amplitudes relating one observable in terms of another observable
We never encountered divergences
We never have what are known as ultraviolet divergences?
That is to say short distance divergences that would appear when that cutoff lambda is taken to zero
Question
Sorry infinity yeah
Yes absolutely, I mean this is gonna happen again and again and again, yeah
Well I mean you know in I
Mean this sort of happened this sort of happens all of the time
so
You know an example would be safety value of the electric charge
Dealing, okay? You're not I mean
Yeah, okay, I mean you found it you did an example of this in one of your problems
That's last semester where you studied the Kasmir energy
So you take a pair of parallel plate capacitors, and then then strictly speaking?
There's a divergence if you calculate the energy in
the quantized electromagnetic field between the two parallel plates
Then you found a divergence
But energies are not observable and when you phrased it in terms of observable quantities like forces, then you kind of find out answers
So that was that's a pretty elementary example
Yeah, I mean
Right I mean
Yeah, I mean yeah, I'm not quick
About this we're gonna find examples of this again and again over the next few weeks, so
Well I mean it did I mean it's a field theory effect
That's right, that's right, I mean we could probably come up with analogies in classical mechanics
But
Yeah, but yeah
I don't know how strong the analogy would be I mean there are very often cases
You know in classical physics we can compute things that are naively infinite
But are not observable like the self energy of an electron in
Electro dynamic electro magnetism is is infinite, but it's not actually observable
So that's the sort of thing
You know what you really observe your energy differences, so that's the sort of thing that can happen
In classical physics as well
Okay and so
If we wanted to we could then proceed to go ahead and continue to study quantum field theory in this vein so in particular we
Could proceed
To study interactions carefully in quantum field theory
By introducing physical that is to say we normalized couplings and
Then rewriting everything
In terms of these were normalized couplings
But in practice this tends to be a bit cumbersome and
In practice what we're going to use is somewhat simpler algorithm
Using water known as counter terms
So counter terms of course are something that we encountered in a couple different contexts last semester
But I wanted to spend a few minutes just to introduce them a little bit more
systematically
So in particular
Let's consider again the case of massless five of the four theory
So
You could imagine
Studying the theory with a coupling constant lambda
That's now equal to its realized value its physical value where it's defined to be equal to the two to two scattering amplitude at
some particular value of the energy and
Then we could introduce another coupling
That I'll call Delta sub lambda
Which is I'm going to include
Just as another coupling constant in the theory. This is known as a counter term and
The sole purpose of this term Delta Lambda is just to order by order in perturbation theory
enforce our definition of lambda sub R as the value of
the Tuda two scattering amplitude at s naught and
so in particular
This Delta Sablan de
You know if you write it in terms of the cutoff scale capital lambda is going to be formally divergent
But we will treat it
As being a warder lambda squared
So in particular we
Could go ahead and compute say this two to two scattering amplitude and
Then if you work at order lambda squared
then at tree level
You'll just get
The tree level coupling so that's lambda R
- plus Delta and
Then you'll get this quadratic term
but we're gonna treat Delta as
of order lambda squared
So that if will only work to order lambda squared, then these are the only two
contributions to that scattering amplitude and
Then we fix the Renault realization constant delta lambda
By setting lambda per normalized to be the value of the scattering amplitude at energy S naught
So that means that
that second order in the coupling Delta Lambda has to be equal to
minus lambda R squared over 32 PI squared log s not
over lambda squared
Just so that when you evaluate this scattering amplitude at energy s naught you get minus
Lambda sub R and
Then plugging that back into our answer. We get exactly what we found earlier
for the sub leading correction
At order lambda squared to the two to two scattering amplitude
So the introduction of this counter term is just a sort of polite fiction
That allows us to slightly more efficiently keep track of the various divergences that appear in quantum field theory
when we write things in terms of unphysical quantities like parameters in lagrangian as opposed to physical quantities such as
normalized couplings or physical couplings and
The general strategy
That we use
when
considering these counter terms is
that for each possible coupling constant in
the Lagrangian
We introduce a counter term
Which is whose only role is to quote unquote absorb the divergence
That appears in the loop expansion of quantum field theory and
We fix the counter term
Order by order and perturbation theory
To enforce a physical condition
Such as the definition of a physical coupling
So it's this strategy more or less clear
One of the things that we're going to do in the next
couple of lectures is understand a
Somewhat more systematic strategy where we determine exactly what counter terms we need to introduce in a quantum field theory
How many counter terms we need to introduce and how this affects the general ability of a given quantum field theory to make useful predictions?
But hopefully at least the general strategy is clear and how its implemented in this specific case is clear
Any questions on this?
No questions
Yeah, yes
That's correct, that's right. I mean so this yeah, okay?
Well so there's a sense in what we're doing here is we're balancing different orders in a perturbation series against each other in the sense
that
You know Delta Lambda is
A term that's appearing in the Lagrangian so you might naively think it's a tree-level quantity
But we're setting it to be of order lambda squared
It's a second order and the coupling constant the real coupling constant
And so what we're doing is we're taking this tree-level term, and we're balanced balancing it against this quadratic term
so in that so I mean I
Understand what I mean, so that's maybe not quite what you're thinking I guess
Yeah, I mean
I mean certainly in the case of anomalies there are cases where there are tree
Level effects that need to be balanced against one-loop effects and things like that
Those are really I guess I would say well what we're doing here is my Jupiter. Okay, because what we're doing here is really just
Introducing Delta as a bookkeeping device to keep track of who normalized vs. Bear coupling constants in the Lagrangian
Whereas when we talk about anomalies
It is literally that's like a physical fact. You know where we really have so I would sort of draw contrast there
Other questions good question
Well the answer is that it works out if we say that maybe a slightly better answer is that remember that at treat you know
We want to define our
coupling constant lambda sub R to be the two to two scattering amplitude at some value of the energy and that
Only works if Delta lamda is of order lambda square
so
For exam so it may well, so there may well be let's put it this way there are going to be other
Sorts there could be other sorts of theories where loop diagrams only appear at order lambda cubed
Lambda to the fourth there's a bank that in that case
You could introduce a counter term and say well. It's only lambda squared, but then if you start trying to enforce these physical conditions
You'll see that actually the coefficient. That's of order lambda squared is zero, and then it would be some higher order I
Don't want to set it to be some lower order because then it would be competing against the classical effect
I want it to be a quantum effect, so it has to be at least quadratic
But there's no rule that says has to be nonzero at quadratic order. That's when you should buy the square. You don't mean like it
dimension
That's correct
Yeah, so one of the that's actually an excellent point the coupling constant lambda that I've introduced here is dimensionless
Hey, that's why I started out by talking about by to the four theory when we talk about dimensional couplings
We need to be a little bit more careful about the scales that we introduced
but yeah
What I mean when I say it's of order lambda squared
I mean I treat it as higher one order higher in the perturbation expansion
When I do the find the diagram expansion so that for example I don't need to in this term include a delta squared term
Now if I want it so now what I really want to do so here. I've only done this
Calculated this counter term at leading a quarter
You know if I wanted to if I was particularly powerful
I could try and compute
two to two scattering at higher order at order lambda to the fourth say and
Then I would need to go ahead and compute this counter term at higher order okay, and so I really need to enforce this condition
physical condition the definition of the normalized coupling order by order and perturbation theory and
So for example there would be
You know there would be contributions Delta Lambda that are the water lambda the fourth that would be needed to cancel other terms that appear
in
This scattering amplitude m at order lambda to the floor
Hopefully, that's more than square
Other questions
And okay so the question then so this general strategy
Too many R's in that word the general strategy
Then is that we introduce these counter terms in order to enforce these physical
conditions and
The question then is precisely what sort of physical conditions are we trying to enforce?
In the present case I introduced one physical condition namely that the value of the coupling should be
Defined to be the scattering amplitude at some particular value of the energy
but more generally you can ask what exactly is it that I mean by these physical conditions and
in fact
We have already introduced
These physical conditions
When we defined interacting quantum field theory
So let me just remember remind you of something that we touched on last semester, so
remember that and free quantum field theory
So let's just take for definiteness a I don't know a massive scalar field
Then in free quantum field theory the Hilbert space is perfectly easy to study
it's a fox space with no particle 1 particle to particle States, and so forth the observables are this field Phi and
The crucial properties
That define the no particle state which is the vacuum state and?
The one particle state which we denoted K. That defines a state with momentum K
Are
given by these two formulas
So the vacuum state 0 is essentially defined so that the one-point function of the field in the vacuum
vanishes
And the one-particle state k is
Defined so that the operator v creates a one particle state with momentum k and
What do we mean by that we mean that if you take Phi of X and you act on the vacuum State?
You take the overlap with a one particle state k then you get the wave function e to the ikx
describing a particle with momentum k in position space
So and here
k
Is on shell so it obeys the
equation of motion that k squared is equal to m squared and
So when we the point is that when we study an interacting quantum field theory?
We can't really
Describe the hilbert space in terms of a simple font space
But we still wish to impose these two conditions
Which essentially define the no particle and the one-particle states?
So
In particular
What we argued is that if you add to this free Lagrangian here some interaction terms?
Of order of I cubed Y to the fourth or something like that
Then we no longer know exactly what a particle means in terms of the Fox based operators the raising and lowering operators
but
We can always preserve these two conditions here
By shifting
Or rescaling our field variable by
So in particular
if you added some interaction term
And then you discovered that in terms of your field variable Phi
There was a one-point function in the vacuum state you would just shift your field variable to some new value
for I prime which is five plus a constant such that the one-point function of this new variable Phi prime is equal to zero and
likewise if you found that say
this overlap with the one particle state was not e to the ikx, but
12 times e to the ikx, then you would just rescale Phi by a factor of 12 in order to remove that
So
What I want to do is try and consider that process of shifting and rescaling the field v a little bit more systematically
So explicitly
Let's just consider phi hugh berry
Just because it's really the simplest case all of the calculations are going to be rather simple
So in this case the Lagrangian well, let's start with
The free Lagrangian
So we have an md5 squared term and an M squared Phi squared term and then let's write a
Coupling term
And
What we wish to do is allow ourselves to rescale or
Shift the value of the field Phi in order to enforce these conditions
So what does that mean?
that means that really when we study the
interaction Theory the interacting theory we should put some new constant there Z which is known as the
wave function normalization
constant
And we could put a new constant there a C sub M
which will describe the renewal ization of
That coupling constant M. So that's the mass for normalization constant
If we allow ourself to shift the field
In order to remove the one-point function
Then we should add another coupling term Y times, Phi
So this is to remove the one-point function
And
as we saw in the case of the fight of the four theory we should allow the
counter term of order Phi cubed so
In order to set the physical value of the coupling so we can introduce a new constant there that I'll call Z sub G
Which is the coupling constant from o?
So we now are considering a theory with four undetermined constants
The wave function or normalization the master normalization the
tadpole or one-point functional mobilization and the coupling constant or normalization and these four constants
are fixed
By four physical conditions
In particular
Z Phi is fixed by the normalization of the one-particle states
So
The fact that this goes like e to the I KX is just a consequence of translation invariance
But the overall constant out front the fact that it's 1 times e to the ikx is not and so
What you we need to do is
Rescale order by order and perturbation theory or change that adjusts the value of Z sub Phi
Ordered by order and perturbation theory to enforce this condition
The coupling constant Y is fixed
by the condition that the one-point function and
the vacuum is equal to zero
So again by translation invariance the one-point function in the vacuum has to be a constant
But that constant is not necessarily equal to zero and so you have a 1 constants worth
So you have one equation that you need to enforce?
On this constant lambda this constant Y to set this one-point function equal to zero
Similarly Z sub G
Is
fixed by the statement that
G should be equal to the physical value of the coupling
That is they the physical say three point function
at
some fiducial value of the energy
Just as we were considering for the four-point function earlier and finally the master normalization
Z sub M is. Just fixed by the condition that the mass of a one particle state
Should be M Squared
Or. I should say the mass squared of a one particle state should be M squared the M appearing in the Lagrangian
does not
Necessarily have anything to do with the physical masses of the states in the theory so of course as we saw in the free Theory
The parameter M appearing on the Lagrangian is the physical mass, but once you start considering perturbative Corrections?
That's no longer going to be the case. Let's do that explicitly
So what we have here are four conditions on these four unknowns and
These conditions should be enforced order by order in perturbation theory in the coupling constant G of the theory
So for example at tree level
Order G to the zero we just have
The sort of trivial case where there's no wave function or mass or normalization
The one-point functions are zero and
the coupling constant G is
Just the three-point function evaluated. Whatever scale you want it to be so there's no normalization of the coupling constant
But at higher order
Inje
These renormalization constants are nonzero
and
May contain luke divergences and
So they need to be fixed
Order by order in perturbation theory
In the coupling constant sheet
And in fact we already saw this
Earlier in the class when we were considering tadpole diagrams in quantum field theory
We also saw this when we considered the Casimir effect and one of the problems on the problem
So for example
Y is fixed
By the condition that there are no tadpoles by
The canceled by the vanishing of the one point function
So in particular the term Y Phi in the Lagrangian
Means that if you are considering the perturbation expansion in terms of Fineman diagrams
including this new coupling Wi-Fi
Then there would be
some new vertex
interaction vertex in your Fineman diagram expansion when you just had one
insertion of Y with a 5 particle coming off of that and
This counter term is fixed
By the condition that
The one point function so
Which if we wanted to compute it at first order in the coupling constant G would look like that
Is equal to zero?
So
Typically when we talk about these counter terms and we represent them
So these are interactions that appear in Fineman diagrams, and we typically represent them in phiman diagrams by a new interaction vertex
That's labeled by an X
So here I've written down the very simplest interaction vertex
Just describing Wi-Fi
which is just a single X for the interaction Y with one leg one particle leg coming off of it and
It's defined so that it cancels all of the contributions to the one point function coming from all of the other diagrams of the theory
So in practice of course we never need to actually compute the value of the coupling constant Y
So we don't actually need to compute Y
We just need to remember
That it will cancel
Any diagram that has only one leg that is to say any tadpole
So this actually has a very interesting consequence, which is something that we actually mentioned last semester, but I'll remind you of again
So it means that not only do the one-point functions all vanish because we've adjusted
the value of this parameter Y to make it so
But it means that in any Fineman diagram
Expansion of a scattering amplitude
We can ignore any diagram, which has the property that
If you cut it in two or
That is to say if you cut one line
Then it will form two pieces
One of which is not correct connected to any external vertex
So let's just
See that a little bit more explicitly so for example we saw that
The coefficient is the coefficient Y is defined such that this counter term
At order at leading order in the coupling constant G
will cancel
Will cancel the one loop contribution so that order G
So that means for example that if we were considering two to two scatterings
Then you might naively think that you would have say a 1 loop correction that looked like this
But you would also have a counter term involving the new coupling Y
that looks like this and
Then these two terms would cancel against each other
And so you can see that you can just effectively ignore any of these
diagrams on the diagrams that you get just by
Attaching on an extra tag pole or a tap taken one leg and attaching on some other
Fineman diagnose some other piece of the Fineman diagram
Is this clear more or less we actually discussed this last semester, but I wanted to return to it
Just to remind you of it because it's a another very simple example of how we use
counter terms in order to enforce physical conditions in
The Fineman diagram expansion and in this case it's a very easy physical condition to enforce just the condition that there are no tadpoles
Okay, so this is sometimes called as the no tadpole result through the no tadpoles theorem
Any questions on this
Okay
so
We saw that enforcing the physical condition that the one-point function vanishes as a condition on the
coupling constant Y is very easy
But for the other counter terms
We actually have to do a calculation
And
So starting next class what we're going to do is we're going to develop some of the
technology to compute some of these loop diagrams
And we're going to start computing some of these other counter terms in Phi cubed theory to start
Explicitly, okay, so we'll start by enforcing the condition that
all of the physical excitations have mass m and that
The one-particle states are normalized appropriately and see explicitly what that means how we enforce those conditions on these counter terms
And in practice often we don't end up
actually computing the values of Z
Order by order and perturbation theory, but instead just as we did in the case of the tadpole diagrams just
Noting that by enforcing these we
Enforced some sort of physical condition that we can then just impose directly on our results and as we'll see this will automatically remove
Many of the divergences while all of the divergences that will appear in quantum field theory
So starting next time. We'll calculate what's called the self energy of the particle or the
Well really the self energy of the scalar field
Before moving on to study the self energy of photons
Dynamics
Any questions so maybe I should stop here. I guess well, okay, and I've run out of time
So maybe I should just stop and see if there any questions
Any questions, this is more or less clear so this class was a bit of review, but next class will start into
some new stuff
Yes
Yeah, well gee now in five cubes theory G is a dimensional constant
So well we'll get back to how to deal with dimension for constants in fact is easier all Z
Disease disease or dementia
Yeah, well I'll get back to that
Yeah, I mean the dimensional. Okay, what we're gonna spend a lot of time doing dimensional analysis in fact. Yeah
normalization that is basically dimension well
There's a lot. It's basically dimensional analysis, but there's a little bit more
There's a few integrals one has to do as well, but it's mostly dimensional analysis
With a strong knowledge of dimensional analysis you can do just about anything
Good any other questions
Good, I will see you on Thursday
