Um well, it's a really a tremendous

pleasure and honor uh to be here and I'd

say a surprise. Um you know, if you

asked me uh 10 or 15 years ago whether

I'd be giving fancy cow lectures to a

group of distinguished mathematicians,

some of you are distinguished for sure.

Well, there's one at least. uh in

Leipzig I would have thought you're uh

totally nuts. Uh I'm obviously not a

mathematician. Uh I'm not even a

mathematical physicist. I'm a physicist

who loves math. That's not the same

thing. Uh um but uh this has happened um

15 years ago. I've spent uh two years of

my life where my dominant collaborators

were experimental particle physicists at

the Large Adron Collider. And somehow

it's come to be that I've spent the last

10 years of my life where my dominant

collaborators, my most exciting

collaborations have been with

mathematicians. Uh and this is a

reflection of the grandeur of the

physical universe and the platonic

mathematical universe and their still

mysterious and wonderful connection uh

that it managed to uh bring these groups

of people together. I I suspect it would

have also been a shock to a

combinatorialists or uh you know

algebraic geometers to think that

anything that they're thinking about

would have to do with very basic physics

you know fundamental very basic process

that happen in the world all the time

you know not uh esoteric questions about

the partition function of complicated

theory X on complicated manifold Y but

what happens when things are going on in

the real world right outside our window.

Um it's remarkable. It's a surprise and

uh so um I'm I'm delighted to be here uh

to be able to say something about what's

going on. So indeed that's what's going

on. There's been over the past 15 years,

really longer, but from this sort of

particular point of view that we're

talking about for the past 10 or 15

years, uh accelerating uh through to the

present, some indication of a deep

connection between very basic physics uh

physics having to do with the scattering

of elementary particles, more recently

physics having to do with uh uh the

creation of structure in the universe

and cosmology on the one hand and uh I

would say what seems to me as an

outsider very basic in deep parts of

mathematics. Um and uh another

interesting aspect of this uh connection

is it's uh not like at least the most

stereotypical version of the interaction

between math and physics that that we

learn about in uh you know in school in

in lay history. Um the the stereotypical

interaction is either that math is a

century ahead of physics. uh

mathematicians come up with group

theory, they come up with differential

geometry and 50 or 100 years later uh

the physicists find this exactly what

they need in order to make progress in

physics. Nor is it the other way around.

That's sort of been the situation I

would say with the interaction between

quantum field theory and modern

mathematics over the past 40 years that

because physicists uh well namely Edward

Whitten but some other physicists also

have an amazing intuition for what the

path integral does uh that leads to lots

of amazing mathematical predictions that

then mathematicians have to go try to

understand from their own way. So if

there the sort of physicists are ahead

in some sense and the mathematicians are

catching up. This has been a different

subject. This has been a subject where

somehow again rather mysteriously

physicists and mathematicians are

running into roughly the same structures

roughly simultaneously and so it's been

actually good not for bureaucratic

reasons. So you can say we sit together

in the same room and have

interdisciplinary conversations but uh

when it's actually useful it's actually

useful to uh uh to to uh to to to

discuss the kind of the same objects

being seen from different points of

view. And it's been a persistent and

wonderful shock to me over and over

again in a number of different settings

how how this has happened.

All right. So um uh so anyway that's the

uh that's uh the topic for these uh

lectures is tell you something about

that. Now before I proceed I just want

to have a rough idea just a show of

hands. How many people at least before

this morning's lectures how many people

knew what a positive grossmanian was?

Okay excellent. Very good. How many

people uh know what spinner holicity

variables are? Okay, fewer. Momentum

twisters.

Okay. Um

um that's good. Okay. So um so th this

this uh this talk is not going to be uh

targeted uh at the experts. I'll have

some things uh I'll have some things I

mean this set of talks not going to be

targeted only at the experts. Um and I

don't have any very fixed uh I have a

rough set of things that I want to talk

about but um it it'll be uh

exponentially better to uh have uh have

a real interaction. So stop me to do

concrete examples. Anything you want to

do we can slow down arbitrarily and talk

about arbitrarily elementary points. I

love nothing more than elementary points

because I understand them well. So it's

enjoyable to explain things I understand

well. Um uh so please do uh uh stop me

with the uh questions about any subject

as we're as we're going.

Okay. So um so let me just begin by

setting the stage and I'll I'll then get

in a bit to what the lectures are going

to be about. But a picture that you've

seen uh a number of times um uh to at

least begin with we're talking about the

very basic physics of elementary

particle uh scattering processes. So we

have scattering uh processes and

amplitudes

and their amplitudes.

Okay. And uh the sort of most basic

picture is you might imagine sort of two

particles come in. So there's some

instate uh something happens and there's

a bunch of outstates.

Um and so uh this this first picture

well looks like what it is. It's a it's

an evolution of of a process in space

and time, right? Things start at minus

infinity, something happens and you

measure something at uh plus infinity.

The world is quantum mechanical. So, uh

when you uh when you prepare exactly the

same initial states, something different

comes out every time. So, we can't

calculate uh uh we can't say exactly

what happens next, but we can calculate

the probabilities for uh for this

particular process to happen. The

probability of in to out is the mod

squar of this amplitude of a scattering

amplitude for in to out.

One of the very very first things that

you do in textbooks uh when you learn

this in in in grad school is already uh

change this formula so that you don't

actually have a distinction between

who's in and out. So one thing that we

say is that uh we have energy and

momentum conservation. So, so each one

of the particles I'll denote the

particle label by a. A will be one to

the total number of particles involved

in the scattering process. And let's say

if we're in four dimensions, these would

be four vectors. So these have an energy

component and a spatial momentum

component.

And if the particles have a mass m,

these are constrained by

this equation. Uh that the particles

might even have different masses. So

that's the that's Einstein's formula for

the energy of the particle uh given its

momentum. This is sometimes called the

mass shell because uh as you can see in

energy momentum space there's some

hyperbooid. Okay. So you have to lie on

a hyperbooid in uh energy momentum uh

space. So this is called being on the

mass shell. So the particles are on the

mass shell and we also have energy and

momentum conservation. Let me just write

it in the small here. that says the sum

over the P ins

uh is equal to the sum over the P outs.

Okay. Well, one of the first things that

you learn when thinking about uh

amplitudes is that uh we can um let me

write this up here. Uh we can trade this

for a function that only depends uh

democratically on a on a bunch of p

where we don't have to say who's in and

out. Okay, so there's a function that's

actually an analytic function of an

analytic enough function of the momenta

where you just get to decide whether pa

zero. If the if the energy component is

less than zero by conventional we say

it's incoming or outgoing and if it's

bigger than zero we say it's outgoing.

Okay. So right from the start you

actually deal with a completely

symmetric function right at least we

don't don't make any distinctions

between in and out and you get to decide

who's in and who's out by controlling

the uh the sign of the energy component.

OKAY THAT'S ALREADY A NON-TRIVIAL FACT

that this is true. This is a statement

about what this function looks like. It

has it's analytic enough that you can do

an analytic continuation from from some

set of one set of signs to another set

of signs. This is a property that's not

obvious. It's called crossing symmetry

in quantum field theory. It's not

obvious uh but it ends up being being

true. It has very uh not uh satisfying

proofs. Uh but in any case, we're going

to just take it uh take it to be true.

When you work uh in perturbation theory

uh and especially at uh in the simplest

tree amplitudes, it's an obvious

statement about the what the functions

look like. They're simple rational

functions. This sort of analytic

continuation is obvious.

But right away I want you to observe

from here something interesting has

happened in in this is the year 1950.

Okay that that this idea of the

amplitude is a matrix that takes you

from the past to the future is already a

little bit dissolved right you know so

it does not already does not have an

interpretation necessarily of an in to

out. It's something that's uh has all

the particles on an equal footing. Okay.

So this is the object that we're going

to be talking about and often we label

it by the uh number of particles

participating in the scattering process.

Now one of the famous things in this

subject is that the amplitudes are I'm

going to use this thing. So um we can uh

the the the sort of textbook picture for

for calculating amplitudes is using

fineman diagrams. Okay. So, so we draw

these pictures

um

where again you know what what's

happening I I want to stress this point

uh maybe mathematicians will appreciate

it more than physicists do um because

vis have just gotten so used to thinking

about these pictures um but what's

really happening is the experimentalist

wakes up in the morning in Geneva

collides a bunch of particles goes and

has their espresso and comes back in the

evening to see what the output of the

collisions is right that's what's

actually happening. In other words, the

measurements are done in some uh

idealized way at infinity infinite past

and infinite future.

If you then ask what IS A STORY FOR WHAT

HAPPENED? HOW CAN I COME up with a

theory that explains what what what

happens? That's when the theoretical

physicist comes along and says you know

what's happening. These particles are

moving through space and time and this

crashed into that one and this made this

one and that went that way and this did

that and this did that. Right? This is a

story that theoretical physicists give

to give an accounting a rational

accounting for what the outcome of this

process is. Okay, nobody sees this,

right? None of us are riding along with

the proton at the LHC to see what's

happening with the quarks and gluons

inside the proton as they smash into

each other. Okay, but it's a story we

tell. It's a true story. It's a correct

picture of what is going on. But it is

we're now learning one story you can

tell for what is going on. Okay. So, and

the purpose of this story, the purpose

of the textbook formalism is to make

manifest two very important properties.

The two important properties are the

two, you know, big principles of modern

physics. We want to uh manifest that we

have a picture of space-time processes

that are local.

In other words that we don't have

interactions between random points

separated in in uh in space and time. So

locality is one of the big principles uh

Lorent invariance the picture of special

relativity and locality is one of the

big principles of uh modern physics and

secondly quantum mechanics. Okay. So,

uh, and this is where Richard Fineman

taught us that the way to think about

quantum mechanics is by drawing all

possible histories and summing over all

possible histories. Again, in a

classical world, there would be a single

history for what happened. Classical

physics is deterministic. Quantum

mechanics is not. And so, you have to

sum over all the possible histories for

what's happened. That's why we have to

sum over all the uh diagrams. So, the

sort of technical word for this is

locality. And the technical word for

this is unitarity. Okay? that that that

there there there's a matrix this this

amplitude if I go back to in to out this

amplitude for in to out is the uh uh is

the

is some uh matrix or an operator that's

sandwiched between the in andout states

and unitarity the statement this has a

consistent interpretation as quantum

mechanical probabilities that says that

this matrix has to satisfy that it's

unitary that saguras equals one okay so

those are the two big principles of 20th

century physics and um this textbook

picture is built to make them as

manifest as possible. Right? So when you

have the picture of finding diagrams,

setting overall diagrams, you then go to

you know grad school for a semester or

or a year and you learn how these

principles are sort of translated into

the rules that uh precisely take us from

these pictures to the final answer.

Okay. So what I want to stress is that

the standard picture is completely

correct. Nothing wrong with it. Um it's

totally universal unlike the things

we're talking about now which are not

yet totally universal. Okay? So we're

far from being totally universal. Okay.

Um but the reason to be looking for an

alternate picture is first of all I mean

just purely logically no one said I mean

we don't know all WE ALL WE HAVE ACCESS

to is in and out right? So it's entirely

logically possible that there's a

different picture for what might be

going on inside a different machine that

whose crank you turn to produce the

answer that does not involve drawing

these pictures. Okay,

we have various reasons for looking

looking for such a picture. We have

various uh conceptual reasons internally

within physics for looking for such an

alternate picture because for a whole

variety of reasons we expect that the

notion of space and time eventually

breaks down at very very short short

distances because of uh questions

involving putting quantum mechanics and

gravity together. We even expect for

very subtle questions involving applying

quantum mechanics to the entire universe

that quantum mechanics might have to be

extended in some way in very very

extreme situations. So these two giant

principles of the 20th century are both

under question. They're both likely to

be superseded in whatever next picture

of physics uh uh we end up getting. Uh

that transcension is going to be at

least as dramatic as whatever took us

from classical physics to quantum

mechanics because we have to figure out

how to get rid of spaceime and how to

move beyond the quantum mechanics.

And it's such a radical thing that if

it's true, it's very unlikely it'll

leave all the rest of physics untouched.

And so it must be there's al there must

be if if spacetime is not really there

if quantum mechanics is somehow

transcended it cannot be that uh that

when we think about these these simple

basic processes where there's no reason

to question the presence of a spaceime

no reason to expect quantum mechanics is

breaking down. Nonetheless if they're

not really there in in in the true

theory of nature there must be a way of

talking about these things also uh

without putting them in as sort of

hardwired into the description of the

physics. And this has happened to us

before in physics when we went precisely

when we went from classical physics to a

quantum mechanics. We underwent such a

transition. It seems as shocking as

possible to lose determinism. If you

tell a classical physicist in the year

1850 that in 1930 determinism is gone.

They'd be totally shocked. How could it

be the Newtonian clockwork universe is

working perfectly. Everything is great.

The industrial revolution depends on it.

Right? All this stuff. How could it be

that uh determinism is lost? And the

answer is in a subtle way. Right? U so

if you if you told that to a a physicist

in 1850, they might wonder whether

there's some way of talking about

physics in such a way that determinism

is not the star of the show. Uh that

they're getting this clue from the

future. They might wonder if there's a

way of thinking about physics where

determinism is not the star of the show.

And they're not going to guess quantum

mechanics. It's crazy that they're going

to guess quantum mechanics just from the

clue that determinism is gone. But they

can take uh inspiration and try to

recast classical physics in such a way

that determinism is not the star of the

show. Of course, it turns out there is

such a way of recasting classical

physics. It's the principle of least

action. The lrangeian uh picture of

physics. When you first learn it in

school, it's very surprising. It looks

like the particle goes from A to B, not

by slavishly following Fals MA at every

point, but by sort of sniffing out all

the paths it can take from A to B and

choosing the one that minimizes the

action. That's a way of talking about

physics that definitely does not look

deterministic on the face of it. Right?

Of course, it turns out to be uh

deterministic. It turns out to be a

rewriting of Newton's laws in some

cases, but it's a way of thinking about

Newton's laws where the idea of

determinism is not central. And of

course, we now know today that this

other way of thinking about classical

physic action is the way that uplifts

the quantum mechanics. Okay? So, uh not

Newton's laws. You know you don't begin

with Newton's law and do a small

modification to it to get quantum

mechanics. You do start from the

principle of least action and do a small

modification of it to find this picture

of summing over all the history. So this

is a mysterious thing about physics that

at any given moment in time the laws can

be cast in numerous different ways

starting from seemingly radically

different starting points. Some of them

are better suited for the leap to the

next level of description than others.

And so if you have a clue for what

you're going to lose in the future, it's

inspiration to go back and dislodge that

from your way of thinking now. Okay. So

this is very abstract, very highfulutin,

almost content-free seeming uh uh

motivation for trying to learn how to

fill this blob with something else. We

don't want to have a picture of diagrams

and space-time processes. We want to

think that there's some other machine in

here, some some question mark that we're

looking for that's going to give us the

answer without drawing pictures of

space-time uh trajectories. That's the

something that that we're looking for.

>> So, do you mean something like

non-connected graphs?

>> Sorry.

>> Do you mean something like a

non-connected graph?

>> It will have nothing to do with graphs.

>> Okay. So, we that's that's that's I mean

it's it's a big question mark. Okay. So,

we're going to fill that with uh with

something else, but we're going to get

more concrete about this as as as we go

on. Yes, there was a question I ignored

all this time. Yes, sorry.

>> Um, could you please explain a bit more

locality thing that you said and did you

at some point mentioned the change

between in and out?

>> What do you mean by that? Can you

explain this?

>> Yes. The the the the change between in

and out is just the following cool fact

that you can have let's say two

processes where particles A and B go in

and particle CDE E F go out. Okay,

that's one process. That's clearly

different than the process particle ABC

goes in, particles DEF, F go out. Those

are two obviously different processes.

The remarkable thing IS THAT THERE'S ONE

AND THE SAME function that calculates

both of them. Okay, there's just a

single function that depends on six

momenta P1 P A B CDE E F. And then

depending on whether the signs of the

energies, the zero components of those

P's are positive or negative, you get to

interpret that as a process where some

things are in and some things are out.

So you don't need reversing in and out.

You changing.

>> No, no, just changing in and out. Not

reversing in and out. Just just

declaring who's in and and uh who's out.

That's not at all obvious that those

things should be related to each other.

I'm saying at a super early point this

picture that all we're doing is making a

matrix that's unitary is somehow already

missing something very basic because the

object doesn't even have this matrix

interpretation really, right? It's just

something that democratically depends on

lots of uh uh lots of momenta.

>> What about locality? Well, locality is a

statement that uh you can't reach out

and touch someone, right? That uh that

uh that there's uh that that that things

can only interact when they're on on top

of each other. You can't pick up a phone

and talk to your friend in Alpha

Centtory uh as quickly as you like. Uh

and that's a very basic uh principle. Um

uh it's even a basic principle before

relativity, but but but acquires much

more significance after relativity when

you believe there's finite speed of

propagation for any signals. Okay. an

upper bound for the speed of propagation

for any signal. So fundamentally things

should only uh interact with each other

when they're on on top of each other.

Okay. All right.

So that's one set of reasons for looking

for a different way. I mean the zero

thing is that there might be a different

way. The first one is these highulutin

abstract statements about uh learning to

do physics uh without uh the rules of

spacetime and quantum mechanics as being

king. But the most practical thing is

that when you use the standard way of

thinking about things in terms of

refinement diagrams, it for sure gives

you the CORRECT ANSWERS. NO DOUBT IT

CORRECT GIVES you the correct answers,

BUT IT GIVES YOU THE CORRECT ANSWER, BUT

but it leads you to qualitatively

incorrect inconclusions about what the

answer should look like, right? And so

this is that's one what's been uh uh

alluded to a number of times is the

explosive complexity

uh of finding diagram calculations

and uh uh for very simple reason first

of all the number of diagrams depending

on the kind of theory we're talking

about either grows factorially or

exponentially with n. Okay. So can just

uh um uh the number of terms similarly

can grow factorially or uh exponentially

but very practically if we talk about

sort of a collision between two protons

at the uh uh at either at the LHC or

again just you know when a cosmic ray

hits the upper atmosphere and this is

happening gazillions of times a second

uh everywhere around us right so uh so

the the the the proton is made out of

quarks they're held together by gluons

uh Um so for example one process that

happens here is that one gluon from this

proton collides with another gluon from

this proton and something happens and

let's say two gluons go out. Um this is

something that you put on a problem set

and you know if the students don't do

well you don't take them as graduate

students. Okay. So um the next uh

problem is if you have uh two gluons

come in and three gluons go out. And

this you would be very cruel to put on

the problem set because this already is

like 30 pages of algebra. Just a

horrendous horrendous mess. You would

never torture someone with this

calculation. Uh you put four gluons out.

That was the 220 diagrams that were

referred to in one of the talks uh uh

this morning. 100 pages of algebra,

right? Five. If you write big like I do,

500 pages of algebra, right? Uh and

okay. Well, that's just what it looks

like. Uh you see not EVERY QUESTION IN

PHYSICS IS guaranteed to have a simple

answer and it's part of the chauvinism

of fundamental physics as to declare

interesting only those questions that

have simple answers and all the other

questions to call engineering. Okay. So

this looks like a classic example of

engineering. Right. However also physics

has a wonderful way of punishing people

who look down on engineers and rewarding

morally good behavior. Okay. Because

some people JUST HAVE TO DO THIS

CALCULATION. They they just have to do

it. They had to do it because

experimentalists needed the answer. By

the way, experimentalists need answers

for up to like eight gluons coming out

today if you want to like you know

interpret data from the uh from the from

the LHC. So this is not esoterica. These

are we we really need these uh to

compare theory uh and experiment.

Okay. But in this famous calculation by

Park and Taylor in the late 80s I don't

remember uh the year was it 87? Uh

anyway uh sometime in the uh late late '

80s people found an amazing uh surprise.

So these gluons are particles like

photons. They have a spin or

polarization. So you can have photons

that are positively or negatively uh

circularly polarized. Similarly, you can

think of the gluons as being positively

or negatively circularly polarized. And

the shocking result is that as it turns

out for any number of particles, if you

have the amplitude for gluons one

through n and let's say they're all

positively polarized or all negatively

polarized, the amplitude is zero.

Not at all obvious. Those hundreds of

pages are adding up to zero, right? Or

if one of them is positive and the rest

are negative, it's also zero.

And the amazing thing is if two of them

are uh negative then it has this famous

uh can you see down here or should I

maybe I'll use it I'll I'll do here

there is this famous result

that

uh by park and Taylor oops

this sort of one term expression okay

So,

so we know that that in fact in this

case

and I'll be more precise about this uh

uh in a bit but uh but if if two of them

are negative and everything else is

positive and they're negative in

locations i and j one two up to n that

this amplitude I'll define what these

symbols mean later. Those of you who

know what they know them know them, but

we'll be talking about them in a bit. Is

this like studying oneline expression?

Okay.

So, this is not this is not supposed to

be obvious. Okay. Um and uh in fact, for

five particles, uh nothing you can do,

right? At least two of them have to have

they're either all the same sign, one of

them is one, the rest or it's this. So

for for all those 30 pages of algebra

either give you zero or this incredible

one term expression and similarly the

few hundred 500 pages of algebra for

this elicitity configuration collapsed

to a single term and okay now this is

this is very very striking. Now back

then in the late 80s when this was

discovered this was not universally

recognized as the tip of a giant

iceberg. Today we know it's a tip of a

giant iceberg. Um and that's the it's

the iceberg that uh we want to be uh uh

exploring uh in these lectures. So but

we now know I mean you could have

already taken this as a strong hint back

then but we now know that this is

happening because while the standard

picture is definitely correct once again

nothing wrong with it. It leads you to

expect that these processes have 500page

uh long computations when the result is

a single term. Right? And so it strongly

suggests the existence of another

picture of the world, a different

picture of the world where the fact that

this is the answer is obvious. And

presumably there's no free lunch, right?

So what's not going to be obvious in

that picture? What's not going to be as

obvious anyway is that there's something

like spacetime and quantum mechanics

going on. Right? You make space time and

quantum mechanics obvious.

Thank you. Right? So if you're going to

see the oneline expression, presumably

the space time in quantum mechanics is

not going to be as obvious and it's

going to have to sort of come out from

some other set of ideas, right?

And ahead of time it's clear before you

embark on trying to discover what this

question mark is or this other way of

thinking about things are it should be

totally obvious ahead of time that new

physical ideas will be needed. That's

obvious physical question. You're trying

to reformulate physics in some way. I

think it's equally clear that new

mathematical ideas are needed. Why?

BECAUSE THE ACTUAL FUNCTIONS THAT WE GET

IN the end, no matter how you DO THE

COMPUTATION, THE ACTUAL FUNCTIONS that

we get for the amplitudes, they're not

these like Mickey Mouse functions.

They're not signs and cosiness or little

polomials. They're are very rich

complicated functions with all kinds of

there might be rational functions with

very complicated intricate patterns of

poles. They might be transcendental

functions with all sorts of remarkable

properties. They're normally the output

of this giant machine that's called

quantum field theory, textbook quantum

field theory. if they're going to be the

output of some other machine that cannot

be some Mickey Mouse thing. It also has

to have some, you know, interesting

growth to it to uh to be able to uh uh

uh accomplish this. Okay. So, uh so it's

clear that we're we're looking for

things that have uh uh both new physical

and new mathematical ideas. And I'll say

a little bit about what those uh ideas

are in a bit.

Um but

so let me just say what what our sort of

program is that we'll be talking about

in these lectures. Um so we want to find

you know uh sometimes people describe

uh this research as you know simplifying

calculations for amplitudes. Um and that

is does not interest me one iota. Okay.

I in fact if this question mark turns

out to be more COMPLICATED I WOULDN'T

CARE that much. uh uh the the fact the

answers are so simp shockingly SIMPLE IS

A CLUE THAT this other picture exists

but for me is definitely not an an end

to itself. I actually don't personally

care about these amplitudes that much.

Um

what I care about is the story they're

telling us about the principles of space

time and quantum mechanics. That's what

I really really care about. Um

um and so that's what we're what we're

so we're not looking to find some tricks

to make the calculations simpler. That's

incredibly important for getting

theoretical data, right? You know, had

these people not there done this

unbelievably heroic work to get to that

oneline answer. We would not have gone

down this uh this line of work. So

there's absolutely nothing wrong with

tricks. It's just not what I'm

personally interested in. It's not

tricks. It's not starting with the

expression and manipulating it, changing

variables until you get something a

little simpler at the end. The goal is

to sort of turn everything on its head

and find a different question to which

the amplitudes are the answer. That's

what we want to do. Okay. So, we want to

find so that's been the mantra for a

long time. You know, what is a new

question

to which the amplitudes are the answer?

Okay, that's what we're interested in.

So again we want to fill in this

question mark here right there's some

other thing which is going on to produce

uh uh crappy question mark okay to

produce uh the uh uh amplitudes

and okay now where is this question

going to live

right this question is going to live

you see the conventional uh the

conventional thing is to fill in the

interior here with space-time

trajectories Okay. So definitely we

should not have any picture of

space-time trajectories.

On the other hand, THE AMPLITUDES DEPEND

on things. They depend on the moment of

the particles. Those momenta are things

that you see at infinity. Okay. So the

amplitudes depend on something you can

call the sort of kinematic space. Okay.

So um

so this question that we're looking for

should live in the kinematical space

that that specifies the amplitude. Okay.

So we're looking for a question

in the kinematical space

that specifies the scattering process.

whose answer

is equal to the amplitude whose answer

is a okay

what is this kinematic space again the

the the amplitudes sometimes I'm calling

a sometimes I'm calling m sorry for

example in the simplest way they really

depend on n momenta

okay these are these on shell momenta

that we just talked about so pi in a

lorrenian variant we'd say pi^2= m^2 on

shell momenta

So this is my kinematic space. The space

of n momenta and four vectors. Okay. So

that's the first thought about what this

kinematic space is. Okay. So we we we

descend from these lofty heights with a

thud that we somehow have to find a

question in the space of P1, P2, P3 up

to P. Okay. We have to find an

interesting mathematical question in

that space whose answer is the

amplitude.

Right?

we refuse to talk about. So if I uh one

of the physical ideas here, let me put

it here.

See when you draw a finding diagram like

this, even the sort of simplest example

of a finding diagram here, you'd have a

particles one and two and three and

four. And here we'd have this

intermediate particle whose momentum is

given by P1 plus P2.

Okay? And so it's plain that uh it's

plain that if p1 squar let's say is

equal to zero the particles are massive.

If p1 square p2 square is equal to zero

this does not mean that p intermediate

squar is equal to zero is not equal to

zero in general. Okay that's why we call

these things in textbooks virtual

particles. They're not real because

they're not on shell. They're also not

real because they don't go click click

click in an experimentalist detector.

They don't travel over infinite

distances, okay, which real particles

do.

Fman diagrams are littered with these

virtual particles and that's again

they're there in order to make locality

in quantum mechanics manifest. They're

also directly responsible for the

explosion in complexity. Okay, so the

physical idea is to banish virtual

particles. We never want to talk about

virtual particles. We somehow want to do

physics where we never talked about the

notion of a virtual particle.

Okay, that's physical one of the one of

the physical ideas. And so that means

that the the space uh the kinematic

space in this case is just a space of n

momenta. Right?

Now in fact we're going to uh uh we're

going to

>> do a little more. Um let me just mention

a couple of natural choices of the

kinematic space.

But so if you think about momenta, do

you think like about spacetime in the

background or like I mean

>> no no these are just things that are

these are just things that specify

what's being observed at infinity. Okay.

So so uh so these you know these four

moments are really labeled by three

things right because they're they're on

shell. So so uh so that's right. Now

there's no there there's some space

here. It's not space time. It's not the

conventional space. This is just a space

of n4 vectors. Okay. uh that are each on

shelf in the space. There's no notion of

time. There's no notion of two things

being close together. There's no notion

of a metric. This is as arid and boring

a space as it seems like like you might

have. Okay. Um yes.

>> Sorry. Could you please elaborate why uh

the right question for you is to ask

what question the amplitude is an

answer? Well, uh, it's a it's a question

that I'm, uh, it's a question I I I try

to, uh, motivate. Um, but if you like,

it's trying to be as as closely tied

into the actual observable as possible.

This kind of philosophy has had a uh,

sometimes glorious, sometimes tragic

history in physics, okay? Where you take

the attitude to to stick to the

observables as much as you possibly can.

Sometimes it's been an excellent idea,

sometimes it's led people astray. I'm

making a bet this time is going to not

lead us astray and it's a good idea.

Okay, but we want to hold on as much as

possible to the direct observables. The

virtual particles you do not see all

those momenta inside the diagrams you do

not see. What you do see are the asmtoic

momenta. So that's the space we're going

to live in. That's the space in which uh

we're going to ask our questions. So

there's actually a few versions of the

kinematic space. This is going to be

loose. We're going to make it more

precise as you'll see. uh but there's

kind of this is kind of our canvas right

our canvas could be one it could be the

space of for momenta thought of

individually okay so this is really like

a space of you know four vectors it's

like a 4 byn matrix okay so my kinetic

my my kinematic space is a space of 4

byn matrices

we're going to have to ask a question of

the space of 4 byn matrices and get an

amplitude out as as an answer

possibility So this is kinematic space

one. Another sort of qualitatively

different possibility

closely related but not the same is to

take advantage of the fact that we know

the theory is lorent variance. So uh the

amplitudes should not I'm still being a

little bit loose here. They should not

depend on the individual momenta they

should depend on dot products of

momenta. The answer only depends on dot

products of momenta. So here the

kinematic space will be made OUT OF DOT

PRODUCTDUCTS OF MOMENTA.

OKAY. So it's very similar but in one in

one uh in this case the kinematic space

is like you know 4 by matrix d by a

matrix in this case it's like a roughly

n²

and

uh is that we've been finding in this

kinematic space a question

uh to which the amplitude is the answer

and they've really roughly come in two

waves. The first wave was something very

close to this and this is a story of the

amplahedron. Okay, so we're going to

talk about this

a second wave starting in 2017. So this

is going back to starting sort of 2009

up to 2013 or so. uh second wave started

in 2017 thinking about this as the

essential uh uh kinematic space and this

first of all connected to the associ'll

say more what this is about but uh which

developed into a more general story of

amplitudes associated with curves on

surfaces

or some sometimes what we're calling

surfaceology.

Okay,

in both of these cases, there's a sort

of a uniform picture for uh what the

character of the question is. And I want

to uh uh even though the the nature of

the questions end up being quite

different in the two cases. Okay. Um but

the sort of character is basically the

same. And this is going to be a cartoon.

Okay. But here we have our kinematic

space.

Okay.

And in the kinematics space there's

going to be a natural region P that you

can call the positive region. Okay. So

there's going to be some notion of

positivity is is going to make a crucial

appearance in uh in this story. So

there's going to be a positive region in

this kinematic space.

Uh and the scattering amplitudes sorry

the scattering amplitudes are going to

also be associated with certain

differential forms that live in this

kinematic space. They're not even going

to be top forms in general. there going

to be some lower dimensional forms that

live in this uh that live in this

kinematical space. They're going to be

some little d-dimensional forms. Uh

we'll say what little d is later.

But in this big uh kinematical space,

there's also going to be some natural

little d-dimensional subspace. Okay, so

there's some uh little uh uh

d-dimensional

uh subspace.

The intersection of this subspace with

this positive region P is going to be a

shape.

That shape is going to be a positive

geometry.

And this form you see the this form just

lives in the just lives in the in the

kinematic space. It is a form right

everywhere. But it's finally entirely

determined by the requirement that when

you take this form that lives everywhere

and you pull back to this uh uh uh

subspace, let me call it S SD. When I

take this omega that lives everywhere

and I and I pull back onto this subspace

SD, this is the canonical form

for this uh for this uh you can call it

the sort of generalized

which is the intersection of this

positive region with that plane. All

right.

Right. That's the sort of general story

which has emerged.

Now the story of the amplahedrin from

the 2013 PAPER DID NOT YET LOOK LIKE

THIS. In fact uh uh uh if we did not

hear about the B ample earlier today, we

would not have seen it. Okay. So so

there's been some developments to bring

things uh to the form where we can talk

about it in this language directly

formulated in the kinematic space. Okay.

Um and that's what I want to tell you a

little bit about is uh is uh sort of

conceptually what is going on and how

that how that works. But this story ends

up being exactly the same for the two

pictures. Just the nature of the

positive region and the nature of the

subspaces depends on whether it's the

amplahedron or the other story involving

uh involving surfaceology. But it's it's

always the same. Amplitudes are

canonical forms of positive geometries.

Where do those positive geometries live?

They live in kinematic space. What

determines them? that some interesting

intersection of a positive region with

some appropriately chosen uh sub spaces.

Okay.

>> Okay.

>> Excuse me.

>> All right. So um so with that uh with

that yes there was a question. Yes.

>> Repeat the motivation why you expect

amplitude in the dots of

>> oh yes that's just I mean this is again

somewhat loose. uh if you have the

particles with spin there's other uh

things as well you can talk about

polarization vectors but in general all

the data are various vectors or tensors

or even spinners that are uh that

transform into the lorren group but

since the amplitudes of lorren invariant

they only depend on lent in variant

combinations okay so if you only have

momenta then uh they only depend on

these dot dot dot products the main

thing that I want to emphasize is the

difference between things that depend on

the particles individually versus that

depend on pairs of particles

Okay. And you'll see when we do

particles individually, we'll be in the

world of brasmanians, amphetra, blah

blah blah. And here we'll be in the

world of surfaces, curves on surfaces.

Why curves on surfaces? Because a curve,

an open curve on a surface has a a

beginning point and an end point. Okay?

And that tuness of where it starts and

end is reflected in the tuness of the

fact that that our kinematic space

depends on uh dot products here. All

right.

Now so so uh so so with this very rough

guide let me uh just say what my plan is

for the lecture. So uh today what I'd

like to do is talk about the ample talk

about this story and we had uh a very

very nice uh uh uh lecture this morning

on on the cominatorics of the amplrin. I

thought maybe for this audience I would

explain a little bit where the ample

came from. Okay, where it came from, why

it's not such a crazy seeming thing. um

uh and you know what what uh what what

the motivations were just so you see

sort of step by step uh uh how it

emerged and also how it finally leads to

this picture because if you're familiar

with amplitude the y equals c.z picture

looks absolutely nothing like this.

Okay. So um so it was a number of steps

to get to that picture between 2013 and

2017 um that that's associated with the

picture uh that we heard about this

morning that defines the empahedron in

terms of these twister variables which

uh anyway have a very natural uh uh

interpretation in terms of this uh uh

kinematic space. Um and uh and uh

strangely enough only after that picture

for the amplahedron was developed in

2017 DID IT BECOME POSSIBLE TO SEE THIS

IN IN HINDSIGHT much much simpler

picture for these simple scalar

theories. Um that however was missing

the sort of key idea of looking for this

pattern okay of a positive region and a

bigger kinematic space and a subspace on

which the amplitudes uh uh reveal

themselves. Okay. So this really is a

kind of a common uh uh motif in this uh

uh entire story uh uh so far. So today I

I want to tell you uh a little bit about

the the motivation behind the aahedron.

Tomorrow I'll tell you about uh uh uh

surfaceology. We'll start again. We'll

think about the kinematics space again

and uh we'll we'll see uh the story of

uh surfaceology.

Um and uh and for my third lecture I

want to uh talk about um something else

which the uh recent developments with uh

uh over here have made um have made

possible something I'm extremely excited

about um is uh you you know all of these

pictures even these new geometric

pictures for amplitudes they have this

feature I'll I'll stress in a moment

there's something basically recursive

about them. Okay. So, so you build these

geometries. You know, we heard a

definition of positive geometry earlier,

but the definition of positive geometry

has some recursive quality to it. You

know, you go to a boundary, you you see

something that looks like a positive

geometry. Again, um and this this

recursive feature is kind of there in

all of these pictures that have to do

with amplitudes that one way or the

other. Think about amplitudes as very

simple when there's few particles and

you put them together, glue them

together to make more and more

complicated ones. We can think about it

in this super direct way with finding

diagrams. We can think about it in this

much fancier way uh which exposes it's

wonderful exposes all these things

positive geometries blah blah blah. But

still you know if you want to know what

is the million particle amplitude you're

not going to get the million particle

amplitude without first getting the

million minus one particle amplitude

million minus two particle amplitude and

so on. Right? It's so fundamentally as a

feature that you have to build something

complicated out of simple pieces. And

I've long sort of fantasized that uh uh

uh something really new about amplitude

should let you go the other way. Sort of

give you the amplitude for infinitely

many particles first. Kind of a second

theory just a formulation for what the

physics is and limit where the number of

particles is infinity and then give you

a systematic one over n where n is the

number of particles expansion to sort of

correct and bring back to a finite

number of of particles. And uh that's I

think what we're now seeing and I want

to tell you about in the third third

lecture. there's really sort of a

qualitatively different picture for what

the amplitudes are that gives you access

to a regime that we did not know how to

access before in physics where the

number of particles is like literally

infinite. Okay? And then you sort of

back away from that from a systematic

expansion. So that's the rough plan. But

again, I'm I'm in no rush. We don't have

to get through everything. Uh uh so we

can stop uh uh uh for as long as or as

little as uh uh you want on uh any of

these subjects. All right,

let's just stop there. I've barely said

anything, but in the little bit that

I've said, are there any questions?

I'd be surprised if there were.

>> Sorry. Can you say something about

measuring always at like negative and

positive infinity? Like what's the

meaning of that?

>> Yeah. Well, I mean, it's an

idealization. Um uh you know um but uh

let me just say uh practically speaking

when experimentalists collide particles

at the LA experimentalists are about

your size um maybe their brains are a

little smaller but uh I'm kidding.

I'm joking my favorite people in the

world. I love you experimentalists. Um

they're big. They're you know meter

scale creatures. They're scattering

protons that are 10 -4 centimeters big.

Okay. Okay, so the experimentalists are

huge compared to the things that are

being collided and also you know the LX

is 27 km around 27 km is nearly infinity

compared to 10 - 14 cm the size size of

the proton but more conceptually you

know whenever you do any kind of

observation you want to disentangle the

physics of the apparatus from the

physics that you care about. So that's

why you you put them far away, right? So

the apparatus is far away, you know, you

don't put the apparatus right on top

where all the collision is happening

because then the the apparatus is

interfering with what you're trying to

uh trying to measure because it's just a

very important part of the active

observation in quantum mechanics is to

really decouple the system from the uh

uh the apparatus from the system as much

as possible. So the experiments start

and end at infinity. Okay,

>> did that answer your question?

>> Yes.

>> Okay, any other questions? Yes. What is

the number d?

>> Oh, it's a number. It depends on the it

depends on the process depends on n. But

uh so if you specify your scattering

process, this d is going to depend on

that n in some way. Okay. So it it

depends on on on the context. Okay. But

there are some but the but but for any

particular process there is some

particular dimension uh subspace and a

way of choosing that subspace. So that

this is the answer. All right. And is

this subspace the boundary of the

>> No no no no no

>> the intersection with the

>> No no no no no I mean this subspace

intersects with the positive region it

intersects somewhere and this somewhere

as a you know as a region that has it's

an interior and has boundaries of all

co- dimension uh and the intersection of

S with uh this positive region P is a

positive geometry that's uh that's the

idea so so you then want to find a

canonical form for that positive

geometry Okay. Now, the cool thing is,

you see, um, uh, if you're it, if I just

told you loosely, amplitudes are

canonical forms of positive geometries.

It's great. But then you think a little

bit and say, well, what could that mean?

I mean, the amplitudes depend on a bunch

of momenta. But a positive geometry is

like this square, right? Or this

pentagon or this fancy shape with this

particular parameters. What does it what

could it mean that the amplitude which

depends on all possible momenta is the

is the form of like a positive geometry

and this picture solves that. You see it

tells you that the that the amplitude is

a form. THE FORM LIVES EVERYWHERE lives

just defined on the full space. But when

you pull back on a particular subspace

it has a magical property that when you

pull it back to that subspace it becomes

the canonical form of the positive

geometry you see on that subspace. If

you move the sub subspace around well in

detail the the shape of the positive

geometry will change the form will

change but the underlying form that

lives in the whole space is is uniform

is always the same and is literally

interpreted as the amplitude. There's a

little bit of an extra step to go from a

form to a function. Okay, I have not uh

told you that. So that's a little extra

thing how we how we convert these forms

to functions and the nature of this

pullback is going to tell you precisely

how to do that in all cases. Okay, has a

slightly different answer in the amplan

case and in this case but always in the

end of the day the amplitude is really

that object. Okay.

All right.

Okay. Now um maybe before uh

getting into more details I want to say

one more thing about what so um

let me say two more things and then

we'll start into

something more meaningful

maybe As I'm racing, I can say that um

we're going to spend some time kind of

like slightly moving around in what we

mean by this kinematic space. It can be

the space of momentum to begin with.

We'll we'll muck around with it a

little. We'll talk about spinner

holicity variables. We'll talk about

momentum twisters, twisters, momentum

twisters, but they're all just different

ways of labeling the uh end momenta for

the particles. Okay. So um and similarly

that pi.pj

space the pi.pj space naively it's a

space of n squ numbers pi.pj of course

momentum conservation which says that

the sum of the ps adds up to zero is

going to put a few relations on those

guys. So so it's a not quite an n square

dimensional space or an n choose two

dimensional space. It's a little lower

than that. And there's various ways of

coming up with nice bases for that

space. But we really what I want to

emphasize we're really talking about

those two spaces one way or the other up

to a simple basis changes. So it should

really surprise you that there's

anything you can do in these in these

seemingly stupid spaces that is going to

produce all the richness and complexity

of amplitudes. Okay. And uh just to say

now a big word that is uh important here

that that brings almost everything to

life. There are two words that bring

everything to life. The notion of an

ordering that these uh that these n

momenta are not handed in a randomly to

you but are naturally ordered in some

way. P1 followed by p2 followed by p3

and the existence of a cyclic ordering.

Okay, this is related to the fact that

all the physics that we're talking about

here is uh has to do with the scattering

amplitudes for particles with color.

We'll say that a little more precisely

in a second, but I just want want you to

get this in in your mathematical heads

right away that somehow without the

notion of a cyclic ordering, there's

nothing to do with the notion of a

cyclic ordering, poof, all of a sudden a

lot of things become possible. And then

also the word positivity. Okay. And

again as we heard in the in the review

about positivity in positories and

grasmanians uh this is very much

reflects something that if you're just

looking at you know general uh matroid

stratification impossibly complicated

any algebraic variety can be realized uh

in that world if you say the word cyclic

ordering you don't care about the

matroids are about linear dep arbitrar

linear dependence between vectors uh in

arbitrary orderings but you say You only

want to keep track of the linear

dependence between consecutive sets of

vectors. You give a cyclic order and you

say the word consecutive sets of

vectors. Then all of a sudden now

there's a structure. Now there's a very

rigid structure and this posi

stratification which also goes along

with the notion of positivity. So

somehow this very very basic thing which

was seen in the story of going from

random matroids to posits has to do with

ordering and and cyclicity and has to do

with the presence of color. uh when we

talk uh about physics that's kind of

responsible for what can breathe some

life into these seemingly very erid

almost either empty or too complicated

uh

mathematical spaces. Okay, so just uh

keep that in mind. Now something else

that uh I want to say is you know uh

virtual particles in the standard way of

doing field theory has a purpose in

life. Um and of course we said that it's

supposed to make uh space time and

quantum mechanics manifest but actually

does more than that. Uh it it does that

in um uh in a way that's very sharply

reflected in what the actual formulas

look like, what the actual amplitudes uh

look like. So let's say again we have

this sort of simplest process.

Maybe there's a there's a coupling

constant G here. If this vertex G is

nothing complicated, they're just scalar

particles. So the amplitude for this

process following standard final rules

would be g ^2 over p1 + p2 2 minus m^2.

Okay. So that's this momentum here. Uh

this uh intermediate momentum is p1 plus

p2. Okay. Um so

uh so that's that's what this very very

simplest uh amplitude looks like. And um

you might have there might be another

diagram which is this one.

So this would be g ^2 over p2 + p3^ 2

minus m^2. Okay. Now this example is

extremely simple. You can compute the

final rules. No problem. But let's say

you're super lazy. You didn't even do

this. Okay. Uh and someone came to you

and said, "I did the calculation. Here's

the answer. Uh can you check that

they're right or wrong?" Right? There's

a very easy way we can check whether

you're right or wrong. In other words,

the way in which uh locality and

unitarity are reflected uh are they're

reflected in very precise sharp

properties of the amplitude

and uh it's it's in the following very

simple way. Locality

this is true I should say this is true

for these simplest tree amplitudes. Um

I'll say a little bit about what's known

at uh loop level in a bit. Locality is

reflected in the fact that that that the

that you only have simple poles.

So in other words, only things that look

like some 1 over p^2 - m^2. You don't

have things that look like 1 over p^2 -

m^2.

Don't have that. You definitely don't

have things that look like 1 p2 ^2 - r

t^2. You don't have that. Okay. The only

things you have are simple poles and

these P's that show up can at most be

sums of subsets of the momenta of the

external particles. Okay,

so that's a statement of locality.

That's a statement in moment in momentum

space that the particles met at a point

and so the momentum of this thing that

they're making is uh is the sum of the

momenta incoming to them. Okay,

none of these other things are allowed,

right? So, so if someone said I did the

calculation and they produced something

that had this kind of term in it,

they're lying to you, right? Okay. So,

locality is where the poles are. The

poles are simple poles and are located

only in this spot where the PS are sums

of uh subsets of momenta.

Unitarity is the statement

that so this is this is spacetime

locality. This is quantum mechanical

unitarity is a statement that in the

neighborhood of the pole you have to be

able to interpret. You see this is like

a resonance right? So when this is going

to zero literally the amplitude is

blowing up. So it's like the swing

amplitude is blowing up when you go on

resonance. Okay. So the amplitude is

blowing up there's a pole. So as p ^2

goes to m^ 2

the amplitude indeed goes like 1 over p

^2us m^2* something times some residue.

But this residue has to be interpreted

as the product of the in amplitudes

going in and making this intermediate

state. And then the intermediate state

goes out and makes the out guys.

Why? Because you're getting a pole

precisely because the virtual particle

is becoming real in the standard

language. Right? This is now becoming

real. On the pole, this this PART IS

REALLY INTERPRETED AS THAT amplitude

now, right? It's not some unphysical

virtual particle anymore. This is a real

particle and so you have to be able to

interpret the amplitude as this

sequential process where you first made

the intermediate state then it

propagated out a long distance and then

it decayed. Right? It's propagating

along distance in position space is

reflected by the presence of that pole

in momentum space.

>> Okay.

>> So you sense one more time like how if

you make this cut somehow it wants to

take this.

>> Yeah. You see I mean the the the the the

the point is that you see it first of

all in the in the concrete computation

right as P1 plus P2 squ goes to M^2 this

is blowing up what's the residue that's

blowing up it's G * G well G is the

amplitude for just this part and this G

is the amplitude for the other part

right and this happens in total

generality whatever blob is happening

here and whatever's going on there once

that momentum goes on shell as far as

this part of the graph is concerned this

is as if you're calcul calcating the

amplitude for that guy. And this is as

if you're calculating the amplitude for

this guy on the other side. Right? So

that's why it's just obvious in the

that's the point of the standard

formalism to make it obvious that on

these poles the amplitudes factorizes

into the product of a left and a right

piece. Okay.

>> Could you explain again how locality is

related to simple poles? Yeah,

locality is related to uh uh simple

poles because this is really a sort of

translation from what's going on in

momentum space where we're thinking

about these amplitudes to position space

where the notion of locality is sort of

manifest. Right? So in position space um

uh all the kind of all the things that

uh uh either things are totally local in

position space or it can have things

like inverse square law forces say.

Right? Now why why can we have things

like inverse square law forces because

they actually satisfy local equations of

motion things like lelassian sca

potential equals zero right so because

you have local things like that the only

the only way you can get uh any momentum

downstairs is by inverses of things like

leloians which look like uh one over p^

squ okay um so the reason there's no one

over p to the 4th is that we don't have

like uh leloian squared in in our

descript description of the physics and

that has very good reasons that are also

related to a also related in some way to

a unitarity. But essentially if you

imagine any conventional picture of a

lrangeian upstairs um as two derivatives

ordinary equations of motion and so on

classically then when you compute

anything with them quantum mechanically

you'll never get anything other than

poles that are at positions one over p^

squ and the and those one over p^ squ

just come from inverting the uh the uh

the the two time derivatives if you like

in the underlying uh uh lrangeian in

position space. Did I answer your

question? Uh

>> yes thanks. Okay.

Okay. So, all right. So, so now um so so

so this is very beautiful because it

means that that uh uh if we're looking

for some sort of whisbang new question

in kinematics base uh to which the

amplitudes are the answer they have a

job to do. This question has to produce

functions. The functions have to have

poles in special places and only these

places not other places. and they have

to have a reason why on the poles they

factoriize into the product of lower

objects of the same sort. Okay.

So, um now what we've been uh what we've

been

what we've been hearing about earlier

today um in the story of positive

geometries and canonical forms are an

example of what that could look like.

You see, one way of guaranteeing that

you have objects that have poles in the

correct place and THAT FACTORIIZE IS

FINDING DIAGRAMS. That's just that's

where they came from, right? That's uh

that's exactly what that that makes it

as obvious as possible. Okay, but we're

trying to see is there another kind of

object that can give us the same uh the

same kind of structure. Okay.

And so um

you know if we if we think about you

know roughly uh the oops

we have positive geometries

and their canonical forms just uh

don't want to uh I'm not going to repeat

Aaliyah's beautiful lecture but let me

just uh say uh a little about it.

See on this side there are some there

are some real geometries, right? There

are some real geometries

maybe they could even have curvy

boundaries. Okay, but again the the the

the uh the idea is that that they have

boundaries at least loosely. We had a

more precisely more precise definition

before but they're boundaries of all

code dimension.

Okay.

uh the word positive is there because

kind of in the simplest cases like the

inside of a triangle, how do we talk

about the notion of an interior is

intimately related with notions of

positivity, right? If you want to talk

about the interior points of a triangle,

one way to do it is to take a positive

weighted average of the vertices to give

the convex hall of the point. So that's

where the word positive comes from,

right? Another way is is to specify a

bunch of inequalities that you have to

be on the positive sign of all these

lines. Okay, both cases the notion of

interior has to do with notion of

positivity. Right? All right. So on the

one hand we have these uh we have real

geometries uh with with the property of

having boundaries of all code dimension.

On the other hand we have these

canonical forms associated with them uh

which have this feature of putting dlog

singularities. So dx so locally they

look like some dx uh if it's an n

dimensional space locally uh they will

look like this near some near the

boundaries.

and with the with logarithmic

singularities uh

on and only on

uh the boundaries of P.

Okay. And already if we talked about the

simplest cases um um uh I'm just going

to recap some of the things that uh uh

quickly some of the things Aaliyah said.

Um but just just to get an an

appreciation how even very very simple

geometries can only give us rather rich

uh rich functions. Um if we talk about

the sort of we talk about the very

simplest case

let's just do this

which which Lee also talked about. Let's

talk about the sort of very simplest

case of uh an interval right we talked

about an uh an an interval. So let's say

this direction is x. Here's x= a. x= b.

What is the canonical form for this guy?

Well, it has to have logarithmic

singularities at x= a and xals b. So

what can it be? It's a dx over x - a

clearly. And dx over x - b.

But there's something important here. Is

it a plus sign or a minus sign?

It's a minus sign. So these are actually

oriented. It's a minus sign because if

it was a plus sign, this would also have

a residue at infinity. Right? So this

shows very simply already that uh it's

important that you only have residues on

the boundaries and only on the

boundaries, nowhere else. Okay? Um and

that uh and that that dictates the

relative sign. Okay. Let's do the next

example.

The triangle, right? So let's say I I

have a triangle. So this is x and y. So

here's y = 0 x = 0 1 - x - y = 0. Now

what is a canonical form for this guy?

Well, it's a two form. So it's going to

be dx dy. Clearly it should have a

singularity on x= 0, y equals 0 here. So

it's clear that we should have these

factors downstairs.

But now you see what's cool about this

again talked about uh uh uh uh this in

uh even more interesting examples. So

let's just quickly quickly do it again.

So, so this clearly has singularities

you know the correct code dimension one

singularities but you need more the idea

that it has singularities on and only on

means that if you take any residues you

should keep seeing the canonical forms

for the lower dimensional boundaries. So

let's say I take the residue here at x

equals z. So taking the residue at x

equals 0 just in this case just means

erasing dx overx and putting x= z there.

So if I take the the residue of this

form on x= 0, it's going to be dy over y

1 - y. So you see that's very

beautifully here. I'm at x= 0. And this

form is exactly this form for this

interval. Again, it's y 1 - y. Nope,

there's no residue at infinity. Goes

like dy over y^2 infinity. Great. So it

only has residues on and only on the

boundaries of the triangle. Again, you

can check the other ones, but already we

can have some fun with this example.

Even if we do something as dumb as a

square or a

I'll draw it like this. Every attempt of

my life, oh, I I draw them with parallel

sides. I've never not drawn them with

parallel sides. Okay. So, uh All right.

So, so let's say this is a this is a

square. And there are some I'm not going

to write them explicitly, but there's

some lines line one, line two, line

three, line four. Each one of these

things is like an a1x plus b1 y + c one,

right? So this line is where that equals

zero and so on. Okay. So what is the

form for this guy? Well, again it would

be vx dy. Now clearly it has the four

lines downstairs. L1, L2, L3, L4.

But now you see that this cannot be the

answer because this guy would would has

singularities all right when any one of

the L's goes to zero. But let's say I

take the residue where L1 goes to zero.

Then it will also have a singularity

when either L2, L3 or L4 goes to zero.

Okay? And some of them are good and some

of them are bad. So this is L1. And so

you see

there are these two bad points. So this

is where L1 and L4 meet. That's good.

That's that point. When L1 and L uh L1

and L2 meet, good. That's this point.

But we also have where uh we also have

where L1 and L3 meet which is here. This

is L1 intersect L3.

That's not on the square. Okay. So

similarly we have where L2 and L4 meet

which is also not on the square. Right?

And so that cannot be the right form. We

have to have something that puts a

vanishing residue here. But then that

tells us precisely what has to be.

there's this unique line L which is the

line that precisely passes through uh

this point and that point puts a zero

there and so that gives us the uh that

gives us uh the form you see already

something as simple as this has a kind

of rich interesting uh form it's not

just some trivial thing with the product

of all the factors downstairs um uh if

you want to do the pentagon

um then you have this thing right now

you have uh now you have uh this is my

pentagon But I have these sort of five

bad points outside.

And so now now the form

is again we'd have dx dy with the

product of the five lines downstairs.

But now we have to have something that

puts a zero on all of those five points.

And there's a unique conic that passes

through five points. Okay? And that's

the conic that you put upstairs in order

to get the economical form. That always

shows you how interesting this object

is. Right? already for a pentagon knows

about these conics that pass through all

these points and you put almost nothing

in other than that you just have a

pentagon and you start getting some rich

and interesting uh objects out. Okay.

Okay. So um so that's uh that's just

some uh some some feeling um and so one

thing that you you see here is that uh

the boundary structure of positive

geometries is therefore

reflecting the pole structure of the

canonical form. Okay.

So the basic mantra is going to be that

amplitudes

are canonical forms.

Okay.

And so uh so uh poles of the amplitude.

So poles of amplitude

are again of course poles of canonical

forms.

But this is then related to boundaries

of P.

So the amplitude is the canonical form

for some positive geometry P. The poles

of the amplitude are poles of the

canonical form and these are related to

boundaries of P.

And so roughly what we're looking for,

we're looking for some geometries if

we're if we want to find some alternate

origin, some alternate understanding for

where locality and unitarity come from.

Right? Remember, we've now translated

those principles into very sharp

properties of the amplitude. The

amplitude has poles in particular spots.

All those spots it has to factoriize

into lower amplitudes. Right? If we're

now thinking about amplitudes as

canonical forms of geometries, then this

translates into a statement that's

naturally a question that lives within

the world of these geometries. Now,

without making any reference to finding

diagrams and particles and all the rest

of it, they have to have poles in the

right spots means they have to have a

particular kind of facet that we have or

boundaries that we're going to sort of

discover precisely what what that should

mean. Okay? And they should factoriize

on the poles means what? It means that

on those facets the geometries should

look like either precisely or a little

loosely. We'll see the precise meaning

of it products of lower positive

geometries in that same world. Right? So

that's the kind of thing that we are

looking for. We're looking for that kind

of set of mathematical objects uh those

sorts of geometries that have the

property that when you go to their

boundaries they factoriize into products

of other things that look like

themselves. Right? That's the that's the

kind of structure that we are actor.

Okay. So if I yeah anyway I can can say

it just going to write it again

that is not immediately obvious from

refinement diagrams right there

factorization problem. It is. It is.

Yeah. Yeah. Exactly. That that's that's

that's what I was saying. Uh the the the

location of the poles and factoriization

is as obvious as possible in finding

diagrams when I say that as obvious as

possible. Uh for scalar theories, it's

100% obvious. Okay. U there's really

nothing more to say than what I said.

And it's simply that when you go I mean

in in finding diagrams, it's clear that

the only poles are propagators because

that's what they are. And then it's

clear when you go in the neighborhood of

a pole then you know as far as the left

part of the diagram goes it just thinks

that p^ square is equal to m^2. So that

part of the computation is just what you

compute if you're calculating an

amplitude. Similarly for the right part

it's exactly what what you compute if

you're computing an amplitude. Should

have taken the product of the two

amplitudes. right now. Um it's maybe

slightly less obvious when you have

particles with spin because now when you

when you're exchanging them, you're sort

of contracting their uh they have maybe

some vector indices. So those vector

indices kind of like sniff from one side

to the other. Um but anyway, it's a it's

a two-line argument to to essentially

see that you s still get factoriization.