So thank you very much indeed for that for
that introduction and for the forty and

for patient Thank you all for for
coming and for allowing me to speak about

Black History of quantum mechanics rather
than about Einstein and general T.V..

So.

What I'm going to be talking about
his sort of sides interested in so

the general pattern of
how theories change and

the received wisdom of this is still.

The theory that was put into
circulation by Thomas Kuhn it is one

hundred sixty two classic like the
Structure of Scientific Revolutions and

it's about paradigm shifts and
I don't think that fits

like the developments of modern physics
like very well so I worked on a paper

where I looked at a few episodes in the
development of special relativity general

tutti of quantum mechanics where
I want to push like another.

Another idea about how serious
change that that I use for

which I use a metaphor called arch and
scaffolds and explain to you and

I'll talk about like a very specific
like case study did a cover and

paper that has to do with the transition
from what is known as a yard on the rock

transformation theory to fund for
more months Hilbert Space.

So first let me say a little bit
about to about the general project so

this is cool like almost a decade
before he publishes The Structure

of Scientific Revolutions in a grant
application and so he says like science

that's not progress by adding stones to
an initially incomplete structure but

by tearing down one habitable structure
and rebuilding to a new plan with the old

materials perhaps new ones besides So the
picture that you know look it's Is that

what happens in a paradigm shift is
that the old paradigm gets leveled and

on the burning embers of the old
one you were wrecked to do one and.

So I don't think that is a very credible
picture of how science advances and

so to question that it's like our
the scientific revolutions we

don't dispute that you know like
the quantum revolution like it's for

a reason that this is called like
a revolution say for relativity are these

revolutions really as disruptive
as this image suggests and

even if you look at QUT himself
you see that he is very fond of

using these metaphors and they're not all
as there most of them are not nearly as

disruptive feste what he puts like
in that in that grant application so

the basic idea flecked The Structure of
Scientific Revolutions place with the idea

of it's like a political revolution but so
be a look at the chemical revolution to

date had century like very consciously
used the term revolution in that sort

of political sense for the changes
that he was going to bring about.

In chemistry so that's one metaphor but
could also like uses other metaphors to

explain how the rich change so in his book
in the late fifty's about the Copernican

revolution he describes like Copernicus
as sort of a bend in the roads but

so he says like if you if you follow what
happens after Copernicus you look at

Galileo Kepler Newton and you look
back you sort of you know you see that

it looks as if it started with
Copernicus but in fact what happens

is that there is a bend to the road and
if you look forward for a block.

You also she just like not just going off
into infinity but there is nothing really

if you followed all the ground nothing
discontinuous it just bits right so

that that metaphor is not at all
destructive and I think that it's actually

a very apt metaphor to describe
the transition from told me to

prove it could Copernican astronomy
another one that he's very fond off

is the notion of a stalled switch right
where you you know you look at it and

you see a doc and now you look at it and
you see a rabbit again you know there's

something very discontinuous about it but
it's not disruptive.

It's not that you ripped down the dock and
then construct the rabbit No you see

it differently so again like this
continuity but not but not disruptive.

So.

So now left of the question then becomes
like what about what about the quantum

revolution what what what what's
the transition from the old quantum theory

of board some befell to modern quantum
mechanics in the mid twenty's what was

said like so I sampled like
a few statements about this and

so some people say yes this
was very discontinuous so

I don't know what textbook is being
used here at the undergraduate level of

Minnesota we used this book but Griffith
Griffith writes in the preface of his book

he says quantum mechanics is not in my
view something that flows smoothly and

naturally from earliest the research on
the contrary it represents an rapped and

revolutionary departure from classical
ideas and similarly like the historian

of physics now held a crock of his book
about the history of the old quantum

theory towards the end he says matrix
mechanics grew out of what little was left

of the old quantum theory it's roots right
it's almost exactly that metaphor that

that who used in this in
this grant application and

then here so you know we have
like a lot of textbook writer and

historian of physics and he is somebody
who actually lift through like this period

my countryman Hendrik cost and
he writes in his you know well known

autobiography have pastored reality he
says between one hundred twenty four and

nine hundred twenty eight and so
he wrote paraphrased to develop a new

quantum mechanics swept physics like
an enormous wave tearing down professional

structures stripping classical
edifice of illegitimate extensions and

clearing a most fertile soil so
that if you're focused on world metaphors

you see that he's mixing as metaphors
a little bit but you see years also

the emphasis on like
tearing down right but

even here like the tearing down it's like
it's not complete right it's staring down

illegitimate extensions of edifice so
there's also like you can.

Statements.

Knowledgeable people who deny that
theory is that there is this element of

tearing down in this transition so
Max jammer who wrote like

the students databanks the best sort
of single volume of the conceptual

development of quantum mechanics in
one thousand nine hundred sixty six so

he writes in the rights of the Preface He
says the primary objective is this book

is to study off how in the process of
constructing the conceptual edifice

of quantum mechanics each states depended
on those proceeding it without necessarily

falling from them as a logical consequence
but this is very different from

the picture like you know out with
the old and in with the with the new and

here like you know one of the architects
of quantum mechanics Paul the rock

quoted in another book approvingly by
Olivia Donegal a prominent historian

of a more recent historian of quantum
mechanics wonderful book from C.

numbers to Q.

numbers from one thousand nine hundred
two he quotes the rock as saying the new

quantum theory requires very few changes
from the classical theory and then

adding sort of paradoxically these changes
being of a fundamental nature right so

that many of the features of the classical
theory to which it so it's attractiveness

can be taken over and changed into
the quantum theory so you see that there

are there's a variety of opinion on
like this question of whether or

not there was really sort of a destructive
change tried to select I lead very

much to the view here that it was not that
it was not the tearing down of anything

so now what could what did himself have
to say about the quantum revolution right

because like you know it's one thing that
he has to say the structure of something

bigger evolutions he also did a lot of
historical work just all the development

of quantum mechanics and they're
actually like he also does not endorse

this picture that you would
get from structure so the one

thing that he emphasizes is like a crisis
that's very much like part of his model

of a paradigm shifts rightly
the paradigm is abandoned after it's

a masses like a lot of anomalies and
you know would be and

people figure out like this is not
a problem with the theorists but

it's a problem with the theory and
then you know you're looking for

something new but when he didn't talks
about like you know how did we get out of

the crisis he talks about like well
this is a series of connected steps and

in fact like one of his colleagues who did
it like this this has this image of like

he was a radical change he
is very critical of that So

this is the other philosophers
like it were a lot of dos and

he says well you know his account
of the quantum revolution is

like a magician pulling a rabbit from a
hats like he doesn't want any any of that.

So the upshot that is for for.

Despite this this quote that I started
with the new quantum theory was not really

built on the burning embers of
the old quantum theory right it was

a sequence of like complicated steps so

this this this metaphor of tearing
down is not a very good metaphor so

I want to introduce like a different
metaphor and the better for us one of our

church that scaffolds that was we do
original title of my talk and sometimes.

A new theory is built like an arch on the
scaffold provided by the old theory and

once the new theory can support itself
like you know the scaffold like drops out

and I've worked on this like to show that
that this is like the case this is a good

way to understand like to your arrival of
special relativity general to be the envy.

Serious steps in the development.

Of quantum theory and

I'm going to give I'm going to talk
about the specific example of that where

unfortunately I picked this example
because there the metaphor gets caught if

it gets kind of messy and therefore
I think a little more interesting so

the image that you are looking at
the background disposals up here for

is to illustrate this metaphor so this is
the construction of what was originally

noticed the Strand Brits are just across
the fence if you early nineteenth century

which was then read a the Waterloo Bridge
to commemorate the battle it eight hundred

seventy days British worse than the story
at some time in the twentieth century and

rebuilt but it was sitting there for
wartime and

you see you know elect the wood scaffold
to deadlock the arch built on top of that.

Now there's different
uses of this metaphor so

the use that I'm most interested
in is that you use a scaffold to

built like a new arch but there's another
way that you can use a scaffold maybe

you could use a scuffle to prevent an arch
that you've already built from collapsing

this is just like the building metaphor it
turns out you know having looked at this

how this is used by scientists to have
collected some passages where people use

sort of similar sort of metaphors that
among mathematicians this second use is

actually popular and so is here is
Hilbert in a lecture in nineteen zero

five where he says the beliefs of science
are not erected the way residential

property is where the retaining walls
are put in place before wall moves on to

the construction and expansion of living
quarters size prefers to get it habitable

spaces ready as quickly as possible to
conduct its business only afterwards when

it turns out that the loosely on an even
late foundations cannot carry the weight

of some additions to the living quarters
the sides get around to support and

secure those foundations and

then he asked this is not of the fish and
sea but rotted the correct and

healthy developments so the concrete
example that I'm going to be talking about

is going to evolve like the great
mathematician John for Norma and

so she should perhaps not be too
surprising that you can tell the story

sort of both ways as for Lloyd mom like
building is own arch on the scaffold

providing by the physicists like
your Don and the rock or S.

football among providing like
a scuffle to prevent the rickety arch

off the physicist from collapsing
mathematical shortcomings.

So this is the case study looking at
like the transition from your Dom

to full right and remember like what I'm
after like in the big picture is select to

characterize this transition using a very
different metaphor then like you know

the paradigm shifts that most of you
will be familiar with from from who.

So I'm going to start sort of in I'm not
going to give you like a fool history of

quantum mechanics I'm just going to start
like in this period like in the middle

of one thousand nine hundred twenty six
where after a period there where it was

not a single theory that worked there
was now kind of an embarrassment of

riches there were four different
theories that seemed to work and so

they were like there was matrix mechanics
so this was like done by Heisenberg by

borne by your done a good thing
in there is like wave mechanics

by Schroedinger in Vienna there is
cute numbers by Paul the rock in

Cambridge not completely independent of
the basics mechanics but different and

then like the least known of the four
it's of the operator calculus of born and

Norbert Vader Witchboard worked
out when he visited with

her at MIT it early twenty six so
that's the situation and

it was already getting clear at that
point that even though these theories

look very different that they're connected
to each other and Sol Schroeder for

instance had already proved that it
all the interesting applications

like matrix mechanics and waste mechanics
you have to say predictions it also become

clear that this formalism calls for some
sort of probabilistic interpretation right

that this is the work by born at board of
course later one of the Bill Bell price

for what was not clear was OK these
theories are related to each other but

what is the underlying formalism that
connects all these four different theories

together and the other thing that was
not clear is like what is the general

probabilistic interpretation of the
formalism of born it only done this in.

Very specific case it
pollution processes so

this is where where late
one nine hundred twenty six

early one nine hundred twenty seven in
the pen of Lee of one another the Rock and

your Dom proposed like
very similar theories and

I'm going to be focusing on
the way that your done did

this because your dog was Jordan's version
was the one that really influenced for

most in formulating you know the formalism
that we now work with like the Hilbert

space for bows so Jordaan is often
called like sort of the unsung

hero of the development of quantum
mechanics and quite a few theory and

he mostly has himself to blame for this
because like he's the only sort of first

rate German theoretician who fell in very
deep with the Nazis OK He was a very

a to say yes to a supporter of the
ideology even though you know lucky Devore

helped to be eighty eight eighty shape
before black he spent the time in pain and

where he where he where they were building
rockets but as best as we could tell

your dad was writing a textbook
it's despicable Haddix it didn't

help much with the Rockets Heisenberg who
hated the Nazis actually helped him quite

a bit by heading up the bomb project but
Heisenberg was not a doxie right he was

like a very right wing German nationalist
but not a Nazi Jordaan really was.

So but if it hadn't been for
that like your dad were probably be quite

famous because like he did the bench
a very important contributions

to settling some interpretational
issues about quantum mechanics and

to get like quite a few theory off
the ground right to it should i was so

shape with the rock but
probably the way we do quite a few

theory today is closer to what Jordan
was doing that what the rock was doing.

But as I said the guy only has
himself to blame so what was so

he publishes the space that annoy
you have the good do to quote Mikati

like to do foundation of quantum
mechanics to Germany to stalk will be so

simple that you can understand it if you
don't even if you don't know any other

if you don't know German and
the basic ideas.

This it's and
I put in some disclaimers here that.

So I a doing it talking about your Don
absented his atrociously bad notation and

I suppress a Glock saw some complications

especially something called like the
Against to do supplement the supplementary

applet dude you know which is best you
know when it is best like Drew and

best to draw the veil of
charity over that production.

The other thing I should say is that
your dad initially only considered black

quantities with continuous spectra and
was hoping that the extension

to quantities was discrete Specter or
mixed continuous

discrete Specter was going to be very easy
that turned out to be very difficult but

OK with DOS to disclaimers
he had this idea for

a new foundation of quantum mechanics
therefore just based on two basic

ideas first of all the quantum mechanics
this is a theory about it's always

a theory about conditional probabilities
it Quantum mechanics tells you like you

know if you if you have a system
prepared such that some quantity B.

Big B.

has some value little B.

What is the probability that some other
quantity a hassle value little a bit and

so that's the first idea the second idea
is that those probabilities are given by

the absolute squares of complex
probability amplitudes but so.

This probability is like
you know the amplitude for

the probability of finding a given B.

you know a dead you know luck of course
like we're talking about you know we

prepared to be what is the probability
that falls like in a small range for

four A So this I should say like
a the region a through a plus D.

A and so so this may sound
like a little of familiar but

they are all familiar with
the please what example of this and

these are elected energy eigenfunctions of
the time independence Rodier equation so

it was the mention right so here you have
to have all told and acting all this

eigenfunction spits out like the same
thing with like the eigenvalue

an and probably sick interpretation
of this which is actually do or

do not to bore but to poly is that.

If you if you take the the absolute
square this the disc

issued a probability of finding
the system at some position X.

if you know that it is
having luck in energy.

So this then is an example of this that
your dog generalizes And so if you think

about this to specific case as you I
could function to destroy your equation

you would write it like this right it's
the probability amplitude of fighting X.

given up and that's that's that's it and
he just generalize that for

a whole bunch of arbitrary quantities but
keep in mind that like

it only really works if you're talking
about quantities with continuous spectrum.

I should also say that at this point it
had already become clear that you can

write way functions configuration
space you can write any momentum space

you can write it like any space you
want but that that people have had to.

OK so now so these are the basic ideas
of the theory now your Don was very

sympathetic to mathematics so he gave
like an axiomatic set up of his theory so

what he's going to do is introduce
like a set of postulates for

his probability amplitudes and
then provide like a realization of these

postulates and so this is described
very nicely in a paper by Hilbert for

a new mom and lo thought more time about
your doubts theory that they're right that

they only came out in twenty eight but
they wrote of the nineteen twenty seven so

they stay explained a strategy it's like
what imposes certain physical requirements

of these probabilities which are suggested
by earlier experience of developments and

the satisfaction of which calls for

certain relations between
the probabilities

that's part one then secondly one searches
for a simple analytical apparatus

it which quantities occur that
satisfy these relations Exactly.

So now your dog years
after the fact remember

like an exchange that he had to infest
were aired fests like sort of porting out

like you know this this this unusual set
up and said look you know look if you

tell me first what quantities you're going
to use which are going to be Lector answer

mation major cities that are then going
to double as these probability amplitudes

rather the other way round that would be
a lot easier and so he told your Dot I.

Remember this and told it said it is if
you that F.S.S. like well since you wrote

the paper actually magically that only
beans that one has to read back to front

so I'm going to follow
the presentation of your dog but

like you no luck at the end of the day
luck these probability amplitudes

are going to be identified with
transformation matrices which

is why this theory has become known as
the statistical transformation theory.

So the first postulate is that
the basic probability amplitudes for

position and momentum is given
by this heat of the minus I P.

Q.
over age bar and

that's not going to going
to prove that for you but

it turns out that that luck implies
the usual computation relation so

you don't have to make that as
a separate separate assumption.

So good to.

The one thing I want to point out about
this is that if this is your probability

amplitudes as your don't recognize step
beads that for a given value of Q.

all possible values of P.

are actor probable and
this is sort of the gist of luck with

later hands becomes like to assert
the principle and Heisenberg like not

coincidentally was heavily influenced
by this paper by by your daughter.

Now what you are not noticed is that
this probability amplitudes satisfies

trivially like some these differential
equations right so multiplying by P.

is the same as minus H.

bar over I.D.D. Q.

you can see that right here and
similarly like multiplied by Q.

is the same as minus eight bar over I.D.P.
So these things as zero and so we think

that that will allow you to start from
this probability amplitudes between B.

and Q.

and then work your way up to probability
amplitudes between all sorts of other

quantities and so
I show you how this works so

think of flecked introduced
to do quantities A B.

that are related to being Q.

by like a canonical transformation
you know so it's basically

think of these as Matrix C's that you're
going to write sandwiched between T.

and T.

T.
minus one you get something new say with

B.

and the idea is that it's
dead that will give you

luck the probability
amplitudes between A and B.

and.

So as your poured sound like it is it the
same with the few that I just quoted from

you just get out of the transformations
were a daily breath so to tie in the new

results with those as closely as possible
that was something very bashful for

us to try but you see here is that
this is before the days of Hilbert

space that's your doubt is trying
to stretch the classical theory

like as far as he can to get a handle
on these quantum things right then so

it doesn't quite work but
will see that it inspires like for

blood to replace it by the basket
will do what your this tried to do so

here is to give you an idea as to how
this works and also to give you an idea

as to the limitations of this framework
is like start with these trivial.

Trivial.

Equations that we have before
sticking a few decent T.

by this one right so

this is just one I could if if this
acting all this is zero adding a T.

Here is also going to work and you see
here that what you have here over here is

a new quantity and he have a new way of
put two pts right same thing over here so

he thought OK this way we get
from the basic amplitude for

you to the more general amplitude for a B.

hopefully among them like an amplitude for
X.

and event that which is the solutions
of the Schroedinger equation.

So now unfortunately and
again I'm not going to show you

this quickly I'm not going to go through
through the calculation idiot rest of top.

You actually you can't do this and
basically the problem is that

these canonical transformations
just preserve the spectrum so

flashes up for us explain this if you
could think he said quickly like so

much the better otherwise don't worry
about it all you have to trust me on this.

So.

The point is is that you could if you want
to get to a situation but with energy

typically the energy is going to have
like a discrete part of the part of this.

Ector so you will never get there for
a luck piece and

queues with continuous spectrum right
because the spectrum is just preserved and

so your god like you know it after
he did realize this right away but

when he started to like extend his
approach to deal with situations with

discrete specter he read it
to this to this problem so

the rock didn't have this probably
just like the rock was not as hung up

on getting everything from
a few postulate he was happy to

postulate more equations writes as if
you do it that that it did you fly.

It so above or only valid so
this was the first boss as you

can already see that it's problematic the
second postulate is rather straightforward

it's a symmetry property that
the probability amplitudes for

be given a is just a complex budget of
the probability amplitudes for a given B.

and that implies because like he's
probably to get the probabilities you

square it that the probability of be given
a it's the same as the probability of

a given B..

So then he asked a third possibly this is
actually the most interesting wanted this

is what he calls it
the Ference of probability for

reasons that will become
clear in a little bit and so

what what's your nonsense here the strange
thing about quantum mechanics is that

the ordinary rules of probability do not
apply to the probabilities themselves but

to the probability amplitudes and
these ordinary rules are the addition and

multiplication rule it probably
is the theory right so let F.

one F.

two be two outcomes with
probability amplitudes five one and

five two then the multiplication
rule says that if those two things

are independent at the Apple Apple two for
getting F.

one F.

outcome F.

one A and F.

two is just the product of these two
amplitudes and IF IF IF IF one and if

two are mutually exclusive the probability
amplitudes for the outcome if one or F.

two is going to be to some of
these these two amplitudes and

you can sort of understand why you
would call this the interference of

probability theory
because now let's look at

what is the probability that the
probability amplitudes of probability F.

water F.

two Well that's the square
of this year and

you see that you're going to
get like a probability of F.

one the probability of F.

two plus a bunch of it appears there are
less you probably like point here look up

front if you need to look over there so
I can

look at you would have screamed roughly
at same five but you see it here right.

OK.

Now.

So what's what but

your time now ask himself it's like if
you have a probability for a given B.

and for be given see what is
the probability of a given C.

and he says like the rule there is that
this probability amplitudes for A C.

is going to be this into Grohl over B.

flecked the probability of amplitude of
a given be at the probability of amplitude

of be given c And so

what your dad's claim is that he could
derive this expression from these

postulates about like the the rules
of probability are applying

not to the probabilities themselves but
to the probability amplitudes now that

derivation is shaky but if you look
further you see that he doesn't really

what really takes over the role of the
three postulates are these relations and

these relations made up look very
familiar to do you doubt but

they will like it just a few slides of
the days these relational is true today as

they were back in your jobs day so he
looks at one special case namely when a C.

and A are equal to one another and then of
course you know what is the probability.

Of a given that it's a prime
Well you know what you know for

sure is that that's going to be
zero if a prime were different so

they're lucky essentially So
he does not use it but this is

the point at which the rock introduces
the famous the rock delta function OK so

your not sort of talks around this but
essentially does the same C..

And so as I said like these
the postulate da boils down

to just have a decent
relationships of the.

Least of These probability amplitudes now.

If you are like thinking
like well these these these

these postulates look like very strange
you're a very good company because like.

Polly there's a letter of his' of birth
to Polly when he first starts reading

your doubt stuff and he says like I
could understand your doubts paper

the postulates are so intensive all of the
FIDE I cannot make heads or tails of them.

None the less Heisenberg like
freely admits that he heavily

relied of these papers by yard on
writing to famous paper that he wrote

of the uncertainty principle but so so
he somehow you know like was able to do.

Understand this eventually.

So River so
now we have like we have like a bunch of

relations that these probability
amplitudes need to satisfy So

the second part of this action about
approach is now to identify some

mathematical gadgets that extent place the
role of these probability amplitudes and

satisfies these various conditions that
we have put on it and that is going to be

done by transformation matrices that are
written like this and here are borrowing

just your thoughts notation is absolutely
atrocious this is the notation from

the rock that is of course like it is
very reminiscent of the bracket notation

of the rock but in fact like you know what
the rock is doing this is that he twenty

seven he doesn't have Hilbert space he
doesn't think of breaking this up bras and

gets where it is just one unit that
is doing the transformation so

this is the transformation matrix to
get you from like a way function in

the space to a way function in a space and

this is just how you write it and so
but with that identification of D.C.

these things if this if you
identified this as the probability

amplitudes of a give a B.

like all these relations are satisfied and
so you could quickly check this

what about the first one well this is just
that the way from should be space is to

transform of the wave function in Q
space so disk wanted here is a deeds

old the transformation matrix and
that is according to your What's

the what this basic probability
amplitudes should be so that works.

Likewise the symmetry property while
dept are boils down to like that

the in first of this matrix be a like
is just the accomplished God You Get So

that is to say that you have like
you to tear it out of this matrix

that to resolve not to be such a natural
property in your doubts formalism but

he now look sort of limits attention
to like things that are used to theory.

So that works and then finally like this
would look like the hardest one is this

relation satisfied Yes it is so
here you this I could quickly show you so

here you have like
the transformation from B.

to A right get you for B.

day here a you can then write B.

as like you know something
that is transformed from C.

C.
to B.

but that gets you that you can also
write this as directly going from C.

to A but you get this comparison of
these two equation tells you that a C.

is just D.

B.
over the transformation of B.

to A at a transformation the C.

to be so this is exactly that could be
should that that your post and I hope that

your knowledge to stand sort of fish quip
like if you had started there you know it

would have been much easier to understand
like what what this theory is about

it so now it may of ninety twenty
seven after reading all of this.

It produces Hilbert space it a paper
called The body should be good

no rights as I said the German software
heart mathematical foundation.

And so you can have like a realization of
your doubts postulate using the model but

space formalism and

what you going to do then is that you can
identify these probability amplitudes but

just in a products of eigenvectors of her
mission operators in Hilbert Space OK And

so your dorm allows to be using the delta
function and so it's not till one hundred

thirty nine that the rock is finally going
to split its brackets into bras and heads.

Now if you have to side in a fixation that
you can see there e quickly that your

doubts postulates are satisfied so
postulate one has to be this well it's

just the overlap of these two
vectors is easy to divide this IP Q.

over eight so that works.

I have this is completely trivial
it's just like a basic property

of the product and like the final one.

The results that that followed by
the arrive to dissipate parade day

twenty seven which is.

The specter of decomposition theory so you
could stick a lucky you didn't operate or

here and then use the resolution
of the units right is as D B

can't be bra be right this is just
a projection operators would this would

be one right as corresponding to the
spectral decouples you stick that in and

you have to like the results that
you meet but that's the Ritz.

Beautiful you think done right so so
one way to understand like the transition

from your done to following month is
that well you know like we just replaced

like this this identification of
probability amplitudes by the so.

The direct transformation matrix sees
by in a products in a Hilbert space

they were off to the races now
you can feel that there is a but

coming this is not work for minimum bids.

So why not.

So full of on just thought that
this was mathematically ricketty So

this is lovely quote This is
a few years later it is book but

it's the same sort of sentiment it is
nineteen twenty seven paper where he says

direct method same goes for your art on
the stock beat the demands of mathematical

rigor in any way not even when
it's reduced in a natural and

cheap way to the level that is
common in theoretical physics.

The correct formulation is not just
a matter of making the rocks method

mathematically precise and explicit but
right from the start calls for

a different approach related to
Hilbert spectral theory of operators.

So he's going to go like a very
different different routes but

now in hindsight in hindsight we can see
that what I did on the previous flights

you know could be bait mathematically
rigorous but that calls for

a mathematical developments that were
certainly not available at the time so

you know you need to get comfortable with
with with the direct delta function which

dismisses as nonsense so you have to fewer
distributions and you have to do with

the problem that a lot of these vectors
are not actually in Hilbert space so

you need something called like Rick
Hilbert space but so so these these these

developments are well beyond sort of my
bath of magical mastery of the theory and

I think they're well beyond the mastery
of most graduate students but

you know like we're reassured by our
teachers like you know like you just

assumed that you could do this like
a you'll never go but that seems to

work just fine but before for Norman
of course that was not good enough so

what did do so look it should so
he as I said like you know luck.

Twenty seven points out that the reason
he doesn't want to do this is that

some of these ample to.

It's are not in the open space and like
the delta function is not like a function

at all as far as he's concerned he's right
about that of course so what does he do so

what he does is he's using the Hilbert
space formalism to give if like

a mathematically unobjectionable
derivation of the central quantity.

Of a your dogs theory
Dave leaders probability.

To think back sort of this arch and
scaffolds thing that a bushing so

he can now replace like the way
you built the expression for

Ford that probability using not
these probability amplitudes but

using projection operators and so
all show you quickly how this go so this

is a formula that of course is still in
use him on a quantum mechanics so rather

than writing this probability like that to
downright it just a trace of the product.

Of these.

Projection operators so take as an example
to use like you want to buy one and

sort of quickly show you that you offer
million with this this kind of stuff a B.

B.
X.

and H.
right position energy and so

now what you get is that you
know the debt probability is you

know the square absolute square of the
sorting away function that can be written

as this probability amplitudes we can now
write that this just is in a product and

this quantity over here we could
write a little differently so

the projection operator out to exit out to
end would be written like that's right and

now calculate the trace using
again like you know the resolution

of unity with some arbitrary discrete
orthonormal base of Hilbert Space.

You put this in right so this is just
take that from here a stick in this

this this decomposition of resolution
of unity I move things around

a little bit and what I see is that
I just get like to trace the product.

Of these projection operators if that were
a little too fast don't worry too much

about it the point is that following
month to now is a man the math

of it a mathematically unobjectionable
way get that same formula for

the probability that that
that your doubles after.

OK so.

So now.

But before long I was not satisfied for

wall with this with this particular way of
doing things and he writes another paper

but shyly cuts to a to show off though
this may be more challenging for

if you don't speak German sewed so this is
like the probability theoretical building.

Off right.

So and.

I should have mentioned like you know this
is I put it on the opening slide this

is based on work with my collaborator
in history of quantum mechanics like

Tony Duncan physics at
the University of Pittsburgh so

we speculate that what happened
here is that photo of a got

unhappy about that first paper that he
wrote it twenty seven after he read.

Heisenberg Uncertainty Principle
search the paper and

before he had read that paper.

He essentially explicitly admit Dorsey.

Like your God's view of
probability theory but

so he writes the Says like you know
postulate with three is obviously a valid

use the addition of multiplication
theorems of ordinary probability calculus

except that it disguised to hold for
the amplitudes rather than for

the probabilities themselves but so he
says this completely approvingly in this

paper he writes all your doubts
theory with Hilbert and north and

it is open the box should be good to he
again writes without comfort a comment to

multiple patient loft probabilities does
not hold what does hold it's a weaker law

corresponding to your doubts combining
the probability amplitudes now after

he is read Heisenberg he changes
his tune and Heisenberg like.

Really protests against your dogs
use of probability theory saying

look you know the rule the basic rules of
probability theory are not dependent on

physics you know they are what they are
and even quantum mechanics is not going to

change them so it doesn't make a lot
of sense to introduce these new rules

of probability that apply to these
two days to be templates and

follow him on a Greece with with this and
so so he writes now in this new

paper he says the relation to the ordinary
probability calculus was not sufficiently

clarified the basic fully to
the full it if its basic rules were.

Sufficiently stressed that So he's now
back to the you know no we're not going to

best with the basic rules of probability
theory we're going to stick to those and

that's why you know this paper is called
like probability theoretical instruction

because he's now going to construct you
know a century out of nothing you know

this whole new edifice and S..

So the so does the strategy changed a
little bit right what he did in that first

paper as I showed you very quickly was
to rewrite Jordan's expression for

these probabilities in terms of
less objectionable Bassa batiks So

instead of using probability amplitudes
he was writing it in terms of projection

operators what he's doing now is
is is is not like rewriting it but

just derives from scratch this
this this this quantity death

that your doubt introduces And he's doing
that by introducing density OPERATOR So

this is the paper it which for
the very first time the notion

of a density operator is introduced it is
really a brilliant paper Buffalo Bill.

And so like metaphorically speaking so

he is now build again another scaffold
to prevent like you know the original

arch even supported by its first
scaffold from black collapse.

And as luck would have it followed
like you know it was this

at this point is all of twenty four
you know look it's really amazing

what these people were
doing at a very young age.

He has come to know one of
the greats probability theory

namely we should for this and so for
me says like you know it's one of

the guys like the freak with this the
pro-choice to probability theory and so

he says like they don't have
to defied even what it means

to talk about probabilities what you
need to do is talk about all solvable So

if LIKE IT systems and then you pick
like members of that all solvable and

you check for the property that you're
interested in OK you want to know what is

the probability that this flower is red
you know half your half your flower speak

to tell me what percentage of that
all solvable has red flowers so

follow it it's going to do
the same sort of thing but

it's doing this within
the framework of Hilbert Space.

So so so so he doubt it's going to take
the basic open space formalism and

it's going to fall is try and
find an expression for Dex picked a should

value so and so this is what I've
indicated here some function Big E.

F.
So

this is the form of that function
needs to be the target and

this is really like it amazing piece for
Lloyd's work so

the two of the two assumptions that he
makes is that that function is linear

that if you have lots of
quantity that is represented by.

OPERATOR A plus B.

to operate a B.

etc that the expectation value is that
expectation of a beta explication of B..

This this this assumption
would come in for

a lot of criticism later on because
it's also to some sure that he

makes like it is famous no had variable
proofs a quantum mechanics but

it is particular causes before perfectly
fine and the other one is that positive

definite this that if the value of A is
always greater than one of the dead

the expectation value is always greater or
greater Grey is always greater than.

Or equal to zero dead the expectation
value of A is greater dead or

equal to zero it seems like you know
these are very and Oculus assumptions but

it turns out that this uniquely
determines that function and

that function is going to be
given by a density operator types

the quantity of interest and
if you have a uniform.

Where you prepare every
system in the same state

he can he can proof that
then that case the.

Density operator is just a projection
OPERATOR It's just that.

So in this paper like you know like
all the sides he has also luck for

the first time introducing the difference
between a pure state and a big stage right

because so this is a pure state if it's
a big state but you know you would have

liked at all solvable where you have a
system prepared like the different states.

And so

now what you could do is like if you if
you work out what this expression is.

You know show you can evaluate the trace
shuffle this around use like I said again.

The resolution of beauty so this is one so
you see that you just get your familiar

expression for expectation values out
again but now football has succeeded it

deriving is from almost nothing right from
the basic basic agreed it's Hilbert space

at two very innocuous looking assumptions
but so this is deadlock sort of.

Where he leaves it to sing
the praises of this of this of this

paper a little bit more that it would
be there would be one more paper it so

this is become the oldest of the twenty
seven trilogy the final papers about

quantum statistical mechanics I don't I
will talk about this but in this in this

in this first two papers follow but also
all possible is proving the equivalence

of these four versions of quantum
mechanics and he's basically Buckley's

that even claiming a great insight here he
sounds kind of tired say like well if only

these physicists were paying a little bit
of attention to what we'd best imitation

they would know that matrix mechanics is
just like a theory about square solvable.

Sequences and that wave mechanics is a
theory about square it will functions and

those things are isomorphic and
those are two instantiations of something

more general that I'm going to
call the Hilbert space so right so

it only sort of they knew about like you
know the theories of partial fall and

reach Fisher which at this point are old
but they would have they would have

noticed but now you know like I out of the
goodness of my heart as mathematicians of

good to help out these poor physicists and
point out these obvious things to them.

OK now the sort of paper that Tony and

I wrote is called Never mind your piece
and Q.'s were never is it parentheses So

you're Don since he doesn't
have Hilbert Space.

This is very much like my niggaz piece
accuse which is all this business about

canonical transformations followed
by realizes that you don't need

any of that right you just need
to do Hilbert space right and

then you could do everything
your dad could do a dead so.

The follow ups of A should that I want to
make is like something that he calls to

set to boom of the state vector and I
won't reach you the whole the whole quote

but this is the essential the first
mention that I'm aware off of the famous

measurement problem but so he says like
Look sometimes you know you you have

a state that you know like you get your
guarantee to get a certain outcome but

many times you have a stated that it's
a probabilistic issue what outcome

you're going to get at the state in
the process will be sent to demolished

the notion of chip to about the job it is
very much you know like the picture that

I had but the very beginning of like you
know it's a place blown to smithereens.

Conclusions but so this was a quick
tour of how we get to Hilbert Space.

So in terms of my better for
how did we get there right so

one possibility but one metaphorical
way of telling the story is that.

Turned your dogs arch into a scaffold for
his old arch but.

The arch that being
the Hilbert space formalism so

this would be like you know your down at
the bottom and four more model top but

you can also tell a differently which is
a suite you don't have sort of switch back

and forth between these two ways of saying
it you can also take a photo of a building

a scaffold to prevent the arch of your dog
from collapsing so it would still have

your down on top and for law about
building ever better scaffolding to

support that arch off of your doubts and
so you may think that you know we're.

I think of this metaphor it up be very
upset that I can't pick between these two

cases but in fact I don't mind this said
So look I think that the ambiguity here

sort of points to deliberate patience of
the better for Ed sort of advice now to

come up with like a better way to get
beyond dispute be a paradigm shift story

tell stories like this but
using like terminology that is not

is a big US arch and scaffolds right to be
scuffles I think is a sort of nice picture

to get the to get the basic idea across
but to really work on this you need

a saying to you need a better
sort of conceptual tools to

get this if it is conceptual tools like
I'll be happy to talk about this in.

Q.
and A but for

now just want to throw it out is that you
can borrow ideas from evolutionary biology

and then not so much like ideas from
population genetics dockets but

like theories about what is now called
evil Divo where you look at the good

strayed sport of the development of
species that you know come from the way

to terminate right so it's not like silly
dislike denying that natural selection is

to suggest changes species but you that's
only telling a part of the story and

other part of the story is
that you know you can't

though the actual selection is going
to turn like an elephant into a giraffe

you need to know like you know what are
the limits that I could change things and

these limits what you change things to
be that's sort of the interesting part

in a story about like how
you develop a theoretical.

How you develop like theories insides and
so this is be this getting some traction

in the history of philosophy aside
societies will end with this guy like Bill

lives that from you for Chicago who has
you know piety at this approach and

so the paper that I've written about this
will come out in a volume that he edited

with my colleague a love
philosopher of science that you.

The Minnesota but beyond to me.

Is not the internet me but
it's the Beeb from like Dawkins like

the selfish gene right where now you do
Beebs are for culture what genes are for

species right so it's beyond to be
doing this sort of evil Divo style

stuff development of structure
in a cultural evolution so

by story is sort of part of this
broader effort to come up with a way to

characterize cultural evolution including
the evolution of scientific theories

helping ourselves to bottom insights from
like evolutionary biology thank you for.

The.

Way.

On.

Yeah.

Yeah.

Yeah yeah.

OK thank you thank you very much for
that question so the first one.

So so yeah I totally agree if you that
that could be like the cool you defense

right now if I only told you a lot one
of my case studies I've done another one

that is about like how do we
get from the old quantum theory

to bomb the quantum mechanics right so

Dalek it's a little more interesting
if that's not a paradigm shift what is

right to be there you go from make
the dramatic from describing things like.

Classical face space with a few conditions
slept on it right they say they recognize

this a going to work to describing
something like with matrices way functions

what have you and it turns out that that
story too as cooled like in his historical

work would readily admit it is not
a batter of like you know OK somebody

this is now dead we need like a DO idea
you know like that's now the kernel of

the know it's again like a very much like
a continuous development and so did I mean

I can't block substantiated right now but
I can give you like a very sort of nice.

You know sort of hints that this that this
metaphor is going to work because remember

like the title of the Heisenberg paper
that introduces matrix mechanics is called

going to paper which is to durable for
lack reinterpreting So

what this theory does want to be.

What makes mechanics does is not
to reject classical mechanics

to repeal classical mechanics but
to reinterpret it right so

what you have is like all the all
the relations of classical mechanics like

proceed exactly the way they were before
remember that quote from the rock right.

But the fundamental changes is that out
all the terms like it period it are now.

She's right so
again it's a very sort of continuous

that can be described very nicely in this
arch scaffold metaphor where the disk

Ace It's like you know you have to scuttle
this big built dispersion theory and

then what what what what essentially
what Heisenberg does is to show well you

know what we could do for dispersion
theory we could do physics in general and

they you're off to the races so that's the
first question the second question I also

like very much so as I said like
you know like the this particular

example of my arch and scaffolds is
very best and as an aside as I indicated

like I like it because it shows sort of
the the limitations of the better for

like I have a few very clear examples the
one Heisenberg is very clean the one of

relativity is are the world so relativity
even cleaner this is a very messy one and

part of it is that it's unclear whether
the scaffold ever gets taken down

right and so I would say that's even if
it goes even further would you say select

my picture is you have
like wave mechanics matrix

mechanics of top of that you built
to direct your transformation theory

of top of that you built like
the full mobile theory but

it was a version of the better for
now all of that is still with us right so

so yes you are completely agreed if you're
not a philosopher or a quantum mechanics

you typically have very little use for
luck for you're perfectly happy to use

Jews you sort of the back to badly fixed
up version of the direct transformation

theory which the way we teach it
through our grassroots day is kind of

a big A for boy but
at the rock when you read the rock

you don't really know you don't really
realize But the guy originally was

to talk about the Hilbert space at all
you just like factor that in right and

they you trust that this can be bad
sabbatical be done well big There you have

your graduate course quantum mechanics at
the undergraduate level right you don't

even care about that right you go straight
back to the waste mechanics right and so

like you introduce people to the way
functions have what have you write.

And it's only later that you that you say
you know by the way like the way functions

are just one particular instantiation of
Hilbert space but so even like you know so

this is often very fascinating that
almost a century later that you still see

clear Red Sea the way we teach used
the theory that are just an artifact of

how this theory luck a bit to be you
know and so that though in this case

like the the scaffold is NOT TAKE IT
DOWN AT ALL people are very very happy

to live in parts of the building you know
that I just dismissed the scaffolding.

Yeah

OK.

OK

now so

I hope I've made it very clear that I'm
a great admirer of these two papers.

For more a minute if like if
I DID YOU KNOW THAT is just

like indicative of my all my own
limitations in mathematics OK.

OK.

I've heard of her death but
thank thank you for that so.

So so a part like you know like
I'd like to talk about this.

You know about this about this development
precisely because I admired these paper so

much so if you so like you know I gave you

I made some disparaging comments
about like this does your papers.

But so your dad had a few good ideas
right that he had so so and without those

I.D.'s you know like for a lot of stuff
would not have gotten off the ground but

at the by your done our holy mess
OK like I mean I was telling

somebody earlier today that there are
literally equations in that paper where X.

on the left hand side an X.

on the right hand side
completely different things and

where this whole section said that
paper that a complete nonsense and

that our best sort of it Dort now compared
to that like these two papers that have

been talking about by far more of them are
like masterpieces where he's there's not

a step wrong and so the only reason that
it takes from one thousand nine hundred

ninety twenty seven to nine hundred thirty
two before he publishes the book rights

of these three papers are the backbone of
his book at thirty two is that he wants to

nail shot a few mathematical details so
he particularly like you know to proof

the two they will down to proof the
spectral theory for about operators but

so so I've I read through this like
in Iraq a sort of cavalier fashion.

To make the point that the way
that this is being used in physics

now right is that you know like
we don't actually be like.

That's my impression we don't actually
worry about these mathematical niceties

at all right we just treat this like as
a piece of literal to read it might as

well be lucky to find out dimensions and
you devil war right and so.

I've.

Since I'm not a mathematician I can't
really judge as to the how important like.

Contributions.

Compared to these countries about
Rick Hilbert space at the theory of

distributions but I take your point that
you know the latter two are sort of minor

corrections to what or minor additions to
what four Lojban himself put in play but.

Some of the stuff that I did mention
like this is the introduction.

Like the notion of density operators
which I think is that old that

if it were me like this is this I would
teach beginning students that's how you do

quantum mechanics with density operators
the introduces the difference between like

big states.

And pure States he he is
the water who not sure ot or S.

It's often said to prove
the equivalence of matrix mechanics and

wave mechanics by just like porting to
that boy you could the fact that little L.

to a big L.

to R.
so

morphic I mean there's a tremendous
amount of work that is being done in this

in this paper while Johnny Football
is all of twenty four years old.

Yes.

Yeah as far as I'm concerned yes

it was a story.

Yes but they haven't read it but

I do think that the rock comes
out a little earlier dead.

For more and more right the rocks
textbooks in one thousand

nine hundred thirty.

But the way but if you think about it so
so did that go through many editions but

the way that you read the textbook write
whatever you see the like you know like.

Brackets like you just read that there's
a need a product of Hilbert space that is

not the rocks idea right the rock knew
nothing about Hilbert space that is that

is for moment so what I think what you're
teaching your your graduate students like

in their first year is really sort of
like a combination of the rock and

followed where you don't really care all
that much about like you know what's

coming your way after all you're not
teaching history but who do we want and so

like a whole top of his course like for

Norman is also the one who introduces
the measurement problem and.

In a way puts his finger on the you know
the one big it's a potential problem

right that still has not been solved
right in a way that it's much much more.

Pointed than any of the other
characters in this this in this story.

So if I decide I really so the.

A lot of for the moment scholars.

They have to be a typically
very good mathematicians so

they consider all of this like very
elementary and so they don't spend they

don't spend the time talking about this
and this is a shame because like you know

DIA because of that these twenty seven
papers don't get like a lot of attention

the only thing that gets attention is
the one hundred thirty two book and

so I really think it would be a would be
great and I'm hoping to do this like one

of these days to translate these these
twenty seven paper said just make

them available to a broader audience
because they're really quite beautiful

yeah yeah yeah we're.

Here.