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.