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DAVE GOLDBERG: Thank you, first of all, for having me.
I'm going to give a very general talk today.
The talk is one of these things where the table itself
is going to give away, basically, the crux of it,
which is why symmetry matters.
Now I would not be the least bit surprised if in a Google
audience, some small-- perhaps some large-- fraction of you
were physics majors in college.
Just out of curiosity--
one, two, three, et cetera.
I stopped counting after three, perhaps out of ability.
But one of things that is not obvious-- even, by the way, as
a physics major in college--
is the sort of shorthand that physicists especially sort of
use to describe the universe.
They'll say some theory is elegant or beautiful.
Or they'll talk about the importance of symmetry.
And here's a very famous quote by the Nobel laureate Phil
Anderson, which is, it's only slightly overstating the case
that the study of physics is the study of symmetry.
And yet typically, when you see physics in high school,
when you see physics in college, even if you take
advanced courses in graduate school, it's very
equation-driven.
The idea of what it is that underlies our assumptions
about the universe, how it is we got to where we are, why it
is that physics is sort of held up as this paragon of a
beautiful academic discipline, why is it the universe itself
is so beautiful--
that you don't learn about as much, until much, much later,
until you delve much, much deeper.
Indeed, I will say that even as a practicing physicist,
until I started working on this book, I didn't fully
appreciate how much of what we know is really undergirded by
sort of these beautifully symmetric, simple, in many
cases, assumptions about how the universe works and how
much can be drawn from that.
So I'm going to begin with a sort of working definition of
what symmetry is.
This is given by the mathematician Hermann Weyl,
which is basically, a thing is symmetric if there's something
you can do so it stays the same after you've done it.
A wheel, a circle, for example--
I can turn a circle in any given direction, and it is the
same as it was before.
Or just another very simple example, a triangle.
Now a triangle isn't quite as symmetric as a circle is,
because there's only a couple of things you can do.
You can rotate a triangle, but only by a specified amount.
Only by 120 degrees, a third of a turn.
Or you can reflect.
I'm drawing it as a rotation on the bottom there-- or
rather, my great Illustrator, Herb Thornby, who did all of
these, is drawing it is as a rotation.
But it's really a reflection.
We're just taking the triangle and we're
looking at it in a mirror.
And you know, we can talk about geometric objects till
the cows come home, and say, yes, they
are, of course, beautiful.
There is something very nice, pleasing about it.
For those of us who simply enjoy puzzles, there's
something appealing to the human mind.
And I've just picked a few very natural examples here of
symmetric objects in nature, symmetric objects in art.
The Taj Mahal, for example.
Architecture, flowers.
We've got an MC Escher print.
There's the galaxy M81 in the lower right.
And for those of you who don't do these sorts of things, I
occasionally compete in the National Crossword Puzzle
Tournament here in New York.
Crossword puzzles obey, in the US at least, a asymmetry where
you can either rotate by 180 degrees or reflect them, and
that's the rule for how the black squares need to show up.
If you've ever noticed, the grids for crossword puzzles
look also symmetric.
And these things appeal to our mind.
I mean, nature seems to enjoy symmetry.
But symmetry itself seems to appeal to our mind and our
sense of order.
It also shows up in scientific discoveries.
So when Rosalind Franklin, for example, took her images of
the double helix, the fact that DNA had this structure
allowed Watson and Crick-- well, to uncover that it was a
double helix.
But also to start understanding what the
workings of DNA were.
I mean, the double helix of DNA is, of course, one of
these great natural symmetric objects, insofar as you can
take this thing and twist it, and it will look the same as
it did before.
But it's worth noting--
and double helix is particularly
interesting in this way--
that it spirals in a particular direction.
It's got one symmetry--
that is to say, I can twist it as much as I like.
But if I were to look at a double helix in a mirror, you
could tell it was wrong.
You could tell it did not come from living
being here on Earth.
It would have the wrong orientation.
So one of the things that's going to be very interesting
for us about sort of understanding symmetries in
the universe is to recognize, symmetries are very beautiful.
They're going to give us guidance.
They're going to help us understand how things work.
But there's a symmetry breaking here, or an
asymmetry, that's going to, in some sense, give us even more
information.
That's going to allow for, it'll turn out, nothing less
than our existence.
So all will recognize these.
I bet no small number of you will recognize these from a
game-playing context of a name-brand role-playing game.
You also, by the way, notice that there is one missing.
These are the Platonic solids.
They are also applying to "D&D" dice.
The missing one, of course, is the D10, which is not in fact
a Platonic solid.
It is known as an antidypyramid and has the
beautiful name "***'s lozenge," and doesn't have
these properties.
But the Platonic solids all have wonderfully symmetric
properties.
And many, many thinkers, in trying to uncover sort of the
secrets of nature, assume that the Platonic
solids and the sphere--
which is basically the Platonic solid with an
infinite number of faces--
that the Platonic solids must ultimately give some sort of
clue as to how the universe works.
I mean, the sphere, of course, is the most obvious.
The sphere prompted Aristotle, for example.
He assumed that because the sphere is this perfectly
symmetric thing that it must describe nature.
We owe, ultimately, to him and his contemporaries the idea of
the celestial sphere, the idea that the Earth is at the
center of the universe, that the orbits
of the various planets--
and the sun, for that matter-- are embedded in spheres around
us, and that the stars are at the most
distant sphere of all.
And that must somehow, because of it's beautiful simplicity,
that that must somehow be a representation of nature.
Now the spheres were wrong, in a lot of ways--
not least of which, the Earth is not at the center of the
solar system.
The sun is.
But there are other reasons, as well.
I mean, even if you accept that the sun is at the center
of the solar system, something that I'm hoping you will
accept, we are still left with the fact that even the orbits
of planets are not perfectly circular, obviously not
perfectly spherical.
They're elliptical, and it turns out there's going to be
a deeper explanation for that.
Even as we moved forward, even when thinkers started to
recognize that the sun was at the center, they still tried
to exploit a lot of these wonderfully symmetric figures
in order to describe the universe.
This is Kepler's awesomely-named "Mysterium
Cosmographicum." And basically, what he did is he
took the various Platonic solids and the sphere--
so there were five Platonic solids, the
sphere gives us six--
and there were six planets then known, out to Saturn.
And he said, oh, you know, six Platonic solids including the
sphere, six planets--
maybe they're related somehow.
And he kept embedding them in this turducken of a
cosmological model to try to figure out-- he said, if you
embed the one in the other, in the other, in the other, they
represent, basically, distances.
And those might represent the relative distances of the
planets from the sun.
And there might be something profound and
fundamental about that.
It turns out, obviously, that any match you have to this is
entirely coincidental.
Coincidental plus the fact, you know, if you mix and match
them in any given way, you have
6-factorial possible orderings.
You're going to get close with one of them, presumably.
But that was sheer luck.
We owe, of course, to Kepler, we owe our knowledge of the
fact that the planets move in elliptical orbits.
Ellipses, at first glance, do not seem terribly symmetric.
It will turn out--
and we ow this to Isaac Newton-- that sure, the orbits
themselves aren't elliptical, but the force of gravity is.
The force of gravity acts the same in all directions.
And that is a very, very important symmetry of nature.
I should note, by the way, that the only reason that
Kepler did not immediately hit upon the idea of planets going
in elliptical orbits, despite the data that he got from his
mentor, Tycho Brahe, was he just assumed that-- he didn't
even try it.
Because he figured if the orbits of the plants were
something as simple as an ellipse, surely someone would
have come up with it already.
I think it's giving his predecessors a little bit too
much credit, perhaps.
So it is interesting to note that when we talk about
symmetries in the universe, what we really mean is, what
are some ways that you could adjust,
say, the entire universe--
turning the universe, for example, or moving forward or
backwards in time, or moving throughout space--
and the laws themselves don't change?
So the laws of physics are the same here as they are here.
And the only reason anything appears to be different is
because of my relative motion, say, compared to all of you.
Or if I were to go 1,000 miles up in the air, things are
different because of my relative position to the
Earth, but not because anything fundamental about the
laws of physics have changed.
So the laws of physics basically say that all of
these things-- time, orientation,
positions in space--
none of those matter in the physical law.
Now that actually is incredibly helpful.
That's way more helpful than you might think, simply saying
the laws of physics can't be dependent on
any of those things.
It means, for example, that if you were to describe a giant
equation that describes all of the physics of the universe,
where you are in it can't ever appear in that equation.
Or when it is can't ever appear in that equation.
Or there can't be any equation that ever describes things
with an absolute direction.
Won't ever appear.
That makes the equations a lot simpler than
they would be otherwise.
Those things turn out not to matter in our symmetries of
our universe.
Other things seem like they might matter
or might not matter.
One of them is physical scale.
And this is a trope of sort of science-fiction and children's
books and things like that.
You know, "Horton Hears a Who," or the end of "Men in
Black."
The idea that you can take our universe--
you know, we're this giant universe.
We have these things called atoms.
And if you look at the old models of atoms, atoms look a
little bit like a sun with things orbiting around it.
And you can imagine thinking of that and saying, oh, maybe
you could be a tiny little creature living inside of
that, with a life very much like a human being's, except
much, much, much smaller.
Maybe the universe, on various scales, is almost identically
the same as it is on the human scale.
And we're just not able to see it, because our perception
isn't good enough.
And you know, this is the sort of question you have very,
very late at night in your dorm rooms.
I think I'd be prudent, as this is going to go on the
web, to not go any further.
But you get the point.
This is something we might think of as a
property of the universe.
The question is, does physical scale matter?
Well, one of the great thinkers on the subject was
Galileo, who decided to ask the question in sort of an
absurd way, which is to think about the existence of giants.
So there are biblical descriptions
of the ages of giants.
And if we imagine a giant as merely looking like a normal
human being, except scaled up by a factor of 10 or 20 or
however much--
this is the same premises in "Gulliver's Travels"--
would you be able to do that without re-engineering the
entire thing?
And Galileo says no.
If you were to simply scale up a human to giant size, you'd
have all sorts of problems.
I mean, after all, if I make you 10 times bigger in every
direction, you become 1,000 times more massive, 1,000
times more weight to support, for example.
But the strength of your bones are based on the
cross-sectional area.
So if I make you 10 times larger in every direction,
your bones only become 100 times stronger.
You're supporting more weight per unit area, basically, the
larger you are.
And so he said, look, you'd basically have to totally
redesign the bone of a giant until eventually, the thing
was entirely bones, and the bones were incredibly,
incredibly thick, and the thing wouldn't be able to
function at all.
And of course, the opposite is true.
I mean, insects don't require the internal skeletal
structure that we have.
They're of an entirely different design, simply
because again, this is not a symmetry of nature.
Which is again, like, "Spiderman" is not such a
great premise.
Take a spider, scale him up to human size, and
he'll squash himself.
I can't skip this adorable quote by Galileo.
He talks about an oak tree.
You know, "Nature could not produce a horse as large as
twenty ordinary horses or a giant ten times taller than an
ordinary man."
And then he concludes with this adorable imagery, which
unfortunately does not carry with it an illustration.
Galileo's work is filled with illustrations, but this isn't
one of them.
"Thus, a small dog could probably carry on his back two
or three dogs of his own size, but I don't believe that a
horse could even carry one horse of his own size." I
choose to believe that this experiment was
never carried out.
So some things are symmetries, some things aren't.
But what about-- we have a human intuition about what
should be a symmetry of nature.
Antimatter is a very, very important one.
We see antimatter in science-fiction all the time.
We're able to produce antimatter in a lab.
And if you know absolutely one thing, only one thing about
antimatter, it's this--
if you have an antimatter friend, do
not shake their hand.
Why?
Boom.
You will be completely annihilated and
converted into energy.
But antimatter really very much the
same as ordinary matter.
There's an antimatter version of every particle.
An electron, for example, has an antimatter particle called
a positron.
Same mass, but opposite electric charge.
And it is absolutely true that when you take matter and
antimatter of the same particle type and bring them
into contact, they will annihilate completely.
It is also true, by the way, that if you produce matter and
antimatter in a lab, that we are able to produce them in
equal quantities.
This raises kind of an important question, one that
is not immediately obvious how to resolve it.
Given the similarity between matter and antimatter, given a
very important fact, which is that the laws of physics don't
seem to care whether you're talking about matter or
antimatter-- it's just the sign that changes, and given
the fact that we produce and annihilate them in equal
quantities, why are we here and made of matter?
The laws of physics, I should say, are almost completely
identical if we take matter and convert it to antimatter.
I've got a little silly example here, where I've done
two things.
I've got a current-carrying wire, electrons.
Electrons have a negative charge, so they go the
opposite direction of the flow of current.
We owe that convention,
incidentally to Benjamin Franklin.
And it creates a magnetic field.
And likewise, if we look at the same thing in a mirror and
also change all of the matter to antimatter, we get the
exact same magnetic field.
There's something intimately related to mirrors and to
antimatter.
And those two combinations of things, called CP symmetry--
Charge for antimatter and P for parity or reflection--
that seems to almost be a asymmetry of nature.
Almost.
I mean, clearly, as I've said, it can't be a perfect symmetry
of nature, because there is something different between
matter and antimatter.
And by the way, it's not just us.
It's not just that we are all made of matter, and the Earth,
and sun, and the solar system, and our galaxy--
every galaxy seems to be made ordinary matter and not
antimatter.
And we can tell that, because if a galaxy and an anti-galaxy
were to collide with one another, we would see that
across space.
And if there were antimatter galaxies, it would happen
occasionally.
And we do see galaxies colliding, by the way, and
they just have the regular sort of gastrophysics that
you'd expect.
There's only the simplest difference between even the
reflected version of this CP transformation between matter
and antimatter, and that is that we can see little things.
For example, this is the cobalt-60 decay.
So you take cobalt-60, and atoms have this property
called spin, which is, in principle, directly
measurable.
And it turns out that the ordinary matter version has
electrons preferentially given off in the direction that the
thing is spinning.
So you use this thing called the right-hand rule, and
preferentially, electrons are ejected more in the direction
of the spin than opposite the direction of the spin.
We've got little hints like this, that there are slight
violations of symmetry in nature, but we don't know
where they come from.
And they only show up in what is known as the weak force.
The laws of physics are broken down into sort of four
fundamental forces--
gravity, electromagnetism, and the strong and
weak nuclear force.
And every one of them, except for the weak force, seems to
not give any concern whatsoever about the
distinction between matter and antimatter.
It's only the weak force that shows even the slightest
preference.
And that slightest preference seems to be, in very, very
subtle ways-- it doesn't even mean that more matter is
created than antimatter.
It's just in what we can do in a lab, we see very, very tiny
differences that say, somewhere in the equation--
and we can identify those terms--
but somewhere in the equations is a tiny difference.
The universe knows about the difference between the two.
But how did it choose that?
We don't know.
Here's just one other sort of
illustration of the same thing.
This is just a matter of a particle
known as the neutrino.
A neutrino is a relic, generally speaking, of weak
nuclear interactions.
And one of the things that's very interesting about a
neutrino is that if it is created in a reaction, it
always flies out in such a way that it's spinning as given by
your left hand.
Anti-neutrinos are always spinning in a way that would
be given by your right hand, with your thumb giving its
direction of motion.
And as I was implying, they look basically the same.
If you take all anti-neutrinos and turn them into neutrinos,
and vice versa, and take everything and look at it in a
mirror, which makes the spins go the opposite way.
Except that cobalt-60 thing.
It turns out that we can even make that difference between
matter and antimatter go away.
There's an interesting discussion, and I've
excerpted it here.
But the crux of it is, in his Nobel Laureate speech, Richard
Feynman, one of the great physicists of the 20th
century, was relating a conversation that he'd had
with his advisor, John Archibald Wheeler.
And Wheeler had this idea.
And he had a lot more that went into it, but the idea
was, he said, you know, the physics of positrons looks
almost exactly the same as the physics of electrons, if you
assume that positrons are electrons going
backwards in time.
And it's true.
And so we ask the question.
We've got this combination.
We've got three symmetries that I've mentioned so far.
Three symmetries, three very good approximations to almost
perfect symmetries in nature.
Take matter, turn it into antimatter, look at it in a
mirror, and reverse it in time.
CPT, it's called.
And every single reaction in nature that we've ever
discovered--
ever--
will work totally the same if you do those
three reversals, basically.
Incredible.
Surprising.
And also very, very weird, when you think about it.
Because when you think about what that third one is, that
third one, time reversal--
time reversal seems like it is hardwired into our laws of
physics, right?
Time reversal should not be an even
approximate symmetry of nature.
I mean after all, you don't know how this talk is going to
end, but you do know hot it started.
You remember the past.
You do not remember the future.
There's a thing called cause and effect.
I could go on and on.
I mean, we almost don't have the words to explain how weird
it is that there is such a thing as the arrow of time.
And yet there is an arrow of time.
But that said, if you look at microscopic interactions--
and after all, what are macroscopic interactions, but
a collection of microscopic interactions?
A big collection.
You look at these things, and you can reverse them in time.
This is my time mirror that I had my illustrator draw.
You take a movie of, say, two electrons scattering off of
one another and look at that scatter in a mirror.
That process looks equally valid.
I take a ball, I throw it though the air-- well, I throw
it a little bit more professionally.
I throw a ball through the air, it makes an arc.
I take a movie of it, watch that arc in reverse.
It also looks like a valid trajectory that
a ball could make.
The laws of physics seem perfectly comfortable being
run forward or reversed.
But obviously there is a complication--
one that shows up in a fairly complicated way.
And that is with the idea of entropy.
So entropy's one of these words that's thrown around.
It's generally considered to be--
we talk about it as something like disorder, for example.
But entropy, we think about entropy with gas molecules,
for example, and entropy, to a physicist, is really just a
measure of possibilities.
So what we've got here is a little cartoon of gas
molecules in a box.
And we've got the same number of molecules in both
illustrations, 10, jumping around.
In the left box, we've got almost all of the gas
molecules in the right partition.
Now there's very, very few ways to do that.
If I were to number all my atoms, for example, there's
only 10 different atoms that can be the sole atom in the
left partition.
This is what's called low entropy, or very high order.
In other words, it's like putting away your room.
I mean, there's a lot of empty space once you've
cleaned up your room.
On the other hand, in the right box, we've got this
thing called high entropy And high entropy is basically,
everything is much more uniformly distributed.
I cannot stress this strongly enough, because this is a very
popular misconception, even when people have encountered
thermodynamics in high school and college.
But going from low entropy to high entropy, something that
is so well-established that it is known as the second law of
thermodynamics, is not really a law at all.
It's merely a very, very good suggestion.
More or less even distributions like this right
image are just far more likely, because there's far
more ways to distribute your air molecules than
the ones on the left.
It is absolutely possible--
possible, within the realm of physics--
that spontaneously, all of the air molecules on that side of
the room migrate for some short period of time over to
that side of the room.
And all of you asphyxiate, and all of you, I guess, get
crushed from air pressure?
We could probably survive.
We could probably survive two atmospheres.
I think you guys are gonna be fine.
I'm sorry to you.
But it's incredibly unlikely that that would happen.
But possible.
That said, this is our biggest clue as to how the arrow of
time works.
The idea that things go from low entropy/high order to high
entropy/high disorder.
It is within the realm of physics that if I break a set
of cue balls, a racked set of cue balls, that they're going
to scatter around.
If I hit a ball in just the right way, that they might
reassemble into a triangle.
I would not try to make that shot.
It would be almost vanishingly difficult to do so.
But it is within the realm of physics.
It's just there's not that many ways that a set of balls
can be arrayed in a triangle.
It's extremely difficult.
It's extremely unlikely.
So the increase in entropy is a probabilistic statement of
the universe.
It's just there's so many particles out there that those
probabilities become almost certainties.
There are even games that people play--
yeah, I like that.
This is one of my favorites.
There are even games that people play to try to see
whether the second law of thermodynamics can be toyed
with, not just in a probabilistic sense, but if we
might be able to do this.
This is something known as Maxwell's demon, the idea that
we might be able to make a low entropy and a high entropy--
make a low-entropy system merely by having a robot or
some other brain open a box and let particles through
depending on the nature of those particles.
So in this case, we might imagine Maxwell's demon takes
the high-energy particles and tries to sort them to the left
side of the partition and the low-energy
particles into the right.
This would be a zero-energy way of basically creating a
refrigerator.
Put all the hot air on one side, put all the cold air on
the other, and that indeed would be
a decrease in entropy.
It turns out especially philosophers of science have
thought about this.
And one of the problems with a scenario like Maxwell's demon
is that in making these determinations, it isn't a
zero increase in entropy.
You'd basically have to have the demon itself record, say,
the speed of the particle in its brain, and then when it
made the next measurement, it'd have to sort of erase
that measurement and put it again.
And that erasure is going to increase the net entropy of
the universe.
So time itself seems to have an arrow.
And if you notice that I haven't actually resolved why
it is that we have this arrow of time, you'd be right.
We do not presently know, we do not really know, why it is
that there is the arrow of time and that it's one way and
not the other.
We do not know whether entropy increases with time, or
whether entropy is what makes the arrow of time.
You know, one of the great observations that we've made
about the universe is--
tempted to see-- ah, yes, it is a pointer.
This is the microwave background of the universe.
It represents hot points-- those are the reds--
and cold points in the universe.
Hot and cold in this case being
about one part in 100,000.
And the very early universe, the universe was very, very
ordered, very, very low entropy.
This is very, very smooth.
And so the question is, did the universe start with low
entropy, or are we only in a universe where the arrow of
time is defined because we define the past as being
low-entropy?
We do not know.
There are, however, other symmetries
that we can look at.
And in particular, we see that the universe is largely the
same in all directions.
That is a symmetry not just of the laws of physics itself,
but of the universe, apparently.
We can see that not just in the distribution of the
microwave background, but also over here with the
distributions of galaxies.
The question is, what does this mean?
What do all of these symmetries that we've talked
about, what ultimately do they mean?
Many of you could rightly look at them and be like, yeah,
that's an interesting curiosity.
Yeah, sure, that's great.
But why?
Why do they ultimately matter, apart from
saying, isn't this beautiful?
This is, after all-- we're talking with physics.
This is not an art museum.
We can't look at it to be beautiful just for the sake of
being beautiful.
So one of the great breakthroughs with regards to
understanding why symmetry is so important came up with a
mathematician by the name of Emmy Noether Mathematicians
tend to revere her, by the way.
But many physicists either haven't heard of her or
remember very vaguely hearing about Noether's theorem in one
class when they were, say, a sophomore or junior in
college, and then they promptly forgot about her.
She is, to my mind, one of the most important mathematicians
that we haven't really heard of.
And I was in the same boat.
I mean, I'd seen Noether's theorem and promptly forgotten
about it, when I saw it in college.
And yet she is the foundation, in many respects, of things
like supersymmetry and these grand unified theories, all of
our understanding of why we have a fundamentally unified
theory-- a standard model, for that matter--
of particle physics.
And you know, she's an interesting case study for
many reasons.
I'm going to give you a couple slides, because I want to
point this out.
She has this parallel, in many ways, to Einstein.
Einstein very famously toiled in obscurity in a Swiss patent
office until his great breakthroughs, his miracle
year of 1905.
Noether had similar, but the motivations of the problems
were different, around the same time.
She was born in Erlangen.
Her father was a mathematics professor.
And in 1898, 1900, when she was going to school or would
have gone to school, they basically said, no, we can't
admit women.
That would overthrow all academic order.
So she audited all her classes.
She was-- you know, one of these cyber-schools that we'd
now have, I guess.
And she basically went in just to take her final exams in
Nuremberg--
which she, of course, aced.
She pursued her Ph.D. Eventually, of course, this
restriction on women was lifted.
She got her Ph.D. at Erlangen.
And she wasn't able to get a position.
She stayed at home, basically, writing important mathematics
papers, occasionally substitute-teaching for her
father, and that was it--
until Einstein came up with his theory of general
relativity.
So 1915, he came up his theory of general relativity.
Everyone recognized the importance of it almost
immediately.
And Noether was invited by David Hilbert and Klein to go
to Gottingen to explain it.
They said, basically, I'm sorry, the
university won't pay you.
I mean, Hilbert was this incredible advocate for her,
but he was very unsuccessful for a very, very long time.
But she went.
She went, and almost immediately created this
wonderful work which is known as Noether's theorem, which
I'm going to relate to you in just a moment.
But again, it's worth relating a little bit more her story.
Hilbert--
there's a quote here.
You know, "I don't see the sex of a candidate as an argument
against her admission as a Privatdozent." That's
basically an associate professor.
"After all, we're a university, not a bathhouse."
She developed her theorem, and it wasn't for another seven,
eight years that she was able to get any sort of paycheck,
incredibly tiny amount-- by the way, not just because she
was a woman, but also because she was a pacifist and a Jew
and a socialist, as I understand it.
1933, Nazis come to power and she goes to Bryn Mawr College,
and sadly, about 18 months later, passed away due to
complications from cancer surgery.
And here's this wonderful comment by Einstein.
"The most competent living mathematicians, Noether was
the most significant creative mathematical genius thus far
produced since the education of women began." And I mean,
that's understanding it.
I mean, not just amongst women, but among men, as well.
She was an incredible mathematician.
So what is it she told us?
What is it she said?
I mean, Noether's theorem, in words, sounds very, very--
it sounds pithy.
It almost sounds content-free.
But it says that every one of these symmetries that we've
been talking about produces a conserved quantity.
And conserved quantities, I will say, to physics is the
bread and butter of the universe.
We hear about things like conservation of energy.
Conservation of energy is incredibly useful because it
says, look, if you start off with an energy budget, if you
start off with the sun, and the sun does something, the
energy of the sun needs to go somewhere.
Either it heats the Earth, or it's converted into mass, or
whatever it may be--
the universe will contain a constant amount of energy, or
electric charge, or what have you.
And what Noether said is all these symmetries--
symmetries give rise to conserved quantities.
So for example, what she showed was the fact that the
laws of physics are the same everywhere in the universe
immediately gives rise to the conservation of momentum.
This is a big deal.
Conservation of momentum, of course, was known.
It's Newton's first law of motion.
Objects in motion stay in motion, blah, blah, blah.
But that was the starting point.
What Noether's theorem essentially did was she pushed
us back a step.
She said, no, there's something even more
fundamental than that.
Newton's first law itself is built on the idea of a
symmetry, on the fact that the laws of physics
are constant in space.
The fact that the laws of physics are the same in all
directions, that there's no terms that tell you about a
fixed direction of the universe, say that there's a
conservation of angular momentum.
The fact that the laws of physics are constant in time
immediately gives rise to this conservation of energy.
And there's more.
Now I'm sure some of you may have pure mathematics
backgrounds.
I don't expect you to parse this, either way.
I'm only putting this up here just to mention how important
this ends up being in our modern
understanding of physics.
Our modern understanding of physics is all of those
fundamental forces that we talk about are fundamentally
built on symmetries.
And these symmetries basically describe how the quantum
mechanical waves of a system can be changed without any of
the underlying quantities changing and without the
energies of interaction changing.
And it turns out you can describe, for example, U1.
I'll just give this simple example.
That's the phase of a wave.
If you can adjust the phase of a wave without changing
anything, that's a symmetry.
And what Noether--
well, what her successors ultimately showed is that
immediately gives rise to all of Maxwell's equations.
Absolutely incredible.
And show that there are conserved
quantities like charges.
And subsequently predicted the particle that's associated
with that, which is the photon.
In other words, Noether's theorem is the first of an
incredibly important step in showing--
make this very simple assumption about the laws
work, and you get the nature of the interaction, all the
equations, and the particle that relates it.
Same is true for the weak force, which produce the W and
Z particles, and for the strong force, which created
the gluons.
All built on symmetries.
Even if there those symmetries themselves don't look elegant,
putting a little plot of all the particles in the standard
model does start to look symmetric.
I mean, this does look much more orderly, where what we've
got here, ranked from bottom to top, are the charges of the
various particles.
And besides ordinary electrical charge, there are
various weak charges that are associated, as well.
And we can plot these things, and yes, indeed, this forms a
very beautiful pattern.
So one of the things that modern-day particle physicists
do is they come up with models based on symmetries.
And when you hear about things like at the Large Hadron
Collider, or Brookhaven, or elsewhere, discovering a new
particle that they thought to exist, it was because there
was a hole in this diagram of what we actually discovered
versus what had been predicted.
Even things that seem like they are symmetry breaking--
and they are symmetry breaking, things like the
discrete discovery about the Higgs boson, for example--
are built on the idea of a fundamental
symmetry of the universe.
The idea of the Higgs boson is that there is
an additional field.
An additional field that-- you know, you can look at that
little pattern--
was initially symmetric, but at some early time when the
universe became colder, much like ice freezing into a
crystal or anything else, that initial symmetry breaking got
frozen into the universe.
The Higgs--
yet another one of these fundamental discoveries of how
we understand the universe to work--
also built upon this symmetry foundation.
So if there are such beautiful symmetries in the universe--
and there are--
the question is, why isn't
everything perfectly symmetric?
I mean, I've pointed to individual cases--
matter and antimatter don't perfectly annihilate because
there seems to be a slight symmetry
breaking, blah, blah, blah.
But we can ask the question, where did all
of that come from?
And the answer seems to be that for all of these
symmetries in the universe, there's also sort of a
corresponding effect that makes the universe
interesting, that they can give rise to things like us,
that are complicated.
That are, if you don't mind me calling you this, breaks,
mars, in the beautiful simplicity and
symmetry of the universe.
And that is the randomness that comes
from quantum mechanics.
So quantum mechanics has random effects going into it.
And I've got a little illustration here.
You can imagine starting a series of tops,
just perfectly arrayed.
You may recognize these from a popular movie.
And you know that if you start spinning a top, eventually, a
real top will eventually topple.
Or not, depending on how much you choose to read into that.
[LAUGHTER]
DAVE GOLDBERG: But which one starts to topple first, and
which direction it topples in--
you can imagine these things, and you can see a few of them
starting to topple, just in this case.
You start with something perfectly symmetric.
You add a random component, and all of a sudden, you get
some beautiful, beautiful structure.
And in short, that is the story of our universe.
Thanks very much.
[APPLAUSE]
DAVE GOLDBERG: Questions?
This is the point when we have questions, right?
AUDIENCE: You were talking about how all of the particles
have anti versions of themselves.
What about the photon?
Is there an anti-photon?
DAVE GOLDBERG: So you're right.
I did do a little bit of shorthand, because I didn't
want to do too many caveats.
You're absolutely correct.
There are a few-- very few, fewer than you'd think--
particles that are their own anti-particles.
A photon is its own anti-particle, as
is the Higgs boson.
But most other particles are not their own anti-particles.
So for example, if I take an electron and a positron,
particle and anti-particle collide,
it creates two photons.
And there is no distinguishing between which one's the
particle version and which one's the anti-particle.
But you're quite right.
AUDIENCE: What's your personal opinion of supersymmetry,
given that LHC hasn't found it yet?
DAVE GOLDBERG: So that's a really good question.
So supersymmetry involves--
just to give a background to those who are unfamiliar.
Supersymmetry involves the relationship between the two
different fundamental types of particles.
There are particles called fermions that are electrons
that are the quarks that make up our protons and neutrons--
basically, the particles of matter.
And there are particles called bosons, which are essentially
the force-carriers.
Those include photons and gluons and the Higgs.
And the idea is, why should we have two such different groups
of particles in different quantities?
And not just why should we, but there's
other technical reasons.
You end up with, for example, certain particles based on
interactions should be hugely more massive than they are,
because the fermions sort of subtract mass
and bosons add mass.
And unless they cancel or partially cancel, what ends up
happening is you end up with a huge deficit,
one way or the other.
So there's a lot of sort of fundamental reasons.
And the idea that the people come up with is that for every
fermion, there must be a corresponding boson, and every
boson, there must be a corresponding fermion.
And many of these particles are undiscovered.
Many of these particles are likely to be unstable.
One hope, by the way, is that there is such a thing called
the lightest supersymmetric particle, one that probably
doesn't interact very strongly--
which we've sort of have a placeholder called a
neutralino--
that might be this missing dark-matter particle that
we've been looking for.
But we haven't detected any of these partners.
And experiments like Large Hadron Collider are capable of
measuring some of these, in principle.
And hasn't-- that's the upshot of your question.
So now you're caught up with the actual
nature of the question.
So the problem is--
I mean, supersymmetry is such a beautifully elegant theory.
And here's the problem with it.
I mean, it solves a lot of problems in particle physics.
It is almost impossible to disprove, because the number
of parameters that you can keep adjusting--
there's what's called minimal supersymmetric model, which
with reasonable parameters that most physicists would
agree are reasonable, has been essentially ruled out.
But you can keep adding bells and whistles, and you could
put in unreasonable that don't disprove the model.
My personal feeling is it's grasping at
straws, at this point.
I don't know what the better solution is.
There might be a more complicated version of
supersymmetry that could turn out to be correct, but it's
not looking good.
In fact, that's a position I take in the book, as well.
I sort of say, look, here's why we introduced it.
Here's why there are problems.
It hasn't been observed.
And every time every time that the threshold for observation
gets higher and higher, things look worse and worse.
So honestly, my gut is saying, at this point, no.
But I don't know.
I honestly don't.
And I really don't know what a better answer is.
Yes?
AUDIENCE: Hi.
So I have a question about the arrow of time.
So I've heard general arguments about how, like,
things are differentially symmetric in time.
So you could go backwards in time, and it
still looks like physics.
But a lot of arguments I hear about the arrow of time being
biased in the way we perceive it isn't this
differential argument.
It's this argument about ensembles and entropy and
stuff like that.
So to me, those two thoughts in my head are not reconciled
in any constructive way.
So could you elaborate on the relationship
between those two?
DAVE GOLDBERG: Right.
So yeah, I tried to take entire of Chapter Two of my
book and brush it into about 30 seconds of exposition here.
So it's a fair question.
So the issue is we've got two different senses--
at least two different philosophers would probably
give several more of the arrow of time.
One, you could call it, almost, a
psychological arrow, right?
The remembering the past and not the future.
You've got an entropy arrow, which is the increase in
entropy goes toward the future.
We've also got an arrow of time--
at least in the equations, where we're implying them-- in
the weak force.
Because we say the arrow goes this way, and that's the one
that defines how matter works, and so on.
And so part of the problem--
and this is something I try to approach in the book by
getting humans out of the equation entirely.
Like we don't know how brains work.
But thinking about how a robot or a disk drive or something
like that would work.
And you're a robot.
You're awoken.
You look at your disk.
And you do not know--
I mean, this is not about the arrow of time.
This is just sort of about the relationship between
information and entropy and memories.
You look at your disk, and you see that all of your
bits are set to 0.
And you say, OK, this is a clean slate.
I am--
I have no memories.
On the other hand, I wake you up as a robot, and there is
this pattern of zeros and ones.
A complicated pattern, not 0101.
Some complicated thing that you can't figure out,
necessarily, what it means.
You don't know anything else.
The question is, are those legitimate memories, or is
that noise?
And this is something we have no great reconciliation to.
Because our universe did start with low entropy, and yet
there's no physical principle that says why that should be.
And so the assumption is either there's a physical
principle we don't know, or there are two states of the
universe, one at low entropy and one at high entropy, and
there is another physical principle that basically says
the arrow of time is, by definition, moving from the
low to the high.
Now I'll tell you what my problem with that is, the sort
of entropy making time.
It is almost impossible to ever describe a single state
of the universe.
I mean, because the universe is not causally connected at
any point to one another.
So saying "The universe is increasing in entropy" is an
almost meaningless statement, because there's no process by
which something over the entire universe can be
integrated and calculated.
So the fact that the whole universe, or some part of the
multiverse, which is now not causally connected to one
another, where some property defined over that entire
region then defines the arrow of time, I don't see any
mechanism for how that could work.
So it is absolutely an open question, as to why the one
should be related to the other.
The fact that time exists as a dimension that behaves very,
very different from space--
like one could almost an anthropic argument about that.
So time does behave differently from space.
Regardless of which direction the arrow of time is, the fact
that there is dimension means that we can do things like
learn from the past, and make inferences, and so on.
Which we would not be able to do in quite the same way if we
lived in, say, four dimensions in space with a universe with
no time at all.
And it may be that there are part--
if you believe in the multiverse--
there are parts of the multiverse with different
dimensionality, and we are simply here because this is
the most complicated one that is anthropically favorable,
such that we would be able to exist at all, or anything
would be able to exist at all.
So I fully recognize I've given you a non-answer, but
I've hopefully sort of at least laid out the space of
the problem.
Yes?
AUDIENCE: So this is a question about Noether's
theorem and the weak force.
So in Noether's theorem, the things that are in variance,
like space and time, give us various conservation laws,
like momentum and energy.
So in a sort of not-to-mathematical, elaborate
way, what's the non-conserved thing?
What's the variant?
What is the dissipative piece of some equation that gives us
the lack of conservation, of CPT in weak force?
DAVE GOLDBERG: So this is the answer is, it's not a matter
of saying that it's not a conservation.
It's more a matter of saying that there's a very specific
thing that is being conserved, and it's the handedness.
So it's not just a matter of creating, in the weak force,
creating particles of all type.
It's that when you create a particle, you're creating a
left-handed neutrino.
Specifically a left-handed neutrino, and specifically a
right-handed anti-neutrino.
And what that means--
the reason that it's left-handed versus
right-handed means that essentially, you're creating
half as many types of particles.
We think, oh, what does it matter whether the thing is
spinning this way or spinning that way?
But from a physics perspective, those are two
different particles.
And so what it means is that we are essentially--
we are essentially getting half as many particles out of
the equations, half as many particles getting dumped out
of these equations, as we might otherwise have if the
thing was both left-handed and right-handed.
So we're only getting one.
So from a practical perspective, when we think
about all the statistics of the universe, we count in
order to figure out things like pressure and so on, how
many different particles could exist.
Electrons, which can be either spin direction,
we count for two.
Neutrinos, for each species, we only count for one, for
exactly that reason.
Because there's essentially half as many effective species
as there might otherwise be.
Are there other questions?
No.
Thank you.
Oh.
[APPLAUSE]
DAVE GOLDBERG: Thanks so much for coming.