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PROFESSOR: Welcome back to 8701.

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In the second chapter, we
will discuss symmetries

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and the importance of symmetries
in physics in general,

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but also especially in particle
physics and nuclear physics.

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So we start with a short
introductory video,

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and then we'll move on to
more details as we go along.

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The importance of symmetries
cannot be understated

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in physics.

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And there's two aspects
which are important.

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The first one is that
symmetries and conservation

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laws go hand in hand, as
discussed by Noether's theorem.

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To express the theorem
in an informal way,

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you can say that if a system
has a continuous symmetry

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property, then there are
corresponding properties whose

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values do not change with time,
meaning that they're conserved.

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You can express this
more sophisticated,

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and say to every differentiable
symmetry generated

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by local action,
there's correspondence.

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There's a correspondent
conserved current.

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And we're going to look at
those actions and currents

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as we go along.

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The second aspect,
beyond the fact

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that there's
conservation laws, is

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that you can understand
physics experiments and nature

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if you know that physics
has an underlying symmetry,

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without fully
understanding the physics

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or the mathematical
backgrounds in order

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to do calculation in detail.

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So knowing that there
is underlying symmetry

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can help in really expressing
or understanding the physics

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behavior of experiments.

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A few historic remarks
on Emmy Noether--

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Emmy Noether was born in Germany
in the 1880s in Erlangen,

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where she grew up
and also studied

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Mathematics at the University
of Erlangen. After getting

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her degree, she worked
for a full seven years

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at the university in
the Math Department,

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and received zero
dollars, and not

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just because it wasn't the
currency being used there,

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but at that time,
women didn't really

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have a prominent
role in academia.

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And so there was no
job for her to take.

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But her talents and
her qualification

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was seen in the mathematical
world at the time,

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specifically in the center
of the mathematical world,

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which was in Goettingen.
So Hilbert basically

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discovered her, and asked her
to come to Goettingen. In order

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to do habilitation, she did get
an habilitation in Goettingen

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in 1919, and then
stayed in Goettingen

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till the situation in Europe
degraded in the 1930s.

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She was born Jewish and
couldn't stay in Goettingen

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beyond the year
1933, and then had

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to immigrate into
the United States,

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where she worked at
Bryn Mawr College,

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and also with Princeton.

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Her work-- you see here
her habilitation, which

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is in German
[SPEAKING GERMAN],, "Invariant

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Variation of Problems,"
was highly regarded.

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And she had a lot of
influence and impact

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on various strands of
mathematics and physics.

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Unfortunately, she
passed away already

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when she was about 50 years old.

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She was diagnosed with
some sort of cancer,

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and passed away really,
really quickly after this,

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after some surgery.

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Her temperature rose and a few
days later, she passed away.

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To come back to symmetries
and conservation laws,

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every symmetry of nature
uses a conservation law.

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That is what Noether's
theorem tells you.

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And you can reverse
this to saying

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that every conservation
law in physics

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reflects an underlying symmetry.

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And examples for
this are the fact

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that the properties,
the laws of physics

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are invariant on the
time translation, meaning

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that physics is the same
yesterday, the same tomorrow,

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and it's going to be
the same next week.

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And out of this, we can
deduce energy conservation.

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Similarly, translation
in space results

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in a momentum
conservation, angular

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rotations or rotations
without the angular momentum.

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And then a little
bit harder to grasp,

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but we will see
this in more detail,

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internal symmetries can also
lead to conservation laws.

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And gauge transformation leads
to the conservation of charge.

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So there is internal
symmetries as well.

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Before we dive into more
detail, a few things.

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First, in many cases,
symmetry operations

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can be expressed via
matrices or groups.

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And there's a few
rules or operations

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which are rather important
and define symmetry.

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The first one is that
any symmetry operation

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has to have identity,
meaning there

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has to be an operation
which doesn't do anything

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with an element of this group.

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There has to be closure,
meaning that if you apply

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a first transformation
and then a second,

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the resulting transformation
is, again, part

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of the set of transformations.

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And there is an inverse, meaning
that if you rotate in one

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direction, you can rotate back.

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And there's
associativity, meaning

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that if you have a rotation
acting on two other rotations,

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you can regroup and follow
what's shown in this equation

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here.

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It's not clear that
you can reverse

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the order of certain elements
of your group or your symmetry

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operation.

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You can classify them, however.

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Those where you can commute,
those are called abelian

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groups, and those that you
cannot, those are non-abelian.

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All right, so with this,
we have introduced,

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with the first
video, symmetries.

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And now, we just
dive into more detail

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in understanding continuous
symmetries and also

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discrete symmetries, and
what we can learn from them.