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PROFESSOR: So so far
even though these things

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look maybe interesting
or a little familiar,

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we have not yet stated clearly
how they apply to physics.

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We've been talking about vector
spaces, V, for a particle.

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Then V tensor N. And we've
looked at states there.

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We've looked at permutation
operators there.

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Symmetric states there.

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Antisymmetric states there.

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What is missing is
something that connects

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it to quantum mechanics.

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And that is given by the
so-called symmetrization

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postulate.

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So it's a postulate.

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It's something that you
technically can't derive.

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Therefore, you postulate.

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You can say it's an extra
axiom even, quantum mechanics.

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You would say also
that probably there's

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no other way to do things.

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So in some sense, it's forced.

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But I think it's more honest to
admit it's an extra postulate.

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So here it goes.

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I'll read it first.

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Then I'll write it.

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So it says the following.

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If you have a system of
N identical particles,

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arbitrary states in V tensor
N are not physical states.

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The physical states
are the states

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that are totally symmetric.

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In that case, the particles
are called bosons.

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Or totally antisymmetric,
in which case

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the particles are
called fermions.

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That's basically it.

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The arbitrary state that
is neither symmetric

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nor antisymmetric,
and all that is not

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a physical state of a system
of identical particles.

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So it's stated here.

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In a system with N
identical particles,

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physical states are not
arbitrary states in V tensor N.

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Rather, they are
totally symmetric.

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In parentheses, they
belong to SymN of V.

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In which case, the particles
are said to be bosons

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or they are totally
antisymmetric.

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Are totally antisymmetric.

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They belong to anti NV.

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In which case, they are
said to be fermions.

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All right.

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So this is really
pretty fundamental.

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It's a statement that
we began our discussion

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by saying we have an
electron at this up

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and another electron
at this down.

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Is it the plus minus state or
is it the minus plus state?

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And we said it's neither one.

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You cannot declare those
two to be equivalent.

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And that's what this says.

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It's not arbitrary
states in this.

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But they have to become
totally symmetric,

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if they're bosons or
totally antisymmetric

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if they are fermions.

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So we have to make
a few comments.

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And that's what will keep
us busy for the next 20,

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25 minutes.

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And to understand this
and see what it implies

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and how we use it.

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So here it is.

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The statement we have
made is a statement

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of fact in three dimensions.

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There's no particles that
we like and we study.

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There are further
possibilities in worlds

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of lower dimension two, three
dimensions, space, time,

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or so two dimensional
space, that

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allows for further
kinds of statistics

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that are interesting.

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But this is the
general statement.

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So the first comment is this
is the general statement.

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But further statistics,
other types of space,

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exist in two dimensions,
not in three, where we live.

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We say three in
the Galilean ways.

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3 plus 1 time.

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But this is spatial dimensions.

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This is the
statistical definition

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of bosons and fermions.

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But then comes the spin
statistics theorem.

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That applies then you understand
with quantum mechanics.

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And that's a deep
theorem that ends up

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telling you that
particles with spin 0, 1,

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2 integer are bosons.

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And particles with spin 1/2,
3/2 [INAUDIBLE] are fermions.

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So it's a great achievement
of quantum field

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theory associating the
statistical properties

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with the spin.

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It's a deep connection.

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It's valid for
elementary particles

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or composite particles.

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The list of particles
is not that big.

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Particles of spin 1,
we know a lot of them.

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A photon is spin 1.

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Gluons are spin 1.

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The w's, the c's are
spin 1 particles.

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Particles spin 0.

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We didn't know an
elementary one for a while.

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We finally know the
[INAUDIBLE] is there, spin 0.

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Particles of spin 2.

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Just [INAUDIBLE] along.

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Higher spin unknown.

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It may exist.

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String theory has them.

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All kinds of theories
postulate them.

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They may exist.

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They may not.

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Spin of 1/2, fractional spin.

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Well, all the matter particles,
quarks, muons, leptons,

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neutrinos are spin 1/2.

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Spin 3/2 is hypothetical
as an elementary particle.

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If it exists, it would
be called the gravitino.

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There are composite particles.

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You could [INAUDIBLE]
with quarks,

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or with other particles,
mesons, baryons.

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You can get several spins.

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Not with elementary particles.

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Many time unstable particles.

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So anyway, those are
very interesting things.

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Now next fact is that you can
go from elementary particles

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to composite particles.

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So if you have, for
example, the hydrogen atom.

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That this has a proton
and an electron.

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And you want to figure out
if it's a boson or a fermion.

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Well, you think of another
hydrogen atom here, p and e.

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And you write a
wave function that

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involves the protons, the first
proton, the second proton,

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the first electron, and
the second electron.

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That's a little bit
of a funny notation.

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This would correspond
to maybe the coordinates

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of the first proton, the second
proton, the first electron,

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the second electron.

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Since protons are identical
particles that are fermions,

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the wave function
must change sign

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if P1 and P2 are exchanged.

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There would be a minus sign.

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It's antisymmetric.

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Since the electrons
are fermions,

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it should antisymmetric
by exchanging this 2.

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So it's also antisymmetric
under this exchange.

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So finally, if you exchange
this item with this item,

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you must exchange the two
protons and the two electrons

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at the same time.

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Two minuses and you get a plus.

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Therefore, the hydrogen
atom is a boson.

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So these statistical
properties build up.

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So an object built
with a number of bosons

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and a number of fermions
will be a boson or a fermion,

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depending on those numbers.

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So that's a nice thing.

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So the hydrogen atom is a boson.