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PROFESSOR: So one more thing
let's do with this operator.

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So we're getting accustomed
to these operators,

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and these permutation
operators can also

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act on operators themselves.

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So that's important.

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So consider the
action on operators.

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So for example,
an operator B that

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acts belonging to
the linear operator

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some V, when acting
on V tensor V,

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we define two
operators, B1 and B2.

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And you define them
in an obvious way,

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like B1 acting on
Ui 1 tensor Uj 2.

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OK, B as an operator
knows how to act

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on every vector on the
vector space capital V.

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So when you say 1, you're
meaning that this operator acts

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on the first Hilbert space.

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So this is equal to B
times Ui 1 tensor Uj 2.

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So it just acts on
the first state.

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How does it act?

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Via B, that is an operator,
in the vector space V.

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Similarly, if you have
B2 of Ui tensor Uj 2

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you have Ui 1
tensor BUj 2 to OK.

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So these are operators that
act either on the first state

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or in the second state.

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So the permutation operators can
do things to these operators,

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as well.

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So we can ask a
question, what is P21B--

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should I start with one?

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Yes, one-- P21 dagger.

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Remember, when you ask how an
operator acts on an operator,

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you always have the
operator that you're

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acting with come from the
left and from the right.

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That is the natural way
in which an operator

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acts on an operator.

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You can think of this
thing as your operator

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is being acted upon
as having surrounded

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by a [INAUDIBLE]
and a [INAUDIBLE]..

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And then when the
states transform,

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one transforms with U, one
transforms with U dagger.

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So always the action
on an operator

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is with a U and a U dagger.

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So if you ask how does the
permutation operator act on B,

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you don't ask generally what's
the product of P times B.

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You ask this question.

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This is the question that
may have a nice answer.

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Then we'll see that there's
other ways of doing this.

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So we want to investigate
this operator.

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So what I can do is
let it act Ui, Uj.

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So what do we get?

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We get P21B1.

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Now, P21 dagger, we saw
that it's Hermitian anyway,

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so it's just P21.

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And now it acts on Ui 1.

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I'll put the j here, and Uj 2.

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So I let up the
P21 on that state,

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and that the moves
the i's and the j's.

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Now, B1 acts on the
first Hilbert space.

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So now we have P21 and we
have BUj 1 and tensor Ui 2.

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Now, P21 is supposed to put the
second state in position one

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and the first state
in position two.

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So this is Ui 1 BUj 2.

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I could put this thing--

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BUj 2.

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And then you see, oh,
this term is here.

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So this is nothing else than
B2 acting on the same state

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of the Ui Uj, which means--

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I guess I could use
this blackboard--

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that P21 B1 P21 dagger is B2.

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So it has moved you.

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The operator used to act
on the first particle.

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Two and one changes the first
particle with the second.

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It moved it into the other one.

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Similarly, you
could do this also.

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Would not be a surprise
to you that P21 B2 P21

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dagger is equal to B1.

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And you don't have to do
the same argument again.

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You could multiply
this equation by P21

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from the left and P21
dagger from the right.

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These things become one
and one, and the operators

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remain on the other
side and gives you this.

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So this second equation comes
directly from the first.

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You don't have to go
through the arguments.

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So what is the
use of this thing?

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You may have a
Hamiltonian, and you

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want to understand
what it means to have

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a symmetric Hamiltonian.

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And these operators
allow you to do that.

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So for example, you may
have an operator O 1,2.

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What is an operator O 1,2?

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It's an operator build on things
that act on one or act on two.

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So if you want to
imagine it, it could

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be an O that depends
on the operator

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A acting on the first
label, an operator B acting

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on the second label, an
operator C on the first label,

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an operator D on
the second label.

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Could be a very complicated
product of those operators

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acting on all kinds of labels.

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Now suppose you act with
P21 O 1,2 P21 dagger.

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Now, the great advantage of
having a P and a P dagger

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acting on a string
of operators is

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that it is the same as
having a P and a P dagger

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acting on each one.

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Remember, if you have
like P and P dagger,

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and it's a unitary
operator on ABC,

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it's the same as PAP dagger,
PABP dagger, PCP dagger.

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It's like acting on each one.

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So when you have this P21
P21 dagger acting on this,

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it's as if each one of those is
surrounded by a P21 P21 dagger.

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So each label one will
become a label two,

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and each label two will
become a label one.

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And therefore, this operation
is going to give you O2,1

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for an arbitrary operator
acting on these two labels.

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Now, it may happen that
the operator is symmetric

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if O is symmetric.

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By that, we mean O
2,1 is equal to O 1,2.

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If that happens--
if that happens--

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then from this equation you
would have P21 O 1,2 P21 dagger

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is O1,2 is itself.

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And you could multiply
by a P21 from the right,

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giving you P21 O 1,2 equal O
1,2 You're multiplying by a P21

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from the right that
cancels this P21 dagger.

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P21.

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And there you see that
an operator is symmetric

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if it commutes with the
permutation operator.

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So if always symmetric, this
is true, and this is true,

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and then finally, P21 with
O 1,2 commutator is 0.

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Oops.

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Too low.

129
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Let me see.

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It's a commutator.

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It's 0.

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So that's basically how you
manipulate these operators

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on this Hilbert space.