1
00:00:00,500 --> 00:00:02,930
PROFESSOR: We can go
a step forward now.

2
00:00:07,740 --> 00:00:10,830
So we're trying to
understand these operators.

3
00:00:10,830 --> 00:00:17,270
And the next claim is
that if you multiply

4
00:00:17,270 --> 00:00:23,130
an arbitrary upward
permutation operator times S,

5
00:00:23,130 --> 00:00:26,690
or you multiply on the
other side, it's the same.

6
00:00:26,690 --> 00:00:32,390
And in fact, it's just S. On
the other hand, if you multiply

7
00:00:32,390 --> 00:00:38,750
by a permutation operator
A, it is still the same

8
00:00:38,750 --> 00:00:43,940
as multiplying A from the right
with that permutation operator.

9
00:00:43,940 --> 00:00:50,420
But this time you get
epsilon alpha 0 times A.

10
00:00:50,420 --> 00:00:56,390
So this is a claim
of the simple way

11
00:00:56,390 --> 00:01:00,640
in which permutation
operators interact with your A

12
00:01:00,640 --> 00:01:14,190
and S. So here is, again,
the same kind of argument.

13
00:01:14,190 --> 00:01:15,880
We'll say it first.

14
00:01:15,880 --> 00:01:33,390
So P alpha 0 acting on the
list of permutation operators

15
00:01:33,390 --> 00:01:40,920
just rearranges the list.

16
00:01:40,920 --> 00:01:44,350
We just argue that taking the
inverse of the permutation

17
00:01:44,350 --> 00:01:46,810
operators just
rearranges the list.

18
00:01:46,810 --> 00:01:50,650
Multiplying by P
alpha 0 the list

19
00:01:50,650 --> 00:01:53,430
of all permutation operators--

20
00:01:53,430 --> 00:01:55,840
here you have all the
permutation operators.

21
00:01:55,840 --> 00:02:00,520
Now you multiply by P alpha 0.

22
00:02:00,520 --> 00:02:05,200
And then the claim is that
they just go somewhere

23
00:02:05,200 --> 00:02:07,430
but you get the same
list at the end.

24
00:02:17,180 --> 00:02:20,900
Now that, again, you can
prove by the same argument.

25
00:02:20,900 --> 00:02:23,520
Show that two elements
that are different

26
00:02:23,520 --> 00:02:26,090
here go to two elements
that are different.

27
00:02:26,090 --> 00:02:28,820
And any element
here can be written

28
00:02:28,820 --> 00:02:32,780
as P alpha 0 of something here.

29
00:02:32,780 --> 00:02:36,270
I think you can do that.

30
00:02:36,270 --> 00:02:40,430
And it's a repeat of this kind
of argument we've been doing.

31
00:02:40,430 --> 00:02:43,320
So P alpha 0
rearranges the list.

32
00:02:43,320 --> 00:02:46,430
So that's a first thing.

33
00:02:46,430 --> 00:02:49,440
So therefore, let's
try one of these.

34
00:02:49,440 --> 00:02:55,790
So the first one, P alpha
0 on S. So all we can do

35
00:02:55,790 --> 00:02:58,580
is use the formula
for S. So you have 1

36
00:02:58,580 --> 00:03:02,060
over N factorial, the sum of.

37
00:03:02,060 --> 00:03:06,470
S is the sum over
alpha of P alpha.

38
00:03:06,470 --> 00:03:09,300
But now we have P alpha 0.

39
00:03:15,700 --> 00:03:19,780
And then we don't really
do much at this stage.

40
00:03:19,780 --> 00:03:28,330
We say, look, by this claim
that P alpha 0 rearranges

41
00:03:28,330 --> 00:03:34,210
all the operators, this is
just another sum, P beta,

42
00:03:34,210 --> 00:03:37,410
of all the operators.

43
00:03:37,410 --> 00:03:42,260
This P alpha 0 times P
alpha is another operator.

44
00:03:42,260 --> 00:03:47,160
But at the end of the job,
when you sum over alpha,

45
00:03:47,160 --> 00:03:49,860
you get just the
list rearranged.

46
00:03:49,860 --> 00:03:52,980
So here it is, the
list, rearranged.

47
00:03:52,980 --> 00:04:01,800
So this is back to S.

48
00:04:01,800 --> 00:04:03,450
For the A case--

49
00:04:03,450 --> 00:04:04,380
let's do it.

50
00:04:04,380 --> 00:04:05,710
P alpha 0.

51
00:04:05,710 --> 00:04:07,440
We have to work a little more.

52
00:04:07,440 --> 00:04:10,160
A is always a
little more subtle.

53
00:04:10,160 --> 00:04:17,290
1 over N factorial
alpha P alpha 0.

54
00:04:17,290 --> 00:04:25,565
But this time I have the epsilon
alpha of the A. P alpha 0 P

55
00:04:25,565 --> 00:04:26,065
alpha.

56
00:04:30,155 --> 00:04:30,655
OK.

57
00:04:35,420 --> 00:04:42,120
To make this clear, I'm going to
put the sine factor two times.

58
00:04:42,120 --> 00:04:44,020
So we'll have sum over alpha.

59
00:04:44,020 --> 00:04:45,920
I'll put epsilon alpha.

60
00:04:45,920 --> 00:04:49,190
And then I'll put
epsilon alpha 0.

61
00:04:52,170 --> 00:04:56,070
Then another epsilon alpha 0.

62
00:04:56,070 --> 00:05:01,840
Whatever epsilon alpha 0 is
I can do this because I'm

63
00:05:01,840 --> 00:05:03,780
inserting just the 1 here.

64
00:05:06,410 --> 00:05:10,670
And then we have
P alpha 0 P alpha.

65
00:05:13,990 --> 00:05:19,675
Now look at this operator
and look at this sine factor.

66
00:05:23,640 --> 00:05:27,330
A few things are clear here.

67
00:05:27,330 --> 00:05:31,440
This epsilon alpha 0 has
nothing to do with the sum.

68
00:05:31,440 --> 00:05:38,190
So this epsilon alpha 0
can go out of the sum.

69
00:05:38,190 --> 00:05:40,200
And then you have a sum--

70
00:05:40,200 --> 00:05:47,260
so 1 over N factorial
epsilon alpha 0.

71
00:05:47,260 --> 00:05:50,730
The sum of this
epsilon alpha epsilon

72
00:05:50,730 --> 00:05:54,765
alpha 0 P alpha 0 P alpha.

73
00:05:59,240 --> 00:06:07,590
But here, it's clear that
if this is some P beta,

74
00:06:07,590 --> 00:06:12,030
this sine factor is,
in fact, epsilon beta,

75
00:06:12,030 --> 00:06:15,570
because if P beta
is made by these two

76
00:06:15,570 --> 00:06:20,550
permutations of the sine
factor of that permutation,

77
00:06:20,550 --> 00:06:23,670
it's the product of
these two sine factors.

78
00:06:23,670 --> 00:06:27,390
This is kind of clear
because it's just

79
00:06:27,390 --> 00:06:29,920
a number of
transpositions again.

80
00:06:29,920 --> 00:06:34,530
This is counting the number
of transpositions mode two.

81
00:06:34,530 --> 00:06:37,170
This is counting the number
of transpositions mode two.

82
00:06:37,170 --> 00:06:40,350
And this works like that.

83
00:06:40,350 --> 00:06:42,600
If you have two
even permutations,

84
00:06:42,600 --> 00:06:44,790
the product is an
even permutation.

85
00:06:44,790 --> 00:06:47,370
And therefore, the two
pluses give you a plus.

86
00:06:47,370 --> 00:06:50,760
If one is even and one
is odd, the product

87
00:06:50,760 --> 00:06:54,230
is an odd permutation, and
you have a 1 and a minus 1.

88
00:06:54,230 --> 00:06:56,440
This product is minus 1.

89
00:06:56,440 --> 00:07:03,470
So this is an epsilon
beta P beta sum over beta.

90
00:07:03,470 --> 00:07:08,090
And that is the antisymmetrizer.

91
00:07:08,090 --> 00:07:11,720
So you've got epsilon
alpha 0 in here,

92
00:07:11,720 --> 00:07:16,210
and the rest is, again,
the antisymmetrizer.

93
00:07:16,210 --> 00:07:19,470
So these two properties
have been proven.

94
00:07:19,470 --> 00:07:23,130
I've proven them when
P is from the left.

95
00:07:23,130 --> 00:07:27,390
The proof when P is from
the right is very similar.

96
00:07:27,390 --> 00:07:28,320
Almost no changes.

97
00:07:33,181 --> 00:07:33,680
OK.

98
00:07:33,680 --> 00:07:35,520
Little by little,
these operators

99
00:07:35,520 --> 00:07:39,770
are a little complicated
because they have lots of terms.

100
00:07:39,770 --> 00:07:43,680
And therefore, manipulating
them requires some care.

101
00:07:43,680 --> 00:07:49,700
But now we're ready for the main
claim about these operators.

102
00:07:49,700 --> 00:07:54,140
We can do the
arithmetic with them,

103
00:07:54,140 --> 00:08:03,700
and finally say that A and
S are orthogonal projectors.

104
00:08:09,810 --> 00:08:11,410
So what does that mean?

105
00:08:11,410 --> 00:08:18,390
It means that S squared is equal
to S, A squared is equal to A,

106
00:08:18,390 --> 00:08:23,420
and we'll also show that
they're orthogonal to each other

107
00:08:23,420 --> 00:08:31,210
in that AS equal SA
equals 0 as well.

108
00:08:31,210 --> 00:08:36,270
So if you first apply the
symmetrization operator

109
00:08:36,270 --> 00:08:40,140
to a state and then you
apply the antisymmetrization,

110
00:08:40,140 --> 00:08:41,070
you get 0.

111
00:08:41,070 --> 00:08:45,420
So once you symmetrize,
that symmetric state

112
00:08:45,420 --> 00:08:49,140
has no component along
the antisymmetric state.

113
00:08:49,140 --> 00:08:50,800
They're complementary.

114
00:08:50,800 --> 00:08:55,030
So this is really what you
want from these operators.

115
00:08:55,030 --> 00:08:57,400
They're doing the right job.

116
00:08:57,400 --> 00:09:01,800
So this is the last thing
we need to prove about them.

117
00:09:01,800 --> 00:09:07,060
But by now, we really
have worked quite well.

118
00:09:07,060 --> 00:09:09,520
So it's going to be simple.

119
00:09:09,520 --> 00:09:12,840
The other statement, of course,
of orthogonal projectors

120
00:09:12,840 --> 00:09:16,920
also includes the fact that
they are Hermitian operators.

121
00:09:16,920 --> 00:09:21,000
So we already showed
that they are Hermitian,

122
00:09:21,000 --> 00:09:23,010
so we're almost there.

123
00:09:23,010 --> 00:09:26,980
We just need to prove that
these two things happen.

124
00:09:26,980 --> 00:09:31,080
So let's see how
difficult or easy this is.

125
00:09:31,080 --> 00:09:36,960
So first case, the S squared
equal S. S squared is

126
00:09:36,960 --> 00:09:46,920
S times S, so I will write
it as one S, here, acting

127
00:09:46,920 --> 00:09:52,670
on the other S.
Now, the next step

128
00:09:52,670 --> 00:10:02,675
is just erasing this
parenthesis over alpha.

129
00:10:05,630 --> 00:10:09,580
And now we have it there.

130
00:10:09,580 --> 00:10:19,990
The product of any
projector times S is S.

131
00:10:19,990 --> 00:10:23,380
So projector in alpha
is just S. So we'll

132
00:10:23,380 --> 00:10:29,590
have 1 over N factorial,
the sum over alpha of S.

133
00:10:29,590 --> 00:10:34,150
But S has no alpha index.

134
00:10:34,150 --> 00:10:37,370
So it goes out of the sum.

135
00:10:37,370 --> 00:10:43,200
It's equal to S times 1 over N
factorial, the sum over alpha

136
00:10:43,200 --> 00:10:43,900
of 1.

137
00:10:46,770 --> 00:10:50,790
And the sum over alpha
of 1 is a 1 added

138
00:10:50,790 --> 00:10:55,140
for each element of the
permutation group, which

139
00:10:55,140 --> 00:10:57,690
is N factorial elements.

140
00:10:57,690 --> 00:11:01,080
Therefore, this
sum is N factorial.

141
00:11:01,080 --> 00:11:04,990
And therefore, this cancels
with that N factorial

142
00:11:04,990 --> 00:11:09,290
and we get S, as we wanted.

143
00:11:09,290 --> 00:11:14,480
So S squared is S. It
is a projector operator.

144
00:11:14,480 --> 00:11:16,250
That's good.

145
00:11:16,250 --> 00:11:18,620
Let's do the second.

146
00:11:18,620 --> 00:11:28,400
A squared is going to be 1 over
N factorial sum epsilon alpha P

147
00:11:28,400 --> 00:11:39,788
alpha acting on an antisymmetric
A. So this is the first A,

148
00:11:39,788 --> 00:11:45,050
and there's the second A. I
should put the sum over alpha.

149
00:11:45,050 --> 00:11:51,860
But we already
calculated P alpha on A.

150
00:11:51,860 --> 00:11:55,430
This is sum over
alpha epsilon alpha.

151
00:11:55,430 --> 00:11:59,390
And P alpha on A was
calculated on that board.

152
00:11:59,390 --> 00:12:04,730
It's just another
epsilon alpha times A.

153
00:12:04,730 --> 00:12:07,220
Well, epsilon alpha
times epsilon alpha,

154
00:12:07,220 --> 00:12:12,900
whether is 1 or minus 1,
the product is just plus 1.

155
00:12:12,900 --> 00:12:16,130
So you get 1 over N factorial.

156
00:12:16,130 --> 00:12:20,900
And again, the sum over
alpha of A, the A goes out.

157
00:12:20,900 --> 00:12:29,710
Sum over alpha of 1,
and we're back to A.

158
00:12:29,710 --> 00:12:34,920
So A squared is also A.

159
00:12:34,920 --> 00:12:41,520
Finally, let's do one of
those mixed products, three.

160
00:12:41,520 --> 00:12:46,950
A multiplied with
S. So as usual,

161
00:12:46,950 --> 00:12:49,950
we write the first
operator explicitly.

162
00:12:49,950 --> 00:13:02,330
Epsilon alpha P alpha on S. But
P alpha on S, we know already,

163
00:13:02,330 --> 00:13:08,060
is just S. So 1
over N factorial sum

164
00:13:08,060 --> 00:13:15,840
over alpha epsilon
alpha times S.

165
00:13:15,840 --> 00:13:17,610
The S goes out.

166
00:13:17,610 --> 00:13:27,600
1 over N factorial S sum
over alpha of epsilon alpha.

167
00:13:27,600 --> 00:13:31,890
That's as far as we
can simplify this.

168
00:13:31,890 --> 00:13:38,380
And then we recall, in
any permutation group Sn,

169
00:13:38,380 --> 00:13:40,810
the number of even
permutations is

170
00:13:40,810 --> 00:13:43,790
equal to the number
of odd permutations.

171
00:13:43,790 --> 00:13:47,500
So there is equal
number of pluses

172
00:13:47,500 --> 00:13:51,660
and equal number of
minuses in that sum.

173
00:13:51,660 --> 00:13:54,160
So this sum is equal to 0.

174
00:13:57,220 --> 00:13:59,230
And therefore, this result is 0.

175
00:13:59,230 --> 00:14:05,500
The operators are orthogonal,
and they will do the job.

176
00:14:09,870 --> 00:14:10,480
So good.

177
00:14:10,480 --> 00:14:12,650
We have orthogonal operators.

178
00:14:12,650 --> 00:14:19,910
And therefore, we can do with
these projectors what you would

179
00:14:19,910 --> 00:14:23,510
expect we should be able to do.

180
00:14:23,510 --> 00:14:26,030
So here it is, the
last statement.

181
00:14:26,030 --> 00:14:40,220
S takes V tensor
N to Sym N of V.

182
00:14:40,220 --> 00:14:43,250
It's a projector to
the symmetric states.

183
00:14:43,250 --> 00:14:44,960
Why is that?

184
00:14:44,960 --> 00:14:52,070
Well, take a psi belonging
to V tensor N. Then

185
00:14:52,070 --> 00:14:57,870
consider S on psi.

186
00:14:57,870 --> 00:15:02,120
That's the operator, S,
acting on the state, psi.

187
00:15:02,120 --> 00:15:05,120
And that should be
a symmetric state.

188
00:15:05,120 --> 00:15:13,790
And to check that it's symmetric
we apply a P alpha on S psi.

189
00:15:13,790 --> 00:15:18,980
But P alpha on S is S,
so that's just S psi.

190
00:15:18,980 --> 00:15:25,820
So the state of psi
is independent--

191
00:15:25,820 --> 00:15:28,720
unchanged-- by the
action of the P alpha.

192
00:15:28,720 --> 00:15:32,380
So the state S psi is, indeed--

193
00:15:32,380 --> 00:15:42,670
this implies that S psi
belongs to Sym N V, as claimed.

194
00:15:42,670 --> 00:15:49,990
So the operator does
what you wanted it to do.

195
00:15:49,990 --> 00:15:54,500
This property helps
for the other one.

196
00:15:54,500 --> 00:16:02,440
So A is an operator that takes
arbitrary states in the tensor

197
00:16:02,440 --> 00:16:09,340
product to anti N V.
So indeed, if, again,

198
00:16:09,340 --> 00:16:16,270
you have some psi in V N,
you can then form A psi

199
00:16:16,270 --> 00:16:18,050
and see where it lies.

200
00:16:18,050 --> 00:16:22,255
For that you apply
P alpha on A psi.

201
00:16:25,390 --> 00:16:29,800
But P alpha on A from
that middle blackboard

202
00:16:29,800 --> 00:16:36,490
is epsilon alpha A psi.

203
00:16:36,490 --> 00:16:40,300
But that is precisely
the definition

204
00:16:40,300 --> 00:16:42,310
of an antisymmetric state.

205
00:16:47,780 --> 00:16:55,670
So indeed, A takes states
to antisymmetric states.

206
00:16:55,670 --> 00:17:06,849
If you think of the vector
space V tensor N here,

207
00:17:06,849 --> 00:17:15,780
there's some subspace here, Sym
N V, and some subspace here,

208
00:17:15,780 --> 00:17:23,040
anti N V. And
these two subspaces

209
00:17:23,040 --> 00:17:26,859
don't contain any
common element.

210
00:17:26,859 --> 00:17:32,310
Maybe I should--
well, I don't know.

211
00:17:32,310 --> 00:17:36,150
Maybe I should really make
them touch at one point.

212
00:17:36,150 --> 00:17:39,100
That point would be what vector?

213
00:17:39,100 --> 00:17:40,330
AUDIENCE: 0?

214
00:17:40,330 --> 00:17:41,040
PROFESSOR: 0.

215
00:17:41,040 --> 00:17:42,520
Yes.

216
00:17:42,520 --> 00:17:48,010
If you have a vector subspace
it has to have the 0 vector.

217
00:17:48,010 --> 00:17:51,450
So here is the 0 vector.

218
00:17:56,840 --> 00:18:00,140
But they don't fill
the whole thing.

219
00:18:00,140 --> 00:18:02,810
In general, they don't
fill the whole thing.

220
00:18:06,310 --> 00:18:10,730
When you have two particle
states, however, they do.

221
00:18:10,730 --> 00:18:16,550
Remember when we
had N equals 2, we

222
00:18:16,550 --> 00:18:24,290
had the symmetrizer
was 1 plus P 2 1.

223
00:18:24,290 --> 00:18:30,440
And A was 1/2 1 minus P 2 1.

224
00:18:30,440 --> 00:18:36,620
In that case, the symmetrizer
plus the antisymmetrizer

225
00:18:36,620 --> 00:18:38,270
was equal to 1.

226
00:18:38,270 --> 00:18:42,560
But this is only
for two particles.

227
00:18:42,560 --> 00:18:47,760
If you have three particles,
this will not work.

228
00:18:47,760 --> 00:18:52,772
Suppose I get three particles--
maybe I'll use this blackboard.

229
00:19:13,960 --> 00:19:20,660
And the fact that S plus A
is equal to the unit operator

230
00:19:20,660 --> 00:19:22,880
means something, of course.

231
00:19:22,880 --> 00:19:27,880
It means that still
for N equals 2,

232
00:19:27,880 --> 00:19:30,520
means that any
vector can be written

233
00:19:30,520 --> 00:19:37,660
as S plus A times the
vector, because this is 1.

234
00:19:37,660 --> 00:19:43,210
And therefore, it's SV plus AV.

235
00:19:43,210 --> 00:19:50,470
So any vector where V belongs to
the tensor product, because you

236
00:19:50,470 --> 00:19:54,760
have two particles, can be
written as a symmetric state

237
00:19:54,760 --> 00:19:56,980
plus an antisymmetric state.

238
00:19:56,980 --> 00:20:00,430
That is the statement
that the symmetric states

239
00:20:00,430 --> 00:20:04,550
plus the antisymmetric
state span the space.

240
00:20:04,550 --> 00:20:08,650
So that is true
for N equals to 2.

241
00:20:08,650 --> 00:20:15,520
So in N equals to 2, the
two states, the two spaces,

242
00:20:15,520 --> 00:20:16,615
span the space.

243
00:20:19,770 --> 00:20:26,730
On the other hand, for N
equals to 3, remember we had--

244
00:20:26,730 --> 00:20:30,420
and this is a good way to
write some things explicitly.

245
00:20:30,420 --> 00:20:34,290
If you have N equals to 3,
you have the permutation 1 2

246
00:20:34,290 --> 00:20:39,990
3, which is the identity,
the permutation 3 1 2,

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00:20:39,990 --> 00:20:44,640
the permutation of 2 3 1.

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00:20:44,640 --> 00:20:49,360
All these were
even permutations.

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00:20:49,360 --> 00:20:55,930
This is original, and this can
be built with two permutations.

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00:20:55,930 --> 00:21:02,125
Then you have P 1 3 2,
where you flip just 2 and 3.

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This has one transposition.

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00:21:05,240 --> 00:21:07,060
You can cycle them.

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00:21:07,060 --> 00:21:09,354
3 2 1.

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00:21:09,354 --> 00:21:10,180
P 2 1 3.

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00:21:14,365 --> 00:21:20,890
These are the even, and here
are the odd permutations.

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00:21:26,490 --> 00:21:28,940
So this was for three particles.

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00:21:28,940 --> 00:21:32,420
Six operators, three
even, three odd.

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00:21:32,420 --> 00:21:35,580
So what is the symmetrizer?

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00:21:35,580 --> 00:21:39,570
The symmetrizer, 1
over 3 factorial.

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00:21:39,570 --> 00:21:40,590
And you should sum.

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00:21:40,590 --> 00:21:44,990
So it's 1 over 3
factorial, which is 6.

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00:21:48,520 --> 00:21:54,700
The 1 plus P 3 1 2 plus P 2 3 1.

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00:21:54,700 --> 00:21:55,750
I keep adding.

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00:21:55,750 --> 00:21:58,280
You should add all of them.

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00:21:58,280 --> 00:22:02,255
Plus P 3 2 1 plus P 2 1 3.

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00:22:02,255 --> 00:22:03,860
Boom.

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00:22:03,860 --> 00:22:04,600
That's it.

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00:22:04,600 --> 00:22:07,690
All the ones, you
have to add them.

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00:22:07,690 --> 00:22:12,560
What is A is 1/6.

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00:22:12,560 --> 00:22:16,320
You add the ones that are even.

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00:22:16,320 --> 00:22:22,920
1 plus P 3 1 2 plus P
2 1 3, and you subtract

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00:22:22,920 --> 00:22:24,660
all the ones that are odd.

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00:22:24,660 --> 00:22:31,560
So you minus P 1 3 2 minus
P 3 2 1 minus P 2 1 3.

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00:22:35,000 --> 00:22:42,130
So those are the two operators
in case you wanted to see them.

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00:22:42,130 --> 00:22:53,350
But now notice that A plus S
is equal to 1/3 1 plus P 3 1 2

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00:22:53,350 --> 00:22:56,560
plus P 2 3 1.

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00:22:56,560 --> 00:23:02,380
And that's not the equal to
the identity, as it was before.

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00:23:02,380 --> 00:23:04,090
It's equal to something else.

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00:23:04,090 --> 00:23:11,440
So you cannot say that the
vector is equal to A plus S

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00:23:11,440 --> 00:23:12,430
times the vector.

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00:23:12,430 --> 00:23:15,820
That's not true here.

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00:23:15,820 --> 00:23:18,020
Doesn't happen.

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00:23:18,020 --> 00:23:22,390
So there is more to
the triple tensor

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00:23:22,390 --> 00:23:24,580
product than the
symmetric states

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00:23:24,580 --> 00:23:27,100
and the antisymmetric states.

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00:23:27,100 --> 00:23:31,780
That's a nice subject for a
slightly more advanced course

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00:23:31,780 --> 00:23:36,220
in quantum mechanics where you
studying the Young tableaux

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00:23:36,220 --> 00:23:38,540
of the permutation group.

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00:23:38,540 --> 00:23:44,650
And while for two things, two
objects that are represented

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00:23:44,650 --> 00:23:46,600
with Young tableaux
with little squares,

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00:23:46,600 --> 00:23:48,190
you can form something
that is called

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00:23:48,190 --> 00:23:54,570
symmetric and antisymmetric,
for three objects you

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00:23:54,570 --> 00:23:58,440
can form a symmetric
object, you can

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00:23:58,440 --> 00:24:01,710
form an antisymmetric
object, but you can

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00:24:01,710 --> 00:24:07,800
form a mixed symmetry object.

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00:24:07,800 --> 00:24:10,450
And this Young
tableaux represent

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00:24:10,450 --> 00:24:13,160
the tensors that enter here.

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00:24:13,160 --> 00:24:16,610
There are other projectors
to this kind of object.

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00:24:16,610 --> 00:24:20,180
These objects are not
symmetric nor antisymmetric.

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00:24:20,180 --> 00:24:24,610
They have partial symmetry
and partial antisymmetry.

301
00:24:24,610 --> 00:24:29,620
And they take a while to get
around to visualize them.

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00:24:29,620 --> 00:24:34,060
But that's why the permutation
group is interesting

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00:24:34,060 --> 00:24:39,290
and why we try to understand
it better and better.

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00:24:39,290 --> 00:24:46,840
So that's basically
the mathematics

305
00:24:46,840 --> 00:24:50,510
that we need for
understanding permutations.

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00:24:50,510 --> 00:24:52,960
So we're going to do
some of the physics that

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00:24:52,960 --> 00:24:55,680
follows from this now.