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PROFESSOR: So today, I'd
like to wrap together

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00:00:25,490 --> 00:00:29,390
and summarize everything that
we have been doing the last 10,

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00:00:29,390 --> 00:00:31,520
12 lectures.

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00:00:31,520 --> 00:00:37,320
So the idea started by saying
that you take something

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like a magnet, you
change its temperature.

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00:00:41,450 --> 00:00:44,080
You go from one phase
that is paramagnet

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00:00:44,080 --> 00:00:49,182
at some critical temperature
Tc to some other phase that

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00:00:49,182 --> 00:00:50,430
is a ferromagnet.

16
00:00:50,430 --> 00:00:52,570
Naturally, the
direction of magnetism

17
00:00:52,570 --> 00:00:57,520
depends whether you put on a
magnetic field and went to 0.

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00:00:57,520 --> 00:01:00,239
So there's a lot of
these transitions

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00:01:00,239 --> 00:01:01,280
that involve ferromagnet.

20
00:01:05,530 --> 00:01:09,570
There were a set of other
transitions that involved,

21
00:01:09,570 --> 00:01:11,823
for example, superfluids
or superconductivity.

22
00:01:15,530 --> 00:01:19,820
And the most common
example of phase

23
00:01:19,820 --> 00:01:23,990
transition being
liquid gas, which

24
00:01:23,990 --> 00:01:26,390
has a coexistence line
that also terminates

25
00:01:26,390 --> 00:01:28,160
at the critical point.

26
00:01:28,160 --> 00:01:30,820
So you have to turn
it around a little bit

27
00:01:30,820 --> 00:01:36,200
to get a coexistence
line like this.

28
00:01:36,200 --> 00:01:37,260
Fine.

29
00:01:37,260 --> 00:01:38,870
So there are phase transitions.

30
00:01:38,870 --> 00:01:44,160
The interesting thing was that
when people did successively

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00:01:44,160 --> 00:01:46,120
better and better
experiments, they

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00:01:46,120 --> 00:01:48,810
found that the singularities
in the vicinity

33
00:01:48,810 --> 00:01:54,910
of these critical
points are universal.

34
00:01:54,910 --> 00:01:58,150
That is, it doesn't matter
whether you have iron, nickel,

35
00:01:58,150 --> 00:02:01,410
or some other thing that is
undergoing ferromagnetism.

36
00:02:01,410 --> 00:02:05,690
You can characterize the
divergence of the heat capacity

37
00:02:05,690 --> 00:02:12,180
to an exponent alpha, which
ferromagnets is minus 0.12.

38
00:02:12,180 --> 00:02:17,770
For superfluids, we said
that people even take things

39
00:02:17,770 --> 00:02:21,670
on satellites to calculate
this exponent to much

40
00:02:21,670 --> 00:02:24,240
higher accuracy than
I have indicated.

41
00:02:24,240 --> 00:02:27,710
For the case of liquid gas,
there is a true divergence

42
00:02:27,710 --> 00:02:33,790
and the exponent is around 0.11.

43
00:02:33,790 --> 00:02:35,810
There are a whole set
of other exponents

44
00:02:35,810 --> 00:02:37,505
that I also mentioned.

45
00:02:37,505 --> 00:02:45,050
There is the exponent beta for
how the magnetization vanishes.

46
00:02:45,050 --> 00:02:52,106
And the values here
were 0.37, 0.35, 0.33.

47
00:02:52,106 --> 00:02:58,130
There is the divergence of
the susceptibility gamma that

48
00:02:58,130 --> 00:03:01,240
is characterized
through exponents

49
00:03:01,240 --> 00:03:10,950
that are 1.39, 1.32, 1.24.

50
00:03:10,950 --> 00:03:14,900
And there is a divergence
of the correlation length

51
00:03:14,900 --> 00:03:26,050
characterized by exponents
[INAUDIBLE] 0.71, 0.67, 0.63.

52
00:03:26,050 --> 00:03:32,400
So there is this
table of pure numbers

53
00:03:32,400 --> 00:03:36,590
that don't depend on the
property of the material

54
00:03:36,590 --> 00:03:38,250
that you are looking at.

55
00:03:38,250 --> 00:03:45,240
And the fact that you don't
have this material dependence

56
00:03:45,240 --> 00:03:49,800
suggests that these pure
numbers are some characteristics

57
00:03:49,800 --> 00:03:52,900
of the collective
behavior that gives rise

58
00:03:52,900 --> 00:03:56,730
to what's happening at
this critical point.

59
00:03:56,730 --> 00:04:00,380
And we should be able to
devise some kind of a theory

60
00:04:00,380 --> 00:04:02,600
to understand
that, maybe extract

61
00:04:02,600 --> 00:04:07,970
these nice, pure numbers,
which are certainly embedded

62
00:04:07,970 --> 00:04:12,420
in the physics of the problem
that we are looking at.

63
00:04:12,420 --> 00:04:19,220
So the first idea that
we explored conceptually

64
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due to lambda probably,
probably others,

65
00:04:24,760 --> 00:04:28,205
is that we should construct
the statistical field.

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00:04:32,426 --> 00:04:37,530
That is, what is
happening is irrespective

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00:04:37,530 --> 00:04:41,070
of whether we are dealing with
nickel or iron, et cetera.

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00:04:41,070 --> 00:04:43,690
So the properties of
the microscopic elements

69
00:04:43,690 --> 00:04:45,560
should disappear
and we should be

70
00:04:45,560 --> 00:04:50,280
focusing on the quantity that is
undergoing a phase transition.

71
00:04:50,280 --> 00:04:55,130
And that quantity, we said, is
some kind of a magnetization.

72
00:04:55,130 --> 00:04:59,330
And what distinguishes
the different systems is

73
00:04:59,330 --> 00:05:01,340
that for ferromagnet,
it's certainly

74
00:05:01,340 --> 00:05:03,160
a three-component system.

75
00:05:03,160 --> 00:05:07,190
But in general, for superfluid
we would have a phase.

76
00:05:07,190 --> 00:05:09,135
And that's a
two-component object,

77
00:05:09,135 --> 00:05:12,550
so we introduced
this parameter n

78
00:05:12,550 --> 00:05:17,300
that characterized the symmetry
of the order parameter.

79
00:05:17,300 --> 00:05:20,490
And in the same way,
we said let's look

80
00:05:20,490 --> 00:05:24,150
at things that are
embedded in space.

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00:05:24,150 --> 00:05:27,410
That is, in general,
d dimensional.

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00:05:27,410 --> 00:05:34,430
So our specification of
the statistical field

83
00:05:34,430 --> 00:05:37,830
was on the basis
of these things.

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00:05:37,830 --> 00:05:42,930
And the idea of lambda was
to construct a probability

85
00:05:42,930 --> 00:05:47,380
for this field,
configurations across space.

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00:05:47,380 --> 00:05:52,530
Once we had that, we could
calculate a partition function,

87
00:05:52,530 --> 00:05:55,735
let's say, by integrating
over all configurations

88
00:05:55,735 --> 00:05:59,860
of this field of some
kind of a weight.

89
00:06:05,800 --> 00:06:10,540
And we constructed the weight.

90
00:06:10,540 --> 00:06:18,460
We wrote something as beta
H was an integral d dx.

91
00:06:18,460 --> 00:06:21,800
And then we put a whole
bunch of things in here.

92
00:06:21,800 --> 00:06:26,630
We said we could have something
like t over 2m squared.

93
00:06:26,630 --> 00:06:30,530
We had gradient of m squared.

94
00:06:30,530 --> 00:06:33,860
Potentially, we could
have higher derivatives

95
00:06:33,860 --> 00:06:37,850
staying at the
order of m squared.

96
00:06:37,850 --> 00:06:39,630
And so this list
of things that I

97
00:06:39,630 --> 00:06:45,090
could put that are all order of
m squared is quite extensive.

98
00:06:45,090 --> 00:06:48,710
Then I could have things
that are fourth order,

99
00:06:48,710 --> 00:06:51,940
like u m to the fourth.

100
00:06:51,940 --> 00:06:56,250
And we saw that when we
performed this renormalization

101
00:06:56,250 --> 00:07:01,520
group last time around, that a
term that we typically had not

102
00:07:01,520 --> 00:07:04,050
paid attention to was generated.

103
00:07:04,050 --> 00:07:06,430
Something that was,
again, order of m

104
00:07:06,430 --> 00:07:11,440
to the fourth, but had
a structure maybe like m

105
00:07:11,440 --> 00:07:14,870
squared gradient of m
squared, or some other form

106
00:07:14,870 --> 00:07:18,040
of the two-derivative operator.

107
00:07:18,040 --> 00:07:20,190
This is OK.

108
00:07:20,190 --> 00:07:21,880
There could be other
types of things

109
00:07:21,880 --> 00:07:24,490
that have four m's in it.

110
00:07:24,490 --> 00:07:29,350
And you could have something
that is m to the sixth, m

111
00:07:29,350 --> 00:07:30,500
to the eighth, et cetera.

112
00:07:34,060 --> 00:07:42,650
So the idea of lambda was to
include all kinds of terms

113
00:07:42,650 --> 00:07:44,760
that you can put.

114
00:07:44,760 --> 00:07:49,200
But actually, we have
already constrained terms.

115
00:07:49,200 --> 00:07:53,830
So the idea of
lambda is all terms

116
00:07:53,830 --> 00:07:59,220
consistent with some
constraints that you put.

117
00:07:59,220 --> 00:08:02,690
What are the
constraint that we put?

118
00:08:02,690 --> 00:08:10,480
We put locality in that we wrote
this as an integral over x.

119
00:08:10,480 --> 00:08:14,040
We considered symmetry.

120
00:08:14,040 --> 00:08:17,770
So if I am at the 0
field limit, I only

121
00:08:17,770 --> 00:08:21,670
have terms that are proportional
to m squared, rotationally

122
00:08:21,670 --> 00:08:22,170
symmetric.

123
00:08:28,490 --> 00:08:33,200
And there is something
else that is implicit,

124
00:08:33,200 --> 00:08:34,330
which is analyticity.

125
00:08:40,890 --> 00:08:42,369
What do I mean?

126
00:08:42,369 --> 00:08:44,910
I mean that there is,
in some sense here,

127
00:08:44,910 --> 00:08:49,210
a space of parameters
composed of all

128
00:08:49,210 --> 00:08:58,150
of these coefficients--
t, k, u, v.

129
00:08:58,150 --> 00:09:00,520
And these are
supposed to represent

130
00:09:00,520 --> 00:09:05,390
what is happening to my
system that I have in mind

131
00:09:05,390 --> 00:09:08,730
as I change the temperature.

132
00:09:08,730 --> 00:09:13,650
And so in principle, if I
do some averaging procedure

133
00:09:13,650 --> 00:09:18,200
and arrive at this description,
all of these parameters

134
00:09:18,200 --> 00:09:21,100
presumably will be
functions of temperature.

135
00:09:25,260 --> 00:09:29,190
And the statement
is that the process

136
00:09:29,190 --> 00:09:31,780
of coarse gaining the
degrees of freedom

137
00:09:31,780 --> 00:09:35,380
and averaging to arrive
at this description

138
00:09:35,380 --> 00:09:38,210
and the corresponding
parameters involves

139
00:09:38,210 --> 00:09:40,570
finite number of
degrees of freedom.

140
00:09:40,570 --> 00:09:43,360
And adding and
integrating finite numbers

141
00:09:43,360 --> 00:09:48,410
of degrees of freedom can only
lead to analytical functions.

142
00:09:48,410 --> 00:09:51,980
So the statement here is
that this set of parameters

143
00:09:51,980 --> 00:09:53,555
are analytical functions.

144
00:09:56,630 --> 00:10:03,210
So given this
construction by lambda,

145
00:10:03,210 --> 00:10:07,300
we should be able to figure
out if this is correct,

146
00:10:07,300 --> 00:10:10,760
what is happening and where
these numbers come from.

147
00:10:10,760 --> 00:10:13,450
So what did we attempt?

148
00:10:13,450 --> 00:10:15,400
The first thing
that we attempted

149
00:10:15,400 --> 00:10:16,585
was to do saddle point.

150
00:10:19,910 --> 00:10:23,570
And we saw that
doing so just looking

151
00:10:23,570 --> 00:10:27,850
at the most probable state
fails because fluctuations

152
00:10:27,850 --> 00:10:30,750
were important.

153
00:10:30,750 --> 00:10:35,760
We tried to break
the Hamiltonian

154
00:10:35,760 --> 00:10:39,740
into a part that was
quadratic and Gaussian.

155
00:10:39,740 --> 00:10:46,930
And we could calculate
everything about it,

156
00:10:46,930 --> 00:10:53,820
and then treating everybody
else as a perturbation.

157
00:10:53,820 --> 00:10:57,030
And when we
attempted to do that,

158
00:10:57,030 --> 00:10:59,550
we found that
perturbation theory

159
00:10:59,550 --> 00:11:00,870
failed below four dimensions.

160
00:11:10,660 --> 00:11:15,570
So at this stage, it
was kind of an impasse

161
00:11:15,570 --> 00:11:19,310
in that as far as
physics is concerned,

162
00:11:19,310 --> 00:11:22,620
we feel that this thing
captures all of the properties

163
00:11:22,620 --> 00:11:24,750
that you need in
order to somehow

164
00:11:24,750 --> 00:11:27,520
be able to explain
those phenomena.

165
00:11:27,520 --> 00:11:31,000
Yet, we don't have
the mathematical power

166
00:11:31,000 --> 00:11:36,950
to carry out the integrations
that are implicit in this.

167
00:11:36,950 --> 00:11:42,060
So then the idea was, can
we go around it somehow?

168
00:11:42,060 --> 00:11:46,450
And so the next set of
things that we introduced

169
00:11:46,450 --> 00:11:50,920
were basically
versions of scaling.

170
00:11:50,920 --> 00:11:55,025
And quite a few
statistical physicists

171
00:11:55,025 --> 00:11:56,840
were involved with that.

172
00:11:56,840 --> 00:12:03,950
Names such as [INAUDIBLE],
Fisher, and number of others.

173
00:12:03,950 --> 00:12:10,510
And the idea is that
if we also consider,

174
00:12:10,510 --> 00:12:17,610
let's say, introducing a
magnetic field direction here

175
00:12:17,610 --> 00:12:21,670
and look at the singularities
in the plane that involves

176
00:12:21,670 --> 00:12:25,640
those two that is necessary
to also characterize

177
00:12:25,640 --> 00:12:29,300
some of these other
things such as gamma,

178
00:12:29,300 --> 00:12:34,480
then you have a singular
part for the free energy that

179
00:12:34,480 --> 00:12:38,340
is a function of how
far you go away from Tc.

180
00:12:38,340 --> 00:12:42,820
So this t now stands
for T minus Tc.

181
00:12:42,820 --> 00:12:48,070
And how far you go along the
direction that breaks symmetry.

182
00:12:48,070 --> 00:12:51,920
And the statement was
that all of the results

183
00:12:51,920 --> 00:12:59,170
were consistent with the form
that depended on really two

184
00:12:59,170 --> 00:13:04,020
exponents that could
be bonded together

185
00:13:04,020 --> 00:13:05,650
into the behavior
of the singular

186
00:13:05,650 --> 00:13:08,630
part of the free
energy or the singular

187
00:13:08,630 --> 00:13:11,110
part of the
correlation function.

188
00:13:11,110 --> 00:13:17,520
And essentially, this
approach immediately

189
00:13:17,520 --> 00:13:20,550
leads to exponent identities.

190
00:13:23,570 --> 00:13:26,510
And these exponent identities
we can go back and look

191
00:13:26,510 --> 00:13:31,890
at the table of numbers
that we have up there.

192
00:13:31,890 --> 00:13:36,035
And we see that they
are correct and valid.

193
00:13:38,940 --> 00:13:44,060
But well, what are the
two primary exponents?

194
00:13:44,060 --> 00:13:46,820
How can we obtain them?

195
00:13:46,820 --> 00:13:51,830
Well, going and looking at this
scaling behavior a little bit

196
00:13:51,830 --> 00:14:01,840
further, one could trace back to
some kind of a self-similarity

197
00:14:01,840 --> 00:14:05,890
that should exist right
at the critical point.

198
00:14:05,890 --> 00:14:08,280
That is, the
correlation functions,

199
00:14:08,280 --> 00:14:11,010
et cetera at the
critical point should

200
00:14:11,010 --> 00:14:14,150
have this kind of
scaling variance.

201
00:14:14,150 --> 00:14:19,700
And then the question
is, can we somehow manage

202
00:14:19,700 --> 00:14:22,470
to use that property,
that looking at things

203
00:14:22,470 --> 00:14:25,210
at different scales at
the critical points,

204
00:14:25,210 --> 00:14:28,470
should give you the
same thing to divine

205
00:14:28,470 --> 00:14:30,850
what these exponents are.

206
00:14:30,850 --> 00:14:34,230
So the next stage
in this progression

207
00:14:34,230 --> 00:14:45,678
was the work of Kadanoff in
introducing the idea of RG.

208
00:14:45,678 --> 00:14:52,130
And the idea of RG was to
basically average things

209
00:14:52,130 --> 00:14:53,976
further.

210
00:14:53,976 --> 00:14:56,910
Here, implicit in
the calculation

211
00:14:56,910 --> 00:15:02,440
that we had was some kind of a
short distance [INAUDIBLE] a.

212
00:15:02,440 --> 00:15:07,070
And if we average
between a and b a,

213
00:15:07,070 --> 00:15:12,390
then presumably these parameters
mu would change to something

214
00:15:12,390 --> 00:15:15,520
else-- mu prime--
that correspond

215
00:15:15,520 --> 00:15:18,970
to rescaling by a factor of b.

216
00:15:18,970 --> 00:15:22,410
And these mu primes
would be a function

217
00:15:22,410 --> 00:15:27,770
of the original set
of parameters mu.

218
00:15:27,770 --> 00:15:32,870
And then Kadanoff's idea was
that the scaling variant points

219
00:15:32,870 --> 00:15:37,400
would correspond to the points
where you have no changed.

220
00:15:44,250 --> 00:15:51,840
And that if you then
deviated from that point,

221
00:15:51,840 --> 00:15:54,175
you would have some
characteristic scale

222
00:15:54,175 --> 00:15:55,830
in the problem.

223
00:15:55,830 --> 00:15:58,930
And you could capture
what was happening

224
00:15:58,930 --> 00:16:07,370
by looking at essentially how
the changes, delta mu prime,

225
00:16:07,370 --> 00:16:10,980
were related to the
changes delta mu.

226
00:16:10,980 --> 00:16:15,270
So essentially, linearizing
these relationships.

227
00:16:15,270 --> 00:16:17,753
So there would be some kind of
a linearized transformation.

228
00:16:20,380 --> 00:16:26,990
And then the eigenvalues
of this transformation

229
00:16:26,990 --> 00:16:31,230
would determine how many
relevant quantities you should

230
00:16:31,230 --> 00:16:33,580
have.

231
00:16:33,580 --> 00:16:36,350
Now, the physics, the entire
physics of the process,

232
00:16:36,350 --> 00:16:38,610
then comes into play here.

233
00:16:38,610 --> 00:16:43,130
That the experiments
tell us that you can,

234
00:16:43,130 --> 00:16:48,050
let's say, take superfluid and
it has this phase transition.

235
00:16:48,050 --> 00:16:50,510
You change the pressure of
it, it still has that phase

236
00:16:50,510 --> 00:16:52,570
transition-- slightly
different temperature,

237
00:16:52,570 --> 00:16:54,650
but it's the same
phase transition.

238
00:16:54,650 --> 00:16:56,710
You can add some
impurities to it

239
00:16:56,710 --> 00:16:58,650
as you did in one
of the problems.

240
00:16:58,650 --> 00:17:01,920
You still have the
phase transition.

241
00:17:01,920 --> 00:17:06,430
So basically, the existence
of a phase transition

242
00:17:06,430 --> 00:17:10,930
as a function of one parameter
that is temperature-like

243
00:17:10,930 --> 00:17:13,270
is pretty robust.

244
00:17:13,270 --> 00:17:17,099
And if we think about that in
the language of fixed point,

245
00:17:17,099 --> 00:17:20,730
it meant that along
the symmetry direction,

246
00:17:20,730 --> 00:17:25,020
there should be only
one relevant eigenvalue.

247
00:17:25,020 --> 00:17:29,550
So this construction
that Kadanoff proposed

248
00:17:29,550 --> 00:17:34,200
is nice and fine, but
one has to demonstrate

249
00:17:34,200 --> 00:17:40,700
that, indeed, this infinite
number of parameters

250
00:17:40,700 --> 00:17:43,100
can be boiled down
to a fixed point.

251
00:17:43,100 --> 00:17:46,200
And that fixed point has
only one relevant direction.

252
00:17:49,230 --> 00:17:54,050
So the next step
in this progression

253
00:17:54,050 --> 00:18:01,570
was Wilson who did perturbative
version of this procedure.

254
00:18:08,690 --> 00:18:17,650
So the idea was that we can
certainly solve beta H0.

255
00:18:17,650 --> 00:18:23,020
And beta H0 is really a bunch
of Gaussian independent modes

256
00:18:23,020 --> 00:18:26,690
as long as we look at
things in Fourier space.

257
00:18:26,690 --> 00:18:29,460
So in Fourier space,
we have a bunch

258
00:18:29,460 --> 00:18:33,630
of modes that exist over
some [INAUDIBLE] zone.

259
00:18:33,630 --> 00:18:39,910
And as long as we are looking
at some set of wavelengths

260
00:18:39,910 --> 00:18:43,360
and no fluctuation shorter
than that wavelength has been

261
00:18:43,360 --> 00:18:46,390
allowed, there is
a maximum to here.

262
00:18:46,390 --> 00:18:51,680
And the procedure of
averaging and increasing

263
00:18:51,680 --> 00:18:54,490
this minimum wavelength
then corresponds

264
00:18:54,490 --> 00:19:01,100
to integrating out modes that
are sitting outside lambda

265
00:19:01,100 --> 00:19:06,870
over v and keeping modes
that we call m tilde that

266
00:19:06,870 --> 00:19:10,333
live within 0 to lambda over p.

267
00:19:13,300 --> 00:19:20,250
And if we integrate
these modes--

268
00:19:20,250 --> 00:19:24,570
so if I rewrite this integration
as an integration over Fourier

269
00:19:24,570 --> 00:19:28,300
modes, do this
decomposition, et cetera--

270
00:19:28,300 --> 00:19:34,660
what I find is that
my-- after I integrate.

271
00:19:34,660 --> 00:19:36,805
So step 1, I do a
coarse graining.

272
00:19:41,210 --> 00:19:45,510
I find a Hamiltonian
that governs

273
00:19:45,510 --> 00:19:46,890
the coarse-grained modes.

274
00:19:49,580 --> 00:19:53,120
Well, if I integrate
out the sigma modes

275
00:19:53,120 --> 00:19:55,850
and treat them as
Gaussians, then there

276
00:19:55,850 --> 00:19:58,640
will be a contribution to
the free energy trivially

277
00:19:58,640 --> 00:20:03,360
from those modes proportional
to volume, presumably.

278
00:20:03,360 --> 00:20:05,710
Since these modes
and these modes

279
00:20:05,710 --> 00:20:09,290
don't couple at the Gaussian
level, at the Gaussian level

280
00:20:09,290 --> 00:20:18,370
we also have beta H0 acting on
these modes that I have kept.

281
00:20:18,370 --> 00:20:22,390
And the hard part is, of
course, the interaction

282
00:20:22,390 --> 00:20:25,390
between these
types of modes that

283
00:20:25,390 --> 00:20:29,650
are governed by all of
these non-linearities

284
00:20:29,650 --> 00:20:31,400
that we have over here.

285
00:20:31,400 --> 00:20:36,660
And we kind of can formally
write that as minus log of e

286
00:20:36,660 --> 00:20:42,440
to the minus u that depends
on m tilde and sigma

287
00:20:42,440 --> 00:20:45,470
when I integrate out
in the Gaussian weight

288
00:20:45,470 --> 00:20:47,760
the modes that are
the sigma parameters.

289
00:20:52,740 --> 00:20:55,490
So that's formally correct.

290
00:20:55,490 --> 00:20:59,120
This is some complicated
function of m tilde

291
00:20:59,120 --> 00:21:02,690
after I get rid of
the sigma variables.

292
00:21:02,690 --> 00:21:07,370
But presumably, if I
were to expand and write

293
00:21:07,370 --> 00:21:11,570
this in powers of m tilde
and powers of gradient,

294
00:21:11,570 --> 00:21:15,830
it will reproduce back
the original series.

295
00:21:15,830 --> 00:21:19,830
Because I said that the original
series includes everything

296
00:21:19,830 --> 00:21:22,550
that could possibly
be generated.

297
00:21:22,550 --> 00:21:25,650
This is presumably, after
I do all of these things,

298
00:21:25,650 --> 00:21:28,500
still consistent with
symmetries and so will

299
00:21:28,500 --> 00:21:31,641
generate those kinds of terms.

300
00:21:31,641 --> 00:21:33,930
But of course to
evaluate it, then we

301
00:21:33,930 --> 00:21:36,190
have to do perturbation.

302
00:21:36,190 --> 00:21:39,870
And so we can start expanding
this in powers of u.

303
00:21:39,870 --> 00:21:43,770
And the first term
would be u, assuming

304
00:21:43,770 --> 00:21:45,540
that u is a small quantity.

305
00:21:45,540 --> 00:21:55,942
Then minus 1/2 u squared minus
u average squared and so forth.

306
00:22:04,130 --> 00:22:09,150
And of course, this RG
has two other steps.

307
00:22:09,150 --> 00:22:13,020
After I have
performed this step,

308
00:22:13,020 --> 00:22:20,610
I have to do rescaling,
which in Fourier space

309
00:22:20,610 --> 00:22:25,040
means I blow up q.

310
00:22:25,040 --> 00:22:32,690
So q I will replace
with b inverse q prime.

311
00:22:32,690 --> 00:22:40,980
And renormalize, which
meant that in Fourier space

312
00:22:40,980 --> 00:22:45,265
I replace m with z m prime.

313
00:22:49,750 --> 00:22:53,660
So after I do these
procedures, what do I find?

314
00:22:53,660 --> 00:22:56,720
I find that to
whatever order I go,

315
00:22:56,720 --> 00:23:01,120
I start with some
original Hamiltonian

316
00:23:01,120 --> 00:23:04,840
that includes all terms
consistent with symmetries.

317
00:23:04,840 --> 00:23:10,490
And I generate a new
log of probability.

318
00:23:10,490 --> 00:23:11,830
It's not really a Hamiltonian.

319
00:23:11,830 --> 00:23:14,205
It's the kind of
effective free energy.

320
00:23:14,205 --> 00:23:16,790
It's the log of the probability
of these configurations.

321
00:23:20,010 --> 00:23:24,450
So now I should be able to
read this transformation of how

322
00:23:24,450 --> 00:23:29,110
I go from mu to mu prime.

323
00:23:29,110 --> 00:23:35,660
So let's go through
this list and do it.

324
00:23:35,660 --> 00:23:42,320
So t prime is something
that in Fourier space

325
00:23:42,320 --> 00:23:45,260
went with one
integration over q.

326
00:23:45,260 --> 00:23:47,295
So I got b to the minus d.

327
00:23:47,295 --> 00:23:50,760
There's two factors
of m, so it's

328
00:23:50,760 --> 00:23:54,290
a z square type of contribution.

329
00:23:54,290 --> 00:24:02,270
And at the 0 order
from here, I have my t.

330
00:24:02,270 --> 00:24:05,880
And then when I did the
average, from the average

331
00:24:05,880 --> 00:24:11,220
of u I got a contribution
that was proportional to u.

332
00:24:11,220 --> 00:24:13,610
There was a degeneracy of 4.

333
00:24:13,610 --> 00:24:17,560
There were two kinds of diagrams
that were contributing to it,

334
00:24:17,560 --> 00:24:22,545
and then I had the integral
from lambda over b to lambda,

335
00:24:22,545 --> 00:24:29,740
d dk 2 pi to the d 1 over
t plus t plus k k squared.

336
00:24:29,740 --> 00:24:33,260
And just to remind you
the kind of diagrams

337
00:24:33,260 --> 00:24:36,220
that were contributing
to this, one of them

338
00:24:36,220 --> 00:24:40,030
was something like
this and the other one

339
00:24:40,030 --> 00:24:44,420
was something like this.

340
00:24:44,420 --> 00:24:48,020
This one that had a closed
loop gave me the factor of n

341
00:24:48,020 --> 00:24:50,810
and the other gave me
what was eventually

342
00:24:50,810 --> 00:24:54,350
the eighth that I
have observed here.

343
00:24:54,350 --> 00:25:00,040
So this is what we
found at order of u.

344
00:25:00,040 --> 00:25:03,540
We went on and
calculated the u squared.

345
00:25:03,540 --> 00:25:08,880
And the u squared will give
me another contribution.

346
00:25:08,880 --> 00:25:12,910
There is a coefficient out here
that also similarly involves

347
00:25:12,910 --> 00:25:14,330
integrals.

348
00:25:14,330 --> 00:25:17,000
The integrals will depend on t.

349
00:25:17,000 --> 00:25:18,770
They will depend on k.

350
00:25:18,770 --> 00:25:21,790
They will depend on this
lambda that I'm integrating.

351
00:25:21,790 --> 00:25:24,430
They will depend
on t, et cetera.

352
00:25:24,430 --> 00:25:27,910
So there is some function here.

353
00:25:27,910 --> 00:25:35,180
And we argued last time that
we don't need to evaluate it,

354
00:25:35,180 --> 00:25:36,305
but let's write it down.

355
00:25:36,305 --> 00:25:39,710
And let's make sure that
it doesn't contribute.

356
00:25:39,710 --> 00:25:44,890
But this is only looking
at the effect of this u.

357
00:25:44,890 --> 00:25:48,330
And I know that I have
all of these other terms.

358
00:25:48,330 --> 00:25:51,150
So presumably, I
will get something

359
00:25:51,150 --> 00:25:57,800
that will be of the order
of, let's say, v squared uv.

360
00:25:57,800 --> 00:26:01,850
I will certainly get something
that is of the order of u6.

361
00:26:01,850 --> 00:26:05,950
If I think of u6 as something
that has six legs associated

362
00:26:05,950 --> 00:26:08,370
with it, I can
certainly join two

363
00:26:08,370 --> 00:26:10,790
of these legs, two of these
legs and have something

364
00:26:10,790 --> 00:26:16,670
that is two legs leftover, just
as I did getting from the u4 m

365
00:26:16,670 --> 00:26:17,920
to the 4 2m squared.

366
00:26:17,920 --> 00:26:19,620
I certainly can do that.

367
00:26:19,620 --> 00:26:24,380
So there is all kinds of higher
order terms here in principle

368
00:26:24,380 --> 00:26:26,352
that we have to keep track of.

369
00:26:29,870 --> 00:26:34,780
Now, the next term in
the series is the k.

370
00:26:34,780 --> 00:26:38,120
Compared to the t-terms, it
had two additional factor

371
00:26:38,120 --> 00:26:39,000
of gradients.

372
00:26:39,000 --> 00:26:43,030
When we do everything, it turns
out that it will be minus 2

373
00:26:43,030 --> 00:26:47,530
because of the two gradients
that became q squared.

374
00:26:47,530 --> 00:26:51,480
It's still second order
in m, so I would get this.

375
00:26:51,480 --> 00:26:55,200
And then I start with k.

376
00:26:55,200 --> 00:26:58,240
Now, the interesting
and important thing

377
00:26:58,240 --> 00:27:02,920
is that when we do the
calculation at order of u,

378
00:27:02,920 --> 00:27:07,170
we don't get any
correction to k.

379
00:27:07,170 --> 00:27:10,470
The only diagrams that
could have contributed to k

380
00:27:10,470 --> 00:27:14,480
were diagrams of this variety.

381
00:27:14,480 --> 00:27:16,690
But diagrams of
this variety, we saw

382
00:27:16,690 --> 00:27:20,040
that when I performed those
integrals the result just

383
00:27:20,040 --> 00:27:21,480
doesn't depend on q.

384
00:27:21,480 --> 00:27:24,010
It only corrected
the constant term

385
00:27:24,010 --> 00:27:26,990
that is proportional
to t squared.

386
00:27:26,990 --> 00:27:30,960
But that structure
will not preserve.

387
00:27:30,960 --> 00:27:34,510
If I go to order
of u squared, there

388
00:27:34,510 --> 00:27:37,760
will be some kind
of a correction that

389
00:27:37,760 --> 00:27:40,020
is order of u squared.

390
00:27:40,020 --> 00:27:44,410
And we kind of had in the
table in the table of 6

391
00:27:44,410 --> 00:27:47,260
by 6 things that I
had a diagram that

392
00:27:47,260 --> 00:27:49,650
gives contribution such as this.

393
00:27:49,650 --> 00:27:51,810
That, in fact, look
something like this.

394
00:27:59,500 --> 00:28:02,420
So basically, this is
a four-point vertex,

395
00:28:02,420 --> 00:28:04,170
a four-point vertex.

396
00:28:04,170 --> 00:28:05,060
I joined them.

397
00:28:05,060 --> 00:28:07,650
I make these kinds
of calculation.

398
00:28:07,650 --> 00:28:10,560
Now, once you do
that, you'll find

399
00:28:10,560 --> 00:28:12,710
that the difference
between this diagram

400
00:28:12,710 --> 00:28:17,840
and let's say that diagram
is that the momentum that

401
00:28:17,840 --> 00:28:21,720
goes in here, q, will have
to just go through here

402
00:28:21,720 --> 00:28:25,120
and there is no influence
on it on the momentum

403
00:28:25,120 --> 00:28:27,620
that I'm integrating.

404
00:28:27,620 --> 00:28:30,170
Whereas, if you look
at this diagram,

405
00:28:30,170 --> 00:28:32,730
you will find that it
is possible to have

406
00:28:32,730 --> 00:28:37,970
a momentum that going in here
and gets a contribution over

407
00:28:37,970 --> 00:28:38,780
here.

408
00:28:38,780 --> 00:28:43,350
And so the calculation, if I
were to do at higher order,

409
00:28:43,350 --> 00:28:46,800
will have in the
denominator a product of two

410
00:28:46,800 --> 00:28:48,890
of these factors,
but one of them

411
00:28:48,890 --> 00:28:51,780
will explicitly depend on q.

412
00:28:51,780 --> 00:28:54,390
And if I expanded
in powers of q,

413
00:28:54,390 --> 00:28:58,100
I will get a correction
that will appear here.

414
00:28:58,100 --> 00:28:59,945
It will not change
our life as we

415
00:28:59,945 --> 00:29:03,835
will see what it's good
to know that it is there.

416
00:29:03,835 --> 00:29:07,702
And there be higher-order
corrections here, too.

417
00:29:07,702 --> 00:29:08,630
AUDIENCE: Question.

418
00:29:08,630 --> 00:29:10,460
PROFESSOR: Yes.

419
00:29:10,460 --> 00:29:14,510
AUDIENCE: Are the
functions a1 and a2

420
00:29:14,510 --> 00:29:18,875
just labeled as such to
make our lives easier

421
00:29:18,875 --> 00:29:24,240
or because they don't have any
sort of universality with them?

422
00:29:24,240 --> 00:29:25,550
PROFESSOR: Both.

423
00:29:25,550 --> 00:29:29,500
They don't carry at this
order in the expansion

424
00:29:29,500 --> 00:29:32,320
any information for
what we will need,

425
00:29:32,320 --> 00:29:35,810
but I have to show
it to you explicitly.

426
00:29:35,810 --> 00:29:39,740
So right now, I keep
them as placeholders.

427
00:29:39,740 --> 00:29:41,800
If, at the end of
the day, we find

428
00:29:41,800 --> 00:29:45,660
that our answers will
depend on these quantities,

429
00:29:45,660 --> 00:29:48,270
then we have to go back
and calculate them.

430
00:29:48,270 --> 00:29:52,170
But ultimately, the reason
that we won't need them

431
00:29:52,170 --> 00:29:53,940
is what I described last time.

432
00:29:53,940 --> 00:29:59,320
That we will calculate
exponents only to lowest order

433
00:29:59,320 --> 00:30:03,220
in 4 minus epsilon-- 4
minus d, which is epsilon.

434
00:30:03,220 --> 00:30:06,460
And that all of these
u's at the fixed point

435
00:30:06,460 --> 00:30:08,510
will be of the order of epsilon.

436
00:30:08,510 --> 00:30:10,650
So both of these
terms are terms that

437
00:30:10,650 --> 00:30:13,206
are order of epsilon
squared and will

438
00:30:13,206 --> 00:30:14,715
be ignorable at
level of epsilon.

439
00:30:18,640 --> 00:30:21,850
But so far I haven't talked
anything about epsilon,

440
00:30:21,850 --> 00:30:25,060
so I may as well keep it.

441
00:30:25,060 --> 00:30:29,050
And similarly, l prime
would be something

442
00:30:29,050 --> 00:30:31,360
that goes with q
to the fourth if I

443
00:30:31,360 --> 00:30:33,450
were to Fourier transform it.

444
00:30:33,450 --> 00:30:35,670
So this would be b
to the d minus 4.

445
00:30:35,670 --> 00:30:37,550
It is still z squared.

446
00:30:37,550 --> 00:30:40,010
It will be proportional to l.

447
00:30:40,010 --> 00:30:44,350
It will have exactly these
kinds of corrections also.

448
00:30:48,160 --> 00:30:52,645
And I can keep going with the
list of all second-order terms.

449
00:30:57,700 --> 00:31:00,980
Now, then we got to u prime.

450
00:31:00,980 --> 00:31:04,050
u prime was a fourth
order, fourth power of m.

451
00:31:04,050 --> 00:31:06,600
So it carried z to the fourth.

452
00:31:06,600 --> 00:31:09,640
It involved three
derivatives in q space,

453
00:31:09,640 --> 00:31:14,330
so it gave me b to the minus 3d.

454
00:31:14,330 --> 00:31:20,300
And then to the lowest order
when I did the expansion

455
00:31:20,300 --> 00:31:23,460
from u, one of the
terms that I got

456
00:31:23,460 --> 00:31:26,570
was the original
potential evaluated

457
00:31:26,570 --> 00:31:29,570
for m tilde rather
than the original m.

458
00:31:29,570 --> 00:31:33,360
So I always will get this term.

459
00:31:33,360 --> 00:31:36,100
And then I noticed that when
I go and calculate things

460
00:31:36,100 --> 00:31:39,100
at second order--
and we explicitly

461
00:31:39,100 --> 00:31:45,110
did that-- we got a
term that was minus 4 u

462
00:31:45,110 --> 00:31:51,425
squared n plus 8 integral
lambda over b to lambda

463
00:31:51,425 --> 00:31:58,560
d dk 2 pi to the d 1
over t plus k k squared.

464
00:31:58,560 --> 00:32:02,780
And presumably, these
series also continue.

465
00:32:02,780 --> 00:32:03,830
The whole thing squared.

466
00:32:03,830 --> 00:32:08,010
It was a squared propagator
that was appearing over here.

467
00:32:08,010 --> 00:32:11,310
And again, reminding
you that these

468
00:32:11,310 --> 00:32:18,330
came from diagrams that were--
some of it was like this.

469
00:32:18,330 --> 00:32:22,230
There was a loop that
gave us the factor of n.

470
00:32:22,230 --> 00:32:31,840
And then there were things
like this one or this one.

471
00:32:41,850 --> 00:32:42,350
Yeah.

472
00:32:48,350 --> 00:32:56,470
And those gave out this huge
number compared to this.

473
00:32:56,470 --> 00:33:03,740
Now, if I had included this
term that is proportional to v,

474
00:33:03,740 --> 00:33:07,270
presumably I would
have gotten corrections

475
00:33:07,270 --> 00:33:10,070
that are order of uv.

476
00:33:10,070 --> 00:33:12,560
If I go to higher
orders, I will certainly

477
00:33:12,560 --> 00:33:15,665
get things that are
of the order of u6.

478
00:33:15,665 --> 00:33:21,200
I will certainly get things
that are of the order of u cubed

479
00:33:21,200 --> 00:33:23,080
and so forth.

480
00:33:23,080 --> 00:33:26,270
So there is a whole
bunch of corrections

481
00:33:26,270 --> 00:33:29,140
that in principle,
if I am supposed

482
00:33:29,140 --> 00:33:31,920
to include everything and
keep track of everything,

483
00:33:31,920 --> 00:33:34,038
I should include.

484
00:33:34,038 --> 00:33:40,160
Now, the v itself, v prime,
has two additional derivatives

485
00:33:40,160 --> 00:33:41,800
with respect to u.

486
00:33:41,800 --> 00:33:45,770
So it will be b to
the minus 3d minus 2.

487
00:33:45,770 --> 00:33:48,540
It's a z to the
fourth type of term.

488
00:33:48,540 --> 00:33:54,500
It is v, and then it will
certainly get corrections at,

489
00:33:54,500 --> 00:33:57,430
say, at order of
uv and so forth.

490
00:34:04,680 --> 00:34:08,130
And what else did I
write down in the series?

491
00:34:08,130 --> 00:34:12,449
I can write as many as we
like. u prime to the 6.

492
00:34:12,449 --> 00:34:17,659
This is something that
goes with 6 powers of z

493
00:34:17,659 --> 00:34:20,975
and will have 5 derivatives.

494
00:34:20,975 --> 00:34:23,760
Sorry, 5 integrations in q.

495
00:34:23,760 --> 00:34:26,320
So it will give me
b to the minus 5d.

496
00:34:26,320 --> 00:34:31,150
And again, presumably I
will have u6 minus order

497
00:34:31,150 --> 00:34:35,489
of something like u squared
v and all kinds of things.

498
00:34:40,350 --> 00:34:42,090
All right.

499
00:34:42,090 --> 00:34:43,492
Yes.

500
00:34:43,492 --> 00:34:46,648
AUDIENCE: So in the calculation
of terms like k prime and l

501
00:34:46,648 --> 00:34:50,534
prime, you have factors
like b minus d minus 2.

502
00:34:50,534 --> 00:34:53,427
Is it minus 2 or plus 2?

503
00:34:53,427 --> 00:34:54,010
PROFESSOR: OK.

504
00:34:54,010 --> 00:34:59,870
So these came from Fourier
transforming this entity.

505
00:34:59,870 --> 00:35:03,860
When I Fourier transform,
I get integral dd q.

506
00:35:03,860 --> 00:35:07,640
I will have t plus
k q squared plus l q

507
00:35:07,640 --> 00:35:10,010
to the fourth, et cetera.

508
00:35:10,010 --> 00:35:12,820
And my task is that
whenever I see q,

509
00:35:12,820 --> 00:35:16,380
I replace it with
p inverse q prime.

510
00:35:16,380 --> 00:35:18,710
So this would be
b to the minus d.

511
00:35:18,710 --> 00:35:21,410
This would be b to the minus 2.

512
00:35:21,410 --> 00:35:23,150
This would be b to the minus 4.

513
00:35:27,100 --> 00:35:28,199
So that's how it comes.

514
00:35:28,199 --> 00:35:28,740
AUDIENCE: OK.

515
00:35:28,740 --> 00:35:29,240
Thank you.

516
00:35:33,749 --> 00:35:34,790
PROFESSOR: Anything else?

517
00:35:40,390 --> 00:35:42,220
OK.

518
00:35:42,220 --> 00:35:54,030
So then we had to choose
what this factor of z is.

519
00:35:54,030 --> 00:35:58,590
And we said, let's
choose it such

520
00:35:58,590 --> 00:36:05,392
that k prime is the same as k.

521
00:36:08,730 --> 00:36:16,410
But k prime over k we can see
is z squared b to the minus

522
00:36:16,410 --> 00:36:18,620
d minus 2.

523
00:36:18,620 --> 00:36:24,740
If I divide through by
this k, then I will get 1,

524
00:36:24,740 --> 00:36:29,350
and then something here
which is order of u squared.

525
00:36:35,540 --> 00:36:42,040
Now, we will justify later
why u in order to be small

526
00:36:42,040 --> 00:36:45,320
so that I can make
a construction that

527
00:36:45,320 --> 00:36:49,910
is perturbative in u will
be of the order of epsilon.

528
00:36:49,910 --> 00:36:53,500
But in any case, if I
want to in some sense

529
00:36:53,500 --> 00:36:56,540
keep the lowest order
in u, at this order

530
00:36:56,540 --> 00:37:01,250
I am justified to
get rid of this term.

531
00:37:01,250 --> 00:37:04,100
And when I do that
to this order,

532
00:37:04,100 --> 00:37:09,220
I will find that z is b
to the 1 plus d over 2.

533
00:37:09,220 --> 00:37:11,870
And probably in
principle, corrections

534
00:37:11,870 --> 00:37:19,010
that will be of the order of
this epsilon to the squared.

535
00:37:21,930 --> 00:37:27,570
So I do that choice.

536
00:37:27,570 --> 00:37:32,340
Secondly, I'll make my
b to be infinitesimal.

537
00:37:35,100 --> 00:37:39,530
And so that means that
mu prime at scale b,

538
00:37:39,530 --> 00:37:42,380
the set of parameters--
each parameter

539
00:37:42,380 --> 00:37:49,170
would be basically mu plus
a small shift d mu by dl.

540
00:37:49,170 --> 00:37:56,120
And then I can recast
these jumps by factors of b

541
00:37:56,120 --> 00:38:00,820
that I have up there
to flow equations.

542
00:38:00,820 --> 00:38:02,960
And so what do I get?

543
00:38:02,960 --> 00:38:07,350
For the first one,
we got dt by dl.

544
00:38:07,350 --> 00:38:12,380
And I had chosen z squared to
be b to the minus d minus 2.

545
00:38:12,380 --> 00:38:14,500
So compared to the
original one, it's

546
00:38:14,500 --> 00:38:16,820
just two more factors of b.

547
00:38:16,820 --> 00:38:21,780
Two more factors of
b will give me 2t.

548
00:38:21,780 --> 00:38:26,720
And then I will have to deal
with that integration evaluated

549
00:38:26,720 --> 00:38:28,810
when b is very
small, which means

550
00:38:28,810 --> 00:38:32,140
that I have to just
evaluate it on the shell.

551
00:38:32,140 --> 00:38:43,750
So I will have 4u n plus 2
kd lambda to the d divided

552
00:38:43,750 --> 00:38:48,900
by t plus k lambda squared
plus higher-orders terms

553
00:38:48,900 --> 00:38:52,050
in this propagator.

554
00:38:52,050 --> 00:38:54,540
And then I will
have, presumably,

555
00:38:54,540 --> 00:38:58,260
some a1-looking
quantity, but evaluated

556
00:38:58,260 --> 00:39:02,160
on the shell that
depends on u squared,

557
00:39:02,160 --> 00:39:04,230
and then I will have
higher-order terms.

558
00:39:12,340 --> 00:39:13,272
Yes.

559
00:39:13,272 --> 00:39:15,660
AUDIENCE: I'm kind
of curious on why

560
00:39:15,660 --> 00:39:18,010
we choose k equals
k prime instead

561
00:39:18,010 --> 00:39:20,510
of the constant in front of any
of the other gradient terms.

562
00:39:20,510 --> 00:39:24,967
Why is k equals k prime better
than l equals l prime or--

563
00:39:24,967 --> 00:39:25,550
PROFESSOR: OK.

564
00:39:25,550 --> 00:39:30,220
We discussed this in the
context of the Gaussian model.

565
00:39:30,220 --> 00:39:32,570
So what we saw for
the Gaussian model is

566
00:39:32,570 --> 00:39:36,150
that if I choose
l prime to be l,

567
00:39:36,150 --> 00:39:39,840
then I will have k
prime being b squared k

568
00:39:39,840 --> 00:39:42,215
and t prime will be
b to the fourth t.

569
00:39:42,215 --> 00:39:47,190
So I will have two
relevant directions.

570
00:39:47,190 --> 00:39:51,120
So I want to have, in some
sense, the minimal levels

571
00:39:51,120 --> 00:39:54,770
of direction guided by
the experimental fact

572
00:39:54,770 --> 00:39:57,170
that you do whatever you
like and you see the phase

573
00:39:57,170 --> 00:40:01,530
transition, except that you
have to change one parameter.

574
00:40:01,530 --> 00:40:03,120
There could be something else.

575
00:40:03,120 --> 00:40:06,310
There could very well--
somebody comes to me later

576
00:40:06,310 --> 00:40:08,680
and describes some kind
of a phase transition

577
00:40:08,680 --> 00:40:12,010
that requires two
relevant directions.

578
00:40:12,010 --> 00:40:16,210
And the physics of it may guide
me to make the other choice.

579
00:40:16,210 --> 00:40:18,840
But for the problem that
I'm telling you right now,

580
00:40:18,840 --> 00:40:20,860
the physics guides me
to make this choice.

581
00:40:26,450 --> 00:40:28,950
There is no equation
for k because we already

582
00:40:28,950 --> 00:40:31,930
set that to be 0.

583
00:40:31,930 --> 00:40:33,665
Let's write the equation for u.

584
00:40:36,430 --> 00:40:41,460
So four u, z to the
fourth becomes 4 plus 2d.

585
00:40:41,460 --> 00:40:50,120
And then minus 3d
becomes 4 minus d u.

586
00:40:50,120 --> 00:40:52,980
And then the next
order term becomes

587
00:40:52,980 --> 00:41:02,930
minus 4 u squared n plus 8 kd
lambda to the d t plus k lambda

588
00:41:02,930 --> 00:41:08,440
squared and so
fourth squared when

589
00:41:08,440 --> 00:41:10,840
I evaluate that
integral on the shell.

590
00:41:10,840 --> 00:41:12,670
And then I will have
higher-order terms.

591
00:41:21,560 --> 00:41:28,710
So this is where this idea
of making an expansion

592
00:41:28,710 --> 00:41:31,660
in dimensions come into play.

593
00:41:31,660 --> 00:41:34,450
Because we want to have
these sets of equations

594
00:41:34,450 --> 00:41:38,040
somehow under control, we
need to have a small parameter

595
00:41:38,040 --> 00:41:40,850
in which we are
making an expansion.

596
00:41:40,850 --> 00:41:44,610
And ultimately, we will be
looking at the fixed point.

597
00:41:44,610 --> 00:41:46,510
And the fixed point
occurs at u star.

598
00:41:46,510 --> 00:41:49,950
That is, of the
order of 4 minus d.

599
00:41:49,950 --> 00:41:54,380
Otherwise, there is no
small control parameter.

600
00:41:54,380 --> 00:41:58,170
So the suggestion, actually,
that goes to Fisher

601
00:41:58,170 --> 00:42:07,470
was to organize the expansion as
a power series in this quantity

602
00:42:07,470 --> 00:42:09,340
epsilon.

603
00:42:09,340 --> 00:42:14,600
And eventually then, ask what
the properties of these series

604
00:42:14,600 --> 00:42:15,950
are as a function of epsilon.

605
00:42:19,410 --> 00:42:21,870
So then I have all those others.

606
00:42:21,870 --> 00:42:28,230
I forgot, actually, to
write l, by dt by dl.

607
00:42:28,230 --> 00:42:31,880
Well, compared to k, it
has two additional factor

608
00:42:31,880 --> 00:42:38,680
of gradients, which means that
it will start with minus 2l.

609
00:42:38,680 --> 00:42:41,330
And then we said that
it will get corrections

610
00:42:41,330 --> 00:42:44,135
that are of the
order of u squared,

611
00:42:44,135 --> 00:42:46,210
and uv, and such things.

612
00:42:49,180 --> 00:42:50,220
dv by dl.

613
00:42:54,540 --> 00:42:58,440
I mean, compared to this
term, compared to u,

614
00:42:58,440 --> 00:43:02,150
it has two more gradients
in the construction.

615
00:43:02,150 --> 00:43:07,740
So it's dimension will
be minus 2 plus epsilon.

616
00:43:07,740 --> 00:43:10,975
And then we'll get directions
of the order of, presumably,

617
00:43:10,975 --> 00:43:12,441
u squared, uv, [INAUDIBLE].

618
00:43:16,370 --> 00:43:18,870
And then, what else did I write?

619
00:43:18,870 --> 00:43:24,710
I wrote something
about d u6 by dl.

620
00:43:24,710 --> 00:43:31,565
d u6 by dl, I have to substitute
for z 1 plus d over 2.

621
00:43:31,565 --> 00:43:34,400
Subtract 5d.

622
00:43:34,400 --> 00:43:38,290
Rewrite d as 4 minus epsilon.

623
00:43:38,290 --> 00:43:41,400
Once you do that, you will
find that it becomes minus 2

624
00:43:41,400 --> 00:43:44,360
plus 2 epsilon.

625
00:43:48,889 --> 00:43:55,725
Let me just make sure that I
am not saying something wrong.

626
00:43:55,725 --> 00:44:02,451
Yeah, u6 plus order
of uv and so forth.

627
00:44:05,820 --> 00:44:10,930
So there is this whole
set of parameters

628
00:44:10,930 --> 00:44:24,280
that are being changed as a
function of-- going away--

629
00:44:24,280 --> 00:44:29,065
changing the rescaling
by a factor of b

630
00:44:29,065 --> 00:44:32,050
is 1 plus an infinitesimal.

631
00:44:32,050 --> 00:44:37,130
So this is the flow of
parameter in this space.

632
00:44:37,130 --> 00:44:42,540
So then to confirm
the ideas of Kadanoff,

633
00:44:42,540 --> 00:44:44,180
we have to find the fixed point.

634
00:44:53,440 --> 00:44:55,200
And there is clearly
a fixed point

635
00:44:55,200 --> 00:44:57,410
when all of these
parameters are 0.

636
00:44:57,410 --> 00:44:59,530
If they are 0, nothing changes.

637
00:44:59,530 --> 00:45:03,800
And I'm back to
the Gaussian model,

638
00:45:03,800 --> 00:45:07,790
which is described
by just gradient of m

639
00:45:07,790 --> 00:45:09,910
squared type of theory.

640
00:45:09,910 --> 00:45:12,610
So this is the fixed
point that corresponds

641
00:45:12,610 --> 00:45:17,406
to t star, u star, l star,
v star, all of the things

642
00:45:17,406 --> 00:45:20,290
that I can think
of, are equal to 0.

643
00:45:20,290 --> 00:45:22,650
It's a perfectly good fixed
point of the transformation.

644
00:45:26,300 --> 00:45:28,940
It doesn't suit us
because it actually

645
00:45:28,940 --> 00:45:32,310
has still two
relevant directions.

646
00:45:32,310 --> 00:45:36,610
It's obvious that if I
make a small change in u,

647
00:45:36,610 --> 00:45:42,660
then in dimensions less than
4, u is a relevant direction

648
00:45:42,660 --> 00:45:44,220
and t is a relevant direction.

649
00:45:44,220 --> 00:45:49,580
Two directions does not describe
the physics that I want.

650
00:45:49,580 --> 00:45:53,160
But there is fortunately
another fixed point,

651
00:45:53,160 --> 00:45:57,020
the one that we call the
O n fixed point because it

652
00:45:57,020 --> 00:46:00,150
explicitly depends
on the parameter n.

653
00:46:00,150 --> 00:46:06,860
And what I need to do is
to set this equal to 0.

654
00:46:06,860 --> 00:46:10,830
And if I set that equal
to 0, what do I get?

655
00:46:10,830 --> 00:46:16,470
I get u star just
manipulating this.

656
00:46:16,470 --> 00:46:18,110
It is proportional to epsilon.

657
00:46:18,110 --> 00:46:20,500
1u drops out.

658
00:46:20,500 --> 00:46:25,710
It is proportional to
epsilon The coefficient has

659
00:46:25,710 --> 00:46:33,350
a factor of 1 over 4 n plus 8.

660
00:46:33,350 --> 00:46:35,910
Basically, the inverse of this.

661
00:46:35,910 --> 00:46:38,320
And then also, the
inverse of all of that.

662
00:46:38,320 --> 00:46:45,970
So I have t star plus k lambda
squared and so forth squared

663
00:46:45,970 --> 00:46:48,625
divided by kd lambda to the d.

664
00:46:56,060 --> 00:47:02,550
Then, what I need to do is to
set the second equation to 0.

665
00:47:02,550 --> 00:47:04,360
You can see that
this is a term that

666
00:47:04,360 --> 00:47:07,310
is order of epsilon squared now.

667
00:47:07,310 --> 00:47:10,380
Whereas, this is a term
that is order of epsilon.

668
00:47:10,380 --> 00:47:13,470
So for calculating the
position of the fixed point,

669
00:47:13,470 --> 00:47:15,560
I don't need this parameter.

670
00:47:15,560 --> 00:47:16,680
And what do I get?

671
00:47:16,680 --> 00:47:31,010
I will get that t star is minus
2 n plus 2 kd lambda to the d

672
00:47:31,010 --> 00:47:35,730
divided by t star plus k
lambda squared and so forth.

673
00:47:35,730 --> 00:47:38,450
Times u star.

674
00:47:38,450 --> 00:47:46,290
u star is epsilon 4 n plus
8 t star plus k lambda

675
00:47:46,290 --> 00:47:48,440
squared and so forth squared.

676
00:47:48,440 --> 00:47:52,006
Divided by kd lambda to the d.

677
00:47:56,040 --> 00:47:59,770
Again, I'm calculating
everything correctly

678
00:47:59,770 --> 00:48:01,920
to order of epsilon.

679
00:48:01,920 --> 00:48:04,350
So since t star is
order of epsilon,

680
00:48:04,350 --> 00:48:06,510
I can drop it over here.

681
00:48:06,510 --> 00:48:14,020
So my u star is, in fact,
epsilon divided by 4 n plus 8.

682
00:48:14,020 --> 00:48:17,470
And then this
combination k lambda

683
00:48:17,470 --> 00:48:24,270
squared and so forth squared
divided by kd lambda to the d.

684
00:48:24,270 --> 00:48:26,590
And doing the same
thing up here,

685
00:48:26,590 --> 00:48:36,515
my t star is epsilon n plus
2 divided by 2 n plus 8.

686
00:48:36,515 --> 00:48:40,680
It has an overall minus sign.

687
00:48:40,680 --> 00:48:43,460
The kd parts cancel.

688
00:48:43,460 --> 00:48:46,050
And one of these
factors cancel, so I

689
00:48:46,050 --> 00:48:50,285
will get k lambda
squared squared.

690
00:48:50,285 --> 00:48:51,940
Sorry, no square here.

691
00:48:51,940 --> 00:48:53,610
And both of these
will get corrections

692
00:48:53,610 --> 00:48:56,250
that are order of
epsilon squared

693
00:48:56,250 --> 00:48:57,694
that I haven't calculated.

694
00:49:03,410 --> 00:49:06,630
Now, let's make sure that
it was justified for me

695
00:49:06,630 --> 00:49:12,390
to focus on these two parameters
and look at everything else

696
00:49:12,390 --> 00:49:15,576
as being not important before.

697
00:49:15,576 --> 00:49:18,390
Well, look at these equations.

698
00:49:18,390 --> 00:49:21,650
This equation says that
if I had a term that

699
00:49:21,650 --> 00:49:25,890
was order of u squared
evaluated at a fixed point,

700
00:49:25,890 --> 00:49:27,830
it would be epsilon squared.

701
00:49:27,830 --> 00:49:32,220
So l star would be of the
order of epsilon squared.

702
00:49:32,220 --> 00:49:34,320
You can check that
v star would be

703
00:49:34,320 --> 00:49:37,310
of the order of epsilon squared.

704
00:49:37,310 --> 00:49:40,625
A lot of those things will be
of the order of epsilon squared.

705
00:49:44,670 --> 00:49:47,390
v star.

706
00:49:47,390 --> 00:49:49,520
And actually, if you
look at it carefully,

707
00:49:49,520 --> 00:49:52,850
you'll find that things
like u6 will be even worse.

708
00:49:52,850 --> 00:49:56,473
They would start at order of
epsilon cubed and so forth.

709
00:50:00,550 --> 00:50:06,330
So quite systematically in this
small parameter that Fisher

710
00:50:06,330 --> 00:50:14,400
introduced, we see that what has
happened is that we have a huge

711
00:50:14,400 --> 00:50:18,770
set of parameters, these mu's.

712
00:50:18,770 --> 00:50:22,090
But we can focus
on the projection

713
00:50:22,090 --> 00:50:28,290
in the parameter space t and u.

714
00:50:28,290 --> 00:50:33,030
And in that parameter
space, we certainly always

715
00:50:33,030 --> 00:50:36,320
have the Gaussian fixed point.

716
00:50:36,320 --> 00:50:42,040
But as long as I am in
dimensions less than 4,

717
00:50:42,040 --> 00:50:45,360
the Gaussian fixed
point is not only

718
00:50:45,360 --> 00:50:51,440
relevant in the t-direction
by a factor of 2,

719
00:50:51,440 --> 00:50:56,190
but it also is relevant
in another direction.

720
00:50:56,190 --> 00:51:00,900
There is an eigen-direction
that is slightly shifted

721
00:51:00,900 --> 00:51:04,350
with respect to t equals to 0.

722
00:51:04,350 --> 00:51:06,280
It's not just the u-axis.

723
00:51:06,280 --> 00:51:09,550
Along that direction,
it moves away.

724
00:51:09,550 --> 00:51:11,240
Here you have an
eigenvalue of 2.

725
00:51:11,240 --> 00:51:15,370
Here you have an
eigenvalue of epsilon.

726
00:51:15,370 --> 00:51:17,115
So that's the
Gaussian fixed point.

727
00:51:20,030 --> 00:51:23,400
But now we found
another fixed point,

728
00:51:23,400 --> 00:51:25,550
which is occurring
for some positive u

729
00:51:25,550 --> 00:51:29,050
star and some negative u star.

730
00:51:29,050 --> 00:51:30,705
This is the o n fixed point.

731
00:51:39,350 --> 00:51:43,170
Kind of just by
the continuity, you

732
00:51:43,170 --> 00:51:47,420
would expect that if
things are going into here,

733
00:51:47,420 --> 00:51:50,560
it probably makes sense that
it should be going like here

734
00:51:50,560 --> 00:51:54,280
and this should be a
negative eigenvalue.

735
00:51:54,280 --> 00:51:57,260
But one can
explicitly check that.

736
00:51:57,260 --> 00:52:01,250
So basically, the
procedure to check that

737
00:52:01,250 --> 00:52:04,610
is to do what I told you.

738
00:52:04,610 --> 00:52:09,550
I have to construct a
linearized matrix that

739
00:52:09,550 --> 00:52:19,390
relates delta mu going
away from this fixed point

740
00:52:19,390 --> 00:52:23,795
to what happens under rescaling.

741
00:52:26,320 --> 00:52:35,250
So basically, under rescaling
I will find that if I set my mu

742
00:52:35,250 --> 00:52:41,270
to be mu star plus a
small change delta mu,

743
00:52:41,270 --> 00:52:43,890
then it will be moving away.

744
00:52:43,890 --> 00:52:47,250
And I can look at how,
let's say, delta t changes,

745
00:52:47,250 --> 00:52:50,880
how delta u changes,
how delta l changes,

746
00:52:50,880 --> 00:52:54,950
the whole list of parameters
that I have over here.

747
00:52:54,950 --> 00:52:59,190
The linearized matrix
will relate them

748
00:52:59,190 --> 00:53:02,850
to the vector that
corresponds to delta t,

749
00:53:02,850 --> 00:53:04,660
delta u, blah, blah.

750
00:53:09,120 --> 00:53:15,390
So I have to go back to
these recursion relations,

751
00:53:15,390 --> 00:53:19,380
make small changes in
all of the parameters,

752
00:53:19,380 --> 00:53:23,370
linearize the result,
construct that matrix,

753
00:53:23,370 --> 00:53:27,250
and then evaluate the
eigenvalues of that matrix.

754
00:53:27,250 --> 00:53:32,090
Again, consistency to the
order that I have done things.

755
00:53:32,090 --> 00:53:36,360
And for example, one of the
things that we saw last time

756
00:53:36,360 --> 00:53:38,720
is that there will
be an element here

757
00:53:38,720 --> 00:53:40,740
that corresponds
to the change in u

758
00:53:40,740 --> 00:53:43,430
if I make a change in delta t.

759
00:53:43,430 --> 00:53:45,370
There is such a contribution.

760
00:53:45,370 --> 00:53:49,430
If I make a change delta t with
respect to the fixed point,

761
00:53:49,430 --> 00:53:52,780
I will get a
derivative from here.

762
00:53:52,780 --> 00:53:56,930
But that derivative
multiplies u squared.

763
00:53:56,930 --> 00:53:58,950
Evaluated at the
fixed point means

764
00:53:58,950 --> 00:54:01,040
that I will get a
term down here that

765
00:54:01,040 --> 00:54:02,610
is order of epsilon squared.

766
00:54:06,420 --> 00:54:11,080
And then the second
element here,

767
00:54:11,080 --> 00:54:14,630
what happens if I
make a change in u?

768
00:54:14,630 --> 00:54:18,060
Well, I will get a epsilon here.

769
00:54:18,060 --> 00:54:22,110
And then I get a
subtraction from here.

770
00:54:22,110 --> 00:54:25,530
And this subtraction
we evaluated last time

771
00:54:25,530 --> 00:54:30,190
and it turned out to be
epsilon minus 2 epsilon.

772
00:54:30,190 --> 00:54:33,960
So the relevance
that we had over here

773
00:54:33,960 --> 00:54:36,820
became an irrelevance
that I wanted.

774
00:54:39,560 --> 00:54:43,690
So there is some matrix
element in this corner.

775
00:54:43,690 --> 00:54:47,770
But since this is 0, as we
discussed if I look at this 2

776
00:54:47,770 --> 00:54:52,380
by 2 block, it doesn't
affect this eigenvalue.

777
00:54:52,380 --> 00:54:56,250
Since I did not evaluate
this eigenvalue last time,

778
00:54:56,250 --> 00:54:58,110
I'll do it now.

779
00:54:58,110 --> 00:55:01,650
So in order to
calculate the yt, what

780
00:55:01,650 --> 00:55:03,740
I need to do is to
see what happens

781
00:55:03,740 --> 00:55:06,740
if I change t to t plus delta t.

782
00:55:06,740 --> 00:55:09,500
So I have to take a
derivative with respect to t.

783
00:55:09,500 --> 00:55:11,980
From the first
one, I will get 2.

784
00:55:11,980 --> 00:55:19,200
From the second term, I will
get minus 4 u n plus 2 kd

785
00:55:19,200 --> 00:55:23,310
lambda to the d-- what is
in the denominator squared.

786
00:55:23,310 --> 00:55:29,540
So t plus k lambda squared
and so forth squared.

787
00:55:29,540 --> 00:55:33,720
But I have to evaluate
this at the fixed point.

788
00:55:33,720 --> 00:55:36,420
So I put a u star here.

789
00:55:36,420 --> 00:55:38,560
I put a u star here.

790
00:55:38,560 --> 00:55:41,600
Since u star is already
order of epsilon,

791
00:55:41,600 --> 00:55:45,700
this order of epsilon
term I can ignore.

792
00:55:45,700 --> 00:55:54,170
And so what I have here is 2
minus 4 n plus 2 epsi-- OK.

793
00:55:54,170 --> 00:55:56,210
Now, let's put u star.

794
00:55:56,210 --> 00:55:58,310
u star I have up here.

795
00:55:58,310 --> 00:56:06,600
It is epsilon divided
by 4 n plus 8.

796
00:56:06,600 --> 00:56:11,380
I have k lambda squared
and so forth squared.

797
00:56:11,380 --> 00:56:14,846
kd lambda to the d.

798
00:56:14,846 --> 00:56:18,020
So I substituted the u star.

799
00:56:18,020 --> 00:56:19,260
I had the n plus 2.

800
00:56:19,260 --> 00:56:24,540
So now I have the
kd lambda to the d.

801
00:56:24,540 --> 00:56:28,310
And I have this
whole thing squared.

802
00:56:31,460 --> 00:56:37,370
And you see that all of
these things cancel out.

803
00:56:37,370 --> 00:56:43,040
And the answer is
simply 2 minus n

804
00:56:43,040 --> 00:56:48,260
plus 2 epsilon
divided by n plus 8.

805
00:56:48,260 --> 00:56:52,630
And somehow, I feel that
I made a factor of-- no,

806
00:56:52,630 --> 00:56:53,890
I think that's fine.

807
00:56:56,680 --> 00:56:57,295
Double check.

808
00:57:06,315 --> 00:57:06,815
Yep.

809
00:57:12,640 --> 00:57:13,230
All right.

810
00:57:26,730 --> 00:57:29,990
So let's see what happened.

811
00:57:29,990 --> 00:57:37,080
We have identified two
fixed points, Gaussian

812
00:57:37,080 --> 00:57:40,457
and in dimensions
less than 4, the O n.

813
00:57:43,140 --> 00:57:46,230
Associated with
this are a number

814
00:57:46,230 --> 00:57:50,370
of operators that tell me-- or
eigen-directions that tell me

815
00:57:50,370 --> 00:57:56,750
if I go away from the fixed
point, whether I would go back

816
00:57:56,750 --> 00:57:58,980
or I would go away.

817
00:57:58,980 --> 00:58:03,860
And so for the
Gaussian, these just

818
00:58:03,860 --> 00:58:08,940
have the names of the parameters
that we have to set non-zero.

819
00:58:08,940 --> 00:58:19,860
So their names are things like
t, u, l, v, u6, and so forth.

820
00:58:19,860 --> 00:58:23,820
And the value of these
exponents we can actually

821
00:58:23,820 --> 00:58:28,250
get without doing anything
because what is happening here

822
00:58:28,250 --> 00:58:30,560
is just dimensional analysis.

823
00:58:30,560 --> 00:58:35,010
So if I simply replace
m by something,

824
00:58:35,010 --> 00:58:38,430
m prime, gradients or
integrations with some power

825
00:58:38,430 --> 00:58:41,760
of distance b, I can
very easily figure out

826
00:58:41,760 --> 00:58:44,740
what the dimensions of
these quantities are.

827
00:58:44,740 --> 00:58:54,250
They are 2, epsilon, minus 2,
minus 2 plus epsilon, minus 2

828
00:58:54,250 --> 00:58:57,900
plus 2 epsilon, and so forth.

829
00:58:57,900 --> 00:59:00,840
So this is simple
dimensional analysis.

830
00:59:00,840 --> 00:59:06,020
And in some sense, these
correspond to their dimensions

831
00:59:06,020 --> 00:59:10,480
of the theory of the
variables that you have.

832
00:59:10,480 --> 00:59:14,390
Problem with it as a description
of what I see in experiments

833
00:59:14,390 --> 00:59:18,460
is the presence of two
relevant directions.

834
00:59:18,460 --> 00:59:22,550
Now, we found a new
fixed point that

835
00:59:22,550 --> 00:59:27,320
is under control to
order of epsilon.

836
00:59:27,320 --> 00:59:30,310
And what we find is
that this exponent

837
00:59:30,310 --> 00:59:38,340
for what was analogous to t
shifted to be 2 minus n plus 2

838
00:59:38,340 --> 00:59:39,860
over n plus 8 epsilon.

839
00:59:43,460 --> 00:59:47,010
While the one that was
epsilon shifted by minus 2

840
00:59:47,010 --> 00:59:49,310
epsilon and became
minus epsilon.

841
00:59:54,380 --> 00:59:57,560
What I see is a pattern
that essentially all that

842
00:59:57,560 --> 01:00:01,590
can happen, since I'm doing
a perturbation in epsilon,

843
01:00:01,590 --> 01:00:05,800
is that these quantities
can at most change

844
01:00:05,800 --> 01:00:07,760
by order of epsilon.

845
01:00:07,760 --> 01:00:12,360
So this was minus 2 becomes
minus 2 plus order of epsilon.

846
01:00:12,360 --> 01:00:15,730
This becomes minus 2
plus order of epsilon.

847
01:00:15,730 --> 01:00:18,385
It will not necessarily
be minus 2 plus epsilon.

848
01:00:18,385 --> 01:00:21,980
It could be minus 2 minus
7 epsilon plus 11 epsilon.

849
01:00:21,980 --> 01:00:24,610
Maybe even epsilon
squared, I don't know.

850
01:00:24,610 --> 01:00:32,340
But the point is
that clearly, even

851
01:00:32,340 --> 01:00:35,070
if I put all the
infinity of parameters,

852
01:00:35,070 --> 01:00:40,320
as long as I am in 3.999
dimension, at this fixed point

853
01:00:40,320 --> 01:00:45,540
I only have one
relevant direction.

854
01:00:45,540 --> 01:00:50,330
So it does describe the
physics that I want, at least

855
01:00:50,330 --> 01:00:53,074
in this perturbative sense
of the epsilon expansion.

856
01:00:56,770 --> 01:01:02,460
And so I have my yt.

857
01:01:02,460 --> 01:01:06,000
Actually, in order to
get all of the exponents,

858
01:01:06,000 --> 01:01:07,771
I really need two.

859
01:01:07,771 --> 01:01:10,820
I need yt and maybe yh.

860
01:01:10,820 --> 01:01:12,310
But yh is very simple.

861
01:01:12,310 --> 01:01:20,560
If I were to add to this
magnetic field term,

862
01:01:20,560 --> 01:01:27,840
then in Fourier representation
it just goes and sits over here

863
01:01:27,840 --> 01:01:30,170
at q equals to 0.

864
01:01:30,170 --> 01:01:32,970
And when I do all
of my rescalings,

865
01:01:32,970 --> 01:01:35,650
et cetera, the only
thing that happens to it

866
01:01:35,650 --> 01:01:37,930
is that it just picks
the factor of z.

867
01:01:41,930 --> 01:01:47,120
And we've shown that z is
b to the 1 plus d over 2

868
01:01:47,120 --> 01:01:50,590
plus order of epsilon squared.

869
01:01:50,590 --> 01:01:55,380
And so essentially,
we also have our yh.

870
01:01:55,380 --> 01:01:57,980
So I can even add
it to this table.

871
01:01:57,980 --> 01:01:59,900
There is a yh.

872
01:01:59,900 --> 01:02:01,710
There is a magnetic field.

873
01:02:01,710 --> 01:02:06,080
The corresponding yh is 1 plus
d over 2 for the Gaussian.

874
01:02:06,080 --> 01:02:14,235
It is 1 plus d over 2 plus order
of epsilon squared for the O n

875
01:02:14,235 --> 01:02:14,734
model.

876
01:02:17,470 --> 01:02:21,880
So now we have
everything that we need.

877
01:02:21,880 --> 01:02:26,840
We can compute
things in principle.

878
01:02:26,840 --> 01:02:29,730
We find, first of
all, that if I look

879
01:02:29,730 --> 01:02:35,780
at the divergence of the
correlation length, essentially

880
01:02:35,780 --> 01:02:42,310
we saw that under
rescaling yt tells us

881
01:02:42,310 --> 01:02:47,050
that if our magnetic field
is 0, how I get thrown away

882
01:02:47,050 --> 01:02:49,870
from the fixed point over here.

883
01:02:49,870 --> 01:02:51,870
There is a relevant
direction out here

884
01:02:51,870 --> 01:02:57,770
that we've discovered whose
eigenvalue here is no longer 2.

885
01:02:57,770 --> 01:03:02,540
It is 2 minus this formula
that we've calculated.

886
01:03:02,540 --> 01:03:06,450
And presumably again,
if I go and look

887
01:03:06,450 --> 01:03:11,070
at my set of parameters,
what I have is that

888
01:03:11,070 --> 01:03:16,245
in this infinite dimensional
space, as I reduce temperature,

889
01:03:16,245 --> 01:03:21,090
I will be going from,
say, one point here,

890
01:03:21,090 --> 01:03:23,350
then lower temperature
would be here,

891
01:03:23,350 --> 01:03:25,220
lower temperature would be here.

892
01:03:25,220 --> 01:03:30,020
So there would be a trajectory
as a function of shifting

893
01:03:30,020 --> 01:03:36,100
temperature, which at some
point that trajectory hits

894
01:03:36,100 --> 01:03:40,990
the basin of attraction of this
O n fixed point that we found.

895
01:03:40,990 --> 01:03:45,680
And then being
away from here will

896
01:03:45,680 --> 01:03:48,780
have a projection
along this axis.

897
01:03:48,780 --> 01:03:52,020
And we can relate that to
the divergence of, say,

898
01:03:52,020 --> 01:03:54,810
the correlation length
to the free energy.

899
01:03:54,810 --> 01:03:57,550
Our u was 1 over yt.

900
01:03:57,550 --> 01:04:01,550
It is the inverse
of this object.

901
01:04:01,550 --> 01:04:07,550
So let's say this
is-- if I divide,

902
01:04:07,550 --> 01:04:12,900
I have 1/2 1 minus
n plus 2 over 2 n

903
01:04:12,900 --> 01:04:18,320
plus 8 epsilon raised
to the minus 1 power.

904
01:04:18,320 --> 01:04:21,260
And to be consistent, I
should really only expand this

905
01:04:21,260 --> 01:04:23,310
to the order of epsilon.

906
01:04:23,310 --> 01:04:31,420
So I have 1/2 plus 1/4 n
plus 2 n plus 8 epsilon.

907
01:04:35,080 --> 01:04:37,500
So what does it tell me?

908
01:04:37,500 --> 01:04:40,600
Well, it tells me that
Gaussian fixed point,

909
01:04:40,600 --> 01:04:43,010
the correlation length
exponent was 1/2.

910
01:04:43,010 --> 01:04:44,990
We already saw that.

911
01:04:44,990 --> 01:04:47,780
We see that when we
go to this O n model,

912
01:04:47,780 --> 01:04:51,820
the correlation length exponent
becomes larger than 1/2.

913
01:04:51,820 --> 01:04:54,460
I guess that agrees
with our table.

914
01:04:54,460 --> 01:04:57,690
And I guess we can
try to estimate

915
01:04:57,690 --> 01:05:02,630
what values we would get if
we were to put n equals to 1,

916
01:05:02,630 --> 01:05:05,300
n equals to 2, et cetera.

917
01:05:05,300 --> 01:05:09,460
So this is n equals to 1.

918
01:05:09,460 --> 01:05:15,100
If I put epsilon equals to
1, what do I get for mu?

919
01:05:15,100 --> 01:05:20,580
I will get 1/2 plus 1/4 of 3/9.

920
01:05:20,580 --> 01:05:22,460
So that's 1/12.

921
01:05:22,460 --> 01:05:30,152
So that would give me
something like 0.58.

922
01:05:30,152 --> 01:05:31,060
All right.

923
01:05:31,060 --> 01:05:33,030
Not bad for a
low-order expansion

924
01:05:33,030 --> 01:05:36,680
coming from 4 to something
that's in three dimensions.

925
01:05:36,680 --> 01:05:40,140
What happens if I
go to n equals to 2?

926
01:05:40,140 --> 01:05:44,090
OK, so correction is
4/10 divided by 4.

927
01:05:44,090 --> 01:05:45,090
So it's 0.1.

928
01:05:45,090 --> 01:05:49,280
So I would get 0.6.

929
01:05:49,280 --> 01:05:51,930
What happens if I
put n equals to 3?

930
01:05:51,930 --> 01:05:56,040
I will get 5 divided by 44.

931
01:05:56,040 --> 01:06:01,860
And I believe that gives
me something like 61.

932
01:06:06,380 --> 01:06:12,250
So it gets worse when I
go to larger values of n,

933
01:06:12,250 --> 01:06:14,690
but it does capture a trend.

934
01:06:14,690 --> 01:06:18,740
Experimentally, we
see that mu becomes

935
01:06:18,740 --> 01:06:23,960
larger as you go from 1 to 2
or 3-component order parameter.

936
01:06:23,960 --> 01:06:29,920
That trend is already captured
by this low-order expansion.

937
01:06:29,920 --> 01:06:35,990
Once you have mu, you can,
for example, calculate alpha.

938
01:06:35,990 --> 01:06:39,480
Alpha is 2 minus d mu.

939
01:06:39,480 --> 01:06:43,530
So you do 2 minus your
d is 4 minus epsilon.

940
01:06:43,530 --> 01:06:54,795
Your mu is this 1/2 1 plus
1/8 n plus 2 n plus 8 epsilon.

941
01:06:59,860 --> 01:07:02,050
1/2, sorry.

942
01:07:02,050 --> 01:07:05,420
And you do the algebra
and I'll write the answer.

943
01:07:05,420 --> 01:07:11,199
It is 4 minus n epsilon
divided by 2 n plus 8.

944
01:07:16,089 --> 01:07:17,556
OK, and let me check.

945
01:07:32,200 --> 01:07:36,480
So if I now substitute
epsilon equals

946
01:07:36,480 --> 01:07:40,890
to 1 for these different
values of n, what I get

947
01:07:40,890 --> 01:07:56,220
are for alpha 0.17, 0.11, 0.06.

948
01:07:56,220 --> 01:07:59,200
I don't know, maybe I have
a factor of 2 missing,

949
01:07:59,200 --> 01:07:59,830
or whatever.

950
01:07:59,830 --> 01:08:04,710
But these numbers, I
think, are correct.

951
01:08:04,710 --> 01:08:10,590
So you can see that in
reality alpha is positive

952
01:08:10,590 --> 01:08:13,495
for the liquid gas
system n equals to 1.

953
01:08:13,495 --> 01:08:15,220
It is more or less 0.

954
01:08:15,220 --> 01:08:19,310
This is the logarithmic
lambda point for superfluids.

955
01:08:19,310 --> 01:08:24,260
And then it becomes negative,
clearly, for magnets.

956
01:08:24,260 --> 01:08:27,560
The formula that we have
predicts all of these numbers

957
01:08:27,560 --> 01:08:30,779
to be positive, but it
gets the right trend

958
01:08:30,779 --> 01:08:34,470
that as you go to
larger values of n,

959
01:08:34,470 --> 01:08:37,990
the value of the
exponent alpha calculated

960
01:08:37,990 --> 01:08:43,430
at this order in epsilon
expansion becomes lower.

961
01:08:43,430 --> 01:08:46,090
So that trend is captured.

962
01:08:46,090 --> 01:08:51,210
So at this stage,
I guess I would

963
01:08:51,210 --> 01:08:54,882
say that the
problem that I posed

964
01:08:54,882 --> 01:08:58,100
is solved in the same
sense that I would

965
01:08:58,100 --> 01:09:05,399
say we have solved for the
energy levels of the helium

966
01:09:05,399 --> 01:09:06,560
atom.

967
01:09:06,560 --> 01:09:09,930
Certainly, you can sort
of ignore the interaction

968
01:09:09,930 --> 01:09:13,990
between electrons and
calculate hydrogenic energies.

969
01:09:13,990 --> 01:09:16,790
And then you can do
perturbation in the strength

970
01:09:16,790 --> 01:09:21,330
of the interaction and
get corrections to that.

971
01:09:21,330 --> 01:09:24,810
So essentially, you know the
trends and you know everything.

972
01:09:24,810 --> 01:09:28,310
And we have been
able to sort of find

973
01:09:28,310 --> 01:09:31,609
the physical structure
that would give us

974
01:09:31,609 --> 01:09:36,229
a root to calculating
what these exponents are.

975
01:09:36,229 --> 01:09:38,170
We see that the
exponents are really

976
01:09:38,170 --> 01:09:41,920
a function of dimensionality
and the symmetry of the order

977
01:09:41,920 --> 01:09:43,130
parameter.

978
01:09:43,130 --> 01:09:47,410
All of the trends are captured,
but the numerical values,

979
01:09:47,410 --> 01:09:50,920
not surprisingly,
at this low order

980
01:09:50,920 --> 01:09:54,060
have not been
captured very well.

981
01:09:54,060 --> 01:09:56,540
So presumably, you would
need to do the same thing

982
01:09:56,540 --> 01:09:58,270
that you would do
for the helium atom.

983
01:09:58,270 --> 01:10:01,360
You could do to higher and
higher order calculations.

984
01:10:01,360 --> 01:10:02,670
You could do simulations.

985
01:10:02,670 --> 01:10:05,320
You could do all
kinds of other things.

986
01:10:05,320 --> 01:10:08,360
But the conceptual
foundation is basically

987
01:10:08,360 --> 01:10:10,364
what we have laid out here.

988
01:10:12,970 --> 01:10:19,210
OK, there is one thing
that-- well, many things that

989
01:10:19,210 --> 01:10:22,350
remain to be answered.

990
01:10:22,350 --> 01:10:25,730
One of them is--
well, how do you

991
01:10:25,730 --> 01:10:29,710
know that there isn't a
fixed point somewhere else?

992
01:10:29,710 --> 01:10:33,185
You calculated things
perturbatively.

993
01:10:33,185 --> 01:10:38,590
The answer is that once you
do higher-order calculations,

994
01:10:38,590 --> 01:10:41,880
et cetera, you find that
your results converge,

995
01:10:41,880 --> 01:10:45,970
more or less, better and better
to the results of simulations

996
01:10:45,970 --> 01:10:48,030
or experiments, et cetera.

997
01:10:48,030 --> 01:10:51,990
So there is no
evidence from whatever

998
01:10:51,990 --> 01:10:55,640
we know that there
is need for something

999
01:10:55,640 --> 01:10:58,970
that I would call
a strong coupling,

1000
01:10:58,970 --> 01:11:01,090
non-perturbative fixed point.

1001
01:11:01,090 --> 01:11:02,240
It's not a proof.

1002
01:11:02,240 --> 01:11:04,880
We can't prove that
there isn't such a thing.

1003
01:11:04,880 --> 01:11:09,270
But there is apparently
no need for such a thing

1004
01:11:09,270 --> 01:11:12,450
to discuss what is
observed experimentally.

1005
01:11:12,450 --> 01:11:14,712
Yes.

1006
01:11:14,712 --> 01:11:18,250
AUDIENCE: You told us last time
that in order for the mu fixed

1007
01:11:18,250 --> 01:11:21,056
point to make sense, we must
have epsilon very small.

1008
01:11:21,056 --> 01:11:22,520
PROFESSOR: Yes.

1009
01:11:22,520 --> 01:11:26,920
AUDIENCE: But now we're
putting epsilon back to 1.

1010
01:11:26,920 --> 01:11:28,370
PROFESSOR: OK.

1011
01:11:28,370 --> 01:11:32,790
So when people inevitably
ask me this question,

1012
01:11:32,790 --> 01:11:37,670
I give them the following
two functions of epsilon.

1013
01:11:37,670 --> 01:11:41,010
One of them is e to
the epsilon over 100

1014
01:11:41,010 --> 01:11:44,960
and the other is e
to the 100 epsilon.

1015
01:11:47,760 --> 01:11:53,310
Do I know if I put epsilon
of 1 a priori whether or not

1016
01:11:53,310 --> 01:11:57,720
putting epsilon equals to 1 is
a good thing for the expansion

1017
01:11:57,720 --> 01:11:58,830
or not?

1018
01:11:58,830 --> 01:12:00,050
I don't.

1019
01:12:00,050 --> 01:12:03,690
And so I don't know
whether it is bad

1020
01:12:03,690 --> 01:12:05,910
and I don't know
whether it is good,

1021
01:12:05,910 --> 01:12:09,150
unless I calculate many
more terms in the series

1022
01:12:09,150 --> 01:12:13,284
and discuss what the
convergence of the series is.

1023
01:12:13,284 --> 01:12:16,670
AUDIENCE: Epsilon is 4 minus d
is supposed to be an integer.

1024
01:12:16,670 --> 01:12:17,670
We just--

1025
01:12:17,670 --> 01:12:20,440
PROFESSOR: Oh, you're worried
about its integerness as

1026
01:12:20,440 --> 01:12:25,150
opposed to treating
it as a continuum?

1027
01:12:25,150 --> 01:12:26,066
OK.

1028
01:12:26,066 --> 01:12:31,898
AUDIENCE: It's OK when you
first assume it [INAUDIBLE].

1029
01:12:31,898 --> 01:12:35,300
At the end of the day,
it must be an integer.

1030
01:12:35,300 --> 01:12:36,120
PROFESSOR: OK.

1031
01:12:36,120 --> 01:12:39,800
So the example somebody was
asking me also last time

1032
01:12:39,800 --> 01:12:43,310
that I have in mind is
you've learned n factorial

1033
01:12:43,310 --> 01:12:47,020
to be the product of 1, 2, 3, 4.

1034
01:12:47,020 --> 01:12:50,360
And you know it for whatever
integer that you would like,

1035
01:12:50,360 --> 01:12:52,770
you do the multiplication.

1036
01:12:52,770 --> 01:12:57,610
But we've also established that
n factorial is an integral 0

1037
01:12:57,610 --> 01:13:03,120
to infinity dx x to
the n e to the minus x.

1038
01:13:03,120 --> 01:13:07,340
And so the question
is now, can I

1039
01:13:07,340 --> 01:13:11,770
talk about 4.111
factorial or not?

1040
01:13:11,770 --> 01:13:16,340
Can I expand 4.11
factorial close

1041
01:13:16,340 --> 01:13:21,580
to what I have 4 assuming the
form of this so-called gamma

1042
01:13:21,580 --> 01:13:23,390
function?

1043
01:13:23,390 --> 01:13:29,510
So the gamma function
is a function of n

1044
01:13:29,510 --> 01:13:34,140
that at integer values it
falls on whatever we know,

1045
01:13:34,140 --> 01:13:37,240
but it has a perfect
analytic continuation

1046
01:13:37,240 --> 01:13:41,750
and I can in principle evaluate
4 factorial by expanding

1047
01:13:41,750 --> 01:13:45,750
around 3 and the derivatives
of the gamma function

1048
01:13:45,750 --> 01:13:47,386
evaluated at 3.

1049
01:13:47,386 --> 01:13:48,885
AUDIENCE: But these
kind of integers

1050
01:13:48,885 --> 01:13:51,700
don't have anything
with dimensionality.

1051
01:13:51,700 --> 01:13:54,770
PROFESSOR: OK, where did our
dimensionality come from?

1052
01:13:54,770 --> 01:13:57,460
Our dimensionality
appears in our expressions

1053
01:13:57,460 --> 01:14:00,070
because we have to do
integrals of this form.

1054
01:14:05,746 --> 01:14:07,660
And what do we do?

1055
01:14:07,660 --> 01:14:10,780
We replace this with a
surface area, which actually

1056
01:14:10,780 --> 01:14:13,460
involves the
factorial by the way.

1057
01:14:13,460 --> 01:14:17,620
And then we have k
to the d minus 1 dk.

1058
01:14:17,620 --> 01:14:22,600
So these integrals are
functions of dimension

1059
01:14:22,600 --> 01:14:25,740
that have exactly the same
properties as the gamma

1060
01:14:25,740 --> 01:14:27,740
function and the n factorial.

1061
01:14:27,740 --> 01:14:31,690
They're perfectly
well expandable.

1062
01:14:31,690 --> 01:14:33,300
And they do have singularities.

1063
01:14:33,300 --> 01:14:36,060
Actually, it turns out
the gamma functions also

1064
01:14:36,060 --> 01:14:39,340
have singularities at
minus 1, things like that.

1065
01:14:39,340 --> 01:14:42,991
Our functions have singularities
at two dimensions and so forth.

1066
01:14:46,470 --> 01:14:51,300
But the issue of convergence
is very important.

1067
01:14:51,300 --> 01:14:55,610
So let's say that there were
some powerful field theories.

1068
01:14:55,610 --> 01:14:59,020
And in order to do
calculations at higher orders,

1069
01:14:59,020 --> 01:15:02,273
you need to go and
do field theory.

1070
01:15:02,273 --> 01:15:07,350
And you calculate
the exponent gamma.

1071
01:15:07,350 --> 01:15:12,220
And I will write the gamma
exponent for the case of n

1072
01:15:12,220 --> 01:15:14,610
equals to 1.

1073
01:15:14,610 --> 01:15:20,580
And the series for that is 1
plus-- if we sort of go and do

1074
01:15:20,580 --> 01:15:25,510
all of our calculations to
lowest order, gamma is 2 mu.

1075
01:15:25,510 --> 01:15:29,690
So it will simply be twice
what we have over here.

1076
01:15:29,690 --> 01:15:35,065
And the first correction is,
indeed, 167 times epsilon.

1077
01:15:40,580 --> 01:15:49,160
The next one is 0.077
epsilon squared.

1078
01:15:49,160 --> 01:15:53,870
The next one is
minus-- problematic--

1079
01:15:53,870 --> 01:15:58,080
049 epsilon cubed.

1080
01:15:58,080 --> 01:16:04,130
Next one, 180 epsilon
to the fourth.

1081
01:16:04,130 --> 01:16:06,550
Next one-- and I think
this is as far as people

1082
01:16:06,550 --> 01:16:10,220
have calculated things--
epsilon to the fifth.

1083
01:16:13,680 --> 01:16:17,900
Then, let's put
epsilon equals to 1

1084
01:16:17,900 --> 01:16:21,180
and see what we get
at the various orders.

1085
01:16:21,180 --> 01:16:23,250
So clearly, I start with 1.

1086
01:16:23,250 --> 01:16:27,750
Next order I will get 1.167.

1087
01:16:27,750 --> 01:16:34,355
At the next order,
I will get 1.244.

1088
01:16:34,355 --> 01:16:36,880
It's getting there, huh?

1089
01:16:36,880 --> 01:16:43,850
And the next order
I will get 1.195.

1090
01:16:43,850 --> 01:16:47,990
Then I will get 1.375.

1091
01:16:47,990 --> 01:16:51,270
And then I will get 0.96.

1092
01:16:51,270 --> 01:16:54,820
[LAUGHTER]

1093
01:16:54,820 --> 01:17:01,730
So this is the signature of what
is called an asymptotic series,

1094
01:17:01,730 --> 01:17:06,230
something that as you evaluate
more therms gets closer

1095
01:17:06,230 --> 01:17:08,740
to the expected
result, but then starts

1096
01:17:08,740 --> 01:17:11,880
to move away and oscillate.

1097
01:17:11,880 --> 01:17:14,400
Yet, there are tricks.

1098
01:17:14,400 --> 01:17:18,585
And if you know your tricks,
you can put epsilon equals to 1

1099
01:17:18,585 --> 01:17:22,070
in that series and do
clever enough terms

1100
01:17:22,070 --> 01:17:35,730
to get 1.2385 minus plus 0.0025.

1101
01:17:35,730 --> 01:17:39,760
So the trick is called
Borel summation.

1102
01:17:45,770 --> 01:17:53,470
So one can show that if you go
to high orders in this series,

1103
01:17:53,470 --> 01:17:57,770
asymptotically the terms in
the series scale like, say,

1104
01:17:57,770 --> 01:18:04,670
the p-th term in the series will
scale as p factorial, something

1105
01:18:04,670 --> 01:18:08,980
like a to the power of p and
some coefficient in front.

1106
01:18:08,980 --> 01:18:13,250
So if I write the general
term in this series

1107
01:18:13,250 --> 01:18:17,540
f sub p epsilon to
the p, my statement

1108
01:18:17,540 --> 01:18:23,620
is that the magnitude of f sub
p asymptotically for p much

1109
01:18:23,620 --> 01:18:26,270
larger than 1 going to
infinity has this form.

1110
01:18:29,570 --> 01:18:33,570
So clearly, because
of this p factorial,

1111
01:18:33,570 --> 01:18:34,935
this is growing too rapidly.

1112
01:18:38,490 --> 01:18:44,750
But what you can do is you
can rewrite this series, which

1113
01:18:44,750 --> 01:18:52,230
is sum over p f of p epsilon
of p using this integral that I

1114
01:18:52,230 --> 01:18:54,835
had over here for p factorial.

1115
01:18:58,430 --> 01:19:02,380
So I multiply and
divide by p factorial.

1116
01:19:02,380 --> 01:19:09,110
So it becomes a sum over p f
of p epsilon of p integral 0

1117
01:19:09,110 --> 01:19:14,100
to infinity dx x to
the p e to the minus x

1118
01:19:14,100 --> 01:19:18,080
divided by p factorial.

1119
01:19:18,080 --> 01:19:22,940
And the fp over p factorial
gets rid of this factor.

1120
01:19:22,940 --> 01:19:25,930
So then you can recast
this as the integral

1121
01:19:25,930 --> 01:19:30,400
from 0 to infinity
dx e to the minus x.

1122
01:19:30,400 --> 01:19:35,310
And what you do is you sum
the series f of p divided by p

1123
01:19:35,310 --> 01:19:39,250
factorial epsilon x
raised to the power of p.

1124
01:19:42,560 --> 01:19:50,060
And it turns out that this is
called some kind of a Borel

1125
01:19:50,060 --> 01:19:53,900
function corresponding
to this series.

1126
01:19:53,900 --> 01:19:58,890
And as long as the terms in your
series only diverge this badly,

1127
01:19:58,890 --> 01:20:04,180
people can make sense of this
function, Borel function.

1128
01:20:04,180 --> 01:20:06,310
And then you perform
the integration,

1129
01:20:06,310 --> 01:20:07,906
and then you come
up with this number.

1130
01:20:12,690 --> 01:20:17,760
So that's one thing to note.

1131
01:20:17,760 --> 01:20:25,310
The other thing
to note is I said

1132
01:20:25,310 --> 01:20:29,680
that what I want to do
for my perturbation theory

1133
01:20:29,680 --> 01:20:35,270
to make sense is for
this u star to be small.

1134
01:20:35,270 --> 01:20:38,100
And I said that the
knob that we have

1135
01:20:38,100 --> 01:20:41,120
is for epsilon to be small.

1136
01:20:41,120 --> 01:20:44,810
But there is, if you look at
that expression, another knob.

1137
01:20:44,810 --> 01:20:49,380
I can make n going to infinity.

1138
01:20:49,380 --> 01:20:54,930
So if n becomes very
large-- so that also

1139
01:20:54,930 --> 01:20:57,940
can make the thing small.

1140
01:20:57,940 --> 01:21:01,180
So there is an
alternative expansion.

1141
01:21:01,180 --> 01:21:03,410
Rather than going with
epsilon going to 0,

1142
01:21:03,410 --> 01:21:06,670
you go to what is called
a spherical model.

1143
01:21:06,670 --> 01:21:08,800
That is, an infinite
number of components,

1144
01:21:08,800 --> 01:21:11,540
and then do in
expansion in 1 over n.

1145
01:21:11,540 --> 01:21:15,795
And so then you
basically-- what you

1146
01:21:15,795 --> 01:21:19,920
are interested is things
that are happening

1147
01:21:19,920 --> 01:21:24,455
as a function of d and n.

1148
01:21:24,455 --> 01:21:28,960
And you have-- above
4, you know that you

1149
01:21:28,960 --> 01:21:30,320
are in the Gaussian world.

1150
01:21:36,100 --> 01:21:43,790
At n goes to infinity, you
have this O n type of models.

1151
01:21:43,790 --> 01:21:47,670
And you find that these models
actually only make sense

1152
01:21:47,670 --> 01:21:52,520
in dimensions that
are larger than 2.

1153
01:21:52,520 --> 01:21:58,640
So you can then perturbatively
come either from here

1154
01:21:58,640 --> 01:22:01,780
or you can come from
here and you try to get

1155
01:22:01,780 --> 01:22:05,930
to the exponents you
are interested over here

1156
01:22:05,930 --> 01:22:09,060
or over here.

1157
01:22:09,060 --> 01:22:13,850
So basically, that's the story.

1158
01:22:13,850 --> 01:22:16,330
And for this work,
I think, as I said,

1159
01:22:16,330 --> 01:22:21,620
Wilson did this pertubative RG.

1160
01:22:21,620 --> 01:22:24,680
Michael Fisher
was the person who

1161
01:22:24,680 --> 01:22:28,250
focused it into an
epsilon expansion.

1162
01:22:28,250 --> 01:22:34,140
And in 1982, Wilson got the
Nobel Prize for the work.

1163
01:22:34,140 --> 01:22:35,850
Potentially, it
could have been also

1164
01:22:35,850 --> 01:22:39,590
awarded to Fisher and Kadanoff
for their contributions

1165
01:22:39,590 --> 01:22:41,680
to this whole story.

1166
01:22:41,680 --> 01:22:45,590
So that's the end of
this part of the course.

1167
01:22:45,590 --> 01:22:48,700
And now that we have
established this background,

1168
01:22:48,700 --> 01:22:54,500
we will try to get the exponents
and the statistical behavior

1169
01:22:54,500 --> 01:22:57,410
by a number of
other perspectives.

1170
01:22:57,410 --> 01:23:00,050
So basically, this
was a perspective

1171
01:23:00,050 --> 01:23:03,020
and a route that gave an answer.

1172
01:23:03,020 --> 01:23:05,960
And hopefully, we'll be
able to complement it

1173
01:23:05,960 --> 01:23:09,143
with other ways of
looking at the story.