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PROFESSOR: OK, let's start.

9
00:00:25,770 --> 00:00:31,060
So recapping what
we have been doing,

10
00:00:31,060 --> 00:00:34,800
we said that many systems that
undergo phase transition-- so

11
00:00:34,800 --> 00:00:38,830
there's some material that
undergoes phase transition-- we

12
00:00:38,830 --> 00:00:41,130
could look at it
and characterize it

13
00:00:41,130 --> 00:00:43,900
through a statistical field.

14
00:00:43,900 --> 00:00:49,290
But my analogy in the case
of magnetization of magnet

15
00:00:49,290 --> 00:00:53,890
will be noted by m that
varies as a function

16
00:00:53,890 --> 00:00:56,350
of the position on the sample.

17
00:00:56,350 --> 00:00:59,210
And it's a vector.

18
00:00:59,210 --> 00:01:01,950
And this vector
has n components.

19
00:01:01,950 --> 00:01:04,879
And we said that basically
we could distinguish

20
00:01:04,879 --> 00:01:09,470
different types of systems by
the number of components of n.

21
00:01:09,470 --> 00:01:13,600
And for the case of
things like liquid gas,

22
00:01:13,600 --> 00:01:17,060
we had a scale or
density difference,

23
00:01:17,060 --> 00:01:18,980
which is one component.

24
00:01:18,980 --> 00:01:27,590
For the case of superfluid,
we had the phase

25
00:01:27,590 --> 00:01:31,670
of a quantum mechanical
wave function, which

26
00:01:31,670 --> 00:01:36,450
had, therefore, two components
when we included the magnitude.

27
00:01:36,450 --> 00:01:45,860
And for the case of, say,
[INAUDIBLE] ferromagnet,

28
00:01:45,860 --> 00:01:47,589
we had n equals 3.

29
00:01:52,080 --> 00:01:57,380
We said that basically all of
these systems in the vicinity

30
00:01:57,380 --> 00:02:02,180
of the transition point
where the field n of x

31
00:02:02,180 --> 00:02:05,970
is presumably fluctuating
around a small quantity

32
00:02:05,970 --> 00:02:08,259
and the correlation
lengths are large,

33
00:02:08,259 --> 00:02:12,640
we could describe in
terms of weight that

34
00:02:12,640 --> 00:02:15,570
was constructed on
the basis of symmetry

35
00:02:15,570 --> 00:02:20,240
and a form of
locality which allowed

36
00:02:20,240 --> 00:02:24,970
us to express the
weight in powers of m

37
00:02:24,970 --> 00:02:33,760
squared integrated in the
vicinity of some point x.

38
00:02:33,760 --> 00:02:38,040
Then the connection between
the different clients

39
00:02:38,040 --> 00:02:42,514
was captured through terms
that involves gradients of m.

40
00:02:46,600 --> 00:02:51,850
And higher order derivatives
are also possible.

41
00:02:51,850 --> 00:02:56,250
And that easy back to deviate
from the symmetry axis,

42
00:02:56,250 --> 00:02:58,590
we could add a term that is h.m.

43
00:03:04,350 --> 00:03:07,860
So that was this
statistical weight

44
00:03:07,860 --> 00:03:12,370
that we assigned to
configurations of this field.

45
00:03:12,370 --> 00:03:15,810
Now we said that when
you do measurements

46
00:03:15,810 --> 00:03:18,770
of these kinds of
systems, for example,

47
00:03:18,770 --> 00:03:21,325
you will see singularities
in heat capacity.

48
00:03:28,730 --> 00:03:33,580
And those in the vicinity
of the phase transitions

49
00:03:33,580 --> 00:03:35,950
were characterized
by an exponent alpha.

50
00:03:44,730 --> 00:03:50,250
Now the value of alpha, you can
go and look at various system,

51
00:03:50,250 --> 00:03:54,000
you find for liquid gas
systems many different versions

52
00:03:54,000 --> 00:03:57,820
of carbon dioxide, et
cetera, and other systems

53
00:03:57,820 --> 00:04:00,600
that you would
correspond to n equals 1,

54
00:04:00,600 --> 00:04:03,560
like binary mixtures
such as the one that

55
00:04:03,560 --> 00:04:07,440
is in the first
problem set, correspond

56
00:04:07,440 --> 00:04:17,430
to a value of alpha divergence
that is roughly around 0.11.

57
00:04:17,430 --> 00:04:21,250
For the case of
superfluid, we saw

58
00:04:21,250 --> 00:04:24,220
curves that described
this lambda point.

59
00:04:24,220 --> 00:04:27,620
There is, again, a
divergence, or the divergence

60
00:04:27,620 --> 00:04:29,010
is weaker than the [INAUDIBLE].

61
00:04:29,010 --> 00:04:34,370
It is approximately a
logarithmic divergence.

62
00:04:34,370 --> 00:04:37,520
Whereas for ferromagnets,
there is a cost singularity.

63
00:04:37,520 --> 00:04:39,420
There's no divergence.

64
00:04:39,420 --> 00:04:42,400
And the singularity
can be expressed

65
00:04:42,400 --> 00:04:45,610
in terms of a negative alpha.

66
00:04:45,610 --> 00:04:47,920
So there are these
classes, depending

67
00:04:47,920 --> 00:04:53,860
on the value of this parameter
n, which are all the same.

68
00:04:53,860 --> 00:04:58,300
And in our case, they are all
described by this same field

69
00:04:58,300 --> 00:05:04,350
theory, with different number of
components of this quantity n.

70
00:05:04,350 --> 00:05:09,140
So we asked whether or not
we could get that result.

71
00:05:09,140 --> 00:05:15,600
So what we did was we said, OK,
let's calculate the partition

72
00:05:15,600 --> 00:05:18,560
function that corresponds
to this system

73
00:05:18,560 --> 00:05:22,260
by integrating over all
configurations of this field.

74
00:05:25,340 --> 00:05:29,300
And this is actually
just the singular part,

75
00:05:29,300 --> 00:05:32,400
because in the process
of going from whatever

76
00:05:32,400 --> 00:05:36,850
microscopic variables we have
to these variables that describe

77
00:05:36,850 --> 00:05:40,320
the statistical field,
we have to integrate

78
00:05:40,320 --> 00:05:43,090
over many microscopic
configurations.

79
00:05:43,090 --> 00:05:46,510
So there could be a
non-singular part that emerges.

80
00:05:46,510 --> 00:05:50,960
But the singularities are due to
the appearance of magnetization

81
00:05:50,960 --> 00:05:52,480
spontaneously.

82
00:05:52,480 --> 00:05:56,660
So they should be reflected
in calculating the partition

83
00:05:56,660 --> 00:05:58,010
function of this component.

84
00:06:01,370 --> 00:06:05,830
Now what we did
was, then, to say,

85
00:06:05,830 --> 00:06:08,490
OK, this is difficult thing.

86
00:06:08,490 --> 00:06:18,240
What I am going to do is do
a subtle point approximation,

87
00:06:18,240 --> 00:06:22,415
which really amounted to
finding the most probable state.

88
00:06:31,280 --> 00:06:39,280
And that most probable
state corresponded to the m

89
00:06:39,280 --> 00:06:47,700
being uniform across the system,
value m bar, that potentially

90
00:06:47,700 --> 00:06:55,640
would be directed along
the magnetic field.

91
00:06:55,640 --> 00:06:58,650
But there's only the limit
that magnetic field goes to 0,

92
00:06:58,650 --> 00:07:02,780
spontaneously select
some kind of a direction.

93
00:07:02,780 --> 00:07:07,210
But of course, this
m bar would be 0

94
00:07:07,210 --> 00:07:10,320
if you are above the
transition, which

95
00:07:10,320 --> 00:07:13,180
in this most
probable state occurs

96
00:07:13,180 --> 00:07:17,255
for t's that are
positive at h equals 0.

97
00:07:20,340 --> 00:07:26,000
While for t negative,
minimizing tm squared plus um

98
00:07:26,000 --> 00:07:30,920
to the fourth gave us a value of
square root of minus t over 4.

99
00:07:39,370 --> 00:07:52,490
Then our z singular in the
subtle point approximation

100
00:07:52,490 --> 00:08:00,240
evaluated as a function
of t for h equals 0

101
00:08:00,240 --> 00:08:05,990
is simply related to the value
of this most probable state

102
00:08:05,990 --> 00:08:08,700
at this particular point.

103
00:08:08,700 --> 00:08:14,480
And we found that the answer
was exponential of minus because

104
00:08:14,480 --> 00:08:17,450
of the integration over space.

105
00:08:17,450 --> 00:08:18,695
But everything is uniform.

106
00:08:18,695 --> 00:08:21,740
It will be
proportional to volume.

107
00:08:21,740 --> 00:08:24,870
And then multiplied
by a function

108
00:08:24,870 --> 00:08:30,140
that was either 0, if you
were looking at t positive.

109
00:08:30,140 --> 00:08:33,679
Whereas for t negative,
substituting that value

110
00:08:33,679 --> 00:08:38,790
of h bar, gave us minus
t squared over 16.

111
00:08:45,476 --> 00:08:45,975
Yeah.

112
00:08:55,640 --> 00:08:58,820
So essentially, it's a function.

113
00:08:58,820 --> 00:09:04,860
There is really no magnetization
above the critical point.

114
00:09:04,860 --> 00:09:06,690
And you get 0.

115
00:09:06,690 --> 00:09:09,740
Below the critical
point, what you

116
00:09:09,740 --> 00:09:13,550
have is this quadratic
behavior in t.

117
00:09:13,550 --> 00:09:17,280
So if I were to take two
derivatives of it, which

118
00:09:17,280 --> 00:09:19,950
would give me something that
is proportional to the heat

119
00:09:19,950 --> 00:09:27,450
capacity-- so from here I would
get a heat capacity evaluated

120
00:09:27,450 --> 00:09:33,570
in the subtle point method
as a function of t for h

121
00:09:33,570 --> 00:09:39,545
equals 0, which would be
either 0 or 1 over 8u,

122
00:09:39,545 --> 00:09:42,440
if I'm taking this derivative.

123
00:09:42,440 --> 00:09:48,440
So the prediction is that
you have an alpha which is 0

124
00:09:48,440 --> 00:09:52,520
because there is no
power law dependence.

125
00:09:52,520 --> 00:09:56,795
And what it really
reflects is that there

126
00:09:56,795 --> 00:10:00,750
is a discontinuity
in heat capacity.

127
00:10:00,750 --> 00:10:02,870
So none of the
examples that I showed

128
00:10:02,870 --> 00:10:07,670
you aboved-- the liquid gas,
the superfluid, ferromagnet--

129
00:10:07,670 --> 00:10:09,820
have a discontinuous
heat capacity.

130
00:10:09,820 --> 00:10:12,560
So this does not seem to work.

131
00:10:12,560 --> 00:10:15,320
On the other hand,
this discontinuity

132
00:10:15,320 --> 00:10:21,323
is observed for
superconductor transitions.

133
00:10:29,080 --> 00:10:32,070
So that's the state
of the affairs.

134
00:10:32,070 --> 00:10:36,250
What we have to understand
now is, first of all,

135
00:10:36,250 --> 00:10:39,060
why doesn't it work in general?

136
00:10:39,060 --> 00:10:43,620
Secondly, why does it
work for superconductors?

137
00:10:43,620 --> 00:10:46,978
So that's the task for today.

138
00:10:46,978 --> 00:10:49,740
All right?

139
00:10:49,740 --> 00:10:53,780
So the one thing
that is certainly

140
00:10:53,780 --> 00:10:57,220
a glaring approximation
is to replace

141
00:10:57,220 --> 00:11:01,550
this integration over all
configuration by just the one

142
00:11:01,550 --> 00:11:03,900
most probable state.

143
00:11:03,900 --> 00:11:07,190
But we did precisely
that when we

144
00:11:07,190 --> 00:11:11,870
were talking about the subtle
point method of integration

145
00:11:11,870 --> 00:11:17,110
in the previous class in 8 333.

146
00:11:17,110 --> 00:11:22,770
So let's examine why it
was legitimate to do so

147
00:11:22,770 --> 00:11:24,690
at that point.

148
00:11:24,690 --> 00:11:29,020
So there we are evaluating
essentially an integration

149
00:11:29,020 --> 00:11:31,360
that involved one variable.

150
00:11:31,360 --> 00:11:33,360
Let's call it m.

151
00:11:33,360 --> 00:11:42,410
And we had a large number that
was appearing in the exponent.

152
00:11:42,410 --> 00:11:53,120
And we had some function
that we were looking at,

153
00:11:53,120 --> 00:11:54,870
depending on the
variable of integration.

154
00:11:57,640 --> 00:12:00,490
Now the most probable
value of this

155
00:12:00,490 --> 00:12:04,500
occurs for some
particular m bar.

156
00:12:04,500 --> 00:12:09,520
And what we can do,
without essentially

157
00:12:09,520 --> 00:12:13,980
doing any approximation
at this point,

158
00:12:13,980 --> 00:12:19,130
is to make a Taylor
expansion of the function

159
00:12:19,130 --> 00:12:21,130
around its maximum.

160
00:12:21,130 --> 00:12:30,520
So the function I can write as
psi evaluated at this extremum.

161
00:12:30,520 --> 00:12:33,180
But since I am looking
at an extremum,

162
00:12:33,180 --> 00:12:35,630
if I make a Taylor
expansion, the term

163
00:12:35,630 --> 00:12:38,650
that is proportional to the
first derivative is absent.

164
00:12:38,650 --> 00:12:41,910
I'm expanding
around an extremum.

165
00:12:41,910 --> 00:12:52,020
The term that is proportional to
the second derivative evaluated

166
00:12:52,020 --> 00:12:57,600
at m bar will go with
m minus m bar squared.

167
00:12:57,600 --> 00:13:00,370
And in principle, there are
higher and higher order terms

168
00:13:00,370 --> 00:13:01,710
I can put in this expansion.

169
00:13:05,610 --> 00:13:10,410
Now the value at the
most probable position,

170
00:13:10,410 --> 00:13:13,020
which is the subtle point
value, is a constant.

171
00:13:13,020 --> 00:13:16,730
I can put it outside.

172
00:13:16,730 --> 00:13:20,410
So essentially,
terminating here is exactly

173
00:13:20,410 --> 00:13:24,470
like what I was doing
over there, more or less.

174
00:13:24,470 --> 00:13:28,140
But then I have
fluctuations around

175
00:13:28,140 --> 00:13:30,540
this most probably state.

176
00:13:30,540 --> 00:13:33,840
So I can do the
integration, let's say,

177
00:13:33,840 --> 00:13:38,100
in the variable delta n.

178
00:13:38,100 --> 00:13:44,176
I have the
differential of delta m

179
00:13:44,176 --> 00:13:51,070
into the minus 1/2
psi double m bar

180
00:13:51,070 --> 00:13:57,060
m minus delta m bar
delta m squared.

181
00:13:57,060 --> 00:13:59,680
And then I have
higher order terms.

182
00:13:59,680 --> 00:14:02,790
And principle, those
higher order terms I

183
00:14:02,790 --> 00:14:05,330
can start expanding over here.

184
00:14:14,600 --> 00:14:17,510
And I forgot the very
important factor,

185
00:14:17,510 --> 00:14:21,530
which is that this whole
thing is proportional to n.

186
00:14:21,530 --> 00:14:25,200
And indeed, all of
these terms over here

187
00:14:25,200 --> 00:14:29,500
will also be proportional to n.

188
00:14:29,500 --> 00:14:30,870
OK?

189
00:14:30,870 --> 00:14:33,650
But the first term
in the series is just

190
00:14:33,650 --> 00:14:36,080
the Gaussian integration.

191
00:14:36,080 --> 00:14:43,680
And so I know that the leading
correction to the subtle point

192
00:14:43,680 --> 00:14:52,520
comes from this factor of root 2
pi n psi double prime of m bar.

193
00:14:52,520 --> 00:14:58,370
And then, in principle, there
will be higher order terms.

194
00:14:58,370 --> 00:15:03,120
And if you keep track of
how many factors of delta m

195
00:15:03,120 --> 00:15:05,160
allowed-- delta m
cubed is certainly not

196
00:15:05,160 --> 00:15:07,890
allowed because of
the evenness of what

197
00:15:07,890 --> 00:15:09,690
I'm integrating against.

198
00:15:09,690 --> 00:15:13,040
So the next order term will
be delta m to the fourth.

199
00:15:13,040 --> 00:15:15,770
Evaluated against
this Gaussian, it

200
00:15:15,770 --> 00:15:19,330
will give you something that
is order of 1 over n squared.

201
00:15:19,330 --> 00:15:22,420
Multiplied by n, you will get
corrections of the order of 1

202
00:15:22,420 --> 00:15:25,680
over n.

203
00:15:25,680 --> 00:15:29,770
So very systematically,
we could see

204
00:15:29,770 --> 00:15:33,900
that if I called the result
of this integration i,

205
00:15:33,900 --> 00:15:40,440
that log of i has a
term that is dominated

206
00:15:40,440 --> 00:15:45,580
by the most probable
value of the integrant.

207
00:15:45,580 --> 00:15:52,260
And then there are corrections,
such as this factor of log n

208
00:15:52,260 --> 00:15:57,170
psi double prime
m bar over 2 pi,

209
00:15:57,170 --> 00:16:02,190
and lower order corrections
of order of 1 over n.

210
00:16:02,190 --> 00:16:09,400
Basically all of these
terms in the limit of n

211
00:16:09,400 --> 00:16:13,300
being much larger than
1, you can ignore.

212
00:16:13,300 --> 00:16:16,340
And essentially, this term
will dominate everything.

213
00:16:20,840 --> 00:16:24,660
So what we did over there
kind of looks the same.

214
00:16:24,660 --> 00:16:29,700
So let's repeat that for
our functional integral.

215
00:16:29,700 --> 00:16:31,970
So I have z.

216
00:16:31,970 --> 00:16:35,250
Actually it is the singular
part of the partition

217
00:16:35,250 --> 00:16:40,790
function, which is obtained
by integrating over

218
00:16:40,790 --> 00:16:44,560
all functions m of x.

219
00:16:44,560 --> 00:16:47,210
And for the time being,
let's just focus on the h

220
00:16:47,210 --> 00:16:49,400
equals 0 part.

221
00:16:49,400 --> 00:16:53,450
So I have exponential
of minus integral

222
00:16:53,450 --> 00:17:00,420
over x, t over 2m
square, um to the fourth,

223
00:17:00,420 --> 00:17:05,595
k over 2 gradient of m
squared, and so forth.

224
00:17:12,130 --> 00:17:18,700
And repeat what
we did over there.

225
00:17:18,700 --> 00:17:22,530
So what we need over here
was to basically pick

226
00:17:22,530 --> 00:17:27,819
the most probable state
and then expand around

227
00:17:27,819 --> 00:17:30,280
the most probable state.

228
00:17:30,280 --> 00:17:37,300
So going beyond just picking the
contribution of most probable

229
00:17:37,300 --> 00:17:41,670
state involves including
these fluctuations.

230
00:17:41,670 --> 00:17:50,810
So let me write my m of x
to be essentially m bar,

231
00:17:50,810 --> 00:17:53,130
but allowing a little
bit of fluctuation.

232
00:17:53,130 --> 00:17:57,080
And we saw that we could
divide the fluctuations

233
00:17:57,080 --> 00:17:59,100
into a longitudinal part.

234
00:17:59,100 --> 00:18:02,290
Let's call it e1 hat.

235
00:18:02,290 --> 00:18:09,610
And the transfers part, which
is an n minus 1 component

236
00:18:09,610 --> 00:18:13,570
vector in the n minus
1 transfers directions.

237
00:18:17,280 --> 00:18:21,320
And then I substitute
this over here.

238
00:18:21,320 --> 00:18:24,200
So what do I get?

239
00:18:24,200 --> 00:18:26,920
Just like here, I
can pull out the term

240
00:18:26,920 --> 00:18:29,160
that corresponds to
the subtle point.

241
00:18:29,160 --> 00:18:31,770
In fact, I had
calculated it up there.

242
00:18:31,770 --> 00:18:37,646
So I have exponential
of minus v,

243
00:18:37,646 --> 00:18:40,520
the value of this thing
at the subtle point.

244
00:18:48,400 --> 00:18:52,960
And then I have essentially
replaced the variable m

245
00:18:52,960 --> 00:18:56,040
with the integration
over fluctuations.

246
00:18:56,040 --> 00:19:02,890
So now I have to integrate over
the longitudinal fluctuations

247
00:19:02,890 --> 00:19:05,260
and the transfers fluctuations.

248
00:19:08,590 --> 00:19:15,720
And what I need to do is
to expand this quantity up

249
00:19:15,720 --> 00:19:18,130
to second order.

250
00:19:18,130 --> 00:19:20,310
But that's exactly
what we did last time,

251
00:19:20,310 --> 00:19:24,880
where you were looking at how
the system was scattering.

252
00:19:24,880 --> 00:19:31,510
So we can rely on the
result from last time

253
00:19:31,510 --> 00:19:34,290
for what the quadratic part is.

254
00:19:34,290 --> 00:19:39,130
So we saw that the answer
could be written as minus k

255
00:19:39,130 --> 00:19:43,942
over 2, integral ddx.

256
00:19:43,942 --> 00:19:49,270
Well, actually, let's
keep it this way.

257
00:19:49,270 --> 00:20:02,790
We have cl to the minus
2 plus phi l squared

258
00:20:02,790 --> 00:20:11,560
plus gradient of phi l squared.

259
00:20:11,560 --> 00:20:14,700
So this is what I did.

260
00:20:14,700 --> 00:20:21,300
What we had to do was to
replace this function.

261
00:20:21,300 --> 00:20:24,035
The only part that has a
contribution from variation

262
00:20:24,035 --> 00:20:29,260
in space, and hence contributes
to gradient, comes from phi.

263
00:20:29,260 --> 00:20:32,830
So from here, we will
get a k over 2 gradient

264
00:20:32,830 --> 00:20:33,790
of phi l squared.

265
00:20:37,090 --> 00:20:41,930
Then there is a
contribution from t,

266
00:20:41,930 --> 00:20:44,350
and one that comes
from expanding m

267
00:20:44,350 --> 00:20:47,720
to the fourth to
quadratic folder, that

268
00:20:47,720 --> 00:20:51,190
are proportional
to phi l squared.

269
00:20:51,190 --> 00:20:53,430
And the coefficient
of both of them

270
00:20:53,430 --> 00:20:57,190
we combine to write
as cl to the minus 2.

271
00:20:57,190 --> 00:21:01,070
And if I go back to
what we had last time,

272
00:21:01,070 --> 00:21:09,510
our result was that
k over cl squared

273
00:21:09,510 --> 00:21:17,120
was either t, if I
was for t positive,

274
00:21:17,120 --> 00:21:20,055
or minus 2t if I
was for t negative.

275
00:21:22,850 --> 00:21:30,050
Whereas, when I expanded
the transfers component,

276
00:21:30,050 --> 00:21:35,750
what I got above
tc, for t positive

277
00:21:35,750 --> 00:21:38,255
there is no difference between
longitudinal transfers,

278
00:21:38,255 --> 00:21:40,380
so I had the same result.

279
00:21:40,380 --> 00:21:45,330
Below, there was no cost
for these Goldstone modes,

280
00:21:45,330 --> 00:21:48,860
and the answer was 0.

281
00:21:48,860 --> 00:21:51,740
But essentially, I have a
similar expression, then,

282
00:21:51,740 --> 00:22:05,050
to write for the
transfers component.

283
00:22:11,020 --> 00:22:16,130
So this part amounts
to essentially

284
00:22:16,130 --> 00:22:18,711
what I have over here.

285
00:22:18,711 --> 00:22:24,880
And in principle, I can put
a whole bunch of other things

286
00:22:24,880 --> 00:22:30,470
that would correspond to higher
order fluctuations, effects

287
00:22:30,470 --> 00:22:33,710
beyond the quadratic.

288
00:22:33,710 --> 00:22:37,420
But again, our
anticipation is that, just

289
00:22:37,420 --> 00:22:41,200
like what is happening
here, the leading correction

290
00:22:41,200 --> 00:22:47,340
to the subtle point will already
come from the quadratic part.

291
00:22:47,340 --> 00:22:51,080
So let's evaluate that.

292
00:22:51,080 --> 00:22:53,460
So let's continue.

293
00:22:53,460 --> 00:22:59,155
So this is exponential of
the subtle point phi energy.

294
00:23:05,460 --> 00:23:11,370
And then I have to do all of
these integrations over phi

295
00:23:11,370 --> 00:23:12,560
l and phi q.

296
00:23:18,210 --> 00:23:23,350
Now what I can do, and I already
did this also last time around,

297
00:23:23,350 --> 00:23:30,120
is we introduced an
expansion of phi.

298
00:23:30,120 --> 00:23:34,120
We said each phi
of x I can write

299
00:23:34,120 --> 00:23:37,510
as a sum over
Fourier components--

300
00:23:37,510 --> 00:23:44,790
e to the iq.x phi tilda
of q, and with a root

301
00:23:44,790 --> 00:23:48,200
phi for normalization
convenience.

302
00:23:48,200 --> 00:23:53,560
So I can certainly replace
both phi l and phi t,

303
00:23:53,560 --> 00:23:58,610
just as I did last time, in
terms of Fourier component.

304
00:23:58,610 --> 00:24:03,660
And then the integration over
all configurations of phi

305
00:24:03,660 --> 00:24:05,720
is equivalent to
integrating over

306
00:24:05,720 --> 00:24:10,760
all configurations
of the phi tilda

307
00:24:10,760 --> 00:24:12,513
of q's, sll the
Fourier amplitudes.

308
00:24:16,320 --> 00:24:19,280
But the advantage is that
when we look at the Fourier

309
00:24:19,280 --> 00:24:23,680
amplitudes, the
different q's are

310
00:24:23,680 --> 00:24:26,650
completely independent
of each other.

311
00:24:26,650 --> 00:24:30,190
So this integration
over here that

312
00:24:30,190 --> 00:24:33,430
was not the
one-dimensional integration

313
00:24:33,430 --> 00:24:37,880
becomes a product of
one-dimensional integrations

314
00:24:37,880 --> 00:24:42,410
when we go to the Fourier
component representation.

315
00:24:42,410 --> 00:24:45,360
So now I have to
integrate for each q.

316
00:24:45,360 --> 00:24:49,190
I have either phi
l of q, or I have

317
00:24:49,190 --> 00:24:52,620
the n minus 1
component phi p of q.

318
00:24:52,620 --> 00:24:58,190
So these are whole bunch
of one dimensional Gaussian

319
00:24:58,190 --> 00:24:59,710
integrations.

320
00:24:59,710 --> 00:25:03,720
Because when I look at
what these rates are doing,

321
00:25:03,720 --> 00:25:11,800
I get e to the minus k over 2,
q squared plus cl to the minus 2

322
00:25:11,800 --> 00:25:18,060
phi l of q squared for
the longitudinal mode,

323
00:25:18,060 --> 00:25:22,620
and a very similar
factor k over 2 q

324
00:25:22,620 --> 00:25:27,930
squared plus ct to the
minus 2, phi t of q

325
00:25:27,930 --> 00:25:30,460
squared for the
transfers vectors.

326
00:25:33,352 --> 00:25:35,830
I have a whole bunch of
these different things.

327
00:25:39,080 --> 00:25:43,870
Now we can, again, follow
like what we had before.

328
00:25:43,870 --> 00:25:51,200
The leading behavior is minus v
t m bar squared over 2, u m bar

329
00:25:51,200 --> 00:25:53,510
to the fourth.

330
00:25:53,510 --> 00:26:00,150
And then I have a product
of Gaussian integrations.

331
00:26:00,150 --> 00:26:04,860
For each one of these
longitudinal modes,

332
00:26:04,860 --> 00:26:08,090
just like here, I
will get a factor

333
00:26:08,090 --> 00:26:18,140
of 2 pi divided by
k q squared plus cl

334
00:26:18,140 --> 00:26:23,040
to the minus 2 square root.

335
00:26:23,040 --> 00:26:26,950
And for each one of the
transfers components,

336
00:26:26,950 --> 00:26:34,170
I will get 2 pi k q 2
plus ct to the minus 2.

337
00:26:34,170 --> 00:26:37,940
And there are n
minus 1 of these.

338
00:26:37,940 --> 00:26:39,460
So I will get that factor.

339
00:26:42,250 --> 00:26:45,370
And then presume,
again, there will

340
00:26:45,370 --> 00:26:50,110
be corrections due
to higher orders that

341
00:26:50,110 --> 00:26:52,200
will be multiplying
the whole thing.

342
00:27:00,220 --> 00:27:04,880
So the quantity that
we are interested

343
00:27:04,880 --> 00:27:11,510
is, in fact, something
like phi energy.

344
00:27:11,510 --> 00:27:13,710
So we take log of z.

345
00:27:13,710 --> 00:27:16,700
Let's look at the singular part.

346
00:27:16,700 --> 00:27:18,930
Let's divide it by
volume, because we

347
00:27:18,930 --> 00:27:21,580
expect this to be an
extensive quantity,

348
00:27:21,580 --> 00:27:25,156
just like this other result
was proportional to n.

349
00:27:25,156 --> 00:27:28,010
And let's put a minus
sign-- typically

350
00:27:28,010 --> 00:27:30,850
you have to change
sign in any case--

351
00:27:30,850 --> 00:27:36,890
so that the leading term
then becomes this tm squared

352
00:27:36,890 --> 00:27:42,700
plus um to the fourth,
which, let me remind you,

353
00:27:42,700 --> 00:27:46,250
is-- actually,
let's just write it.

354
00:27:46,250 --> 00:27:50,140
tm bar squared over 2 plus
u m bar to the fourth.

355
00:27:52,940 --> 00:28:00,122
And then when I take the
log, this product over q

356
00:28:00,122 --> 00:28:04,162
will go to a sum over q.

357
00:28:07,858 --> 00:28:13,040
And the sum over q in
the continuum limit,

358
00:28:13,040 --> 00:28:19,926
I will replace by v integral
over q divided by 3 pi

359
00:28:19,926 --> 00:28:20,708
to the d.

360
00:28:23,640 --> 00:28:27,150
So then the next
step of the process,

361
00:28:27,150 --> 00:28:29,540
I will have a sum
over q which I replace

362
00:28:29,540 --> 00:28:31,830
with v times the integration.

363
00:28:31,830 --> 00:28:37,230
But the volume will go
away, and what I'm left with

364
00:28:37,230 --> 00:28:39,230
is the integration.

365
00:28:39,230 --> 00:28:46,530
So I have the integral
vdq 2 pi to the d.

366
00:28:46,530 --> 00:28:50,030
And I have the log of whatever
is appearing over here.

367
00:28:53,810 --> 00:28:59,930
So what I have
there is log of k q

368
00:28:59,930 --> 00:29:06,900
squared plus k cl to
the minus 2 with 1/2.

369
00:29:06,900 --> 00:29:07,960
Why the 1/2?

370
00:29:07,960 --> 00:29:10,120
Because it's the square root.

371
00:29:10,120 --> 00:29:12,750
I take it to the
exponential because it

372
00:29:12,750 --> 00:29:15,200
becomes one half of the log.

373
00:29:15,200 --> 00:29:17,160
In fact, it is in
the denominator.

374
00:29:17,160 --> 00:29:18,440
So there's a minus sign.

375
00:29:18,440 --> 00:29:22,690
And the minus sign cancels
the minus sign out here.

376
00:29:22,690 --> 00:29:25,580
And then the next term from
the transfers component,

377
00:29:25,580 --> 00:29:31,310
I will get n minus 1
over 2, integral dbq 2 pi

378
00:29:31,310 --> 00:29:39,650
to the d log of kq squared
plus kct to the minus 2.

379
00:29:39,650 --> 00:29:43,470
And presumably, if I go ahead
with higher and higher order

380
00:29:43,470 --> 00:29:45,258
corrections, there
will be other things.

381
00:29:45,258 --> 00:29:45,758
Yes, Carter.

382
00:29:45,758 --> 00:29:47,252
AUDIENCE: So [INAUDIBLE].

383
00:29:52,740 --> 00:29:53,980
PROFESSOR: No.

384
00:29:53,980 --> 00:29:56,860
It's just, like
the subtle point,

385
00:29:56,860 --> 00:30:00,790
I'm trying to calculate
a systematic expansion

386
00:30:00,790 --> 00:30:02,420
around the subtle point.

387
00:30:02,420 --> 00:30:07,390
So I've calculated so far
the lowest order term,

388
00:30:07,390 --> 00:30:11,120
although I haven't explicitly
told you what its behavior is.

389
00:30:11,120 --> 00:30:14,790
Once I'm satisfied with what
kind of connection that this,

390
00:30:14,790 --> 00:30:18,900
I need to go beyond and include
higher and higher order terms,

391
00:30:18,900 --> 00:30:22,510
and maybe show you that they
are explicitly unimportant,

392
00:30:22,510 --> 00:30:24,580
like they are in the
ordinary subtle point,

393
00:30:24,580 --> 00:30:25,920
or that they are important.

394
00:30:25,920 --> 00:30:28,160
At this stage, we are agnostic.

395
00:30:28,160 --> 00:30:29,341
We don't say anything.

396
00:30:37,060 --> 00:30:39,330
One thing to note--
of course, there

397
00:30:39,330 --> 00:30:42,310
are all of these
factors of 2 pi.

398
00:30:42,310 --> 00:30:46,900
Now if you go to a
mathematician and show them

399
00:30:46,900 --> 00:30:51,570
a functional integral, they
say it's an undefined quantity.

400
00:30:51,570 --> 00:30:54,190
And part of the reason
for undefined quantity

401
00:30:54,190 --> 00:30:57,440
is, well, how many factors
of 2 pi do you have?

402
00:30:57,440 --> 00:31:01,270
And what are the limits
of this integration?

403
00:31:01,270 --> 00:31:03,970
So from the perspective
of mathematics,

404
00:31:03,970 --> 00:31:05,680
a functional
integral is something

405
00:31:05,680 --> 00:31:08,610
that is very sick
and ill-behaved.

406
00:31:08,610 --> 00:31:11,190
In our case, there
is no problem,

407
00:31:11,190 --> 00:31:14,110
because we know that our
field, although I wrote it

408
00:31:14,110 --> 00:31:16,830
as a continuous
function, it is really

409
00:31:16,830 --> 00:31:20,100
a continuous function that
has a limited set of Fourier

410
00:31:20,100 --> 00:31:21,390
components.

411
00:31:21,390 --> 00:31:24,430
This product over
q will not extend

412
00:31:24,430 --> 00:31:26,452
to arbitrary short wavelength.

413
00:31:26,452 --> 00:31:27,910
There's a characteristic
wavelength

414
00:31:27,910 --> 00:31:31,370
which is the scale over which
I did the coarse graining,

415
00:31:31,370 --> 00:31:33,710
and I don't have
anything beyond that.

416
00:31:33,710 --> 00:31:36,110
So these are finite
number of Fourier modes

417
00:31:36,110 --> 00:31:37,440
that I'm integrating here.

418
00:31:37,440 --> 00:31:41,080
There is a finite number of 2,
2 pi, et cetera, that one has.

419
00:31:45,780 --> 00:31:47,310
All right.

420
00:31:47,310 --> 00:31:50,320
So fine, so this
is the behavior.

421
00:31:50,320 --> 00:31:55,640
Again, I have looked only
as a function of t setting h

422
00:31:55,640 --> 00:31:58,230
equals to 0.

423
00:31:58,230 --> 00:32:01,300
I didn't include
the effect of h.

424
00:32:01,300 --> 00:32:07,110
And let's explicitly look at
what these things are for t

425
00:32:07,110 --> 00:32:12,850
positive that I will write
above, and t negative

426
00:32:12,850 --> 00:32:14,402
that I will write below.

427
00:32:14,402 --> 00:32:17,810
We saw that above, this is 0.

428
00:32:17,810 --> 00:32:20,510
Below, it is minus
t squared over 16u.

429
00:32:24,020 --> 00:32:29,950
That this quantity kcl to
the minus 2, it is t above

430
00:32:29,950 --> 00:32:33,660
and it is minus 2t below.

431
00:32:33,660 --> 00:32:39,820
This quantity kc to the minus t
squared is t above and 0 below.

432
00:32:43,650 --> 00:32:46,710
Why do I bother to write that?

433
00:32:46,710 --> 00:32:52,250
Because I want to go and address
this question of heat capacity.

434
00:32:52,250 --> 00:32:57,370
And we said that heat
capacity is ultimately

435
00:32:57,370 --> 00:33:04,620
related to taking two
derivatives of this log c

436
00:33:04,620 --> 00:33:08,850
singular with respect
to temperature and beta

437
00:33:08,850 --> 00:33:09,740
and all of that.

438
00:33:09,740 --> 00:33:11,740
But let's write it
as a proportionality.

439
00:33:11,740 --> 00:33:12,850
It goes like this.

440
00:33:15,716 --> 00:33:16,215
Yes?

441
00:33:19,470 --> 00:33:20,900
Yes?

442
00:33:20,900 --> 00:33:23,850
AUDIENCE: So the third
line on the top board,

443
00:33:23,850 --> 00:33:30,020
you have under this continuous
product over all elements of q.

444
00:33:30,020 --> 00:33:31,510
PROFESSOR: So this
product over q

445
00:33:31,510 --> 00:33:33,818
goes all the way to the
end of the line, yes.

446
00:33:33,818 --> 00:33:34,442
AUDIENCE: Yeah.

447
00:33:34,442 --> 00:33:38,387
So can [INAUDIBLE]
be in the exponents,

448
00:33:38,387 --> 00:33:39,470
or are they still outside?

449
00:33:42,630 --> 00:33:44,593
PROFESSOR: What infinitesimals?

450
00:33:44,593 --> 00:33:46,485
AUDIENCE: d phi l and d phi t.

451
00:33:49,330 --> 00:33:51,140
PROFESSOR: OK, so
what I have left out,

452
00:33:51,140 --> 00:33:53,285
and you're quite
right, is the integral.

453
00:33:56,310 --> 00:34:02,730
So for each q, I have
to do n integrations

454
00:34:02,730 --> 00:34:06,335
over this variable and
this n minus 1 component.

455
00:34:06,335 --> 00:34:08,670
So I forgot the integral
sign, so that's correct.

456
00:34:18,530 --> 00:34:20,100
All right.

457
00:34:20,100 --> 00:34:21,280
So what do we have?

458
00:34:24,040 --> 00:34:29,400
So for t positive, if I take two
derivative of this with respect

459
00:34:29,400 --> 00:34:35,410
to t-- and actually there is
a minus sign involved here,

460
00:34:35,410 --> 00:34:35,909
sorry.

461
00:34:38,719 --> 00:34:42,350
Above the transition,
I will get 0.

462
00:34:42,350 --> 00:34:46,020
Below the transition, I
will get this 1 over 8u.

463
00:34:46,020 --> 00:34:49,710
So this is the discontinuity
that I had calculated before.

464
00:34:52,850 --> 00:34:56,400
Now above the transition,
I have to take

465
00:34:56,400 --> 00:35:03,490
a derivative of log of tkq
squared plus t with respect

466
00:35:03,490 --> 00:35:04,780
to t.

467
00:35:04,780 --> 00:35:07,960
Taking the derivative
of log will give me

468
00:35:07,960 --> 00:35:11,010
1 over its argument.

469
00:35:11,010 --> 00:35:14,820
Taking the second
derivative will give me

470
00:35:14,820 --> 00:35:16,740
the argument squared.

471
00:35:16,740 --> 00:35:20,310
Because of the minus sign, I
forget about the minus sign.

472
00:35:20,310 --> 00:35:23,310
So two derivatives of this
object with respect to t

473
00:35:23,310 --> 00:35:26,430
will bring down a factor of
kq squared plus t squared.

474
00:35:29,030 --> 00:35:31,800
And I have to
integrate that over q.

475
00:35:36,760 --> 00:35:43,540
And there is one from here, and
there's n minus 1 from here.

476
00:35:43,540 --> 00:35:47,070
So there is a total
of n over 2 of that.

477
00:35:54,070 --> 00:35:57,810
Below the transition, I
have to take a derivative,

478
00:35:57,810 --> 00:36:02,090
except that plus thing is
replaced with minus 2t.

479
00:36:02,090 --> 00:36:04,200
So every time I
take a derivative,

480
00:36:04,200 --> 00:36:07,380
I will get an
additional factor of 2.

481
00:36:07,380 --> 00:36:14,765
So rather than 1/2, I will
end up with 2 integral over q

482
00:36:14,765 --> 00:36:21,950
2 pi to the d 1 over kq
squared minus 2t squared

483
00:36:21,950 --> 00:36:24,360
from the longitudinal part.

484
00:36:24,360 --> 00:36:26,860
And the transfers part
has no t dependence,

485
00:36:26,860 --> 00:36:28,137
so it doesn't contribute.

486
00:36:35,300 --> 00:36:41,090
So the entire
thing, you can see,

487
00:36:41,090 --> 00:36:45,740
is what I had calculated
at the subtle point.

488
00:36:45,740 --> 00:36:51,990
And to this order in expansions
around the subtle point, which

489
00:36:51,990 --> 00:36:57,190
corresponds, essentially,
only to the quadratic part,

490
00:36:57,190 --> 00:36:58,430
I have found a correction.

491
00:37:01,440 --> 00:37:06,070
And generically, we see
that these corrections

492
00:37:06,070 --> 00:37:15,512
are proportional to an
integral over q 2 pi to the d.

493
00:37:15,512 --> 00:37:17,690
I can actually
pull out one factor

494
00:37:17,690 --> 00:37:24,220
of k squared outside so that the
integral more looks more nice

495
00:37:24,220 --> 00:37:29,430
with some characteristic
lengths scale, which is either

496
00:37:29,430 --> 00:37:32,000
coming from t or from minus 2t.

497
00:37:32,000 --> 00:37:34,160
So I can write it as cl squared.

498
00:37:39,640 --> 00:37:47,120
So in order to understand how
important these corrections

499
00:37:47,120 --> 00:37:50,350
are-- and here the corrections
were under control.

500
00:37:50,350 --> 00:37:54,920
So really, I'm asking question,
are these corrections small

501
00:37:54,920 --> 00:37:57,270
compared to what I
started in the same sense

502
00:37:57,270 --> 00:38:00,080
that log n is small
compared to n?

503
00:38:00,080 --> 00:38:02,060
Well, what I need to
do is to understand

504
00:38:02,060 --> 00:38:03,920
how this integral behaves.

505
00:38:03,920 --> 00:38:06,650
There is no factor
of log n versus n,

506
00:38:06,650 --> 00:38:08,650
because you can see
both of those terms

507
00:38:08,650 --> 00:38:10,921
have the factor of volume.

508
00:38:10,921 --> 00:38:14,770
So the issue is not that you
have something like square,

509
00:38:14,770 --> 00:38:17,950
log of the volume that will
give you small quantity.

510
00:38:17,950 --> 00:38:20,080
You have to hope
that for some reason

511
00:38:20,080 --> 00:38:25,890
or other this whole integral
here is not important.

512
00:38:25,890 --> 00:38:32,520
So if I look at the integrand--
well, I can do one more thing.

513
00:38:32,520 --> 00:38:37,260
I can note that it behaves
as 1 over k squared.

514
00:38:37,260 --> 00:38:38,880
There is a combination
that you will

515
00:38:38,880 --> 00:38:43,230
see appearing many, many
times in this course.

516
00:38:43,230 --> 00:38:45,380
Because this is
spherically symmetric,

517
00:38:45,380 --> 00:38:49,030
I can write it as
some solid angle q

518
00:38:49,030 --> 00:38:51,340
to the d minus 1 with q.

519
00:38:51,340 --> 00:38:54,310
And so the whole thing is
proportional to the ratio

520
00:38:54,310 --> 00:38:59,430
of solid angle divided by 2 pi
to the d, which will occur so

521
00:38:59,430 --> 00:39:03,150
many times in this course that
we will give it a name k sub d.

522
00:39:05,760 --> 00:39:09,230
And then the eventual
integral is simply

523
00:39:09,230 --> 00:39:16,350
an integral over one variable
q, q to the d minus 1.

524
00:39:16,350 --> 00:39:21,370
And then I have q squared
plus c to the minus 2 squared.

525
00:39:26,910 --> 00:39:27,540
Yes?

526
00:39:27,540 --> 00:39:30,040
AUDIENCE: Should that 1
over kd be 1 over k squared?

527
00:39:34,840 --> 00:39:37,855
PROFESSOR: There is a q
over k-- Yeah, that's right.

528
00:39:37,855 --> 00:39:39,090
I already had it.

529
00:39:39,090 --> 00:39:40,410
Yes, 1 over k squared.

530
00:39:40,410 --> 00:39:41,701
And then there's 1 over k.

531
00:39:41,701 --> 00:39:42,200
Thank you.

532
00:39:48,260 --> 00:39:52,770
So how does this
integrand look like,

533
00:39:52,770 --> 00:39:56,120
the thing that I
have to integrate?

534
00:39:56,120 --> 00:39:58,250
As a function of q,
I have to integrate

535
00:39:58,250 --> 00:40:01,680
a function that at
least that small q has

536
00:40:01,680 --> 00:40:06,320
no problem of singularity,
divergence, et cetera.

537
00:40:06,320 --> 00:40:10,390
It is simply q to
the minus something

538
00:40:10,390 --> 00:40:12,280
with the coefficient
that is like c.

539
00:40:15,510 --> 00:40:21,131
At large distances, however,
let's say three dimensions,

540
00:40:21,131 --> 00:40:27,906
it would fall off as q to the
power of d minus 1 minus 4.

541
00:40:27,906 --> 00:40:30,960
At large q, I can ignore
whatever is from here

542
00:40:30,960 --> 00:40:34,720
and just look at
the powers of q.

543
00:40:34,720 --> 00:40:38,020
But then if I'm at
sufficiently large dimension,

544
00:40:38,020 --> 00:40:42,030
the function will keep growing.

545
00:40:42,030 --> 00:40:45,100
So basically, depending
on which dimensions

546
00:40:45,100 --> 00:40:49,040
you are, and the borderline
dimension is clearly for,

547
00:40:49,040 --> 00:40:51,190
it's an integration
that you can either

548
00:40:51,190 --> 00:40:55,130
perform without any difficulty
going all the way to infinity

549
00:40:55,130 --> 00:41:00,010
in q, or you have to worry
about the upper column.

550
00:41:00,010 --> 00:41:01,450
OK?

551
00:41:01,450 --> 00:41:07,600
So if you are in
dimensions greater than 4,

552
00:41:07,600 --> 00:41:16,070
what you find is that this cf
in dimensions that are larger

553
00:41:16,070 --> 00:41:22,640
than 4, as you go to
larger and larger q,

554
00:41:22,640 --> 00:41:24,770
you are integrating
something that

555
00:41:24,770 --> 00:41:27,550
is getting bigger and bigger.

556
00:41:27,550 --> 00:41:30,690
And you have to worry
about that being infinity.

557
00:41:30,690 --> 00:41:33,110
Except, as I told you,
we don't have any worries

558
00:41:33,110 --> 00:41:36,690
about infinity,
because our q has

559
00:41:36,690 --> 00:41:41,160
to be cut off by the inverse of
the character wavelength, which

560
00:41:41,160 --> 00:41:44,460
is the length scale over which
I am doing the coarse grain.

561
00:41:44,460 --> 00:41:48,020
So let's call that cut
off lambda, presumably

562
00:41:48,020 --> 00:41:53,510
this inverse of some kind
of lattice-like spacing.

563
00:41:53,510 --> 00:41:55,060
It's not the lattice spacing.

564
00:41:55,060 --> 00:41:58,580
It's the coarse graining scale.

565
00:41:58,580 --> 00:42:02,010
So if I'm doing this,
then this integral,

566
00:42:02,010 --> 00:42:05,500
I can really forget about
what's happening here.

567
00:42:05,500 --> 00:42:06,930
Most of the integral
contribution

568
00:42:06,930 --> 00:42:10,350
will come from the large
lambda, and so the answer

569
00:42:10,350 --> 00:42:13,600
will be proportional
to 1 over k squared

570
00:42:13,600 --> 00:42:16,093
and whatever this
other cut off is

571
00:42:16,093 --> 00:42:20,336
raised to the
power of t minus 4.

572
00:42:20,336 --> 00:42:21,920
It's proportion.

573
00:42:21,920 --> 00:42:24,670
I don't care about constants
of proportionality, et cetera.

574
00:42:27,700 --> 00:42:32,430
However, if I am at dimensions
that is 3 less than 4,

575
00:42:32,430 --> 00:42:36,670
any dimension less than 4, I can
as well say the upper cut off

576
00:42:36,670 --> 00:42:40,440
go all the way to infinity,
because the contribution that I

577
00:42:40,440 --> 00:42:44,180
get by replacing
lambda to infinity

578
00:42:44,180 --> 00:42:47,150
is going to be very small.

579
00:42:47,150 --> 00:42:50,790
So then it becomes like
a definite integral.

580
00:42:50,790 --> 00:42:53,700
And it becomes more
like a definite integral

581
00:42:53,700 --> 00:42:57,370
if I scale q by c inverse.

582
00:42:57,370 --> 00:43:02,920
And then what you have to do
is you have 1 over k squared.

583
00:43:02,920 --> 00:43:06,410
You have c inverse to
the power of t minus 4

584
00:43:06,410 --> 00:43:09,980
or c to the power of
4 minus t, and then

585
00:43:09,980 --> 00:43:13,700
some definite integral,
which is 0 to infinity dx,

586
00:43:13,700 --> 00:43:18,000
x to the d minus 1 divided by x
squared plus 1 to the squared.

587
00:43:18,000 --> 00:43:20,340
I don't really care
what the number is.

588
00:43:20,340 --> 00:43:23,606
It's just some number that
goes in this proportionality.

589
00:43:23,606 --> 00:43:27,504
So this is what happens
for d less than 4.

590
00:43:32,690 --> 00:43:38,397
So let's see what
all of this means.

591
00:43:38,397 --> 00:43:41,180
So we are trying to understand
the behavior of the heat

592
00:43:41,180 --> 00:43:49,270
capacity of the system as a
function of this parameter t.

593
00:43:49,270 --> 00:43:55,820
And actually, only the
part that corresponds

594
00:43:55,820 --> 00:43:57,466
to integrating the
magnetization field.

595
00:43:57,466 --> 00:44:00,230
As I said, there's
phonon contributions,

596
00:44:00,230 --> 00:44:02,700
all kinds of other
phonon contributions

597
00:44:02,700 --> 00:44:05,020
that give you some
kind of a background.

598
00:44:05,020 --> 00:44:08,870
Let's subtract that background
and see what we have.

599
00:44:08,870 --> 00:44:14,410
So what we have is that
from the subtle point part,

600
00:44:14,410 --> 00:44:17,130
we get this continuity.

601
00:44:17,130 --> 00:44:19,100
So let's draw the
subtle point part.

602
00:44:19,100 --> 00:44:22,816
So the subtle point
part is-- oops.

603
00:44:22,816 --> 00:44:25,140
Wrong direction.

604
00:44:25,140 --> 00:44:31,550
Above 0, it's 0.

605
00:44:31,550 --> 00:44:36,370
Below 0, it jumps to 1 over 8u.

606
00:44:36,370 --> 00:44:40,330
So it's a behavior such as this.

607
00:44:40,330 --> 00:44:44,120
So this part is the c
of the subtle point.

608
00:44:50,770 --> 00:44:54,390
But to that, I have
to add a correction.

609
00:44:54,390 --> 00:44:56,200
So let's look at the correction.

610
00:44:56,200 --> 00:44:58,390
First of all, if I'm
looking at the correction

611
00:44:58,390 --> 00:45:03,790
above four dimensions,
whether I'm above or below,

612
00:45:03,790 --> 00:45:08,100
I have to add one
of these quantities.

613
00:45:08,100 --> 00:45:11,700
These quantities don't have
any explicit dependence

614
00:45:11,700 --> 00:45:14,150
on t itself.

615
00:45:14,150 --> 00:45:17,510
So what happens is that if
I add that, presumably there

616
00:45:17,510 --> 00:45:21,505
is a correction that I will
get from below and a correction

617
00:45:21,505 --> 00:45:23,680
that I will get from above.

618
00:45:23,680 --> 00:45:30,760
So this is cf for d
that is larger than 4.

619
00:45:30,760 --> 00:45:36,000
So what it certainly does is
when I add this part to what

620
00:45:36,000 --> 00:45:42,450
I had before, I will change the
magnitude of the discontinuity.

621
00:45:42,450 --> 00:45:43,430
But so what?

622
00:45:43,430 --> 00:45:47,220
The discontinuity itself was not
something that was important,

623
00:45:47,220 --> 00:45:50,030
because u was not
a universal number.

624
00:45:50,030 --> 00:45:55,260
So there was some singularity
before, some singularity above.

625
00:45:55,260 --> 00:46:00,230
We see that the corrections
for dimensions greater than 4

626
00:46:00,230 --> 00:46:03,620
do not change the qualitative
statement that the heat

627
00:46:03,620 --> 00:46:07,800
capacity should have
a discontinuity.

628
00:46:07,800 --> 00:46:12,580
But if I go to
dimensions less than 4

629
00:46:12,580 --> 00:46:16,780
and I realize that my c goes
like the square root of t--

630
00:46:16,780 --> 00:46:22,060
there is the formulas for c over
there, or t to the minus 1/2--

631
00:46:22,060 --> 00:46:26,245
we find that this quantity
is proportional to t

632
00:46:26,245 --> 00:46:29,715
to the minus 4 minus d over 2.

633
00:46:32,900 --> 00:46:38,850
So below four
dimensions, what we get

634
00:46:38,850 --> 00:46:44,310
is that the correction
that we calculated

635
00:46:44,310 --> 00:46:45,435
is actually divergent.

636
00:46:48,270 --> 00:46:53,570
So this is cf for d less than 4.

637
00:46:53,570 --> 00:46:57,360
There is a divergence as t
goes to 0 that, let's say,

638
00:46:57,360 --> 00:47:00,030
if you're sitting
three dimensions

639
00:47:00,030 --> 00:47:02,160
would be an exponent
t to the minus 1/2.

640
00:47:07,140 --> 00:47:09,850
So you started with a
subtle point prediction

641
00:47:09,850 --> 00:47:14,220
that the heat capacity
should be discontinuous.

642
00:47:14,220 --> 00:47:16,640
You add the analog
of these corrections

643
00:47:16,640 --> 00:47:19,340
to the subtle point
calculation, and you

644
00:47:19,340 --> 00:47:24,620
find that the correction is
much, much more important

645
00:47:24,620 --> 00:47:26,245
than the original discontinuity.

646
00:47:26,245 --> 00:47:30,530
It completely changes
your conclusions.

647
00:47:30,530 --> 00:47:33,860
So once we go beyond
this approximation

648
00:47:33,860 --> 00:47:36,840
that we did over here,
the subtle point,

649
00:47:36,840 --> 00:47:40,240
and the difference between
our problematic and the one

650
00:47:40,240 --> 00:47:44,705
that we did in 8 333 is that
we don't have one variable

651
00:47:44,705 --> 00:47:46,510
that we are integrating.

652
00:47:46,510 --> 00:47:48,630
We are integrating
over fluctuations

653
00:47:48,630 --> 00:47:51,170
over the entirety of the system.

654
00:47:51,170 --> 00:47:53,330
And we see that
these fluctuations

655
00:47:53,330 --> 00:47:57,580
over the entirety of the system
are so severe, at least close

656
00:47:57,580 --> 00:48:00,230
to the transition point,
that they completely

657
00:48:00,230 --> 00:48:03,670
invalidate the results that
you had from the subtle point.

658
00:48:03,670 --> 00:48:04,340
Yes?

659
00:48:04,340 --> 00:48:08,780
AUDIENCE: So obviously you have
some high order [INAUDIBLE].

660
00:48:08,780 --> 00:48:11,140
And here you're basically
completing [INAUDIBLE].

661
00:48:11,140 --> 00:48:12,540
PROFESSOR: Exactly.

662
00:48:12,540 --> 00:48:16,070
AUDIENCE: Is there
an easy way to argue

663
00:48:16,070 --> 00:48:19,140
that for b greater than
4 there is no divergence

664
00:48:19,140 --> 00:48:22,850
lurking in the
higher order terms?

665
00:48:22,850 --> 00:48:25,080
PROFESSOR: Actually,
the answer is no.

666
00:48:25,080 --> 00:48:29,750
If I look at this integral
that I have over here,

667
00:48:29,750 --> 00:48:31,610
it depends on t.

668
00:48:31,610 --> 00:48:34,760
If I take sufficiently
high derivatives of it,

669
00:48:34,760 --> 00:48:37,520
I will encounter a singularity.

670
00:48:37,520 --> 00:48:41,810
So indeed, what I
have focused here

671
00:48:41,810 --> 00:48:43,690
is at the level of
the heat capacity.

672
00:48:43,690 --> 00:48:46,410
But if I were to look at the
fifth derivative of the phi

673
00:48:46,410 --> 00:48:48,585
energy, I will
see singularities.

674
00:48:48,585 --> 00:48:50,793
AUDIENCE: No, I'm talking
about the second derivative

675
00:48:50,793 --> 00:48:52,342
for higher order terms.

676
00:48:56,590 --> 00:48:59,160
PROFESSOR: These higher
order terms, the phis?

677
00:48:59,160 --> 00:49:00,090
OK, all right.

678
00:49:00,090 --> 00:49:02,100
So that was my next one.

679
00:49:02,100 --> 00:49:09,040
So you may be tempted to say,
OK, I found the divergence.

680
00:49:09,040 --> 00:49:14,160
Let's say that the heat capacity
diverges with exponent of 1/2.

681
00:49:14,160 --> 00:49:15,710
And no.

682
00:49:15,710 --> 00:49:19,470
The only thing that it says
is that your starting point

683
00:49:19,470 --> 00:49:21,710
was wrong.

684
00:49:21,710 --> 00:49:23,930
Any conclusion that
you want to make

685
00:49:23,930 --> 00:49:29,120
based on what we are
doing here is wrong.

686
00:49:29,120 --> 00:49:33,050
There is no point in my
going beyond and calculating

687
00:49:33,050 --> 00:49:35,040
the higher order term,
because I already

688
00:49:35,040 --> 00:49:38,430
see that the lowest order
correction is invalidating

689
00:49:38,430 --> 00:49:39,190
my result.

690
00:49:39,190 --> 00:49:42,200
AUDIENCE: So you [INAUDIBLE]
conclude that mean field theory

691
00:49:42,200 --> 00:49:44,340
is good for bigger than 4.

692
00:49:53,870 --> 00:49:55,960
PROFESSOR: From what
I have told you,

693
00:49:55,960 --> 00:50:00,000
I've shown you that the
discontinuity in the heat

694
00:50:00,000 --> 00:50:04,030
capacity is maintained.

695
00:50:04,030 --> 00:50:07,870
It is true that if I look at
sufficiently high derivatives,

696
00:50:07,870 --> 00:50:11,760
I may encounter some
difficulty in justifying

697
00:50:11,760 --> 00:50:18,880
why d greater than 4 or less
that 4 is making a difference.

698
00:50:18,880 --> 00:50:23,590
But certainly, as we will
build on what we know later

699
00:50:23,590 --> 00:50:27,420
on in the course, I will
be able to convince you

700
00:50:27,420 --> 00:50:29,220
that the mean field
theory is certainly

701
00:50:29,220 --> 00:50:32,790
valid in dimensions
greater than 4.

702
00:50:32,790 --> 00:50:39,270
But right now, I guess the only
thing that we can say for sure

703
00:50:39,270 --> 00:50:43,880
is that the subtle point method
cannot be applied when you are

704
00:50:43,880 --> 00:50:46,955
dealing with a field that is
varying all over the space.

705
00:50:50,490 --> 00:50:56,980
So we have this situation.

706
00:50:56,980 --> 00:51:00,730
On the other hand,
you say, well,

707
00:51:00,730 --> 00:51:02,522
if it is so bad,
why does it work

708
00:51:02,522 --> 00:51:03,980
for the case of
the superconductor?

709
00:51:06,940 --> 00:51:11,290
So let's see if we can
try to understand that.

710
00:51:11,290 --> 00:51:15,320
Again, sticking with the
language of the heat capacity,

711
00:51:15,320 --> 00:51:21,140
we see that if I am, let's
say, sitting in some dimensions

712
00:51:21,140 --> 00:51:26,760
below 4, to the
lowest order I will

713
00:51:26,760 --> 00:51:31,390
predict that there is a
discontinuity in the singular

714
00:51:31,390 --> 00:51:40,898
part and that the fluctuations
lead to a correction

715
00:51:40,898 --> 00:51:42,235
where it should be divergent.

716
00:51:47,024 --> 00:51:49,130
Now it is
mathematically correct.

717
00:51:49,130 --> 00:51:52,540
But let's see how
you would go and see

718
00:51:52,540 --> 00:51:53,940
that in the experiments.

719
00:51:53,940 --> 00:51:57,890
So presumably in the experiment,
in the analog of your t going

720
00:51:57,890 --> 00:52:03,560
to 0 is that you have a
t that passes through tc.

721
00:52:03,560 --> 00:52:06,370
And what you are doing
in the experiment

722
00:52:06,370 --> 00:52:09,140
is that you are
making measurements,

723
00:52:09,140 --> 00:52:11,789
let's say, at this point, at
this point, at this point,

724
00:52:11,789 --> 00:52:13,247
and then you are
going all the way.

725
00:52:15,770 --> 00:52:18,380
Now we can see that
there could potentially

726
00:52:18,380 --> 00:52:23,600
be a difference, depending on
the amplitude of this term.

727
00:52:23,600 --> 00:52:27,490
If it is like that, and I can
resolve things at this scale

728
00:52:27,490 --> 00:52:29,940
that I have indicated
here, there's no problem.

729
00:52:29,940 --> 00:52:32,650
I should see the divergence.

730
00:52:32,650 --> 00:52:37,020
But suppose the amplitude
is much, much smaller

731
00:52:37,020 --> 00:52:41,110
and it is something that
is looking like this,

732
00:52:41,110 --> 00:52:43,320
and you are taking
measurements that

733
00:52:43,320 --> 00:52:47,240
correspond to, essentially,
intervals such as this,

734
00:52:47,240 --> 00:52:50,420
then you really
integrate across this.

735
00:52:50,420 --> 00:52:53,740
You don't see the peak.

736
00:52:53,740 --> 00:52:55,820
You don't sufficient resolution.

737
00:52:55,820 --> 00:53:00,200
It's kind of searching for a
delta function more or less.

738
00:53:00,200 --> 00:53:03,860
And so whether or not
you are in one situation

739
00:53:03,860 --> 00:53:07,360
or another situation
could tell you

740
00:53:07,360 --> 00:53:11,820
about the result of
experimental observation.

741
00:53:11,820 --> 00:53:16,250
So how do I find out
something about that?

742
00:53:16,250 --> 00:53:21,410
Well, I want the
amplitude of this

743
00:53:21,410 --> 00:53:24,950
to be at least as large
as the discontinuity

744
00:53:24,950 --> 00:53:27,330
for me to be able to state it.

745
00:53:27,330 --> 00:53:31,540
That is, I want
to have a c that I

746
00:53:31,540 --> 00:53:35,910
have from the subtle point,
which is a discontinuity that

747
00:53:35,910 --> 00:53:38,900
is of the order of one
over 8u, so there's

748
00:53:38,900 --> 00:53:41,900
at a discontinuity
heat capacity.

749
00:53:41,900 --> 00:53:48,015
This discontinuity should be
of the order of this quantity 1

750
00:53:48,015 --> 00:53:54,830
over k squared c to
the power of 4 minus t.

751
00:53:54,830 --> 00:54:00,240
But now it becomes
kind of non-universal

752
00:54:00,240 --> 00:54:05,870
because I really want to compare
things, compare amplitudes.

753
00:54:05,870 --> 00:54:12,330
I know that my c is predicted
from the subtle point

754
00:54:12,330 --> 00:54:18,570
to go like t to the minus
1/2, where t is kind

755
00:54:18,570 --> 00:54:20,856
a rescaled version
of temperature.

756
00:54:20,856 --> 00:54:25,370
So t is, let's say,
tc minus t over tc.

757
00:54:25,370 --> 00:54:28,530
It is something that
is dimensionless.

758
00:54:28,530 --> 00:54:30,600
And so all of the
dimensions should

759
00:54:30,600 --> 00:54:33,940
be carried by some kind
of a prefactor here,

760
00:54:33,940 --> 00:54:36,610
that is some kind
of a landscape.

761
00:54:36,610 --> 00:54:40,260
So the correlation,
then, is a length scale.

762
00:54:40,260 --> 00:54:43,060
There is some prefactor
that is also a length scale,

763
00:54:43,060 --> 00:54:45,460
and then this
reduced temperature

764
00:54:45,460 --> 00:54:49,960
that controls the
functional divergence.

765
00:54:49,960 --> 00:54:55,730
Actually, I can read off what
this c0 should depend on.

766
00:54:55,730 --> 00:55:03,240
You can see that c0 should
scale like k square root of k.

767
00:55:09,050 --> 00:55:12,750
So then you can see
that this object

768
00:55:12,750 --> 00:55:16,070
k scales like c0 squared.

769
00:55:16,070 --> 00:55:20,570
So this scales like 1
over c0 to fourth power.

770
00:55:20,570 --> 00:55:24,910
And this scales like c0
to the power of 4 minus t.

771
00:55:24,910 --> 00:55:28,090
And then I have this
reduced temperature

772
00:55:28,090 --> 00:55:31,766
to the power of
d minus 4 over 2.

773
00:55:37,200 --> 00:55:43,080
So you can see that for these
things to be compatible,

774
00:55:43,080 --> 00:55:48,610
I should reduce my
t to a value such

775
00:55:48,610 --> 00:55:53,400
that this divergence
compensates for the combination

776
00:55:53,400 --> 00:56:01,220
c0 to the d delta csp,
should be of the order

777
00:56:01,220 --> 00:56:05,210
of some minimal value of t.

778
00:56:05,210 --> 00:56:08,280
Let's call it tc.

779
00:56:08,280 --> 00:56:15,930
Actually, let's call it tg to
the power of d minus 4 over 2.

780
00:56:20,440 --> 00:56:34,960
Or tg is of the
order of delta csp c0

781
00:56:34,960 --> 00:56:38,250
to the d, the whole
thing to the power of 2

782
00:56:38,250 --> 00:56:41,369
divided by d minus 4.

783
00:56:47,357 --> 00:56:50,040
Let me wrote that
slightly better.

784
00:56:50,040 --> 00:56:57,660
So tg goes off the order
of delta cp, delta csp

785
00:56:57,660 --> 00:57:03,510
to the power of
minus 2 4 minus t,

786
00:57:03,510 --> 00:57:07,290
since we are going to be
looking at dimensions such as 3,

787
00:57:07,290 --> 00:57:13,240
and then c0 to the
power of minus 2

788
00:57:13,240 --> 00:57:17,752
divided by 2d
divided by 4 minus d.

789
00:57:23,470 --> 00:57:28,700
So we can see that the
resolution that you need,

790
00:57:28,700 --> 00:57:32,720
how close you have to go
to the critical point,

791
00:57:32,720 --> 00:57:35,900
very much depends
on this quantity c0.

792
00:57:35,900 --> 00:57:38,870
It does depend
also on delta csp.

793
00:57:38,870 --> 00:57:43,230
But we can argue that that is
a less important contribution.

794
00:57:43,230 --> 00:57:49,770
Let's focus, for the time being,
on the dependence on this c0.

795
00:57:49,770 --> 00:57:54,060
So c0 presumably
has something to do

796
00:57:54,060 --> 00:57:57,260
with the physics of the system
that you are looking at.

797
00:57:57,260 --> 00:58:01,390
So then we are leaving the realm
of things that were universal.

798
00:58:01,390 --> 00:58:05,900
And we have to think about the
system under consideration.

799
00:58:05,900 --> 00:58:09,390
And we have to identify
a length scale associated

800
00:58:09,390 --> 00:58:12,770
with the system that
is under consideration.

801
00:58:12,770 --> 00:58:20,000
Now if I think about
something like liquid gas,

802
00:58:20,000 --> 00:58:25,980
well, one kind of length scale
that immediately comes to mind

803
00:58:25,980 --> 00:58:31,880
is the length scale over which
the particles are interacting.

804
00:58:31,880 --> 00:58:35,290
Also I can look at the
kind of phase diagrams

805
00:58:35,290 --> 00:58:40,990
that we were looking get, and
there was some critical volume

806
00:58:40,990 --> 00:58:45,770
where this transition
from one type of isotherm

807
00:58:45,770 --> 00:58:48,500
to another type of
isotherm occurs,

808
00:58:48,500 --> 00:58:54,700
I can ask that critical volume
how many angstroms it is.

809
00:58:54,700 --> 00:58:57,080
But again, everything
here, we have

810
00:58:57,080 --> 00:59:00,410
to try to be as
dimensionless as possible.

811
00:59:00,410 --> 00:59:04,350
So let's say this critical
volume corresponds

812
00:59:04,350 --> 00:59:06,830
to how many particles.

813
00:59:06,830 --> 00:59:08,840
And let's take the
cube root of that

814
00:59:08,840 --> 00:59:11,980
and convert it to a
length scale over which

815
00:59:11,980 --> 00:59:15,820
these number of particles are
confined in three dimensions.

816
00:59:15,820 --> 00:59:18,950
And what we find is,
for liquid gas systems,

817
00:59:18,950 --> 00:59:24,980
that number c0 that you get
in units of atomic spacing

818
00:59:24,980 --> 00:59:30,490
is of the order of 1
to 10 atomic spacings.

819
00:59:37,224 --> 00:59:38,186
Yes.

820
00:59:38,186 --> 00:59:43,010
AUDIENCE: Scale on which atoms
interact with each other?

821
00:59:43,010 --> 00:59:44,800
PROFESSOR: Well, it could be.

822
00:59:44,800 --> 00:59:48,660
But for the case of, say,
particles in this room,

823
00:59:48,660 --> 00:59:54,290
the range of interaction is not
that different than the size

824
00:59:54,290 --> 00:59:56,180
of the particles
coming together.

825
00:59:56,180 --> 00:59:58,510
It's maybe a few times that.

826
00:59:58,510 --> 01:00:02,060
So that's basically a few times
of [INAUDIBLE] saying here.

827
01:00:02,060 --> 01:00:04,170
And I'm not going
to argue whether it

828
01:00:04,170 --> 01:00:06,493
is twice that or 10 times that.

829
01:00:06,493 --> 01:00:07,742
It really makes no difference.

830
01:00:10,340 --> 01:00:11,870
The thing is that
when I'm looking

831
01:00:11,870 --> 01:00:19,040
about the problem of
superconductivity,

832
01:00:19,040 --> 01:00:23,130
this is the only place
where we introduce

833
01:00:23,130 --> 01:00:25,050
a little bit of physics.

834
01:00:25,050 --> 01:00:28,410
When one is looking at
something like aluminum

835
01:00:28,410 --> 01:00:32,400
that goes into being
a superconductor,

836
01:00:32,400 --> 01:00:38,320
it is an ordering of
bosons in the same sense

837
01:00:38,320 --> 01:00:41,050
that we have for liquid helium.

838
01:00:41,050 --> 01:00:43,640
But the difference is
that what is ordering

839
01:00:43,640 --> 01:00:45,960
in superconductivity
is not bosons,

840
01:00:45,960 --> 01:00:48,780
but it is fermions or electrons.

841
01:00:48,780 --> 01:00:51,480
And electrons have
Coulomb repulsion.

842
01:00:51,480 --> 01:00:53,570
So what has to
happen is that there

843
01:00:53,570 --> 01:00:56,850
is some mechanism,
phonons or whatever, that

844
01:00:56,850 --> 01:01:01,960
gives an effective attraction
between electrons and pairs

845
01:01:01,960 --> 01:01:04,960
them together into
a Cooper pair.

846
01:01:04,960 --> 01:01:08,060
The characteristic
size of a Cooper pair,

847
01:01:08,060 --> 01:01:12,820
because of the
repulsion that you

848
01:01:12,820 --> 01:01:17,350
have between electrons, rather
than being 1 to 10, say,

849
01:01:17,350 --> 01:01:28,350
angstroms, is c0 is suddenly of
the order of 1,000 angstroms.

850
01:01:28,350 --> 01:01:32,520
Now note that if you
are in three dimensions,

851
01:01:32,520 --> 01:01:35,110
this is something that is
raised to the sixth power.

852
01:01:38,400 --> 01:01:44,140
So if I think of this after
dividing by an atomic size

853
01:01:44,140 --> 01:01:46,860
or whatever, to a number that
is of the order of, let's say,

854
01:01:46,860 --> 01:01:51,510
100 or even 1,000 and I
raise it to the sixth power,

855
01:01:51,510 --> 01:01:53,670
you can see that the
kind of resolution

856
01:01:53,670 --> 01:01:58,100
that you need when you raise
something large to a huge power

857
01:01:58,100 --> 01:02:03,200
corresponds to t that is of the
order of 10 to the minus 12,

858
01:02:03,200 --> 01:02:05,490
10 to the minus 15, et cetera.

859
01:02:05,490 --> 01:02:07,090
And that's just
not the resolution

860
01:02:07,090 --> 01:02:08,640
that you have in experiment.

861
01:02:08,640 --> 01:02:12,880
So basically experiment
will go over this

862
01:02:12,880 --> 01:02:14,760
without really seeing it.

863
01:02:14,760 --> 01:02:19,800
Essentially the units
are so big that you

864
01:02:19,800 --> 01:02:25,210
don't have that many of them
to fluctuate across the system.

865
01:02:25,210 --> 01:02:28,760
The effect of fluctuations
is much diminished

866
01:02:28,760 --> 01:02:32,850
compared to superfluid helium
or compared to liquid gas,

867
01:02:32,850 --> 01:02:35,110
where over the
size of the system,

868
01:02:35,110 --> 01:02:39,860
you have many, many fluctuations
that can take place.

869
01:02:39,860 --> 01:02:43,170
This condition,
whether or not you're

870
01:02:43,170 --> 01:02:47,320
going to be able to see
the effects of fluctuations

871
01:02:47,320 --> 01:02:49,910
and something that is
[INAUDIBLE] field like,

872
01:02:49,910 --> 01:02:52,441
I'll call it t sub g, because
it's called a Ginzburg

873
01:02:52,441 --> 01:02:52,940
criterion.

874
01:03:06,440 --> 01:03:12,150
So this basically
answers the questions

875
01:03:12,150 --> 01:03:15,520
that we had over here.

876
01:03:15,520 --> 01:03:17,770
For all of our
phase transitions,

877
01:03:17,770 --> 01:03:21,110
we constructed the
Landau-Ginzburg theory,

878
01:03:21,110 --> 01:03:23,380
and we evaluated
its consequences

879
01:03:23,380 --> 01:03:25,250
for phase transition,
such as divergence

880
01:03:25,250 --> 01:03:29,120
of heat capacity using
the subtle point method.

881
01:03:29,120 --> 01:03:31,360
We saw that the results
worked extremely well

882
01:03:31,360 --> 01:03:36,000
for superconductors, but
not for anything else.

883
01:03:36,000 --> 01:03:39,890
And the answer to that is
that for superconductors,

884
01:03:39,890 --> 01:03:42,530
fluctuations are
not so important.

885
01:03:42,530 --> 01:03:45,090
And the most probable
state gives you

886
01:03:45,090 --> 01:03:47,410
a good idea of
what is happening.

887
01:03:47,410 --> 01:03:49,990
Whereas for super
helium, for liquid gas,

888
01:03:49,990 --> 01:03:53,320
et cetera, fluctuations
are very important,

889
01:03:53,320 --> 01:03:56,550
and the starting point that is
the subtle point, most probable

890
01:03:56,550 --> 01:04:00,695
state, is simply
not good enough.

891
01:04:00,695 --> 01:04:01,665
Yes.

892
01:04:01,665 --> 01:04:05,545
AUDIENCE: So when
you were giving us

893
01:04:05,545 --> 01:04:09,920
the system of different phase
transitions [INAUDIBLE],

894
01:04:09,920 --> 01:04:12,096
you only talked about
the critical exponents,

895
01:04:12,096 --> 01:04:18,370
because, for instance, there is
a discontinuity of [INAUDIBLE]

896
01:04:18,370 --> 01:04:21,040
heat capacity for all
phase transitions.

897
01:04:21,040 --> 01:04:24,372
But it's often masked with
fixed singularity, right?

898
01:04:29,040 --> 01:04:31,340
PROFESSOR: Once you
have a divergence,

899
01:04:31,340 --> 01:04:36,064
I don't know how you would be
talking about a singularity.

900
01:04:36,064 --> 01:04:40,297
AUDIENCE: If you roughly measure
the heat capacity further away

901
01:04:40,297 --> 01:04:42,920
from singularity,
wouldn't it kind of

902
01:04:42,920 --> 01:04:47,300
converges left and right
of two different values?

903
01:04:47,300 --> 01:04:48,010
PROFESSOR: OK.

904
01:04:48,010 --> 01:04:54,070
So if I draw a random
function that has divergence,

905
01:04:54,070 --> 01:04:57,390
the chances are very, very
good that, if I go a little bit

906
01:04:57,390 --> 01:04:59,200
further, the two
of them will not

907
01:04:59,200 --> 01:05:00,990
be exactly at the same height.

908
01:05:00,990 --> 01:05:02,970
There will be an asymmetry.

909
01:05:02,970 --> 01:05:06,690
So are you talking about
the asymmetry in amplitudes?

910
01:05:06,690 --> 01:05:10,230
Because I know the
amplitudes are not symmetric.

911
01:05:10,230 --> 01:05:14,250
If I go very, very far away,
then all kinds of other things

912
01:05:14,250 --> 01:05:15,110
come in to play.

913
01:05:15,110 --> 01:05:18,350
There's the phonon, heat
capacity, et cetera.

914
01:05:18,350 --> 01:05:21,640
So the statement
that you make, I

915
01:05:21,640 --> 01:05:24,390
have never heard
before, in fact.

916
01:05:24,390 --> 01:05:26,290
But I'm trying to
see whether or not

917
01:05:26,290 --> 01:05:28,150
it's even mathematically
conceivable.

918
01:05:31,366 --> 01:05:36,390
AUDIENCE: Another
question with this series

919
01:05:36,390 --> 01:05:38,580
you just wrote out with
[INAUDIBLE] singularity,

920
01:05:38,580 --> 01:05:44,111
doesn't it give you that
exponent for the singularity?

921
01:05:44,111 --> 01:05:44,694
PROFESSOR: No.

922
01:05:44,694 --> 01:05:47,630
AUDIENCE: It's a
[INAUDIBLE] number.

923
01:05:47,630 --> 01:05:50,850
PROFESSOR: It is 1/2, yes.

924
01:05:50,850 --> 01:05:52,870
So there is a theory.

925
01:05:52,870 --> 01:05:54,920
There is a mathematical
theory that

926
01:05:54,920 --> 01:05:58,270
has this 1/2
exponent divergence.

927
01:05:58,270 --> 01:05:59,880
What is that theory?

928
01:05:59,880 --> 01:06:06,550
It's a theory that is cut
off at the Gaussian level.

929
01:06:06,550 --> 01:06:11,770
So if we had some system
for which we were sure

930
01:06:11,770 --> 01:06:16,110
that when we write our
statistical field theory,

931
01:06:16,110 --> 01:06:19,980
I can terminate at the
level of Gaussian terms,

932
01:06:19,980 --> 01:06:22,170
m squared, gradient of
m squared, et cetera.

933
01:06:22,170 --> 01:06:24,300
If such a theory
existed, it would

934
01:06:24,300 --> 01:06:27,140
have exactly this divergence.

935
01:06:27,140 --> 01:06:31,305
But I don't see any reason
for eliminating all those--

936
01:06:31,305 --> 01:06:33,694
AUDIENCE: So we still
have not found the reason

937
01:06:33,694 --> 01:06:36,210
why the actual experimental
exponents are--

938
01:06:36,210 --> 01:06:38,072
PROFESSOR: No, we
have not found.

939
01:06:38,072 --> 01:06:39,940
Yes.

940
01:06:39,940 --> 01:06:43,265
AUDIENCE: So how do we
interpret the larger

941
01:06:43,265 --> 01:06:46,800
signal of superconductity?

942
01:06:46,800 --> 01:06:50,260
Does that mean the correlation
actually is longer?

943
01:06:50,260 --> 01:06:51,310
PROFESSOR: Yes, yes.

944
01:06:51,310 --> 01:06:54,340
AUDIENCE: But then
why are we saying

945
01:06:54,340 --> 01:06:57,690
that the fluctuation
there is not so important?

946
01:06:57,690 --> 01:07:00,566
We have longer correlation,
then usually that

947
01:07:00,566 --> 01:07:02,440
means we have bigger
fluctuation [INAUDIBLE].

948
01:07:26,950 --> 01:07:30,710
PROFESSOR: OK, so let's
see if we can unpack that.

949
01:07:30,710 --> 01:07:44,570
So our correlation length is
some c0 t to the minus 1/2.

950
01:07:44,570 --> 01:07:49,520
And indeed, what
that says is that

951
01:07:49,520 --> 01:07:54,130
at some particular same
value of how far I am away

952
01:07:54,130 --> 01:08:00,860
from the critical point, the
correlations are longer ranged.

953
01:08:00,860 --> 01:08:05,906
If I go and look at the
amplitudes of the fluctuations

954
01:08:05,906 --> 01:08:21,649
that I have, then I am
closer as a function of q

955
01:08:21,649 --> 01:08:25,460
to a situation such as this.

956
01:08:25,460 --> 01:08:33,330
So c0 is large c0.

957
01:08:33,330 --> 01:08:34,979
Inverse would be smaller.

958
01:08:34,979 --> 01:08:38,270
So that's correct.

959
01:08:38,270 --> 01:08:43,620
And then in real space,
what it would mean

960
01:08:43,620 --> 01:08:49,439
is that if I look at my
system, what I would have

961
01:08:49,439 --> 01:08:55,930
is that there are parts that
are of the order of c0 t

962
01:08:55,930 --> 01:09:00,689
to the minus 1/2 that
are doing the same thing.

963
01:09:14,496 --> 01:09:16,260
Let me understand your question.

964
01:09:16,260 --> 01:09:19,920
So it is true that
the superconductor

965
01:09:19,920 --> 01:09:26,319
you have more correlations,
and what that means is

966
01:09:26,319 --> 01:09:34,350
that the number of independent
modes that you have that

967
01:09:34,350 --> 01:09:39,770
can contribute and
fluctuate is less.

968
01:09:39,770 --> 01:09:46,290
And what we will
see ultimately is

969
01:09:46,290 --> 01:09:52,470
that the reason for all of
these exponents being different

970
01:09:52,470 --> 01:09:56,810
from what we have
in superconductivity

971
01:09:56,810 --> 01:10:02,280
is that there is essentially
a much more broader range

972
01:10:02,280 --> 01:10:04,700
of the influence
that is contributing

973
01:10:04,700 --> 01:10:07,740
to the whole thing.

974
01:10:07,740 --> 01:10:12,670
So I'm not sure if I'm
answering your question.

975
01:10:12,670 --> 01:10:15,790
Let's go back and think
about your question.

976
01:10:15,790 --> 01:10:20,620
So basically for superconductor,
certainly everything

977
01:10:20,620 --> 01:10:23,740
that we said, including
being able to express it

978
01:10:23,740 --> 01:10:26,440
in terms of this
statistical field theory,

979
01:10:26,440 --> 01:10:29,830
having large correlation lengths
close to the critical point,

980
01:10:29,830 --> 01:10:32,480
all of that is correct.

981
01:10:32,480 --> 01:10:34,800
The only thing
that is not correct

982
01:10:34,800 --> 01:10:43,060
is that the diversity of
fluctuations here is less.

983
01:10:43,060 --> 01:10:47,130
And this lack of diversity
of fluctuations compared

984
01:10:47,130 --> 01:10:51,620
to something like
liquid gas gives you

985
01:10:51,620 --> 01:10:53,745
more subtle point,
like exponents.

986
01:10:56,760 --> 01:11:05,720
AUDIENCE: So you mean the limit
for my integrand with respect

987
01:11:05,720 --> 01:11:07,291
to q is smaller?

988
01:11:07,291 --> 01:11:07,790
[INAUDIBLE]

989
01:11:10,940 --> 01:11:14,500
So the q space I'm
integrating is smaller.

990
01:11:14,500 --> 01:11:16,210
PROFESSOR: Yes.

991
01:11:16,210 --> 01:11:21,875
AUDIENCE: But if I calculate
fluctuation function,

992
01:11:21,875 --> 01:11:22,716
something?

993
01:11:22,716 --> 01:11:23,340
PROFESSOR: Yes.

994
01:11:23,340 --> 01:11:28,060
So this is what I was trying
to calculate here, yes.

995
01:11:28,060 --> 01:11:30,505
AUDIENCE: Then it
should be larger than--

996
01:11:30,505 --> 01:11:31,130
PROFESSOR: Yes.

997
01:11:31,130 --> 01:11:36,100
But it is larger for a
smaller limit of q's.

998
01:11:36,100 --> 01:11:37,860
So I guess what
you are saying is

999
01:11:37,860 --> 01:11:39,960
that if I look at
the superconductor,

1000
01:11:39,960 --> 01:11:42,030
I will see something like this.

1001
01:11:42,030 --> 01:11:47,010
If I look at the liquid gas, I
will see something like this.

1002
01:11:47,010 --> 01:11:50,560
AUDIENCE: And [INAUDIBLE]
just intuitively interpret

1003
01:11:50,560 --> 01:11:53,280
what's the behavior of the
heat capacity from this--

1004
01:11:57,665 --> 01:11:59,560
PROFESSOR: [INAUDIBLE],
because if you

1005
01:11:59,560 --> 01:12:03,160
look at something like this, and
particular its t dependence--

1006
01:12:03,160 --> 01:12:05,720
after all, everything
that we are interested

1007
01:12:05,720 --> 01:12:10,880
is how things change as
a function of t minus tc.

1008
01:12:10,880 --> 01:12:13,900
So presumably, when we do
that for superconductor,

1009
01:12:13,900 --> 01:12:16,770
if you do some kind of
a scattering experiment,

1010
01:12:16,770 --> 01:12:19,410
you will see some peak
like this emerging,

1011
01:12:19,410 --> 01:12:21,890
but the peak never
expanding as much

1012
01:12:21,890 --> 01:12:24,090
as it would do for these things.

1013
01:12:24,090 --> 01:12:27,130
You should be able,
based on that,

1014
01:12:27,130 --> 01:12:30,350
to deduce that the range
of wavelengths that

1015
01:12:30,350 --> 01:12:32,680
are fluctuating in
the superconductor

1016
01:12:32,680 --> 01:12:35,860
is less compared to
the liquid gas system.

1017
01:12:35,860 --> 01:12:42,730
And so there is not much range
in the diversity of length

1018
01:12:42,730 --> 01:12:46,000
scales that are contributing
to the fluctuations

1019
01:12:46,000 --> 01:12:46,945
in a superconductor.

1020
01:12:51,130 --> 01:12:54,460
AUDIENCE: So that explains why
we have only very narrow peak

1021
01:12:54,460 --> 01:12:55,430
in a cp?

1022
01:12:59,410 --> 01:13:00,140
PROFESSOR: Yes.

1023
01:13:00,140 --> 01:13:02,950
You have to go
very close in order

1024
01:13:02,950 --> 01:13:05,360
to expand the range
of wavelengths.

1025
01:13:08,024 --> 01:13:11,170
But then you go a little
bit one side or the other,

1026
01:13:11,170 --> 01:13:13,200
and then you are
passed that range

1027
01:13:13,200 --> 01:13:14,920
that you can see very
large wavelengths.

1028
01:13:17,500 --> 01:13:18,230
Yes.

1029
01:13:18,230 --> 01:13:20,420
AUDIENCE: So in our subtle
point, the approximation

1030
01:13:20,420 --> 01:13:23,610
we found our maximum
when we looked

1031
01:13:23,610 --> 01:13:25,855
at the second derivative,
if we had considered more

1032
01:13:25,855 --> 01:13:28,640
derivatives, would we have
captured those exponents?

1033
01:13:32,080 --> 01:13:34,970
PROFESSOR: So if we
think about things,

1034
01:13:34,970 --> 01:13:40,230
and mathematical consistence,
here we have a parameter.

1035
01:13:40,230 --> 01:13:45,360
And we can explicitly calculate
higher and higher order terms

1036
01:13:45,360 --> 01:13:49,630
and how they are smaller and
become more and more small

1037
01:13:49,630 --> 01:13:53,140
as the parameter becomes
larger and larger.

1038
01:13:53,140 --> 01:13:58,140
Now what we have here is
the following situation.

1039
01:13:58,140 --> 01:14:02,140
If I presumably stick at
some value that is away

1040
01:14:02,140 --> 01:14:09,611
from the critical point, let's
say t of 10 to the minus 1,

1041
01:14:09,611 --> 01:14:12,260
at that point I
calculate subtle point.

1042
01:14:12,260 --> 01:14:13,940
And then I calculate
fluctuations

1043
01:14:13,940 --> 01:14:17,140
around subtle point, and
I add more and more term,

1044
01:14:17,140 --> 01:14:19,070
eventually, I think,
I will converge

1045
01:14:19,070 --> 01:14:21,970
to some value for
the heat capacity.

1046
01:14:21,970 --> 01:14:26,010
The problem is that I don't
want to stick to one value of t.

1047
01:14:26,010 --> 01:14:29,990
I want to see what's the
singularity as I approach 0.

1048
01:14:32,530 --> 01:14:37,150
Now we can see that
the problem here

1049
01:14:37,150 --> 01:14:43,610
is that this correction
gives a functional form that

1050
01:14:43,610 --> 01:14:45,750
is divergent.

1051
01:14:45,750 --> 01:14:54,130
And then I would say that if I
go to from t of minus 1 to 10

1052
01:14:54,130 --> 01:15:00,870
to the minus 3, then I'm less
sure about the first correction

1053
01:15:00,870 --> 01:15:03,680
and maybe I will do many,
many more corrections,

1054
01:15:03,680 --> 01:15:05,880
and then I would
get something else.

1055
01:15:05,880 --> 01:15:09,590
And presumably, the closer
I get to t equals to 0,

1056
01:15:09,590 --> 01:15:13,130
I have to go further and
further down in the series.

1057
01:15:13,130 --> 01:15:15,513
And so that becomes
essentially useless.

1058
01:15:20,250 --> 01:15:23,140
Now we will actually do
later on another version

1059
01:15:23,140 --> 01:15:27,020
of this problem, where
we say the following.

1060
01:15:27,020 --> 01:15:31,030
What I did for you here
was calculating essentially

1061
01:15:31,030 --> 01:15:33,220
Gaussian integrals.

1062
01:15:33,220 --> 01:15:36,390
And I know how to do
Gaussian integrals.

1063
01:15:36,390 --> 01:15:40,165
And for Gaussian theory,
this result is exact.

1064
01:15:40,165 --> 01:15:43,610
I will get alpha equals to 1/2.

1065
01:15:43,610 --> 01:15:47,550
Maybe what I can do, instead
of doing subtle points

1066
01:15:47,550 --> 01:15:49,690
approximation,
approach the problem

1067
01:15:49,690 --> 01:15:52,130
completely in a
different fashion.

1068
01:15:52,130 --> 01:15:54,590
I will start with
the Gaussian part,

1069
01:15:54,590 --> 01:15:59,800
and then I do a perturbation
in all of these nonlinearities.

1070
01:15:59,800 --> 01:16:01,190
That's another approach.

1071
01:16:01,190 --> 01:16:06,190
You can say, OK, I know
the problem for u equals 0,

1072
01:16:06,190 --> 01:16:10,570
and so let's say I got this
result for u equals to 0.

1073
01:16:10,570 --> 01:16:13,170
And I want to calculate
what the correction will

1074
01:16:13,170 --> 01:16:17,850
be in proportion to u,
u squared, et cetera.

1075
01:16:17,850 --> 01:16:21,960
But what we find is that
we start expanding in u

1076
01:16:21,960 --> 01:16:24,180
and calculate the
first correction.

1077
01:16:24,180 --> 01:16:25,860
And the first
correction, you'll find,

1078
01:16:25,860 --> 01:16:36,980
is proportional to uc to the
power of t minus 4 over 2.

1079
01:16:36,980 --> 01:16:40,210
So exactly the same
problem over here

1080
01:16:40,210 --> 01:16:43,490
reappears when we try to
do preservation theory.

1081
01:16:43,490 --> 01:16:47,600
You think you are preserving
around a small quantity,

1082
01:16:47,600 --> 01:16:51,140
but as you go to
t equals to 0, you

1083
01:16:51,140 --> 01:16:55,810
find that the coefficient of the
first term in the preservation

1084
01:16:55,810 --> 01:17:00,010
theory actually blows up.

1085
01:17:00,010 --> 01:17:03,920
So we will try a
number of these methods

1086
01:17:03,920 --> 01:17:10,690
to try to extract the right
answer out of this expression.

1087
01:17:10,690 --> 01:17:13,160
This expression is,
in fact, correct.

1088
01:17:13,160 --> 01:17:15,385
The difficulty is mathematical.

1089
01:17:15,385 --> 01:17:19,830
We don't know how to deal
with this kind of integration.

1090
01:17:19,830 --> 01:17:26,520
And I was just listening to the
story of Oppenheimer and Pauli.

1091
01:17:26,520 --> 01:17:28,540
And Oppenheimer,
when he was young,

1092
01:17:28,540 --> 01:17:33,530
goes to-- actually, not
Pauli but [INAUDIBLE].

1093
01:17:33,530 --> 01:17:36,270
And he says, I am
working on some problem,

1094
01:17:36,270 --> 01:17:38,285
and I'm not having any progress.

1095
01:17:38,285 --> 01:17:41,120
He says is the problem,
the difficulty,

1096
01:17:41,120 --> 01:17:43,560
mathematical or physical?

1097
01:17:43,560 --> 01:17:46,360
And Oppenheimer is
flustered because he

1098
01:17:46,360 --> 01:17:48,590
didn't know the answer.

1099
01:17:48,590 --> 01:17:52,520
So here, we know the
problem is mathematical,

1100
01:17:52,520 --> 01:17:56,570
because the physics is
entirely captured here.

1101
01:17:56,570 --> 01:17:58,790
We haven't done anything.

1102
01:17:58,790 --> 01:18:01,790
Now the question,
however, is whether

1103
01:18:01,790 --> 01:18:05,120
the mathematical
problem will be resolved

1104
01:18:05,120 --> 01:18:08,820
by mathematical insights
or physics insights.

1105
01:18:08,820 --> 01:18:11,500
And the interesting thing is
that, in a number of cases

1106
01:18:11,500 --> 01:18:14,940
where the problem originates
from physics, eventually

1107
01:18:14,940 --> 01:18:19,040
the mathematical solution
is provided also by physics.

1108
01:18:19,040 --> 01:18:23,450
So ultimately, people develop
this idea of a normalization

1109
01:18:23,450 --> 01:18:28,190
group that I will be developing
for you in future lectures,

1110
01:18:28,190 --> 01:18:32,590
which is how to solve this
mathematical problem, which we

1111
01:18:32,590 --> 01:18:34,820
have addressed from
this perspective.

1112
01:18:34,820 --> 01:18:39,330
We will try to approach from
the perturbative perspective.

1113
01:18:39,330 --> 01:18:43,045
And it just doesn't
work until we introduce

1114
01:18:43,045 --> 01:18:46,470
a more physical way
of looking at it.