ABSTRACT
For the moment, let us forget about critical phe-
nomena and the RG. We begin with the Gaussian probability
- K q distribution P « e with our old notation
o
K o ' Ï / dd* [ ( ,σ )2 + r o " ! ]
Σ ì k j 2 « · ^ 2 » ■ I9· 1» k, i
'λ Note that σ.. = σ. , since CT.(x) is rea l. Let a., and
ik ι -k l ik
ßik be, respectively, the real and the imaginary parts of
of wave vectors (k, -k) and for eve ry component i we have
an independent Gaussian probability distribution
- a 2. /G (k) - β 2 /G (k) ik o ik ο ^
e e (9 . 2 )
2 - 1 where G (k) = (r + k ) . The averages over the proba-
G IN Z B U R G -LANDAU M O D E L 281
< I σ., I2 > = G (k) , 1 ik 1 ο ο
<°-b. °-u/ > 3 δ , , , 6 .. G (k) . (9. 3)ik jk o -kk ij o
It is a good exerc ise to show, using (9.2), that
z m _ m v / , ,2 m v( σ σ > = < σ. , ) ik ι -k ο 1 ik 1 o
(2ml! G (k)m . (9 .4) m ! 2
The factor (2m )!/ (m ! 2m ) happens to be just the number
of ways of dividing up 2m objects into m pairs. More
generally we can figure out the average
A = < σ ΐ k σ ί k · · · ° i k ) (9-5) ' f l 2 2 h i / o
for any i^ , . . . , i , k ^ , . . . , k , by the following rule s :
(a) i f I is odd, A = 0, and
(b) i f i is even, A is the sum of products of pa ir -
w ise average, summing over ail possible ways of pairing
up the I a 's . Each pair gives a factor shown in the last
line of 49-3).
