Pictures Paths Particles Processes

November 7, 2013 271
Coarse-graining effects
n N n (exact) N n (asymptotic)
1 1 0.85
2 1 1.07
3 4 4.01
4 25 25.17
5 220 220.94
6 2485 2493.60
7 34300 34397.35
8 559405 560754.85
9 10525900 10547973.57
In the above we have assumed that there is only a single J 0 . This is indeed
usually the case ; for pure ϕ p theories, however, Eq.(10.65) reads
and this has solutions
φ n = (q!) 1/q exp
the corresponding values for J are
J n = 1 −
1
q! ϕq = 1 , q = p − 2 , (10.73)
(
2iπ n )
q
, n = 1, 2, . . . , q ; (10.74)
1
(q + 1)! φ n q+1 = q
q + 1 φ n , n = 1, 2, . . . , q , (10.75)
and these have all the same absolute value. The thing to do is therefore to take
the asymptotic contributions from all these q singular points into account, and
sum them. We then obtain
√
q∑ (2k − 2)!
N k ≈
(k − 1)! (4J n) −k 8(q − 1)!J n
=
n=1
(2k − 2)!
(k − 1)!
( ) k √ q + 1 8
4q q
φ n
q−1
q∑
φ −(k−1) n . (10.76)
The sum over the n values of φ will vanish completely, except when k − 1 is a
multiple of q, and then it evaluates to q/(q!) k−1 ; this is exactly the behaviour
we found using Legendre expansion.
We might have proceeded otherwise, by simply taking the single real solution
φ q = (q!) 1/q as the only singular point. The number of diagrams N k will then
be nonvanishing for every k value, while in the asymptotic expression (10.76)
the sum over n φ’s is replaced by φ q −(k−1) , that is precisely q times smaller than
the nonvanishing sums of Eq.(10.76). We see that the taking into account of
only the single, real solution causes the asymptotic values of N k to be ‘smeared
out’ ; N k is then never zero anymore, but its average value 18 is still correct.
18 For the correct definition of ‘average’.
n=1