QUANTUM STRINGS AND RANDOM SURFACES
163
Now we are ready to compute (9.6). Combining (9.45), (9.22) and
(9.7) we obtain:
ou
1
G(x, x') = const. d r exp( —(mo — const./£)T
X T
exp( —(x — x 'Y / e T )
(9.46)
00
= const. J dT exp( —|I£T)
exp( —(x — x 'Y ls T )
= const.
i
d >
+ H
exp(i/>(jc - Ac')),
ii = £ ‘(mo - mo.cr) = s ‘ mo -
const
All our derivations make sense only if relevant values of T in the
integral (9.46) are much larger than the cut-off (or “lattice spacing”) £.
Therefore, the integral (9.6) has a continuum limit only if we adjust Mq
to be very close to the critical value, or, in other words we have to take
the limit £ -► 0 for the cut-off simultaneously with mo(£) ^ Wo,cr- Then
the terms like e^^® will be compensated and we obtain the continuum
theory. This result is quite easy to understand from the lattice point of
view. On a lattice we have:
G (x,x')= X N ^(x,x')e“"’«^
(9.47)
where A^(x, x') is the number of paths of the length L connecting points
X and x'. We know that for large L, N^(x, x') ^ (c)^ where c depends on
the type of the lattice. We see, that for Mq > log c, the relevant paths in
(9.47) have a length of the order of the lattice spacing and no continuum
limit is possible. But as we approach ytiq -► Wo,cr = 1^8 G then a typical
L ~ (mo — ^W o,cr) ” ^ ^ 1 and the theory becomes continuous and lattice
independent. Our description refers precisely to this limit.
We see from (9.46) that the correlation length or the physical mass
have a nontrivial critical exponent:
“ phys = yl P lin to- W o , „ ) ‘
(9.48)
The procedure of renormalization in this case consists of expressing
everything in terms of the physical mass and eliminating the irrelevant