152
GAUGE FIELDS AND STRINGS
where itiq is some parameter (connected with the mass of the particle)
and L is the length of the path
Our aim now is to define properly the sum in (9.1) and to proceed to
its computation. Had we had a lattice in the x space, the definition of
the sum would be easy, because on the lattice there are a finite number
of
having a fixed length. So, one respectable way of doing things is
to start on a lattice, compute the sum (9.1), and then take the mesh
a -► 0, simultaneously choosing mo(a) so as to make the amplitude
G(x, x') finite. Since the problem of the random walk on a lattice is
exactly solvable, the way described will readily give the desired answer.
It has only one flaw (apart from the aesthetic one): there exists no
generalization of this approach to the case of surfaces. Because of this
we shall not take this standard route but try to develop continuum
theory ab initio.
In the continuum theory the action (9.2) takes the form:
S = mo
• r/d x „ \n
dr ( — 1
LWt ; J
1/2
(9.3)
(here we have parametrized the path by
= xj^x) with x^(0) = x^;
x^(l) = x^). This action is invariant under the “gauge transformations”
or “difleomorphisms” or “reparametrizations”, given by
(9.4)
with the function / (t) satisfying the conditions
/(0) = 0 /(1 )= 1 ^ > 0
(9.5)
At this point we encounter two difficulties. First, we have to manage the
integration of the action (9.3) containing a square root. Second, the
measure of integration should be defined so as to count only x ^ (t )
modulo reparametrization (9.4) (it is obvious that two functions x ^ (t )
and x^(/(t)) describe the same single path). We shall now show how to
resolve these difficulties which, to some extent, compensate each other.
We wish to compute the integral:
G(x, x') = I
^ x(t)\
(x^(t))^^^ dr
(9.6)
where we have denoted by the ratio 3)x{x)IS)f{x) the measure on the
coset space obtained from the space of all x ^ (t ) by identifying functions