190
GAUGE FIELDS AND STRINGS
solved. The simplest way to find the Jacobian is to consider the norm in
the space of all metrics:
\\S h f =
(9.178)
and to substitute
^Kb = i^(p + ^ \)K b + (^^)ab
(where L was defined in (9.113).
We obtain:
\\S h f
From here it follows
- J e ,(S(P + ^ \ ) \ 2 + 4c)
+ (Lc, U )
(9.179)
(9.180)
m = 9(p^ e dQi^’\L^L)
(L^Lco), =
We perform now the computation of det (L^L) in a way similar to the
one we used for the ordinary Laplacian. Namely, let us again take a
weak gravitational field. The formal quadratic form, from which the
operator (9.180) arises as a kernel, can be written as:
W=
(l>ab = ¥ y a ^ b +
- /l^ftVH)
(9.181)
Now, we shall compute the polarization operator
as in
(9.164). For doing this it is sufficient to take h^b in the form:
2 0
(9.182)
with small /i+ + . As we have seen in the scalar case, the general
determinant can be reconstructed from this small /i-field limit. Standard
formulae for covariant derivatives give:
(/>++ V+W+ -K V + io_ -h V_w+)/i+^
-i(d + h^+)co_
— |/i++(^ + o;_ +d_(o^)
(9.183)