We consider quantum fluctuations about a given classical background metricg which satisfies Einstein’s field equation.32To this end, we carry out the following replacement of the classical metricg
g ! g C h; (1.4.1)
in the Einstein-Hilbert action in (1.2.1), suppressing the depence of the action on the parameter, withh now describing quantum fluctuations. This leads to the new action denoted byW:
W DZ p
g.x/.dx/R.x/!Z p
g.x/.dx/R.x/ˇˇˇŒg!.gCh/ W: (1.4.2) From (1.2.10), we may infer that for a given background metricg, the new action Wis invariant under infinitesimal transformation ofh
•h D.gCh/ @C.gCh/ @ C @.gCh/: (1.4.3) Introducing the covariant derivativer defined in terms of the background metric g, we may rewrite the above transformation as33
•h D.gCh/rC.gCh/r C rh; (1.4.4) where in writing the last term we have used the fact thatrgD0.
We now develop the theory of quantum fluctuations, about a given background metricg, with the latter satisfying Einstein’s field equation, as follows. We carry out the analysis up to second order inhstarting from the actionW. In particular, we note that indices are lowered via the metricg, and raised via its inverseg. That is, covariance is defined in terms of the background metric. Also we note that the covariant derivativeris defined in terms of the background metricgand is independentofh. For simplicity of the notation, any coupling parameter has been absorbed inh.
As in the entries of Box1.1(Sect.1.3) we have, up to second order, in particular, pg !pg.1C 1
2hC 1 8h21
4hh/; (1.4.5)
g !g hC hh; (1.4.6)
32The background field method was introduced by DeWitt [21].
33See Problem1.13
wheregg D ı, and from Problem1.14, in particular, !C1
2
rhC rh rh 1
2h
rhC rh rh : We will see, however, only the first correction in• will contribute as given in (1.4.10) below.
The investigation of the transformation rule of the Ricci tensor R is more involved. This has been considered in Problem1.7, part (iii), to first order. Let us consider this afresh for additional clarity and for generalization to second order.
Let•1 C•2 denote the correction of the connection up to second order.
From (1.1.49), we explicitly have up to second order, using the notation:R ! RC•R,
•R D
h@.•1C•2/C.•1C•2/.•1C•2/
.•1C •2/i h
@.•1 C•2/.•1C•2/i Ch
•1•1•1•1i
; (1.4.7)
by conveniently re-arranging the order of the various terms, and by relabeling dummy indices. We recognize the expressions within the first two pairs of square brackets, as the difference between two covariant derivatives: [r.•1C•2/ r.•1 C•2/]. Accordingly the explicit change in the Ricci tensor, up to second order, may be rewritten as
•R D r.•1C•2/ r.•1 C•2/
C•1•1•1•1: (1.4.8) To first order, this reduces to the expression given in Problem1.7(iii) as expected.
The first two terms give rise to the following contributions to the new action pg gr
•1C•2
; p
g gr
•1 C•2
: (1.4.9) From part (i) of Problem1.5, both of these terms have the general structure
pg gr D @.p g/;
and as total derivatives do not contribute to the new action. We may thus restrict the change in the connection in (1.4.8) to the first order
•1 D 1 2
rhC rh rh
: (1.4.10)
The expression whichreplacesthe background termp
g R, taking into account of the quantum fluctuations, is now readily obtained from (1.4.5)–(1.4.8), (1.4.10), by omitting, in the process, the corrections terms in (1.4.9) coming from•R, and is given by
pg LD pgh
R.R1
2gR/h 1
4R h.h1
2gh/CRh.h1 2gh/i C 1
2 pgh
1
2rhrhC rhrh rhrhC1
2rhrh i:
(1.4.11) As expected, the expression within the second pair of square brackets coincides, up to total derivatives, with the one within the round brackets in (1.3.7) when the covariant derivatives are replaced by partial ones, i.e., for a Minkowski metric.
We consider the covariant gauge generalization of the one in (1.3.11) r.h1
2gh/ D ; (1.4.12)
referred to as the DeWitt gauge, which may be implemented by adding the term 1
2
pg Π2 r.h1
2gh/ ; (1.4.13)
topg L in (1.4.11), leading to pgL C 1
2
pgΠ2 r.h1
2gh/ : (1.4.14) We may use the gauge condition (1.4.12) to eliminatein (1.4.14) in favor of the fieldh. This gives for the gauge fixing term
1 2
pgΠ2 r.h1 2gh/
D 1 2
pg h
grhrh rhrhC 1
4rhrh
i: (1.4.15)
We apply the result of partial integration rules spelled out in (1.2.5), and more generally in Problem1.10, to (1.4.14) to obtain
pgL C 1 2
pg h
2 r.h1 2gh/i Dpg h
R.R1
2gR/h
iCpg 1
2 hM; h; (1.4.16) where.Dg@@/,
M; D 1
2gg 1
4gg gŒr;r C 2Rg RgC1
4R gg1
2R gg; (1.4.17) where symmetrization of M; over $ , $ , .; / $ ./ is understood. From Problem1.2(ii),
Œr;rhDRhCRh DRhR h: (1.4.18) Accordingly, for the term quadratic inhin (1.4.16), we obtain
pgLh2 Dp g 1
2 h M; h; (1.4.19)
M; D 1
2gg 1
4gg C R C Rg RgC1
4R gg1
2R gg; (1.4.20)
and upon symmetrization M; D 1
4
ggCgggg
C 1 2
RCR
C1 4
RgCRgCRgCRg 1
2
RgCRg
1 4R
ggCgggg
: (1.4.21)
Note that we may write
M; DE
hı Q
i; (1.4.22)
where E;D 1
4
ggCgggg
; h
ı Q
i F ; (1.4.23) ı D 1
2
ııCıı
; (1.4.24)
Q D
RCR C 1
2R
ııCıı g g CRg CR g1
2
ıRCıRCıRCıR ; (1.4.25) andı is the symmetric Kronecker delta. We have also definedF in (1.4.23) which will play an important role in the sequel.
The first term pg R in (1.4.11) corresponds to the classical background.
We may invoke the Einstein field equation satisfied by g, to set the second termpg
R .g=2/R
equal to zero. Thus, in addition to the ghost field contribution to be given below, only the quadratic partpg Lh2 inh will be considered in the sequel.
The Faddeev-Popov ghost field contribution may be inferred from the gauge function r
h .g=2/h
and the gauge transformation of the field h in (1.4.4), as done for the non-abelian theory case in (6.3.32) in Volume I as follows.
Restricting to the term giving only the “kinetic part” of the ghost contribution r
h1 2gh
! •
•
gC rr rr
; (1.4.26)
and using the identity
rrrr
DRin Problem1.2(i), the “kinetic part” (KP) of the ghost field follows to be,
pg Lghostˇˇˇ
KPDp g
gCR
; (1.4.27)
where the ghost vector fieldobeys the “wrong” statistics.
The action integral for consideration, including the ghost field, may be then written as
A D Z
.dx/p g h 1
2hŒM;hCŒgR i
; (1.4.28) whereM; is defined in (1.4.22).
To study the one loop contribution to the effective action of quantum general relativity based on the description of quantum fluctuation about a classic metric just developed, we consider first a useful technique, referred to as the Schwinger-DeWitt technique, to this end.