The Schwinger technique34 of a parametric representation of a propagator and the generation of an effective action was introduced and applied for a particular case in Sect. 3.7 of Vol. I. In the present section, we consider further aspects of this technique for wider applications with additional generalizations due to Bryce DeWitt.35 This powerful and elegant technique is referred to as the Schwinger- DeWitt technique. This will be used in the next section for studying the one loop contribution to the effective action of quantum general relativity based on the description of quantum fluctuation about a classic metric developed in the previous section.
For the simplest treatment of this technique, consider first the propagator associated with a free scalar field inn-dimensional Minkowski spacetime
4C.xx0/D
Z .dp/
.2 /n
eip.xx0/
.p2Cm2i/; .dp/Ddp0dp1: : : dpn1: (1.5.1) By working inn-dimensional spacetime, we write36
1
.p2Cm2i/ Di Z 1
0 dsexpŒis.p2Cm2i/;
(1.5.2) Z 1
1dp0expŒis.p0/2D r
s ei4; (1.5.3)
Z
dn1k exp i
.k1/2C C.kn1/2 Dr
s ei4 n1
: (1.5.4)
Hence from (1.5.2), the propagator in (1.5.1) may be rewritten as 4C.xx0/Di
Z 1
0 dseis.m2i/
Z .dp/
.2 /n eip.xx0/eis p2: (1.5.5) Upon completing the squares in the expressionŒsp2p.xx0/ , in thep-integral of the above equation,
sp2p.xx0/Ds
p.xx0/ 2s
2.xx0/2
4s ; (1.5.6)
34Schwinger [55].
35See the monumental work of DeWitt, e.g., in [21] and references therein.
36Such useful integrals are also considered in details in Appendix II of Vol. I [43].
and using the integrals in (1.5.3), (1.5.4), we obtain 4C.xx0/D .i/2
.4 i/n=2 Z 1
0
ds sn=2 exp
h
i.xx0/2 4s
i
eis.m2i/; (1.5.7) .xx0/2D.xx0/2.x0x00/2.
An equally important parametric relation involving the propagator, and in view of applications to the effective action in quantum gravity, is the evaluation of the simple functional integral of the form in (II.13)–(II.18) in AppendixIIat the end of this volume, involving a scalar field:
ei D Z
.D/exp hi1
2
Cm2i i D p 1
detŒCm2i Dexp 1
2Tr lnŒCm2i
; (1.5.8)
where the trace Tr is over the spacetime variablex, and here D @@, with denoting the (inverse of the) Minkowski metric. From the above expression, we have
D i
2Tr lnŒCm2i: (1.5.9)
From Appendix B of this chapter, dealing with a parametric representation of the logarithm of a matrix (operator), we may rewrite the latter as
D i 2
Z 1
0
ds s Tr
h exp
is
Cm2ii
; (1.5.10)
up to an additive constant independent of the propagator. Upon introducing the spacetime matrix elements
˝xˇˇexp
isCm2i ˇˇx0˛DK.x;x0Is/; (1.5.11) K.x;x0Is/ˇˇˇs!
0Dı.n/.xx0/; (1.5.12) we may finally rewrite (1.5.10) as
D 1 2
Z .dx/
Z 1
0
ds
isK.x;xIs/: (1.5.13) A closed expression for K.x;x0Is/ may be explicitly obtained. To this end by inserting the identity operator in p-representation in which becomes a
multiplicative operator, we obtain K.x;x0Is/DZ ˝
xjp˛ .dp/
.2 /n
˝pˇˇexp is
Cm2iˇˇx0˛ D
Z .dp/
.2 /neip.xx0/exp
is.p2Cm2i/
: (1.5.14)
From (1.5.5)–(1.5.7), we obtain K.x;x0Is/ D i
.4 i/n=2 1 sn=2exp
h
i.xx0/2 4s
i
eis.m2i/: (1.5.15) The above equation allows us to re-express in the following manner
D 1 2
Z .dx/
Z 1 0
ds is
i
.4 is/n=2exp h
i.xx0/2 4s
i
eis.m2i/ˇˇ
ˇˇx0Dx: (1.5.16) Note the damping produced by the es factor fors ! 1. On the other hand, a singularity arises forsin the neighborhood of the origin. We will see in Sect.1.7 how such a singularity is handled in the quantum general relativity via dimensional regularization.
As a generalization, now consider the equation of a Green functionG.x;x0/ satisfying an equation of the type
pg.x/ ˝xG.x;x0/Dı.n/.x;x0/: (1.5.17)
where˝xis some operator involving covariant derivatives. The above equation may be rewritten in a matrix notation as
˝G D 1; ˝xhxjGjx0i D hxj1jx0i D hxjx0i D ıp.n/.x;x0/
g.x/ ; (1.5.18) g.x/jxi D jxig.x/.
We may express the matrixGin the following convenient form G D i
Z 1
0 dseis.˝i/; (1.5.19)
G.x;x0/ D hxjGjx0i Di Z 1
0 dshxjeis.˝i/jx0i: (1.5.20) Upon setting
K.x;x0Is/D hxjeis.˝i/jx0i; (1.5.21)
we have
i@
@sK.x;x0Is/D˝xK.x;x0Is/: (1.5.22) Because of the similarity with the heat equation, without the i factor,K.x;x0Is/is referred to as the “heat kernel” .
From (1.5.20), (1.5.22)
˝xG.x;x0/D Z 1
0 ds @
@sK.x;x0Is/DK.x;x0Is/ˇˇˇ
s!0D hxjx0i; (1.5.23) assuming thatK.x;x0Is/ˇˇˇ
s!1D0. This gives the normalization condition K.x;x0Is/ˇˇˇ
s!0D ıp.n/.x;x0/
g.x/ : (1.5.24)
In studying the one loop contribution, in a loop expansion carried out in the next section, we will be particularly interested in an expression as
i
2TrŒln˝ D i 2
Z .dx/p
g.x/
Z 1 0
ds
s hxjeis.˝i/jxi D 1
2 Z
.dx/p g.x/
Z 1 0
ds
isK.x;xI s/; (1.5.25) up to an additive constant independent of˝, where we have used the parametric representation of the logarithm of a matrix (operator) derived in the Appendix B of this chapter. Tr involves a trace over the spacetime variable as well, and it is understood that the trace over the other variables pertinent to the problem in hand have been carried out.
Guided by the expression in (1.5.15),K.x;xIs/is written as K.x;xIs/ D i
.4 is/n=2F.xI s/; (1.5.26) for someF.xIs/, leading to
TrŒln˝ D 1 2 Z
.dx/p g.x/
Z 1 0
ds is
i
.4 is/n=2F.xI s/: (1.5.27) Assuming that the integrand vanishes sufficiently rapidly fors! 1, singularities may, nevertheless, arise fromsin the neighborhood of the origin. To investigate the nature of such singularities near the origin, one carries out an expansion ofF.xIs/
of the form37
F.xI s/ D X
k0
.is/kak.x/; (1.5.28)
in the neighborhood of the origin. The coefficientsak.x/, referred to as DeWitt coefficients, are obtained from the coincident limitx0 ! xof recursion relations derived from the application in (1.5.22).38From power counting, forn D 4, it is easy to see that fors ! 0, the divergences occur fork D 0; 1; 2. Dimensional regularization, however, wipes out the singularities forkD0; 1, and the divergence of the expression of TrŒln˝ in (1.5.27) comes solely from the DeWitt coefficient a2.x/. The corresponding analysis will be carried out in Sect.1.7.
In the next section, the one loop contribution to the effective action of quantum general relativity is obtained, and the Schwinger-DeWitt technique is applied to it to rewrite it in a manageable form for further analysis. Dimensional regularization is applied to the one loop contribution in Sect.1.7. This is followed by Sect.1.8which deals, in general, with renormalization aspects of quantum gravity.