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Nonlocal heat kernel with separated points
Physics Letters B 273 ( 1991 ) 471-475 North-Holland
PHYSICS LETTERS B
Nonlocal heat kernel with separated points Andrei Zelnikov P.N. LebedevPhysicalInstitute, Leninskyprospect 53, SU-117 924 Moscow, USSR Received 25 September 1991
The covariant perturbation theory proposed by Barvinsky and Vilkovisky is applied to the calculation of the heat kernel up to second order in field strengths. A nonlocal expression for the heat kernel is obtained.
1. Introduction
The effective action approach is one o f the most powerful tools in q u a n t u m field theory. In the general case the effective action o f a q u a n t u m field on an arbitrary external background is nonlocal. The calculation of nonlocal terms in the effective action for spinor and vector fields in flat spacetime and a self-interacting scalar field on a gravitational background was p e r f o r m e d in a n u m b e r of works by Parker and Toms (see ref. [ 1 ], and references therein ). They d e m o n s t r a t e d that the existence of nonlocal terms in the effective action is responsible for the running behavior o f the q u a n t u m field theory o f effective charges. Nonlocality arises from the dependence o f the effective action on the choice o f the q u a n t u m state o f the field and, hence, on the b o u n d a r y conditions. Usually, the effective action is represented in the p r o p e r time formalism as an integral o f the trace o f a heat kernel. The central object in this a p p r o a c h is the heat kernel in coinciding points:
W=-½
i'
dssTrK(s)=-
½ ds
0
dxtrK(slx, x),
(1)
0
where the heat kernel K(slx, z) depends on the q u a n t u m state and must satisfy the corresponding b o u n d a r y conditions. The point o f this work is to calculate the one-loop heat kernel for the q u a n t u m field q~propagating in the background 2~o - d i m e n s i o n a l geometry described by the metric g¢,Vand arbitrary external fields Fop=0,
F=P(V)=iD+P-i.~R,
D=g;'"V;,Vv,
(2)
where f'(V)~0=FA(V)~0 ",
fi=P}~,
i=~)~,
~0=~0"=~0"(x~).
(3)
The field strengths are
(V,,V#-V,,V,,)~=~,,#~.
~,,, = / ~ ~,,,,
(4)
The potential term in the operator F is an arbitrary matrix, which for convenience has been splitted into two terms in eq. ( 2 ) , an arbitrary matrix/~ and one sixth o f the scalar r i e m a n n i a n curvature. The spacetime is supposed to be asymptotically flat, and the Green function
G=-F-~=
f dsK(s)
(5)
0
Elsevier Science Publishers B.V.
471
Volume 273, number 4
PHYSICS LETTERSB
26 December 1991
is fixed by the requirement that it vanish at infinity. The corresponding heat kernel is
K(s) = e x p ( s F ) , K(s) =KA(s)=K(slx (m), z(B)) , T r K ( s ) = I dxtrK(slx(m)'x(')): f dxK(s[x(A)'X(A))"
(6)
The boundary conditions imposed on the Green function correspond to the variational rule 8G= G 3FG, which is exactly the variational rule for a euclidean Green function. The following consideration is applicable both to asymptotically flat euclidean and Minkowski spacetimes of arbitrary dimension 20) that are asymptotically flat and empty in the remote past. One can derive from this one euclidean Green functions on manifolds with boundary or satisfying other boundary conditions by contraction with appropriate formfactors [2 ]. Covariant perturbation theory [ 3-5 ] is the nonlocal generalization of the Schwinger-De Witt expansion [ 6 ] and provides the possibility of summing all derivative terms in this expansion in each order in curvatures.
2. Perturbation theory for the heat kernel
According to the covariant perturbation theory [3 ] one can expand the heat kernel in powers of the field strengths P, Ru. and ~?u.. The coefficients in this expansion are nonlocal operators, acting on these curvatures. One can write K ( s ) = ~ Kn(s).
(7)
n=0
Here K. (six, z) is the heat kernel of the nth order in curvatures. Note that Ru.~ p can be expressed in terms of Ru. and nonlocal operators like the inverse d'alembertian due to the Bianchi identity. We take as the zero-order approximation for K(s) the heat kernel in empty, flat spacetime with vacuum external fields (6, Ru., and ~u. equal to zero)
Ko(s)=exp(s~),
I£o(slx, z)=~'/4(x)~'/4(z)(47rs)-~°exp(-~-~s#~du)gto(X,Z).
(8)
A tilde marks the quantities and operators corresponding to the flat metric gu.:
g""=~¢'"+h u", V . = V . + P . ,
F=~+Ig(V),
V(V)=fi'~V.V.+21~'Vu+P-~k,
/~'=(g'~+h"~)/~,
(9)
fi'~=Th "v, /~..=]R,,~.
(10)
The heat kernel of the nth order in powers of perturbations V(V) then takes the form Sn
0
Au(So, Sl ..... Sn)= exp(so 71) I ? e x p ( & D ) . . . I ? e x p ( s . r l ) , A.(so, sl ..... s.)
=dn(So,
S 1.....
SnlXl,X2) •
(11)
Here Ao(so)= exp(so~)=Ko(so) , do(SolX, Z)=gW4(x)~l/4(z)(4rCSo)-'°exp - ~So ~ a~ ao(x,z) , where ~io(x, z) is the operator of parallel displacement. The operator A~(so, s~[ x, z) reads
472
(12)
Volume 273, number 4
PHYSICS LETTERS B
26 December 1991
A, (So, s, ) = exp(so~ ) Vexp(s, ~ ) , "~1 (So, SI Ix, Z) -~.gl/4(x)gl/4(Z)
× d2'°8~g'/Z(x,)
(4ZrSo)-°)(4ns,)
exp-SZ-8"f,-vy-_
,o
(~"-t/")(ff~-t/,)-OuV,,
(/(XItl,~-6~;s,) ~o(X,Z), (13)
_
1 h""
_ 1
+P(x)).
(14)
The covariant derivative V~ acts only on the first argument (x) of Uand does not act on t/and 40,
~,'=~'a,,
~,,=~,A(x,y), ~,,=~,A(x,z).
(15)
After gaussian integration we obtain ~ZlI ( S o , S I I x ,
I(
× exp
Z) =g:~l/4(x)gl/4(Z) [4zt (So +s, ) ] - , o
1
4(So + s j ) f/~f/~
'°s
s~° o" V,,+ -(So+Sj) - Q (So+S1)
Al(So, S, Ix, z)=~J/a(x)~J/a(z)[47r(So+Sl)] ×(
2(So '+ s l ) h . ~ . +
exp
) O(xltl"+O/OV,,;s,) I ~o(X,Z) 4(So+S1)
fll'fl" (So+Sl~
(16) (So+Sl)
~ l 4(So+S1) 2h='(rlu+2s°Wl')(rl=+2s°W')-~RI+i- (So+Sj~ (tl=+2s°~u)p= )1
XCTo(X, z) .
(17)
Note that the derivatives act only on the arguments of the quantities h i'",/%', R, and/3. Let us now calculate the operator A2 (So, sl, s~_lx,z). By definition in operator designations we have A2(so, s,, s2) = exp(so[~ ) Vexp(sl [~ ) Vexp(s2 71 ) . This means /i2(So, s,, sz Ix,
[
(
z) =gl/4(x)gl/4(z)(4ZrSo)-'°(4zts, )-~°(4ZrS2)-*'
1
1
x O(x, I~_ ~ - ~ , ~; s, ) O(x~
f d2'°a,
~'/2(x, ) f
dZ°~a2~1/2(x2)
1
~ ~
)
Itl. -~: .; s2)] do(X. z).
(18)
Here V~ ~, and V2 ~ act only on the points x~ and x2 respectively. After differentiation we set Xl = x 2 = x . Let us denote
(exp)=(4nSo)-O,(4z~s,)-,o(4ns2) .... f
d2"#, g ' / 2 ( & )
× exp (~--~o (T~'61 l , - ~ 1 ( # t ' - 6 ~ ) ( 6 , ,,-#2~,)
f dZ'°a2g'/2(x2)
1 ( # 5 ' - q ~ ' ) ( 6 2 ~ ' - r h ' ) - # f V~ " - ~ 4ss2
~72u) '
(19)
473
Volume273, number4
PHYSICS LETTERSB
26 December 1991
(exp) = (4ns) -~
Xexp[l ((so-l-Sl)S2~-12-.F2SoS2VfV2,-t-So(S,-t-s2)~-']l--(So-{-Sl)~lt~21t--SoqCtVl.u--l~l'~iz)],
(20)
where s=so+&
+Sz. The kernel de takes the form Ae(so, s,, s2 Ix, z) =~l/4(x)~l/4(z) [ (exp) 0(xl I 3/OV? - 0/0V~; sj ) U(x: I~/. + 0/0V~; s2) ]~o(X, z) . In this formula the derivatives 0/0V" act on the left and should be considered as the partial derivatives on the parameter V,~ at the point x, (i = 1, 2 ). Pushing ( exp ) through the operators 0 gives Az(so,
s,, s2 Ix, z) =~'/4(x)~l/4(z) (exp)
i + ~ hl'"dJ'd~"- s2d~,At,_ s1 rl'Af- ~5 × [ ( - ~s h, +13, - gR, 4s1 h1,.q.t/.)
dl,dtJ
1
1
i 1 X ( -- ~ssh2-1-132 - gR2~y1h 2 .. 2 2 2d2,A,_sri.A,_~szh~.q.rl. )
- ~ ~ . . ( - A f + l hl"d~)(-A, + l h;'d2)+ ~--~5~..~whf'h~']£to(X,z) ,
(21)
where
h,=h~%,., A ' = P ' d],= [ - & V . +
i
Tssh'°,7~.
(So+&) D . I = [so V , . - & V2.] , d~= [(So+S,) V~,-s, D . I = [So V,~.+ (So+S,) V2.1 ,
V.=VI.+V2.,
D~=V~ ~,
(22)
and (exp) = (4~s) -°~
×exp[l ((so+S,)Sz V"V.-2s,s2 V'D.+s,(so+s2) D"D.-(So+S,)q'V.+s,~l'D.-lq"tl.)],
(23)
XexpI~((So+S,)S2E]2+2SoS2 ~qV21,'t-So(SI"~S2)E]I--(So"~-SI)qltV2It--So?]PmVI#--IP~U..u)].
(23')
A remarkable property of the derived heat kernel at separated points is worth emphasizing: it can be obtained from the heat kernel in coinciding points by the substitutions
~'--' A " = J'e"- 2ssh "¢tl*' 13-'13- -s eft%+ --~szhU"tluq., ( e x p ) ~ ( e x p ) e x p The heat kernel K(s) finally reads
474
s(d'q
+~q q.) .
(24)
Volume 273, number 4
PHYSICS LETTERS B
26 December 1991
K(s) = Ko(s) + K, (s) + K2(s) + ....
0
K(s)=
i ; dt
0
0
l()
dr~l/4(x)~l/4(z) ( e x p ) ~ 5
-
(25)
0
Here
( e x p ) : ( 4 n s ) - ' ° e x p [ l ( z ( s - z ) E]z+2tzV'{V~,,+t(s-t):,-(s-r)~luV2,,-t~l"V,,,-~tl~tl~)], d~'=tV~'-rV~,
d~=tV~'+(s-r) V~,
(26)
1, , -tic.A1 - l h , + -s (hu'~ dud.11 ]h~,V//ur/v) ) - gR~ ) -2duA ×(l+s(P2
- ~ R' 2 )
+l_2s~,,(_~{,+
- 2d~,~i5' - q , / i ~ ' - ½h2 + -S1 ,I ~' I,t~2 . ~ ~, ~/ t t ~~,~ v _ kh'~"q~q~) )
1 a, d,, ) ( _ ~ , + -h~ S
1 p 2 +~g~,,g,t,h~ )t~ ~ a, h2,p] ~to(X,Z). -h~Pd
(27)
S
This heat kernel corresponds to the euclidean Green function which vanishes at infinity. To obtain the heat kernel for other boundary conditions one should contract this kernel K(sl x, y) with the appropriate source J(y, z). For example, for a manifold which is compact in one direction, with period fl and periodic boundary conditions, one gets
KP(slx, y)=
f dyg(slx, y)J(y,z),
J/J(y,z)=
~
6(y~-z~+fln~m),
where n" is the unit vector in the direction of the compact coordinate. Applications of the obtained formulas will be discussed in future publications.
Acknowledgement I am very grateful to A.O. Barvinsky, A.V. Leonidov, and G.A. Vilkovisky for stimulating and useful conversations.
References [ 1] L. Parker and D.J. Toms, Phys. Rev. D 32 (1985) 1409. [2] J.S. Dowker and J.P. Schofield, Nucl. Phys. B 327 (1989) 267. [3] A.O. Barvinsky and G.A. Vilkovisky,Nucl. Phys. B 282 (1987) 163. [4] A.O. Barvinsky and G.A. Vilkovisky,Nucl. Phys. B 333 (1990) 471. [5] A.O. Barvinsky and G.A. Vilkovisky,Nucl. Phys. B 333 (1990) 512. [ 6 ] B.S. DeWiu, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965).
475
PHYSICS LETTERS B
Nonlocal heat kernel with separated points Andrei Zelnikov P.N. LebedevPhysicalInstitute, Leninskyprospect 53, SU-117 924 Moscow, USSR Received 25 September 1991
The covariant perturbation theory proposed by Barvinsky and Vilkovisky is applied to the calculation of the heat kernel up to second order in field strengths. A nonlocal expression for the heat kernel is obtained.
1. Introduction
The effective action approach is one o f the most powerful tools in q u a n t u m field theory. In the general case the effective action o f a q u a n t u m field on an arbitrary external background is nonlocal. The calculation of nonlocal terms in the effective action for spinor and vector fields in flat spacetime and a self-interacting scalar field on a gravitational background was p e r f o r m e d in a n u m b e r of works by Parker and Toms (see ref. [ 1 ], and references therein ). They d e m o n s t r a t e d that the existence of nonlocal terms in the effective action is responsible for the running behavior o f the q u a n t u m field theory o f effective charges. Nonlocality arises from the dependence o f the effective action on the choice o f the q u a n t u m state o f the field and, hence, on the b o u n d a r y conditions. Usually, the effective action is represented in the p r o p e r time formalism as an integral o f the trace o f a heat kernel. The central object in this a p p r o a c h is the heat kernel in coinciding points:
W=-½
i'
dssTrK(s)=-
½ ds
0
dxtrK(slx, x),
(1)
0
where the heat kernel K(slx, z) depends on the q u a n t u m state and must satisfy the corresponding b o u n d a r y conditions. The point o f this work is to calculate the one-loop heat kernel for the q u a n t u m field q~propagating in the background 2~o - d i m e n s i o n a l geometry described by the metric g¢,Vand arbitrary external fields Fop=0,
F=P(V)=iD+P-i.~R,
D=g;'"V;,Vv,
(2)
where f'(V)~0=FA(V)~0 ",
fi=P}~,
i=~)~,
~0=~0"=~0"(x~).
(3)
The field strengths are
(V,,V#-V,,V,,)~=~,,#~.
~,,, = / ~ ~,,,,
(4)
The potential term in the operator F is an arbitrary matrix, which for convenience has been splitted into two terms in eq. ( 2 ) , an arbitrary matrix/~ and one sixth o f the scalar r i e m a n n i a n curvature. The spacetime is supposed to be asymptotically flat, and the Green function
G=-F-~=
f dsK(s)
(5)
0
Elsevier Science Publishers B.V.
471
Volume 273, number 4
PHYSICS LETTERSB
26 December 1991
is fixed by the requirement that it vanish at infinity. The corresponding heat kernel is
K(s) = e x p ( s F ) , K(s) =KA(s)=K(slx (m), z(B)) , T r K ( s ) = I dxtrK(slx(m)'x(')): f dxK(s[x(A)'X(A))"
(6)
The boundary conditions imposed on the Green function correspond to the variational rule 8G= G 3FG, which is exactly the variational rule for a euclidean Green function. The following consideration is applicable both to asymptotically flat euclidean and Minkowski spacetimes of arbitrary dimension 20) that are asymptotically flat and empty in the remote past. One can derive from this one euclidean Green functions on manifolds with boundary or satisfying other boundary conditions by contraction with appropriate formfactors [2 ]. Covariant perturbation theory [ 3-5 ] is the nonlocal generalization of the Schwinger-De Witt expansion [ 6 ] and provides the possibility of summing all derivative terms in this expansion in each order in curvatures.
2. Perturbation theory for the heat kernel
According to the covariant perturbation theory [3 ] one can expand the heat kernel in powers of the field strengths P, Ru. and ~?u.. The coefficients in this expansion are nonlocal operators, acting on these curvatures. One can write K ( s ) = ~ Kn(s).
(7)
n=0
Here K. (six, z) is the heat kernel of the nth order in curvatures. Note that Ru.~ p can be expressed in terms of Ru. and nonlocal operators like the inverse d'alembertian due to the Bianchi identity. We take as the zero-order approximation for K(s) the heat kernel in empty, flat spacetime with vacuum external fields (6, Ru., and ~u. equal to zero)
Ko(s)=exp(s~),
I£o(slx, z)=~'/4(x)~'/4(z)(47rs)-~°exp(-~-~s#~du)gto(X,Z).
(8)
A tilde marks the quantities and operators corresponding to the flat metric gu.:
g""=~¢'"+h u", V . = V . + P . ,
F=~+Ig(V),
V(V)=fi'~V.V.+21~'Vu+P-~k,
/~'=(g'~+h"~)/~,
(9)
fi'~=Th "v, /~..=]R,,~.
(10)
The heat kernel of the nth order in powers of perturbations V(V) then takes the form Sn
0
Au(So, Sl ..... Sn)= exp(so 71) I ? e x p ( & D ) . . . I ? e x p ( s . r l ) , A.(so, sl ..... s.)
=dn(So,
S 1.....
SnlXl,X2) •
(11)
Here Ao(so)= exp(so~)=Ko(so) , do(SolX, Z)=gW4(x)~l/4(z)(4rCSo)-'°exp - ~So ~ a~ ao(x,z) , where ~io(x, z) is the operator of parallel displacement. The operator A~(so, s~[ x, z) reads
472
(12)
Volume 273, number 4
PHYSICS LETTERS B
26 December 1991
A, (So, s, ) = exp(so~ ) Vexp(s, ~ ) , "~1 (So, SI Ix, Z) -~.gl/4(x)gl/4(Z)
× d2'°8~g'/Z(x,)
(4ZrSo)-°)(4ns,)
exp-SZ-8"f,-vy-_
,o
(~"-t/")(ff~-t/,)-OuV,,
(/(XItl,~-6~;s,) ~o(X,Z), (13)
_
1 h""
_ 1
+P(x)).
(14)
The covariant derivative V~ acts only on the first argument (x) of Uand does not act on t/and 40,
~,'=~'a,,
~,,=~,A(x,y), ~,,=~,A(x,z).
(15)
After gaussian integration we obtain ~ZlI ( S o , S I I x ,
I(
× exp
Z) =g:~l/4(x)gl/4(Z) [4zt (So +s, ) ] - , o
1
4(So + s j ) f/~f/~
'°s
s~° o" V,,+ -(So+Sj) - Q (So+S1)
Al(So, S, Ix, z)=~J/a(x)~J/a(z)[47r(So+Sl)] ×(
2(So '+ s l ) h . ~ . +
exp
) O(xltl"+O/OV,,;s,) I ~o(X,Z) 4(So+S1)
fll'fl" (So+Sl~
(16) (So+Sl)
~ l 4(So+S1) 2h='(rlu+2s°Wl')(rl=+2s°W')-~RI+i- (So+Sj~ (tl=+2s°~u)p= )1
XCTo(X, z) .
(17)
Note that the derivatives act only on the arguments of the quantities h i'",/%', R, and/3. Let us now calculate the operator A2 (So, sl, s~_lx,z). By definition in operator designations we have A2(so, s,, s2) = exp(so[~ ) Vexp(sl [~ ) Vexp(s2 71 ) . This means /i2(So, s,, sz Ix,
[
(
z) =gl/4(x)gl/4(z)(4ZrSo)-'°(4zts, )-~°(4ZrS2)-*'
1
1
x O(x, I~_ ~ - ~ , ~; s, ) O(x~
f d2'°a,
~'/2(x, ) f
dZ°~a2~1/2(x2)
1
~ ~
)
Itl. -~: .; s2)] do(X. z).
(18)
Here V~ ~, and V2 ~ act only on the points x~ and x2 respectively. After differentiation we set Xl = x 2 = x . Let us denote
(exp)=(4nSo)-O,(4z~s,)-,o(4ns2) .... f
d2"#, g ' / 2 ( & )
× exp (~--~o (T~'61 l , - ~ 1 ( # t ' - 6 ~ ) ( 6 , ,,-#2~,)
f dZ'°a2g'/2(x2)
1 ( # 5 ' - q ~ ' ) ( 6 2 ~ ' - r h ' ) - # f V~ " - ~ 4ss2
~72u) '
(19)
473
Volume273, number4
PHYSICS LETTERSB
26 December 1991
(exp) = (4ns) -~
Xexp[l ((so-l-Sl)S2~-12-.F2SoS2VfV2,-t-So(S,-t-s2)~-']l--(So-{-Sl)~lt~21t--SoqCtVl.u--l~l'~iz)],
(20)
where s=so+&
+Sz. The kernel de takes the form Ae(so, s,, s2 Ix, z) =~l/4(x)~l/4(z) [ (exp) 0(xl I 3/OV? - 0/0V~; sj ) U(x: I~/. + 0/0V~; s2) ]~o(X, z) . In this formula the derivatives 0/0V" act on the left and should be considered as the partial derivatives on the parameter V,~ at the point x, (i = 1, 2 ). Pushing ( exp ) through the operators 0 gives Az(so,
s,, s2 Ix, z) =~'/4(x)~l/4(z) (exp)
i + ~ hl'"dJ'd~"- s2d~,At,_ s1 rl'Af- ~5 × [ ( - ~s h, +13, - gR, 4s1 h1,.q.t/.)
dl,dtJ
1
1
i 1 X ( -- ~ssh2-1-132 - gR2~y1h 2 .. 2 2 2d2,A,_sri.A,_~szh~.q.rl. )
- ~ ~ . . ( - A f + l hl"d~)(-A, + l h;'d2)+ ~--~5~..~whf'h~']£to(X,z) ,
(21)
where
h,=h~%,., A ' = P ' d],= [ - & V . +
i
Tssh'°,7~.
(So+&) D . I = [so V , . - & V2.] , d~= [(So+S,) V~,-s, D . I = [So V,~.+ (So+S,) V2.1 ,
V.=VI.+V2.,
D~=V~ ~,
(22)
and (exp) = (4~s) -°~
×exp[l ((so+S,)Sz V"V.-2s,s2 V'D.+s,(so+s2) D"D.-(So+S,)q'V.+s,~l'D.-lq"tl.)],
(23)
XexpI~((So+S,)S2E]2+2SoS2 ~qV21,'t-So(SI"~S2)E]I--(So"~-SI)qltV2It--So?]PmVI#--IP~U..u)].
(23')
A remarkable property of the derived heat kernel at separated points is worth emphasizing: it can be obtained from the heat kernel in coinciding points by the substitutions
~'--' A " = J'e"- 2ssh "¢tl*' 13-'13- -s eft%+ --~szhU"tluq., ( e x p ) ~ ( e x p ) e x p The heat kernel K(s) finally reads
474
s(d'q
+~q q.) .
(24)
Volume 273, number 4
PHYSICS LETTERS B
26 December 1991
K(s) = Ko(s) + K, (s) + K2(s) + ....
0
K(s)=
i ; dt
0
0
l()
dr~l/4(x)~l/4(z) ( e x p ) ~ 5
-
(25)
0
Here
( e x p ) : ( 4 n s ) - ' ° e x p [ l ( z ( s - z ) E]z+2tzV'{V~,,+t(s-t):,-(s-r)~luV2,,-t~l"V,,,-~tl~tl~)], d~'=tV~'-rV~,
d~=tV~'+(s-r) V~,
(26)
1, , -tic.A1 - l h , + -s (hu'~ dud.11 ]h~,V//ur/v) ) - gR~ ) -2duA ×(l+s(P2
- ~ R' 2 )
+l_2s~,,(_~{,+
- 2d~,~i5' - q , / i ~ ' - ½h2 + -S1 ,I ~' I,t~2 . ~ ~, ~/ t t ~~,~ v _ kh'~"q~q~) )
1 a, d,, ) ( _ ~ , + -h~ S
1 p 2 +~g~,,g,t,h~ )t~ ~ a, h2,p] ~to(X,Z). -h~Pd
(27)
S
This heat kernel corresponds to the euclidean Green function which vanishes at infinity. To obtain the heat kernel for other boundary conditions one should contract this kernel K(sl x, y) with the appropriate source J(y, z). For example, for a manifold which is compact in one direction, with period fl and periodic boundary conditions, one gets
KP(slx, y)=
f dyg(slx, y)J(y,z),
J/J(y,z)=
~
6(y~-z~+fln~m),
where n" is the unit vector in the direction of the compact coordinate. Applications of the obtained formulas will be discussed in future publications.
Acknowledgement I am very grateful to A.O. Barvinsky, A.V. Leonidov, and G.A. Vilkovisky for stimulating and useful conversations.
References [ 1] L. Parker and D.J. Toms, Phys. Rev. D 32 (1985) 1409. [2] J.S. Dowker and J.P. Schofield, Nucl. Phys. B 327 (1989) 267. [3] A.O. Barvinsky and G.A. Vilkovisky,Nucl. Phys. B 282 (1987) 163. [4] A.O. Barvinsky and G.A. Vilkovisky,Nucl. Phys. B 333 (1990) 471. [5] A.O. Barvinsky and G.A. Vilkovisky,Nucl. Phys. B 333 (1990) 512. [ 6 ] B.S. DeWiu, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965).
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