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On covariant scalar green functions in a weak gravitational field
ANNALS
OF PHYSICS:
55,
301-321 (1969)
On Covariant Scalar Green in a Weak Gravitational
Functions Field
J. R. CARVER* Department of Physics, Seton Hail University, South Orange, New Jersey 07079
By applying the proper-time method to a scalar field in interaction with a weak gravitational field, the coordinate arbitrariness of the linearized gravitational theory is treated as the counterpart of electromagnetic gauge. The calculation of the scalar Green function is developed and shown to possess manifest coordinate-gauge covariance; the explicit contribution of the gauge function isolated by such a procedure is evaluated. A line-integral choice of form for the gauge function when reexpressed in terms of the fields produces the path-dependent Mandelstam line integral upon a change of base gauge. The space-like path averaging of gauge-independent line-integral expressions of the first-order fields does not correspond to a familiar gravitational gauge. Gauge-invariant gravitational potentials are constructed in the case of constant curvature. The exact expression obtained for the corresponding Green function exhibits the expected singularities of the flat space-time case.
I. INTRODUCTION
Significant efforts have been made in the development and application of covariant Green functions in curved space-time, and these have been concerned principally with the construction of such Green functions from the standpoints of coincidence limits of power series expansions (I) following Hadamard and of functional power series with respect to arbitrary variations in the metric (2). Although such Green functions have been related to the very general so-called proper-time method (I), its usefulness with regard to perturbation techniques has neither been demonstrated nor exploited. Since the proper-time method was developed and used by J. Schwinger (3) for the electromagnetic field, its applicability seems especially evident when one notes the analogies between electromagnetic theory and the linearized general theory. In particular, we are to take advantage of these analogies by referring to those coordinate transformations permitted by the linearized gravitational theory as gauge transformations in the spirit of R. Amowitt and S. Deser (4) as well as by confining any such gauge dependence that is to arise. Additionally, the divergences * Present address: 524 East 13th Street, New York, New York 10009.
301
302
CARVER
which occur in the Green functions become isolated with respect to the proper-time parameter in a way that is independent of that coordinate gauge. The notation that we are to employ is for the most part that of J. Schwinger (5). Coordinates of a general Riemann manifold are denoted by xu (p = 0, 1, 2, 3), and, in particular, we write for notational compactness e(x) = d-g(x) where g(x) is the determinate of the covariant metric tensor. The contravariant Minkowski metric tensor vu” is to have the value - 1 for its temporal component. In those situations where 3 + 1 decompositions are made, components of a three-vector are designated by xk (k = 1,2, 3). Since the prototype of the Green functions of interest in quantum field theory have been those of the neutral scalar meson, we are to begin in this paper with considerations of scalar Green functions in curved space-time. As we are to be concerned in particular with the curved space-time analog of the Feynman propagator in flat space-time, we retain the following postulates of R. Utiyama (6) and R. Utiyama and B. Dewitt (7): in the limits of x0 -+ fco the metric tensor possess a canonical form identical to the Minkowski metric tensor, while asyptotic flatness obtains for both x0 + foe at all points x and x + &co at any time x”; the metric tensor be a suitably well-behaved global function of xu such that every hypersurface x0 = constant is space-like. In Section II we provide the extension of the flat space-time formalism for the neutral scalar meson to arbitrary coordinate frames for the purpose of obtaining a covariant Green function equation suitable to our methods. In Section III we review and clarify for our purposes that very elegant, useful, general proper-time method for solving Green function equations in which one views the Green function as the matrix element of an operator in a continuous space-time coordinate Hilbert space. Here we use the interaction scheme within the proper-time method. After investigating all the pertinent mathematical formalism to first order in the deviation of the covariant metric tensor from its flat space-time value in the beginning of Section IV, we proceed to develop for the scalar field a general expression for the space-time coordinate matrix element of the proper-time evolution operator corresponding to such small deviations from flatness. Considerations of the group of infinitesimal coordinate transformations permitted by the linearized general theory then provide the concept of gauge that we are to emp1oy.l Assuming also in Section IV a decomposition of the linearized gravitational field into gauge-independent and gauge-dependent portions, we obtain a gauge-covariant expression for the Green function and calculate the explicit 1 Although the gauge invariance group permitted by the full general theory is non-Abelian, the subgroup allowed by the linearized theory is an Abelian one.
COVARIANT
SCALAR
GREEN
303
FUNCTIONS
contribution of the gauge function. We conclude Section IV by verifying the transformation properties of the Green function. Section V deals in general with path-dependent line-integral choices of the gauge function, which are subsequently re-expressed in terms of the fields and related to the explicit gauge dependence of the scalar Green function. Further exploitation is then made of the analogies between the electromagnetic and linearized gravitational fields whereby we parallel the procedure that F. Rohrlich and F. Strocchi (8) carried out for the Maxwell field. In particular, gauge-invariant line-integral expressions of the first-order gravitational fields are constructed, and gauge properties are investigated after the space-like path averaging of these integrals is performed. Finally, in Section VI we consider the case of a gravitational field of uniform curvature in order to, in part, illustrate the construction of gauge-invariant “potentials.” The Green function is calculated to first order by the proper-time method and its singularities are duly noted.
II. THE COVARLANT
SCALAR
GREEN
FUNCTION
EQUATION
Since the quantity d4x e(x),
d4x = dx” dxl dx2 dx3,
is an invariant for general coordinate transformations, the correct generalization of the flat space-time action functional describing a scalar field 4(x) in interaction with an external scalar source to arbitrary coordinate frames, subject to the requirement of coordinate invariance, is then
W41 = 1 d4xMx) CC4- M.W,&)
f”(x) h#(x> + m2#2(41)
where p(x) is now a scalar density and V, is the appropriate covariant derivative; in particular, V,A’ = &A’ + r;,A” where r is the affinity. The Principle of Stationary Action then yields the inhomogeneous equation of motion
44[-- 0 + m21+(x> = ~(4 where the d’Alembertian
595l55lw
operator in curved space-time takes the forms
(1)
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CARVER
The various global bi-scalar (I) Green functions LI(x, x’) satisfying e(x)[- 0 + m”] d(x, x’) = 6(x - x’),
(2)
where 8(x - x’) is a scalar density, serve to give integral definitions of particular solutions of Eq. (1) as 4(x) = p”‘(x)
+ 1 @x’Ll(x, x’) #0(x’).
We are interested in that Green function which in the limit of asymptotic flatness satisfies the boundary conditions corresponding to the propagation of positive frequencies into the future and negative frequencies into the past. Such a Green function is just what one means by the so-called Feynman propagator when space-time is curved (I) as &)
= 1 d4X’dF(X, 4 p(x’>,
since there are no homogeneous scalar fields satisfying those boundary conditions. III. METHOD
OF CALCULATION
Consider a Hilbert space with basis vectors 1x’) that are eigenvectors of a set of commuting Hermitian operators xu where x” 1x’) = X’” 1x’),
(x’ 1x”) = 8(x’ - x”),
and
I d4x’ 1x’)(x’
/ = 1.
The concomitant definition of the additional set of Hermitian which are to act upon pure scalars in this case, according to -ia;(x’I
leads one to the usual commutation
operators pu ,
= (x’Ip,
relations
Lx”, PJ = ih”,
[P, 3PYI = 03
and an operator form of Eq. (2) using the density form d’Alembertian operator as [e-11apueguvpye-1/2 + m”] G = 1
of the covariant
COVARIANT
SCALAR
GREEN
305
FUNCTIONS
where after B. Dewitt we have defined the operator G, normalized of performing intermediate sums (9), as
for the purpose
G(x, x’) = c+/~(x) d(x, x’) elia( Since the Feynman propagator regarded as a continuous by analytic continuation from its inverse, we have GF = [*
matrix can be obtained
+ m2 - iO+J-l = i fm ds exp(--i&s)
U(s)
where the negative imaginary part attached to m2 is to be understood in the righthand-side of the definition (3), U(s) = exp(-iX.s), and &f = e-Vp,eguvp,e-1/2.
The matrix element of Eq. (3) is then G&G x’) = i 1,” ds exp(--imax)
(x 1 U(s) 1x’).
Since the quantity U(s) obeys the equation i(a/as) U(s) = imq..), one is led by such a procedure to an associated dynamical description governed by the “hamiltonian” &’ in regard to the so-called proper-time parameters and U(s) can be regarded as a proper-time evolution operator. If one separates out of the hamiltonian a portion Z. for which the exact, known solution is given by G$“(x, x’) = i /r ds exp(-im?s)
(x I U,(s) I x’)
where U,(s) = exp(-iHos),
then a proper-time order, i.e.,
interaction
x=-%J+*,
scheme may be constructed (3) where to first
= + , one has W(s)
= &(--is)
1’ -1
dv U,[Q(l -
v) s] ~IUo[~(l
+ v) s].
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CARVER
If Z0 is chosen to be p2, then the insertions of the unit factors J d4p 1p)(p for p1 and pz into the expression of the matrix element of SV(s), since (x I P& = G-F2
ew(ip,
* 4,
(P2
I x’>
=
w
1= 1
I Pz)*,
gives
(x 1 W(s) X
1x’) = +(-is)
exp[-
s1, dv / d4p, d4p2(2r)-4
&(l - v) p12s - *i( 1 + 0) p22s + ip1 * x - ip2 * x’] ( p1 I & I pz).
With the changes of variables given by ~1 =
P +
#,
= r + $5,
x
~2 = P -
ik,
x’ = r -
86,
(5) (8)
one has further that (x I au(s) 1x’) = $(-is)
/’
dv l d4k(2rr)-4 exp(ik * r - $ik2s) -1
x
I
d4p exp{--i[p2s
-
P e @Sk
+
&I>
+
iik
I & IP
-
%k),
(7) and the matrix elements of functions of x, which depend only upon momentum difference, are to have Fourier transforms defined according to the convention f(k)
= / d4x exp(--ik
. x)f(x).
IV. THE COVARIANT SCALAR GREEN TO FIRST ORDER AND THE GAUGE
FUNCTION PROBLEM
For sufficiently weak gravitational fields, coordinate systems can be introduced for which the metric tensor takes the form &w(x) = %Y + 24”(4
(8)
where h,,“(x) represents the deviation of the covariant metric tensor from its flat-space value and E = d(c = l), where G, Is Newton’s gravitational constant. Utilizing the normalization of the metric tensor and the well-known variational relationship Gdx) = g(x) gw
&4”(4~
307
COVARIANT SCALAR GREEN FUNCTIONS
one readily finds that to first order in E gqx)
= 77”’ - 2&P”(X)
(9)
= 1 + 2&(x)
(10)
and -g(x) where we, in addition,
have written h(x) = yhuy(x).
Accordingly,
with the definitions of the Riemann-Christoffel
tensor as
and the curvature scalar as R(x) = gd-4 g&4
R““Yx),
we have, in view of the connection cd4
= mh(x)
+ avgudx) - aAguv(4i9
to first order in E
and e)(x)
= 2[-a,a,hqX)
where we have written for uniformity,
+ a2h(x)l
(12)
e.g.,
R(x) = l R"'(x). Raising and lowering of indices may henceforth be done with the Minkowski metric since this is clearly sufficient. IV. 1. LINEARIZATION OF THE “HAMILTONIAN"
AND THE PROPER-TIME METHOD
With the insertions of Eqs. (8), (9), and (10) into Eq. (4) for the hamiltonian operator, one obtains to first order in E the operator a = p2 -
4Mp2
+ p2h) -p&p"
+
2p,h”“p,l.
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CARVER
Corresponding to the zeroth-order part, p2, one readily obtains the flat space-time scalar Green function Gg)(x, x’) = i ILo ds exp(- itYz”s)(x 1 U,(s) 1x’), = Q’(x
- x’)
0
where (x 1 U,(s) 1x’)o = --i(4~~)-~ The accompanying
choice of the interaction sf’p’
exp[i(4s)-l (x - x’)2]. hamiltonian
(13)
operator is then
= -•E[$(hp2 + p2h) - p,,hp” + 2p,h”‘p,]
(13)
to first order in E. In order to employ the general expression (7) of the space-time coordinate matrix element of SU(S) in our case, we need merely form the indicated momentum matrix element of Eq. (14) as
I P - Bk) = W4-*F-pupJ+”
+ W&
-
quyk21 h”“W. (15)
However, recognizing in this equation the appearance of the Fourier transform of Eq. (12) for the “curvature scalar” as R”‘(k)
one then has for that matrix operator in this application (x 1 iSI
1x’) = &(--is) x
s
= 2[k,ky
- q,J2] h““(k),
element of the first-order proper-time
J-1 du j d4k(2?r)-4 exp(ik * r -
d“p(2n))-* exp{-i[p2s
-p
evolution
fik2s)
* (vsk + t)]}[-2p,pJz““(k)
+ @‘(WI.
We thereby encounter the elementary integrals
s s
d*p exp(--i[p2s - p . (vsk + ()I> = (is”)” a2 exp [; i (vk + f,”
d4ppupyexp{--ib2s
=
- P * @Sk +
(16)
011
--(vs)-~ (a/ak”)(a/ak”) $ d% exp(-i[p”s
= -ir2sm2 [i (vk + +),
s]
(vk + 4).
- p * (vsk + 511)
- i(2s)-1 quv] exp [i i (vk + 4,” s].
(17)
COVARIANT
SCALAR
GREEN
309
FUNCTIONS
With the aid of Eqs. (16) and (17), we have finally for the space-time coordinate matrix element of au(s)
(x I SW Ix’>=ll;,: dfi x I #k(2?r)-4 exp[ik . r - fik% + @(4k%“s + ok * 01 x ([Qs)-’
(vks + 0, (vks + 0, + qLtyl h”“(k)
- fisR”‘(k)}, (18)
where the appearance of the multiplicative flat space-time matrix element as given by Eq. (13) has been indicated. Since the reconstruction of the scalar Green function follows from AF(x, x’; h) = i y ds exp(-in?s)[(x I
I U,(s) I x’),
+ (x I Ws) I x’)lU - &VW + W)lI to the order of our calculations, it is more convenient for our purposes to include the multiplicative factors of h within the v-integration to first order in E. With the realization that it is possible to write $[h(x) + h(x’)]
= 4 /’
du f d4k(2?r)-4 exp[ik * r -
$ik2s]
-1
x exp[+i(&k2u2s+ vk . f)J{l + $iu[k2us+ k . t]} h(k),
this may be accomplished by writing
A&, x’; h) = i 1: ds exp(- im2s)
x exp[+i($k2u?s + vk * @]{[i(2s)-l (vks + 0, (oks + & - $i(k2v2s+ uk * 5) vuy] h”“(k) - #R(‘)(k)).
(20)
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CARVER
IV. 2. COORDINATE-GAUGE DEPENDENCE From the general transformation properties of the components of the metric tensor under an arbitrary change of coordinates, one can readily deduce for the infinitesimal coordinate transformation X” = x’1+ 6x11 that the corresponding to order 6x,
variations
of the components
of the metric tensor are,
= -guA(x) a,62 - gvA(x>a,sxA - ~x~aAguvw. Since we have defined hJx) according to the definition (8), the analogous transformation properties of the components of h”“(x) follow from those of the metric tensor as
%&) = f;,“(X) - L(x) = -(2+1
(a,8 X,
- a,sx,) - buy(x)a,6xA - hvA(x) a,6xA - sxAaAh,,(x).
Therefore, in order to examine the gauge properties of our calculations to first order in E we need consider only the simple coordinate-gauge transformation of hUY(x) as km
-+ km
where we have, in addition,
= 4d4
+ mw
+ avu4i
written A,(x) = -(c)-l
sx, .
To the end of carrying out gauge-covariant calculations of the covariant function in curved space-time, we further assume the separations u4
= fL(x)
+ &(X)~
where the Huy are to be independent of coordinate transformations order), and A,(x) - ir,(x> = A,(x) + &(x).
Green (22)
(in lowest (24)
COVARIANT
IV. 3. THE CONTRIBUTION The substitution
SCALAR
GREEN
OF THE GAUGE
311
FUNCTIONS
FUNCTION
of the separations (22) into Eq. (21) gives
Z(x, x’; s) = ZH(x, x’; s) + ; ,‘, dv j d4k (277)-4
(25)
x
exp[ik * r - $ik2s + &i(&k2v2s+ vk . [)I
x
i2(2s)-’ (vsk2 + t . k) 5 * A(k),
where by Z&x, x’; s) we mean exactly expression (21) with the P(k) replaced by the Hu”(k), since W)(k) is gauge invariant (a gauge scalar). The v-integration in the second member of Eq. (25) can easily be effected as Z(x, x’; s) = Z,(x, x’; s) + i(2s)-1 (x - x’) * [A(x) - l&x’)],
(26)
where we have, in addition, taken the appropriate Fourier transforms. Thus, using Eq. (26) in conjunction with Eq. (20), a gauge-covariant expression can be written for the scalar Green function in curved space-time as dF(x, x’) = i Irn ds exp(--im2s) (x 1 U,(s) 1x’),
x “r1 + ia(2s)-l (x - x’) - [A(x) - A(x’)] + eZH(x, x’; s)}, (27) since all gauge dependence is included in the second bracketed term, and the matrix element of U,(s) is given by Eq. (13). Alternatively, since a, = I-1
-
(X
= -a:
x’),
when acting upon Eq. (13), the factors (x - x’),, appearing in Eq. (27) can be eliminated in favor of the derivatives whereby the Green function can be re-expressed as A,(x,
XT)
= (1 +
aa +
+4(x)
A(x’)
. a/j}
&)(x
-
XJ)
+ dt)(~,
x’;
H) (28)
where dj?‘(x, x’; H) = i
ds exp( -im2s) (X / U,,(s) I x’>o Zdx, x’; s). I O” 0
(29)
If we write further that dF(X, x’; A) = {I + @t(x) * a + 4(x’) * a’‘I} dc,o)(x - x’) _N exp{+I(x)
. a + A(d) . a’]) L$?(x -
x’),
then Eq. (28) becomes to first order in E Ll,(x, x’) = Ll,(x, x’; A) + Ed(F1)(X,x’; H),
(30)
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CARVER
which displays the gauge covariance even more clearly. In terms of the space-time matrix elements such a procedure is equivalent to including all gauge dependence within the matrix element corresponding to the “unperturbed hamiltonian” A$ , for one has from Eq. (27) (x 1 U,(s) 1x’) = (x 1 U,(s) 1x’), (1 + iC(2S))l (X - X’) * [A(X) - A(.+)]} N (x 1 U,(s) 1x’)~ exp{ic(2s)-l (x - x’) . [A(x) - A(Y)]}
= expWNx) * a + 4x’) - a’11(x I W) I x’h , which therefore method.
exhibits
the coordinate-gauge
IV. 4. THE TRANSFORMATION
covariance
of the proper-time
PROPERTIES OF THE GREEN FUNCTION
Suppose that we temporarily suppress the dependence of the Green function on the Hu’, since they are gauge scalars, in order make the gauge dependence explicit, then under a gauge transformation A(x) + A(x) = A(x) + A(x), X(x) = -(e)-1 6x, we have AAX, x’; 4 -+ Ap(X, x’; A) = Ap(x, x’; A) - [6x * a + 6x’ * a’] Ap(x, x’; A). Since under an infinitesimal
general coordinate transformation
AAx, x’; A) + A@, x’; A) = AF(x, x’; A) + [Sx - a + 6x’ - a’] A,(x, x’; A), we verify the transformation A&,
properties of the covariant scalar Green function (6) as
x’; A) --+ A&,
2’; (I> = AF(x, x’; A),
whence AF(x, x’; h) -+ A&, V. CHOICES
x’; /i) = A&
OF THE GAUGE
V. 1. PATH- AND GAUGE-DEPENDENT
x’; h).
FUNCTION
LINE INTEGRALS
One may write for the gauge function A,(x) = Jr dx’“a;A,,(x’)
COVARIANT
SCALAR
GREEN
313
FUNCTIONS
where P is an arbitrary reference point at which A,(P) is to vanish and the path is, of course, arbitrary; alternatively, we write further that
i&(x) = J; dx’9l\ll,,(x’) + 4 J‘: dx’“[a&(x’) - ql,(x’)] where the definitions are rewritten as
(31)
(23) have been noted. If the second integrals of Eqs. (29) $ 1 dx’A[a;A,(x’) s
- a;A,(x’)]
31(x’ - x)
and integrated by parts, then we obtain
A,(x) = 1; dx’A&&(x’,x)
(32)
where we have defined
F,,(x’,x) = AA, + [Q.l,,(x?- wL(x’Mx - x? and have made the additional [a&(P)
assumption that - a:n,(P)](P
- xy = 0.
Through Eqs. (22), the line-integral form of the gauge function re-expressed in terms of the fields h,, and HUy as A,(x) = A,(x; h) - A,(x; H)
can now be (33)
where in accordance with Eqs. (32) we write, e.g., A,(x; h) - J:, dx’A\F,,(x’, x; h)
(34)
where F&(X’, x; h) = h&&(X’) + (x’ - xy [iQn,(x’)
- qz,,(x’)].
(35)
It is interesting to inquire at this juncture as to the conditions under which such line integrals become path independent; for the general expressions as given by Eqs. (32) we find kyF,,(x’, x) - a;F”,(x’, x) = (x’ - xy R&(x’). Thus, path-independence obtains if the Riemann-Christoffel every point along the path.
tensor vanishes at
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CARVER
V. 2. THE MANDELSTAM
LINE INTEGRAL
The realization that the gauge function can be written as the sum of terms involving the gauge-independent or gauge-dependent fields as shown in Section V. 1 affords us the opportunity to demonstrate the ambiguity that exists even in defining gauge-invariant fields, for we can return to Eqs. (22), insert Eqs. (33) into the tensor gauge functions as defined by Eqs. (23), and obtain
hi”(4 = Ww(4 - A&; ml + A&; 4
(36)
where, e.g.,
f&Jx; h) = 4[qA(x; 4 + %4(x; h)l.
(37)
The bracketed members of the right-hand-side of Eqs. (36) simply provide an alternate definition of a gauge-invariant field, say, HLy as h&)
= m4
+ L”k
4.
(38)
The corresponding effect on our result for the scalar Green function as given by Eq. (28) is simply to cause it to be rewritten as dF(X, X’) = (1 + +l(x;
h) * a + A@‘; h) * Z]} &‘(x
- x’) + E&)(X, x’; H’ )
where L$‘(x, x’; H’) is Eq. (29) with H replaced by H’. If we now examine the path-dependent line-integral form of a gauge term appearing in Eq. (39), say, A(x; h) * a, then with some additional symmetrization of Eqs. (35) we have with the aid of Eqs. (34)
+ 4 J5pdx'l[a:hAp(Xy - a;h,,(xy[(x
Eq. (40) is exactly the result of S. Mandlestam our case of a scalar field. V. 3. GAUGE-INVARIANT
PATH-DEPENDENT
- x?, au - (X - x'y aK]. (40)
(20) that he has made explicit for
LINE INTEGRALS
Following F. Rohrlich and F. Strocchi (8) we may use the four arbitrary singlevalued differential four-vector functions xl(x, 5) of the space-time variables xU and a parameter 5, which satisfy the boundary conditions xxx, 0) = -% , !jliW X:(X, 5) = spatial infinity
COVARIANT
SCALAR
GREEN
315
FUNCTIONS
in order to carry out the development of Section V.1. By straightforward manipulation expressions analogous to Eqs. (34) and (35) can be obtained as2 A,(x) = 1” d( y --m
{Ad,(X’) - (x’ - x>” [a:n,,(x’)
- ayAK(x
(41)
and the reference point P is now spatial infinity. A gauge-invariant line integral can now be formed by constructing the tensor gauge functions as given by Eqs. (37) in accordance with Eqs. (41) and using them in conjunction with the separations as given by Eqs. (33). By such a procedure we obtain &(x)
= 3 so d&x - x’y F --m
($$
Rz;Jx’)
+ $$
R$~,(x’)),
(42)
which therefore uniquely determines the gauge-independent potentials once the curvature tensor RuAKv is specified. Although Eqs. (42) are no longer gauge dependent, they are still path dependent because the choice of path x:(x, 5) along which the integration is to be performed has yet to be made. V. 4. THE SPACE-LIKE
PATH AVERAGE
Following F. J. Belinfante (12) and F. Rohrlich the straight-line paths
and F. Strocchi (a), we choose
xl = x, + Eu5 and select the Lorentz frame in which
co = 0f
x’ - x “r-.d--.-; 5
in this Lorentz frame the straight lines converge to the field point x in arbitrary directions given by the unit vectors E, and Eqs. (42) reduce to f&(x)
= - so d~~
(43)
Averaging Eqs. (43) over all the possible directions of E as +%(x)) where d252 is an infinitesimal (H;,(x))
= j d252(47+l f&“(X)
solid angle in the direction of E, we have = f d3x’(4nC)-l ~“~zR~~k(~o, x’)
*This alternate procedure also justifies the insertion of the gradient delta-function in the precedingsubsection.
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CARVER
since dsx’ = -d2Q12 d<. Gauge properties of the fields are easily investigated in terms of the recent notation due to J. Schwinger (22) whereby the four-vector functions f”(x) = vu ( ;*y;
) c -V”.qX)
with the consequence are used. With the shorthand notation f”fY(x
- x”) E j d4x’f”(x
- x’) fv(x’
-
x”),
these functions obey fY”(4
= fY”W
The result of the path averaging as given by Eqs. (44) may then be written as (H;“(x))
= 2 ( d4x’[flf”(x
- x’) - iq’Kfif’(x
- x’)] R;$(x’).
(45)
The insertion of that part of the curvature tensor symmetric in E and k, written in the form
into Eqs. (45) produces after some manipulation
= L(x)
- 4”(X)
- x’) -
&jz~f’(x
(46)
where now d,(x) = 2 j- d4x’[flfk(x
- x’)] F/$(x’),
(47)
and we have written for the sake of uniformity ~U”KW = 4%x>. If the gauge terms in Eqs. (46) can be made to vanish, then
=
aZhk,(x)
+
akhZ,(x)
-
%hZktX)
the coordinate conditions on the h,, then give the gauge conditions Therefore, by requiring that Eqs. (47) vanish we have v2&hk,(X)
-
&%%hdX)
+
&V’a,,hkk(X)
=
0.
on the (HLJ. (48)
COVAFUANT SCALAR GREEN FUNCTIONS
317
For p = j Eqs. (48) yield (6,jV2 - ajaz) ~k~~~(~)= - Bv2ajhkk(x)*
(49)
Since the left-hand-sides of Eqs. (49) are transverse in j and the right-hand-sides are longitudinal, both sides must vanish as @WV2 - w,> w&l(X) = 0, h7&) = 0.
(50) (51)
For p = 0 Eqs. (48) yield,3 after some manipulation, (a& - &V2) where 7@(x) is the second fundamental C. Misner (13) in lowest order as &‘(x) Eqs. (50x52)
7&)(x) = 0 form of R. Arnowitt,
= ~rcl$jr~$(x)
(52) S. Deser, and
- rg.
then furnish the gauge conditions on the (HL,). VI. THE EXAMPLE
OF CONSTANT
CURVATURE
Since it can be readily verified that the expression (11) of the “hrst-order Riemann-Christoffel tensor” is invariant4 under the simple coordinate-gauge transformations discussed in Subsection IV.2, one can employ the concept of constant curvature as a gauge-invariant one and construct the corresponding gauge-invariant “potentials”S (53) Direct substitution of these potentials6 into Eqs. (11) then yields the constant tensor R,$i, . The insertion of the Fourier transform of Eqs. (22) with those potentials H,, , viz, 8 We thank the referee for pointing this out to us. 4 Although the Riemann-Christoffel tensor is not a gauge scalar in general, it is in the case of the linearized theory. 5 We approach the example of constant curvature from the same standpoint that one views constant electromagnetic fields, e. g., Ref. (3); formally, in order to prevent && from becoming arbitrarily large the Riemann-Christoffel tensor should be a slowly varying function of space-time that vanishes at both temporal and (now) spatial infhrity, whether this be achieved by a modification of l or Rc,jAK. * Since the fields given by Eqs. (53) satisfy the conditions a,Hfiv(x) = ~@H(.x), we are now in the so-called Lorentz-Hilbert gauge.
318
CARVER
into Eq. (21) produces, of course, a gauge-independent gauge-dependent part already considered as
portion in addition
to the
Z&x, x’; s) = Z&f, r; s) - $isRcl)
(54)
where we have written K([, r; s) E - QR“‘““““(&)
j’,
dv j- d4kfu,,(k; v, s)(~/~k”)(~/~kK)
S(k)
and &(k;
v, s) = exp[ik * (r + &I[) + aik2(v2 - 1) s] x [i(2s)-l
(vks + [),, (vks + 0, - $i(k2v2s + vk . 6) r),,].
Somewhat lengthy calculations yield
Then, since
K(f, r; s) = $&(s)-’
R$,&‘f’rArx
- R$‘~“tJ” + isRcl’].
(55)
The combinations of Eqs. (55), (54), (29), (13), and (6) give the first-order contribution to the scalar Green function in curved space-time as &‘(x,
x’; Z-Z) = i /,” ds exp(--i&s) x &[i(s)-’
R$$‘x’~x”x’~
- tRz(x
- x’)“ (x - x’)” - @R”‘], (56)
where the singularities to be encountered in the limit as x approaches x’ are contained in the lower limit of the s-integration (see Eq. (5.20) of Ref. (3)).
COVARIANT
SCALAR
GREEN
319
FUNCT’IONS
Alternatively, if line integrals of the Hpy with a range of integration are separated from the gauge functions as’
from x’ to x
4(x) - 4dx’) = V&(x> - ~:Wl - /:, WK,(Y), then with the straight-line
choice of path given by
the first member of Eq. (56) may be dispensed with entirely and the covariant scalar Green function takes the simpler form
03 d.(x, x’) = d.(x, x’; A’) + ic I ds exp( - im2s) 0 x ~[s2R(1wway- $isR’l’] (x 1U,(s) 1x’), . VII. CONCLUDING
(57)
REMARKS
In Sections III and IV we have demonstrated in an unambiguous fashion that the proper-time approach is applicable to the treatment of a scalar field in a linearized classical gravitational field. In those cases in which the first-order gravitational fields can be written as the sum of a gauge scalar and gauge-dependent terms in accordance with Eqs. (22)-(24), the proper-time method when utilized for the calculation of scalar Green functions has been shown to exhibit manifest gauge covariance as illustrated by Eq. (27), (30), or even (39). In fact, the transition from Eq. (28) to (39) represents a shift in “base” gauge; that is, Eqs. (22) relate the fields Huy to all other gauges obtainable from them just as Eqs. (38) relate the fields HLy to all other gauges obtainable from them. In particular, the latter choice of base gauge in conjunction with the line-integral representations of Subsection V.l allowed us to produce the Mandelstam line integral and in conjunction with those of Subsection V.2 to pass from a class of gauge-dependent fields to the gauge-independent field variables given by Eqs. (42). Since we had first written the gauge functions (1, as path-dependent line integrals and then proceeded to split-up the gauge functions according to Eqs. (33), the connection between the fields Huy and HLy has been made path-dependent. In fact, if one 7 Although such separations of the gauge functions may appear to be an ad hoc one at this jucture, it is a natural choice in the case of spin fields, which we are to make clear in a future publication. 595/55/2-8
320
CARVER
chooses to use a line-integral form of a gauge function and proceeds to split it up into two portions, then the corresponding transformation between any two base gauges, say, zy(x) --f HT(x) = H,“‘(x) + A,““(x), has in that manner been reduced to one of path selection, for in principle one can obtain other base gauges by choosing particular paths for the gauge functions involved. In particular, if there exists some class of paths for which, say, the /lt vanish, then we associate this class of paths with the Hr. That such an association, indeed, exists has already been demonstrated in the case of electromagnetism by F. Rohrlich and F. Strocchi (8). The space-like path averaging of the gauge-invariant line-integral expressions of the first-order fields produced the path-independent expressions as given by Eqs. (44) or (45). Although such a procedure produced field variables in the radiation gauge in the case of electromagnetism (8), we find that the gauge conditions as expressed by Eqs. (50)--(52) do not appear to be identical with those of any of the usual gauges used in gravitation in the presence of matter. In particular, it is obvious that we have produced coordinate conditions that are a variant of the radiation gauge used by J. Schwinger (I2), since the vanishing of our Eqs. (47) imposes new and different conditions on the metric (compare with Eqs. (94) of Ref. (12)). The gauge-invariant first-order contribution to the Green function of a scalar field coupled to a weak gravitational field of constant curvature is provided by Eq. (56). The transition from Eq. (56) to (57) evidently allows one to display clearly that the singularities to be encountered in the limit as x approaches x’ are no different than in the case of flat space-time. That identical divergences are to be expected has already been pointed out in general by B. Dewitt (1). Additionally, one may deduce from Section VI that the concept of zero curvature is a gauge-invariant one to first order in E; in such a case the Green function is dependent only on the gauge function, which was made explicit in Subsection IV.4. In passing, we cite some roles that the Riemann-Christoffel tensor has played in this paper which strongly resemble those played by the electromagnetic-field tensor of electromagnetic theory. Note, in particular, the striking similarities of the gravitational potentials for the example of constant curvature in Section VI to gauge-invariant four-vector potentials for a constant electromagnetic field. As in the preceding comparison the gauge-invariant line-integral expressions of the fields which were developed in Subsection V.3 possess the necessary increase in indices (compare our Eqs. (42) with Eqs. (5) of Ref. (8)). Aspects of this paper are at present being extended to systems of higher-order spin.
COVARIANT
SCALAR GREEN FUNCTIONS
321
The author finds it a pleasure to thank Dr. A. F. Radkowski for valuable discussions and critical remarks and Prof. M. J. Tausner for his interest, encouragement, and remarks that have led to the constant curvature considerations.
REFERENCES I. B. S. DEWITT, “Dynamical Theory of Groups and Fields.” Gordon and Breach, New York (1965); B. S. DEW~I-~ AND R. W. Basrrhm, Ann. Phys. 9,220 (1960). 2. C. M. DEWXTT AND B. S. DEWI-IT, Physics 1, 3 (1964); C. M. DEWITT, Compt. Rend. Acud. Sci. (Fmnce) 256,3827 (1963). 3. J. SCHWINGER,Phys. Rev. 82,664 (1951). 4. R. ARNowrrr AND S. DESER,Phys. Rev. 113,745 (1959). 5. J. SCHWPIGER, Phys. Rev. 130, 800 (1963). 6. R. UTIYAMA, Phys. Rev. 125, 1727 (1962). 7. R. Umm AND B. S. DEWITT, J. Math. Phys. 3,608 (1962). 8. F. Rornu..tc~ AM) F. Sraoccrrt, Phys. Rev. 139, B476 (1965). 9. B. S. DEWY, Rev. Mod. Phys. 29, 377 (1957). 10. S. MANDELSTAM, Ann. Phys. (N.Y.) 19,25 (1962). II. F. J. BELINFANTE, Phys. Rev. 128,2832 (1962). 12. J. SCHWINGER, Phys. Rev. 173, 1264 (1968). 13. R. ARNOWITT, S. DESW, AND C. MISNER, in “Gravitation” (E. Witten, Ed.), Sec. 7-3.2, Wiley, New York, (1962).
OF PHYSICS:
55,
301-321 (1969)
On Covariant Scalar Green in a Weak Gravitational
Functions Field
J. R. CARVER* Department of Physics, Seton Hail University, South Orange, New Jersey 07079
By applying the proper-time method to a scalar field in interaction with a weak gravitational field, the coordinate arbitrariness of the linearized gravitational theory is treated as the counterpart of electromagnetic gauge. The calculation of the scalar Green function is developed and shown to possess manifest coordinate-gauge covariance; the explicit contribution of the gauge function isolated by such a procedure is evaluated. A line-integral choice of form for the gauge function when reexpressed in terms of the fields produces the path-dependent Mandelstam line integral upon a change of base gauge. The space-like path averaging of gauge-independent line-integral expressions of the first-order fields does not correspond to a familiar gravitational gauge. Gauge-invariant gravitational potentials are constructed in the case of constant curvature. The exact expression obtained for the corresponding Green function exhibits the expected singularities of the flat space-time case.
I. INTRODUCTION
Significant efforts have been made in the development and application of covariant Green functions in curved space-time, and these have been concerned principally with the construction of such Green functions from the standpoints of coincidence limits of power series expansions (I) following Hadamard and of functional power series with respect to arbitrary variations in the metric (2). Although such Green functions have been related to the very general so-called proper-time method (I), its usefulness with regard to perturbation techniques has neither been demonstrated nor exploited. Since the proper-time method was developed and used by J. Schwinger (3) for the electromagnetic field, its applicability seems especially evident when one notes the analogies between electromagnetic theory and the linearized general theory. In particular, we are to take advantage of these analogies by referring to those coordinate transformations permitted by the linearized gravitational theory as gauge transformations in the spirit of R. Amowitt and S. Deser (4) as well as by confining any such gauge dependence that is to arise. Additionally, the divergences * Present address: 524 East 13th Street, New York, New York 10009.
301
302
CARVER
which occur in the Green functions become isolated with respect to the proper-time parameter in a way that is independent of that coordinate gauge. The notation that we are to employ is for the most part that of J. Schwinger (5). Coordinates of a general Riemann manifold are denoted by xu (p = 0, 1, 2, 3), and, in particular, we write for notational compactness e(x) = d-g(x) where g(x) is the determinate of the covariant metric tensor. The contravariant Minkowski metric tensor vu” is to have the value - 1 for its temporal component. In those situations where 3 + 1 decompositions are made, components of a three-vector are designated by xk (k = 1,2, 3). Since the prototype of the Green functions of interest in quantum field theory have been those of the neutral scalar meson, we are to begin in this paper with considerations of scalar Green functions in curved space-time. As we are to be concerned in particular with the curved space-time analog of the Feynman propagator in flat space-time, we retain the following postulates of R. Utiyama (6) and R. Utiyama and B. Dewitt (7): in the limits of x0 -+ fco the metric tensor possess a canonical form identical to the Minkowski metric tensor, while asyptotic flatness obtains for both x0 + foe at all points x and x + &co at any time x”; the metric tensor be a suitably well-behaved global function of xu such that every hypersurface x0 = constant is space-like. In Section II we provide the extension of the flat space-time formalism for the neutral scalar meson to arbitrary coordinate frames for the purpose of obtaining a covariant Green function equation suitable to our methods. In Section III we review and clarify for our purposes that very elegant, useful, general proper-time method for solving Green function equations in which one views the Green function as the matrix element of an operator in a continuous space-time coordinate Hilbert space. Here we use the interaction scheme within the proper-time method. After investigating all the pertinent mathematical formalism to first order in the deviation of the covariant metric tensor from its flat space-time value in the beginning of Section IV, we proceed to develop for the scalar field a general expression for the space-time coordinate matrix element of the proper-time evolution operator corresponding to such small deviations from flatness. Considerations of the group of infinitesimal coordinate transformations permitted by the linearized general theory then provide the concept of gauge that we are to emp1oy.l Assuming also in Section IV a decomposition of the linearized gravitational field into gauge-independent and gauge-dependent portions, we obtain a gauge-covariant expression for the Green function and calculate the explicit 1 Although the gauge invariance group permitted by the full general theory is non-Abelian, the subgroup allowed by the linearized theory is an Abelian one.
COVARIANT
SCALAR
GREEN
303
FUNCTIONS
contribution of the gauge function. We conclude Section IV by verifying the transformation properties of the Green function. Section V deals in general with path-dependent line-integral choices of the gauge function, which are subsequently re-expressed in terms of the fields and related to the explicit gauge dependence of the scalar Green function. Further exploitation is then made of the analogies between the electromagnetic and linearized gravitational fields whereby we parallel the procedure that F. Rohrlich and F. Strocchi (8) carried out for the Maxwell field. In particular, gauge-invariant line-integral expressions of the first-order gravitational fields are constructed, and gauge properties are investigated after the space-like path averaging of these integrals is performed. Finally, in Section VI we consider the case of a gravitational field of uniform curvature in order to, in part, illustrate the construction of gauge-invariant “potentials.” The Green function is calculated to first order by the proper-time method and its singularities are duly noted.
II. THE COVARLANT
SCALAR
GREEN
FUNCTION
EQUATION
Since the quantity d4x e(x),
d4x = dx” dxl dx2 dx3,
is an invariant for general coordinate transformations, the correct generalization of the flat space-time action functional describing a scalar field 4(x) in interaction with an external scalar source to arbitrary coordinate frames, subject to the requirement of coordinate invariance, is then
W41 = 1 d4xMx) CC4- M.W,&)
f”(x) h#(x> + m2#2(41)
where p(x) is now a scalar density and V, is the appropriate covariant derivative; in particular, V,A’ = &A’ + r;,A” where r is the affinity. The Principle of Stationary Action then yields the inhomogeneous equation of motion
44[-- 0 + m21+(x> = ~(4 where the d’Alembertian
595l55lw
operator in curved space-time takes the forms
(1)
304
CARVER
The various global bi-scalar (I) Green functions LI(x, x’) satisfying e(x)[- 0 + m”] d(x, x’) = 6(x - x’),
(2)
where 8(x - x’) is a scalar density, serve to give integral definitions of particular solutions of Eq. (1) as 4(x) = p”‘(x)
+ 1 @x’Ll(x, x’) #0(x’).
We are interested in that Green function which in the limit of asymptotic flatness satisfies the boundary conditions corresponding to the propagation of positive frequencies into the future and negative frequencies into the past. Such a Green function is just what one means by the so-called Feynman propagator when space-time is curved (I) as &)
= 1 d4X’dF(X, 4 p(x’>,
since there are no homogeneous scalar fields satisfying those boundary conditions. III. METHOD
OF CALCULATION
Consider a Hilbert space with basis vectors 1x’) that are eigenvectors of a set of commuting Hermitian operators xu where x” 1x’) = X’” 1x’),
(x’ 1x”) = 8(x’ - x”),
and
I d4x’ 1x’)(x’
/ = 1.
The concomitant definition of the additional set of Hermitian which are to act upon pure scalars in this case, according to -ia;(x’I
leads one to the usual commutation
operators pu ,
= (x’Ip,
relations
Lx”, PJ = ih”,
[P, 3PYI = 03
and an operator form of Eq. (2) using the density form d’Alembertian operator as [e-11apueguvpye-1/2 + m”] G = 1
of the covariant
COVARIANT
SCALAR
GREEN
305
FUNCTIONS
where after B. Dewitt we have defined the operator G, normalized of performing intermediate sums (9), as
for the purpose
G(x, x’) = c+/~(x) d(x, x’) elia( Since the Feynman propagator regarded as a continuous by analytic continuation from its inverse, we have GF = [*
matrix can be obtained
+ m2 - iO+J-l = i fm ds exp(--i&s)
U(s)
where the negative imaginary part attached to m2 is to be understood in the righthand-side of the definition (3), U(s) = exp(-iX.s), and &f = e-Vp,eguvp,e-1/2.
The matrix element of Eq. (3) is then G&G x’) = i 1,” ds exp(--imax)
(x 1 U(s) 1x’).
Since the quantity U(s) obeys the equation i(a/as) U(s) = imq..), one is led by such a procedure to an associated dynamical description governed by the “hamiltonian” &’ in regard to the so-called proper-time parameters and U(s) can be regarded as a proper-time evolution operator. If one separates out of the hamiltonian a portion Z. for which the exact, known solution is given by G$“(x, x’) = i /r ds exp(-im?s)
(x I U,(s) I x’)
where U,(s) = exp(-iHos),
then a proper-time order, i.e.,
interaction
x=-%J+*,
scheme may be constructed (3) where to first
= &(--is)
1’ -1
dv U,[Q(l -
v) s] ~IUo[~(l
+ v) s].
306
CARVER
If Z0 is chosen to be p2, then the insertions of the unit factors J d4p 1p)(p for p1 and pz into the expression of the matrix element of SV(s), since (x I P& = G-F2
ew(ip,
* 4,
(P2
I x’>
=
w
1= 1
I Pz)*,
gives
(x 1 W(s) X
1x’) = +(-is)
exp[-
s1, dv / d4p, d4p2(2r)-4
&(l - v) p12s - *i( 1 + 0) p22s + ip1 * x - ip2 * x’] ( p1 I & I pz).
With the changes of variables given by ~1 =
P +
#,
= r + $5,
x
~2 = P -
ik,
x’ = r -
86,
(5) (8)
one has further that (x I au(s) 1x’) = $(-is)
/’
dv l d4k(2rr)-4 exp(ik * r - $ik2s) -1
x
I
d4p exp{--i[p2s
-
P e @Sk
+
&I>
+
iik
I & IP
-
%k),
(7) and the matrix elements of functions of x, which depend only upon momentum difference, are to have Fourier transforms defined according to the convention f(k)
= / d4x exp(--ik
. x)f(x).
IV. THE COVARIANT SCALAR GREEN TO FIRST ORDER AND THE GAUGE
FUNCTION PROBLEM
For sufficiently weak gravitational fields, coordinate systems can be introduced for which the metric tensor takes the form &w(x) = %Y + 24”(4
(8)
where h,,“(x) represents the deviation of the covariant metric tensor from its flat-space value and E = d(c = l), where G, Is Newton’s gravitational constant. Utilizing the normalization of the metric tensor and the well-known variational relationship Gdx) = g(x) gw
&4”(4~
307
COVARIANT SCALAR GREEN FUNCTIONS
one readily finds that to first order in E gqx)
= 77”’ - 2&P”(X)
(9)
= 1 + 2&(x)
(10)
and -g(x) where we, in addition,
have written h(x) = yhuy(x).
Accordingly,
with the definitions of the Riemann-Christoffel
tensor as
and the curvature scalar as R(x) = gd-4 g&4
R““Yx),
we have, in view of the connection cd4
= mh(x)
+ avgudx) - aAguv(4i9
to first order in E
and e)(x)
= 2[-a,a,hqX)
where we have written for uniformity,
+ a2h(x)l
(12)
e.g.,
R(x) = l R"'(x). Raising and lowering of indices may henceforth be done with the Minkowski metric since this is clearly sufficient. IV. 1. LINEARIZATION OF THE “HAMILTONIAN"
AND THE PROPER-TIME METHOD
With the insertions of Eqs. (8), (9), and (10) into Eq. (4) for the hamiltonian operator, one obtains to first order in E the operator a = p2 -
4Mp2
+ p2h) -p&p"
+
2p,h”“p,l.
308
CARVER
Corresponding to the zeroth-order part, p2, one readily obtains the flat space-time scalar Green function Gg)(x, x’) = i ILo ds exp(- itYz”s)(x 1 U,(s) 1x’), = Q’(x
- x’)
0
where (x 1 U,(s) 1x’)o = --i(4~~)-~ The accompanying
choice of the interaction sf’p’
exp[i(4s)-l (x - x’)2]. hamiltonian
(13)
operator is then
= -•E[$(hp2 + p2h) - p,,hp” + 2p,h”‘p,]
(13)
to first order in E. In order to employ the general expression (7) of the space-time coordinate matrix element of SU(S) in our case, we need merely form the indicated momentum matrix element of Eq. (14) as
I P - Bk) = W4-*F-pupJ+”
+ W&
-
quyk21 h”“W. (15)
However, recognizing in this equation the appearance of the Fourier transform of Eq. (12) for the “curvature scalar” as R”‘(k)
one then has for that matrix operator in this application (x 1 iSI
1x’) = &(--is) x
s
= 2[k,ky
- q,J2] h““(k),
element of the first-order proper-time
J-1 du j d4k(2?r)-4 exp(ik * r -
d“p(2n))-* exp{-i[p2s
-p
evolution
fik2s)
* (vsk + t)]}[-2p,pJz““(k)
+ @‘(WI.
We thereby encounter the elementary integrals
s s
d*p exp(--i[p2s - p . (vsk + ()I> = (is”)” a2 exp [; i (vk + f,”
d4ppupyexp{--ib2s
=
- P * @Sk +
(16)
011
--(vs)-~ (a/ak”)(a/ak”) $ d% exp(-i[p”s
= -ir2sm2 [i (vk + +),
s]
(vk + 4).
- p * (vsk + 511)
- i(2s)-1 quv] exp [i i (vk + 4,” s].
(17)
COVARIANT
SCALAR
GREEN
309
FUNCTIONS
With the aid of Eqs. (16) and (17), we have finally for the space-time coordinate matrix element of au(s)
(x I SW Ix’>=
(vks + 0, (vks + 0, + qLtyl h”“(k)
- fisR”‘(k)}, (18)
where the appearance of the multiplicative flat space-time matrix element as given by Eq. (13) has been indicated. Since the reconstruction of the scalar Green function follows from AF(x, x’; h) = i y ds exp(-in?s)[(x I
I U,(s) I x’),
+ (x I Ws) I x’)lU - &VW + W)lI to the order of our calculations, it is more convenient for our purposes to include the multiplicative factors of h within the v-integration to first order in E. With the realization that it is possible to write $[h(x) + h(x’)]
= 4 /’
du f d4k(2?r)-4 exp[ik * r -
$ik2s]
-1
x exp[+i(&k2u2s+ vk . f)J{l + $iu[k2us+ k . t]} h(k),
this may be accomplished by writing
A&, x’; h) = i 1: ds exp(- im2s)
x exp[+i($k2u?s + vk * @]{[i(2s)-l (vks + 0, (oks + & - $i(k2v2s+ uk * 5) vuy] h”“(k) - #R(‘)(k)).
(20)
310
CARVER
IV. 2. COORDINATE-GAUGE DEPENDENCE From the general transformation properties of the components of the metric tensor under an arbitrary change of coordinates, one can readily deduce for the infinitesimal coordinate transformation X” = x’1+ 6x11 that the corresponding to order 6x,
variations
of the components
of the metric tensor are,
= -guA(x) a,62 - gvA(x>a,sxA - ~x~aAguvw. Since we have defined hJx) according to the definition (8), the analogous transformation properties of the components of h”“(x) follow from those of the metric tensor as
%&) = f;,“(X) - L(x) = -(2+1
(a,8 X,
- a,sx,) - buy(x)a,6xA - hvA(x) a,6xA - sxAaAh,,(x).
Therefore, in order to examine the gauge properties of our calculations to first order in E we need consider only the simple coordinate-gauge transformation of hUY(x) as km
-+ km
where we have, in addition,
= 4d4
+ mw
+ avu4i
written A,(x) = -(c)-l
sx, .
To the end of carrying out gauge-covariant calculations of the covariant function in curved space-time, we further assume the separations u4
= fL(x)
+ &(X)~
where the Huy are to be independent of coordinate transformations order), and A,(x) - ir,(x> = A,(x) + &(x).
Green (22)
(in lowest (24)
COVARIANT
IV. 3. THE CONTRIBUTION The substitution
SCALAR
GREEN
OF THE GAUGE
311
FUNCTIONS
FUNCTION
of the separations (22) into Eq. (21) gives
Z(x, x’; s) = ZH(x, x’; s) + ; ,‘, dv j d4k (277)-4
(25)
x
exp[ik * r - $ik2s + &i(&k2v2s+ vk . [)I
x
i2(2s)-’ (vsk2 + t . k) 5 * A(k),
where by Z&x, x’; s) we mean exactly expression (21) with the P(k) replaced by the Hu”(k), since W)(k) is gauge invariant (a gauge scalar). The v-integration in the second member of Eq. (25) can easily be effected as Z(x, x’; s) = Z,(x, x’; s) + i(2s)-1 (x - x’) * [A(x) - l&x’)],
(26)
where we have, in addition, taken the appropriate Fourier transforms. Thus, using Eq. (26) in conjunction with Eq. (20), a gauge-covariant expression can be written for the scalar Green function in curved space-time as dF(x, x’) = i Irn ds exp(--im2s) (x 1 U,(s) 1x’),
x “r1 + ia(2s)-l (x - x’) - [A(x) - A(x’)] + eZH(x, x’; s)}, (27) since all gauge dependence is included in the second bracketed term, and the matrix element of U,(s) is given by Eq. (13). Alternatively, since a, = I-1
-
(X
= -a:
x’),
when acting upon Eq. (13), the factors (x - x’),, appearing in Eq. (27) can be eliminated in favor of the derivatives whereby the Green function can be re-expressed as A,(x,
XT)
= (1 +
aa +
+4(x)
A(x’)
. a/j}
&)(x
-
XJ)
+ dt)(~,
x’;
H) (28)
where dj?‘(x, x’; H) = i
ds exp( -im2s) (X / U,,(s) I x’>o Zdx, x’; s). I O” 0
(29)
If we write further that dF(X, x’; A) = {I + @t(x) * a + 4(x’) * a’‘I} dc,o)(x - x’) _N exp{+I(x)
. a + A(d) . a’]) L$?(x -
x’),
then Eq. (28) becomes to first order in E Ll,(x, x’) = Ll,(x, x’; A) + Ed(F1)(X,x’; H),
(30)
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which displays the gauge covariance even more clearly. In terms of the space-time matrix elements such a procedure is equivalent to including all gauge dependence within the matrix element corresponding to the “unperturbed hamiltonian” A$ , for one has from Eq. (27) (x 1 U,(s) 1x’) = (x 1 U,(s) 1x’), (1 + iC(2S))l (X - X’) * [A(X) - A(.+)]} N (x 1 U,(s) 1x’)~ exp{ic(2s)-l (x - x’) . [A(x) - A(Y)]}
= expWNx) * a + 4x’) - a’11(x I W) I x’h , which therefore method.
exhibits
the coordinate-gauge
IV. 4. THE TRANSFORMATION
covariance
of the proper-time
PROPERTIES OF THE GREEN FUNCTION
Suppose that we temporarily suppress the dependence of the Green function on the Hu’, since they are gauge scalars, in order make the gauge dependence explicit, then under a gauge transformation A(x) + A(x) = A(x) + A(x), X(x) = -(e)-1 6x, we have AAX, x’; 4 -+ Ap(X, x’; A) = Ap(x, x’; A) - [6x * a + 6x’ * a’] Ap(x, x’; A). Since under an infinitesimal
general coordinate transformation
AAx, x’; A) + A@, x’; A) = AF(x, x’; A) + [Sx - a + 6x’ - a’] A,(x, x’; A), we verify the transformation A&,
properties of the covariant scalar Green function (6) as
x’; A) --+ A&,
2’; (I> = AF(x, x’; A),
whence AF(x, x’; h) -+ A&, V. CHOICES
x’; /i) = A&
OF THE GAUGE
V. 1. PATH- AND GAUGE-DEPENDENT
x’; h).
FUNCTION
LINE INTEGRALS
One may write for the gauge function A,(x) = Jr dx’“a;A,,(x’)
COVARIANT
SCALAR
GREEN
313
FUNCTIONS
where P is an arbitrary reference point at which A,(P) is to vanish and the path is, of course, arbitrary; alternatively, we write further that
i&(x) = J; dx’9l\ll,,(x’) + 4 J‘: dx’“[a&(x’) - ql,(x’)] where the definitions are rewritten as
(31)
(23) have been noted. If the second integrals of Eqs. (29) $ 1 dx’A[a;A,(x’) s
- a;A,(x’)]
31(x’ - x)
and integrated by parts, then we obtain
A,(x) = 1; dx’A&&(x’,x)
(32)
where we have defined
F,,(x’,x) = AA, + [Q.l,,(x?- wL(x’Mx - x? and have made the additional [a&(P)
assumption that - a:n,(P)](P
- xy = 0.
Through Eqs. (22), the line-integral form of the gauge function re-expressed in terms of the fields h,, and HUy as A,(x) = A,(x; h) - A,(x; H)
can now be (33)
where in accordance with Eqs. (32) we write, e.g., A,(x; h) - J:, dx’A\F,,(x’, x; h)
(34)
where F&(X’, x; h) = h&&(X’) + (x’ - xy [iQn,(x’)
- qz,,(x’)].
(35)
It is interesting to inquire at this juncture as to the conditions under which such line integrals become path independent; for the general expressions as given by Eqs. (32) we find kyF,,(x’, x) - a;F”,(x’, x) = (x’ - xy R&(x’). Thus, path-independence obtains if the Riemann-Christoffel every point along the path.
tensor vanishes at
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CARVER
V. 2. THE MANDELSTAM
LINE INTEGRAL
The realization that the gauge function can be written as the sum of terms involving the gauge-independent or gauge-dependent fields as shown in Section V. 1 affords us the opportunity to demonstrate the ambiguity that exists even in defining gauge-invariant fields, for we can return to Eqs. (22), insert Eqs. (33) into the tensor gauge functions as defined by Eqs. (23), and obtain
hi”(4 = Ww(4 - A&; ml + A&; 4
(36)
where, e.g.,
f&Jx; h) = 4[qA(x; 4 + %4(x; h)l.
(37)
The bracketed members of the right-hand-side of Eqs. (36) simply provide an alternate definition of a gauge-invariant field, say, HLy as h&)
= m4
+ L”k
4.
(38)
The corresponding effect on our result for the scalar Green function as given by Eq. (28) is simply to cause it to be rewritten as dF(X, X’) = (1 + +l(x;
h) * a + A@‘; h) * Z]} &‘(x
- x’) + E&)(X, x’; H’ )
where L$‘(x, x’; H’) is Eq. (29) with H replaced by H’. If we now examine the path-dependent line-integral form of a gauge term appearing in Eq. (39), say, A(x; h) * a, then with some additional symmetrization of Eqs. (35) we have with the aid of Eqs. (34)
+ 4 J5pdx'l[a:hAp(Xy - a;h,,(xy[(x
Eq. (40) is exactly the result of S. Mandlestam our case of a scalar field. V. 3. GAUGE-INVARIANT
PATH-DEPENDENT
- x?, au - (X - x'y aK]. (40)
(20) that he has made explicit for
LINE INTEGRALS
Following F. Rohrlich and F. Strocchi (8) we may use the four arbitrary singlevalued differential four-vector functions xl(x, 5) of the space-time variables xU and a parameter 5, which satisfy the boundary conditions xxx, 0) = -% , !jliW X:(X, 5) = spatial infinity
COVARIANT
SCALAR
GREEN
315
FUNCTIONS
in order to carry out the development of Section V.1. By straightforward manipulation expressions analogous to Eqs. (34) and (35) can be obtained as2 A,(x) = 1” d( y --m
{Ad,(X’) - (x’ - x>” [a:n,,(x’)
- ayAK(x
(41)
and the reference point P is now spatial infinity. A gauge-invariant line integral can now be formed by constructing the tensor gauge functions as given by Eqs. (37) in accordance with Eqs. (41) and using them in conjunction with the separations as given by Eqs. (33). By such a procedure we obtain &(x)
= 3 so d&x - x’y F --m
($$
Rz;Jx’)
+ $$
R$~,(x’)),
(42)
which therefore uniquely determines the gauge-independent potentials once the curvature tensor RuAKv is specified. Although Eqs. (42) are no longer gauge dependent, they are still path dependent because the choice of path x:(x, 5) along which the integration is to be performed has yet to be made. V. 4. THE SPACE-LIKE
PATH AVERAGE
Following F. J. Belinfante (12) and F. Rohrlich the straight-line paths
and F. Strocchi (a), we choose
xl = x, + Eu5 and select the Lorentz frame in which
co = 0f
x’ - x “r-.d--.-; 5
in this Lorentz frame the straight lines converge to the field point x in arbitrary directions given by the unit vectors E, and Eqs. (42) reduce to f&(x)
= - so d~~
(43)
Averaging Eqs. (43) over all the possible directions of E as +%(x)) where d252 is an infinitesimal (H;,(x))
= j d252(47+l f&“(X)
solid angle in the direction of E, we have = f d3x’(4nC)-l ~“~zR~~k(~o, x’)
*This alternate procedure also justifies the insertion of the gradient delta-function in the precedingsubsection.
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since dsx’ = -d2Q12 d<. Gauge properties of the fields are easily investigated in terms of the recent notation due to J. Schwinger (22) whereby the four-vector functions f”(x) = vu ( ;*y;
) c -V”.qX)
with the consequence are used. With the shorthand notation f”fY(x
- x”) E j d4x’f”(x
- x’) fv(x’
-
x”),
these functions obey fY”(4
= fY”W
The result of the path averaging as given by Eqs. (44) may then be written as (H;“(x))
= 2 ( d4x’[flf”(x
- x’) - iq’Kfif’(x
- x’)] R;$(x’).
(45)
The insertion of that part of the curvature tensor symmetric in E and k, written in the form
into Eqs. (45) produces after some manipulation
= L(x)
- 4”(X)
- x’) -
&jz~f’(x
(46)
where now d,(x) = 2 j- d4x’[flfk(x
- x’)] F/$(x’),
(47)
and we have written for the sake of uniformity ~U”KW = 4%x>. If the gauge terms in Eqs. (46) can be made to vanish, then
=
aZhk,(x)
+
akhZ,(x)
-
%hZktX)
the coordinate conditions on the h,, then give the gauge conditions Therefore, by requiring that Eqs. (47) vanish we have v2&hk,(X)
-
&%%hdX)
+
&V’a,,hkk(X)
=
0.
on the (HLJ. (48)
COVAFUANT SCALAR GREEN FUNCTIONS
317
For p = j Eqs. (48) yield (6,jV2 - ajaz) ~k~~~(~)= - Bv2ajhkk(x)*
(49)
Since the left-hand-sides of Eqs. (49) are transverse in j and the right-hand-sides are longitudinal, both sides must vanish as @WV2 - w,> w&l(X) = 0, h7&) = 0.
(50) (51)
For p = 0 Eqs. (48) yield,3 after some manipulation, (a& - &V2) where 7@(x) is the second fundamental C. Misner (13) in lowest order as &‘(x) Eqs. (50x52)
7&)(x) = 0 form of R. Arnowitt,
= ~rcl$jr~$(x)
(52) S. Deser, and
- rg
then furnish the gauge conditions on the (HL,). VI. THE EXAMPLE
OF CONSTANT
CURVATURE
Since it can be readily verified that the expression (11) of the “hrst-order Riemann-Christoffel tensor” is invariant4 under the simple coordinate-gauge transformations discussed in Subsection IV.2, one can employ the concept of constant curvature as a gauge-invariant one and construct the corresponding gauge-invariant “potentials”S (53) Direct substitution of these potentials6 into Eqs. (11) then yields the constant tensor R,$i, . The insertion of the Fourier transform of Eqs. (22) with those potentials H,, , viz, 8 We thank the referee for pointing this out to us. 4 Although the Riemann-Christoffel tensor is not a gauge scalar in general, it is in the case of the linearized theory. 5 We approach the example of constant curvature from the same standpoint that one views constant electromagnetic fields, e. g., Ref. (3); formally, in order to prevent && from becoming arbitrarily large the Riemann-Christoffel tensor should be a slowly varying function of space-time that vanishes at both temporal and (now) spatial infhrity, whether this be achieved by a modification of l or Rc,jAK. * Since the fields given by Eqs. (53) satisfy the conditions a,Hfiv(x) = ~@H(.x), we are now in the so-called Lorentz-Hilbert gauge.
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CARVER
into Eq. (21) produces, of course, a gauge-independent gauge-dependent part already considered as
portion in addition
to the
Z&x, x’; s) = Z&f, r; s) - $isRcl)
(54)
where we have written K([, r; s) E - QR“‘““““(&)
j’,
dv j- d4kfu,,(k; v, s)(~/~k”)(~/~kK)
S(k)
and &(k;
v, s) = exp[ik * (r + &I[) + aik2(v2 - 1) s] x [i(2s)-l
(vks + [),, (vks + 0, - $i(k2v2s + vk . 6) r),,].
Somewhat lengthy calculations yield
Then, since
K(f, r; s) = $&(s)-’
R$,&‘f’rArx
- R$‘~“tJ” + isRcl’].
(55)
The combinations of Eqs. (55), (54), (29), (13), and (6) give the first-order contribution to the scalar Green function in curved space-time as &‘(x,
x’; Z-Z) = i /,” ds exp(--i&s) x &[i(s)-’
R$$‘x’~x”x’~
- tRz(x
- x’)“ (x - x’)” - @R”‘], (56)
where the singularities to be encountered in the limit as x approaches x’ are contained in the lower limit of the s-integration (see Eq. (5.20) of Ref. (3)).
COVARIANT
SCALAR
GREEN
319
FUNCT’IONS
Alternatively, if line integrals of the Hpy with a range of integration are separated from the gauge functions as’
from x’ to x
4(x) - 4dx’) = V&(x> - ~:Wl - /:, WK,(Y), then with the straight-line
choice of path given by
the first member of Eq. (56) may be dispensed with entirely and the covariant scalar Green function takes the simpler form
03 d.(x, x’) = d.(x, x’; A’) + ic I ds exp( - im2s) 0 x ~[s2R(1wway- $isR’l’] (x 1U,(s) 1x’), . VII. CONCLUDING
(57)
REMARKS
In Sections III and IV we have demonstrated in an unambiguous fashion that the proper-time approach is applicable to the treatment of a scalar field in a linearized classical gravitational field. In those cases in which the first-order gravitational fields can be written as the sum of a gauge scalar and gauge-dependent terms in accordance with Eqs. (22)-(24), the proper-time method when utilized for the calculation of scalar Green functions has been shown to exhibit manifest gauge covariance as illustrated by Eq. (27), (30), or even (39). In fact, the transition from Eq. (28) to (39) represents a shift in “base” gauge; that is, Eqs. (22) relate the fields Huy to all other gauges obtainable from them just as Eqs. (38) relate the fields HLy to all other gauges obtainable from them. In particular, the latter choice of base gauge in conjunction with the line-integral representations of Subsection V.l allowed us to produce the Mandelstam line integral and in conjunction with those of Subsection V.2 to pass from a class of gauge-dependent fields to the gauge-independent field variables given by Eqs. (42). Since we had first written the gauge functions (1, as path-dependent line integrals and then proceeded to split-up the gauge functions according to Eqs. (33), the connection between the fields Huy and HLy has been made path-dependent. In fact, if one 7 Although such separations of the gauge functions may appear to be an ad hoc one at this jucture, it is a natural choice in the case of spin fields, which we are to make clear in a future publication. 595/55/2-8
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chooses to use a line-integral form of a gauge function and proceeds to split it up into two portions, then the corresponding transformation between any two base gauges, say, zy(x) --f HT(x) = H,“‘(x) + A,““(x), has in that manner been reduced to one of path selection, for in principle one can obtain other base gauges by choosing particular paths for the gauge functions involved. In particular, if there exists some class of paths for which, say, the /lt vanish, then we associate this class of paths with the Hr. That such an association, indeed, exists has already been demonstrated in the case of electromagnetism by F. Rohrlich and F. Strocchi (8). The space-like path averaging of the gauge-invariant line-integral expressions of the first-order fields produced the path-independent expressions as given by Eqs. (44) or (45). Although such a procedure produced field variables in the radiation gauge in the case of electromagnetism (8), we find that the gauge conditions as expressed by Eqs. (50)--(52) do not appear to be identical with those of any of the usual gauges used in gravitation in the presence of matter. In particular, it is obvious that we have produced coordinate conditions that are a variant of the radiation gauge used by J. Schwinger (I2), since the vanishing of our Eqs. (47) imposes new and different conditions on the metric (compare with Eqs. (94) of Ref. (12)). The gauge-invariant first-order contribution to the Green function of a scalar field coupled to a weak gravitational field of constant curvature is provided by Eq. (56). The transition from Eq. (56) to (57) evidently allows one to display clearly that the singularities to be encountered in the limit as x approaches x’ are no different than in the case of flat space-time. That identical divergences are to be expected has already been pointed out in general by B. Dewitt (1). Additionally, one may deduce from Section VI that the concept of zero curvature is a gauge-invariant one to first order in E; in such a case the Green function is dependent only on the gauge function, which was made explicit in Subsection IV.4. In passing, we cite some roles that the Riemann-Christoffel tensor has played in this paper which strongly resemble those played by the electromagnetic-field tensor of electromagnetic theory. Note, in particular, the striking similarities of the gravitational potentials for the example of constant curvature in Section VI to gauge-invariant four-vector potentials for a constant electromagnetic field. As in the preceding comparison the gauge-invariant line-integral expressions of the fields which were developed in Subsection V.3 possess the necessary increase in indices (compare our Eqs. (42) with Eqs. (5) of Ref. (8)). Aspects of this paper are at present being extended to systems of higher-order spin.
COVARIANT
SCALAR GREEN FUNCTIONS
321
The author finds it a pleasure to thank Dr. A. F. Radkowski for valuable discussions and critical remarks and Prof. M. J. Tausner for his interest, encouragement, and remarks that have led to the constant curvature considerations.
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