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Trace anomaly of the energy-momentum tensor of quantized scalar fields in Robertson-Walker spacetime
Volume 71A, number 2,3
PHYSICS LETTERS
30 April 1979
TRACE ANOMALY OF THE ENERGY—MOMENTUM TENSOR OF QUANTIZED SCALAR FIELDS IN ROBERTSON—WALKER SPACETIME * B.L. HU Department ofPhysics, University of California, Santa Barbara, CA 93106, USA
Received
16 October 1978 Revised manuscript received 28 February 1979
The energy—momentum tensors of quantized massless scalar fields minimally or conformaily coupled to a homogeneous and isotropic spacetime are calculated by adiabatic regularization. In our derivation, the trace anomaly results simply from a reduction of logarithmic divergent integrals before taking the massless limit.
A non-zero trace in the quantum stress tensor of a conformally invariant field in curved spacetime has recently been shown to exist by various regularization methods [1, 5—14] We show in this letter how one can obtain these results in a mathematically simple and physically intuitive way by yet another method that of adiabatic regularization [2—4] This method has been used by Parker, Fulling and Hu to calculate regularized energy—momentum tensors in the Robertson—Walker and Kasner-type universes with relative ease. It is our aim here to show how the trace anomaly arises naturally from the formulae already presented in refs. [2] and [4] provided that proper care is exerted in taking the massless limit. Bunch [5] has used the results in ref. [4] to derive the conformal anomaly and Birrel [6] has likewise derived the stress tensor for a quantized scalar field in RW spacetime by using cut-off functions in evaluating the mode integrals. The method presented here instead employs a recursion relation for logarithmic divergent integrals, the reduction of which leads to the trace anomaly. The avoidance of the use of cut-off functions eliminates possible ambiguities in the integrated results. We obtain the same answer for the conformal anomaly as before [9—14] In the minimal coupling case, the integrated trace of the stress tensor can be calculated but differs from those derived earlier by proper-time quantization [8] or perturbation expansion [12] methods. The application of this method to the more complicated Kasner-type spacetimes, which include the present (flat) RW spacetime as a special case, will be published elsewhere [7]. .
—
.
,
.
Energy—momentum tensor. We consider a massive (m) scalar field 4(t, x) in a flat Robertson—Walker (RW) universe with metric
p,v0,1,2,3,
~
(1)
i,j1,2,3,
obeying the field equation (0 + ~R + m2)’t(t, x) = 0, where 0 denotes the d’alembertian, R = 6[~’a + (a/a)2] is the scalar curvature and the constant ~ assumes the value 0 for “minimal coupling” and 1/6 for “conformal coupling”. The energy—momentum tensor of the scalar field is given by =
(1
—
2~)(I~,,.t4~,v + (2~
—
~
+
~g,
24,2
—
~VM(~I~2)~, + ~ggA0V~(cI2)
—
~42GMV, (2)
2~m *
Supported in part by the National Science Foundation.
169
Volume 71A, number 2,3
where
PHYSICS LETTERS
30 April 1979
is the Einstein tensor. The quantized solution to the field equation can be represented in the form
~I(t,x)= (27r)_31’2 fd3k [AkØk(t)e~’~’ +A~c~(t)e”~]
(3)
where the operators Ak satisfy the equal-time commutation relations and the amplitude function satisfies the wave equation in the form ~ + ~2 ~ = 0. Here i~frdenotes ~ or x a~,~2denotes ~m V~(V a3) or &2c ao~,and a prime denotes d/dr Vd/dt or d/di~~ad/dt for the cases ~ = 0 and 1/6, respectively. The frequency is given by w2 p2 +m2,wherep~k/aandk~Ikl Let 10A ) be a normalized state annihilated by all the Ak, then (TM V) in that state are given by ~
(4a)
P=—(OAIT
(4b)
~=~:
2VY’ fp2dp {Iø,rI2_2~+2m2V2)IøI2}, 1’IOA)=(4Ir p(4ir2a)~fp2dp {Ix,~I2+~IxI2},
(5a)
~0:
P~(4ir2a~1 fp2dp~Ix~!2 +(cZ~—2m2a2)IxI2}.
(Sb)
These integrals are formally divergent. To obtain physical results, some meaningful way of subtracting or recasting the infinities are necessary.
4
Adiabatic regularization. In this scheme, the operators Ak and are chosen in such a way that in the adiabatic limit (of arbitrarily slow time variation of a, as characterized by the inverse of a large adiabatic parameter T~ —, (a/a)) they become the annihilation and creation operators for physical particles. In an asymptotic expansion of the integrand in powers of T~the terms after the first three subtractions (of order T0, T2 and T—4) will be finite and correspond to contributions from real physical particles. Thus, one assumes a fourth-order WKB positivefrequency solution in the form =
[2v(~)W~(~)] 112exp
[_~f
u(~’)Wk(~’)d~’].
(6)
Here, we have written ~ for either r or r~,and v for either V or a in the two cases ~ = 0 or 1/6, respectively; Wk = + ~2 + 64) with = ~ 6~= ~ + 62) 1{~2 (1 + 2)’[(l + 62)1/4] ~}‘ ‘,
x~4kn)is regarded as an exact solution to ~k(r) or Xk(fl), and expansion in T~(as measured by the power of time derivatives) is carried out to the fourth order one obtains from eqs. (4) and (5) the divergent terms in the energy density and pressure as [2,4] When 44~(r)or
=
(8n.2~y1f p2dp &Z~[2~2 + ~ (~ ~/~~)2 ~ (~l/~)2+ ~ (~Z’/~)e~ + ~~
‘~divPdiv
(~ = 0 or ~),
i(4~2~t)lfp2dpa1(2~2+m2v2)[1 —~e 2—~(e~ _~e~)] (~=O),
~dv~Pdiv
i(4~2a)1fp2dp~_1(m2a2)[l —j~e2—~(e~ —~e~)] (~=~).
The regularized energy density and pressure are given by (TM I)) 170
(7)
=
(TM~’)o (TMz~)dj,,. —
(8a)
(8b)
Volume 71A, number 2,3
PHYSICS LEUERS
30 April 1979
The calculation of Pdw is relatively simple for RW spacetimes and the results for both the minimal (~ = 0) and conformal (~ = 1/6) cases have been given before [3,4]. These earlier results were however incomplete in that the existence of an anomalous trace in the massless case was not identified. This resulted from overlooking some subtle points in evaluating the divergent integrals and taking the massless limits. We shall show in the following how one can recover the trace anomaly from the results obtained earlier by adiabatic regularization. Conformal anomaly. For the conformal case (~ = 1/6), the trace TCCQ TMM = m2Ø2 is zero for classical massless fields. It would appear to be true also for quantized fields if one sets m = 0 formally in the integrand of eq. (8b). However, if one performs the integration before setting m = 0, one sees that the fourth-order part ~ contains a constant term independent of m, resulting from the reduction of integrals. This is the conformal anomaly T~ (T,~>reg= —(p 3p)~).To illustrate how one calculates this, write m2 = w2 and —
—
5
Ac
8(aw)4(e 4
E
=
—
n1
then eq. (8b) can be written as 3(1 _p2cr2)Ac. (9) Tc = (641r2a4Y1fp2dp w In evaluating the integrals, one observes that any p-integral can be reduced to successively lower order by the recursion formula pM~jp
~,
‘~
(p2
+
p121
=
m~)Vfl
—
(v2)(p2
+
+ p
—
1
v
—
2
m2)0’2~2
r -~
For the logarithmic integrals L
(10)
pM2dp
~
+
2~l/w2~~+l and p = v 1. In the massless case, eq. (10) becomes L 2~mf~dpp 2~ = —(2n 1)~ + L2n2. One can thus reduce all L2n into a sum of the logaritlunic integralL2 and a constant X2n, i.e.,L2~= L2 X2n, where X2n = 1/3, 8/15, 71/105, 248/315, 3043/2465 for n = 2 to 6. Writing eq. (9) as —
—
—
4)’ EB~nL Tc=(64?a
(11)
2n(64~2a4)_1(T~L2 —Ti),
whereB2n A2~ A2 n—il (forn the new finite constant ¶~‘~= ~2X2~
=
2 to 5)andB2 =A2,B12 = —A10,one findsthat T~= .~iB~n0 and B~nyields the conformal anomaly:
5 A~~/(2n 2a4)~ E +1) (480ir2a4)1(D” —4D’D2)a
=
(64ir
2
(~ ~),
(12)
n1
where
2)~ [—(R~6R~.~ _R~~aRag 2R2+ (6 30~)OR] (13) a2 = (2880ir 3) ~(1 6~) This result agrees with those obtained by other methods of regularization [8—14]. For the minimally coupled (~= 0) case, to calculate the divergent quantities, one can adopt the formulae presented in ref. [4] for the conformal case with appropriate changes d/dr, fl Vw, a V, D -÷Cm dV/dt in (BI) to (B4), and D C/3 in (B5) and (BlO) to (813)]. The result for the energy density is —
—
.
—
[‘-+
-~
-~
-~
p~)= (64ir~V~)~ [~(—2ãC+ 4C’C2
+
C2
—
C4)L
2
2 rh(l 1CC 32CC For the divergent pressure, one first calculates from eq. (8a) the quantity ~ —
—
~
+
~
C’~)J-
(14)
—
E (P
—
~)div and then adds on to
eq. (14) to obtain P~f.1 It can also be obtained more simply by using the divergence equation, but proper care must -
171
Volume 71A, number 2,3
PHYSICS LETTERS
30 April 1979
be taken to include the contributions from L2 through the integration variables p = k/a [3] Proceeding in this way, the total trace is given by 2a4)~[2(D” 6D’D2)L Tm’ —(p 3p)~= (32ir 2 1~(—11D”+60D”D+l84D’D2+45D’2_45D4)] (~=0), (15) —jwhere ‘ d/di~and D = a~da/di~.This result agrees with that of ref. [6] which uses L 0, related to our L2 by L2 L0 1 Since the coefficient of L2 is a total divergence, the integral of Twill contain only the finite terms given by -
—
—
—
~‘.
,,
2
fT~(_g)hI2d4x=__-!~ 2fD4dn+f(-~_~) di~ (16) 32ir 4 term (R 6a ‘ía). The discrepancy could arise from the difThis disagrees the result ofrefs. [8,9] by D the expectation values of TM’~(we calculate (in I TM~I in> ferent choiceswith of vacuum states with which onethe takes while refs. [8,9] calculate (in I TM~’lout>). It could also be an indication of intrinsic ambiguities associated with the definition of trace anomalies for non-conformal fields. Indeed, if one defines the anomalous trace T~to be the difference between the vacuum expectation values of the total trace Tm and that of the classical trace = 0), i.e., .
T~m
—
lim (2m2~2>~=(64ir2V4) fp2dpw3(l m—~0
_p2~,_2)Am,
one would get instead T~=(64~2V~~1 EA~n/(2fl+l)(480~2V4)1(6D”—64D’D2
—
1SD’2
—
l5D4)a (p0),
2 where A’~= ~
~
and the coefficients AT~are given by: 2—~C2—VC4, A_~(C_~CC+~fCC2—l1C2_1-~5C4),
A~C—6CC+YCC A~=~(—14CC+~CC2 _~i’2—~4~5C~),A~=
2+~C4), A~ ‘—~-C4
.
(17)
(18a,b) (18c,d,e)
1~(—i’C The integrated anomalous trace defined above would give: ,,
2
f T~(—g)~’2d’~x —~f(-~--) dt~, 32ir =
(19)
—
which differs from the result of refs. [8,9] by a sign. These calculations show that for non-conformal fields different choices of vacuum states and different definitions of the trace anomaly generally lead to different results. Only for conformally invariant fields will the definition of trace anomaly be unambiguous and meaningful. In comparing eqs. (8a) and (8b), one sees that the quantity F = p p in the minimal case plays a similar role as T = p 3p in the conformal case. Classically, a conformally invariant field has an effective equation of state = ~ (or TCQ = 0) whereas a minimally coupled scalar field behaves like a PcQ = PcQ fluid (or Fc 2 = 0) [15]. When the fields are quantized, (TMM> no longer vanishes. Just as the conformal anomaly T~is to be regarded as a quanturn correction to the relativistic (p = p13) radiation fluid, F could be regarded as a correction to the ultrarelativistic (p = p) fluid. The quantum effects associated with T (as in the conformal scalar field) [12a] and F (as in the gravi—
—
I thank T.S. Bunch for this observation.
172
PCQ
Volume 71A, number 2,3
PHYSICS LETTERS
30 April 1979
ton field) in influencing the dynamics of the early universe should be of considerable interest in relativistic cosmology. I would like to thank N.D. Birrell, S.A. Fulling, J.B. Hartle and L. Parker for helpful comments. References See, e.g., M.J. Duff, Nuci. Phys. B125 (1977) 334. [2] L. Parker and S.A. Fulling, Phys. Rev. D9 (1974) 341. [1]
[3] S.A. Fulling and L. Parker, Ann. Phys. (NY) 87 (1974) 176. [4] S.A. Fulling, L. Parker and B.L. Hu, Phys. Rev. D10 (1974) 3905. [5] T.S. Bunch, J. Phys. All (1978) 603. [6] N.D. Birrell, Proc. Roy. Soc. 361 (1978) 513. [7] B. L. Hu, Phys. Rev. D18 (1978) 4460. [8] B.S. DeWitt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965); Phys. Rep. 19C (1975) 295. [9] S.W. Hawking, Commun. Math. Phys. 55 (1977) 133.
[10] [11] [12] [13] [14] [15]
J.S. Dowker and R. Critchley, Phys. Rev. D16 (1977) 3390. L.S. Brown and J.P. Cassidy, Phys. Rev. Di5 (1977) 2810; D16 (1977) 1712. MV. Fischetti, J.B. Hartle and B.L. Hu, Quantum effects in the early universe, (a) paper I, (b) paper IV. P.C.W. Davies, S.A. Fulling, S.M. Christensen and T.S. Bunch, Ann. Phys. (NY) 109 (1978) 108. C. Bernard and A. Duncan, Ann. Phys. (NY) 107 (1977) 201. R~Tabensky and A.H. Taub, Commun. Math. Phys. 29 (1973) 61.
173
PHYSICS LETTERS
30 April 1979
TRACE ANOMALY OF THE ENERGY—MOMENTUM TENSOR OF QUANTIZED SCALAR FIELDS IN ROBERTSON—WALKER SPACETIME * B.L. HU Department ofPhysics, University of California, Santa Barbara, CA 93106, USA
Received
16 October 1978 Revised manuscript received 28 February 1979
The energy—momentum tensors of quantized massless scalar fields minimally or conformaily coupled to a homogeneous and isotropic spacetime are calculated by adiabatic regularization. In our derivation, the trace anomaly results simply from a reduction of logarithmic divergent integrals before taking the massless limit.
A non-zero trace in the quantum stress tensor of a conformally invariant field in curved spacetime has recently been shown to exist by various regularization methods [1, 5—14] We show in this letter how one can obtain these results in a mathematically simple and physically intuitive way by yet another method that of adiabatic regularization [2—4] This method has been used by Parker, Fulling and Hu to calculate regularized energy—momentum tensors in the Robertson—Walker and Kasner-type universes with relative ease. It is our aim here to show how the trace anomaly arises naturally from the formulae already presented in refs. [2] and [4] provided that proper care is exerted in taking the massless limit. Bunch [5] has used the results in ref. [4] to derive the conformal anomaly and Birrel [6] has likewise derived the stress tensor for a quantized scalar field in RW spacetime by using cut-off functions in evaluating the mode integrals. The method presented here instead employs a recursion relation for logarithmic divergent integrals, the reduction of which leads to the trace anomaly. The avoidance of the use of cut-off functions eliminates possible ambiguities in the integrated results. We obtain the same answer for the conformal anomaly as before [9—14] In the minimal coupling case, the integrated trace of the stress tensor can be calculated but differs from those derived earlier by proper-time quantization [8] or perturbation expansion [12] methods. The application of this method to the more complicated Kasner-type spacetimes, which include the present (flat) RW spacetime as a special case, will be published elsewhere [7]. .
—
.
,
.
Energy—momentum tensor. We consider a massive (m) scalar field 4(t, x) in a flat Robertson—Walker (RW) universe with metric
p,v0,1,2,3,
~
(1)
i,j1,2,3,
obeying the field equation (0 + ~R + m2)’t(t, x) = 0, where 0 denotes the d’alembertian, R = 6[~’a + (a/a)2] is the scalar curvature and the constant ~ assumes the value 0 for “minimal coupling” and 1/6 for “conformal coupling”. The energy—momentum tensor of the scalar field is given by =
(1
—
2~)(I~,,.t4~,v + (2~
—
~
+
~g,
24,2
—
~VM(~I~2)~, + ~ggA0V~(cI2)
—
~42GMV, (2)
2~m *
Supported in part by the National Science Foundation.
169
Volume 71A, number 2,3
where
PHYSICS LETTERS
30 April 1979
is the Einstein tensor. The quantized solution to the field equation can be represented in the form
~I(t,x)= (27r)_31’2 fd3k [AkØk(t)e~’~’ +A~c~(t)e”~]
(3)
where the operators Ak satisfy the equal-time commutation relations and the amplitude function satisfies the wave equation in the form ~ + ~2 ~ = 0. Here i~frdenotes ~ or x a~,~2denotes ~m V~(V a3) or &2c ao~,and a prime denotes d/dr Vd/dt or d/di~~ad/dt for the cases ~ = 0 and 1/6, respectively. The frequency is given by w2 p2 +m2,wherep~k/aandk~Ikl Let 10A ) be a normalized state annihilated by all the Ak, then (TM V) in that state are given by ~
(4a)
P=—(OAIT
(4b)
~=~:
2VY’ fp2dp {Iø,rI2_2~+2m2V2)IøI2}, 1’IOA)=(4Ir p(4ir2a)~fp2dp {Ix,~I2+~IxI2},
(5a)
~0:
P~(4ir2a~1 fp2dp~Ix~!2 +(cZ~—2m2a2)IxI2}.
(Sb)
These integrals are formally divergent. To obtain physical results, some meaningful way of subtracting or recasting the infinities are necessary.
4
Adiabatic regularization. In this scheme, the operators Ak and are chosen in such a way that in the adiabatic limit (of arbitrarily slow time variation of a, as characterized by the inverse of a large adiabatic parameter T~ —, (a/a)) they become the annihilation and creation operators for physical particles. In an asymptotic expansion of the integrand in powers of T~the terms after the first three subtractions (of order T0, T2 and T—4) will be finite and correspond to contributions from real physical particles. Thus, one assumes a fourth-order WKB positivefrequency solution in the form =
[2v(~)W~(~)] 112exp
[_~f
u(~’)Wk(~’)d~’].
(6)
Here, we have written ~ for either r or r~,and v for either V or a in the two cases ~ = 0 or 1/6, respectively; Wk = + ~2 + 64) with = ~ 6~= ~ + 62) 1{~2 (1 + 2)’[(l + 62)1/4] ~}‘ ‘,
x~4kn)is regarded as an exact solution to ~k(r) or Xk(fl), and expansion in T~(as measured by the power of time derivatives) is carried out to the fourth order one obtains from eqs. (4) and (5) the divergent terms in the energy density and pressure as [2,4] When 44~(r)or
=
(8n.2~y1f p2dp &Z~[2~2 + ~ (~ ~/~~)2 ~ (~l/~)2+ ~ (~Z’/~)e~ + ~~
‘~divPdiv
(~ = 0 or ~),
i(4~2~t)lfp2dpa1(2~2+m2v2)[1 —~e 2—~(e~ _~e~)] (~=O),
~dv~Pdiv
i(4~2a)1fp2dp~_1(m2a2)[l —j~e2—~(e~ —~e~)] (~=~).
The regularized energy density and pressure are given by (TM I)) 170
(7)
=
(TM~’)o (TMz~)dj,,. —
(8a)
(8b)
Volume 71A, number 2,3
PHYSICS LEUERS
30 April 1979
The calculation of Pdw is relatively simple for RW spacetimes and the results for both the minimal (~ = 0) and conformal (~ = 1/6) cases have been given before [3,4]. These earlier results were however incomplete in that the existence of an anomalous trace in the massless case was not identified. This resulted from overlooking some subtle points in evaluating the divergent integrals and taking the massless limits. We shall show in the following how one can recover the trace anomaly from the results obtained earlier by adiabatic regularization. Conformal anomaly. For the conformal case (~ = 1/6), the trace TCCQ TMM = m2Ø2 is zero for classical massless fields. It would appear to be true also for quantized fields if one sets m = 0 formally in the integrand of eq. (8b). However, if one performs the integration before setting m = 0, one sees that the fourth-order part ~ contains a constant term independent of m, resulting from the reduction of integrals. This is the conformal anomaly T~ (T,~>reg= —(p 3p)~).To illustrate how one calculates this, write m2 = w2 and —
—
5
Ac
8(aw)4(e 4
E
=
—
n1
then eq. (8b) can be written as 3(1 _p2cr2)Ac. (9) Tc = (641r2a4Y1fp2dp w In evaluating the integrals, one observes that any p-integral can be reduced to successively lower order by the recursion formula pM~jp
~,
‘~
(p2
+
p121
=
m~)Vfl
—
(v2)(p2
+
+ p
—
1
v
—
2
m2)0’2~2
r -~
For the logarithmic integrals L
(10)
pM2dp
~
+
2~l/w2~~+l and p = v 1. In the massless case, eq. (10) becomes L 2~mf~dpp 2~ = —(2n 1)~ + L2n2. One can thus reduce all L2n into a sum of the logaritlunic integralL2 and a constant X2n, i.e.,L2~= L2 X2n, where X2n = 1/3, 8/15, 71/105, 248/315, 3043/2465 for n = 2 to 6. Writing eq. (9) as —
—
—
4)’ EB~nL Tc=(64?a
(11)
2n(64~2a4)_1(T~L2 —Ti),
whereB2n A2~ A2 n—il (forn the new finite constant ¶~‘~= ~2X2~
=
2 to 5)andB2 =A2,B12 = —A10,one findsthat T~= .~iB~n0 and B~nyields the conformal anomaly:
5 A~~/(2n 2a4)~ E +1) (480ir2a4)1(D” —4D’D2)a
=
(64ir
2
(~ ~),
(12)
n1
where
2)~ [—(R~6R~.~ _R~~aRag 2R2+ (6 30~)OR] (13) a2 = (2880ir 3) ~(1 6~) This result agrees with those obtained by other methods of regularization [8—14]. For the minimally coupled (~= 0) case, to calculate the divergent quantities, one can adopt the formulae presented in ref. [4] for the conformal case with appropriate changes d/dr, fl Vw, a V, D -÷Cm dV/dt in (BI) to (B4), and D C/3 in (B5) and (BlO) to (813)]. The result for the energy density is —
—
.
—
[‘-+
-~
-~
-~
p~)= (64ir~V~)~ [~(—2ãC+ 4C’C2
+
C2
—
C4)L
2
2 rh(l 1CC 32CC For the divergent pressure, one first calculates from eq. (8a) the quantity ~ —
—
~
+
~
C’~)J-
(14)
—
E (P
—
~)div and then adds on to
eq. (14) to obtain P~f.1 It can also be obtained more simply by using the divergence equation, but proper care must -
171
Volume 71A, number 2,3
PHYSICS LETTERS
30 April 1979
be taken to include the contributions from L2 through the integration variables p = k/a [3] Proceeding in this way, the total trace is given by 2a4)~[2(D” 6D’D2)L Tm’ —(p 3p)~= (32ir 2 1~(—11D”+60D”D+l84D’D2+45D’2_45D4)] (~=0), (15) —jwhere ‘ d/di~and D = a~da/di~.This result agrees with that of ref. [6] which uses L 0, related to our L2 by L2 L0 1 Since the coefficient of L2 is a total divergence, the integral of Twill contain only the finite terms given by -
—
—
—
~‘.
,,
2
fT~(_g)hI2d4x=__-!~ 2fD4dn+f(-~_~) di~ (16) 32ir 4 term (R 6a ‘ía). The discrepancy could arise from the difThis disagrees the result ofrefs. [8,9] by D the expectation values of TM’~(we calculate (in I TM~I in> ferent choiceswith of vacuum states with which onethe takes while refs. [8,9] calculate (in I TM~’lout>). It could also be an indication of intrinsic ambiguities associated with the definition of trace anomalies for non-conformal fields. Indeed, if one defines the anomalous trace T~to be the difference between the vacuum expectation values of the total trace Tm and that of the classical trace = 0), i.e., .
T~m
—
lim (2m2~2>~=(64ir2V4) fp2dpw3(l m—~0
_p2~,_2)Am,
one would get instead T~=(64~2V~~1 EA~n/(2fl+l)(480~2V4)1(6D”—64D’D2
—
1SD’2
—
l5D4)a (p0),
2 where A’~= ~
~
and the coefficients AT~are given by: 2—~C2—VC4, A_~(C_~CC+~fCC2—l1C2_1-~5C4),
A~C—6CC+YCC A~=~(—14CC+~CC2 _~i’2—~4~5C~),A~=
2+~C4), A~ ‘—~-C4
.
(17)
(18a,b) (18c,d,e)
1~(—i’C The integrated anomalous trace defined above would give: ,,
2
f T~(—g)~’2d’~x —~f(-~--) dt~, 32ir =
(19)
—
which differs from the result of refs. [8,9] by a sign. These calculations show that for non-conformal fields different choices of vacuum states and different definitions of the trace anomaly generally lead to different results. Only for conformally invariant fields will the definition of trace anomaly be unambiguous and meaningful. In comparing eqs. (8a) and (8b), one sees that the quantity F = p p in the minimal case plays a similar role as T = p 3p in the conformal case. Classically, a conformally invariant field has an effective equation of state = ~ (or TCQ = 0) whereas a minimally coupled scalar field behaves like a PcQ = PcQ fluid (or Fc 2 = 0) [15]. When the fields are quantized, (TMM> no longer vanishes. Just as the conformal anomaly T~is to be regarded as a quanturn correction to the relativistic (p = p13) radiation fluid, F could be regarded as a correction to the ultrarelativistic (p = p) fluid. The quantum effects associated with T (as in the conformal scalar field) [12a] and F (as in the gravi—
—
I thank T.S. Bunch for this observation.
172
PCQ
Volume 71A, number 2,3
PHYSICS LETTERS
30 April 1979
ton field) in influencing the dynamics of the early universe should be of considerable interest in relativistic cosmology. I would like to thank N.D. Birrell, S.A. Fulling, J.B. Hartle and L. Parker for helpful comments. References See, e.g., M.J. Duff, Nuci. Phys. B125 (1977) 334. [2] L. Parker and S.A. Fulling, Phys. Rev. D9 (1974) 341. [1]
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