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A new solution of the Wheeler-De Witt equation
Volume 261, number 4
PHYSICS LETTERS B
6 June 1991
A new solution of the Wheeler-De Witt equation Giacomo Giampieri Dipartimento di Fisica Teorica, Via Bassi 6, 1-27100 Pavia, Italy
Received 30 January 1991
We consider the problem of rotation in a homogeneous Bianchi type IX cosmological model. Studying the Wheeler-De Witt equation corresponding to this minisuperspace, and adopting a particular choice of the factor ordering, we are able to find a particular solution which is strongly peaked about isotropy. This result confirms all the previous investigations in this field, and suggests the conclusion that the machian nature of our universe is the natural, i.e. most likely, outcome of the Planck epoch.
I. Introduction It is generally believed that the list o f cosmological p r o b l e m s to be investigated includes the rotation o f the present-day universe. This "rotation p r o b l e m " can be f o r m u l a t e d in this very simple way: if the universe can rotate, why does it rotate so slowly? Various exp l a n a t i o n o f this p r o b l e m should be given, for example invoking M a c h ' s principle or postulating the occurrence o f an inflationary epoch, but none o f t h e m can be considered completely satisfactory. In a recent p a p e r [ 1 ] we investigated the possibility o f solving this p r o b l e m assuming Hawking's proposal for the b o u n d a r y conditions o f the universe. In particular we analyzed the wave function for a Bianchi type IX rotating minisuperspace, using the F e y n m a n path-integral a p p r o a c h to q u a n t u m gravity [21. The result was that the first o r d e r correction to the u n p e r t u r b e d (i.e. non-rotating) wave function confirms some previous qualitative or numerical results about the wave function o f an anisotropic universe [ 3 ]. In this way we were able to conclude that in the limit o f small angular velocity a general Bianchi type IX model is not m o r e likely than a F r i e d m a n n cosmological solution, in spite o f the fact that the set o f cosmological models which a p p r o a c h isotropy at infinite times is o f measure zero in the space o f all spatially homogeneous models. The p e r t u r b a t i v e expansion o f the F e y n m a n path integral was p u r s u e d in o r d e r to o v e r c o m e the m a t h e m a t i c a l difficulties en-
d o w e d with the W h e e l e r - D e W i t t ( W D W ) equation. However, from the physical point o f view, in order to solve the problem o f rotation as stated above, it would be necessary to accomplish a m o r e detailed analysis. This can be done either by allowing arbitrary values for the vorticity or by extending the minisuperspace to different 3-geometries, possibly even not homogeneous. In this short c o m m u n i c a t i o n we will present a particular exact solution o f the W D W equation for the same minisuperspace considered above, that is a Bianchi type IX universe filled with dust, but without any hypothesis about the magnitude o f the rotation. This result makes use o f an interesting method, due to M o n c r i e f and Ryan [4], to solve an equation o f the f o r m / 4 ~ = 0, given an imaginary solution o f the classical H a m i l t o n - J a c o b i equation. Let {x A, PA} be canonical variables. The hamiltonian constraint can be written as H = G A B ( x ) p a PB + 3V'(x) = O,
( 1)
while the classical H a m i l t o n - J a c o b i equation is 0S 0S
GA s - + f~(x) =0 Ox A Ox B
(2)
Suppose S = i5e is a pure imaginary solution o f ( 2 ) . Then we can write H in the form
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
41 1
Volume 26 l, number 4
H
PHYSICS LETTERS B
.-.an.. -AB 05° 05e =lr tX)PAPn+G ~XA OXB =GAB(x)Tr]ItB ,
(3)
where we have defined the new momenta
In ( 11 ) we have introduced, for later use, the Misner parametrization of the spatial metric; 12=12(t) is a scalarandfl=fl(t) is a 3 × 3 symmetric trace-free matrix. In the simplest rotating case we can in turn parametrize fl by f l = A - I fldA ,
• 05e IrA = P A - - 1 0 X A .
(4)
(The expression (3) for the hamiltonian is similar to Ashtekar's form, but does not use spinor variables. ) The quantized version ofeq. (4) is 0 .06e ~A = - i 0 - ~ - , 0x A .
(5)
~u= exp ( - S e ) ,
(6)
then it is straightforward to verify that ~n~=0.
(7)
Thus the wave function (6) is a particular solution of the WDW equation, supposing that one chooses the factor ordering with the operator ~8 on the right; for example 1
[ ~ ( ~ / I GI GA'~8)] •
(8)
Incidentally the choice (8), apart from the use of the old momenta, is the same one made in ref. [ 1 ], on the basis of minisuperspace arguments. Therefore, generally speaking, the very difficult task of solving the WDW equation is reduced to the much easier problem of finding a purely imaginary solution of the Hamilton-Jacobi equation (2).
2. M i n i s u p e r s p a c e
model
In this section we will consider a minisuperspace model very similar to other ones previously studied in the literature ([ 1,3 ] ), that is a Bianchi type IX rotating universe, whose metric is (i, j = 1, 2, 3 ) dS 2= - d t 2 + hij( t )o)io) j , d o ) i = ~ejkOY I i "^ o)k
,
h~j= e x p ( - 2 1 2 ) [exp(2fl) ]o. 412
(9) (10) (11 )
(12)
where fld=diag(--Zfl+,fl+ + x / ~ f l _ , f l + - - X / ~ f l _ ) ,
A=
If we take a wave function of the form
~=
6 June 1991
1
0
0
0
1
- s i n 20 1 + cos 20
0
1 - c o s 20 sin 20
1
(13)
(14)
The metric (9) is determined by the four functions of time 12, fl+, r _ and 0. r+ and r _ have a direct physical interpretation as a measure of the anisotropy of the universe, as can be seen by putting r + = r _ = 0. The variable 0 is the additional degree of freedom due to the presence of rotation (0 = 0 ~ r = fld). Since fl is trace-free, 12 is a measure of the "volume" of the universe ( h = e - 6 a ) . The physical situation requires a hypothesis on the matter content of our universe. In other works, usually, it has been assumed that the matter is represented by a scalar field [ 1,3 ]. In our case, since a scalar field cannot drive rotation, we will put in our model a pressureless fluid. Thus the hamiltonian of our system is [ 5 ] H = Gijkt~z°n kt- h 1/2
3R
+ M [ 1 + exp (2£2+ 4fl+ ) ]1/2,
(15)
where G~jkt is the minisuperspace metric [2 ], 3R is the intrinsic scalar curvature of the homogeneous hypersurfaces, and M is a constant. With Misner's parametrization, eq. (15) becomes H = ~ e x p ( 3g2)GAnpApn +exp(--g2) V(fl+, r _ ) + M [ 1 + exp (2g2+ 4fl+ ) ],/2, where 3 GAB=diag ( - 1 , 1, 1, sinh2 ~-x/~fl_) • In (16) we make use of the fact that
(16)
Volume 261, number 4
-- 3R =
PHYSICS LETTERS B 0(0) = A exp( + ikO).
exp (212). ½Tr [ exp (4#) - 2exp ( - 2//) ]
- exp (212) V(//+,//_ ) .
6 June 1991
( 17 )
It is very important to stress that 3R is given by eq. (17) also in the non-rotating case. This could be verified directly, but we are able to check it considering the relation (12) b e t w e e n / / a n d / / a , for a fixed time, as a similarity transformation which obviously does not change (17). We note also that the coordinate 0 is cyclic, thus the conjugate m o m e n t u m Po is constant both in space and time. Classically the field equation of t i m e - t i m e components is the hamiltonian constraint ( 1 )
(22)
The degree of freedom 0, corresponding to the presence of rotation in the semiclassical limit, contributes to the wave function only with a phase factor, but it leaves behind itself an additional potential term. In fact eq. (20) is a W D W equation corresponding to the classical hamiltonian
H'=Ho +G°°k 2 +24Mexp(-3g2)
[ 1 + e x p ( 2 g 2 + 4 / / + ) ]1/2.
(23)
H=0.
Close to the Planck epoch, during which quantum effects are dominant, we can safely ignore the first term in the dust potential and approximate
The quantized version of ( 1 ) is the W D W equation
2 4 M exp ( - 3 £ 2 ) [ 1 + exp (2g2+ 4p+ ) ]1/2
{Gan~A~n + 2 4 exp(--412) V(//+,//_ )
2 4 M exp ( - 212 + 2p+ ) .
+ 24M exp( - 312) [ 1 + e x p ( 2 0 + 4p+ ) ] i/2}
=0.
(18)
The kinetic term GAn~a~n presents the well-known problem of factor ordering. We have seen in section 1 that an opportune choice of this ordering allows finding an exact solution of eq. ( 18 ) in a very easy way. First of all we put
~ = ~/(t~,//+,//_)0(0).
of the H a m i l t o n - J a c o b i equation for the new hamiltonian (23), that is G,~p OS OS ---+ 2 4 e x p ( - 4 1 2 ) V(fl+,//_) Ox,~ Ox p
=0,
(24)
where x " - (O, fl+, r _ ). If we write down V(fl+, r_ ) explicitly we find
+ 2 4 M e x p ( - 312) [ 1 + exp (2g2+ 4~+ ) ] 1/2}~
V(//+, r _ ) = ½exp( - 8//+ )
(20)
The operator/qo which appears in eq. (20) is the hamiltonian for the non-rotating v a c u u m case, ~o = G"P#,d~# + 24 e x p ( - 4 g 2 ) V(//+, fl_ ),
6e~9~
(19)
{/~o + C ° ° k 2 =0.
S = iSe,
+ G°°k 2 + 2 4 M exp ( - 2g2+ 2l/+ )
Thus eq. (18) separates in /)o/~o0(0) = k 2 0 ( 0 ) ,
As explained above, in order to find a solution of eq. ( 18 ), we have to find a classical imaginary solution
(21)
where G'~P=diag( - 1, 1, 1 ) and p,-= (Pa, p+, p - ) . The general solution of eq. (19) has an oscillatory or exponential behaviour according to the sign o f k 2. Since 0 appears in Misner's parametrization as an angular variable, the former possibility is better than the latter (otherwise the wave function would have been many-valued). Therefore we have
+ exp ( 4 p + ) (cosh 4x/~ r _ - 1 ) - 2 exp ( - 2//+ )cosh 2x/~ p_ , so we look for a solution of the form ow= exp ( -- 2t2) ~ ( f l + , r _ ) + exp ( -- g2) f¢(fl+ )
+~(p_).
(25)
Substituting (25) in (24) gives / op_/ o-Z-+) + \|-Z-g--~
-4~2=24V(fl+,fl-)
,
(26) 413
Volume 261, number 4
PHYSICSLETTERSB
=
03--2-
0'
(27)
+ tV-2/t o-L-_/-
= 24M exp (2fl+), (0~
2
(28)
3k 2
Off_/' - sinh 2 2v/3 fl_ "
(29)
For the first equation we can try
~(fl+, fl_ ) =a(fl_ )exp( - 4 f l + ) +b(fl_ )exp (2fl+).
(30)
Substituting (30) in (26) we see that a particular solution is a(fl_)=l,
(31)
b (fl_) = 2 cosh 2x/~fl_.
(32)
From (31 ) and (32) we find
6 June 1991
We can see that if we take out the matter from our model (i.e. put M = 0 ) , then we recover the wave function of the non-rotating universe, that is the exponential part of expression (3 5 ) (the solution of the WDW equation for the diagonal case has been given to the author by Moncrief [ 4] ). This is not surprising because we already know that classically the field equations of time-space components for a non-diagonal 3-metric ( 11 ) cannot be satisfied in vacuum [ 5 ]. In the general (i.e. rotating) case it is interesting to note that the additional term does not contain the volume g2; furthermore its behaviour enhances the tendency of the wave function to be strongly peaked about the point fl+ = fl_ = 0, that is about isotropy. At this point it should not be difficult to extend our analysis to different minisuperspaces, at least considering the other Bianchi models. If these investigations gave analogous results, then we would be led to the conclusion that the highly machian nature of our universe appears as the most probable outcome of the Planck epoch.
0 ~ - / \ 0--~-/= lZk exp(Zfl+ ) ,
Acknowledgement therefore, if we choose for k the value
k=M, eqs. (27) and (28) can be satisfied putting simply f¢=0.
(33)
Finally integrating (29) we obtain
References
M sinh 2v/3fl_ ~. o~¢(fl_) = -~- In (1 +cosh 2x/~fl_ /
(34)
From eqs. (6), (22), (25), (30), (33) and (34) we obtain the final result
2
--MI2
~ = A e x p ( + i M 0 ) (1 sinh x/~fl+ cosh 2x/~fl_ / x e x p { - e x p ( - 2 / 2 ) [exp( - 4 f l + ) + 2 cosh 2x/~fl_ exp(2fl+ ) ] } .
414
The author would thank Professor V. Moncrief for very helpful discussions during his stay in Pavia, in particular for the hints given in the text, and Dr. R. Bergamini for valuable conversations and suggestions.
(35)
[1] R. Bergamini and G. Giampieri, Phys. Rev. D 40 (1989) 3960. [2] J.B. Hartle and S.W. Hawking,Phys. Rev. D 28 (1983) 2960. [3] S.W. Hawking and J.C. Luttrell, Phys. Lett. B 143 (1984) 83; W.A. Wright and I.G. Moss, Phys. Lett. B 154 ( 1985 ) 115; P. Amsterdamski,Phys. Rev. D 31 (1985) 3073. [4] V. Moncrief and M. Ryan, Amplitude-real-phase exact solutions for quantum mixmaster universes,preprint. [ 5 ] M. Ryan, Hamiltonian cosmology(Springer, Berlin, 1982).
PHYSICS LETTERS B
6 June 1991
A new solution of the Wheeler-De Witt equation Giacomo Giampieri Dipartimento di Fisica Teorica, Via Bassi 6, 1-27100 Pavia, Italy
Received 30 January 1991
We consider the problem of rotation in a homogeneous Bianchi type IX cosmological model. Studying the Wheeler-De Witt equation corresponding to this minisuperspace, and adopting a particular choice of the factor ordering, we are able to find a particular solution which is strongly peaked about isotropy. This result confirms all the previous investigations in this field, and suggests the conclusion that the machian nature of our universe is the natural, i.e. most likely, outcome of the Planck epoch.
I. Introduction It is generally believed that the list o f cosmological p r o b l e m s to be investigated includes the rotation o f the present-day universe. This "rotation p r o b l e m " can be f o r m u l a t e d in this very simple way: if the universe can rotate, why does it rotate so slowly? Various exp l a n a t i o n o f this p r o b l e m should be given, for example invoking M a c h ' s principle or postulating the occurrence o f an inflationary epoch, but none o f t h e m can be considered completely satisfactory. In a recent p a p e r [ 1 ] we investigated the possibility o f solving this p r o b l e m assuming Hawking's proposal for the b o u n d a r y conditions o f the universe. In particular we analyzed the wave function for a Bianchi type IX rotating minisuperspace, using the F e y n m a n path-integral a p p r o a c h to q u a n t u m gravity [21. The result was that the first o r d e r correction to the u n p e r t u r b e d (i.e. non-rotating) wave function confirms some previous qualitative or numerical results about the wave function o f an anisotropic universe [ 3 ]. In this way we were able to conclude that in the limit o f small angular velocity a general Bianchi type IX model is not m o r e likely than a F r i e d m a n n cosmological solution, in spite o f the fact that the set o f cosmological models which a p p r o a c h isotropy at infinite times is o f measure zero in the space o f all spatially homogeneous models. The p e r t u r b a t i v e expansion o f the F e y n m a n path integral was p u r s u e d in o r d e r to o v e r c o m e the m a t h e m a t i c a l difficulties en-
d o w e d with the W h e e l e r - D e W i t t ( W D W ) equation. However, from the physical point o f view, in order to solve the problem o f rotation as stated above, it would be necessary to accomplish a m o r e detailed analysis. This can be done either by allowing arbitrary values for the vorticity or by extending the minisuperspace to different 3-geometries, possibly even not homogeneous. In this short c o m m u n i c a t i o n we will present a particular exact solution o f the W D W equation for the same minisuperspace considered above, that is a Bianchi type IX universe filled with dust, but without any hypothesis about the magnitude o f the rotation. This result makes use o f an interesting method, due to M o n c r i e f and Ryan [4], to solve an equation o f the f o r m / 4 ~ = 0, given an imaginary solution o f the classical H a m i l t o n - J a c o b i equation. Let {x A, PA} be canonical variables. The hamiltonian constraint can be written as H = G A B ( x ) p a PB + 3V'(x) = O,
( 1)
while the classical H a m i l t o n - J a c o b i equation is 0S 0S
GA s - + f~(x) =0 Ox A Ox B
(2)
Suppose S = i5e is a pure imaginary solution o f ( 2 ) . Then we can write H in the form
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
41 1
Volume 26 l, number 4
H
PHYSICS LETTERS B
.-.an.. -AB 05° 05e =lr tX)PAPn+G ~XA OXB =GAB(x)Tr]ItB ,
(3)
where we have defined the new momenta
In ( 11 ) we have introduced, for later use, the Misner parametrization of the spatial metric; 12=12(t) is a scalarandfl=fl(t) is a 3 × 3 symmetric trace-free matrix. In the simplest rotating case we can in turn parametrize fl by f l = A - I fldA ,
• 05e IrA = P A - - 1 0 X A .
(4)
(The expression (3) for the hamiltonian is similar to Ashtekar's form, but does not use spinor variables. ) The quantized version ofeq. (4) is 0 .06e ~A = - i 0 - ~ - , 0x A .
(5)
~u= exp ( - S e ) ,
(6)
then it is straightforward to verify that ~n~=0.
(7)
Thus the wave function (6) is a particular solution of the WDW equation, supposing that one chooses the factor ordering with the operator ~8 on the right; for example 1
[ ~ ( ~ / I GI GA'~8)] •
(8)
Incidentally the choice (8), apart from the use of the old momenta, is the same one made in ref. [ 1 ], on the basis of minisuperspace arguments. Therefore, generally speaking, the very difficult task of solving the WDW equation is reduced to the much easier problem of finding a purely imaginary solution of the Hamilton-Jacobi equation (2).
2. M i n i s u p e r s p a c e
model
In this section we will consider a minisuperspace model very similar to other ones previously studied in the literature ([ 1,3 ] ), that is a Bianchi type IX rotating universe, whose metric is (i, j = 1, 2, 3 ) dS 2= - d t 2 + hij( t )o)io) j , d o ) i = ~ejkOY I i "^ o)k
,
h~j= e x p ( - 2 1 2 ) [exp(2fl) ]o. 412
(9) (10) (11 )
(12)
where fld=diag(--Zfl+,fl+ + x / ~ f l _ , f l + - - X / ~ f l _ ) ,
A=
If we take a wave function of the form
~=
6 June 1991
1
0
0
0
1
- s i n 20 1 + cos 20
0
1 - c o s 20 sin 20
1
(13)
(14)
The metric (9) is determined by the four functions of time 12, fl+, r _ and 0. r+ and r _ have a direct physical interpretation as a measure of the anisotropy of the universe, as can be seen by putting r + = r _ = 0. The variable 0 is the additional degree of freedom due to the presence of rotation (0 = 0 ~ r = fld). Since fl is trace-free, 12 is a measure of the "volume" of the universe ( h = e - 6 a ) . The physical situation requires a hypothesis on the matter content of our universe. In other works, usually, it has been assumed that the matter is represented by a scalar field [ 1,3 ]. In our case, since a scalar field cannot drive rotation, we will put in our model a pressureless fluid. Thus the hamiltonian of our system is [ 5 ] H = Gijkt~z°n kt- h 1/2
3R
+ M [ 1 + exp (2£2+ 4fl+ ) ]1/2,
(15)
where G~jkt is the minisuperspace metric [2 ], 3R is the intrinsic scalar curvature of the homogeneous hypersurfaces, and M is a constant. With Misner's parametrization, eq. (15) becomes H = ~ e x p ( 3g2)GAnpApn +exp(--g2) V(fl+, r _ ) + M [ 1 + exp (2g2+ 4fl+ ) ],/2, where 3 GAB=diag ( - 1 , 1, 1, sinh2 ~-x/~fl_) • In (16) we make use of the fact that
(16)
Volume 261, number 4
-- 3R =
PHYSICS LETTERS B 0(0) = A exp( + ikO).
exp (212). ½Tr [ exp (4#) - 2exp ( - 2//) ]
- exp (212) V(//+,//_ ) .
6 June 1991
( 17 )
It is very important to stress that 3R is given by eq. (17) also in the non-rotating case. This could be verified directly, but we are able to check it considering the relation (12) b e t w e e n / / a n d / / a , for a fixed time, as a similarity transformation which obviously does not change (17). We note also that the coordinate 0 is cyclic, thus the conjugate m o m e n t u m Po is constant both in space and time. Classically the field equation of t i m e - t i m e components is the hamiltonian constraint ( 1 )
(22)
The degree of freedom 0, corresponding to the presence of rotation in the semiclassical limit, contributes to the wave function only with a phase factor, but it leaves behind itself an additional potential term. In fact eq. (20) is a W D W equation corresponding to the classical hamiltonian
H'=Ho +G°°k 2 +24Mexp(-3g2)
[ 1 + e x p ( 2 g 2 + 4 / / + ) ]1/2.
(23)
H=0.
Close to the Planck epoch, during which quantum effects are dominant, we can safely ignore the first term in the dust potential and approximate
The quantized version of ( 1 ) is the W D W equation
2 4 M exp ( - 3 £ 2 ) [ 1 + exp (2g2+ 4p+ ) ]1/2
{Gan~A~n + 2 4 exp(--412) V(//+,//_ )
2 4 M exp ( - 212 + 2p+ ) .
+ 24M exp( - 312) [ 1 + e x p ( 2 0 + 4p+ ) ] i/2}
=0.
(18)
The kinetic term GAn~a~n presents the well-known problem of factor ordering. We have seen in section 1 that an opportune choice of this ordering allows finding an exact solution of eq. ( 18 ) in a very easy way. First of all we put
~ = ~/(t~,//+,//_)0(0).
of the H a m i l t o n - J a c o b i equation for the new hamiltonian (23), that is G,~p OS OS ---+ 2 4 e x p ( - 4 1 2 ) V(fl+,//_) Ox,~ Ox p
=0,
(24)
where x " - (O, fl+, r _ ). If we write down V(fl+, r_ ) explicitly we find
+ 2 4 M e x p ( - 312) [ 1 + exp (2g2+ 4~+ ) ] 1/2}~
V(//+, r _ ) = ½exp( - 8//+ )
(20)
The operator/qo which appears in eq. (20) is the hamiltonian for the non-rotating v a c u u m case, ~o = G"P#,d~# + 24 e x p ( - 4 g 2 ) V(//+, fl_ ),
6e~9~
(19)
{/~o + C ° ° k 2 =0.
S = iSe,
+ G°°k 2 + 2 4 M exp ( - 2g2+ 2l/+ )
Thus eq. (18) separates in /)o/~o0(0) = k 2 0 ( 0 ) ,
As explained above, in order to find a solution of eq. ( 18 ), we have to find a classical imaginary solution
(21)
where G'~P=diag( - 1, 1, 1 ) and p,-= (Pa, p+, p - ) . The general solution of eq. (19) has an oscillatory or exponential behaviour according to the sign o f k 2. Since 0 appears in Misner's parametrization as an angular variable, the former possibility is better than the latter (otherwise the wave function would have been many-valued). Therefore we have
+ exp ( 4 p + ) (cosh 4x/~ r _ - 1 ) - 2 exp ( - 2//+ )cosh 2x/~ p_ , so we look for a solution of the form ow= exp ( -- 2t2) ~ ( f l + , r _ ) + exp ( -- g2) f¢(fl+ )
+~(p_).
(25)
Substituting (25) in (24) gives / op_/ o-Z-+) + \|-Z-g--~
-4~2=24V(fl+,fl-)
,
(26) 413
Volume 261, number 4
PHYSICSLETTERSB
=
03--2-
0'
(27)
+ tV-2/t o-L-_/-
= 24M exp (2fl+), (0~
2
(28)
3k 2
Off_/' - sinh 2 2v/3 fl_ "
(29)
For the first equation we can try
~(fl+, fl_ ) =a(fl_ )exp( - 4 f l + ) +b(fl_ )exp (2fl+).
(30)
Substituting (30) in (26) we see that a particular solution is a(fl_)=l,
(31)
b (fl_) = 2 cosh 2x/~fl_.
(32)
From (31 ) and (32) we find
6 June 1991
We can see that if we take out the matter from our model (i.e. put M = 0 ) , then we recover the wave function of the non-rotating universe, that is the exponential part of expression (3 5 ) (the solution of the WDW equation for the diagonal case has been given to the author by Moncrief [ 4] ). This is not surprising because we already know that classically the field equations of time-space components for a non-diagonal 3-metric ( 11 ) cannot be satisfied in vacuum [ 5 ]. In the general (i.e. rotating) case it is interesting to note that the additional term does not contain the volume g2; furthermore its behaviour enhances the tendency of the wave function to be strongly peaked about the point fl+ = fl_ = 0, that is about isotropy. At this point it should not be difficult to extend our analysis to different minisuperspaces, at least considering the other Bianchi models. If these investigations gave analogous results, then we would be led to the conclusion that the highly machian nature of our universe appears as the most probable outcome of the Planck epoch.
0 ~ - / \ 0--~-/= lZk exp(Zfl+ ) ,
Acknowledgement therefore, if we choose for k the value
k=M, eqs. (27) and (28) can be satisfied putting simply f¢=0.
(33)
Finally integrating (29) we obtain
References
M sinh 2v/3fl_ ~. o~¢(fl_) = -~- In (1 +cosh 2x/~fl_ /
(34)
From eqs. (6), (22), (25), (30), (33) and (34) we obtain the final result
2
--MI2
~ = A e x p ( + i M 0 ) (1 sinh x/~fl+ cosh 2x/~fl_ / x e x p { - e x p ( - 2 / 2 ) [exp( - 4 f l + ) + 2 cosh 2x/~fl_ exp(2fl+ ) ] } .
414
The author would thank Professor V. Moncrief for very helpful discussions during his stay in Pavia, in particular for the hints given in the text, and Dr. R. Bergamini for valuable conversations and suggestions.
(35)
[1] R. Bergamini and G. Giampieri, Phys. Rev. D 40 (1989) 3960. [2] J.B. Hartle and S.W. Hawking,Phys. Rev. D 28 (1983) 2960. [3] S.W. Hawking and J.C. Luttrell, Phys. Lett. B 143 (1984) 83; W.A. Wright and I.G. Moss, Phys. Lett. B 154 ( 1985 ) 115; P. Amsterdamski,Phys. Rev. D 31 (1985) 3073. [4] V. Moncrief and M. Ryan, Amplitude-real-phase exact solutions for quantum mixmaster universes,preprint. [ 5 ] M. Ryan, Hamiltonian cosmology(Springer, Berlin, 1982).