- Email: [email protected]
Supersymmetric Bianchi class A models
Physics Letters B 314 (1993) 303-307 North-Holland
PHYSICS LETTERS B
Supersymmetric Bianchi class A models Masako Asano ~, Masayuki Tanimoto 2 and Noriaki Yoshino
Department of Physics, Tokyo Institute of Technology, Oh-Okayama Meguroku, Tokyo 152, Japan Received 8 June 1993; revised manuscript received 31 July 1993 Editor: H. Georgi
The canonical theory of N = 1 supergravity is applied to Bianchi class A spatially homogeneous cosmologies. The full set of quantum constraints are then solved with the possible ordering ambiguity taken into account by introducing a free parameter. The wave functions are explicitly given for all the Bianchi class A models in a unified way. Some comments are made on the Bianchi type IX cases.
1. I n t r o d u c t i o n
Recently, supersymmetry has been attracting many people in particle physics (see, e.g., ref. [ I ] ). It seems worth studying supersymmetric q u a n t u m cosmology by expecting the existence of supersymmetry in the early universe. Mini-superspace models, i.e. spatially homogeneous relativistic models, have been helpful to understand q u a n t u m cosmology. Among them, those of Bianchi class A [2] (its definition is explained in section 2) are of particular importance when one investigates them in canonical form, because for Bianchi class "B" models, the rest of the Bianchi types, there is an inconvenience that the equations of motion reduced from the full Einstein equations differ from the ones derived from the reduced action. Since the origin of this disaccord is purely geometrical we expect the same problem will happen also in supergravity. So we restrict our attention to the supersymmetric extension of the class A models. Special cases of such models have been investigated by some authors. G r a h a m studied Bianchi type IX in refs. [3,4], where the full q u a n t u m constraints are not explicitly solved. D'Eath, Hawking and Obreg6n [5] #l solved the Bianchi type I case, but the Bianchi type I super-
l E-mail address: [email protected]. 2 E-mail address: [email protected]. #1 Our result presented below disagrees with ref. [5].
symmetric model is so simple that the wave function does not have any interesting structure #2. We shall first restrict the canonical theory of supergravity proposed by D'Eath [6] to spatially homogeneous cases, in particular, Bianchi class A models, and then solve the q u a n t u m constraints. Then we will obtain the wave function
~u =
const, hS/2exP ( ½7mabhab)
+const. h~2-s)/2 exp (-½Ymabhab)
(~Aa~l/Aa)3, (1)
where hab is a spatial metric with respect to an invariant basis, h its determinant and m ab the structure constants of a Bianchi class A group. Here, ~, is a real constant and ~UAaa spin -3 field. The parameter s specifies the ambiguity of the operator ordering [see eqs. (22), (23) ]. This is a direct generalization of the solutions found in the literature [3-5 ].
#2
After the first submission of this paper it came to our attention that related work [ 12,13] by some other authors had appeared. They treat the Bianchi type IX cases and the type II cases, respectively. Here, we emphasize it is worth studying all class A models even when considering compactification [14], contrary to the reason explained in ref. [ 12 ] for which the author did not study Bianchi models except types I and IX.
0370-2693/93/$ 06.00 (~) 1993-Elsevier Science Publishers B.V. All rights reserved
303
Volume 314, number 3,4
PHYSICS LETTERS B
23 September 1993
The metric-like matrix hab defined in (3) is associated with QAA'a through
2. Spatially homogeneous Bianchi class A models We write the four-metric as
hat, ds 2 = (NiNi - N 2 ) d t 2 + 2 N i d t d x i +hij d x i d x j,
(2)
where N is the lapse function, N i the shift vector, and hij the spatial metric. In spatially homogeneous cases, hij is expanded by an invariant basis of a Bianchi group, (3)
hij = h a b ( l ) E a i E b j ,
where the indices a, b . . . . as well as the world indices i, j . . . . run from 1 to 3. The invariant basis Eai obeys the following Maurer-Cartan relation: OliEajl = CabcEbiECj,
(5)
In supergravity, the basic variables are taken to be the spinor version of the tetrad components eAa'u and the spinor valued forms ~uAu,~a'U (for spinor conventions, see ref. [6] ). eAA'u are Grassmann even and ~uAu and v A ' are Grassmann odd. In accordance with (2), we take eAa' 0 = N n AA' + N i e A A ' i .
(9)
-- ~'2AA' aff'2AAt b.
We will u s e hab and its inverse h ab to raise and lower the indices a, b, c , . . . . Further, the spin -3 field ~nu will take the following spatially homogeneous form: ~tAo = ~ t A 0 ( t ) ,
~ A i = ~ A a ( t ) E a i.
(10)
Similar restrictions are to be imposed on ~A' u. The full theory of supergravity reduces to a theory with a finite number of degrees of freedom and its Lagrangian is given by the integration of the Lagrangian density £ ( t , x ) over the spatial coordinates, L(t) =
f d3xC(t,x),
(ll)
(4)
where C%c are the structure constants of the Bianchi group. C%c is antisymmetric in b and c and satisfies the Jacobi identity Cab[cCbde I ---- 0. We can decompose Cabc a s Cabc = madedbc + O~bac], where m ad is symmetric and ac = Caac . The Bianchi class A models are characterized by ac = 0, so that we have Cabc -~- mad edbo
=
(6)
with N, Na, ~-2AA'a, IllAa, ~tlAo, ~A'a, and ~A'0 being the variables of the reduced theory. The reduced action S =- f d t L ( t ) remains invariant under the three homogeneity preserving local transformations Lorentz, coordinate, and supersymmetry transformations. Thus, it is possible to get the reduced Hamiltonian H (t) by integrating the Hamiltonian density. This results in a form formally equivalent to the original one: H ( t ) = N H ± + N a a a + ~ A o S A + -SA,VA'O _ --ArB ~ --O)ABO JAB -- O)A,B,OJ .
(12)
Here, N, N a, ~aO, ~A'o, ¢-.OABO,and (OA,B,O act as Lagrange multipliers which enforce the constraints H i = O, Ha = O, SA = O,-SA, = O, jAB = 0, and 7 ~'B' = o.
Here, nan, is the spinor version of the unit vector normal to the surface t =const., and it is defined by nAA,eAA'i = O,
rlAA, rlAA' = 1.
(7)
From homogeneity the variables N, Ni, and eAA'i are restricted to
Using the same procedure as shown in ref. [6], we can perform canonical quantization of this system. A quantum state is described by a wave function ~, which we choose as an eigenstate of tuAa and ~AA'a, i.e., ~V = ~-,t(QAAta,~llAa)" Corresponding to this choice, W A ' is given by --A'
N = N(t),
eaA' i = g2AA'a(t)Ea i. 304
~b¢ a =
Ni = Na(t)Eai,
(8)
_ ihDa A,
0
ba o lff A b
,
( 13 )
and the momentum conjugate t o ~-2aA'a, PaA, a, by
Volume 3 14,
PHYSICS
number 3,4
a
PAA!’ = -ifi=
LETTERS
sA =
23 September
B
1993
umub~AA’a~A’b
B
a + $ifiuEabcyAbDBA’dc, aWBd
where D AA’
ab
=
+
&h.bnAA’
;f,b,Q
AA’c
(15)
and u = 5 d3x E. We have chosen the operator ordering in PAA!’ so that it becomes hermitian with respect to the formal inner product [6]. Note that we can also take the wave function as an eigenstate of VA’* and S;ZAAiO:Y = Y(QAA’n,WA’a), which relates to Y (QAA’a, vA,) by the fermionic Fourier transform = D-‘(Q)
Y(QAA’a,WA’,)
s
(1 - .S)PAA!‘~~‘,J + .S~A’~P~~~a
(14)
where K* = 8n and s parametrizes the ambiguity of the operator ordering, which comes from noncommutativity of WAa, VA’a and pAAjn. Note that the orderings in refs. [ 6 ] and [ 51 can be obtained by taking s to be 0 and i, respectively. Now we write, without specifying the value of s, the constraint equations (20) in an explicit form
+
SlyAOSZ~~la Y = >
-ymabQAA’aDBA’cbT +&
where ab =
D(Q)
abc -06
= det
(
-~CAA~
ab
.
(18)
>
With this relation, we can obtain Y from Y, and vice versa. Physically allowed wave functions are obtained by solving quantum versions of the constraint equations: JAB!? = 0 , SAY==,
-sQAA~‘D
(17)
>
QAA’c
TA’BJY=O, s,,Y=O, H,Y=O.
BA’
(25)
-ymab&,&AB’b
a CdWB)c
( a
(19)
-m
(20)
a
AB’
(
D
abdvBlb >
(22)
!P = 0,
(2 - s)~,&DAB’Bb-$ aw
(21)
Becauseofthe relation {SA,~A~} 3: H~rzA~l+ H~OAAI~ up to terms proportional to J and 7, eqs. (21) hold automatically if the conditions ( 19) and (20) are satisfied. The constraints ( 19) describe the invariance of Y under Lorentz transformations. We therefore only have to solve SAY = 0 and ??A’Y = 0 under the assumption that Y is Lorentz invariant. Let us write the explicit forms of SA and S,+ :
+ fiK* (SpAA’UV/A,+ (1 - S)WA~p,Wa) ,
(DBAkd;
where y E 2o/fz~*.Moreover, using the Fourier transform (16), eqs. (24) and (25) can be recast into the conditions for Y(QAA’,,vA’a):
+
HlY=O,
(24)
0,
Y(QAA’II, WA,) (16)
CAA’
(23)
(26)
b
d ymabQAA’avA’
b -
v”‘a
m
(
(2 +S~A’aQAACa
)
P
=
0.
(27)
3. Solving quantum constraints Because of the Lorentz invariant property, Y can only contain an even number of vAa. Thus, we can decompose Y into I,v’, t,u*, w4, @ parts, which we write as Yo, Y2, Y4, Y6, respectively, so that we have 305
Volume 314, number 3,4
PHYSICS LETTERS B
= ~0 + ~2 + ~4 + ~u6. These terms are related to the o f ~' by the Fourier transform, respectively. We write ~ = ~'6 + ~4 + 5P2 + ~'0, where the subscripts represent the number of~A'~ contained in each term. We shall give these terms separately by solving the constraint equations along lines similar to ref. [5]. F o r ~u0, eq. (25) is trivial because ~0 has no fermionic variables. Bearing in m i n d that Lorentz invariance imposes a condition ~0 = ~o(h~b), eq. (24) reduces to
~-6, ~4, ~2, ~-0 parts
( 7m~b _2 0__~ Ohab "{-shab) ~10 =0,
(28)
SO that we obtain
~o = const, hS/2exP(½Ymabhab),
~v6 = const, h (2-s)/2
× exp (--½~mabhab) (~lAa~'Aa) 3,
~U2 = 0
(30)
as solutions except for s = O, - 4 / 3 and m ab = O. In these exceptional cases, there are the formal solutions
where ~/Aa~lAa = IgAa~ZAbt~ ab. It is rather remarkable that all the solutions ( 1 ) for Bianchi class A models are found in a unified way. We emphasize that these solutions are actually exact. Recently, D'Eath claimed that the full N = 1 supergravity has no quantum corrections [7]. The reduced N = 1 supergravity presented above also has no quantum corrections and looks like a free theory. If one wants to see the effect of interactions, one probably has to explore N = 1 supergravity coupled to supermatter.
4. An example: Bianchi type IX ease
One specific case would be o f particular interest, i.e. Bianchi type IX representing a homogeneous S 3 universe. Giving the invariant basis for it as E l = s i n f l c o s y d c ~ - sinTdfl,
E2 (s = 0),
= sin fl sin y d~ + cos y dfl,
(31) E 3 = c o s f l d a + dT,
~v2 = const. (~u'2 + 2~u''2) where ~u'2
=
(s = - 4 / 3 ) ,
~Aa~lyAaand g ''2
-
ih-l/2eabc~"2AA,c ×
~4 = 0.
(33)
Although there are also exceptional formal solutions when s = 0, 10/3 and m ab = 0, we discard them for the same reason as above. For ~0, the solution is easily found to be ~0 = const, h -s/2 exp(--½~'mabhab) as in the case of ~0. In other words, we have
(36)
(32)
nB'¥~dal/IB b. However, since they cease to be solutions when considering a small perturbation, we exclude them from our consideration. For ~4, or equivalently, ~2, using a similar method to the case of~2, eqs. (26) and (27) can be solved to yield ~2 = 0, i.e.,
306
(35)
(29)
where h = det(h~b). Next, let us look at ~ . Solving eqs. (24) and (25) under the assumption that ~u2 is Lorentz invariant, we get
~v2 = const.~"2
23 September 1993
(34)
w h e r e E a - Eaidx i, x I = a, x 2 = f l a n d x 3 = y, the structure constants are given by m ab = diag(1, l, 1 ). A n d taking the range of variables as ( 0 ~ < a ~<4zr,
0~< fl ~< zr,
0~<9,~< 2rr),
(37)
we can easily find cr = 167r2, so that we have
I = ½Ymabhab = 2---ffTr(hab).
(38)
For the semi-classical forms of the quantum states, it seems natural to ask if the states ~v0 and ~v6 are the H a r t l e - H a w k i n g state [8 ] and the wormhole state [9], respectively. To answer this question, we note that vacuum Bianchi type IX metrics are diagonalizable [2 ]. In such cases it is fortunately known [ 10,11 ] that the asymptotically Euclidian four-metrics which correspond to the classical flow o f I are found to be
PHYSICS LETTERS B
Volume 314, number 3,4
23 September 1993
Acknowledgement
/94'] We would like to thank Prof. A. Hosoya for helpful discussions and Dr. P.D. D'Eath for correspondence. -- /94']
(E2)2 +
/94']
' (39)
where g = (1-
aa14~ 4 ( 1~ - , p4'] a24~ O 4( 1 -] p4']
(40)
and a~,a2, a3 are constants. At large distances, the metrics (39) are well-behaved and ~6 dumps rapidly. Hence ~6 is certainly the ground quantum wormhole state. On the other hand, though ~0 itself is regular at small three-geometries, the classical flow of I is singular when p ~ 0. This means the quantum state ~0 is not the Hartle-Hawking state [12]. The "filled" state ~6 corresponds to the wormhole state in our description above and this seems to be in contradiction with the literature [3,4,12], where the bosonic state ~0 corresponds to the wormhole state. However, another possibility exists because of the indefiniteness of the sign of a, which is defined as f d3x E, not f d3x IEI. In fact, by the following antipodal transformation fl~fl+~z,
(41)
we have a ~ - a , m ab ~ m ab, so that I ~ - I . Correspondingly this time the bosonic state ~0 becomes the wormhole state. We conclude that there are two distinct solutions of the constraints in the case of Bianchi type IX, which are mutually related by the antipodal transformation.
References [1] K. Hagiwara and S. Komamiya, in: High energy electron-positron physics, eds. A. All and P. S6ding (World Scientific, Singapore, 1988). [2] M.P. Ryan and L.C. Shepley, Homogeneous relativistic cosmologies (Princeton U.P., Princeton, NJ, 1975). [3] R. Graham, Phys. Rev. Lett. 67 (1991) 1381. [4] R. Graham, Phys. Lett. B 277 (1992) 393. [5] P.D. D'Eath, S.W. Hawking and O. Obreg6n, Phys. Lett. B 300 (1993) 44. [6] P.D. D'Eath, Phys. Rev. D 29 (1984) 2199. [7] P.D. D'Eath, preprint, Physical states in N = 1 supergravity ( 1993 ). [ 8 ] J.B. Hartle and S.W. Hawking, Phys. Rev. D 28 ( 1983 ) 2960. [9] S.W. Hawking and D.N. Page, Phys. Rev. D 42 (1990) 2655. [10] V.A. Belinskii, G.W. Gibbons, D.N. Page and C.N. Pope, Phys. Lett. B 76 (1978) 433. [ l 1] G.W. Gibbons and C.N. Pope, Commun, Math. Phys. 66 (1979) 267. [ 12] P.D. D'Eath, preprint, Quantization of the Bianchi IX model in supergravity (1993), to appear in Phys. Rev. D. [13] O. Obreg6n, J. Socorro and J. Ben~tez, Phys. Rev. D 47 (1993) 4471. [ 14 ] T. Koike, M. Tanimoto and A. Hosoya, preprint, Compact homogeneous universes, TIT/HEP208/COSMO-26 (1992).
307
PHYSICS LETTERS B
Supersymmetric Bianchi class A models Masako Asano ~, Masayuki Tanimoto 2 and Noriaki Yoshino
Department of Physics, Tokyo Institute of Technology, Oh-Okayama Meguroku, Tokyo 152, Japan Received 8 June 1993; revised manuscript received 31 July 1993 Editor: H. Georgi
The canonical theory of N = 1 supergravity is applied to Bianchi class A spatially homogeneous cosmologies. The full set of quantum constraints are then solved with the possible ordering ambiguity taken into account by introducing a free parameter. The wave functions are explicitly given for all the Bianchi class A models in a unified way. Some comments are made on the Bianchi type IX cases.
1. I n t r o d u c t i o n
Recently, supersymmetry has been attracting many people in particle physics (see, e.g., ref. [ I ] ). It seems worth studying supersymmetric q u a n t u m cosmology by expecting the existence of supersymmetry in the early universe. Mini-superspace models, i.e. spatially homogeneous relativistic models, have been helpful to understand q u a n t u m cosmology. Among them, those of Bianchi class A [2] (its definition is explained in section 2) are of particular importance when one investigates them in canonical form, because for Bianchi class "B" models, the rest of the Bianchi types, there is an inconvenience that the equations of motion reduced from the full Einstein equations differ from the ones derived from the reduced action. Since the origin of this disaccord is purely geometrical we expect the same problem will happen also in supergravity. So we restrict our attention to the supersymmetric extension of the class A models. Special cases of such models have been investigated by some authors. G r a h a m studied Bianchi type IX in refs. [3,4], where the full q u a n t u m constraints are not explicitly solved. D'Eath, Hawking and Obreg6n [5] #l solved the Bianchi type I case, but the Bianchi type I super-
l E-mail address: [email protected]. 2 E-mail address: [email protected]. #1 Our result presented below disagrees with ref. [5].
symmetric model is so simple that the wave function does not have any interesting structure #2. We shall first restrict the canonical theory of supergravity proposed by D'Eath [6] to spatially homogeneous cases, in particular, Bianchi class A models, and then solve the q u a n t u m constraints. Then we will obtain the wave function
~u =
const, hS/2exP ( ½7mabhab)
+const. h~2-s)/2 exp (-½Ymabhab)
(~Aa~l/Aa)3, (1)
where hab is a spatial metric with respect to an invariant basis, h its determinant and m ab the structure constants of a Bianchi class A group. Here, ~, is a real constant and ~UAaa spin -3 field. The parameter s specifies the ambiguity of the operator ordering [see eqs. (22), (23) ]. This is a direct generalization of the solutions found in the literature [3-5 ].
#2
After the first submission of this paper it came to our attention that related work [ 12,13] by some other authors had appeared. They treat the Bianchi type IX cases and the type II cases, respectively. Here, we emphasize it is worth studying all class A models even when considering compactification [14], contrary to the reason explained in ref. [ 12 ] for which the author did not study Bianchi models except types I and IX.
0370-2693/93/$ 06.00 (~) 1993-Elsevier Science Publishers B.V. All rights reserved
303
Volume 314, number 3,4
PHYSICS LETTERS B
23 September 1993
The metric-like matrix hab defined in (3) is associated with QAA'a through
2. Spatially homogeneous Bianchi class A models We write the four-metric as
hat, ds 2 = (NiNi - N 2 ) d t 2 + 2 N i d t d x i +hij d x i d x j,
(2)
where N is the lapse function, N i the shift vector, and hij the spatial metric. In spatially homogeneous cases, hij is expanded by an invariant basis of a Bianchi group, (3)
hij = h a b ( l ) E a i E b j ,
where the indices a, b . . . . as well as the world indices i, j . . . . run from 1 to 3. The invariant basis Eai obeys the following Maurer-Cartan relation: OliEajl = CabcEbiECj,
(5)
In supergravity, the basic variables are taken to be the spinor version of the tetrad components eAa'u and the spinor valued forms ~uAu,~a'U (for spinor conventions, see ref. [6] ). eAA'u are Grassmann even and ~uAu and v A ' are Grassmann odd. In accordance with (2), we take eAa' 0 = N n AA' + N i e A A ' i .
(9)
-- ~'2AA' aff'2AAt b.
We will u s e hab and its inverse h ab to raise and lower the indices a, b, c , . . . . Further, the spin -3 field ~nu will take the following spatially homogeneous form: ~tAo = ~ t A 0 ( t ) ,
~ A i = ~ A a ( t ) E a i.
(10)
Similar restrictions are to be imposed on ~A' u. The full theory of supergravity reduces to a theory with a finite number of degrees of freedom and its Lagrangian is given by the integration of the Lagrangian density £ ( t , x ) over the spatial coordinates, L(t) =
f d3xC(t,x),
(ll)
(4)
where C%c are the structure constants of the Bianchi group. C%c is antisymmetric in b and c and satisfies the Jacobi identity Cab[cCbde I ---- 0. We can decompose Cabc a s Cabc = madedbc + O~bac], where m ad is symmetric and ac = Caac . The Bianchi class A models are characterized by ac = 0, so that we have Cabc -~- mad edbo
=
(6)
with N, Na, ~-2AA'a, IllAa, ~tlAo, ~A'a, and ~A'0 being the variables of the reduced theory. The reduced action S =- f d t L ( t ) remains invariant under the three homogeneity preserving local transformations Lorentz, coordinate, and supersymmetry transformations. Thus, it is possible to get the reduced Hamiltonian H (t) by integrating the Hamiltonian density. This results in a form formally equivalent to the original one: H ( t ) = N H ± + N a a a + ~ A o S A + -SA,VA'O _ --ArB ~ --O)ABO JAB -- O)A,B,OJ .
(12)
Here, N, N a, ~aO, ~A'o, ¢-.OABO,and (OA,B,O act as Lagrange multipliers which enforce the constraints H i = O, Ha = O, SA = O,-SA, = O, jAB = 0, and 7 ~'B' = o.
Here, nan, is the spinor version of the unit vector normal to the surface t =const., and it is defined by nAA,eAA'i = O,
rlAA, rlAA' = 1.
(7)
From homogeneity the variables N, Ni, and eAA'i are restricted to
Using the same procedure as shown in ref. [6], we can perform canonical quantization of this system. A quantum state is described by a wave function ~, which we choose as an eigenstate of tuAa and ~AA'a, i.e., ~V = ~-,t(QAAta,~llAa)" Corresponding to this choice, W A ' is given by --A'
N = N(t),
eaA' i = g2AA'a(t)Ea i. 304
~b¢ a =
Ni = Na(t)Eai,
(8)
_ ihDa A,
0
ba o lff A b
,
( 13 )
and the momentum conjugate t o ~-2aA'a, PaA, a, by
Volume 3 14,
PHYSICS
number 3,4
a
PAA!’ = -ifi=
LETTERS
sA =
23 September
B
1993
umub~AA’a~A’b
B
a + $ifiuEabcyAbDBA’dc, aWBd
where D AA’
ab
=
+
&h.bnAA’
;f,b,Q
AA’c
(15)
and u = 5 d3x E. We have chosen the operator ordering in PAA!’ so that it becomes hermitian with respect to the formal inner product [6]. Note that we can also take the wave function as an eigenstate of VA’* and S;ZAAiO:Y = Y(QAA’n,WA’a), which relates to Y (QAA’a, vA,) by the fermionic Fourier transform = D-‘(Q)
Y(QAA’a,WA’,)
s
(1 - .S)PAA!‘~~‘,J + .S~A’~P~~~a
(14)
where K* = 8n and s parametrizes the ambiguity of the operator ordering, which comes from noncommutativity of WAa, VA’a and pAAjn. Note that the orderings in refs. [ 6 ] and [ 51 can be obtained by taking s to be 0 and i, respectively. Now we write, without specifying the value of s, the constraint equations (20) in an explicit form
+
SlyAOSZ~~la Y = >
-ymabQAA’aDBA’cbT +&
where ab =
D(Q)
abc -06
= det
(
-~CAA~
ab
.
(18)
>
With this relation, we can obtain Y from Y, and vice versa. Physically allowed wave functions are obtained by solving quantum versions of the constraint equations: JAB!? = 0 , SAY==,
-sQAA~‘D
(17)
>
QAA’c
TA’BJY=O, s,,Y=O, H,Y=O.
BA’
(25)
-ymab&,&AB’b
a CdWB)c
( a
(19)
-m
(20)
a
AB’
(
D
abdvBlb >
(22)
!P = 0,
(2 - s)~,&DAB’Bb-$ aw
(21)
Becauseofthe relation {SA,~A~} 3: H~rzA~l+ H~OAAI~ up to terms proportional to J and 7, eqs. (21) hold automatically if the conditions ( 19) and (20) are satisfied. The constraints ( 19) describe the invariance of Y under Lorentz transformations. We therefore only have to solve SAY = 0 and ??A’Y = 0 under the assumption that Y is Lorentz invariant. Let us write the explicit forms of SA and S,+ :
+ fiK* (SpAA’UV/A,+ (1 - S)WA~p,Wa) ,
(DBAkd;
where y E 2o/fz~*.Moreover, using the Fourier transform (16), eqs. (24) and (25) can be recast into the conditions for Y(QAA’,,vA’a):
+
HlY=O,
(24)
0,
Y(QAA’II, WA,) (16)
CAA’
(23)
(26)
b
d ymabQAA’avA’
b -
v”‘a
m
(
(2 +S~A’aQAACa
)
P
=
0.
(27)
3. Solving quantum constraints Because of the Lorentz invariant property, Y can only contain an even number of vAa. Thus, we can decompose Y into I,v’, t,u*, w4, @ parts, which we write as Yo, Y2, Y4, Y6, respectively, so that we have 305
Volume 314, number 3,4
PHYSICS LETTERS B
= ~0 + ~2 + ~4 + ~u6. These terms are related to the o f ~' by the Fourier transform, respectively. We write ~ = ~'6 + ~4 + 5P2 + ~'0, where the subscripts represent the number of~A'~ contained in each term. We shall give these terms separately by solving the constraint equations along lines similar to ref. [5]. F o r ~u0, eq. (25) is trivial because ~0 has no fermionic variables. Bearing in m i n d that Lorentz invariance imposes a condition ~0 = ~o(h~b), eq. (24) reduces to
~-6, ~4, ~2, ~-0 parts
( 7m~b _2 0__~ Ohab "{-shab) ~10 =0,
(28)
SO that we obtain
~o = const, hS/2exP(½Ymabhab),
~v6 = const, h (2-s)/2
× exp (--½~mabhab) (~lAa~'Aa) 3,
~U2 = 0
(30)
as solutions except for s = O, - 4 / 3 and m ab = O. In these exceptional cases, there are the formal solutions
where ~/Aa~lAa = IgAa~ZAbt~ ab. It is rather remarkable that all the solutions ( 1 ) for Bianchi class A models are found in a unified way. We emphasize that these solutions are actually exact. Recently, D'Eath claimed that the full N = 1 supergravity has no quantum corrections [7]. The reduced N = 1 supergravity presented above also has no quantum corrections and looks like a free theory. If one wants to see the effect of interactions, one probably has to explore N = 1 supergravity coupled to supermatter.
4. An example: Bianchi type IX ease
One specific case would be o f particular interest, i.e. Bianchi type IX representing a homogeneous S 3 universe. Giving the invariant basis for it as E l = s i n f l c o s y d c ~ - sinTdfl,
E2 (s = 0),
= sin fl sin y d~ + cos y dfl,
(31) E 3 = c o s f l d a + dT,
~v2 = const. (~u'2 + 2~u''2) where ~u'2
=
(s = - 4 / 3 ) ,
~Aa~lyAaand g ''2
-
ih-l/2eabc~"2AA,c ×
~4 = 0.
(33)
Although there are also exceptional formal solutions when s = 0, 10/3 and m ab = 0, we discard them for the same reason as above. For ~0, the solution is easily found to be ~0 = const, h -s/2 exp(--½~'mabhab) as in the case of ~0. In other words, we have
(36)
(32)
nB'¥~dal/IB b. However, since they cease to be solutions when considering a small perturbation, we exclude them from our consideration. For ~4, or equivalently, ~2, using a similar method to the case of~2, eqs. (26) and (27) can be solved to yield ~2 = 0, i.e.,
306
(35)
(29)
where h = det(h~b). Next, let us look at ~ . Solving eqs. (24) and (25) under the assumption that ~u2 is Lorentz invariant, we get
~v2 = const.~"2
23 September 1993
(34)
w h e r e E a - Eaidx i, x I = a, x 2 = f l a n d x 3 = y, the structure constants are given by m ab = diag(1, l, 1 ). A n d taking the range of variables as ( 0 ~ < a ~<4zr,
0~< fl ~< zr,
0~<9,~< 2rr),
(37)
we can easily find cr = 167r2, so that we have
I = ½Ymabhab = 2---ffTr(hab).
(38)
For the semi-classical forms of the quantum states, it seems natural to ask if the states ~v0 and ~v6 are the H a r t l e - H a w k i n g state [8 ] and the wormhole state [9], respectively. To answer this question, we note that vacuum Bianchi type IX metrics are diagonalizable [2 ]. In such cases it is fortunately known [ 10,11 ] that the asymptotically Euclidian four-metrics which correspond to the classical flow o f I are found to be
PHYSICS LETTERS B
Volume 314, number 3,4
23 September 1993
Acknowledgement
/94'] We would like to thank Prof. A. Hosoya for helpful discussions and Dr. P.D. D'Eath for correspondence. -- /94']
(E2)2 +
/94']
' (39)
where g = (1-
aa14~ 4 ( 1~ - , p4'] a24~ O 4( 1 -] p4']
(40)
and a~,a2, a3 are constants. At large distances, the metrics (39) are well-behaved and ~6 dumps rapidly. Hence ~6 is certainly the ground quantum wormhole state. On the other hand, though ~0 itself is regular at small three-geometries, the classical flow of I is singular when p ~ 0. This means the quantum state ~0 is not the Hartle-Hawking state [12]. The "filled" state ~6 corresponds to the wormhole state in our description above and this seems to be in contradiction with the literature [3,4,12], where the bosonic state ~0 corresponds to the wormhole state. However, another possibility exists because of the indefiniteness of the sign of a, which is defined as f d3x E, not f d3x IEI. In fact, by the following antipodal transformation fl~fl+~z,
(41)
we have a ~ - a , m ab ~ m ab, so that I ~ - I . Correspondingly this time the bosonic state ~0 becomes the wormhole state. We conclude that there are two distinct solutions of the constraints in the case of Bianchi type IX, which are mutually related by the antipodal transformation.
References [1] K. Hagiwara and S. Komamiya, in: High energy electron-positron physics, eds. A. All and P. S6ding (World Scientific, Singapore, 1988). [2] M.P. Ryan and L.C. Shepley, Homogeneous relativistic cosmologies (Princeton U.P., Princeton, NJ, 1975). [3] R. Graham, Phys. Rev. Lett. 67 (1991) 1381. [4] R. Graham, Phys. Lett. B 277 (1992) 393. [5] P.D. D'Eath, S.W. Hawking and O. Obreg6n, Phys. Lett. B 300 (1993) 44. [6] P.D. D'Eath, Phys. Rev. D 29 (1984) 2199. [7] P.D. D'Eath, preprint, Physical states in N = 1 supergravity ( 1993 ). [ 8 ] J.B. Hartle and S.W. Hawking, Phys. Rev. D 28 ( 1983 ) 2960. [9] S.W. Hawking and D.N. Page, Phys. Rev. D 42 (1990) 2655. [10] V.A. Belinskii, G.W. Gibbons, D.N. Page and C.N. Pope, Phys. Lett. B 76 (1978) 433. [ l 1] G.W. Gibbons and C.N. Pope, Commun, Math. Phys. 66 (1979) 267. [ 12] P.D. D'Eath, preprint, Quantization of the Bianchi IX model in supergravity (1993), to appear in Phys. Rev. D. [13] O. Obreg6n, J. Socorro and J. Ben~tez, Phys. Rev. D 47 (1993) 4471. [ 14 ] T. Koike, M. Tanimoto and A. Hosoya, preprint, Compact homogeneous universes, TIT/HEP208/COSMO-26 (1992).
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