fulltext

(ricevuto il 20 Maggio 1996; revisionato il 2 Agosto 1996; approvato il 30 Agosto 1996)

Summary. — We re-examine the gravitational constant, cosmological constant and phases of quantum gravity based on the Wheeler-DeWitt equation, considering higher order contributions for the wave function in positive or negative (WKB) power expansions in terms of the Planck mass. It is found that, in an unstable state, the cosmological constant decreases with increasing aA( m), and the gravitational constant increases with increasing aA( m) for aA( m) G aA

*( m*) and decreases with increasing aA( m) for aA( m) F aA

*( m*), where aA( m) is a renormalized cosmic-scale factor and aA

*( m*) is a point in which the value of the gravitational constant becomes the maximum. This result may propose a new mechanism, in quantum gravity, which explain the facts that the observed cosmological constant is very small and gravitational forces are remarkably weak in atomic physics, by the old age of our expanding universe. We also briefly discuss the relation between our results and Dirac’s large-number hypothesis.

PACS 04.60 – Quantum gravity.

PACS 04.20 – Classical general relativity.

PACS 04.40 – Self-gravitating systems; continuous media and classical fields in curved spacetime.

PACS 04.90 – Other topics in general relativity and gravitation. PACS 05.70 – Thermodynamics.

1. – Introduction

In a quantum theory of gravity, a wave function C is a function of the spatial metric and matter fields. This wave function obeys the Wheeler DeWitt (WDW) equation (HC 40) [1] which plays a central role in the theory. This equation is expected to be a good framework of quantum gravity which may answer questions of naturalness such as smallness of observed values of cosmological constant and gravitational constant (see sect. 4), and also describes the quantum space-time structure of our universe.

In our previous paper [2], we have explored phases of quantum gravity based on the WDW equation by using our approximate wave functions [3-7]. These approximate wave functions have been obtained by applying heat-kernel regularization [8, 9] to the

properties (the free energy, the entropy, the average curvature, fluctuations of the average curvature and so on) of quantum gravity for approximate wave functions up to sixth order in the WKB expansion and up to second order in the spatial gradient expansion, both for stable states and for unstable states. In sect. 3 we discuss the renormalization group equation and thermodynamic properties of quantum gravity for approximate wave functions up to second order in positive power expansion in terms of the Planck mass and up to second order in spatial gradient expansion, both for stable states and for unstable stables. In sect. 4 we briefly discuss the relation between our results for the behaviors of the gravitational and the cosmological constants and Dirac’s large number hypothesis. In appendix we study qualitative features of beta-functions for the approximate wave function up to fourth order in positive power expansion in terms of the Planck mass and up to second order in spatial gradient expansion. In sect. 5 we give summary and discussion.

2. – Analysis of the renormalization group equation and thermodynamic properties kof quantum gravity for approximate wave functions in the WKB expansion

Firstly, we discuss the case with the approximate WKB wave function. The wave function in the WKB expansion is a good approximation in the region which is larger than the Planck scale [6, 7]. In our previous paper [2], we have discussed the case with an approximate wave function up to fourth order in the WKB expansion and up to terms that contain second order spatial gradients in order to explore the phases of quantum gravity. In this paper, we discuss the case with an approximate wave function up to sixth order in the WKB expansion. According to the method given in ref. [6, 7], we get the approximate wave function up to terms that contain second order spatial gradients: (1) C_{6C exp}

k

_2mP2W12 W22 1 mP2 W32 1 mP4 W42 1 mP6 W52 1 mP8 W6

l

C C exp

k

2A6



d 3_x kh 2B6



d 3_x kh R( 3 )

l

_, (2) A_{64 6im}P2c0L1 /21 g11 mP22d11 mP24v11 mP26e11 mP28s1, (3) B_{64 6im}P2c2L21 /21 g21 mP22d21 mP24v21 mP26e21 mP28s2,

(10) e1C 23.0 Q 102102 3.1 Q 10211k01 2.3 Q 10212k021 2.0 Q 10213k032 6.7 Q 10215k04, (11) e2C 1.5 Q 102101 9.6 Q 10211k02 6.7 Q 10213k022 4.4 Q 10213k031 1.1Q10214k04, (12) s1C 21.7 Q 10213k02 2.8 Q 10214k021 6.8 Q 10216k031 1.1 Q 10216k042 2.9 Q 10218k05, (13) s2C 22.9 Q 102122 6.8 Q 10214k01 6.6 Q 10214k021

15.4 Q 10216k032 2.5Q10216k0415.1Q10218k05, where C₆ are two solutions to the WDW equation, mP is the Planck mass, L is the cosmological constant. m is a mass parameter

(

m f

(

f( 3 )_{( 0 )}

₎

1 /3

₎

_{. This mass parameter m} is needed in our renormalization scheme [3-7], and we have assumed f( 5 )

( 0 ) 4 k0Lf( 3 )( 0 ) [5-7]. We discuss the cases with L D0 throughout this paper. For the expressions of coefficients (g1, g2, d1, d2, v1 and v2), see eqs. (1)-(4) in appendix in ref. [2] and also see ref. [7]. We discuss the case with k04 1 throughout this paper (see appendix in ref. [2]). In this case, we have

g1C 1.3 Q 1022, g2C 24.5 Q 1023, (14) d1C 23.8 Q 1025, d2C 1.7 Q 1025, (15) v1C 27.3 Q 1029, v2C21.1 Q 1027, (16) e1C 23.3 Q 10210, e2C 2.5 Q 10210, (17) s1C 22.0 Q 10213, s2C 22.9 Q 10212. (18)

We discuss the renormalization group equation. The equation for the wave function of the WDW equation is [5-7] D A NC6N 4 0 , (19) D A_fm ¯ ¯m 1 b1 ¯ ¯gw 1 b2 ¯ ¯L , (20) b1fm ¯gw ¯m , (21) b2fm ¯L ¯m , (22)

b14 a3 , (25) b24 a2 a3 , (26) where (27) a1f3 g1g2L22m61 6(g2v11 g1v2) gw2L23m121 19(v1v21 g2s11 g1s2) gw4L24m181 12(v2s11 v1s2) gw6L25m241 15 s1s2gw8L26m30, (28) a2f(26 g2v11 6 g1v2) gwL22m121 (212 g2s11 12 g1s2) gw3L23m181 1(26 v2s11 6 v1s2) gw5L24m24, (28) a3f 22 g2v1gwL23m122 ( 2 v1v21 4 g2s1) gw3L24m182 2( 4 v2s11 2 v1s2) gw5L25m242 4 s1s2gw7L26m30. a1, a2and a3can be expressed as

(30) a1fL22m6] 15 s2s1X41 12(v2s11 v1s2) X31 9(v1v21 g2s11 g1s2) X21 16(g2v11 g1v2) X 13 g1g2( , (31) a2fgwL22m12](26 v2s11 6 v1s2) X21 (212 g2s11 12 g1s2) X 1 1(26 g2v11 6 g1v2)( , (32) a3fgwL23m12]24 s1s2X32 ( 4 v2s11 2 v1s2) X22 2( 2 v1v21 4 g2s1) X 22 g2v1( , where X f gw2L21m6and X D0 for L D0 . (33)

The equation a14 0 has a positive solution, and the equations a24 0 and a34 0 have no positive solutions. Thus, from (25) and (26), we have

b1D 0 for m Em*, (37) b1E 0 for m Dm*, (38) b2D 0 , (39)

where gw*2 L21_*m6_{* 4}X1 (34). This result qualitatively agrees with our previous one within the approximation up to fourth order in the WKB expansion [2, 7]. The curve L

A₄ 1

X1 g

A_w2 _{separates the phases of quantum gravity into the phase I}

1f ](gAw, LA)NLA D

1

X1 g

A_w2 _{and gA}

wD 0 ( and the phase II1f ](gAw, LA)N0 E LA E

1

X1

gAw2 and gAwD 0 ( as discussed in ref. [2], and X1E Xg 1 (see eq. (33) in ref. [2]), where LA4

1

X_{g 1} g A_w2

is the curve which separates the phases of quantum gravity into the phase I₁ and the phase II₁ within the approximation up to fourth order in the WKB expansion (see eq. (39) in ref. [2]). As dealt with in our previous paper [2], if we change the vari-ables (gw, L) as gAwfm2gw, (40) L A_fm22_L, (41)

eqs. (21) and (22) become

m ¯gAw ¯m 4 b A 1, (42) m ¯ L A ¯m 4 b A 2, (43) where (44) bA14 a A₁ a A₃ , b A 24 a A₂ a A₃ ,

g

A_{w, L}A, bA_{1and b}A_{2are dimensionless quantities. We solved eqs. (42) and (43) numerically.} The behaviors of the gravitational constant and the cosmological constant are shown in figs. NH1, NH2, NH3, where we have used (40) and (41) in the numerical plots.

We consider the following scale transformation: L Kr1L, (48) gwK r2gw, (49) m Kr3m , (50) and a constraint r121r22r364 1 , (51)

where r1, r2 and r3are dimensionless constants. Equations (21) and (22)

(

eqs. (42) and (43)

)

for the beta-functions are invariant under the transformations (48)-(50) and the constraint equation (51). We can use these transformations

(

(48)-(50)

)

and (51) in order to get more realistic values of the gravitational constant and the cosmological constant than those in the numerical calculations.

In our previous paper [2], we have calculated several thermodynamic quantities of quantum gravity (average curvature, fluctuations of average curvature, entropy) from our approximate wave functions up to fourth order in the WKB expansion, using together the minisuperspace approximation. We re-examine these thermodynamic quantities, considering the sixth-order contribution. From the wave function (1), we have (52) _{NC6N C exp [2f}L] , (53) fLf(g1m31 v1gw2L21m91 s1gw4L22m15)



d3xkh 1 1(g2L21m31 v2gw2L22m91 s2gw4L23m15)



d3xkh R( 3 ), or (54) fLf

(

g1m31 2 v1(gw21)21l21m91 4 s1(gw21)22l22m15)



d3xkh 1 1( 2 g2gw21l21m31 4 v2l22m91 8 s2(gw21)21l23m15)



d3xkh R( 3 ),

that the values of v for the flat and open models are finite and rescaled appropriately to 2 p2_{. Several thermodynamic quantities per volume}

₍

_{(64), (65), (67)-(69), (71),} S0 L( 0 S )OaV b0, SL( S )OaV b) are independent of the value of v, so that the limit v K Q can be taken at the end for the usual flat or open model. The value of v is not very important in our following discussions. From (55) and (56), eq. (53) becomes

fL4 fL3a31 fL1a , (57) where fL3f(g1m31 v1gw2L21m91 s1gw4L22m15) v, (58) fL1f(g2L21m31 v2gw2L22m91 s2gw4L23m15)(26kv) , (59) or fL3f(g1m31 2 v1(gw21)21l21m91 4 s1(gw21)22l22m15) v, (60) fL1f( 2 g2gw21l21m31 4 v2l22m91 8 s2(gw21)21l23m15)(26kv) . (61)

In Euclidean quantum gravity, the partition function is [12, 13] Z 4



Dg exp [2IE], (62) IE4 1 16 pGN



d4_x kh R( 4 ) 1 l



d4_x kh , (63)

where l f 2 L/16 pGN, kg is the determinant of the metric gmn, and R( 4 ) is the scalar

curvature in four dimension. From the partition function (62), we have (1_{) [2]} aR( 4 )b 42 1 aV b ¯ ¯

(

( 16 pGN)21

)

ln Z , (64) xlR4 1 aV b ¯2 ¯

(

( 16 pGN)21

)

2 ln Z , (65)

(1) For more discussions about thermodynamic quantities (average curvature, fluctuations of the average curvature and so on), see ref. [2].

N

Einstein action (63), from the partition function (62), we have aR( 4 ) b 42 1 aV b

y

¯ ¯( 16 pGN)21 ln Z 2L(16pGN) ¯ ¯Lln Z

z

, (68) xLRC 1 aV b

y

¯2 ¯

(

( 16 pGN)21

)

2 ln Z 2L2( 16 pGN)2 ¯2 ¯L2ln Z

z

, (69) aV b 42 16 pGN 2 ¯ ¯Lln Z , (70) xV4 1 aV b ( 16 pGN)2 4 ¯2 ¯L2 ln Z , (71)

where we have assumed aR( 3 )

Vb C aR( 3 )baV b in eq. (69). As we shall see later, the expressions (64)-(67) and the expressions (68)-(71) lead to the same ones except for (65) and (69) for our approximate wave functions. Looking on NC6N as the partition function Z approximately (2_{) [2], from (64)-(71) and (57), we have}

(72) aR( 4 ) b 4 4 (2v1gw 3_L21_m9 2 2 s1gw5L22m15)(v a3) 1 (g2gwL21m32 s2gw5L23m15)(26kva) aV b , (73) xlR4 (22 v1gw 4 L21_m9 2 6 s1gw6L22m15)(v a3) 1 (22 s2gw6L23m15)(26kva) aV b , (74) xLRC 1 aV b[ (24 v1gw 4_L21_m9 2 14 s1gw6L22m15)(v a3) 1 1( 2 g2gw2L21m32 8 s2gw6L23m15)(26kva) ] ,

(2_{) For discussions about the relation between the partition function Z in Euclidean quantum}

Our results for stable states are as follows.

i) The case with trajectories from the phase II₁to the phase I₁(3_{) (a stable state).} In this case, the gravitational constant increases with increasing m for m Em*, and decreases with increasing m for m Dm* (see trajectories A, B, C in fig. NH1 and fig. NH3), and the cosmological constant increases with increasing m (see fig. NH2).

In our previous paper [2], from (57), we have defined FLfmfL,

(77)

and we have regarded FL(77) as a free energy in our thermodynamic system, and the cosmic-scale factor a has been regarded as an order parameter in our thermodynamic system. The behavior of the free energy (77) for k 421 is shown in fig. NH4. The free energy (77) is minimized by the solution [2]

a0( m) 4

o

2 fL1 3 fL3

for the open model (k 421). (78)

a0( m) is constant since dfL/dm 40 (dfL1/dm 40 and dfL3/dm 40) and a0( m) A105 as an example in our numerical calculations (see fig. NH5). For the flat model (k 40), from (59), 2fL1/3 fL34 0 and, for the closed model (k 4 1 ), the value of 2fL1/3 fL3is negative (see fig. NH6), so that, for the flat and closed models, the free energy does not have the minimum value for a D0. The open model is therefore realistic for our approximate wave function in the WKB expansion in the sense of the minimum of the free energy. These results agree with our previous ones in ref. [2].

If we use a0( m) as the cosmic-scale factor a in eqs. (72)-(76), the behaviors of several thermodynamic quantities are as follows. For the open model (k 421), the value of aR(4)b is positive (negative in the ADM notation [10]) and NaR(4)bN decreases with decreas-ing m (see fig. NH7). This would mean that the universe approaches a flat one for mK0.

(3_{) For the behaviors of trajectories in the phase I}

1and the phase II1, see fig. 1 in ref. [2]. The

case i) corresponds to the trajectory A in fig. 1 in ref. [2]. We note that trajectories with opposite directions are the same curves as those given in fig. 1 in ref. [2] and in figures in this paper since the beta-functions b1, b2 are invariant under the transformation m K2m, and eqs. (21), (22)(or

eqs. (42), (43))are also invariant. This case corresponds to the one with r14 1, r24 1, r34 21 in

Fig. NH1.

Fig. NH2.

Fig. NH4.

Fig. NH5.

Fig. NH7.

Fig. NH8.

Fig. NH10.

Fig. NH11.

Figs. NH1-NH11. – The case i) with trajectories from the phase II₁to the phase I₁(a stable state) (see sect. 2). We have solved eqs. (42) and (43) numerically for bA1, bA2 (44) and k04 1 by

starting from m 40.1 at gAw4 gAiand LA4 1028, where gAi4 1026, 1022, 1021.8, 1021.5, ( 1021.2). These

initial values gAi are shown in the figures. These numerical solutions of gAw and LA from m40.1 to

m 41 were used for numerical plots of various thermodynamic quantities. This case i) exists in the region gAiE 1021.37. This condition was obtained from gAw2LA21E X1, X1C 1.8 Q 105 (see (34)).

This equation gAw2LA214 X1 separates the phases of quantum gravity into the phase I1and the

phase II₁within the approximation up to sixth order in the WKB expansion.

The fluctuations of the average curvature (xlR, xLR) become large for m  m_* ( m_*A0.2) (see figs. NH8, NH9). These results agree with our previous ones qualitatively [2]. In these numerical plots, we have used aV b04

8p2

3 a0

4_{( m) for the de Sitter} minisuper-space model in the 4-dimensional Euclidean minisuper-space, and aV b0 is constant since a0( m) is

correspond to heating up quantum manifolds (quantum universe) with a fixed volume.

iia) The case with trajectories in the phase II₁only (4_{) (a stable state).}

In this case, the gravitational constant only decreases with increasing m (see the trajectory D in fig. NH1), and the cosmological constant increases with increasing m (see fig. NH2). The behavior of the free energy for the closed model (k 41) is shown in figs. NHII-1, NHII-2. For the open model (k 421), the value of 2fL1/3 fL3is negative (see fig. NHII-3), so that, for the open model, the free energy does not have the “minimum” value for a D0. The closed model is therefore realistic in the sense of the “minimum” of the free energy. There are cases that the value of the average curvature is negative for m Gm2_*and is positive for m Fm2_*, where m2_*is the value where aR( 4 )b is zero (see fig. NHII-4). xlR and NxLRN increase with decreasing m (see figs. NHII-5, NHII-6). The entropy per volume s0 L has negative values, and s0 LA 21029 as an example in our numerical calculations (see fig. NHII-7). The fluctuations of the average volume xVincrease with decreasing m (see fig. NHII-8).

We also calculated several thermodynamic quantities for the case up to forth order in the WKB expansion. The results are as follows.

iib) The case with trajectories in the phase II₁only (5_{) (a stable state).}

The free energy for the closed model (k 41) is shown in fig. NII-1 (see also fig. NII-2). For the open model (k 421), the value of 2fL1/3 fL3 is negative (see fig. NII-3). These results are the same as the case iia) up to sixth order in the WKB expansion, and a0( m) A104 as an example in our numerical calculations. The value of the average curvature is positive (see fig. NII-4). The fluctuations of the average curvature (xlR, xLR) increase with decreasing m (see figs. NII-5, NII-6). The entropy has negative values (see fig. NII-7) and s0 LA 21026 as an example in our numerical calculations. The fluctuations of the average volume xV increase with decreasing m (see

fig. NII-8).

We think that, in the case ii)

(

iia) and iib)

)

with trajectories in the phase II₁only, we should regard 2FL as the free energy in order to define the entropy as SLD 0 (or sLD 0 ). In this viewpoint, the sign of the average curvature and the sign of the

(4_{) The case iia) corresponds to trajectories B, C in fig. 1 in ref. [2].}

Fig. NHII1.

Fig. NHII2.

Fig. NHII4.

Fig. NHII5.

Fig. NHII7.

Fig. NHII8.

Figs. NHII-1–NHII-8. – The case iia) with trajectories in the phase II₁only (a stable state) (see sect. 2). The same as the case i) but with gAi4 1021.2, 1021, 1020.9, 1020.8, 1020.7, and this case iia)

exists in the region gAiD 1021.37.

fluctuations of the average curvature change. We think that these strange behaviors of several thermodynamic quantities relating to the sign are probably connected with negative probabilities which arise from the usual definition of conserved currents for second order hyperbolic differential equations such as the Klein-Gordon equation, since the WDW equation is also the same. Negative probabilities would not arise at a large universe in the classically allowed region [15], and would arise at a small universe in the classically forbidden region [16, 17].

If we consider results in the case i) and the case ii), we conclude that, in the sense of the minimum of the free energy, the open model is realistic in the case i), and the closed model is realistic in the case ii). In the case i) and the case ii), the fluctuations of the average curvature become large around m

Fig. NII-1.

Fig. NII-2.

Fig. NII-4.

Fig. NII-5.

Fig. NHII-7.

Fig. NHII-8.

Figs. NII-1–NII-8. – The case iib) with trajectories in the phase II₁only (a stable state) (see sect. 2). We have solved eqs. (42) and (43) numerically for bA1and bA2(44) with s14 0 and s24 0 and k04 1

by starting from m 40.1 at gAw4 gAiand LA4 1025, where gAi4 100.7, 100.9, 101.5, 102, 102.3. These initial

values gAi are shown in the figures. These numerical solutions of gAwand LA from m40.1 to m41

were used for numerical plots of various thermodynamic quantities. This case iib) exists in the region gAiD 100.54. This condition was obtained from gAw2LA21D Xg 1, Xg 1C 1.2 Q 106(see eq. (34) in

ref. [2]). This equation gAw2LA214 Xg 1separates the phases of quantum gravity into the phase I1

and the phase II₁within the approximation up to fourth order in the WKB expansion.

calculations. These results mean that a phase transition (a phase change) in quantum gravity occurs at the transition point m

*.

(83) aA( m) 4 v3 3 1 2 v3 3]27v11 2 v331

k

24 v361 ( 27 v11 2 v33)2(1 /3 1 1 1 3 Q 21 /3 ] 27 v11 2 v3 3 1

k

24 v361 ( 27 v11 2 v33)2(1 /3. This is a renormalized cosmic-scale factor. For simplicity of our analysis, we consider the case with an approximate wave function up to fourth order in the WKB expansion. This approximation corresponds to the case with s14 0 and s24 0 in eqs. (81), (82) and eqs. (72)-(76). The results are as follows.

iii) The case with trajectories from the phase II₁to the phase I₁for the open model (k 421) (7_{) (an unstable state).}

The value of the renormalized cosmic-scale factor aA( m) increases with decreasing m (see fig. NU1). This means that the universe expands with decreasing m. The gravitational constant GN increases with increasing aA( m) for aA( m) G aA_*( m_*) and decreases with increasing aA( m) F aA

*( m*), where aA*( m*) is the value in which the value of the gravitational constant becomes the maximum (see fig. NU2). The cosmological constant decreases with increasing aA( m) (see fig. NU3) [18].

If we use aA( m) as the cosmic-scale factor a in eqs.(72)-(76), the behaviors of the thermodynamic quantities are as follows. The average curvature aR( 4 )

b is positive (negative in the ADM notation) and the value of NaR( 4 )

b N decreases with decreasing m

(

or increasing aA( m)

)

(see figs. NU4, NU5). The fluctuations of the average curvature (xlR, xLR) increase with increasing m

(

or decreasing aA( m)

)

(see figs. NU6-NU9). The correlators a(R( 4 )₎2

b 2 aR( 4 )_b2 _{have clearly a peak (see figs. NU10-NU13) around m}

U1_*

(mU1_*A0.15) or aA_*( m_*), where aA_*( m_*) is the value in which the values of the

cor-relators become the maximum, and the fluctuations of the average volume xVhave clearly

a peak around mU2_*, where mU2_*A0.12 in our numerical calculations (see trajectories

A, B, C, D in fig. NOUII-14) or have clearly a peak around aA

*( m*) (see trajectories A8, B8, C8, D8 in fig. NOUII-15). These results mean that, in this unstable state, a phase transition (a phase change) in quantum gravity occurs at the phase transition

(6_{) The other two solutions to eq. (80) do not become real solutions but become complex solutions.}

(7_{) This case iii) corresponds to a trajectory A in fig. 1 in ref. [2]. We could not find the real}

solution to eq. (80) for the closed model (k 41) in this unstable state in our numerical calculations.

Fig. NU1.

Fig. NU2.

Fig. NU4.

Fig. NU5.

Fig. NU7.

Fig. NU8.

Fig. NU10.

Fig. NU11.

Fig. NU13.

Fig. NU14.

Fig. NU15.

Figs. NU1-NU15. – The case iii) with trajectories from the phase II₁to the phase I₁for the open model (k 421) (an unstable state) (see sect. 2). The same as the case iib), but with gAi4 1022,

1020.5_{, 10}20.2_{, 10}0_{, 10}0.1_{, 10}0.2_{, 10}0.3_{, and the case iii) exists in the region gA}

Fig. NOUII-1.

Fig. NOUII-2.

Fig. NOUII-4.

Fig. NOUII-5.

Fig. NOUII-7.

Fig. NOUII-8.

Fig. NOUII-10.

Fig. NOUII-11.

Fig. NOUII-13.

Fig. NOUII-14.

Fig. NOUII-16.

Fig. NOUII-17.

Figs. NOUII-1–NOUII-17. – The case iv) with trajectories in the phase II₁ only for the open model (k 421) (an unstable state) (see sect. 2). The same as the case iib), but with gAi4 100.6, 100.7,

101_{, 10}1.2_{, 10}1.4_{, 10}1.5_{, 10}1.6_{, 10}1.7_{, 10}1.8_{, 10}2_{, 10}2.2_{, 10}2.3_.

(the phase change) point mU_*, where mU_{* 4}0.12–15 in our numerical calculations,

where we have thought that mU1_{* A}mU2_* approximately. The entropy per volume sL decreases with decreasing m

(

or increasing aA( m)

)

(see figs. NU14, NU15), where sLfSL/aV b, SLf 2fLNa 4 aA, aV b f

8 3p

2_aA4_{( m). This implies that the entropy of} quantum space-time (quantum geometrical entropy) per volume decreases as the universe expands.

iv) The case with trajectories in the phase II_{1only for the open model (k 421) (}8_{) (an} unstable state).

Fig. NCUII-1.

Fig. NCUII-2.

Fig. NCUII-4.

Fig. NCUII-5.

Fig. NCUII-7.

Fig. NCUII-8.

Fig. NCUII-10.

Fig. NCUII-11.

Fig. NCUII-13.

Fig. NCUII-14.

Fig. NCUII-16.

Fig. NCUII17.

Figs. NCUII-1–NCUII-17. – The case v) with trajectories in the phase II₁ only for the closed model (k 41) (an unstable state) (see sect. 2). The same as the case iib), but with gAi4 101, 101.2,

101.4_{, 10}1.5_{, 10}1.6_{, 10}1.7_{, 10}1.8_{, 10}2_{, 10}2.2_{, 10}2.3_.

The renormalized cosmic-scale factor aA( m) increases with decreasing m (see fig. NOUII-1). The gravitational constant GN only increases with increasing aA( m) (see fig. NOUII-2), and the cosmological constant decreases with increasing aA( m) (see fig. NOUII-3). The average curvature is positive, and NaR( 4 )

b N decreases with decreasing m

(

or increasing aA( m)

)

(see figs. NOUII-4, NOUII-5). The fluctuations of the average curvature become large for small m

(

or large aA( m)

)

or large m

(

or small aA( m)

)

in our numerical calculations (see figs. NOUII-6–NOUII-9). The cor-relators a(R( 4 )₎2

b 2 aR( 4 )_b2 _{increase with decreasing m}

₍

_{or increasing aA( m)}

₎

_(see fig. NOUII-10–NOUII-13). Fluctuations of average volume xV increase with

fluctuations of the average volume only increase with increasing aA( m).

3. – Analysis of the renormalization group equation and thermodynamic properties of quantum gravity for approximate wave functions in positive power expansion in terms of the Planck mass

Next we discuss the case with the approximate wave function in the positive power expansion in terms of the Planck mass. The approximate wave function in this expansion is a good approximation in the region beyond the Planck scale [3-5]. In our previous paper [2], we have discussed the thermodynamic nature of the small universe in the “new phase” using the approximate wave function up to leading order, and we have seen that the average curvature and fluctuations of the average curvature become very small for large m (aR( 4 )b Pm23 /2_{, x}

RP m23), (see eqs. (139), (140), fig. 14 and fig. 15

in ref. [2]). Therefore, spatial gradients expansion would be a good approximation in the region beyond the Planck scale. We consider the approximation up to terms that contain second-order spatial gradients, and we have the wave function up to next leading order [5]: (84) _{C Cexp [2m}P4S12 mP8S2] C C exp [2(mP4c01 mP8c 80)



d3xkh 2 (mP4c11 mP8c 81)



d3xkh R( 3 )] , where (85) c04 c0L m23, c14 c1m23, (86) _{c 8}04 c80L2m29, c 814 c81L m29, (87) c04 c001 k0c01, (88) _{C 2.2 Q 10}2 _{for k} 04 1 , (89) c14 2 3 ! b2 C 5.8 Q 102,

(93) _c80c81C 4.5 Q 1013 for k04 1 , (94) c1c802 c0c81C 22.7 Q 109 for k04 1 , (95) a14 2 21 8 1 ( 4 p)3 /2 , b14 3 2 1 ( 4 p)3 /2 , b24 2 11 24 1 ( 4 p)3 /2 , and we have assumed f( 5 )

( 0 ) 4k0Lf( 3 )( 0 ) [5-7].

We discuss the renormalization group equation for the wave function (84). Then, eq. (19) becomes D A(gw22c01 gw24c 80) 40, (96) D A(gw22c11 gw24c 81) 40, (97)

where gwf1 /mP2. Equations (96) and (97) give b14 2 3 2gw 3 c80c81m212gw24L21 2(c0c811 c80c1) m26gw22L 1c0c1 2 c80c81m212gw24L21 (c0c811 2 c80c1) m26gw22L 1c0c1 , (98) b24 L 3(c1c802 c0c81) m26gw22L 2 c80c81m212gw24L21 (c0c811 2 c80c1) m26gw22L 1c0c1 . (99)

Equations (98) and (99) can be expressed as b14 2 3 2gw 3 c80c81XA21 2(c0c811 c80c1) XA1 c0c1 2 c80c81XA21 (c0c811 2 c80c1) XA1 c0c1 , (100) b24 L 3(c1c802 c0c81) XA 2 c80c81XA21 (c0c811 2 c81c1) XA1 c0c1 , (101) where XAfm26_g w22L. b14 0 gives 3 c80c81XA21 2(c0c811 c80c1) XA1 c0c14 0 . (102)

The solutions to eq. (102) are

XA_{1 1C 26.8 Q 10}25 _{for k} 04 1 , (103) XA_{1 2C 21.4Q10}25 _{for k} 041. (104)

This means that the values of the gravitational constant and the cosmological constant decrease with increasing m in the “new phase” (see figs. P1, P2, P3). We also examined the behaviors of the beta-functions up to fourth order in positive power expansion in terms of the Planck mass and up to second order in spatial gradients expansion, and we had the same result as eq. (108) (see appendix). We would have the same result as eq. (108), even if we consider higher orders than the fourth order within our approximation. Hereafter we will discuss the case with the wave function (84) for simplicity. If we change the variables (gw, L) as eqs. (40) and (41), eqs. (98) and (99) become b A 14 2 1 2 g A_w c80c81gAw24LA21 2(c0c812 c80c1) gAw22LA2 c0c1 2 c80c81gAw24LA21 (c0c811 2 c80c1) gAw22LA1 c0c1 , (109) b A 24 LA 24 c8 0c81gAw24LA21 (25 c0c812 c80c1) gAw22LA22 c0c1 2 c80c81gAw24LA21 (c0c811 2 c80c1) gAw22LA1 c0c1 . (110)

We integrated eqs. (42) and (43) for bA1(109) and bA2(110) numerically, and we used (40) and (41) for the numerical plots (see figs. P1, P2, P3). Equations (21) and eq. (22)

(

or eqs. (42) and (43)

)

for the beta-functions

(

(98), (99)

) (

or (109), (110)

)

are invariant under the scale transformation (48)-(50) and the constraint (51).

Next we discuss the thermodynamic properties in the “new phase”. The wave function (84) is expressed as (111) _{C Cexp [2f}S], (112) fS4 (c0gw22L m231 c80gw24L2m29)



d3xkh 1 1(c1gw22m231 c81gw24L m29)



d3xkh R( 3 ), or (113) fS4

g

1 2 c0gw 21_lm23 1 1 4 c80gw 22_l2 m29

h



_d3 xk_{h 1} 1

g

c1gw22m231 1 2 c81gw 23_lm29

h



_d3_{xkh R}( 3 )_, where gw214mP241/16pGN, lf2L/16pGN. If we use the minisuperspace approximation,

Fig. P1.

Fig. P2.

Fig. P4.

Fig. P5.

Fig. P7.

Fig. P8.

Fig. P10.

Fig. P11.

Fig. P13.

Fig. P14.

Fig. P16.

Figs. P1-P16. – The case vi) with trajectories in the “new phase” (stable states). We have solved eqs. (42) and (43) numerically for bA1(109) and bA2(110) and k04 1 by starting from m 4 10 at gAw4 gAi,

and LA4 102, where gAi4 100.6, 100.8, 101, 101.2, 101.4. These initial values gAiare shown in the figures.

These numerical solutions of gAwand LA from m410 to 0.2 were used for numerical plots of various

thermodynamic quantities. In figs. P7, P9, P11, P13, P15 we have used aV b04 ( 8 O3 ) p2a04( m). In

figs. P8, P10, ,P12, P14, P16 we have used aV b04 2 p2a03( m). In our new scenario of the evolution

of the universe [4], it is thought that aV b A2p2_a

03( m) in the “new phase”.

from (55) and (56), eq. (112) becomes

fS4 fS3a31 fS1a , (114) where fS3f(c0gw22L m231 c80gw24L2m29) v, (115) fS1f(c1gw22m231 c81gw24L m29)(26kv) , (116) or fS34

g

1 2 c0gw 21_lm23 1 1 4 c80gw 22_l2_m29

h

_v, (117) fS14

g

c1gw22m231 1 2 c81gw 23_lm29

h

_(26kv). (118)

From (64)-(67) and (68)-(71), we have (119) aR( 4 ) b 4 1 aV b[ (c0gw 21_{L m}23 1 2 c80gw23L2m29)(v a3) 1 1( 2 c1gw21m231 3 c81gw23L m29)(26kva) ], (120) xlR4 1 aV b[ (22 c80gw 22_L2_m29_{)(v a}3 ) 2 (2 c1m231 6 c81gw22L m29)(26ka) ],

The results for the thermodynamic quantities are as follows. vi) The case with trajectories in the “new phase” (a stable state).

In ref. [2], from (112), we have defined FSfmfS, (124)

and we have regarded FS (124) as a free energy in the “new phase”. FS4 2m ln Z , where Z f C [13]. The behavior of the free energy (124) is shown in fig. P4 for the closed model. The free energy (124) is minimized by the solution [2]

a0( m) 4

o

2 fS1 3 fS3

for the closed model (k 41). (125)

a0( m) is constant and a0( m) A0. 04 as an example in our numerical calculations (see fig. P5). For the flat model (k 40), from (116), 2fS1/3 fS34 0 and, for the open model (k 41), the value of 2fS1/3 fS3 is negative (see fig. P6), so that, for the flat and open models, the free energy (124) does not have the minimum value for a D0. The closed model is therefore realistic in the “new phase” in the sense of the minimum of the free energy. This result agrees with our previous one [2].

If we use a0( m) as a cosmic-scale factor in eqs. (119)-(123), the behaviors of the thermodynamic quantities are as follows. For the closed model (k 41), the value of aR( 4 )b is negative (positive in the ADM notation [10]) and increases with decreasing m (see figs. P7, P8), and the fluctuations of the average curvature (xlR, xLR) become large with decreasing m (see figs. P9-P12). These results agree with our previous ones in ref. [2] qualitatively. From the free energy (124), the corresponding entropy S0 S is given by [2]

S0 S4 2fSNa 4a0.

(126)

S0 Sis constant in the sense of the minimum of the free energy, the entropy per volume s0 S is also constant and s0 SA 1011as an example in our numerical calculations, where s0 SfS0 S/aV b0 (see figs. P13, P14). The fluctuations of the average volume NxVN

increase with decreasing m (see figs. P15, P16).

Next, we consider an unstable state. We use eq. (80) for the approximate wave function (111) in order to see the qualitative behavior of the renormalized cosmic-scale factor and other quantities in our quantum gravitational system. In this case, from

Using eq. (80), we will investigate the region which is slightly larger than the region in which a dimensional reduction

(

( 3 11)-dimK3-dim

)

occurs [3-5]. We can use transformations (48)-(50) and (51) to get more realistic values of aA( m) and other quantities in the numerical calculations. The results for the thermodynamic quantities are as follows.

vii) The case with trajectories in near “new phase” (10_{) for the open model (k 421) (an} unstable state).

The renormalized cosmic-scale factor aA( m) increases with decreasing m (see fig. PU1). The gravitational constant and the cosmological constant increase with increasing aA( m) (see figs. PU2, PU3). NaR( 4 )

b N, NxlRN, NxLRN, a(R( 4 ))2bl2 aR( 4 )b2l,

a(R( 4 )₎2_b

L2 aR( 4 )b2L, NxVN and NSSN increase with increasing aA( m) (or decreasing m) (see figs. PU4-PU15), and the entropy sS per volume has negative values (see figs. PU16, PU17), and NsSN decreases with increasing aA( m) (or decreasing m).

viii) The case with trajectories in near “new phase” for closed model (k 41) (an unstable state).

In this case, we had the same figures as those in the case vii) in our numerical calculations. We therefore do not show figures in the case viii). This result means that the coefficient v1(128) is neglected in eq. (80), at least for our initial conditions (gAi, LA4 102_{) in our numerical calculations.}

We think that we should regard 2FSas a free energy in the cases vii), viii) as in the case ii) in sect. 2 since entropy should have positive values. Large fluctuations of the average curvature NxRN and large fluctuations of the average volume NxVN for m K mN_*

in the cases vii), viii) mean that, in an unstable state, a phase transition (a phase change) occurs near the point mN_*, where mN_{* 4}1 –5 in our numerical calculations. The

qualitative behaviors of various thermodynamic quantities in the case ii) are similar to those in the case vi), except for the signs of the thermodynamic quantities, and the qualitative behaviors of various thermodynamic quantities in the cases iv), v) are similar to those in the cases vii), viii) except for the signs of the thermodynamic quantities. This may mean that the “new phase” (the dynamical system described by the three-dimensional Euclidean quantum Einstein gravity exists inside of the phase II₁. We think that, in the cases i)-viii), m

* AmU_{* A}mN_* approximately in the range

0 EmEQ.

Fig. PU1.

Fig. PU2.

Fig. PU4.

Fig. PU5.

Fig. PU7.

Fig. PU8.

Fig. PU10.

Fig. PU11.

Fig. PU13.

Fig. PU14.

Fig. PU16.

Fig. PU17.

Figs. PU1-PU17. – The case vii) with trajectories in near “new phase” (an unstable state) (see sect. 3). The same as the case vi), but with aV b 4 (8/3) p2_aA4_{( m).}

Under the scale transformation (48)-(50) and the constraint (51), various quantities in our quantum gravitational system change as

a0( m) Kr121 /2a0( m) , aA( m) Kr121 /2aA( m) , (129) aV b0K r122aV b0, aV b Kr122aV b , (130) xVK r121r2xV, (131) aR( 4 ) b Kr1aR( 4 )b , (132) xlR(LR)K r1r2xlR(LR), (133) S0 L( 0 S )K (r1r2)21S0 L( 0 S ), SL( S )K (r1r2)21SL( S ), (134) s0 L( 0 S )K r1r221s0 L( 0 S ), sL( S )K r1r221sL( S ). (135)

xlR(LR)K 1018xlR(LR),

(140)

sSK 106sS. (141)

Then, the values of various quantities in figs. PU1-PU17 are rescaled according to (136)-(141).

4. – The relation between our results and Dirac’s large-number hypothesis

Next we discuss Dirac’s large number hypothesis [19]. Dirac’s large-number hypothesis is that any two of the very large dimensionless number occurring in Nature are connected by a simple mathematical relation, in which the coefficients are of order of magnitude unity. Dirac noticed the agreement between the age of our universe in atomic units and the ratio of the gravitational to the electric force between elementary particles [19]: tAU t C GNmempr e2 , (142)

where tAUis an atomic unit of time and tAUfe2/mec3, meis the electron mass, mpris the proton mass, e is the electric charge, c is the light velocity, t represents the age of our universe, t A1010 years, and tAU/t A10239. From eq. (142), Dirac thought (11) that the value of the gravitational constant decreases under the requirement that the equality in eq.(142) is conserved in the course of the expansion of the universe. That is to say,

GNP t21. (143)

Then, the fact that the gravitational force is very weak in comparison with other forces in elementary-particles physics is explained by the old age of our universe. We consider the behavior of the cosmological constant according to Dirac’s large-number hypothesis

(11_{) Dirac thought that G}

N is varying with time and other constants (me, mpr, e) are not almost

(143) and (145), we have l Pt2(n 2 2 ) _{for n F3,} (146) or L Pt2(n 2 1 ) _{for n F3.} (147)

The relation (146)

(

(147)

)

has been discussed in ref. [20] (in ref. [21]) for n 43 [22]. From (143) and (147), we have

L_{(t) PG}N(t)n 21 for n F3. (148)

This behavior (148) of the gravitational constant and the cosmological constant is similar to that around the point (GN, L) 4 (0, 0) in our analysis of the renormalization group equation for the WDW equation (see trajectories A, B, C in fig. NH3 in this paper and also see fig. 1e in ref. [7], fig. 1 in ref. [2]) [23, 24].

The number of nucleons in the visible universe is [19] NBA ( 1039)2P t2. (149)

From (146) and (149), we have

NB(t) Pl(t)22 /(n 2 2 ) for n F3 . (150)

The cosmological constant is strongly connected with the vacuum energy of quantum field theory [13]. Thus, the relation (150) would mean that the decay of the vacuum energy (the decay of the cosmological constant) leads to nucleon production in the course of the expansion of the universe. We expect that a similar relation to (150) can be obtained from the WDW equation with matter.

5. – Summary and discussion

In summary, we re-examined phases of quantum gravity based on the WDW equation from the analysis of the renormalization group equation and thermodynamic analysis for the approximate wave functions. We discussed the cases with L D0 throughout this paper.

In sect. 2, we discussed the case with the approximate wave function up to sixth order in the WKB expansion and up to second order in spatial gradients expansion.

i) with trajectories from the phase II₁ to the phase I₁, our results agree with our previous ones in ref. [2] in qualitative features, i.e. the average curvature is negative in the ADM notation in the sense of the minimum of the free energy, and NaR( 4 )

b N decreases with decreasing m and the fluctuations of the average curvature become large with increasing m ( mKm*) (see figs. NH7-NH9), where we have used together the minisuperspace approximation (a closed, flat and open deSitter minisuperspace model). In the case ii) with trajectories in the phase II₁ only, fluctuations of the average curvature xlR, NxLRN) become large with decreasing m ( m K m *) (see figs. NHII-5, NHII-6, NII-5, NII-6). In this case, m

* C0.1–0.6. For other results, see the case ii) in sect. 2. These large fluctuations of the average curvature around m

* in the case i) and the case ii) mean that a phase transition (a phase change) occurs in quantum gravity. In these stable states

(

i), ii), and iv)

)

, increasing m would correspond to heating up the quantum manifold (quantum universe) with a fixed volume.

We also discussed the behaviors of the cosmic-scale factor, the average curvature and so on in unstable states using eq. (80). It was shown that, for the case iii) with trajectories from the phase II₁ to the phase I₁for the open model, the renormalized cosmic-scale factor aA( m) increases with decreasing m (mK0) (see fig. NU1), the gravitational constant GN increases with increasing aA( m) for aA( m) G aA_*( m_*) and decreases with increasing aA( m) for aA( m) F aA

*( m*) (see fig. NU2), and the cosmological constant decreases with increasing aA( m) (see fig. NU3), where aA

*( m*) is a point in which the gravitational constant becomes the maximum [18]. We think that these results propose a new mechanism, in quantum gravity, which explains the facts that the gravitational forces are remarkably weak in atomic physics and the cosmological constant is very small in the present world [23, 24]. The fluctuations of the average curvature decreases with decreasing m (or increasing aA( m)) (see figs. NU6-NU9). We think that this result coincides with our experience that the space-time in which we live is now classical. The correlators a(R( 4 )₎2

b 2 aR( 4 )b2and the fluctuations of the average volume xV have clearly a peak (see figs. NU10-NU13, and see A, B, C, D in

fig. NOUII-14 and A8, B8, C8, D8 in fig. NOUII-15). These results mean that, in this unstable state, a phase transition (a phase change) in quantum gravity occurs at the transition point mU_*, where mU_{* 4}0.12–0.15 in our numerical calculations. The entropy

per volume decreases with decreasing m

(

or increasing aA( m)

)

(see figs. NU14, NU15). This means that the entropy of quantum space-time (quantum geometrical entropy) per volume decreases as the universe expands. We think that this case iii) is most realistic for our universe. For other results for the stable state iii) or unstable states iv), v), see sect. 2. In sect. 3, we discussed mainly the case with the approximate wave function up to second order in the positive power expansion in terms of the Planck mass and up to

Fig. Q1. – A conceptual figure of phases of quantum gravity and the behaviors of the gravitational constant GNand the cosmological constant L. We think that the value of m_**is near the value of

m

* in the range 0 EmEQ. The existence of m** was suggested from our analysis of the renormalization group equation (see sect. 5).

second order in spatial gradients expansion. This wave function is a good approximation beyond the Planck scale [3-5]. The results are as follows. The gravitational constant decreases with increasing m (see fig. P1), and the cosmological constant decreases with increasing m (see fig. P2). This result suggests that a phase transition (a phase change) point m

** exists and the cosmological constant increases with increasing m for m Em** and decreases with increasing m for mDm**, and the value of m

**is different from that of m*, i.e. m* Em**, by considering our result about the behavior of the beta-function b2 in the WKB expansion. It is expected that there are two phase transition (phase change) points ( m

* Em**) in quantum gravity based on the WDW equation (see fig. Q1) [27]. The results for the stable state vi) in thermodynamic analysis agree with our previous ones in ref. [2] qualitatively, i.e. the average curvature is positive in the ADM notation in the sense of the minimum of the free energy (see figs. P4, P5, P6), and NaR( 4 )b N and the fluctuations of the average curvature (xlR, xLR) become large with decreasing m ( m Km*2) (see figs. P7-P12), where the value of m

*2is roughly of order one in the region 0 EmEQ in our numerical calculations.

The results for the unstable states vii), viii) in thermodynamic analysis are as follows. The renormalized cosmic-scale factor aA( m) increases with decreasing m (see fig. PU1). The gravitational constant and the cosmological constant increase with increasing aA( m) (see fig. PU2 and fig. PU3). NaR( 4 )

b N and the fluctuations of the average curvature (NxlRN, NxLRN), the fluctuations of the average volume NxVN and

entropy NSSN increase with increasing aA( m) (or decreasing m) (see figs. PU4-PU15). The entropy per volume NsSN decreases with increasing aA( m) (or decreasing m), (see figs. PU16, PU17). Strange behaviors in the signs of several thermodynamic quan-tities in cases ii), iv), v), vi), vii), viii) are probably connected with negative probabili-ties which arise from the usual definition of conserved currents for the WDW equation,

(

see eq. (129)-(141)

)

.

In sect. 4, we mainly discussed the behaviors of the gravitational constant and the cosmological constant according to Dirac’s large-number hypothesis. It was pointed out that the behavior (148) is similar to that around the point (GN, L) 4 (0, 0) in our analysis of the renormalization group equation (19) (see trajectories A, B, C in fig. NH3). We think that Dirac’s large-number hypothesis is not correct in classical Einstein gravity, but correct approximately in quantum Einstein gravity (the WDW equation) [23, 24].

AP P E N D I X

In this appendix, qualitative features of beta-functions (b1 and b2) for the approximate wave function up to fourth order in positive power expansion in terms of the Planck mass and up to second order in spatial gradient expansion are discussed. According to the method given in ref. [3, 5], we get

(A.1) _{C Cexp}

k

_2(gw22c01 gw24c 801 gw26p01 gw28q0)



d3xkh 2 2(gw22c11 gw24c 811 gw26p11 gw28q1)



d3xkh R( 3 )

l

, where p04 p0L3m215, p14 p1L2m215, (A.2) q04 q0L4m221, q14 q1L3m221, (A.3) p0C 6.3 Q 1010, p1C 8.0 Q 1011 for k04 1 , (A.4) q0C 2.4 Q 1015, q1C 3.8 Q 1016 for k04 1 . (A.5)

In this case, eq. (19) gives

b14 g1 g3 , b24 g2 g3 , (A.6) (12_{) We think that m}

* Am*2A mU_*approximately in the range 0 EmEQ, where mU_*is a phase

(A.9) g3fgw25m26] 8 q0q1XA61 ( 8 p1q01 6 p0q1) XA51 ( 6 p0p11 8 c81q01 4 c80q1) XA41 1( 6 c81p01 4 c80p11 8 c1q01 2 c0q1) XA31 ( 4 c80c811 6 c1p01 2 c0p1) XA21 1( 4 c80c11 2 c0c81) XA12 c0c1(, where XAfgw22L m26. g14 0 gives (A.10) XA1C 23.9 Q 1025, XA2C 21.8 Q 1025, XA3C 5.7 Q 10276 1.4 Q 1025i , (A.11) XA4C 7.6 Q 10266 3.8 Q 1025i . g24 0 gives X A 5C 28.8 Q 10256 1.8 Q 1024i , XA6C 26.6 Q 10266 9.5 Q 1026i . (A.12) g34 0 gives X A 7C 22.7 Q 10256 3.0 Q 1026i , XA84 2.6 Q 10266 2.8 Q 1025i , (A.13) X A 9C 3.8 Q 10266 2.4 Q 1025i . (A.14)

g14 0, g24 0 and g34 0 have no positive solutions. Therefore, we have b1E 0 and b2E 0. This is the same result as eq. (108).

R E F E R E N C E S

[1] DEWITTB. S., Phys. Rev., 160 (1967) 1113. [2] HORIGUCHIT., Nuovo Cimento B, 112 (1997) 1107.

[3] HORIGUCHI T., Nuovo Cimento B, 110 (1995) 839; preprints KIFR-94-01, Dimensional reduction in a quantum universe; KIFR-94-03, On the renormalizability of the Wheeler-DeWitt Equation.

[4] HORIGUCHIT., Nuovo Cimento B, 111 (1996) 49. [5] HORIGUCHIT., Nuovo Cimento B, 111 (1996) 165. [6] HORIGUCHIT., Nuovo Cimento B, 111 (1996) 85. [7] HORIGUCHIT., Nuovo Cimento B, 111 (1996) 293.

[8] DEWITTB. S., Dynamical Theory of Groups and Fields (Gordon and Breach, Inc., N. Y.) 1965.

STROMINGERA., Nucl. Phys. B, 321 (1989) 481; FISHLERW., KLEBANOVI., POLCHINSKIJ. and SUSSKIND L., Nucl. Phys. B, 327 (1989) 157; DUNCANM. J., Nucl. Phys. B, 361 (1991) 695; HORIGUCHIT., Mod. Phys. Lett. A, 8 (1993) 777; Phys. Rev. D, 48 (1993) 5764.

[17] The description based on third-quantized universe theory [16] may be of importance at the early stage of the evolution of the universe since third quantization can formally avoid negative probabilities in the WDW equation such as second quantization of the Klein-Gordon equation.

[18] For the case with an approximate wave function (1) up to sixth order(s1c0, s2c0, see

eq. (18)), we had qualitatively same behaviors of aA( m), GNand L in our numerical calculations

as those in figs. NU1, NU2, NU3. We do not show those figures in this paper. [19] DIRACP. A. M., Nature, 139 (1937) 323; Proc. R. Soc. London, Ser. A, 165 (1938) 199. [20] ZEE A., in High Energy Physics: Proceedings of the 20th Annual Orbis Scientiae, 1983,

edited by S. L. MINTZand A. PERLMUTTER(Plenum, New York) 1985. [21] BERMANM. S., Int. J. Theor. Phys., 31 (1992) 1217.

[22] In ref. [21], Berman got L Pt22_{from another large number}

g

_cˇ

g

mprme

L

h

1 /2

A 1039

h

, where ˇ stands for Planck’s constant, and he insisted that, in a future “theory of everything”, the relations (L Pt22_{, G}

NP t21) must be given as sound foundations.

[23] Strictly speaking, we need to consider the WDW equation with matter fields.

[24] In ref. [25], Dirac thought that a time t in atomic units in eqs. (142)-(149) differs from a dynamical time t in classical theory of Einstein gravity; in other words, the classical Einstein theory is valid in a different system of units from those provided by the atomic constants. In our opinion an extended real time is introduced through Bohm’s quantum potential interpretation for the WDW equation [26]. And it is thought that decreasing m corresponds to the growth of this real time in unstable states, where m is a mass parameter in our renormalization scheme. It seems that the relations (148), (150) are more accurate than the relations (143), (146), (147), (149) since the relations (148), (150) are independent of the choice of time.

[25] DIRACP. A. M., Proc. R. Soc. London, Ser. A, 365 (1979) 19. [26] HORIGUCHIT., Mod. Phys. Lett. A, 9 (1994) 1429.

[27] In ref. [5], for an approximate wave function up to next leading order in positive power expansion in terms of the Planck mass, we have had b24 g1m3 /2, where g1is a constant(see

eq. (99) in ref. [5]). If we choose k04 1 in the relation f( 5 )( 0 ) 4k0Lf( 3 )( 0 ) as dealt with in

this paper, we have g1E 0 for some values of z and h (for example, z 4 0 and h 4 h1, see eq.

(66) in ref. [5]). z and h are dimensionless parameters which have been introduced in the heat equation, in order to study next leading order contribution [5]. In this case, we have b2E 0.

This means that the cosmological constant decreases with increasing m in the region beyond the Planck scale. This qualitatively agrees with our result (108) in the analysis of an approximate wave function up to second order in the spatial gradient expansion.